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# The Automorphisms of Affine Fusion Rings
Terry Gannon
Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected]
# 1. Introduction
Verlinde’s formula [33]
$$
V_{a^{1}\ldots a^{t}}^{(g)}=\sum_{b\in\Phi}(S_{0b})^{2(1-g)}\frac{S_{a^{1}b}}{S_{0b}}\cdot\cdot\cdot\frac{S_{a^{t}b}}{S_{0b}}
$$
arose first in rational conformal field theory (RCFT) as an extremely useful expression for the dimensions of conformal blocks on a genus $g$ surface with $t$ punctures. $\Phi$ here is the finite set of ‘primary fields’. The matrix $S$ comes from a representation of $\mathrm{SL_{2}}(\mathbb{Z})$ defined by the chiral characters of the theory. Contrary to appearances, these numbers $V_{\star\cdots\star}^{(g)}$ will always be nonnegative integers. See the excellent bibliography in [6] for references to the physics literature.
These numbers are remarkable for also arising in several other contexts: for example, as dimensions of spaces of generalised theta functions; as certain tensor product coefficients in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for $\mathbb{F}_{p}$ ; as certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology; and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein.
The more fundamental of these numbers are those corresponding to a sphere with three punctures. It is more convenient to write these in the form (called fusion coefficients)
$$
N_{a b}^{c}\,{\overset{\mathrm{def}}{=}}\,V_{a,b,C c}^{(0)}=\sum_{d\in\Phi}{\frac{S_{a d}S_{b d}S_{c d}^{*}}{S_{0d}}}
$$
where $C$ is a permutation of $\Phi$ called charge-conjugation and will be defined below. The fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries of the numbers (1.1b), i.e. the permutations $\pi$ of $\Phi$ obeying
$$
{\cal N}_{\pi a,\pi b}^{\pi c}={\cal N}_{a b}^{c}\ ,
$$
are precisely the symmetries of all numbers of the form (1.1a).
|
<html><body>
<h1 data-bbox="140 66 471 87">The Automorphisms of Affine Fusion Rings </h1>
<p data-bbox="260 115 351 129">Terry Gannon </p>
<p data-bbox="147 137 465 185">Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected] </p>
<h1 data-bbox="255 214 355 228">1. Introduction </h1>
<p data-bbox="93 243 216 258">Verlinde’s formula [33] </p>
<div class="equation" data-bbox="209 272 402 308">$$
V_{a^{1}\ldots a^{t}}^{(g)}=\sum_{b\in\Phi}(S_{0b})^{2(1-g)}\frac{S_{a^{1}b}}{S_{0b}}\cdot\cdot\cdot\frac{S_{a^{t}b}}{S_{0b}}
$$</div>
<p data-bbox="70 318 541 408">arose first in rational conformal field theory (RCFT) as an extremely useful expression for the dimensions of conformal blocks on a genus $g$ surface with $t$ punctures. $\Phi$ here is the finite set of ‘primary fields’. The matrix $S$ comes from a representation of $\mathrm{SL_{2}}(\mathbb{Z})$ defined by the chiral characters of the theory. Contrary to appearances, these numbers $V_{\star\cdots\star}^{(g)}$ will always be nonnegative integers. See the excellent bibliography in [6] for references to the physics literature. </p>
<p data-bbox="70 408 541 523">These numbers are remarkable for also arising in several other contexts: for example, as dimensions of spaces of generalised theta functions; as certain tensor product coefficients in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for $\mathbb{F}_{p}$ ; as certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology; and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein. </p>
<p data-bbox="70 523 541 552">The more fundamental of these numbers are those corresponding to a sphere with three punctures. It is more convenient to write these in the form (called fusion coefficients) </p>
<div class="equation" data-bbox="223 567 387 602">$$
N_{a b}^{c}\,{\overset{\mathrm{def}}{=}}\,V_{a,b,C c}^{(0)}=\sum_{d\in\Phi}{\frac{S_{a d}S_{b d}S_{c d}^{*}}{S_{0d}}}
$$</div>
<p data-bbox="70 613 541 657">where $C$ is a permutation of $\Phi$ called charge-conjugation and will be defined below. The fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries of the numbers (1.1b), i.e. the permutations $\pi$ of $\Phi$ obeying </p>
<div class="equation" data-bbox="266 673 344 689">$$
{\cal N}_{\pi a,\pi b}^{\pi c}={\cal N}_{a b}^{c}\ ,
$$</div>
<p data-bbox="70 700 399 715">are precisely the symmetries of all numbers of the form (1.1a). </p>
|
0002044v1
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[{"type": "title", "bbox": [140, 66, 471, 87], "content": "The Automorphisms of Affine Fusion Rings", "index": 0}, {"type": "text", "bbox": [260, 115, 351, 129], "content": "Terry Gannon", "index": 1}, {"type": "text", "bbox": [147, 137, 465, 185], "content": "Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected]", "index": 2}, {"type": "title", "bbox": [255, 214, 355, 228], "content": "1. Introduction", "index": 3}, {"type": "text", "bbox": [93, 243, 216, 258], "content": "Verlinde’s formula [33]", "index": 4}, {"type": "interline_equation", "bbox": [209, 272, 402, 308], "content": "", "index": 5}, {"type": "text", "bbox": [70, 318, 541, 408], "content": "arose first in rational conformal field theory (RCFT) as an extremely useful expression for the dimensions of conformal blocks on a genus surface with punctures. here is the finite set of ‘primary fields’. The matrix comes from a representation of defined by the chiral characters of the theory. Contrary to appearances, these numbers will always be nonnegative integers. See the excellent bibliography in [6] for references to the physics literature.", "index": 6}, {"type": "text", "bbox": [70, 408, 541, 523], "content": "These numbers are remarkable for also arising in several other contexts: for example, as dimensions of spaces of generalised theta functions; as certain tensor product coefficients in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for ; as certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology; and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein.", "index": 7}, {"type": "text", "bbox": [70, 523, 541, 552], "content": "The more fundamental of these numbers are those corresponding to a sphere with three punctures. It is more convenient to write these in the form (called fusion coefficients)", "index": 8}, {"type": "interline_equation", "bbox": [223, 567, 387, 602], "content": "", "index": 9}, {"type": "text", "bbox": [70, 613, 541, 657], "content": "where is a permutation of called charge-conjugation and will be defined below. The fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries of the numbers (1.1b), i.e. the permutations of obeying", "index": 10}, {"type": "interline_equation", "bbox": [266, 673, 344, 689], "content": "", "index": 11}, {"type": "text", "bbox": [70, 700, 399, 715], "content": "are precisely the symmetries of all numbers of the form (1.1a).", "index": 12}]
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[{"bbox": [141, 69, 469, 88], "content": "The Automorphisms of Affine Fusion Rings", "parent_index": 0, "line_index": 0}, {"bbox": [261, 117, 351, 131], "content": "Terry Gannon", "parent_index": 1, "line_index": 0}, {"bbox": [147, 140, 464, 153], "content": "Department of Mathematical Sciences, University of Alberta,", "parent_index": 2, "line_index": 0}, {"bbox": [227, 154, 386, 167], "content": "Edmonton, Canada, T6G 2G1", "parent_index": 2, "line_index": 1}, {"bbox": [215, 173, 396, 185], "content": "e-mail: [email protected]", "parent_index": 2, "line_index": 2}, {"bbox": [257, 216, 355, 229], "content": "1. Introduction", "parent_index": 3, "line_index": 0}, {"bbox": [95, 245, 213, 259], "content": "Verlinde’s formula [33]", "parent_index": 4, "line_index": 0}, {"bbox": [70, 322, 541, 338], "content": "arose first in rational conformal field theory (RCFT) as an extremely useful expression for", "parent_index": 6, "line_index": 0}, {"bbox": [70, 336, 541, 352], "content": "the dimensions of conformal blocks on a genus surface with punctures. here is the", "parent_index": 6, "line_index": 1}, {"bbox": [71, 352, 541, 366], "content": "finite set of ‘primary fields’. The matrix comes from a representation of defined", "parent_index": 6, "line_index": 2}, {"bbox": [68, 361, 545, 386], "content": "by the chiral characters of the theory. Contrary to appearances, these numbers will", "parent_index": 6, "line_index": 3}, {"bbox": [71, 382, 541, 396], "content": "always be nonnegative integers. See the excellent bibliography in [6] for references to the", "parent_index": 6, "line_index": 4}, {"bbox": [70, 397, 165, 410], "content": "physics literature.", "parent_index": 6, "line_index": 5}, {"bbox": [94, 410, 540, 425], "content": "These numbers are remarkable for also arising in several other contexts: for example,", "parent_index": 7, "line_index": 0}, {"bbox": [71, 426, 541, 439], "content": "as dimensions of spaces of generalised theta functions; as certain tensor product coefficients", "parent_index": 7, "line_index": 1}, {"bbox": [69, 439, 542, 455], "content": "in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for ; as", "parent_index": 7, "line_index": 2}, {"bbox": [70, 454, 540, 467], "content": "certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in", "parent_index": 7, "line_index": 3}, {"bbox": [72, 469, 540, 482], "content": "algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator", "parent_index": 7, "line_index": 4}, {"bbox": [70, 482, 541, 497], "content": "algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology;", "parent_index": 7, "line_index": 5}, {"bbox": [70, 496, 541, 511], "content": "and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and", "parent_index": 7, "line_index": 6}, {"bbox": [71, 512, 167, 525], "content": "references therein.", "parent_index": 7, "line_index": 7}, {"bbox": [94, 525, 541, 540], "content": "The more fundamental of these numbers are those corresponding to a sphere with three", "parent_index": 8, "line_index": 0}, {"bbox": [70, 540, 520, 555], "content": "punctures. It is more convenient to write these in the form (called fusion coefficients)", "parent_index": 8, "line_index": 1}, {"bbox": [71, 616, 541, 631], "content": "where is a permutation of called charge-conjugation and will be defined below. The", "parent_index": 10, "line_index": 0}, {"bbox": [70, 630, 542, 645], "content": "fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries", "parent_index": 10, "line_index": 1}, {"bbox": [70, 643, 385, 660], "content": "of the numbers (1.1b), i.e. the permutations of obeying", "parent_index": 10, "line_index": 2}, {"bbox": [70, 703, 398, 717], "content": "are precisely the symmetries of all numbers of the form (1.1a).", "parent_index": 12, "line_index": 0}]
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[{"bbox": [209, 272, 402, 308], "content": "V_{a^{1}\\ldots a^{t}}^{(g)}=\\sum_{b\\in\\Phi}(S_{0b})^{2(1-g)}\\frac{S_{a^{1}b}}{S_{0b}}\\cdot\\cdot\\cdot\\frac{S_{a^{t}b}}{S_{0b}}", "parent_index": 5, "subtype": "interline"}, {"bbox": [322, 342, 328, 349], "content": "g", "parent_index": 6, "subtype": "inline"}, {"bbox": [401, 339, 405, 347], "content": "t", "parent_index": 6, "subtype": "inline"}, {"bbox": [471, 338, 479, 347], "content": "\\Phi", "parent_index": 6, "subtype": "inline"}, {"bbox": [285, 353, 293, 362], "content": "S", "parent_index": 6, "subtype": "inline"}, {"bbox": [461, 352, 498, 364], "content": "\\mathrm{SL_{2}}(\\mathbb{Z})", "parent_index": 6, "subtype": "inline"}, {"bbox": [491, 365, 516, 380], "content": "V_{\\star\\cdots\\star}^{(g)}", "parent_index": 6, "subtype": "inline"}, {"bbox": [506, 441, 519, 453], "content": "\\mathbb{F}_{p}", "parent_index": 7, "subtype": "inline"}, {"bbox": [223, 567, 387, 602], "content": "N_{a b}^{c}\\,{\\overset{\\mathrm{def}}{=}}\\,V_{a,b,C c}^{(0)}=\\sum_{d\\in\\Phi}{\\frac{S_{a d}S_{b d}S_{c d}^{*}}{S_{0d}}}", "parent_index": 9, "subtype": "interline"}, {"bbox": [106, 618, 116, 627], "content": "C", "parent_index": 10, "subtype": "inline"}, {"bbox": [226, 618, 234, 627], "content": "\\Phi", "parent_index": 10, "subtype": "inline"}, {"bbox": [305, 649, 313, 655], "content": "\\pi", "parent_index": 10, "subtype": "inline"}, {"bbox": [330, 646, 339, 655], "content": "\\Phi", "parent_index": 10, "subtype": "inline"}, {"bbox": [266, 673, 344, 689], "content": "{\\cal N}_{\\pi a,\\pi b}^{\\pi c}={\\cal N}_{a b}^{c}\\ ,", "parent_index": 11, "subtype": "interline"}]
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The point of introducing the $N_{a b}^{c}$ in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects $\chi_{a}$ labelled by the $a\in\Phi$ ; the multiplication is defined to have structure constants $N_{a b}^{c}$ :
$$
\chi_{a}\chi_{b}=\sum_{c\in\Phi}N_{a b}^{c}\chi_{c}
$$
As an abstract ring, it is not so interesting (the fusion ring over $\mathbb{C}$ is isomorphic to $\mathbb{C}^{||\Phi||}$ with operations defined component-wise; over $\mathbb{Q}$ it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely $\{\chi_{a}\}$ , and so proper definitions of isomorphisms etc. must respect that.
The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although MooreSeiberg assert that there is a sense in which any torus partition function can be interpreted as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18].
It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries.
A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic.
Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21].
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<html><body>
<p data-bbox="70 70 541 115">The point of introducing the $N_{a b}^{c}$ in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects $\chi_{a}$ labelled by the $a\in\Phi$ ; the multiplication is defined to have structure constants $N_{a b}^{c}$ : </p>
<div class="equation" data-bbox="258 130 353 160">$$
\chi_{a}\chi_{b}=\sum_{c\in\Phi}N_{a b}^{c}\chi_{c}
$$</div>
<p data-bbox="70 173 540 245">As an abstract ring, it is not so interesting (the fusion ring over $\mathbb{C}$ is isomorphic to $\mathbb{C}^{||\Phi||}$ with operations defined component-wise; over $\mathbb{Q}$ it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely $\{\chi_{a}\}$ , and so proper definitions of isomorphisms etc. must respect that. </p>
<p data-bbox="71 246 541 475">The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although MooreSeiberg assert that there is a sense in which any torus partition function can be interpreted as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18]. </p>
<p data-bbox="70 476 541 518">It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries. </p>
<p data-bbox="71 518 541 547">A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic. </p>
<p data-bbox="70 554 540 598">Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21]. </p>
</body></html>
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0002044v1
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{"preproc_blocks": [{"type": "text", "bbox": [70, 70, 541, 115], "lines": [{"bbox": [93, 73, 540, 89], "spans": [{"bbox": [93, 73, 249, 89], "score": 1.0, "content": "The point of introducing the ", "type": "text"}, {"bbox": [249, 75, 268, 88], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [269, 73, 540, 89], "score": 1.0, "content": " in (1.1b) is that they define an algebraic structure,", "type": "text"}], "index": 0}, {"bbox": [70, 87, 541, 104], "spans": [{"bbox": [70, 87, 439, 104], "score": 1.0, "content": "the fusion ring. Consider all formal linear combinations of objects ", "type": "text"}, {"bbox": [440, 93, 453, 101], "score": 0.9, "content": "\\chi_{a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [453, 87, 541, 104], "score": 1.0, "content": " labelled by the", "type": "text"}], "index": 1}, {"bbox": [71, 100, 430, 120], "spans": [{"bbox": [71, 104, 101, 114], "score": 0.91, "content": "a\\in\\Phi", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [101, 100, 405, 120], "score": 1.0, "content": "; the multiplication is defined to have structure constants ", "type": "text"}, {"bbox": [405, 104, 425, 116], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [425, 100, 430, 120], "score": 1.0, "content": ":", "type": "text"}], "index": 2}], "index": 1}, {"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [70, 173, 540, 245], "lines": [{"bbox": [69, 174, 539, 192], "spans": [{"bbox": [69, 174, 412, 192], "score": 1.0, "content": "As an abstract ring, it is not so interesting (the fusion ring over ", "type": "text"}, {"bbox": [412, 178, 421, 187], "score": 0.9, "content": "\\mathbb{C}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [422, 174, 513, 192], "score": 1.0, "content": " is isomorphic to ", "type": "text"}, {"bbox": [513, 176, 539, 187], "score": 0.92, "content": "\\mathbb{C}^{||\\Phi||}", "type": "inline_equation", "height": 11, "width": 26}], "index": 4}, {"bbox": [70, 190, 540, 205], "spans": [{"bbox": [70, 190, 313, 205], "score": 1.0, "content": "with operations defined component-wise; over ", "type": "text"}, {"bbox": [314, 192, 324, 203], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [324, 190, 540, 205], "score": 1.0, "content": " it will be a direct sum of number fields).", "type": "text"}], "index": 5}, {"bbox": [69, 204, 542, 221], "spans": [{"bbox": [69, 204, 542, 221], "score": 1.0, "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "type": "text"}], "index": 6}, {"bbox": [70, 218, 541, 235], "spans": [{"bbox": [70, 218, 541, 235], "score": 1.0, "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "type": "text"}], "index": 7}, {"bbox": [70, 234, 487, 249], "spans": [{"bbox": [70, 234, 113, 249], "score": 1.0, "content": "namely ", "type": "text"}, {"bbox": [113, 235, 138, 247], "score": 0.93, "content": "\\{\\chi_{a}\\}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [138, 234, 487, 249], "score": 1.0, "content": ", and so proper definitions of isomorphisms etc. must respect that.", "type": "text"}], "index": 8}], "index": 6}, {"type": "text", "bbox": [71, 246, 541, 475], "lines": [{"bbox": [94, 247, 541, 263], "spans": [{"bbox": [94, 247, 541, 263], "score": 1.0, "content": "The most important examples of fusion rings are associated to the affine algebras,", "type": "text"}], "index": 9}, {"bbox": [70, 262, 541, 277], "spans": [{"bbox": [70, 262, 541, 277], "score": 1.0, "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "type": "text"}], "index": 10}, {"bbox": [70, 277, 541, 291], "spans": [{"bbox": [70, 277, 541, 291], "score": 1.0, "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "type": "text"}], "index": 11}, {"bbox": [70, 290, 541, 306], "spans": [{"bbox": [70, 290, 541, 306], "score": 1.0, "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "type": "text"}], "index": 12}, {"bbox": [70, 306, 541, 320], "spans": [{"bbox": [70, 306, 541, 320], "score": 1.0, "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "type": "text"}], "index": 13}, {"bbox": [71, 319, 542, 334], "spans": [{"bbox": [71, 319, 542, 334], "score": 1.0, "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "type": "text"}], "index": 14}, {"bbox": [70, 334, 541, 349], "spans": [{"bbox": [70, 334, 541, 349], "score": 1.0, "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "type": "text"}], "index": 15}, {"bbox": [71, 349, 541, 363], "spans": [{"bbox": [71, 349, 541, 363], "score": 1.0, "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "type": "text"}], "index": 16}, {"bbox": [70, 363, 540, 377], "spans": [{"bbox": [70, 363, 540, 377], "score": 1.0, "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "type": "text"}], "index": 17}, {"bbox": [70, 377, 540, 392], "spans": [{"bbox": [70, 377, 540, 392], "score": 1.0, "content": "However most torus partition functions are not automorphism invariants (although Moore-", "type": "text"}], "index": 18}, {"bbox": [71, 392, 540, 406], "spans": [{"bbox": [71, 392, 540, 406], "score": 1.0, "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "type": "text"}], "index": 19}, {"bbox": [71, 407, 540, 420], "spans": [{"bbox": [71, 407, 540, 420], "score": 1.0, "content": "as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "type": "text"}], "index": 20}, {"bbox": [70, 419, 541, 434], "spans": [{"bbox": [70, 419, 541, 434], "score": 1.0, "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "type": "text"}], "index": 21}, {"bbox": [71, 435, 541, 448], "spans": [{"bbox": [71, 435, 541, 448], "score": 1.0, "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "type": "text"}], "index": 22}, {"bbox": [70, 448, 541, 464], "spans": [{"bbox": [70, 448, 541, 464], "score": 1.0, "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "type": "text"}], "index": 23}, {"bbox": [70, 462, 507, 478], "spans": [{"bbox": [70, 462, 507, 478], "score": 1.0, "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "type": "text"}], "index": 24}], "index": 16.5}, {"type": "text", "bbox": [70, 476, 541, 518], "lines": [{"bbox": [94, 477, 541, 492], "spans": [{"bbox": [94, 477, 541, 492], "score": 1.0, "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "type": "text"}], "index": 25}, {"bbox": [70, 491, 541, 506], "spans": [{"bbox": [70, 491, 541, 506], "score": 1.0, "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "type": "text"}], "index": 26}, {"bbox": [70, 506, 498, 520], "spans": [{"bbox": [70, 506, 498, 520], "score": 1.0, "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [71, 518, 541, 547], "lines": [{"bbox": [95, 521, 540, 535], "spans": [{"bbox": [95, 521, 540, 535], "score": 1.0, "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "type": "text"}], "index": 28}, {"bbox": [70, 536, 538, 550], "spans": [{"bbox": [70, 536, 538, 550], "score": 1.0, "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "text", "bbox": [70, 554, 540, 598], "lines": [{"bbox": [95, 556, 541, 572], "spans": [{"bbox": [95, 556, 541, 572], "score": 1.0, "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "type": "text"}], "index": 30}, {"bbox": [70, 571, 540, 585], "spans": [{"bbox": [70, 571, 540, 585], "score": 1.0, "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "type": "text"}], "index": 31}, {"bbox": [72, 586, 354, 600], "spans": [{"bbox": [72, 586, 354, 600], "score": 1.0, "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "type": "text"}], "index": 32}], "index": 31}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 115], "lines": [{"bbox": [93, 73, 540, 89], "spans": [{"bbox": [93, 73, 249, 89], "score": 1.0, "content": "The point of introducing the ", "type": "text"}, {"bbox": [249, 75, 268, 88], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [269, 73, 540, 89], "score": 1.0, "content": " in (1.1b) is that they define an algebraic structure,", "type": "text"}], "index": 0}, {"bbox": [70, 87, 541, 104], "spans": [{"bbox": [70, 87, 439, 104], "score": 1.0, "content": "the fusion ring. Consider all formal linear combinations of objects ", "type": "text"}, {"bbox": [440, 93, 453, 101], "score": 0.9, "content": "\\chi_{a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [453, 87, 541, 104], "score": 1.0, "content": " labelled by the", "type": "text"}], "index": 1}, {"bbox": [71, 100, 430, 120], "spans": [{"bbox": [71, 104, 101, 114], "score": 0.91, "content": "a\\in\\Phi", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [101, 100, 405, 120], "score": 1.0, "content": "; the multiplication is defined to have structure constants ", "type": "text"}, {"bbox": [405, 104, 425, 116], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [425, 100, 430, 120], "score": 1.0, "content": ":", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [70, 73, 541, 120]}, {"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [70, 173, 540, 245], "lines": [{"bbox": [69, 174, 539, 192], "spans": [{"bbox": [69, 174, 412, 192], "score": 1.0, "content": "As an abstract ring, it is not so interesting (the fusion ring over ", "type": "text"}, {"bbox": [412, 178, 421, 187], "score": 0.9, "content": "\\mathbb{C}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [422, 174, 513, 192], "score": 1.0, "content": " is isomorphic to ", "type": "text"}, {"bbox": [513, 176, 539, 187], "score": 0.92, "content": "\\mathbb{C}^{||\\Phi||}", "type": "inline_equation", "height": 11, "width": 26}], "index": 4}, {"bbox": [70, 190, 540, 205], "spans": [{"bbox": [70, 190, 313, 205], "score": 1.0, "content": "with operations defined component-wise; over ", "type": "text"}, {"bbox": [314, 192, 324, 203], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [324, 190, 540, 205], "score": 1.0, "content": " it will be a direct sum of number fields).", "type": "text"}], "index": 5}, {"bbox": [69, 204, 542, 221], "spans": [{"bbox": [69, 204, 542, 221], "score": 1.0, "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "type": "text"}], "index": 6}, {"bbox": [70, 218, 541, 235], "spans": [{"bbox": [70, 218, 541, 235], "score": 1.0, "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "type": "text"}], "index": 7}, {"bbox": [70, 234, 487, 249], "spans": [{"bbox": [70, 234, 113, 249], "score": 1.0, "content": "namely ", "type": "text"}, {"bbox": [113, 235, 138, 247], "score": 0.93, "content": "\\{\\chi_{a}\\}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [138, 234, 487, 249], "score": 1.0, "content": ", and so proper definitions of isomorphisms etc. must respect that.", "type": "text"}], "index": 8}], "index": 6, "bbox_fs": [69, 174, 542, 249]}, {"type": "text", "bbox": [71, 246, 541, 475], "lines": [{"bbox": [94, 247, 541, 263], "spans": [{"bbox": [94, 247, 541, 263], "score": 1.0, "content": "The most important examples of fusion rings are associated to the affine algebras,", "type": "text"}], "index": 9}, {"bbox": [70, 262, 541, 277], "spans": [{"bbox": [70, 262, 541, 277], "score": 1.0, "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "type": "text"}], "index": 10}, {"bbox": [70, 277, 541, 291], "spans": [{"bbox": [70, 277, 541, 291], "score": 1.0, "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "type": "text"}], "index": 11}, {"bbox": [70, 290, 541, 306], "spans": [{"bbox": [70, 290, 541, 306], "score": 1.0, "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "type": "text"}], "index": 12}, {"bbox": [70, 306, 541, 320], "spans": [{"bbox": [70, 306, 541, 320], "score": 1.0, "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "type": "text"}], "index": 13}, {"bbox": [71, 319, 542, 334], "spans": [{"bbox": [71, 319, 542, 334], "score": 1.0, "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "type": "text"}], "index": 14}, {"bbox": [70, 334, 541, 349], "spans": [{"bbox": [70, 334, 541, 349], "score": 1.0, "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "type": "text"}], "index": 15}, {"bbox": [71, 349, 541, 363], "spans": [{"bbox": [71, 349, 541, 363], "score": 1.0, "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "type": "text"}], "index": 16}, {"bbox": [70, 363, 540, 377], "spans": [{"bbox": [70, 363, 540, 377], "score": 1.0, "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "type": "text"}], "index": 17}, {"bbox": [70, 377, 540, 392], "spans": [{"bbox": [70, 377, 540, 392], "score": 1.0, "content": "However most torus partition functions are not automorphism invariants (although Moore-", "type": "text"}], "index": 18}, {"bbox": [71, 392, 540, 406], "spans": [{"bbox": [71, 392, 540, 406], "score": 1.0, "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "type": "text"}], "index": 19}, {"bbox": [71, 407, 540, 420], "spans": [{"bbox": [71, 407, 540, 420], "score": 1.0, "content": "as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "type": "text"}], "index": 20}, {"bbox": [70, 419, 541, 434], "spans": [{"bbox": [70, 419, 541, 434], "score": 1.0, "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "type": "text"}], "index": 21}, {"bbox": [71, 435, 541, 448], "spans": [{"bbox": [71, 435, 541, 448], "score": 1.0, "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "type": "text"}], "index": 22}, {"bbox": [70, 448, 541, 464], "spans": [{"bbox": [70, 448, 541, 464], "score": 1.0, "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "type": "text"}], "index": 23}, {"bbox": [70, 462, 507, 478], "spans": [{"bbox": [70, 462, 507, 478], "score": 1.0, "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "type": "text"}], "index": 24}], "index": 16.5, "bbox_fs": [70, 247, 542, 478]}, {"type": "text", "bbox": [70, 476, 541, 518], "lines": [{"bbox": [94, 477, 541, 492], "spans": [{"bbox": [94, 477, 541, 492], "score": 1.0, "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "type": "text"}], "index": 25}, {"bbox": [70, 491, 541, 506], "spans": [{"bbox": [70, 491, 541, 506], "score": 1.0, "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "type": "text"}], "index": 26}, {"bbox": [70, 506, 498, 520], "spans": [{"bbox": [70, 506, 498, 520], "score": 1.0, "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "type": "text"}], "index": 27}], "index": 26, "bbox_fs": [70, 477, 541, 520]}, {"type": "text", "bbox": [71, 518, 541, 547], "lines": [{"bbox": [95, 521, 540, 535], "spans": [{"bbox": [95, 521, 540, 535], "score": 1.0, "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "type": "text"}], "index": 28}, {"bbox": [70, 536, 538, 550], "spans": [{"bbox": [70, 536, 538, 550], "score": 1.0, "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "type": "text"}], "index": 29}], "index": 28.5, "bbox_fs": [70, 521, 540, 550]}, {"type": "text", "bbox": [70, 554, 540, 598], "lines": [{"bbox": [95, 556, 541, 572], "spans": [{"bbox": [95, 556, 541, 572], "score": 1.0, "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "type": "text"}], "index": 30}, {"bbox": [70, 571, 540, 585], "spans": [{"bbox": [70, 571, 540, 585], "score": 1.0, "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "type": "text"}], "index": 31}, {"bbox": [72, 586, 354, 600], "spans": [{"bbox": [72, 586, 354, 600], "score": 1.0, "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "type": "text"}], "index": 32}], "index": 31, "bbox_fs": [70, 556, 541, 600]}]}
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[{"type": "text", "bbox": [70, 70, 541, 115], "content": "The point of introducing the in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects labelled by the ; the multiplication is defined to have structure constants :", "index": 0}, {"type": "interline_equation", "bbox": [258, 130, 353, 160], "content": "", "index": 1}, {"type": "text", "bbox": [70, 173, 540, 245], "content": "As an abstract ring, it is not so interesting (the fusion ring over is isomorphic to with operations defined component-wise; over it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely , and so proper definitions of isomorphisms etc. must respect that.", "index": 2}, {"type": "text", "bbox": [71, 246, 541, 475], "content": "The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although Moore- Seiberg assert that there is a sense in which any torus partition function can be interpreted as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "index": 3}, {"type": "text", "bbox": [70, 476, 541, 518], "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "index": 4}, {"type": "text", "bbox": [71, 518, 541, 547], "content": "A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "index": 5}, {"type": "text", "bbox": [70, 554, 540, 598], "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21].", "index": 6}]
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[{"bbox": [93, 73, 540, 89], "content": "The point of introducing the in (1.1b) is that they define an algebraic structure,", "parent_index": 0, "line_index": 0}, {"bbox": [70, 87, 541, 104], "content": "the fusion ring. Consider all formal linear combinations of objects labelled by the", "parent_index": 0, "line_index": 1}, {"bbox": [71, 100, 430, 120], "content": "; the multiplication is defined to have structure constants :", "parent_index": 0, "line_index": 2}, {"bbox": [69, 174, 539, 192], "content": "As an abstract ring, it is not so interesting (the fusion ring over is isomorphic to", "parent_index": 2, "line_index": 0}, {"bbox": [70, 190, 540, 205], "content": "with operations defined component-wise; over it will be a direct sum of number fields).", "parent_index": 2, "line_index": 1}, {"bbox": [69, 204, 542, 221], "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "parent_index": 2, "line_index": 2}, {"bbox": [70, 218, 541, 235], "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "parent_index": 2, "line_index": 3}, {"bbox": [70, 234, 487, 249], "content": "namely , and so proper definitions of isomorphisms etc. must respect that.", "parent_index": 2, "line_index": 4}, {"bbox": [94, 247, 541, 263], "content": "The most important examples of fusion rings are associated to the affine algebras,", "parent_index": 3, "line_index": 0}, {"bbox": [70, 262, 541, 277], "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "parent_index": 3, "line_index": 1}, {"bbox": [70, 277, 541, 291], "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "parent_index": 3, "line_index": 2}, {"bbox": [70, 290, 541, 306], "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "parent_index": 3, "line_index": 3}, {"bbox": [70, 306, 541, 320], "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "parent_index": 3, "line_index": 4}, {"bbox": [71, 319, 542, 334], "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "parent_index": 3, "line_index": 5}, {"bbox": [70, 334, 541, 349], "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "parent_index": 3, "line_index": 6}, {"bbox": [71, 349, 541, 363], "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "parent_index": 3, "line_index": 7}, {"bbox": [70, 363, 540, 377], "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "parent_index": 3, "line_index": 8}, {"bbox": [70, 377, 540, 392], "content": "However most torus partition functions are not automorphism invariants (although Moore-", "parent_index": 3, "line_index": 9}, {"bbox": [71, 392, 540, 406], "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "parent_index": 3, "line_index": 10}, {"bbox": [71, 407, 540, 420], "content": "as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "parent_index": 3, "line_index": 11}, {"bbox": [70, 419, 541, 434], "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "parent_index": 3, "line_index": 12}, {"bbox": [71, 435, 541, 448], "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "parent_index": 3, "line_index": 13}, {"bbox": [70, 448, 541, 464], "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "parent_index": 3, "line_index": 14}, {"bbox": [70, 462, 507, 478], "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "parent_index": 3, "line_index": 15}, {"bbox": [94, 477, 541, 492], "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "parent_index": 4, "line_index": 0}, {"bbox": [70, 491, 541, 506], "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "parent_index": 4, "line_index": 1}, {"bbox": [70, 506, 498, 520], "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "parent_index": 4, "line_index": 2}, {"bbox": [95, 521, 540, 535], "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "parent_index": 5, "line_index": 0}, {"bbox": [70, 536, 538, 550], "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "parent_index": 5, "line_index": 1}, {"bbox": [95, 556, 541, 572], "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "parent_index": 6, "line_index": 0}, {"bbox": [70, 571, 540, 585], "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "parent_index": 6, "line_index": 1}, {"bbox": [72, 586, 354, 600], "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "parent_index": 6, "line_index": 2}]
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[]
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[{"bbox": [249, 75, 268, 88], "content": "N_{a b}^{c}", "parent_index": 0, "subtype": "inline"}, {"bbox": [440, 93, 453, 101], "content": "\\chi_{a}", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 104, 101, 114], "content": "a\\in\\Phi", "parent_index": 0, "subtype": "inline"}, {"bbox": [405, 104, 425, 116], "content": "N_{a b}^{c}", "parent_index": 0, "subtype": "inline"}, {"bbox": [258, 130, 353, 160], "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "parent_index": 1, "subtype": "interline"}, {"bbox": [412, 178, 421, 187], "content": "\\mathbb{C}", "parent_index": 2, "subtype": "inline"}, {"bbox": [513, 176, 539, 187], "content": "\\mathbb{C}^{||\\Phi||}", "parent_index": 2, "subtype": "inline"}, {"bbox": [314, 192, 324, 203], "content": "\\mathbb{Q}", "parent_index": 2, "subtype": "inline"}, {"bbox": [113, 235, 138, 247], "content": "\\{\\chi_{a}\\}", "parent_index": 2, "subtype": "inline"}]
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[]
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# 2.1. The affine fusion ring
The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $X_{r}^{(1)}$ [23]. Choose any positive integer $k$ . Consider the (finite) set $P_{+}=P_{+}^{k}(X_{r}^{(1)})$ of level $k$ integrable highest weights:
$$
P_{+}\stackrel{\mathrm{def}}{=}\{\sum_{j=0}^{r}\lambda_{j}\Lambda_{j}\mid\lambda_{j}\in\mathbb{Z},\ \lambda_{j}\geq0,\ \sum_{j=0}^{r}a_{j}^{\vee}\lambda_{j}=k\}\ ,
$$
where $\Lambda_{i}$ denote the fundamental weights, and $a_{j}^{\vee}$ are the co-labels, of $X_{r}^{(1)}$ (the $a_{j}^{\vee}$ will be given for each algebra in §3). We will usually drop the (redundant) component $\lambda_{0}\Lambda_{0}$ . KacPeterson [24] found a natural representation of the modular group $\operatorname{SL_{2}}(\mathbb{Z})$ on the complex space spanned by the affine characters $\chi_{\mu}$ , $\mu\in P_{+}$ : most significantly, $\left(\begin{array}{c c}{{0}}&{{-1}}\\ {{1}}&{{0}}\end{array}\right)$ is sent to the Kac-Peterson matrix $S$ with entries
$$
S_{\mu\nu}=c\,\sum_{w\in\overline{{{W}}}}{\operatorname*{det}(w)}\,\exp[-2\pi\mathrm{i}\,\frac{(w(\mu+\rho)|\nu+\rho)}{\kappa}]\ .
$$
An explicit expression for the normalisation constant $c$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $\overline{W}$ is the (finite) Weyl group of $X_{r}$ , and acts on $P_{+}$ by fixing $\Lambda_{0}$ . The Weyl vector $\rho$ equals $\sum_{i}\Lambda_{i}$ , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.
The matrix $S$ is symmetric and unitary. One of the weights, $k\Lambda_{0}$ , is distinguished and will be denoted ‘ $0^{\circ}$ . It is the weight appearing in the denominator of (1.1). A useful fact is that
$$
S_{\lambda0}>0\qquad\mathrm{for~all~}\lambda\in P_{+}\ .
$$
Equation (2.1a) gives us the important
$$
\chi_{\lambda}[\mu]\stackrel{\mathrm{def}}{=}\frac{S_{\lambda\mu}}{S_{0\mu}}=\mathrm{ch}_{\overline{{{\lambda}}}}(-2\pi\mathrm{i}\frac{\overline{{{\mu}}}+\overline{{{\rho}}}}{\kappa})~,
$$
where $\mathrm{ch}_{{\overline{{\lambda}}}}$ is the Weyl character of the $X_{r}$ -module $L(\overline{{\lambda}})$ . Together with the Weyl denominator formula, it provides a useful expression for the $q$ -dimensions:
$$
{\mathcal D}(\lambda)\,\overset{\mathrm{def}}{=}\frac{S_{\lambda0}}{S_{00}}=\prod_{\alpha>0}\frac{\sin(\pi\left(\lambda+\rho\left|\alpha\right)/\kappa\right)}{\sin(\pi\left(\rho\left|\alpha\right)/\kappa\right)}~,
$$
where the product is over the positive roots $\alpha\in\overline{{\Delta}}_{+}$ of $X_{r}$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4).
|
<html><body>
<h1 data-bbox="70 100 214 116">2.1. The affine fusion ring </h1>
<p data-bbox="70 123 541 171">The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $X_{r}^{(1)}$ [23]. Choose any positive integer $k$ . Consider the (finite) set $P_{+}=P_{+}^{k}(X_{r}^{(1)})$ of level $k$ integrable highest weights: </p>
<div class="equation" data-bbox="174 187 436 226">$$
P_{+}\stackrel{\mathrm{def}}{=}\{\sum_{j=0}^{r}\lambda_{j}\Lambda_{j}\mid\lambda_{j}\in\mathbb{Z},\ \lambda_{j}\geq0,\ \sum_{j=0}^{r}a_{j}^{\vee}\lambda_{j}=k\}\ ,
$$</div>
<p data-bbox="69 239 541 328">where $\Lambda_{i}$ denote the fundamental weights, and $a_{j}^{\vee}$ are the co-labels, of $X_{r}^{(1)}$ (the $a_{j}^{\vee}$ will be given for each algebra in §3). We will usually drop the (redundant) component $\lambda_{0}\Lambda_{0}$ . KacPeterson [24] found a natural representation of the modular group $\operatorname{SL_{2}}(\mathbb{Z})$ on the complex space spanned by the affine characters $\chi_{\mu}$ , $\mu\in P_{+}$ : most significantly, $\left(\begin{array}{c c}{{0}}&{{-1}}\\ {{1}}&{{0}}\end{array}\right)$ is sent to the Kac-Peterson matrix $S$ with entries </p>
<div class="equation" data-bbox="175 338 435 378">$$
S_{\mu\nu}=c\,\sum_{w\in\overline{{{W}}}}{\operatorname*{det}(w)}\,\exp[-2\pi\mathrm{i}\,\frac{(w(\mu+\rho)|\nu+\rho)}{\kappa}]\ .
$$</div>
<p data-bbox="70 389 541 452">An explicit expression for the normalisation constant $c$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $\overline{W}$ is the (finite) Weyl group of $X_{r}$ , and acts on $P_{+}$ by fixing $\Lambda_{0}$ . The Weyl vector $\rho$ equals $\sum_{i}\Lambda_{i}$ , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here. </p>
<p data-bbox="70 452 541 494">The matrix $S$ is symmetric and unitary. One of the weights, $k\Lambda_{0}$ , is distinguished and will be denoted ‘ $0^{\circ}$ . It is the weight appearing in the denominator of (1.1). A useful fact is that </p>
<div class="equation" data-bbox="233 497 378 511">$$
S_{\lambda0}>0\qquad\mathrm{for~all~}\lambda\in P_{+}\ .
$$</div>
<p data-bbox="93 519 300 534">Equation (2.1a) gives us the important </p>
<div class="equation" data-bbox="218 549 393 580">$$
\chi_{\lambda}[\mu]\stackrel{\mathrm{def}}{=}\frac{S_{\lambda\mu}}{S_{0\mu}}=\mathrm{ch}_{\overline{{{\lambda}}}}(-2\pi\mathrm{i}\frac{\overline{{{\mu}}}+\overline{{{\rho}}}}{\kappa})~,
$$</div>
<p data-bbox="69 592 540 622">where $\mathrm{ch}_{{\overline{{\lambda}}}}$ is the Weyl character of the $X_{r}$ -module $L(\overline{{\lambda}})$ . Together with the Weyl denominator formula, it provides a useful expression for the $q$ -dimensions: </p>
<div class="equation" data-bbox="200 636 410 672">$$
{\mathcal D}(\lambda)\,\overset{\mathrm{def}}{=}\frac{S_{\lambda0}}{S_{00}}=\prod_{\alpha>0}\frac{\sin(\pi\left(\lambda+\rho\left|\alpha\right)/\kappa\right)}{\sin(\pi\left(\rho\left|\alpha\right)/\kappa\right)}~,
$$</div>
<p data-bbox="69 685 540 715">where the product is over the positive roots $\alpha\in\overline{{\Delta}}_{+}$ of $X_{r}$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4). </p>
</body></html>
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0002044v1
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[{"type": "text", "text": "2.1. The affine fusion ring ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $X_{r}^{(1)}$ [23]. Choose any positive integer $k$ . Consider the (finite) set $P_{+}=P_{+}^{k}(X_{r}^{(1)})$ of level $k$ integrable highest weights: ", "page_idx": 2}, {"type": "equation", "text": "$$\nP_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{j=0}^{r}\\lambda_{j}\\Lambda_{j}\\mid\\lambda_{j}\\in\\mathbb{Z},\\ \\lambda_{j}\\geq0,\\ \\sum_{j=0}^{r}a_{j}^{\\vee}\\lambda_{j}=k\\}\\ ,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where $\\Lambda_{i}$ denote the fundamental weights, and $a_{j}^{\\vee}$ are the co-labels, of $X_{r}^{(1)}$ (the $a_{j}^{\\vee}$ will be given for each algebra in §3). We will usually drop the (redundant) component $\\lambda_{0}\\Lambda_{0}$ . KacPeterson [24] found a natural representation of the modular group $\\operatorname{SL_{2}}(\\mathbb{Z})$ on the complex space spanned by the affine characters $\\chi_{\\mu}$ , $\\mu\\in P_{+}$ : most significantly, $\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)$ is sent to the Kac-Peterson matrix $S$ with entries ", "page_idx": 2}, {"type": "equation", "text": "$$\nS_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname*{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)|\\nu+\\rho)}{\\kappa}]\\ .\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "An explicit expression for the normalisation constant $c$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $\\overline{W}$ is the (finite) Weyl group of $X_{r}$ , and acts on $P_{+}$ by fixing $\\Lambda_{0}$ . The Weyl vector $\\rho$ equals $\\sum_{i}\\Lambda_{i}$ , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here. ", "page_idx": 2}, {"type": "text", "text": "The matrix $S$ is symmetric and unitary. One of the weights, $k\\Lambda_{0}$ , is distinguished and will be denoted ‘ $0^{\\circ}$ . It is the weight appearing in the denominator of (1.1). A useful fact is that ", "page_idx": 2}, {"type": "equation", "text": "$$\nS_{\\lambda0}>0\\qquad\\mathrm{for~all~}\\lambda\\in P_{+}\\ .\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Equation (2.1a) gives us the important ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\chi_{\\lambda}[\\mu]\\stackrel{\\mathrm{def}}{=}\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\mathrm{ch}_{\\overline{{{\\lambda}}}}(-2\\pi\\mathrm{i}\\frac{\\overline{{{\\mu}}}+\\overline{{{\\rho}}}}{\\kappa})~,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where $\\mathrm{ch}_{{\\overline{{\\lambda}}}}$ is the Weyl character of the $X_{r}$ -module $L(\\overline{{\\lambda}})$ . Together with the Weyl denominator formula, it provides a useful expression for the $q$ -dimensions: ", "page_idx": 2}, {"type": "equation", "text": "$$\n{\\mathcal D}(\\lambda)\\,\\overset{\\mathrm{def}}{=}\\frac{S_{\\lambda0}}{S_{00}}=\\prod_{\\alpha>0}\\frac{\\sin(\\pi\\left(\\lambda+\\rho\\left|\\alpha\\right)/\\kappa\\right)}{\\sin(\\pi\\left(\\rho\\left|\\alpha\\right)/\\kappa\\right)}~,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where the product is over the positive roots $\\alpha\\in\\overline{{\\Delta}}_{+}$ of $X_{r}$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4). ", "page_idx": 2}]
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Choose any positive integer ", "type": "text"}, {"bbox": [394, 144, 401, 153], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [401, 137, 542, 159], "score": 1.0, "content": ". 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We will usually drop the (redundant) component ", "type": "text"}, {"bbox": [480, 259, 506, 271], "score": 0.9, "content": "\\lambda_{0}\\Lambda_{0}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [507, 258, 540, 274], "score": 1.0, "content": ". Kac-", "type": "text"}], "index": 6}, {"bbox": [70, 273, 540, 288], "spans": [{"bbox": [70, 273, 419, 288], "score": 1.0, "content": "Peterson [24] found a natural representation of the modular group ", "type": "text"}, {"bbox": [419, 274, 456, 286], "score": 0.9, "content": "\\operatorname{SL_{2}}(\\mathbb{Z})", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [457, 273, 540, 288], "score": 1.0, "content": " on the complex", "type": "text"}], "index": 7}, {"bbox": [70, 287, 540, 317], "spans": [{"bbox": [70, 293, 277, 311], "score": 1.0, "content": "space spanned by the affine characters ", "type": "text"}, {"bbox": [278, 299, 292, 308], "score": 0.82, "content": "\\chi_{\\mu}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [293, 293, 299, 311], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [299, 296, 338, 308], "score": 0.91, "content": "\\mu\\in P_{+}", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [339, 293, 446, 311], "score": 1.0, "content": ": most significantly,", "type": "text"}, {"bbox": [447, 287, 502, 317], "score": 0.95, "content": "\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)", "type": "inline_equation", "height": 30, "width": 55}, {"bbox": [505, 294, 540, 308], "score": 1.0, "content": "is sent", "type": "text"}], "index": 8}, {"bbox": [70, 315, 295, 330], "spans": [{"bbox": [70, 315, 220, 330], "score": 1.0, "content": "to the Kac-Peterson matrix ", "type": "text"}, {"bbox": [220, 317, 228, 326], "score": 0.87, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [229, 315, 295, 330], "score": 1.0, "content": " with entries", "type": "text"}], "index": 9}], "index": 7}, {"type": "interline_equation", "bbox": [175, 338, 435, 378], "lines": [{"bbox": [175, 338, 435, 378], "spans": [{"bbox": [175, 338, 435, 378], "score": 0.94, "content": "S_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname*{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)|\\nu+\\rho)}{\\kappa}]\\ .", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 389, 541, 452], "lines": [{"bbox": [71, 393, 540, 407], "spans": [{"bbox": [71, 393, 343, 407], "score": 1.0, "content": "An explicit expression for the normalisation constant ", "type": "text"}, {"bbox": [344, 397, 349, 403], "score": 0.87, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [349, 393, 540, 407], "score": 1.0, "content": " is given in e.g. [23, Theorem 13.8(a)].", "type": "text"}], "index": 11}, {"bbox": [70, 406, 540, 421], "spans": [{"bbox": [70, 406, 454, 421], "score": 1.0, "content": "The inner product in (2.1a) is scaled so that the long roots have norm 2. ", "type": "text"}, {"bbox": [455, 407, 468, 417], "score": 0.92, "content": "\\overline{W}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [468, 406, 540, 421], "score": 1.0, "content": " is the (finite)", "type": "text"}], "index": 12}, {"bbox": [70, 420, 541, 436], "spans": [{"bbox": [70, 420, 151, 436], "score": 1.0, "content": "Weyl group of ", "type": "text"}, {"bbox": [151, 423, 167, 433], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [167, 420, 242, 436], "score": 1.0, "content": ", and acts on ", "type": "text"}, {"bbox": [242, 423, 258, 434], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [258, 420, 313, 436], "score": 1.0, "content": " by fixing ", "type": "text"}, {"bbox": [314, 423, 328, 434], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [328, 420, 432, 436], "score": 1.0, "content": ". The Weyl vector ", "type": "text"}, {"bbox": [432, 426, 439, 434], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [439, 420, 480, 436], "score": 1.0, "content": " equals", "type": "text"}, {"bbox": [480, 422, 512, 435], "score": 0.93, "content": "\\sum_{i}\\Lambda_{i}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [512, 420, 541, 436], "score": 1.0, "content": ", and", "type": "text"}], "index": 13}, {"bbox": [68, 435, 473, 457], "spans": [{"bbox": [68, 435, 473, 457], "score": 1.0, "content": "κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.", "type": "text"}], "index": 14}], "index": 12.5}, {"type": "text", "bbox": [70, 452, 541, 494], "lines": [{"bbox": [94, 453, 541, 469], "spans": [{"bbox": [94, 453, 157, 469], "score": 1.0, "content": "The matrix ", "type": "text"}, {"bbox": [158, 456, 166, 465], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [166, 453, 409, 469], "score": 1.0, "content": " is symmetric and unitary. One of the weights, ", "type": "text"}, {"bbox": [409, 456, 430, 467], "score": 0.91, "content": "k\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [430, 453, 541, 469], "score": 1.0, "content": ", is distinguished and", "type": "text"}], "index": 15}, {"bbox": [70, 468, 541, 484], "spans": [{"bbox": [70, 468, 159, 484], "score": 1.0, "content": "will be denoted ‘", "type": "text"}, {"bbox": [159, 470, 169, 479], "score": 0.43, "content": "0^{\\circ}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 468, 541, 484], "score": 1.0, "content": ". It is the weight appearing in the denominator of (1.1). 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", "type": "text"}, {"bbox": [455, 407, 468, 417], "score": 0.92, "content": "\\overline{W}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [468, 406, 540, 421], "score": 1.0, "content": " is the (finite)", "type": "text"}], "index": 12}, {"bbox": [70, 420, 541, 436], "spans": [{"bbox": [70, 420, 151, 436], "score": 1.0, "content": "Weyl group of ", "type": "text"}, {"bbox": [151, 423, 167, 433], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [167, 420, 242, 436], "score": 1.0, "content": ", and acts on ", "type": "text"}, {"bbox": [242, 423, 258, 434], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [258, 420, 313, 436], "score": 1.0, "content": " by fixing ", "type": "text"}, {"bbox": [314, 423, 328, 434], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [328, 420, 432, 436], "score": 1.0, "content": ". 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Another consequence of (2.1b)", "type": "text"}], "index": 24}, {"bbox": [69, 702, 243, 718], "spans": [{"bbox": [69, 702, 243, 718], "score": 1.0, "content": "is the Kac-Walton formula (2.4).", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [69, 688, 540, 718]}]}
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[{"type": "title", "bbox": [70, 100, 214, 116], "content": "2.1. The affine fusion ring", "index": 0}, {"type": "text", "bbox": [70, 123, 541, 171], "content": "The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras [23]. Choose any positive integer . Consider the (finite) set of level integrable highest weights:", "index": 1}, {"type": "interline_equation", "bbox": [174, 187, 436, 226], "content": "", "index": 2}, {"type": "text", "bbox": [69, 239, 541, 328], "content": "where denote the fundamental weights, and are the co-labels, of (the will be given for each algebra in §3). We will usually drop the (redundant) component . Kac- Peterson [24] found a natural representation of the modular group on the complex space spanned by the affine characters , : most significantly, is sent to the Kac-Peterson matrix with entries", "index": 3}, {"type": "interline_equation", "bbox": [175, 338, 435, 378], "content": "", "index": 4}, {"type": "text", "bbox": [70, 389, 541, 452], "content": "An explicit expression for the normalisation constant is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. is the (finite) Weyl group of , and acts on by fixing . The Weyl vector equals , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.", "index": 5}, {"type": "text", "bbox": [70, 452, 541, 494], "content": "The matrix is symmetric and unitary. One of the weights, , is distinguished and will be denoted ‘ . It is the weight appearing in the denominator of (1.1). A useful fact is that", "index": 6}, {"type": "interline_equation", "bbox": [233, 497, 378, 511], "content": "", "index": 7}, {"type": "text", "bbox": [93, 519, 300, 534], "content": "Equation (2.1a) gives us the important", "index": 8}, {"type": "interline_equation", "bbox": [218, 549, 393, 580], "content": "", "index": 9}, {"type": "text", "bbox": [69, 592, 540, 622], "content": "where is the Weyl character of the -module . Together with the Weyl denomi- nator formula, it provides a useful expression for the -dimensions:", "index": 10}, {"type": "interline_equation", "bbox": [200, 636, 410, 672], "content": "", "index": 11}, {"type": "text", "bbox": [69, 685, 540, 715], "content": "where the product is over the positive roots of . Another consequence of (2.1b) is the Kac-Walton formula (2.4).", "index": 12}]
|
[{"bbox": [72, 102, 213, 118], "content": "2.1. The affine fusion ring", "parent_index": 0, "line_index": 0}, {"bbox": [93, 125, 541, 141], "content": "The source of some of the most interesting fusion data are the affine nontwisted", "parent_index": 1, "line_index": 0}, {"bbox": [69, 137, 542, 159], "content": "Kac-Moody algebras [23]. Choose any positive integer . Consider the (finite) set", "parent_index": 1, "line_index": 1}, {"bbox": [71, 156, 347, 174], "content": "of level integrable highest weights:", "parent_index": 1, "line_index": 2}, {"bbox": [69, 241, 542, 260], "content": "where denote the fundamental weights, and are the co-labels, of (the will be", "parent_index": 3, "line_index": 0}, {"bbox": [70, 258, 540, 274], "content": "given for each algebra in §3). We will usually drop the (redundant) component . Kac-", "parent_index": 3, "line_index": 1}, {"bbox": [70, 273, 540, 288], "content": "Peterson [24] found a natural representation of the modular group on the complex", "parent_index": 3, "line_index": 2}, {"bbox": [70, 287, 540, 317], "content": "space spanned by the affine characters , : most significantly, is sent", "parent_index": 3, "line_index": 3}, {"bbox": [70, 315, 295, 330], "content": "to the Kac-Peterson matrix with entries", "parent_index": 3, "line_index": 4}, {"bbox": [71, 393, 540, 407], "content": "An explicit expression for the normalisation constant is given in e.g. [23, Theorem 13.8(a)].", "parent_index": 5, "line_index": 0}, {"bbox": [70, 406, 540, 421], "content": "The inner product in (2.1a) is scaled so that the long roots have norm 2. is the (finite)", "parent_index": 5, "line_index": 1}, {"bbox": [70, 420, 541, 436], "content": "Weyl group of , and acts on by fixing . The Weyl vector equals , and", "parent_index": 5, "line_index": 2}, {"bbox": [68, 435, 473, 457], "content": "κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.", "parent_index": 5, "line_index": 3}, {"bbox": [94, 453, 541, 469], "content": "The matrix is symmetric and unitary. One of the weights, , is distinguished and", "parent_index": 6, "line_index": 0}, {"bbox": [70, 468, 541, 484], "content": "will be denoted ‘ . It is the weight appearing in the denominator of (1.1). A useful fact", "parent_index": 6, "line_index": 1}, {"bbox": [70, 484, 107, 496], "content": "is that", "parent_index": 6, "line_index": 2}, {"bbox": [95, 521, 300, 536], "content": "Equation (2.1a) gives us the important", "parent_index": 8, "line_index": 0}, {"bbox": [70, 595, 540, 611], "content": "where is the Weyl character of the -module . Together with the Weyl denomi-", "parent_index": 10, "line_index": 0}, {"bbox": [70, 610, 424, 625], "content": "nator formula, it provides a useful expression for the -dimensions:", "parent_index": 10, "line_index": 1}, {"bbox": [71, 688, 540, 703], "content": "where the product is over the positive roots of . Another consequence of (2.1b)", "parent_index": 12, "line_index": 0}, {"bbox": [69, 702, 243, 718], "content": "is the Kac-Walton formula (2.4).", "parent_index": 12, "line_index": 1}]
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[]
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[{"bbox": [184, 140, 208, 154], "content": "X_{r}^{(1)}", "parent_index": 1, "subtype": "inline"}, {"bbox": [394, 144, 401, 153], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [71, 156, 152, 173], "content": "P_{+}=P_{+}^{k}(X_{r}^{(1)})", "parent_index": 1, "subtype": "inline"}, {"bbox": [196, 160, 203, 169], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [174, 187, 436, 226], "content": "P_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{j=0}^{r}\\lambda_{j}\\Lambda_{j}\\mid\\lambda_{j}\\in\\mathbb{Z},\\ \\lambda_{j}\\geq0,\\ \\sum_{j=0}^{r}a_{j}^{\\vee}\\lambda_{j}=k\\}\\ ,", "parent_index": 2, "subtype": "interline"}, {"bbox": [105, 246, 117, 257], "content": "\\Lambda_{i}", "parent_index": 3, "subtype": "inline"}, {"bbox": [316, 245, 329, 259], "content": "a_{j}^{\\vee}", "parent_index": 3, "subtype": "inline"}, {"bbox": [436, 241, 460, 256], "content": "X_{r}^{(1)}", "parent_index": 3, "subtype": "inline"}, {"bbox": [488, 245, 501, 259], "content": "a_{j}^{\\vee}", "parent_index": 3, "subtype": "inline"}, {"bbox": [480, 259, 506, 271], "content": "\\lambda_{0}\\Lambda_{0}", "parent_index": 3, "subtype": "inline"}, {"bbox": [419, 274, 456, 286], "content": "\\operatorname{SL_{2}}(\\mathbb{Z})", "parent_index": 3, "subtype": "inline"}, {"bbox": [278, 299, 292, 308], "content": "\\chi_{\\mu}", "parent_index": 3, "subtype": "inline"}, {"bbox": [299, 296, 338, 308], "content": "\\mu\\in P_{+}", "parent_index": 3, "subtype": "inline"}, {"bbox": [447, 287, 502, 317], "content": "\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)", "parent_index": 3, "subtype": "inline"}, {"bbox": [220, 317, 228, 326], "content": "S", "parent_index": 3, "subtype": "inline"}, {"bbox": [175, 338, 435, 378], "content": "S_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname*{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)|\\nu+\\rho)}{\\kappa}]\\ .", "parent_index": 4, "subtype": "interline"}, {"bbox": [344, 397, 349, 403], "content": "c", "parent_index": 5, "subtype": "inline"}, {"bbox": [455, 407, 468, 417], "content": "\\overline{W}", "parent_index": 5, "subtype": "inline"}, {"bbox": [151, 423, 167, 433], "content": "X_{r}", "parent_index": 5, "subtype": "inline"}, {"bbox": [242, 423, 258, 434], "content": "P_{+}", "parent_index": 5, "subtype": "inline"}, {"bbox": [314, 423, 328, 434], "content": "\\Lambda_{0}", "parent_index": 5, "subtype": "inline"}, {"bbox": [432, 426, 439, 434], "content": "\\rho", "parent_index": 5, "subtype": "inline"}, {"bbox": [480, 422, 512, 435], "content": "\\sum_{i}\\Lambda_{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [158, 456, 166, 465], "content": "S", "parent_index": 6, "subtype": "inline"}, {"bbox": [409, 456, 430, 467], "content": "k\\Lambda_{0}", "parent_index": 6, "subtype": "inline"}, {"bbox": [159, 470, 169, 479], "content": "0^{\\circ}", "parent_index": 6, "subtype": "inline"}, {"bbox": [233, 497, 378, 511], "content": "S_{\\lambda0}>0\\qquad\\mathrm{for~all~}\\lambda\\in P_{+}\\ .", "parent_index": 7, "subtype": "interline"}, {"bbox": [218, 549, 393, 580], "content": "\\chi_{\\lambda}[\\mu]\\stackrel{\\mathrm{def}}{=}\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\mathrm{ch}_{\\overline{{{\\lambda}}}}(-2\\pi\\mathrm{i}\\frac{\\overline{{{\\mu}}}+\\overline{{{\\rho}}}}{\\kappa})~,", "parent_index": 9, "subtype": "interline"}, {"bbox": [106, 597, 125, 610], "content": "\\mathrm{ch}_{{\\overline{{\\lambda}}}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [275, 597, 291, 608], "content": "X_{r}", "parent_index": 10, "subtype": "inline"}, {"bbox": [336, 595, 361, 609], "content": "L(\\overline{{\\lambda}})", "parent_index": 10, "subtype": "inline"}, {"bbox": [351, 615, 357, 623], "content": "q", "parent_index": 10, "subtype": "inline"}, {"bbox": [200, 636, 410, 672], "content": "{\\mathcal D}(\\lambda)\\,\\overset{\\mathrm{def}}{=}\\frac{S_{\\lambda0}}{S_{00}}=\\prod_{\\alpha>0}\\frac{\\sin(\\pi\\left(\\lambda+\\rho\\left|\\alpha\\right)/\\kappa\\right)}{\\sin(\\pi\\left(\\rho\\left|\\alpha\\right)/\\kappa\\right)}~,", "parent_index": 11, "subtype": "interline"}, {"bbox": [301, 688, 341, 702], "content": "\\alpha\\in\\overline{{\\Delta}}_{+}", "parent_index": 12, "subtype": "inline"}, {"bbox": [357, 690, 373, 701], "content": "X_{r}", "parent_index": 12, "subtype": "inline"}]
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[]
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Charge-conjugation is the order 2 permutation of $P_{+}$ given by $C\lambda\,=\,^{t}\lambda$ , the weight contragredient to $\lambda$ . For instance $C0=0$ . It has the basic property that
$$
S_{C\lambda,\mu}=S_{\lambda,C\mu}=S_{\lambda\mu}^{*}
$$
and $S^{2}=C$ . $C$ corresponds to a symmetry of the (unextended) Dynkin diagram of $X_{r}$ , as we will see next section.
Related to $C$ are all the other symmetries of the unextended Dynkin diagram. We call these conjugations. The only $X_{r}^{(1)}$ with nontrivial conjugations other than chargeconjugation are $D_{e v e n}^{(1)}$ .
Another important symmetry of the matrix $S$ is called simple-currents. Any weight $j\in P_{+}$ with q-dimension $\mathcal{D}(j)=1$ , is called a simple-current. To any such weight $j$ is associated a permutation $J$ of $P_{+}$ and a function $Q_{j}:P_{+}\to\mathbb{Q}$ , such that $J0=j$ and
$$
S_{J\lambda,\mu}=\exp[2\pi\mathrm{i}\,Q_{j}(\mu)]\,S_{\lambda\mu}
$$
The simple-currents form an abelian group, given by composition of the permutations $J$ .
All simple-currents for the affine algebras were classified in [12], and with one unimportant exception ( ${E}_{8}^{(1)}$ at level 2) correspond to symmetries of the extended Coxeter–Dynkin diagram of $X_{r}^{(1)}$ . The simplest proof would use the methods of Proposition 4.1 below. For a more intrinsically algebraic interpretation of these simple-currents, see [25] where their group is denoted $W_{0}^{+}$ .
Evaluating $S_{J\lambda,j^{\prime}}$ in two ways gives the useful
$$
Q_{j^{\prime}}(J\lambda)\equiv Q_{j}(j^{\prime})+Q_{j^{\prime}}(\lambda)\qquad(\mathrm{mod}\ 1)
$$
and hence the reciprocity $Q_{j}(j^{\prime})=Q_{j^{\prime}}(j)$ .
For each $X_{r}$ , the inner products $(\lambda|\mu)$ of weights are rational; let $N$ denote the least common denominator. E.g. for $A_{r}$ this is $N=r+1$ , while for $E_{8}$ it is $N=1$ . Choose any integer $\ell$ coprime to $\kappa N$ . Then for any $\lambda\in P_{+}$ there is a unique weight $\lambda^{(\ell)}\in P_{+}$ , coroot $\alpha$ , and (finite) Weyl element $\omega$ such that
$$
\ell\left(\lambda+\rho\right)=\omega(\lambda^{\left(\ell\right)}+\rho)+\kappa\alpha\;.
$$
This is simply the statement that the affine Weyl orbit of $\ell\left(\lambda+\rho\right)$ intersects the set $P_{+}+\rho$ at precisely one point (namely $\lambda^{(\ell)}+\rho)$ . Write $\epsilon_{\ell}^{\prime}(\lambda)=\operatorname*{det}\omega=\pm1$ . Then [16]
$$
\epsilon_{\ell}^{\prime}(\lambda)\,S_{\lambda^{(\ell)},\mu}=\epsilon_{\ell}^{\prime}(\mu)\,S_{\lambda,\mu^{(\ell)}}
$$
This has an obvious interpretation as a Galois automorphism [4]: the field generated over $\mathbb{Q}$ by all entries $S_{\lambda\mu}$ lies in the cyclotomic field $\mathbb{Q}[\xi_{4N\kappa}]$ where $\xi_{n}$ denotes the root of unity $\exp[2\pi\mathrm{i}/n]$ ; for any $\sigma_{\ell}\in{\mathrm{Gal}}(\mathbb{Q}[\xi_{4N\kappa}]/\mathbb{Q})\cong\mathbb{Z}_{4N\kappa}^{\times}$ , there will be a function $\epsilon_{\ell}:P_{+}\to\{\pm1\}$ such that
$$
\sigma_{\ell}\bigl(S_{\lambda\mu}\bigr)=\epsilon_{\ell}\bigl(\lambda\bigr)\,S_{\lambda^{(\ell)},\mu}=\epsilon_{\ell}\bigl(\mu\bigr)\,S_{\lambda,\mu^{(\ell)}}\ .
$$
|
<html><body>
<p data-bbox="70 70 540 100">Charge-conjugation is the order 2 permutation of $P_{+}$ given by $C\lambda\,=\,^{t}\lambda$ , the weight contragredient to $\lambda$ . For instance $C0=0$ . It has the basic property that </p>
<div class="equation" data-bbox="250 117 361 132">$$
S_{C\lambda,\mu}=S_{\lambda,C\mu}=S_{\lambda\mu}^{*}
$$</div>
<p data-bbox="70 143 540 172">and $S^{2}=C$ . $C$ corresponds to a symmetry of the (unextended) Dynkin diagram of $X_{r}$ , as we will see next section. </p>
<p data-bbox="70 173 541 219">Related to $C$ are all the other symmetries of the unextended Dynkin diagram. We call these conjugations. The only $X_{r}^{(1)}$ with nontrivial conjugations other than chargeconjugation are $D_{e v e n}^{(1)}$ . </p>
<p data-bbox="70 219 541 263">Another important symmetry of the matrix $S$ is called simple-currents. Any weight $j\in P_{+}$ with q-dimension $\mathcal{D}(j)=1$ , is called a simple-current. To any such weight $j$ is associated a permutation $J$ of $P_{+}$ and a function $Q_{j}:P_{+}\to\mathbb{Q}$ , such that $J0=j$ and </p>
<div class="equation" data-bbox="235 278 376 293">$$
S_{J\lambda,\mu}=\exp[2\pi\mathrm{i}\,Q_{j}(\mu)]\,S_{\lambda\mu}
$$</div>
<p data-bbox="70 306 538 320">The simple-currents form an abelian group, given by composition of the permutations $J$ . </p>
<p data-bbox="70 321 541 397">All simple-currents for the affine algebras were classified in [12], and with one unimportant exception ( ${E}_{8}^{(1)}$ at level 2) correspond to symmetries of the extended Coxeter–Dynkin diagram of $X_{r}^{(1)}$ . The simplest proof would use the methods of Proposition 4.1 below. For a more intrinsically algebraic interpretation of these simple-currents, see [25] where their group is denoted $W_{0}^{+}$ . </p>
<p data-bbox="94 397 338 412">Evaluating $S_{J\lambda,j^{\prime}}$ in two ways gives the useful </p>
<div class="equation" data-bbox="201 426 409 443">$$
Q_{j^{\prime}}(J\lambda)\equiv Q_{j}(j^{\prime})+Q_{j^{\prime}}(\lambda)\qquad(\mathrm{mod}\ 1)
$$</div>
<p data-bbox="70 455 294 469">and hence the reciprocity $Q_{j}(j^{\prime})=Q_{j^{\prime}}(j)$ . </p>
<p data-bbox="70 470 541 527">For each $X_{r}$ , the inner products $(\lambda|\mu)$ of weights are rational; let $N$ denote the least common denominator. E.g. for $A_{r}$ this is $N=r+1$ , while for $E_{8}$ it is $N=1$ . Choose any integer $\ell$ coprime to $\kappa N$ . Then for any $\lambda\in P_{+}$ there is a unique weight $\lambda^{(\ell)}\in P_{+}$ , coroot $\alpha$ , and (finite) Weyl element $\omega$ such that </p>
<div class="equation" data-bbox="228 541 383 557">$$
\ell\left(\lambda+\rho\right)=\omega(\lambda^{\left(\ell\right)}+\rho)+\kappa\alpha\;.
$$</div>
<p data-bbox="70 570 541 600">This is simply the statement that the affine Weyl orbit of $\ell\left(\lambda+\rho\right)$ intersects the set $P_{+}+\rho$ at precisely one point (namely $\lambda^{(\ell)}+\rho)$ . Write $\epsilon_{\ell}^{\prime}(\lambda)=\operatorname*{det}\omega=\pm1$ . Then [16] </p>
<div class="equation" data-bbox="236 614 374 631">$$
\epsilon_{\ell}^{\prime}(\lambda)\,S_{\lambda^{(\ell)},\mu}=\epsilon_{\ell}^{\prime}(\mu)\,S_{\lambda,\mu^{(\ell)}}
$$</div>
<p data-bbox="68 642 541 699">This has an obvious interpretation as a Galois automorphism [4]: the field generated over $\mathbb{Q}$ by all entries $S_{\lambda\mu}$ lies in the cyclotomic field $\mathbb{Q}[\xi_{4N\kappa}]$ where $\xi_{n}$ denotes the root of unity $\exp[2\pi\mathrm{i}/n]$ ; for any $\sigma_{\ell}\in{\mathrm{Gal}}(\mathbb{Q}[\xi_{4N\kappa}]/\mathbb{Q})\cong\mathbb{Z}_{4N\kappa}^{\times}$ , there will be a function $\epsilon_{\ell}:P_{+}\to\{\pm1\}$ such that </p>
<div class="equation" data-bbox="204 701 407 718">$$
\sigma_{\ell}\bigl(S_{\lambda\mu}\bigr)=\epsilon_{\ell}\bigl(\lambda\bigr)\,S_{\lambda^{(\ell)},\mu}=\epsilon_{\ell}\bigl(\mu\bigr)\,S_{\lambda,\mu^{(\ell)}}\ .
$$</div>
</body></html>
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0002044v1
| 3 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Charge-conjugation is the order 2 permutation of $P_{+}$ given by $C\\lambda\\,=\\,^{t}\\lambda$ , the weight contragredient to $\\lambda$ . For instance $C0=0$ . It has the basic property that ", "page_idx": 3}, {"type": "equation", "text": "$$\nS_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{*}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "and $S^{2}=C$ . $C$ corresponds to a symmetry of the (unextended) Dynkin diagram of $X_{r}$ , as we will see next section. ", "page_idx": 3}, {"type": "text", "text": "Related to $C$ are all the other symmetries of the unextended Dynkin diagram. We call these conjugations. The only $X_{r}^{(1)}$ with nontrivial conjugations other than chargeconjugation are $D_{e v e n}^{(1)}$ . ", "page_idx": 3}, {"type": "text", "text": "Another important symmetry of the matrix $S$ is called simple-currents. Any weight $j\\in P_{+}$ with q-dimension $\\mathcal{D}(j)=1$ , is called a simple-current. To any such weight $j$ is associated a permutation $J$ of $P_{+}$ and a function $Q_{j}:P_{+}\\to\\mathbb{Q}$ , such that $J0=j$ and ", "page_idx": 3}, {"type": "equation", "text": "$$\nS_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "The simple-currents form an abelian group, given by composition of the permutations $J$ . ", "page_idx": 3}, {"type": "text", "text": "All simple-currents for the affine algebras were classified in [12], and with one unimportant exception ( ${E}_{8}^{(1)}$ at level 2) correspond to symmetries of the extended Coxeter–Dynkin diagram of $X_{r}^{(1)}$ . The simplest proof would use the methods of Proposition 4.1 below. For a more intrinsically algebraic interpretation of these simple-currents, see [25] where their group is denoted $W_{0}^{+}$ . ", "page_idx": 3}, {"type": "text", "text": "Evaluating $S_{J\\lambda,j^{\\prime}}$ in two ways gives the useful ", "page_idx": 3}, {"type": "equation", "text": "$$\nQ_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "and hence the reciprocity $Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)$ . ", "page_idx": 3}, {"type": "text", "text": "For each $X_{r}$ , the inner products $(\\lambda|\\mu)$ of weights are rational; let $N$ denote the least common denominator. E.g. for $A_{r}$ this is $N=r+1$ , while for $E_{8}$ it is $N=1$ . Choose any integer $\\ell$ coprime to $\\kappa N$ . Then for any $\\lambda\\in P_{+}$ there is a unique weight $\\lambda^{(\\ell)}\\in P_{+}$ , coroot $\\alpha$ , and (finite) Weyl element $\\omega$ such that ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "This is simply the statement that the affine Weyl orbit of $\\ell\\left(\\lambda+\\rho\\right)$ intersects the set $P_{+}+\\rho$ at precisely one point (namely $\\lambda^{(\\ell)}+\\rho)$ . Write $\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname*{det}\\omega=\\pm1$ . Then [16] ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over $\\mathbb{Q}$ by all entries $S_{\\lambda\\mu}$ lies in the cyclotomic field $\\mathbb{Q}[\\xi_{4N\\kappa}]$ where $\\xi_{n}$ denotes the root of unity $\\exp[2\\pi\\mathrm{i}/n]$ ; for any $\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}$ , there will be a function $\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}$ such that ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .\n$$", "text_format": "latex", "page_idx": 3}]
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{"preproc_blocks": [{"type": "text", "bbox": [70, 70, 540, 100], "lines": [{"bbox": [95, 72, 542, 90], "spans": [{"bbox": [95, 72, 360, 90], "score": 1.0, "content": "Charge-conjugation is the order 2 permutation of ", "type": "text"}, {"bbox": [360, 75, 376, 87], "score": 0.93, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [376, 72, 429, 90], "score": 1.0, "content": " given by ", "type": "text"}, {"bbox": [430, 74, 476, 84], "score": 0.93, "content": "C\\lambda\\,=\\,^{t}\\lambda", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [477, 72, 542, 90], "score": 1.0, "content": ", the weight", "type": "text"}], "index": 0}, {"bbox": [70, 88, 452, 104], "spans": [{"bbox": [70, 88, 165, 104], "score": 1.0, "content": "contragredient to ", "type": "text"}, {"bbox": [165, 90, 172, 99], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [173, 88, 248, 104], "score": 1.0, "content": ". For instance ", "type": "text"}, {"bbox": [248, 90, 286, 99], "score": 0.93, "content": "C0=0", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 88, 452, 104], "score": 1.0, "content": ". It has the basic property that", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [250, 117, 361, 132], "lines": [{"bbox": [250, 117, 361, 132], "spans": [{"bbox": [250, 117, 361, 132], "score": 0.93, "content": "S_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{*}", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [70, 143, 540, 172], "lines": [{"bbox": [70, 145, 540, 162], "spans": [{"bbox": [70, 145, 95, 162], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 146, 136, 156], "score": 0.92, "content": "S^{2}=C", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [136, 145, 145, 162], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [146, 148, 155, 157], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [156, 145, 520, 162], "score": 1.0, "content": " corresponds to a symmetry of the (unextended) Dynkin diagram of ", "type": "text"}, {"bbox": [520, 148, 536, 159], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [537, 145, 540, 162], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [71, 161, 212, 174], "spans": [{"bbox": [71, 161, 212, 174], "score": 1.0, "content": "as we will see next section.", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [70, 173, 541, 219], "lines": [{"bbox": [94, 174, 540, 190], "spans": [{"bbox": [94, 174, 155, 190], "score": 1.0, "content": "Related to ", "type": "text"}, {"bbox": [155, 177, 165, 185], "score": 0.9, "content": "C", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [165, 174, 540, 190], "score": 1.0, "content": " are all the other symmetries of the unextended Dynkin diagram. We", "type": "text"}], "index": 5}, {"bbox": [68, 186, 542, 209], "spans": [{"bbox": [68, 186, 256, 209], "score": 1.0, "content": "call these conjugations. The only ", "type": "text"}, {"bbox": [256, 188, 280, 203], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [281, 186, 542, 209], "score": 1.0, "content": "with nontrivial conjugations other than charge-", "type": "text"}], "index": 6}, {"bbox": [70, 203, 192, 226], "spans": [{"bbox": [70, 203, 156, 226], "score": 1.0, "content": "conjugation are ", "type": "text"}, {"bbox": [156, 204, 187, 219], "score": 0.92, "content": "D_{e v e n}^{(1)}", "type": "inline_equation", "height": 15, "width": 31}, {"bbox": [187, 203, 192, 226], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 6}, {"type": "text", "bbox": [70, 219, 541, 263], "lines": [{"bbox": [95, 221, 541, 236], "spans": [{"bbox": [95, 221, 330, 236], "score": 1.0, "content": "Another important symmetry of the matrix ", "type": "text"}, {"bbox": [330, 223, 339, 232], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [339, 221, 541, 236], "score": 1.0, "content": " is called simple-currents. Any weight", "type": "text"}], "index": 8}, {"bbox": [71, 236, 541, 250], "spans": [{"bbox": [71, 238, 110, 249], "score": 0.93, "content": "j\\in P_{+}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [110, 236, 210, 250], "score": 1.0, "content": " with q-dimension ", "type": "text"}, {"bbox": [210, 237, 260, 249], "score": 0.94, "content": "\\mathcal{D}(j)=1", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [261, 236, 520, 250], "score": 1.0, "content": ", is called a simple-current. To any such weight ", "type": "text"}, {"bbox": [520, 238, 527, 249], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [527, 236, 541, 250], "score": 1.0, "content": " is", "type": "text"}], "index": 9}, {"bbox": [71, 250, 520, 264], "spans": [{"bbox": [71, 250, 206, 264], "score": 1.0, "content": "associated a permutation ", "type": "text"}, {"bbox": [207, 252, 214, 261], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [215, 250, 231, 264], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [232, 252, 248, 263], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [248, 250, 331, 264], "score": 1.0, "content": " and a function ", "type": "text"}, {"bbox": [332, 252, 400, 264], "score": 0.93, "content": "Q_{j}:P_{+}\\to\\mathbb{Q}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [401, 250, 459, 264], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [460, 252, 495, 263], "score": 0.94, "content": "J0=j", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [496, 250, 520, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 10}], "index": 9}, {"type": "interline_equation", "bbox": [235, 278, 376, 293], "lines": [{"bbox": [235, 278, 376, 293], "spans": [{"bbox": [235, 278, 376, 293], "score": 0.93, "content": "S_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [70, 306, 538, 320], "lines": [{"bbox": [71, 308, 537, 323], "spans": [{"bbox": [71, 308, 525, 323], "score": 1.0, "content": "The simple-currents form an abelian group, given by composition of the permutations ", "type": "text"}, {"bbox": [525, 310, 533, 319], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [533, 308, 537, 323], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [70, 321, 541, 397], "lines": [{"bbox": [94, 322, 541, 338], "spans": [{"bbox": [94, 322, 541, 338], "score": 1.0, "content": "All simple-currents for the affine algebras were classified in [12], and with one unimpor-", "type": "text"}], "index": 13}, {"bbox": [69, 334, 541, 358], "spans": [{"bbox": [69, 334, 154, 358], "score": 1.0, "content": "tant exception (", "type": "text"}, {"bbox": [154, 336, 176, 353], "score": 0.92, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 17, "width": 22}, {"bbox": [177, 334, 541, 358], "score": 1.0, "content": "at level 2) correspond to symmetries of the extended Coxeter–Dynkin", "type": "text"}], "index": 14}, {"bbox": [69, 353, 542, 372], "spans": [{"bbox": [69, 353, 130, 372], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [131, 354, 154, 368], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [155, 353, 542, 372], "score": 1.0, "content": ". The simplest proof would use the methods of Proposition 4.1 below. For", "type": "text"}], "index": 15}, {"bbox": [70, 371, 540, 385], "spans": [{"bbox": [70, 371, 540, 385], "score": 1.0, "content": "a more intrinsically algebraic interpretation of these simple-currents, see [25] where their", "type": "text"}], "index": 16}, {"bbox": [69, 384, 188, 402], "spans": [{"bbox": [69, 384, 162, 402], "score": 1.0, "content": "group is denoted ", "type": "text"}, {"bbox": [162, 385, 183, 398], "score": 0.93, "content": "W_{0}^{+}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [184, 384, 188, 402], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 15}, {"type": "text", "bbox": [94, 397, 338, 412], "lines": [{"bbox": [95, 399, 336, 414], "spans": [{"bbox": [95, 399, 155, 414], "score": 1.0, "content": "Evaluating ", "type": "text"}, {"bbox": [155, 401, 185, 413], "score": 0.93, "content": "S_{J\\lambda,j^{\\prime}}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [186, 399, 336, 414], "score": 1.0, "content": " in two ways gives the useful", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [201, 426, 409, 443], "lines": [{"bbox": [201, 426, 409, 443], "spans": [{"bbox": [201, 426, 409, 443], "score": 0.89, "content": "Q_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [70, 455, 294, 469], "lines": [{"bbox": [70, 457, 293, 472], "spans": [{"bbox": [70, 457, 208, 472], "score": 1.0, "content": "and hence the reciprocity ", "type": "text"}, {"bbox": [208, 458, 289, 471], "score": 0.93, "content": "Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [290, 457, 293, 472], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 470, 541, 527], "lines": [{"bbox": [94, 471, 541, 487], "spans": [{"bbox": [94, 471, 143, 487], "score": 1.0, "content": "For each ", "type": "text"}, {"bbox": [144, 473, 159, 484], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [159, 471, 267, 487], "score": 1.0, "content": ", the inner products ", "type": "text"}, {"bbox": [268, 473, 295, 485], "score": 0.93, "content": "(\\lambda|\\mu)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [295, 471, 441, 487], "score": 1.0, "content": " of weights are rational; let ", "type": "text"}, {"bbox": [441, 473, 452, 482], "score": 0.89, "content": "N", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [452, 471, 541, 487], "score": 1.0, "content": " denote the least", "type": "text"}], "index": 21}, {"bbox": [69, 484, 541, 502], "spans": [{"bbox": [69, 484, 235, 502], "score": 1.0, "content": "common denominator. E.g. for ", "type": "text"}, {"bbox": [235, 488, 250, 498], "score": 0.92, "content": "A_{r}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [250, 484, 288, 502], "score": 1.0, "content": " this is ", "type": "text"}, {"bbox": [288, 488, 341, 497], "score": 0.9, "content": "N=r+1", "type": "inline_equation", "height": 9, "width": 53}, {"bbox": [341, 484, 396, 502], "score": 1.0, "content": ", while for ", "type": "text"}, {"bbox": [396, 487, 411, 498], "score": 0.86, "content": "E_{8}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [411, 484, 437, 502], "score": 1.0, "content": " it is ", "type": "text"}, {"bbox": [438, 488, 471, 497], "score": 0.92, "content": "N=1", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [471, 484, 541, 502], "score": 1.0, "content": ". Choose any", "type": "text"}], "index": 22}, {"bbox": [69, 499, 542, 515], "spans": [{"bbox": [69, 499, 110, 515], "score": 1.0, "content": "integer ", "type": "text"}, {"bbox": [111, 502, 116, 511], "score": 0.89, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [117, 499, 179, 515], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [180, 502, 198, 511], "score": 0.91, "content": "\\kappa N", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [198, 499, 278, 515], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [279, 502, 316, 514], "score": 0.94, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [316, 499, 449, 515], "score": 1.0, "content": " there is a unique weight ", "type": "text"}, {"bbox": [449, 500, 499, 514], "score": 0.93, "content": "\\lambda^{(\\ell)}\\in P_{+}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [500, 499, 542, 515], "score": 1.0, "content": ", coroot", "type": "text"}], "index": 23}, {"bbox": [71, 514, 286, 529], "spans": [{"bbox": [71, 520, 79, 525], "score": 0.85, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [80, 514, 223, 529], "score": 1.0, "content": ", and (finite) Weyl element ", "type": "text"}, {"bbox": [223, 520, 232, 525], "score": 0.89, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [232, 514, 286, 529], "score": 1.0, "content": " such that", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "interline_equation", "bbox": [228, 541, 383, 557], "lines": [{"bbox": [228, 541, 383, 557], "spans": [{"bbox": [228, 541, 383, 557], "score": 0.91, "content": "\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [70, 570, 541, 600], "lines": [{"bbox": [70, 572, 540, 589], "spans": [{"bbox": [70, 572, 369, 589], "score": 1.0, "content": "This is simply the statement that the affine Weyl orbit of ", "type": "text"}, {"bbox": [370, 573, 412, 586], "score": 0.93, "content": "\\ell\\left(\\lambda+\\rho\\right)", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [412, 572, 505, 589], "score": 1.0, "content": " intersects the set ", "type": "text"}, {"bbox": [505, 575, 540, 586], "score": 0.93, "content": "P_{+}+\\rho", "type": "inline_equation", "height": 11, "width": 35}], "index": 26}, {"bbox": [70, 587, 479, 603], "spans": [{"bbox": [70, 587, 234, 603], "score": 1.0, "content": "at precisely one point (namely ", "type": "text"}, {"bbox": [234, 587, 277, 600], "score": 0.93, "content": "\\lambda^{(\\ell)}+\\rho)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [277, 587, 320, 603], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [320, 587, 420, 601], "score": 0.92, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname*{det}\\omega=\\pm1", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [421, 587, 479, 603], "score": 1.0, "content": ". Then [16]", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "interline_equation", "bbox": [236, 614, 374, 631], "lines": [{"bbox": [236, 614, 374, 631], "spans": [{"bbox": [236, 614, 374, 631], "score": 0.93, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [68, 642, 541, 699], "lines": [{"bbox": [71, 644, 541, 661], "spans": [{"bbox": [71, 644, 541, 661], "score": 1.0, "content": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over", "type": "text"}], "index": 29}, {"bbox": [71, 659, 540, 675], "spans": [{"bbox": [71, 661, 81, 672], "score": 0.88, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [81, 659, 154, 675], "score": 1.0, "content": " by all entries ", "type": "text"}, {"bbox": [154, 661, 174, 673], "score": 0.92, "content": "S_{\\lambda\\mu}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [174, 659, 316, 675], "score": 1.0, "content": " lies in the cyclotomic field ", "type": "text"}, {"bbox": [316, 660, 357, 673], "score": 0.92, "content": "\\mathbb{Q}[\\xi_{4N\\kappa}]", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [358, 659, 395, 675], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [395, 661, 407, 672], "score": 0.92, "content": "\\xi_{n}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [407, 659, 540, 675], "score": 1.0, "content": " denotes the root of unity", "type": "text"}], "index": 30}, {"bbox": [71, 671, 540, 692], "spans": [{"bbox": [71, 674, 127, 687], "score": 0.92, "content": "\\exp[2\\pi\\mathrm{i}/n]", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [127, 671, 173, 692], "score": 1.0, "content": "; for any ", "type": "text"}, {"bbox": [173, 674, 327, 687], "score": 0.91, "content": "\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}", "type": "inline_equation", "height": 13, "width": 154}, {"bbox": [327, 671, 458, 692], "score": 1.0, "content": ", there will be a function ", "type": "text"}, {"bbox": [459, 675, 540, 687], "score": 0.94, "content": "\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}", "type": "inline_equation", "height": 12, "width": 81}], "index": 31}, {"bbox": [71, 688, 122, 701], "spans": [{"bbox": [71, 688, 122, 701], "score": 1.0, "content": "such that", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "interline_equation", "bbox": [204, 701, 407, 718], "lines": [{"bbox": [204, 701, 407, 718], "spans": [{"bbox": [204, 701, 407, 718], "score": 0.91, "content": "\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .", "type": "interline_equation"}], "index": 33}], "index": 33}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [250, 117, 361, 132], "lines": [{"bbox": [250, 117, 361, 132], "spans": [{"bbox": [250, 117, 361, 132], "score": 0.93, "content": "S_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{*}", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [235, 278, 376, 293], "lines": [{"bbox": [235, 278, 376, 293], "spans": [{"bbox": [235, 278, 376, 293], "score": 0.93, "content": "S_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [201, 426, 409, 443], "lines": [{"bbox": [201, 426, 409, 443], "spans": [{"bbox": [201, 426, 409, 443], "score": 0.89, "content": "Q_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [228, 541, 383, 557], "lines": [{"bbox": [228, 541, 383, 557], "spans": [{"bbox": [228, 541, 383, 557], "score": 0.91, "content": "\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "interline_equation", "bbox": [236, 614, 374, 631], "lines": [{"bbox": [236, 614, 374, 631], "spans": [{"bbox": [236, 614, 374, 631], "score": 0.93, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "interline_equation", "bbox": [204, 701, 407, 718], "lines": [{"bbox": [204, 701, 407, 718], "spans": [{"bbox": [204, 701, 407, 718], "score": 0.91, "content": "\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .", "type": "interline_equation"}], "index": 33}], "index": 33}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 540, 100], "lines": [{"bbox": [95, 72, 542, 90], "spans": [{"bbox": [95, 72, 360, 90], "score": 1.0, "content": "Charge-conjugation is the order 2 permutation of ", "type": "text"}, {"bbox": [360, 75, 376, 87], "score": 0.93, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [376, 72, 429, 90], "score": 1.0, "content": " given by ", "type": "text"}, {"bbox": [430, 74, 476, 84], "score": 0.93, "content": "C\\lambda\\,=\\,^{t}\\lambda", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [477, 72, 542, 90], "score": 1.0, "content": ", the weight", "type": "text"}], "index": 0}, {"bbox": [70, 88, 452, 104], "spans": [{"bbox": [70, 88, 165, 104], "score": 1.0, "content": "contragredient to ", "type": "text"}, {"bbox": [165, 90, 172, 99], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [173, 88, 248, 104], "score": 1.0, "content": ". For instance ", "type": "text"}, {"bbox": [248, 90, 286, 99], "score": 0.93, "content": "C0=0", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 88, 452, 104], "score": 1.0, "content": ". It has the basic property that", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [70, 72, 542, 104]}, {"type": "interline_equation", "bbox": [250, 117, 361, 132], "lines": [{"bbox": [250, 117, 361, 132], "spans": [{"bbox": [250, 117, 361, 132], "score": 0.93, "content": "S_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{*}", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [70, 143, 540, 172], "lines": [{"bbox": [70, 145, 540, 162], "spans": [{"bbox": [70, 145, 95, 162], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 146, 136, 156], "score": 0.92, "content": "S^{2}=C", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [136, 145, 145, 162], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [146, 148, 155, 157], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [156, 145, 520, 162], "score": 1.0, "content": " corresponds to a symmetry of the (unextended) Dynkin diagram of ", "type": "text"}, {"bbox": [520, 148, 536, 159], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [537, 145, 540, 162], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [71, 161, 212, 174], "spans": [{"bbox": [71, 161, 212, 174], "score": 1.0, "content": "as we will see next section.", "type": "text"}], "index": 4}], "index": 3.5, "bbox_fs": [70, 145, 540, 174]}, {"type": "text", "bbox": [70, 173, 541, 219], "lines": [{"bbox": [94, 174, 540, 190], "spans": [{"bbox": [94, 174, 155, 190], "score": 1.0, "content": "Related to ", "type": "text"}, {"bbox": [155, 177, 165, 185], "score": 0.9, "content": "C", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [165, 174, 540, 190], "score": 1.0, "content": " are all the other symmetries of the unextended Dynkin diagram. We", "type": "text"}], "index": 5}, {"bbox": [68, 186, 542, 209], "spans": [{"bbox": [68, 186, 256, 209], "score": 1.0, "content": "call these conjugations. The only ", "type": "text"}, {"bbox": [256, 188, 280, 203], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [281, 186, 542, 209], "score": 1.0, "content": "with nontrivial conjugations other than charge-", "type": "text"}], "index": 6}, {"bbox": [70, 203, 192, 226], "spans": [{"bbox": [70, 203, 156, 226], "score": 1.0, "content": "conjugation are ", "type": "text"}, {"bbox": [156, 204, 187, 219], "score": 0.92, "content": "D_{e v e n}^{(1)}", "type": "inline_equation", "height": 15, "width": 31}, {"bbox": [187, 203, 192, 226], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 6, "bbox_fs": [68, 174, 542, 226]}, {"type": "text", "bbox": [70, 219, 541, 263], "lines": [{"bbox": [95, 221, 541, 236], "spans": [{"bbox": [95, 221, 330, 236], "score": 1.0, "content": "Another important symmetry of the matrix ", "type": "text"}, {"bbox": [330, 223, 339, 232], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [339, 221, 541, 236], "score": 1.0, "content": " is called simple-currents. Any weight", "type": "text"}], "index": 8}, {"bbox": [71, 236, 541, 250], "spans": [{"bbox": [71, 238, 110, 249], "score": 0.93, "content": "j\\in P_{+}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [110, 236, 210, 250], "score": 1.0, "content": " with q-dimension ", "type": "text"}, {"bbox": [210, 237, 260, 249], "score": 0.94, "content": "\\mathcal{D}(j)=1", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [261, 236, 520, 250], "score": 1.0, "content": ", is called a simple-current. To any such weight ", "type": "text"}, {"bbox": [520, 238, 527, 249], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [527, 236, 541, 250], "score": 1.0, "content": " is", "type": "text"}], "index": 9}, {"bbox": [71, 250, 520, 264], "spans": [{"bbox": [71, 250, 206, 264], "score": 1.0, "content": "associated a permutation ", "type": "text"}, {"bbox": [207, 252, 214, 261], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [215, 250, 231, 264], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [232, 252, 248, 263], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [248, 250, 331, 264], "score": 1.0, "content": " and a function ", "type": "text"}, {"bbox": [332, 252, 400, 264], "score": 0.93, "content": "Q_{j}:P_{+}\\to\\mathbb{Q}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [401, 250, 459, 264], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [460, 252, 495, 263], "score": 0.94, "content": "J0=j", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [496, 250, 520, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 10}], "index": 9, "bbox_fs": [71, 221, 541, 264]}, {"type": "interline_equation", "bbox": [235, 278, 376, 293], "lines": [{"bbox": [235, 278, 376, 293], "spans": [{"bbox": [235, 278, 376, 293], "score": 0.93, "content": "S_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [70, 306, 538, 320], "lines": [{"bbox": [71, 308, 537, 323], "spans": [{"bbox": [71, 308, 525, 323], "score": 1.0, "content": "The simple-currents form an abelian group, given by composition of the permutations ", "type": "text"}, {"bbox": [525, 310, 533, 319], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [533, 308, 537, 323], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [71, 308, 537, 323]}, {"type": "text", "bbox": [70, 321, 541, 397], "lines": [{"bbox": [94, 322, 541, 338], "spans": [{"bbox": [94, 322, 541, 338], "score": 1.0, "content": "All simple-currents for the affine algebras were classified in [12], and with one unimpor-", "type": "text"}], "index": 13}, {"bbox": [69, 334, 541, 358], "spans": [{"bbox": [69, 334, 154, 358], "score": 1.0, "content": "tant exception (", "type": "text"}, {"bbox": [154, 336, 176, 353], "score": 0.92, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 17, "width": 22}, {"bbox": [177, 334, 541, 358], "score": 1.0, "content": "at level 2) correspond to symmetries of the extended Coxeter–Dynkin", "type": "text"}], "index": 14}, {"bbox": [69, 353, 542, 372], "spans": [{"bbox": [69, 353, 130, 372], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [131, 354, 154, 368], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [155, 353, 542, 372], "score": 1.0, "content": ". The simplest proof would use the methods of Proposition 4.1 below. For", "type": "text"}], "index": 15}, {"bbox": [70, 371, 540, 385], "spans": [{"bbox": [70, 371, 540, 385], "score": 1.0, "content": "a more intrinsically algebraic interpretation of these simple-currents, see [25] where their", "type": "text"}], "index": 16}, {"bbox": [69, 384, 188, 402], "spans": [{"bbox": [69, 384, 162, 402], "score": 1.0, "content": "group is denoted ", "type": "text"}, {"bbox": [162, 385, 183, 398], "score": 0.93, "content": "W_{0}^{+}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [184, 384, 188, 402], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 15, "bbox_fs": [69, 322, 542, 402]}, {"type": "text", "bbox": [94, 397, 338, 412], "lines": [{"bbox": [95, 399, 336, 414], "spans": [{"bbox": [95, 399, 155, 414], "score": 1.0, "content": "Evaluating ", "type": "text"}, {"bbox": [155, 401, 185, 413], "score": 0.93, "content": "S_{J\\lambda,j^{\\prime}}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [186, 399, 336, 414], "score": 1.0, "content": " in two ways gives the useful", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [95, 399, 336, 414]}, {"type": "interline_equation", "bbox": [201, 426, 409, 443], "lines": [{"bbox": [201, 426, 409, 443], "spans": [{"bbox": [201, 426, 409, 443], "score": 0.89, "content": "Q_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [70, 455, 294, 469], "lines": [{"bbox": [70, 457, 293, 472], "spans": [{"bbox": [70, 457, 208, 472], "score": 1.0, "content": "and hence the reciprocity ", "type": "text"}, {"bbox": [208, 458, 289, 471], "score": 0.93, "content": "Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [290, 457, 293, 472], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 20, "bbox_fs": [70, 457, 293, 472]}, {"type": "text", "bbox": [70, 470, 541, 527], "lines": [{"bbox": [94, 471, 541, 487], "spans": [{"bbox": [94, 471, 143, 487], "score": 1.0, "content": "For each ", "type": "text"}, {"bbox": [144, 473, 159, 484], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [159, 471, 267, 487], "score": 1.0, "content": ", the inner products ", "type": "text"}, {"bbox": [268, 473, 295, 485], "score": 0.93, "content": "(\\lambda|\\mu)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [295, 471, 441, 487], "score": 1.0, "content": " of weights are rational; let ", "type": "text"}, {"bbox": [441, 473, 452, 482], "score": 0.89, "content": "N", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [452, 471, 541, 487], "score": 1.0, "content": " denote the least", "type": "text"}], "index": 21}, {"bbox": [69, 484, 541, 502], "spans": [{"bbox": [69, 484, 235, 502], "score": 1.0, "content": "common denominator. E.g. for ", "type": "text"}, {"bbox": [235, 488, 250, 498], "score": 0.92, "content": "A_{r}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [250, 484, 288, 502], "score": 1.0, "content": " this is ", "type": "text"}, {"bbox": [288, 488, 341, 497], "score": 0.9, "content": "N=r+1", "type": "inline_equation", "height": 9, "width": 53}, {"bbox": [341, 484, 396, 502], "score": 1.0, "content": ", while for ", "type": "text"}, {"bbox": [396, 487, 411, 498], "score": 0.86, "content": "E_{8}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [411, 484, 437, 502], "score": 1.0, "content": " it is ", "type": "text"}, {"bbox": [438, 488, 471, 497], "score": 0.92, "content": "N=1", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [471, 484, 541, 502], "score": 1.0, "content": ". Choose any", "type": "text"}], "index": 22}, {"bbox": [69, 499, 542, 515], "spans": [{"bbox": [69, 499, 110, 515], "score": 1.0, "content": "integer ", "type": "text"}, {"bbox": [111, 502, 116, 511], "score": 0.89, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [117, 499, 179, 515], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [180, 502, 198, 511], "score": 0.91, "content": "\\kappa N", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [198, 499, 278, 515], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [279, 502, 316, 514], "score": 0.94, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [316, 499, 449, 515], "score": 1.0, "content": " there is a unique weight ", "type": "text"}, {"bbox": [449, 500, 499, 514], "score": 0.93, "content": "\\lambda^{(\\ell)}\\in P_{+}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [500, 499, 542, 515], "score": 1.0, "content": ", coroot", "type": "text"}], "index": 23}, {"bbox": [71, 514, 286, 529], "spans": [{"bbox": [71, 520, 79, 525], "score": 0.85, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [80, 514, 223, 529], "score": 1.0, "content": ", and (finite) Weyl element ", "type": "text"}, {"bbox": [223, 520, 232, 525], "score": 0.89, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [232, 514, 286, 529], "score": 1.0, "content": " such that", "type": "text"}], "index": 24}], "index": 22.5, "bbox_fs": [69, 471, 542, 529]}, {"type": "interline_equation", "bbox": [228, 541, 383, 557], "lines": [{"bbox": [228, 541, 383, 557], "spans": [{"bbox": [228, 541, 383, 557], "score": 0.91, "content": "\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [70, 570, 541, 600], "lines": [{"bbox": [70, 572, 540, 589], "spans": [{"bbox": [70, 572, 369, 589], "score": 1.0, "content": "This is simply the statement that the affine Weyl orbit of ", "type": "text"}, {"bbox": [370, 573, 412, 586], "score": 0.93, "content": "\\ell\\left(\\lambda+\\rho\\right)", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [412, 572, 505, 589], "score": 1.0, "content": " intersects the set ", "type": "text"}, {"bbox": [505, 575, 540, 586], "score": 0.93, "content": "P_{+}+\\rho", "type": "inline_equation", "height": 11, "width": 35}], "index": 26}, {"bbox": [70, 587, 479, 603], "spans": [{"bbox": [70, 587, 234, 603], "score": 1.0, "content": "at precisely one point (namely ", "type": "text"}, {"bbox": [234, 587, 277, 600], "score": 0.93, "content": "\\lambda^{(\\ell)}+\\rho)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [277, 587, 320, 603], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [320, 587, 420, 601], "score": 0.92, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname*{det}\\omega=\\pm1", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [421, 587, 479, 603], "score": 1.0, "content": ". Then [16]", "type": "text"}], "index": 27}], "index": 26.5, "bbox_fs": [70, 572, 540, 603]}, {"type": "interline_equation", "bbox": [236, 614, 374, 631], "lines": [{"bbox": [236, 614, 374, 631], "spans": [{"bbox": [236, 614, 374, 631], "score": 0.93, "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [68, 642, 541, 699], "lines": [{"bbox": [71, 644, 541, 661], "spans": [{"bbox": [71, 644, 541, 661], "score": 1.0, "content": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over", "type": "text"}], "index": 29}, {"bbox": [71, 659, 540, 675], "spans": [{"bbox": [71, 661, 81, 672], "score": 0.88, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [81, 659, 154, 675], "score": 1.0, "content": " by all entries ", "type": "text"}, {"bbox": [154, 661, 174, 673], "score": 0.92, "content": "S_{\\lambda\\mu}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [174, 659, 316, 675], "score": 1.0, "content": " lies in the cyclotomic field ", "type": "text"}, {"bbox": [316, 660, 357, 673], "score": 0.92, "content": "\\mathbb{Q}[\\xi_{4N\\kappa}]", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [358, 659, 395, 675], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [395, 661, 407, 672], "score": 0.92, "content": "\\xi_{n}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [407, 659, 540, 675], "score": 1.0, "content": " denotes the root of unity", "type": "text"}], "index": 30}, {"bbox": [71, 671, 540, 692], "spans": [{"bbox": [71, 674, 127, 687], "score": 0.92, "content": "\\exp[2\\pi\\mathrm{i}/n]", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [127, 671, 173, 692], "score": 1.0, "content": "; for any ", "type": "text"}, {"bbox": [173, 674, 327, 687], "score": 0.91, "content": "\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}", "type": "inline_equation", "height": 13, "width": 154}, {"bbox": [327, 671, 458, 692], "score": 1.0, "content": ", there will be a function ", "type": "text"}, {"bbox": [459, 675, 540, 687], "score": 0.94, "content": "\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}", "type": "inline_equation", "height": 12, "width": 81}], "index": 31}, {"bbox": [71, 688, 122, 701], "spans": [{"bbox": [71, 688, 122, 701], "score": 1.0, "content": "such that", "type": "text"}], "index": 32}], "index": 30.5, "bbox_fs": [71, 644, 541, 701]}, {"type": "interline_equation", "bbox": [204, 701, 407, 718], "lines": [{"bbox": [204, 701, 407, 718], "spans": [{"bbox": [204, 701, 407, 718], "score": 0.91, "content": "\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .", "type": "interline_equation"}], "index": 33}], "index": 33}]}
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[{"type": "text", "bbox": [70, 70, 540, 100], "content": "Charge-conjugation is the order 2 permutation of given by , the weight contragredient to . For instance . It has the basic property that", "index": 0}, {"type": "interline_equation", "bbox": [250, 117, 361, 132], "content": "", "index": 1}, {"type": "text", "bbox": [70, 143, 540, 172], "content": "and . corresponds to a symmetry of the (unextended) Dynkin diagram of , as we will see next section.", "index": 2}, {"type": "text", "bbox": [70, 173, 541, 219], "content": "Related to are all the other symmetries of the unextended Dynkin diagram. We call these conjugations. The only with nontrivial conjugations other than charge- conjugation are .", "index": 3}, {"type": "text", "bbox": [70, 219, 541, 263], "content": "Another important symmetry of the matrix is called simple-currents. Any weight with q-dimension , is called a simple-current. To any such weight is associated a permutation of and a function , such that and", "index": 4}, {"type": "interline_equation", "bbox": [235, 278, 376, 293], "content": "", "index": 5}, {"type": "text", "bbox": [70, 306, 538, 320], "content": "The simple-currents form an abelian group, given by composition of the permutations .", "index": 6}, {"type": "text", "bbox": [70, 321, 541, 397], "content": "All simple-currents for the affine algebras were classified in [12], and with one unimpor- tant exception ( at level 2) correspond to symmetries of the extended Coxeter–Dynkin diagram of . The simplest proof would use the methods of Proposition 4.1 below. For a more intrinsically algebraic interpretation of these simple-currents, see [25] where their group is denoted .", "index": 7}, {"type": "text", "bbox": [94, 397, 338, 412], "content": "Evaluating in two ways gives the useful", "index": 8}, {"type": "interline_equation", "bbox": [201, 426, 409, 443], "content": "", "index": 9}, {"type": "text", "bbox": [70, 455, 294, 469], "content": "and hence the reciprocity .", "index": 10}, {"type": "text", "bbox": [70, 470, 541, 527], "content": "For each , the inner products of weights are rational; let denote the least common denominator. E.g. for this is , while for it is . Choose any integer coprime to . Then for any there is a unique weight , coroot , and (finite) Weyl element such that", "index": 11}, {"type": "interline_equation", "bbox": [228, 541, 383, 557], "content": "", "index": 12}, {"type": "text", "bbox": [70, 570, 541, 600], "content": "This is simply the statement that the affine Weyl orbit of intersects the set at precisely one point (namely . Write . Then [16]", "index": 13}, {"type": "interline_equation", "bbox": [236, 614, 374, 631], "content": "", "index": 14}, {"type": "text", "bbox": [68, 642, 541, 699], "content": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over by all entries lies in the cyclotomic field where denotes the root of unity ; for any , there will be a function such that", "index": 15}, {"type": "interline_equation", "bbox": [204, 701, 407, 718], "content": "", "index": 16}]
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[{"bbox": [95, 72, 542, 90], "content": "Charge-conjugation is the order 2 permutation of given by , the weight", "parent_index": 0, "line_index": 0}, {"bbox": [70, 88, 452, 104], "content": "contragredient to . For instance . It has the basic property that", "parent_index": 0, "line_index": 1}, {"bbox": [70, 145, 540, 162], "content": "and . corresponds to a symmetry of the (unextended) Dynkin diagram of ,", "parent_index": 2, "line_index": 0}, {"bbox": [71, 161, 212, 174], "content": "as we will see next section.", "parent_index": 2, "line_index": 1}, {"bbox": [94, 174, 540, 190], "content": "Related to are all the other symmetries of the unextended Dynkin diagram. We", "parent_index": 3, "line_index": 0}, {"bbox": [68, 186, 542, 209], "content": "call these conjugations. The only with nontrivial conjugations other than charge-", "parent_index": 3, "line_index": 1}, {"bbox": [70, 203, 192, 226], "content": "conjugation are .", "parent_index": 3, "line_index": 2}, {"bbox": [95, 221, 541, 236], "content": "Another important symmetry of the matrix is called simple-currents. Any weight", "parent_index": 4, "line_index": 0}, {"bbox": [71, 236, 541, 250], "content": "with q-dimension , is called a simple-current. To any such weight is", "parent_index": 4, "line_index": 1}, {"bbox": [71, 250, 520, 264], "content": "associated a permutation of and a function , such that and", "parent_index": 4, "line_index": 2}, {"bbox": [71, 308, 537, 323], "content": "The simple-currents form an abelian group, given by composition of the permutations .", "parent_index": 6, "line_index": 0}, {"bbox": [94, 322, 541, 338], "content": "All simple-currents for the affine algebras were classified in [12], and with one unimpor-", "parent_index": 7, "line_index": 0}, {"bbox": [69, 334, 541, 358], "content": "tant exception ( at level 2) correspond to symmetries of the extended Coxeter–Dynkin", "parent_index": 7, "line_index": 1}, {"bbox": [69, 353, 542, 372], "content": "diagram of . The simplest proof would use the methods of Proposition 4.1 below. For", "parent_index": 7, "line_index": 2}, {"bbox": [70, 371, 540, 385], "content": "a more intrinsically algebraic interpretation of these simple-currents, see [25] where their", "parent_index": 7, "line_index": 3}, {"bbox": [69, 384, 188, 402], "content": "group is denoted .", "parent_index": 7, "line_index": 4}, {"bbox": [95, 399, 336, 414], "content": "Evaluating in two ways gives the useful", "parent_index": 8, "line_index": 0}, {"bbox": [70, 457, 293, 472], "content": "and hence the reciprocity .", "parent_index": 10, "line_index": 0}, {"bbox": [94, 471, 541, 487], "content": "For each , the inner products of weights are rational; let denote the least", "parent_index": 11, "line_index": 0}, {"bbox": [69, 484, 541, 502], "content": "common denominator. E.g. for this is , while for it is . Choose any", "parent_index": 11, "line_index": 1}, {"bbox": [69, 499, 542, 515], "content": "integer coprime to . Then for any there is a unique weight , coroot", "parent_index": 11, "line_index": 2}, {"bbox": [71, 514, 286, 529], "content": ", and (finite) Weyl element such that", "parent_index": 11, "line_index": 3}, {"bbox": [70, 572, 540, 589], "content": "This is simply the statement that the affine Weyl orbit of intersects the set", "parent_index": 13, "line_index": 0}, {"bbox": [70, 587, 479, 603], "content": "at precisely one point (namely . Write . Then [16]", "parent_index": 13, "line_index": 1}, {"bbox": [71, 644, 541, 661], "content": "This has an obvious interpretation as a Galois automorphism [4]: the field generated over", "parent_index": 15, "line_index": 0}, {"bbox": [71, 659, 540, 675], "content": "by all entries lies in the cyclotomic field where denotes the root of unity", "parent_index": 15, "line_index": 1}, {"bbox": [71, 671, 540, 692], "content": "; for any , there will be a function", "parent_index": 15, "line_index": 2}, {"bbox": [71, 688, 122, 701], "content": "such that", "parent_index": 15, "line_index": 3}]
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[]
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[{"bbox": [360, 75, 376, 87], "content": "P_{+}", "parent_index": 0, "subtype": "inline"}, {"bbox": [430, 74, 476, 84], "content": "C\\lambda\\,=\\,^{t}\\lambda", "parent_index": 0, "subtype": "inline"}, {"bbox": [165, 90, 172, 99], "content": "\\lambda", "parent_index": 0, "subtype": "inline"}, {"bbox": [248, 90, 286, 99], "content": "C0=0", "parent_index": 0, "subtype": "inline"}, {"bbox": [250, 117, 361, 132], "content": "S_{C\\lambda,\\mu}=S_{\\lambda,C\\mu}=S_{\\lambda\\mu}^{*}", "parent_index": 1, "subtype": "interline"}, {"bbox": [95, 146, 136, 156], "content": "S^{2}=C", "parent_index": 2, "subtype": "inline"}, {"bbox": [146, 148, 155, 157], "content": "C", "parent_index": 2, "subtype": "inline"}, {"bbox": [520, 148, 536, 159], "content": "X_{r}", "parent_index": 2, "subtype": "inline"}, {"bbox": [155, 177, 165, 185], "content": "C", "parent_index": 3, "subtype": "inline"}, {"bbox": [256, 188, 280, 203], "content": "X_{r}^{(1)}", "parent_index": 3, "subtype": "inline"}, {"bbox": [156, 204, 187, 219], "content": "D_{e v e n}^{(1)}", "parent_index": 3, "subtype": "inline"}, {"bbox": [330, 223, 339, 232], "content": "S", "parent_index": 4, "subtype": "inline"}, {"bbox": [71, 238, 110, 249], "content": "j\\in P_{+}", "parent_index": 4, "subtype": "inline"}, {"bbox": [210, 237, 260, 249], "content": "\\mathcal{D}(j)=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [520, 238, 527, 249], "content": "j", "parent_index": 4, "subtype": "inline"}, {"bbox": [207, 252, 214, 261], "content": "J", "parent_index": 4, "subtype": "inline"}, {"bbox": [232, 252, 248, 263], "content": "P_{+}", "parent_index": 4, "subtype": "inline"}, {"bbox": [332, 252, 400, 264], "content": "Q_{j}:P_{+}\\to\\mathbb{Q}", "parent_index": 4, "subtype": "inline"}, {"bbox": [460, 252, 495, 263], "content": "J0=j", "parent_index": 4, "subtype": "inline"}, {"bbox": [235, 278, 376, 293], "content": "S_{J\\lambda,\\mu}=\\exp[2\\pi\\mathrm{i}\\,Q_{j}(\\mu)]\\,S_{\\lambda\\mu}", "parent_index": 5, "subtype": "interline"}, {"bbox": [525, 310, 533, 319], "content": "J", "parent_index": 6, "subtype": "inline"}, {"bbox": [154, 336, 176, 353], "content": "{E}_{8}^{(1)}", "parent_index": 7, "subtype": "inline"}, {"bbox": [131, 354, 154, 368], "content": "X_{r}^{(1)}", "parent_index": 7, "subtype": "inline"}, {"bbox": [162, 385, 183, 398], "content": "W_{0}^{+}", "parent_index": 7, "subtype": "inline"}, {"bbox": [155, 401, 185, 413], "content": "S_{J\\lambda,j^{\\prime}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [201, 426, 409, 443], "content": "Q_{j^{\\prime}}(J\\lambda)\\equiv Q_{j}(j^{\\prime})+Q_{j^{\\prime}}(\\lambda)\\qquad(\\mathrm{mod}\\ 1)", "parent_index": 9, "subtype": "interline"}, {"bbox": [208, 458, 289, 471], "content": "Q_{j}(j^{\\prime})=Q_{j^{\\prime}}(j)", "parent_index": 10, "subtype": "inline"}, {"bbox": [144, 473, 159, 484], "content": "X_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [268, 473, 295, 485], "content": "(\\lambda|\\mu)", "parent_index": 11, "subtype": "inline"}, {"bbox": [441, 473, 452, 482], "content": "N", "parent_index": 11, "subtype": "inline"}, {"bbox": [235, 488, 250, 498], "content": "A_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [288, 488, 341, 497], "content": "N=r+1", "parent_index": 11, "subtype": "inline"}, {"bbox": [396, 487, 411, 498], "content": "E_{8}", "parent_index": 11, "subtype": "inline"}, {"bbox": [438, 488, 471, 497], "content": "N=1", "parent_index": 11, "subtype": "inline"}, {"bbox": [111, 502, 116, 511], "content": "\\ell", "parent_index": 11, "subtype": "inline"}, {"bbox": [180, 502, 198, 511], "content": "\\kappa N", "parent_index": 11, "subtype": "inline"}, {"bbox": [279, 502, 316, 514], "content": "\\lambda\\in P_{+}", "parent_index": 11, "subtype": "inline"}, {"bbox": [449, 500, 499, 514], "content": "\\lambda^{(\\ell)}\\in P_{+}", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 520, 79, 525], "content": "\\alpha", "parent_index": 11, "subtype": "inline"}, {"bbox": [223, 520, 232, 525], "content": "\\omega", "parent_index": 11, "subtype": "inline"}, {"bbox": [228, 541, 383, 557], "content": "\\ell\\left(\\lambda+\\rho\\right)=\\omega(\\lambda^{\\left(\\ell\\right)}+\\rho)+\\kappa\\alpha\\;.", "parent_index": 12, "subtype": "interline"}, {"bbox": [370, 573, 412, 586], "content": "\\ell\\left(\\lambda+\\rho\\right)", "parent_index": 13, "subtype": "inline"}, {"bbox": [505, 575, 540, 586], "content": "P_{+}+\\rho", "parent_index": 13, "subtype": "inline"}, {"bbox": [234, 587, 277, 600], "content": "\\lambda^{(\\ell)}+\\rho)", "parent_index": 13, "subtype": "inline"}, {"bbox": [320, 587, 420, 601], "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\operatorname*{det}\\omega=\\pm1", "parent_index": 13, "subtype": "inline"}, {"bbox": [236, 614, 374, 631], "content": "\\epsilon_{\\ell}^{\\prime}(\\lambda)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}^{\\prime}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}", "parent_index": 14, "subtype": "interline"}, {"bbox": [71, 661, 81, 672], "content": "\\mathbb{Q}", "parent_index": 15, "subtype": "inline"}, {"bbox": [154, 661, 174, 673], "content": "S_{\\lambda\\mu}", "parent_index": 15, "subtype": "inline"}, {"bbox": [316, 660, 357, 673], "content": "\\mathbb{Q}[\\xi_{4N\\kappa}]", "parent_index": 15, "subtype": "inline"}, {"bbox": [395, 661, 407, 672], "content": "\\xi_{n}", "parent_index": 15, "subtype": "inline"}, {"bbox": [71, 674, 127, 687], "content": "\\exp[2\\pi\\mathrm{i}/n]", "parent_index": 15, "subtype": "inline"}, {"bbox": [173, 674, 327, 687], "content": "\\sigma_{\\ell}\\in{\\mathrm{Gal}}(\\mathbb{Q}[\\xi_{4N\\kappa}]/\\mathbb{Q})\\cong\\mathbb{Z}_{4N\\kappa}^{\\times}", "parent_index": 15, "subtype": "inline"}, {"bbox": [459, 675, 540, 687], "content": "\\epsilon_{\\ell}:P_{+}\\to\\{\\pm1\\}", "parent_index": 15, "subtype": "inline"}, {"bbox": [204, 701, 407, 718], "content": "\\sigma_{\\ell}\\bigl(S_{\\lambda\\mu}\\bigr)=\\epsilon_{\\ell}\\bigl(\\lambda\\bigr)\\,S_{\\lambda^{(\\ell)},\\mu}=\\epsilon_{\\ell}\\bigl(\\mu\\bigr)\\,S_{\\lambda,\\mu^{(\\ell)}}\\ .", "parent_index": 16, "subtype": "interline"}]
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$\epsilon_{\ell}(\lambda)/\epsilon_{\ell}^{\prime}(\lambda)=\sigma_{\ell}(c)/c$ is an unimportant sign independent of $\lambda$ . This Galois action will play a fairly important role in this paper. Note that $\sigma_{-1}=C$ , so this action can be thought of as a generalisation of charge-conjugation. Note also that $\sigma_{\ell}\circ J=J^{\ell}\circ\sigma_{\ell}$ .
The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23 p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities $T_{\lambda\mu}^{\nu}\overset{\mathrm{def}}{=}\mathrm{mult}_{L(\overline{{\lambda}})\otimes L(\overline{{\mu}})}(L(\overline{{\nu}}))$ in $X_{r}$ :
$$
N_{\lambda\mu}^{\nu}=\sum_{w\in W}\operatorname*{det}(w)\,T_{\lambda\mu}^{w.\nu}~,
$$
where $w.\gamma\,{\stackrel{\mathrm{def}}{=}}\,w(\gamma+\rho)-\rho$ and $W$ is the affine Weyl group of $X_{r}^{(1)}$ (the dependence of $N_{\lambda\mu}^{\nu}$ on $k$ arises through the action of $W$ ). We shall see shortly that these fusion coefficients, now manifestly integral, are in fact nonnegative. Let $\mathcal{R}(X_{r,k})$ denote the corresponding fusion ring.
A handy consequence of (2.4) that whenever $k$ is large enough that $\lambda+\mu\in P_{+}^{k}(X_{r}^{(1)})$ (i.e. that $\begin{array}{r}{\sum_{i=1}^{\tau}a_{i}^{\vee}(\lambda_{i}+\mu_{i})\le k)}\end{array}$ , then $N_{\lambda\mu}^{\nu}=T_{\lambda\mu}^{\nu}$ .
It will sometimes be convenient to collect these coefficients in matrix form as the fusion matrices $N_{\lambda}$ , defined by $(N_{\lambda})_{\mu\nu}=N_{\lambda\mu}^{\nu}$ . For instance, $N_{0}=I$ and, more generally, $N_{j}$ is the permutation matrix associated to $J$ .
The importance of (charge-)conjugation and simple-currents for us is that they respect fusions:
$$
\begin{array}{c}{{N_{C\lambda,C\mu}^{C\nu}=N_{\lambda\mu}^{\nu}}}\\ {{{}}}\\ {{N_{J\lambda,J^{\prime}\mu}^{J J^{\prime}\nu}=N_{\lambda\mu}^{\nu}}}\\ {{N_{\lambda\mu}^{\nu}\neq0\ \Rightarrow\ Q_{j}(\lambda){+}Q_{j}(\mu)\equiv Q_{j}(\nu){\qquad}(\mathrm{mod~1})}}\end{array}
$$
for any simple-currents $J,J^{\prime},j$ .
For example, for $\mathcal{R}(A_{1,k})$ we may take $P_{+}\;=\;\{0,1,\ldots,k\}$ (the value of $\lambda_{1}$ ), and then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is trivial here, but $j=k$ is a simple-current corresponding to permutation $J a=k-a$ and function $Q_{j}(a)=a/2$ . The Galois action sends $a$ to the unique weight $a^{(\ell)}\in P_{+}$ satisfying $a^{(\ell)}+1\equiv\pm\ell\left(a+1\right)$ (mod $2k+4$ ), where that sign there equals $\mathrm{i}^{\ell-1}\epsilon_{\ell}^{\prime}(a)$ . The fusion coefficients are given by
$$
N_{a b}^{c}=\left\{\begin{array}{c c}{{1}}&{{\mathrm{if~}c\equiv a\!+\!b\;(\mathrm{mod~}2)\;\mathrm{and~}|a\!-\!b|\leq c\leq\operatorname*{min}\{a\!+\!b,2k\!-\!a\!-\!b\}}}\\ {{0}}&{{\mathrm{otherwise}}}\end{array}\right..
$$
Equation (2.4) tells us the affine fusion rules are the structure constants for the ring $\mathrm{Ch}(X_{r})/\mathcal{I}_{k}$ where $\operatorname{Ch}(X_{r})$ is the character ring for all finite-dimensional $X_{r}$ -modules, and $\mathcal{I}_{k}$ is the subspace spanned by the elements $\mathrm{ch}_{\overline{{\mu}}}-(\operatorname*{det}w)\mathrm{ch}_{\overline{{w}}.\mu}$ . Finkelberg [8] proved that this ring is isomorphic to the K-ring of a “sub-quotient” $\widetilde{\mathcal{O}}_{k}$ of Kazhdan-Lusztig’s category of level $k$ integrable highest weight $X_{r}^{(1)}$ -modules, and t o Gelfand-Kazhdan’s category $\widetilde{\mathcal{O}}_{q}$ coming from finite-dimensional modules of the quantum group $U_{q}X_{r}$ specialised to the root of unity $q\,=\,\xi_{2m\kappa}$ for appropriate choice of $m\in\{1,2,3\}$ . They also arise from the Huang-Lepowsky coproduct [21] for the modules of the VOA $L(k,0)$ . Because of these isomorphisms, we get that the $N_{\lambda\mu}^{\nu}$ do indeed lie in $\mathbb{Z}_{\geq}$ , for any affine algebra.
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<html><body>
<p data-bbox="70 70 541 114">$\epsilon_{\ell}(\lambda)/\epsilon_{\ell}^{\prime}(\lambda)=\sigma_{\ell}(c)/c$ is an unimportant sign independent of $\lambda$ . This Galois action will play a fairly important role in this paper. Note that $\sigma_{-1}=C$ , so this action can be thought of as a generalisation of charge-conjugation. Note also that $\sigma_{\ell}\circ J=J^{\ell}\circ\sigma_{\ell}$ . </p>
<p data-bbox="69 115 541 164">The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23 p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities $T_{\lambda\mu}^{\nu}\overset{\mathrm{def}}{=}\mathrm{mult}_{L(\overline{{\lambda}})\otimes L(\overline{{\mu}})}(L(\overline{{\nu}}))$ in $X_{r}$ : </p>
<div class="equation" data-bbox="239 178 372 209">$$
N_{\lambda\mu}^{\nu}=\sum_{w\in W}\operatorname*{det}(w)\,T_{\lambda\mu}^{w.\nu}~,
$$</div>
<p data-bbox="70 221 541 281">where $w.\gamma\,{\stackrel{\mathrm{def}}{=}}\,w(\gamma+\rho)-\rho$ and $W$ is the affine Weyl group of $X_{r}^{(1)}$ (the dependence of $N_{\lambda\mu}^{\nu}$ on $k$ arises through the action of $W$ ). We shall see shortly that these fusion coefficients, now manifestly integral, are in fact nonnegative. Let $\mathcal{R}(X_{r,k})$ denote the corresponding fusion ring. </p>
<p data-bbox="70 282 541 313">A handy consequence of (2.4) that whenever $k$ is large enough that $\lambda+\mu\in P_{+}^{k}(X_{r}^{(1)})$ (i.e. that $\begin{array}{r}{\sum_{i=1}^{\tau}a_{i}^{\vee}(\lambda_{i}+\mu_{i})\le k)}\end{array}$ , then $N_{\lambda\mu}^{\nu}=T_{\lambda\mu}^{\nu}$ . </p>
<p data-bbox="70 313 541 354">It will sometimes be convenient to collect these coefficients in matrix form as the fusion matrices $N_{\lambda}$ , defined by $(N_{\lambda})_{\mu\nu}=N_{\lambda\mu}^{\nu}$ . For instance, $N_{0}=I$ and, more generally, $N_{j}$ is the permutation matrix associated to $J$ . </p>
<p data-bbox="70 355 540 383">The importance of (charge-)conjugation and simple-currents for us is that they respect fusions: </p>
<div class="equation" data-bbox="176 394 435 455">$$
\begin{array}{c}{{N_{C\lambda,C\mu}^{C\nu}=N_{\lambda\mu}^{\nu}}}\\ {{{}}}\\ {{N_{J\lambda,J^{\prime}\mu}^{J J^{\prime}\nu}=N_{\lambda\mu}^{\nu}}}\\ {{N_{\lambda\mu}^{\nu}\neq0\ \Rightarrow\ Q_{j}(\lambda){+}Q_{j}(\mu)\equiv Q_{j}(\nu){\qquad}(\mathrm{mod~1})}}\end{array}
$$</div>
<p data-bbox="70 463 235 477">for any simple-currents $J,J^{\prime},j$ . </p>
<p data-bbox="70 478 541 572">For example, for $\mathcal{R}(A_{1,k})$ we may take $P_{+}\;=\;\{0,1,\ldots,k\}$ (the value of $\lambda_{1}$ ), and then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is trivial here, but $j=k$ is a simple-current corresponding to permutation $J a=k-a$ and function $Q_{j}(a)=a/2$ . The Galois action sends $a$ to the unique weight $a^{(\ell)}\in P_{+}$ satisfying $a^{(\ell)}+1\equiv\pm\ell\left(a+1\right)$ (mod $2k+4$ ), where that sign there equals $\mathrm{i}^{\ell-1}\epsilon_{\ell}^{\prime}(a)$ . The fusion coefficients are given by </p>
<div class="equation" data-bbox="117 582 496 616">$$
N_{a b}^{c}=\left\{\begin{array}{c c}{{1}}&{{\mathrm{if~}c\equiv a\!+\!b\;(\mathrm{mod~}2)\;\mathrm{and~}|a\!-\!b|\leq c\leq\operatorname*{min}\{a\!+\!b,2k\!-\!a\!-\!b\}}}\\ {{0}}&{{\mathrm{otherwise}}}\end{array}\right..
$$</div>
<p data-bbox="70 624 541 716">Equation (2.4) tells us the affine fusion rules are the structure constants for the ring $\mathrm{Ch}(X_{r})/\mathcal{I}_{k}$ where $\operatorname{Ch}(X_{r})$ is the character ring for all finite-dimensional $X_{r}$ -modules, and $\mathcal{I}_{k}$ is the subspace spanned by the elements $\mathrm{ch}_{\overline{{\mu}}}-(\operatorname*{det}w)\mathrm{ch}_{\overline{{w}}.\mu}$ . Finkelberg [8] proved that this ring is isomorphic to the K-ring of a “sub-quotient” $\widetilde{\mathcal{O}}_{k}$ of Kazhdan-Lusztig’s category of level $k$ integrable highest weight $X_{r}^{(1)}$ -modules, and t o Gelfand-Kazhdan’s category $\widetilde{\mathcal{O}}_{q}$ coming from finite-dimensional modules of the quantum group $U_{q}X_{r}$ specialised to the root of unity $q\,=\,\xi_{2m\kappa}$ for appropriate choice of $m\in\{1,2,3\}$ . They also arise from the Huang-Lepowsky coproduct [21] for the modules of the VOA $L(k,0)$ . Because of these isomorphisms, we get that the $N_{\lambda\mu}^{\nu}$ do indeed lie in $\mathbb{Z}_{\geq}$ , for any affine algebra. </p>
</body></html>
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0002044v1
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[{"type": "text", "text": "$\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\sigma_{\\ell}(c)/c$ is an unimportant sign independent of $\\lambda$ . This Galois action will play a fairly important role in this paper. Note that $\\sigma_{-1}=C$ , so this action can be thought of as a generalisation of charge-conjugation. Note also that $\\sigma_{\\ell}\\circ J=J^{\\ell}\\circ\\sigma_{\\ell}$ . ", "page_idx": 4}, {"type": "text", "text": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23 p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities $T_{\\lambda\\mu}^{\\nu}\\overset{\\mathrm{def}}{=}\\mathrm{mult}_{L(\\overline{{\\lambda}})\\otimes L(\\overline{{\\mu}})}(L(\\overline{{\\nu}}))$ in $X_{r}$ : ", "page_idx": 4}, {"type": "equation", "text": "$$\nN_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname*{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "where $w.\\gamma\\,{\\stackrel{\\mathrm{def}}{=}}\\,w(\\gamma+\\rho)-\\rho$ and $W$ is the affine Weyl group of $X_{r}^{(1)}$ (the dependence of $N_{\\lambda\\mu}^{\\nu}$ on $k$ arises through the action of $W$ ). We shall see shortly that these fusion coefficients, now manifestly integral, are in fact nonnegative. Let $\\mathcal{R}(X_{r,k})$ denote the corresponding fusion ring. ", "page_idx": 4}, {"type": "text", "text": "A handy consequence of (2.4) that whenever $k$ is large enough that $\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})$ (i.e. that $\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}$ , then $N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}$ . ", "page_idx": 4}, {"type": "text", "text": "It will sometimes be convenient to collect these coefficients in matrix form as the fusion matrices $N_{\\lambda}$ , defined by $(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}$ . For instance, $N_{0}=I$ and, more generally, $N_{j}$ is the permutation matrix associated to $J$ . ", "page_idx": 4}, {"type": "text", "text": "The importance of (charge-)conjugation and simple-currents for us is that they respect fusions: ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "for any simple-currents $J,J^{\\prime},j$ . ", "page_idx": 4}, {"type": "text", "text": "For example, for $\\mathcal{R}(A_{1,k})$ we may take $P_{+}\\;=\\;\\{0,1,\\ldots,k\\}$ (the value of $\\lambda_{1}$ ), and then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is trivial here, but $j=k$ is a simple-current corresponding to permutation $J a=k-a$ and function $Q_{j}(a)=a/2$ . The Galois action sends $a$ to the unique weight $a^{(\\ell)}\\in P_{+}$ satisfying $a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)$ (mod $2k+4$ ), where that sign there equals $\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)$ . The fusion coefficients are given by ", "page_idx": 4}, {"type": "equation", "text": "$$\nN_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}|a\\!-\\!b|\\leq c\\leq\\operatorname*{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring $\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}$ where $\\operatorname{Ch}(X_{r})$ is the character ring for all finite-dimensional $X_{r}$ -modules, and $\\mathcal{I}_{k}$ is the subspace spanned by the elements $\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname*{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}$ . Finkelberg [8] proved that this ring is isomorphic to the K-ring of a “sub-quotient” $\\widetilde{\\mathcal{O}}_{k}$ of Kazhdan-Lusztig’s category of level $k$ integrable highest weight $X_{r}^{(1)}$ -modules, and t o Gelfand-Kazhdan’s category $\\widetilde{\\mathcal{O}}_{q}$ coming from finite-dimensional modules of the quantum group $U_{q}X_{r}$ specialised to the root of unity $q\\,=\\,\\xi_{2m\\kappa}$ for appropriate choice of $m\\in\\{1,2,3\\}$ . They also arise from the Huang-Lepowsky coproduct [21] for the modules of the VOA $L(k,0)$ . Because of these isomorphisms, we get that the $N_{\\lambda\\mu}^{\\nu}$ do indeed lie in $\\mathbb{Z}_{\\geq}$ , for any affine algebra. ", "page_idx": 4}]
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Note that ", "type": "text"}, {"bbox": [321, 90, 367, 101], "score": 0.93, "content": "\\sigma_{-1}=C", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [367, 88, 542, 103], "score": 1.0, "content": ", so this action can be thought of", "type": "text"}], "index": 1}, {"bbox": [70, 101, 458, 118], "spans": [{"bbox": [70, 101, 370, 118], "score": 1.0, "content": "as a generalisation of charge-conjugation. Note also that ", "type": "text"}, {"bbox": [371, 102, 453, 115], "score": 0.93, "content": "\\sigma_{\\ell}\\circ J=J^{\\ell}\\circ\\sigma_{\\ell}", "type": "inline_equation", "height": 13, "width": 82}, {"bbox": [453, 101, 458, 118], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [69, 115, 541, 164], "lines": [{"bbox": [94, 115, 541, 132], "spans": [{"bbox": [94, 115, 541, 132], "score": 1.0, "content": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23", "type": "text"}], "index": 3}, {"bbox": [69, 132, 541, 146], "spans": [{"bbox": [69, 132, 541, 146], "score": 1.0, "content": "p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities", "type": "text"}], "index": 4}, {"bbox": [71, 146, 255, 168], "spans": [{"bbox": [71, 146, 215, 166], "score": 0.93, "content": "T_{\\lambda\\mu}^{\\nu}\\overset{\\mathrm{def}}{=}\\mathrm{mult}_{L(\\overline{{\\lambda}})\\otimes L(\\overline{{\\mu}})}(L(\\overline{{\\nu}}))", "type": "inline_equation", "height": 20, "width": 144}, {"bbox": [215, 148, 232, 168], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [232, 151, 248, 162], "score": 0.9, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [248, 148, 255, 168], "score": 1.0, "content": ":", "type": "text"}], "index": 5}], "index": 4}, {"type": "interline_equation", "bbox": [239, 178, 372, 209], "lines": [{"bbox": [239, 178, 372, 209], "spans": [{"bbox": [239, 178, 372, 209], "score": 0.95, "content": "N_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname*{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [70, 221, 541, 281], "lines": [{"bbox": [66, 218, 539, 247], "spans": [{"bbox": [66, 218, 105, 247], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 223, 206, 240], "score": 0.92, "content": "w.\\gamma\\,{\\stackrel{\\mathrm{def}}{=}}\\,w(\\gamma+\\rho)-\\rho", "type": "inline_equation", "height": 17, "width": 101}, {"bbox": [207, 218, 231, 247], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 227, 245, 237], "score": 0.81, "content": "W", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [245, 218, 388, 247], "score": 1.0, "content": " is the affine Weyl group of ", "type": "text"}, {"bbox": [388, 223, 412, 238], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [412, 218, 517, 247], "score": 1.0, "content": "(the dependence of ", "type": "text"}, {"bbox": [518, 228, 539, 242], "score": 0.92, "content": "N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 21}], "index": 7}, {"bbox": [69, 239, 541, 256], "spans": [{"bbox": [69, 239, 88, 256], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [88, 242, 95, 251], "score": 0.84, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [96, 239, 249, 256], "score": 1.0, "content": " arises through the action of ", "type": "text"}, {"bbox": [249, 243, 262, 252], "score": 0.85, "content": "W", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [263, 239, 541, 256], "score": 1.0, "content": "). We shall see shortly that these fusion coefficients,", "type": "text"}], "index": 8}, {"bbox": [69, 254, 541, 270], "spans": [{"bbox": [69, 254, 358, 270], "score": 1.0, "content": "now manifestly integral, are in fact nonnegative. Let ", "type": "text"}, {"bbox": [359, 256, 401, 268], "score": 0.95, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [402, 254, 541, 270], "score": 1.0, "content": " denote the corresponding", "type": "text"}], "index": 9}, {"bbox": [70, 268, 131, 285], "spans": [{"bbox": [70, 268, 131, 285], "score": 1.0, "content": "fusion ring.", "type": "text"}], "index": 10}], "index": 8.5}, {"type": "text", "bbox": [70, 282, 541, 313], "lines": [{"bbox": [93, 283, 539, 301], "spans": [{"bbox": [93, 283, 330, 301], "score": 1.0, "content": "A handy consequence of (2.4) that whenever ", "type": "text"}, {"bbox": [330, 287, 337, 296], "score": 0.91, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [338, 283, 448, 301], "score": 1.0, "content": " is large enough that ", "type": "text"}, {"bbox": [448, 283, 539, 300], "score": 0.92, "content": "\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 91}], "index": 11}, {"bbox": [69, 297, 335, 318], "spans": [{"bbox": [69, 297, 120, 318], "score": 1.0, "content": "(i.e. that", "type": "text"}, {"bbox": [121, 300, 237, 314], "score": 0.91, "content": "\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}", "type": "inline_equation", "height": 14, "width": 116}, {"bbox": [237, 297, 272, 318], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [273, 302, 330, 315], "score": 0.94, "content": "N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [330, 297, 335, 318], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11.5}, {"type": "text", "bbox": [70, 313, 541, 354], "lines": [{"bbox": [93, 313, 540, 327], "spans": [{"bbox": [93, 313, 540, 327], "score": 1.0, "content": "It will sometimes be convenient to collect these coefficients in matrix form as the", "type": "text"}], "index": 13}, {"bbox": [69, 327, 541, 344], "spans": [{"bbox": [69, 327, 154, 343], "score": 1.0, "content": "fusion matrices ", "type": "text"}, {"bbox": [155, 330, 171, 340], "score": 0.91, "content": "N_{\\lambda}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [171, 327, 236, 343], "score": 1.0, "content": ", defined by ", "type": "text"}, {"bbox": [236, 329, 311, 344], "score": 0.9, "content": "(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 15, "width": 75}, {"bbox": [312, 327, 392, 343], "score": 1.0, "content": ". For instance, ", "type": "text"}, {"bbox": [392, 330, 430, 341], "score": 0.94, "content": "N_{0}=I", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [430, 327, 541, 343], "score": 1.0, "content": " and, more generally,", "type": "text"}], "index": 14}, {"bbox": [71, 343, 315, 357], "spans": [{"bbox": [71, 344, 86, 357], "score": 0.91, "content": "N_{j}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [86, 343, 301, 356], "score": 1.0, "content": " is the permutation matrix associated to ", "type": "text"}, {"bbox": [302, 344, 309, 353], "score": 0.82, "content": "J", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [310, 343, 315, 356], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [70, 355, 540, 383], "lines": [{"bbox": [93, 355, 542, 374], "spans": [{"bbox": [93, 355, 542, 374], "score": 1.0, "content": "The importance of (charge-)conjugation and simple-currents for us is that they respect", "type": "text"}], "index": 16}, {"bbox": [70, 371, 112, 387], "spans": [{"bbox": [70, 371, 112, 387], "score": 1.0, "content": "fusions:", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "interline_equation", "bbox": [176, 394, 435, 455], "lines": [{"bbox": [176, 394, 435, 455], "spans": [{"bbox": [176, 394, 435, 455], "score": 0.93, "content": "\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [70, 463, 235, 477], "lines": [{"bbox": [70, 465, 233, 479], "spans": [{"bbox": [70, 465, 195, 479], "score": 1.0, "content": "for any simple-currents ", "type": "text"}, {"bbox": [195, 465, 230, 478], "score": 0.92, "content": "J,J^{\\prime},j", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [230, 465, 233, 479], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [70, 478, 541, 572], "lines": [{"bbox": [93, 479, 542, 495], "spans": [{"bbox": [93, 479, 190, 495], "score": 1.0, "content": "For example, for ", "type": "text"}, {"bbox": [190, 480, 233, 494], "score": 0.92, "content": "\\mathcal{R}(A_{1,k})", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [234, 479, 312, 495], "score": 1.0, "content": " we may take ", "type": "text"}, {"bbox": [313, 479, 414, 493], "score": 0.89, "content": "P_{+}\\;=\\;\\{0,1,\\ldots,k\\}", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [414, 479, 493, 495], "score": 1.0, "content": " (the value of ", "type": "text"}, {"bbox": [493, 482, 506, 492], "score": 0.85, "content": "\\lambda_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [506, 479, 542, 495], "score": 1.0, "content": "), and", "type": "text"}], "index": 20}, {"bbox": [64, 489, 548, 523], "spans": [{"bbox": [64, 489, 548, 523], "score": 1.0, "content": "then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is", "type": "text"}], "index": 21}, {"bbox": [70, 515, 542, 530], "spans": [{"bbox": [70, 515, 159, 530], "score": 1.0, "content": "trivial here, but ", "type": "text"}, {"bbox": [159, 516, 189, 529], "score": 0.9, "content": "j=k", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [190, 515, 456, 530], "score": 1.0, "content": " is a simple-current corresponding to permutation ", "type": "text"}, {"bbox": [456, 517, 516, 527], "score": 0.91, "content": "J a=k-a", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [516, 515, 542, 530], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [69, 527, 542, 547], "spans": [{"bbox": [69, 527, 117, 547], "score": 1.0, "content": "function ", "type": "text"}, {"bbox": [118, 531, 182, 544], "score": 0.92, "content": "Q_{j}(a)=a/2", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [182, 527, 317, 547], "score": 1.0, "content": ". The Galois action sends ", "type": "text"}, {"bbox": [317, 533, 325, 541], "score": 0.69, "content": "a", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [325, 527, 437, 547], "score": 1.0, "content": " to the unique weight ", "type": "text"}, {"bbox": [438, 529, 486, 543], "score": 0.92, "content": "a^{(\\ell)}\\in P_{+}", "type": "inline_equation", "height": 14, "width": 48}, {"bbox": [487, 527, 542, 547], "score": 1.0, "content": " satisfying", "type": "text"}], "index": 23}, {"bbox": [71, 542, 542, 561], "spans": [{"bbox": [71, 544, 185, 558], "score": 0.91, "content": "a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [185, 542, 221, 561], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [221, 544, 257, 557], "score": 0.46, "content": "2k+4", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [258, 542, 425, 561], "score": 1.0, "content": "), where that sign there equals ", "type": "text"}, {"bbox": [425, 545, 471, 558], "score": 0.93, "content": "\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [472, 542, 542, 561], "score": 1.0, "content": ". The fusion", "type": "text"}], "index": 24}, {"bbox": [71, 559, 196, 574], "spans": [{"bbox": [71, 559, 196, 574], "score": 1.0, "content": "coefficients are given by", "type": "text"}], "index": 25}], "index": 22.5}, {"type": "interline_equation", "bbox": [117, 582, 496, 616], "lines": [{"bbox": [117, 582, 496, 616], "spans": [{"bbox": [117, 582, 496, 616], "score": 0.84, "content": "N_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}|a\\!-\\!b|\\leq c\\leq\\operatorname*{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [70, 624, 541, 716], "lines": [{"bbox": [95, 627, 541, 642], "spans": [{"bbox": [95, 627, 541, 642], "score": 1.0, "content": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring", "type": "text"}], "index": 27}, {"bbox": [71, 640, 542, 656], "spans": [{"bbox": [71, 642, 131, 655], "score": 0.91, "content": "\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [131, 640, 168, 656], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [169, 642, 209, 655], "score": 0.9, "content": "\\operatorname{Ch}(X_{r})", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [209, 640, 450, 656], "score": 1.0, "content": " is the character ring for all finite-dimensional ", "type": "text"}, {"bbox": [451, 643, 466, 654], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [466, 640, 542, 656], "score": 1.0, "content": "-modules, and", "type": "text"}], "index": 28}, {"bbox": [71, 656, 541, 671], "spans": [{"bbox": [71, 658, 85, 668], "score": 0.91, "content": "\\mathcal{I}_{k}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [86, 656, 300, 671], "score": 1.0, "content": " is the subspace spanned by the elements ", "type": "text"}, {"bbox": [300, 657, 397, 670], "score": 0.92, "content": "\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname*{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [397, 656, 541, 671], "score": 1.0, "content": ". Finkelberg [8] proved that", "type": "text"}], "index": 29}, {"bbox": [70, 671, 540, 687], "spans": [{"bbox": [70, 672, 363, 687], "score": 1.0, "content": "this ring is isomorphic to the K-ring of a “sub-quotient”", "type": "text"}, {"bbox": [363, 671, 379, 685], "score": 0.91, "content": "\\widetilde{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [379, 672, 540, 687], "score": 1.0, "content": " of Kazhdan-Lusztig’s category", "type": "text"}], "index": 30}, {"bbox": [69, 684, 539, 704], "spans": [{"bbox": [69, 684, 111, 704], "score": 1.0, "content": "of level ", "type": "text"}, {"bbox": [112, 690, 119, 699], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [119, 684, 257, 704], "score": 1.0, "content": " integrable highest weight ", "type": "text"}, {"bbox": [257, 686, 281, 700], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [281, 684, 524, 704], "score": 1.0, "content": "-modules, and t o Gelfand-Kazhdan’s category", "type": "text"}, {"bbox": [525, 687, 539, 702], "score": 0.92, "content": "\\widetilde{\\mathcal{O}}_{q}", "type": "inline_equation", "height": 15, "width": 14}], "index": 31}, {"bbox": [71, 703, 541, 717], "spans": [{"bbox": [71, 703, 412, 717], "score": 1.0, "content": "coming from finite-dimensional modules of the quantum group ", "type": "text"}, {"bbox": [413, 704, 441, 717], "score": 0.93, "content": "U_{q}X_{r}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [442, 703, 541, 717], "score": 1.0, "content": " specialised to the", "type": "text"}], "index": 32}], "index": 29.5}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [239, 178, 372, 209], "lines": [{"bbox": [239, 178, 372, 209], "spans": [{"bbox": [239, 178, 372, 209], "score": 0.95, "content": "N_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname*{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [176, 394, 435, 455], "lines": [{"bbox": [176, 394, 435, 455], "spans": [{"bbox": [176, 394, 435, 455], "score": 0.93, "content": "\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [117, 582, 496, 616], "lines": [{"bbox": [117, 582, 496, 616], "spans": [{"bbox": [117, 582, 496, 616], "score": 0.84, "content": "N_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}|a\\!-\\!b|\\leq c\\leq\\operatorname*{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 114], "lines": [{"bbox": [71, 73, 540, 89], "spans": [{"bbox": [71, 75, 182, 88], "score": 0.92, "content": "\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\sigma_{\\ell}(c)/c", "type": "inline_equation", "height": 13, "width": 111}, {"bbox": [182, 73, 383, 89], "score": 1.0, "content": " is an unimportant sign independent of ", "type": "text"}, {"bbox": [383, 75, 391, 84], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [391, 73, 540, 89], "score": 1.0, "content": ". This Galois action will play", "type": "text"}], "index": 0}, {"bbox": [70, 88, 542, 103], "spans": [{"bbox": [70, 88, 321, 103], "score": 1.0, "content": "a fairly important role in this paper. Note that ", "type": "text"}, {"bbox": [321, 90, 367, 101], "score": 0.93, "content": "\\sigma_{-1}=C", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [367, 88, 542, 103], "score": 1.0, "content": ", so this action can be thought of", "type": "text"}], "index": 1}, {"bbox": [70, 101, 458, 118], "spans": [{"bbox": [70, 101, 370, 118], "score": 1.0, "content": "as a generalisation of charge-conjugation. Note also that ", "type": "text"}, {"bbox": [371, 102, 453, 115], "score": 0.93, "content": "\\sigma_{\\ell}\\circ J=J^{\\ell}\\circ\\sigma_{\\ell}", "type": "inline_equation", "height": 13, "width": 82}, {"bbox": [453, 101, 458, 118], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [70, 73, 542, 118]}, {"type": "text", "bbox": [69, 115, 541, 164], "lines": [{"bbox": [94, 115, 541, 132], "spans": [{"bbox": [94, 115, 541, 132], "score": 1.0, "content": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23", "type": "text"}], "index": 3}, {"bbox": [69, 132, 541, 146], "spans": [{"bbox": [69, 132, 541, 146], "score": 1.0, "content": "p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities", "type": "text"}], "index": 4}, {"bbox": [71, 146, 255, 168], "spans": [{"bbox": [71, 146, 215, 166], "score": 0.93, "content": "T_{\\lambda\\mu}^{\\nu}\\overset{\\mathrm{def}}{=}\\mathrm{mult}_{L(\\overline{{\\lambda}})\\otimes L(\\overline{{\\mu}})}(L(\\overline{{\\nu}}))", "type": "inline_equation", "height": 20, "width": 144}, {"bbox": [215, 148, 232, 168], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [232, 151, 248, 162], "score": 0.9, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [248, 148, 255, 168], "score": 1.0, "content": ":", "type": "text"}], "index": 5}], "index": 4, "bbox_fs": [69, 115, 541, 168]}, {"type": "interline_equation", "bbox": [239, 178, 372, 209], "lines": [{"bbox": [239, 178, 372, 209], "spans": [{"bbox": [239, 178, 372, 209], "score": 0.95, "content": "N_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname*{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [70, 221, 541, 281], "lines": [{"bbox": [66, 218, 539, 247], "spans": [{"bbox": [66, 218, 105, 247], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 223, 206, 240], "score": 0.92, "content": "w.\\gamma\\,{\\stackrel{\\mathrm{def}}{=}}\\,w(\\gamma+\\rho)-\\rho", "type": "inline_equation", "height": 17, "width": 101}, {"bbox": [207, 218, 231, 247], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 227, 245, 237], "score": 0.81, "content": "W", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [245, 218, 388, 247], "score": 1.0, "content": " is the affine Weyl group of ", "type": "text"}, {"bbox": [388, 223, 412, 238], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [412, 218, 517, 247], "score": 1.0, "content": "(the dependence of ", "type": "text"}, {"bbox": [518, 228, 539, 242], "score": 0.92, "content": "N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 21}], "index": 7}, {"bbox": [69, 239, 541, 256], "spans": [{"bbox": [69, 239, 88, 256], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [88, 242, 95, 251], "score": 0.84, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [96, 239, 249, 256], "score": 1.0, "content": " arises through the action of ", "type": "text"}, {"bbox": [249, 243, 262, 252], "score": 0.85, "content": "W", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [263, 239, 541, 256], "score": 1.0, "content": "). We shall see shortly that these fusion coefficients,", "type": "text"}], "index": 8}, {"bbox": [69, 254, 541, 270], "spans": [{"bbox": [69, 254, 358, 270], "score": 1.0, "content": "now manifestly integral, are in fact nonnegative. Let ", "type": "text"}, {"bbox": [359, 256, 401, 268], "score": 0.95, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [402, 254, 541, 270], "score": 1.0, "content": " denote the corresponding", "type": "text"}], "index": 9}, {"bbox": [70, 268, 131, 285], "spans": [{"bbox": [70, 268, 131, 285], "score": 1.0, "content": "fusion ring.", "type": "text"}], "index": 10}], "index": 8.5, "bbox_fs": [66, 218, 541, 285]}, {"type": "text", "bbox": [70, 282, 541, 313], "lines": [{"bbox": [93, 283, 539, 301], "spans": [{"bbox": [93, 283, 330, 301], "score": 1.0, "content": "A handy consequence of (2.4) that whenever ", "type": "text"}, {"bbox": [330, 287, 337, 296], "score": 0.91, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [338, 283, 448, 301], "score": 1.0, "content": " is large enough that ", "type": "text"}, {"bbox": [448, 283, 539, 300], "score": 0.92, "content": "\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 91}], "index": 11}, {"bbox": [69, 297, 335, 318], "spans": [{"bbox": [69, 297, 120, 318], "score": 1.0, "content": "(i.e. that", "type": "text"}, {"bbox": [121, 300, 237, 314], "score": 0.91, "content": "\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}", "type": "inline_equation", "height": 14, "width": 116}, {"bbox": [237, 297, 272, 318], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [273, 302, 330, 315], "score": 0.94, "content": "N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [330, 297, 335, 318], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11.5, "bbox_fs": [69, 283, 539, 318]}, {"type": "text", "bbox": [70, 313, 541, 354], "lines": [{"bbox": [93, 313, 540, 327], "spans": [{"bbox": [93, 313, 540, 327], "score": 1.0, "content": "It will sometimes be convenient to collect these coefficients in matrix form as the", "type": "text"}], "index": 13}, {"bbox": [69, 327, 541, 344], "spans": [{"bbox": [69, 327, 154, 343], "score": 1.0, "content": "fusion matrices ", "type": "text"}, {"bbox": [155, 330, 171, 340], "score": 0.91, "content": "N_{\\lambda}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [171, 327, 236, 343], "score": 1.0, "content": ", defined by ", "type": "text"}, {"bbox": [236, 329, 311, 344], "score": 0.9, "content": "(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 15, "width": 75}, {"bbox": [312, 327, 392, 343], "score": 1.0, "content": ". For instance, ", "type": "text"}, {"bbox": [392, 330, 430, 341], "score": 0.94, "content": "N_{0}=I", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [430, 327, 541, 343], "score": 1.0, "content": " and, more generally,", "type": "text"}], "index": 14}, {"bbox": [71, 343, 315, 357], "spans": [{"bbox": [71, 344, 86, 357], "score": 0.91, "content": "N_{j}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [86, 343, 301, 356], "score": 1.0, "content": " is the permutation matrix associated to ", "type": "text"}, {"bbox": [302, 344, 309, 353], "score": 0.82, "content": "J", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [310, 343, 315, 356], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14, "bbox_fs": [69, 313, 541, 357]}, {"type": "text", "bbox": [70, 355, 540, 383], "lines": [{"bbox": [93, 355, 542, 374], "spans": [{"bbox": [93, 355, 542, 374], "score": 1.0, "content": "The importance of (charge-)conjugation and simple-currents for us is that they respect", "type": "text"}], "index": 16}, {"bbox": [70, 371, 112, 387], "spans": [{"bbox": [70, 371, 112, 387], "score": 1.0, "content": "fusions:", "type": "text"}], "index": 17}], "index": 16.5, "bbox_fs": [70, 355, 542, 387]}, {"type": "interline_equation", "bbox": [176, 394, 435, 455], "lines": [{"bbox": [176, 394, 435, 455], "spans": [{"bbox": [176, 394, 435, 455], "score": 0.93, "content": "\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [70, 463, 235, 477], "lines": [{"bbox": [70, 465, 233, 479], "spans": [{"bbox": [70, 465, 195, 479], "score": 1.0, "content": "for any simple-currents ", "type": "text"}, {"bbox": [195, 465, 230, 478], "score": 0.92, "content": "J,J^{\\prime},j", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [230, 465, 233, 479], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 19, "bbox_fs": [70, 465, 233, 479]}, {"type": "text", "bbox": [70, 478, 541, 572], "lines": [{"bbox": [93, 479, 542, 495], "spans": [{"bbox": [93, 479, 190, 495], "score": 1.0, "content": "For example, for ", "type": "text"}, {"bbox": [190, 480, 233, 494], "score": 0.92, "content": "\\mathcal{R}(A_{1,k})", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [234, 479, 312, 495], "score": 1.0, "content": " we may take ", "type": "text"}, {"bbox": [313, 479, 414, 493], "score": 0.89, "content": "P_{+}\\;=\\;\\{0,1,\\ldots,k\\}", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [414, 479, 493, 495], "score": 1.0, "content": " (the value of ", "type": "text"}, {"bbox": [493, 482, 506, 492], "score": 0.85, "content": "\\lambda_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [506, 479, 542, 495], "score": 1.0, "content": "), and", "type": "text"}], "index": 20}, {"bbox": [64, 489, 548, 523], "spans": [{"bbox": [64, 489, 548, 523], "score": 1.0, "content": "then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is", "type": "text"}], "index": 21}, {"bbox": [70, 515, 542, 530], "spans": [{"bbox": [70, 515, 159, 530], "score": 1.0, "content": "trivial here, but ", "type": "text"}, {"bbox": [159, 516, 189, 529], "score": 0.9, "content": "j=k", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [190, 515, 456, 530], "score": 1.0, "content": " is a simple-current corresponding to permutation ", "type": "text"}, {"bbox": [456, 517, 516, 527], "score": 0.91, "content": "J a=k-a", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [516, 515, 542, 530], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [69, 527, 542, 547], "spans": [{"bbox": [69, 527, 117, 547], "score": 1.0, "content": "function ", "type": "text"}, {"bbox": [118, 531, 182, 544], "score": 0.92, "content": "Q_{j}(a)=a/2", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [182, 527, 317, 547], "score": 1.0, "content": ". The Galois action sends ", "type": "text"}, {"bbox": [317, 533, 325, 541], "score": 0.69, "content": "a", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [325, 527, 437, 547], "score": 1.0, "content": " to the unique weight ", "type": "text"}, {"bbox": [438, 529, 486, 543], "score": 0.92, "content": "a^{(\\ell)}\\in P_{+}", "type": "inline_equation", "height": 14, "width": 48}, {"bbox": [487, 527, 542, 547], "score": 1.0, "content": " satisfying", "type": "text"}], "index": 23}, {"bbox": [71, 542, 542, 561], "spans": [{"bbox": [71, 544, 185, 558], "score": 0.91, "content": "a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [185, 542, 221, 561], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [221, 544, 257, 557], "score": 0.46, "content": "2k+4", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [258, 542, 425, 561], "score": 1.0, "content": "), where that sign there equals ", "type": "text"}, {"bbox": [425, 545, 471, 558], "score": 0.93, "content": "\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [472, 542, 542, 561], "score": 1.0, "content": ". The fusion", "type": "text"}], "index": 24}, {"bbox": [71, 559, 196, 574], "spans": [{"bbox": [71, 559, 196, 574], "score": 1.0, "content": "coefficients are given by", "type": "text"}], "index": 25}], "index": 22.5, "bbox_fs": [64, 479, 548, 574]}, {"type": "interline_equation", "bbox": [117, 582, 496, 616], "lines": [{"bbox": [117, 582, 496, 616], "spans": [{"bbox": [117, 582, 496, 616], "score": 0.84, "content": "N_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}|a\\!-\\!b|\\leq c\\leq\\operatorname*{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [70, 624, 541, 716], "lines": [{"bbox": [95, 627, 541, 642], "spans": [{"bbox": [95, 627, 541, 642], "score": 1.0, "content": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring", "type": "text"}], "index": 27}, {"bbox": [71, 640, 542, 656], "spans": [{"bbox": [71, 642, 131, 655], "score": 0.91, "content": "\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [131, 640, 168, 656], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [169, 642, 209, 655], "score": 0.9, "content": "\\operatorname{Ch}(X_{r})", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [209, 640, 450, 656], "score": 1.0, "content": " is the character ring for all finite-dimensional ", "type": "text"}, {"bbox": [451, 643, 466, 654], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [466, 640, 542, 656], "score": 1.0, "content": "-modules, and", "type": "text"}], "index": 28}, {"bbox": [71, 656, 541, 671], "spans": [{"bbox": [71, 658, 85, 668], "score": 0.91, "content": "\\mathcal{I}_{k}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [86, 656, 300, 671], "score": 1.0, "content": " is the subspace spanned by the elements ", "type": "text"}, {"bbox": [300, 657, 397, 670], "score": 0.92, "content": "\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname*{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [397, 656, 541, 671], "score": 1.0, "content": ". Finkelberg [8] proved that", "type": "text"}], "index": 29}, {"bbox": [70, 671, 540, 687], "spans": [{"bbox": [70, 672, 363, 687], "score": 1.0, "content": "this ring is isomorphic to the K-ring of a “sub-quotient”", "type": "text"}, {"bbox": [363, 671, 379, 685], "score": 0.91, "content": "\\widetilde{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [379, 672, 540, 687], "score": 1.0, "content": " of Kazhdan-Lusztig’s category", "type": "text"}], "index": 30}, {"bbox": [69, 684, 539, 704], "spans": [{"bbox": [69, 684, 111, 704], "score": 1.0, "content": "of level ", "type": "text"}, {"bbox": [112, 690, 119, 699], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [119, 684, 257, 704], "score": 1.0, "content": " integrable highest weight ", "type": "text"}, {"bbox": [257, 686, 281, 700], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [281, 684, 524, 704], "score": 1.0, "content": "-modules, and t o Gelfand-Kazhdan’s category", "type": "text"}, {"bbox": [525, 687, 539, 702], "score": 0.92, "content": "\\widetilde{\\mathcal{O}}_{q}", "type": "inline_equation", "height": 15, "width": 14}], "index": 31}, {"bbox": [71, 703, 541, 717], "spans": [{"bbox": [71, 703, 412, 717], "score": 1.0, "content": "coming from finite-dimensional modules of the quantum group ", "type": "text"}, {"bbox": [413, 704, 441, 717], "score": 0.93, "content": "U_{q}X_{r}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [442, 703, 541, 717], "score": 1.0, "content": " specialised to the", "type": "text"}], "index": 32}, {"bbox": [70, 73, 541, 89], "spans": [{"bbox": [70, 73, 143, 89], "score": 1.0, "content": "root of unity ", "type": "text", "cross_page": true}, {"bbox": [144, 75, 192, 87], "score": 0.94, "content": "q\\,=\\,\\xi_{2m\\kappa}", "type": "inline_equation", "height": 12, "width": 48, "cross_page": true}, {"bbox": [192, 73, 330, 89], "score": 1.0, "content": " for appropriate choice of ", "type": "text", "cross_page": true}, {"bbox": [331, 75, 399, 87], "score": 0.94, "content": "m\\in\\{1,2,3\\}", "type": "inline_equation", "height": 12, "width": 68, "cross_page": true}, {"bbox": [399, 73, 541, 89], "score": 1.0, "content": ". They also arise from the", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [70, 88, 541, 103], "spans": [{"bbox": [70, 88, 403, 103], "score": 1.0, "content": "Huang-Lepowsky coproduct [21] for the modules of the VOA ", "type": "text", "cross_page": true}, {"bbox": [404, 89, 439, 101], "score": 0.94, "content": "L(k,0)", "type": "inline_equation", "height": 12, "width": 35, "cross_page": true}, {"bbox": [440, 88, 541, 103], "score": 1.0, "content": ". Because of these", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [70, 103, 481, 118], "spans": [{"bbox": [70, 103, 233, 117], "score": 1.0, "content": "isomorphisms, we get that the ", "type": "text", "cross_page": true}, {"bbox": [233, 104, 255, 118], "score": 0.93, "content": "N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 22, "cross_page": true}, {"bbox": [255, 103, 343, 117], "score": 1.0, "content": " do indeed lie in ", "type": "text", "cross_page": true}, {"bbox": [343, 104, 360, 117], "score": 0.9, "content": "\\mathbb{Z}_{\\geq}", "type": "inline_equation", "height": 13, "width": 17, "cross_page": true}, {"bbox": [360, 103, 481, 117], "score": 1.0, "content": ", for any affine algebra.", "type": "text", "cross_page": true}], "index": 2}], "index": 29.5, "bbox_fs": [69, 627, 542, 717]}]}
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[{"type": "text", "bbox": [70, 70, 541, 114], "content": "is an unimportant sign independent of . This Galois action will play a fairly important role in this paper. Note that , so this action can be thought of as a generalisation of charge-conjugation. Note also that .", "index": 0}, {"type": "text", "bbox": [69, 115, 541, 164], "content": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23 p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities in :", "index": 1}, {"type": "interline_equation", "bbox": [239, 178, 372, 209], "content": "", "index": 2}, {"type": "text", "bbox": [70, 221, 541, 281], "content": "where and is the affine Weyl group of (the dependence of on arises through the action of ). We shall see shortly that these fusion coefficients, now manifestly integral, are in fact nonnegative. Let denote the corresponding fusion ring.", "index": 3}, {"type": "text", "bbox": [70, 282, 541, 313], "content": "A handy consequence of (2.4) that whenever is large enough that (i.e. that , then .", "index": 4}, {"type": "text", "bbox": [70, 313, 541, 354], "content": "It will sometimes be convenient to collect these coefficients in matrix form as the fusion matrices , defined by . For instance, and, more generally, is the permutation matrix associated to .", "index": 5}, {"type": "text", "bbox": [70, 355, 540, 383], "content": "The importance of (charge-)conjugation and simple-currents for us is that they respect fusions:", "index": 6}, {"type": "interline_equation", "bbox": [176, 394, 435, 455], "content": "", "index": 7}, {"type": "text", "bbox": [70, 463, 235, 477], "content": "for any simple-currents .", "index": 8}, {"type": "text", "bbox": [70, 478, 541, 572], "content": "For example, for we may take (the value of ), and then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is trivial here, but is a simple-current corresponding to permutation and function . The Galois action sends to the unique weight satisfying (mod ), where that sign there equals . The fusion coefficients are given by", "index": 9}, {"type": "interline_equation", "bbox": [117, 582, 496, 616], "content": "", "index": 10}, {"type": "text", "bbox": [70, 624, 541, 716], "content": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring where is the character ring for all finite-dimensional -modules, and is the subspace spanned by the elements . Finkelberg [8] proved that this ring is isomorphic to the K-ring of a “sub-quotient” of Kazhdan-Lusztig’s category of level integrable highest weight -modules, and t o Gelfand-Kazhdan’s category coming from finite-dimensional modules of the quantum group specialised to the root of unity for appropriate choice of . They also arise from the Huang-Lepowsky coproduct [21] for the modules of the VOA . Because of these isomorphisms, we get that the do indeed lie in , for any affine algebra.", "index": 11}]
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[{"bbox": [71, 73, 540, 89], "content": "is an unimportant sign independent of . This Galois action will play", "parent_index": 0, "line_index": 0}, {"bbox": [70, 88, 542, 103], "content": "a fairly important role in this paper. Note that , so this action can be thought of", "parent_index": 0, "line_index": 1}, {"bbox": [70, 101, 458, 118], "content": "as a generalisation of charge-conjugation. Note also that .", "parent_index": 0, "line_index": 2}, {"bbox": [94, 115, 541, 132], "content": "The fusion coefficients (1.1b) are usually computed by the Kac-Walton formula [23", "parent_index": 1, "line_index": 0}, {"bbox": [69, 132, 541, 146], "content": "p. 288, 35] (there are other codiscoverers) in terms of the tensor product multiplicities", "parent_index": 1, "line_index": 1}, {"bbox": [71, 146, 255, 168], "content": "in :", "parent_index": 1, "line_index": 2}, {"bbox": [66, 218, 539, 247], "content": "where and is the affine Weyl group of (the dependence of", "parent_index": 3, "line_index": 0}, {"bbox": [69, 239, 541, 256], "content": "on arises through the action of ). We shall see shortly that these fusion coefficients,", "parent_index": 3, "line_index": 1}, {"bbox": [69, 254, 541, 270], "content": "now manifestly integral, are in fact nonnegative. Let denote the corresponding", "parent_index": 3, "line_index": 2}, {"bbox": [70, 268, 131, 285], "content": "fusion ring.", "parent_index": 3, "line_index": 3}, {"bbox": [93, 283, 539, 301], "content": "A handy consequence of (2.4) that whenever is large enough that", "parent_index": 4, "line_index": 0}, {"bbox": [69, 297, 335, 318], "content": "(i.e. that , then .", "parent_index": 4, "line_index": 1}, {"bbox": [93, 313, 540, 327], "content": "It will sometimes be convenient to collect these coefficients in matrix form as the", "parent_index": 5, "line_index": 0}, {"bbox": [69, 327, 541, 344], "content": "fusion matrices , defined by . For instance, and, more generally,", "parent_index": 5, "line_index": 1}, {"bbox": [71, 343, 315, 357], "content": "is the permutation matrix associated to .", "parent_index": 5, "line_index": 2}, {"bbox": [93, 355, 542, 374], "content": "The importance of (charge-)conjugation and simple-currents for us is that they respect", "parent_index": 6, "line_index": 0}, {"bbox": [70, 371, 112, 387], "content": "fusions:", "parent_index": 6, "line_index": 1}, {"bbox": [70, 465, 233, 479], "content": "for any simple-currents .", "parent_index": 8, "line_index": 0}, {"bbox": [93, 479, 542, 495], "content": "For example, for we may take (the value of ), and", "parent_index": 9, "line_index": 0}, {"bbox": [64, 489, 548, 523], "content": "then the Kac-Peterson matrix is Sab = k2+2 sin(π (a+1k)+ (2b+1) ). Charge-conjugation C is", "parent_index": 9, "line_index": 1}, {"bbox": [70, 515, 542, 530], "content": "trivial here, but is a simple-current corresponding to permutation and", "parent_index": 9, "line_index": 2}, {"bbox": [69, 527, 542, 547], "content": "function . The Galois action sends to the unique weight satisfying", "parent_index": 9, "line_index": 3}, {"bbox": [71, 542, 542, 561], "content": "(mod ), where that sign there equals . The fusion", "parent_index": 9, "line_index": 4}, {"bbox": [71, 559, 196, 574], "content": "coefficients are given by", "parent_index": 9, "line_index": 5}, {"bbox": [95, 627, 541, 642], "content": "Equation (2.4) tells us the affine fusion rules are the structure constants for the ring", "parent_index": 11, "line_index": 0}, {"bbox": [71, 640, 542, 656], "content": "where is the character ring for all finite-dimensional -modules, and", "parent_index": 11, "line_index": 1}, {"bbox": [71, 656, 541, 671], "content": "is the subspace spanned by the elements . Finkelberg [8] proved that", "parent_index": 11, "line_index": 2}, {"bbox": [70, 671, 540, 687], "content": "this ring is isomorphic to the K-ring of a “sub-quotient” of Kazhdan-Lusztig’s category", "parent_index": 11, "line_index": 3}, {"bbox": [69, 684, 539, 704], "content": "of level integrable highest weight -modules, and t o Gelfand-Kazhdan’s category", "parent_index": 11, "line_index": 4}, {"bbox": [71, 703, 541, 717], "content": "coming from finite-dimensional modules of the quantum group specialised to the", "parent_index": 11, "line_index": 5}, {"bbox": [70, 73, 541, 89], "content": "root of unity for appropriate choice of . They also arise from the", "parent_index": 11, "line_index": 6}, {"bbox": [70, 88, 541, 103], "content": "Huang-Lepowsky coproduct [21] for the modules of the VOA . Because of these", "parent_index": 11, "line_index": 7}, {"bbox": [70, 103, 481, 118], "content": "isomorphisms, we get that the do indeed lie in , for any affine algebra.", "parent_index": 11, "line_index": 8}]
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[]
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[{"bbox": [71, 75, 182, 88], "content": "\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\sigma_{\\ell}(c)/c", "parent_index": 0, "subtype": "inline"}, {"bbox": [383, 75, 391, 84], "content": "\\lambda", "parent_index": 0, "subtype": "inline"}, {"bbox": [321, 90, 367, 101], "content": "\\sigma_{-1}=C", "parent_index": 0, "subtype": "inline"}, {"bbox": [371, 102, 453, 115], "content": "\\sigma_{\\ell}\\circ J=J^{\\ell}\\circ\\sigma_{\\ell}", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 146, 215, 166], "content": "T_{\\lambda\\mu}^{\\nu}\\overset{\\mathrm{def}}{=}\\mathrm{mult}_{L(\\overline{{\\lambda}})\\otimes L(\\overline{{\\mu}})}(L(\\overline{{\\nu}}))", "parent_index": 1, "subtype": "inline"}, {"bbox": [232, 151, 248, 162], "content": "X_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [239, 178, 372, 209], "content": "N_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname*{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,", "parent_index": 2, "subtype": "interline"}, {"bbox": [105, 223, 206, 240], "content": "w.\\gamma\\,{\\stackrel{\\mathrm{def}}{=}}\\,w(\\gamma+\\rho)-\\rho", "parent_index": 3, "subtype": "inline"}, {"bbox": [232, 227, 245, 237], "content": "W", "parent_index": 3, "subtype": "inline"}, {"bbox": [388, 223, 412, 238], "content": "X_{r}^{(1)}", "parent_index": 3, "subtype": "inline"}, {"bbox": [518, 228, 539, 242], "content": "N_{\\lambda\\mu}^{\\nu}", "parent_index": 3, "subtype": "inline"}, {"bbox": [88, 242, 95, 251], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [249, 243, 262, 252], "content": "W", "parent_index": 3, "subtype": "inline"}, {"bbox": [359, 256, 401, 268], "content": "\\mathcal{R}(X_{r,k})", "parent_index": 3, "subtype": "inline"}, {"bbox": [330, 287, 337, 296], "content": "k", "parent_index": 4, "subtype": "inline"}, {"bbox": [448, 283, 539, 300], "content": "\\lambda+\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "parent_index": 4, "subtype": "inline"}, {"bbox": [121, 300, 237, 314], "content": "\\begin{array}{r}{\\sum_{i=1}^{\\tau}a_{i}^{\\vee}(\\lambda_{i}+\\mu_{i})\\le k)}\\end{array}", "parent_index": 4, "subtype": "inline"}, {"bbox": [273, 302, 330, 315], "content": "N_{\\lambda\\mu}^{\\nu}=T_{\\lambda\\mu}^{\\nu}", "parent_index": 4, "subtype": "inline"}, {"bbox": [155, 330, 171, 340], "content": "N_{\\lambda}", "parent_index": 5, "subtype": "inline"}, {"bbox": [236, 329, 311, 344], "content": "(N_{\\lambda})_{\\mu\\nu}=N_{\\lambda\\mu}^{\\nu}", "parent_index": 5, "subtype": "inline"}, {"bbox": [392, 330, 430, 341], "content": "N_{0}=I", "parent_index": 5, "subtype": "inline"}, {"bbox": [71, 344, 86, 357], "content": "N_{j}", "parent_index": 5, "subtype": "inline"}, {"bbox": [302, 344, 309, 353], "content": "J", "parent_index": 5, "subtype": "inline"}, {"bbox": [176, 394, 435, 455], "content": "\\begin{array}{c}{{N_{C\\lambda,C\\mu}^{C\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{{}}}\\\\ {{N_{J\\lambda,J^{\\prime}\\mu}^{J J^{\\prime}\\nu}=N_{\\lambda\\mu}^{\\nu}}}\\\\ {{N_{\\lambda\\mu}^{\\nu}\\neq0\\ \\Rightarrow\\ Q_{j}(\\lambda){+}Q_{j}(\\mu)\\equiv Q_{j}(\\nu){\\qquad}(\\mathrm{mod~1})}}\\end{array}", "parent_index": 7, "subtype": "interline"}, {"bbox": [195, 465, 230, 478], "content": "J,J^{\\prime},j", "parent_index": 8, "subtype": "inline"}, {"bbox": [190, 480, 233, 494], "content": "\\mathcal{R}(A_{1,k})", "parent_index": 9, "subtype": "inline"}, {"bbox": [313, 479, 414, 493], "content": "P_{+}\\;=\\;\\{0,1,\\ldots,k\\}", "parent_index": 9, "subtype": "inline"}, {"bbox": [493, 482, 506, 492], "content": "\\lambda_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [159, 516, 189, 529], "content": "j=k", "parent_index": 9, "subtype": "inline"}, {"bbox": [456, 517, 516, 527], "content": "J a=k-a", "parent_index": 9, "subtype": "inline"}, {"bbox": [118, 531, 182, 544], "content": "Q_{j}(a)=a/2", "parent_index": 9, "subtype": "inline"}, {"bbox": [317, 533, 325, 541], "content": "a", "parent_index": 9, "subtype": "inline"}, {"bbox": [438, 529, 486, 543], "content": "a^{(\\ell)}\\in P_{+}", "parent_index": 9, "subtype": "inline"}, {"bbox": [71, 544, 185, 558], "content": "a^{(\\ell)}+1\\equiv\\pm\\ell\\left(a+1\\right)", "parent_index": 9, "subtype": "inline"}, {"bbox": [221, 544, 257, 557], "content": "2k+4", "parent_index": 9, "subtype": "inline"}, {"bbox": [425, 545, 471, 558], "content": "\\mathrm{i}^{\\ell-1}\\epsilon_{\\ell}^{\\prime}(a)", "parent_index": 9, "subtype": "inline"}, {"bbox": [117, 582, 496, 616], "content": "N_{a b}^{c}=\\left\\{\\begin{array}{c c}{{1}}&{{\\mathrm{if~}c\\equiv a\\!+\\!b\\;(\\mathrm{mod~}2)\\;\\mathrm{and~}|a\\!-\\!b|\\leq c\\leq\\operatorname*{min}\\{a\\!+\\!b,2k\\!-\\!a\\!-\\!b\\}}}\\\\ {{0}}&{{\\mathrm{otherwise}}}\\end{array}\\right..", "parent_index": 10, "subtype": "interline"}, {"bbox": [71, 642, 131, 655], "content": "\\mathrm{Ch}(X_{r})/\\mathcal{I}_{k}", "parent_index": 11, "subtype": "inline"}, {"bbox": [169, 642, 209, 655], "content": "\\operatorname{Ch}(X_{r})", "parent_index": 11, "subtype": "inline"}, {"bbox": [451, 643, 466, 654], "content": "X_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 658, 85, 668], "content": "\\mathcal{I}_{k}", "parent_index": 11, "subtype": "inline"}, {"bbox": [300, 657, 397, 670], "content": "\\mathrm{ch}_{\\overline{{\\mu}}}-(\\operatorname*{det}w)\\mathrm{ch}_{\\overline{{w}}.\\mu}", "parent_index": 11, "subtype": "inline"}, {"bbox": [363, 671, 379, 685], "content": "\\widetilde{\\mathcal{O}}_{k}", "parent_index": 11, "subtype": "inline"}, {"bbox": [112, 690, 119, 699], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [257, 686, 281, 700], "content": "X_{r}^{(1)}", "parent_index": 11, "subtype": "inline"}, {"bbox": [525, 687, 539, 702], "content": "\\widetilde{\\mathcal{O}}_{q}", "parent_index": 11, "subtype": "inline"}, {"bbox": [413, 704, 441, 717], "content": "U_{q}X_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [144, 75, 192, 87], "content": "q\\,=\\,\\xi_{2m\\kappa}", "parent_index": 11, "subtype": "inline"}, {"bbox": [331, 75, 399, 87], "content": "m\\in\\{1,2,3\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [404, 89, 439, 101], "content": "L(k,0)", "parent_index": 11, "subtype": "inline"}, {"bbox": [233, 104, 255, 118], "content": "N_{\\lambda\\mu}^{\\nu}", "parent_index": 11, "subtype": "inline"}, {"bbox": [343, 104, 360, 117], "content": "\\mathbb{Z}_{\\geq}", "parent_index": 11, "subtype": "inline"}]
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[]
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A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions and norms. Q-dimensions vary by at most a sign while norms are constant mod $2\kappa$ : ${\cal D}(w.\lambda)\;=\;\operatorname*{det}\left(w\right){\cal D}(\lambda)$ and $(w\lambda|w\lambda)\,\equiv\,(\lambda|\lambda)$ (mod $2\kappa$ ). The point is that for exceptional algebras at small levels, the highest weights can often be distinguished by the pair $(\mathcal{D}(\lambda),(\lambda+\rho|\lambda+\rho)$ (mod $2\kappa$ )). For example this is true of $E_{8,5},E_{8,6},F_{4,4}$ . This is a useful way in practise to use both (2.4) and the Galois action (2.3).
An important property obeyed by the matrix $S$ for any classical algebra $X_{r}$ is ranklevel duality. The first appearance of this curious duality seems to be by Frenkel [9], but by now many aspects and generalisations have been explored in the literature. For $A_{r}^{(1)}$ , it is related to the existence of mutually commutative affine subalgbras $\widehat{\mathrm{sl}(n)}$ and $\widehat{\mathrm{sl}}(\widehat{k})$ in $\widehat{\mathrm{gl}}(n\widehat{k})$ . Witten has another interpretation of it [37]: he found a natural map (a ring homomorphism) from the quantum cohomology of the Grassmannian $G(k,N)$ , to the fusion ring of the algebra $\mathrm{u}(k)\cong\mathrm{su}(k)\oplus\mathrm{u}(1)$ at level $(N-k,N)$ . Witten used the duality between $G(k,N)$ and $G(N-k,N)$ to show that the fusion rings of $\operatorname{u}(k)$ level $(N-k,N)$ and $\mathrm{u}(N-k)$ level $(k,N)$ should coincide. A considerable generalisation, applying to any VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution $\textstyle\sum_{i}[x_{i}]\leftrightarrow\sum_{i}[1-x_{i}]$ of torsion elements of the Bloch group.
The Kac-Peterson matrices of $\widehat{\mathrm{sl}(\ell)}$ level $k$ and $\widehat{\mathrm{sl}(k)}$ level $\ell$ are related, as are those of $C_{r,k}$ and $C_{k,r}$ , and $\widehat{\mathrm{so}(\ell)}$ level $k$ and $\widehat{\mathrm{so}(k)}$ level $\ell$ . We will need only the symplectic one; the details will be given in §3.3.
# 2.2. Symmetries of fusion coefficients
Definition 2.1. By an isomorphism between fusion rings $\mathcal{R}(X_{r,k})$ and $\mathcal{R}(Y_{s,m})$ (with fusion coefficients $N$ and $M$ respectively) we mean a bijection $\pi\ :\ P_{+}^{k}(X_{r}^{(1)})\ \to$ ${\cal P}_{+}^{m}(Y_{s}^{(1)})$ such that
$$
N_{\lambda,\mu}^{\nu}=M_{\pi\lambda,\pi\mu}^{\pi\nu}\qquad\forall\lambda,\mu,\nu\in P_{+}(X_{r,k})\ .
$$
When $X_{r,k}~=~Y_{s,m}$ we call $\pi$ an automorphism or fusion-symmetry. Call the pair of permutations $\pi,\pi^{\prime}$ an $S$ -symmetry if
$$
S_{\pi\lambda,\pi^{\prime}\mu}=S_{\lambda\mu}\qquad\forall\lambda,\mu\in{\cal P}_{+}\ .
$$
The lemma below tells us that fusion- and $S$ -symmetries form two isomorphic groups; the former we will label $\boldsymbol{A}(\boldsymbol{X}_{r,k})$ . Equation (2.5a) says that the charge-conjugation $C$ , and more generally any conjugation, is a fusion-symmetry, while (2.2a) says $(C,C)$ is an $S$ - symmetry. Because $N_{0}=I=M_{\tilde{0}}$ , $N_{\lambda\mu}^{0}=C_{\lambda\mu}$ and $M_{\tilde{\lambda},\tilde{\mu}}^{\tilde{0}}=\widetilde{C}_{\tilde{\lambda},\tilde{\mu}}$ (we use tilde’s to denote quantities in $Y_{s}^{(1)}$ level ${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$ ), any isomorphism $\pi$ must obey $\pi0=\tilde{0}$ and $\widetilde{C}\circ\pi=\pi\circ C$ . More generally, since $N_{\lambda}$ is a permutation matrix of order ${\boldsymbol{n}}$ iff $\lambda$ is a simpl e- current of order $n$ , we see that an isomorphism sends simple-currents to simple-currents of equal order. We get
$$
\pi(J\mu)=\pi(j)\,\pi(\mu)~.
$$
|
<html><body>
<p data-bbox="71 115 545 200">A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions and norms. Q-dimensions vary by at most a sign while norms are constant mod $2\kappa$ : ${\cal D}(w.\lambda)\;=\;\operatorname*{det}\left(w\right){\cal D}(\lambda)$ and $(w\lambda|w\lambda)\,\equiv\,(\lambda|\lambda)$ (mod $2\kappa$ ). The point is that for exceptional algebras at small levels, the highest weights can often be distinguished by the pair $(\mathcal{D}(\lambda),(\lambda+\rho|\lambda+\rho)$ (mod $2\kappa$ )). For example this is true of $E_{8,5},E_{8,6},F_{4,4}$ . This is a useful way in practise to use both (2.4) and the Galois action (2.3). </p>
<p data-bbox="70 201 542 365">An important property obeyed by the matrix $S$ for any classical algebra $X_{r}$ is ranklevel duality. The first appearance of this curious duality seems to be by Frenkel [9], but by now many aspects and generalisations have been explored in the literature. For $A_{r}^{(1)}$ , it is related to the existence of mutually commutative affine subalgbras $\widehat{\mathrm{sl}(n)}$ and $\widehat{\mathrm{sl}}(\widehat{k})$ in $\widehat{\mathrm{gl}}(n\widehat{k})$ . Witten has another interpretation of it [37]: he found a natural map (a ring homomorphism) from the quantum cohomology of the Grassmannian $G(k,N)$ , to the fusion ring of the algebra $\mathrm{u}(k)\cong\mathrm{su}(k)\oplus\mathrm{u}(1)$ at level $(N-k,N)$ . Witten used the duality between $G(k,N)$ and $G(N-k,N)$ to show that the fusion rings of $\operatorname{u}(k)$ level $(N-k,N)$ and $\mathrm{u}(N-k)$ level $(k,N)$ should coincide. A considerable generalisation, applying to any VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution $\textstyle\sum_{i}[x_{i}]\leftrightarrow\sum_{i}[1-x_{i}]$ of torsion elements of the Bloch group. </p>
<p data-bbox="71 366 541 414">The Kac-Peterson matrices of $\widehat{\mathrm{sl}(\ell)}$ level $k$ and $\widehat{\mathrm{sl}(k)}$ level $\ell$ are related, as are those of $C_{r,k}$ and $C_{k,r}$ , and $\widehat{\mathrm{so}(\ell)}$ level $k$ and $\widehat{\mathrm{so}(k)}$ level $\ell$ . We will need only the symplectic one; the details will be given in §3.3. </p>
<h1 data-bbox="71 426 270 441">2.2. Symmetries of fusion coefficients </h1>
<p data-bbox="71 446 541 493">Definition 2.1. By an isomorphism between fusion rings $\mathcal{R}(X_{r,k})$ and $\mathcal{R}(Y_{s,m})$ (with fusion coefficients $N$ and $M$ respectively) we mean a bijection $\pi\ :\ P_{+}^{k}(X_{r}^{(1)})\ \to$ ${\cal P}_{+}^{m}(Y_{s}^{(1)})$ such that </p>
<div class="equation" data-bbox="198 496 413 513">$$
N_{\lambda,\mu}^{\nu}=M_{\pi\lambda,\pi\mu}^{\pi\nu}\qquad\forall\lambda,\mu,\nu\in P_{+}(X_{r,k})\ .
$$</div>
<p data-bbox="71 516 542 545">When $X_{r,k}~=~Y_{s,m}$ we call $\pi$ an automorphism or fusion-symmetry. Call the pair of permutations $\pi,\pi^{\prime}$ an $S$ -symmetry if </p>
<div class="equation" data-bbox="225 556 387 570">$$
S_{\pi\lambda,\pi^{\prime}\mu}=S_{\lambda\mu}\qquad\forall\lambda,\mu\in{\cal P}_{+}\ .
$$</div>
<p data-bbox="70 577 541 700">The lemma below tells us that fusion- and $S$ -symmetries form two isomorphic groups; the former we will label $\boldsymbol{A}(\boldsymbol{X}_{r,k})$ . Equation (2.5a) says that the charge-conjugation $C$ , and more generally any conjugation, is a fusion-symmetry, while (2.2a) says $(C,C)$ is an $S$ - symmetry. Because $N_{0}=I=M_{\tilde{0}}$ , $N_{\lambda\mu}^{0}=C_{\lambda\mu}$ and $M_{\tilde{\lambda},\tilde{\mu}}^{\tilde{0}}=\widetilde{C}_{\tilde{\lambda},\tilde{\mu}}$ (we use tilde’s to denote quantities in $Y_{s}^{(1)}$ level ${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$ ), any isomorphism $\pi$ must obey $\pi0=\tilde{0}$ and $\widetilde{C}\circ\pi=\pi\circ C$ . More generally, since $N_{\lambda}$ is a permutation matrix of order ${\boldsymbol{n}}$ iff $\lambda$ is a simpl e- current of order $n$ , we see that an isomorphism sends simple-currents to simple-currents of equal order. We get </p>
<div class="equation" data-bbox="254 702 358 716">$$
\pi(J\mu)=\pi(j)\,\pi(\mu)~.
$$</div>
</body></html>
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0002044v1
| 5 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "", "page_idx": 5}, {"type": "text", "text": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions and norms. Q-dimensions vary by at most a sign while norms are constant mod $2\\kappa$ : ${\\cal D}(w.\\lambda)\\;=\\;\\operatorname*{det}\\left(w\\right){\\cal D}(\\lambda)$ and $(w\\lambda|w\\lambda)\\,\\equiv\\,(\\lambda|\\lambda)$ (mod $2\\kappa$ ). The point is that for exceptional algebras at small levels, the highest weights can often be distinguished by the pair $(\\mathcal{D}(\\lambda),(\\lambda+\\rho|\\lambda+\\rho)$ (mod $2\\kappa$ )). For example this is true of $E_{8,5},E_{8,6},F_{4,4}$ . This is a useful way in practise to use both (2.4) and the Galois action (2.3). ", "page_idx": 5}, {"type": "text", "text": "An important property obeyed by the matrix $S$ for any classical algebra $X_{r}$ is ranklevel duality. The first appearance of this curious duality seems to be by Frenkel [9], but by now many aspects and generalisations have been explored in the literature. For $A_{r}^{(1)}$ , it is related to the existence of mutually commutative affine subalgbras $\\widehat{\\mathrm{sl}(n)}$ and $\\widehat{\\mathrm{sl}}(\\widehat{k})$ in $\\widehat{\\mathrm{gl}}(n\\widehat{k})$ . Witten has another interpretation of it [37]: he found a natural map (a ring homomorphism) from the quantum cohomology of the Grassmannian $G(k,N)$ , to the fusion ring of the algebra $\\mathrm{u}(k)\\cong\\mathrm{su}(k)\\oplus\\mathrm{u}(1)$ at level $(N-k,N)$ . Witten used the duality between $G(k,N)$ and $G(N-k,N)$ to show that the fusion rings of $\\operatorname{u}(k)$ level $(N-k,N)$ and $\\mathrm{u}(N-k)$ level $(k,N)$ should coincide. A considerable generalisation, applying to any VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution $\\textstyle\\sum_{i}[x_{i}]\\leftrightarrow\\sum_{i}[1-x_{i}]$ of torsion elements of the Bloch group. ", "page_idx": 5}, {"type": "text", "text": "The Kac-Peterson matrices of $\\widehat{\\mathrm{sl}(\\ell)}$ level $k$ and $\\widehat{\\mathrm{sl}(k)}$ level $\\ell$ are related, as are those of $C_{r,k}$ and $C_{k,r}$ , and $\\widehat{\\mathrm{so}(\\ell)}$ level $k$ and $\\widehat{\\mathrm{so}(k)}$ level $\\ell$ . We will need only the symplectic one; the details will be given in §3.3. ", "page_idx": 5}, {"type": "text", "text": "2.2. Symmetries of fusion coefficients ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "Definition 2.1. By an isomorphism between fusion rings $\\mathcal{R}(X_{r,k})$ and $\\mathcal{R}(Y_{s,m})$ (with fusion coefficients $N$ and $M$ respectively) we mean a bijection $\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to$ ${\\cal P}_{+}^{m}(Y_{s}^{(1)})$ such that ", "page_idx": 5}, {"type": "equation", "text": "$$\nN_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "When $X_{r,k}~=~Y_{s,m}$ we call $\\pi$ an automorphism or fusion-symmetry. Call the pair of permutations $\\pi,\\pi^{\\prime}$ an $S$ -symmetry if ", "page_idx": 5}, {"type": "equation", "text": "$$\nS_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "The lemma below tells us that fusion- and $S$ -symmetries form two isomorphic groups; the former we will label $\\boldsymbol{A}(\\boldsymbol{X}_{r,k})$ . Equation (2.5a) says that the charge-conjugation $C$ , and more generally any conjugation, is a fusion-symmetry, while (2.2a) says $(C,C)$ is an $S$ - symmetry. Because $N_{0}=I=M_{\\tilde{0}}$ , $N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}$ and $M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}$ (we use tilde’s to denote quantities in $Y_{s}^{(1)}$ level ${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$ ), any isomorphism $\\pi$ must obey $\\pi0=\\tilde{0}$ and $\\widetilde{C}\\circ\\pi=\\pi\\circ C$ . More generally, since $N_{\\lambda}$ is a permutation matrix of order ${\\boldsymbol{n}}$ iff $\\lambda$ is a simpl e- current of order $n$ , we see that an isomorphism sends simple-currents to simple-currents of equal order. We get ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.\n$$", "text_format": "latex", "page_idx": 5}]
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{"preproc_blocks": [{"type": "text", "bbox": [70, 70, 543, 114], "lines": [{"bbox": [70, 73, 541, 89], "spans": [{"bbox": [70, 73, 143, 89], "score": 1.0, "content": "root of unity ", "type": "text"}, {"bbox": [144, 75, 192, 87], "score": 0.94, "content": "q\\,=\\,\\xi_{2m\\kappa}", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [192, 73, 330, 89], "score": 1.0, "content": " for appropriate choice of ", "type": "text"}, {"bbox": [331, 75, 399, 87], "score": 0.94, "content": "m\\in\\{1,2,3\\}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [399, 73, 541, 89], "score": 1.0, "content": ". They also arise from the", "type": "text"}], "index": 0}, {"bbox": [70, 88, 541, 103], "spans": [{"bbox": [70, 88, 403, 103], "score": 1.0, "content": "Huang-Lepowsky coproduct [21] for the modules of the VOA ", "type": "text"}, {"bbox": [404, 89, 439, 101], "score": 0.94, "content": "L(k,0)", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [440, 88, 541, 103], "score": 1.0, "content": ". Because of these", "type": "text"}], "index": 1}, {"bbox": [70, 103, 481, 118], "spans": [{"bbox": [70, 103, 233, 117], "score": 1.0, "content": "isomorphisms, we get that the ", "type": "text"}, {"bbox": [233, 104, 255, 118], "score": 0.93, "content": "N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [255, 103, 343, 117], "score": 1.0, "content": " do indeed lie in ", "type": "text"}, {"bbox": [343, 104, 360, 117], "score": 0.9, "content": "\\mathbb{Z}_{\\geq}", "type": "inline_equation", "height": 13, "width": 17}, {"bbox": [360, 103, 481, 117], "score": 1.0, "content": ", for any affine algebra.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [71, 115, 545, 200], "lines": [{"bbox": [94, 116, 548, 133], "spans": [{"bbox": [94, 116, 548, 133], "score": 1.0, "content": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions", "type": "text"}], "index": 3}, {"bbox": [70, 131, 540, 146], "spans": [{"bbox": [70, 131, 523, 146], "score": 1.0, "content": "and norms. Q-dimensions vary by at most a sign while norms are constant mod ", "type": "text"}, {"bbox": [523, 133, 536, 142], "score": 0.81, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [537, 131, 540, 146], "score": 1.0, "content": ":", "type": "text"}], "index": 4}, {"bbox": [71, 144, 541, 160], "spans": [{"bbox": [71, 146, 195, 159], "score": 0.92, "content": "{\\cal D}(w.\\lambda)\\;=\\;\\operatorname*{det}\\left(w\\right){\\cal D}(\\lambda)", "type": "inline_equation", "height": 13, "width": 124}, {"bbox": [196, 144, 225, 160], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 146, 317, 159], "score": 0.92, "content": "(w\\lambda|w\\lambda)\\,\\equiv\\,(\\lambda|\\lambda)", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [318, 144, 356, 160], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [356, 147, 369, 156], "score": 0.67, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [370, 144, 541, 160], "score": 1.0, "content": "). The point is that for excep-", "type": "text"}], "index": 5}, {"bbox": [71, 160, 540, 174], "spans": [{"bbox": [71, 160, 540, 174], "score": 1.0, "content": "tional algebras at small levels, the highest weights can often be distinguished by the pair", "type": "text"}], "index": 6}, {"bbox": [71, 173, 541, 190], "spans": [{"bbox": [71, 175, 172, 188], "score": 0.85, "content": "(\\mathcal{D}(\\lambda),(\\lambda+\\rho|\\lambda+\\rho)", "type": "inline_equation", "height": 13, "width": 101}, {"bbox": [172, 173, 207, 190], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [207, 176, 220, 185], "score": 0.65, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 173, 377, 190], "score": 1.0, "content": ")). For example this is true of ", "type": "text"}, {"bbox": [378, 176, 453, 188], "score": 0.93, "content": "E_{8,5},E_{8,6},F_{4,4}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [453, 173, 541, 190], "score": 1.0, "content": ". This is a useful", "type": "text"}], "index": 7}, {"bbox": [70, 189, 393, 203], "spans": [{"bbox": [70, 189, 393, 203], "score": 1.0, "content": "way in practise to use both (2.4) and the Galois action (2.3).", "type": "text"}], "index": 8}], "index": 5.5}, {"type": "text", "bbox": [70, 201, 542, 365], "lines": [{"bbox": [94, 202, 541, 218], "spans": [{"bbox": [94, 202, 338, 218], "score": 1.0, "content": "An important property obeyed by the matrix ", "type": "text"}, {"bbox": [338, 205, 347, 214], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 202, 480, 218], "score": 1.0, "content": " for any classical algebra ", "type": "text"}, {"bbox": [480, 205, 496, 215], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [496, 202, 541, 218], "score": 1.0, "content": " is rank-", "type": "text"}], "index": 9}, {"bbox": [70, 216, 541, 233], "spans": [{"bbox": [70, 216, 541, 233], "score": 1.0, "content": "level duality. The first appearance of this curious duality seems to be by Frenkel [9],", "type": "text"}], "index": 10}, {"bbox": [70, 232, 542, 246], "spans": [{"bbox": [70, 232, 542, 246], "score": 1.0, "content": "but by now many aspects and generalisations have been explored in the literature. For", "type": "text"}], "index": 11}, {"bbox": [71, 244, 542, 265], "spans": [{"bbox": [71, 246, 93, 261], "score": 0.92, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 22}, {"bbox": [93, 244, 489, 265], "score": 1.0, "content": ", it is related to the existence of mutually commutative affine subalgbras", "type": "text"}, {"bbox": [490, 245, 515, 262], "score": 0.91, "content": "\\widehat{\\mathrm{sl}(n)}", "type": "inline_equation", "height": 17, "width": 25}, {"bbox": [515, 244, 542, 265], "score": 1.0, "content": " and", "type": "text"}], "index": 12}, {"bbox": [71, 263, 541, 281], "spans": [{"bbox": [71, 263, 95, 280], "score": 0.9, "content": "\\widehat{\\mathrm{sl}}(\\widehat{k})", "type": "inline_equation", "height": 17, "width": 24}, {"bbox": [96, 265, 114, 281], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [114, 263, 147, 279], "score": 0.91, "content": "\\widehat{\\mathrm{gl}}(n\\widehat{k})", "type": "inline_equation", "height": 16, "width": 33}, {"bbox": [147, 265, 541, 281], "score": 1.0, "content": ". Witten has another interpretation of it [37]: he found a natural map (a", "type": "text"}], "index": 13}, {"bbox": [71, 281, 541, 295], "spans": [{"bbox": [71, 281, 460, 295], "score": 1.0, "content": "ring homomorphism) from the quantum cohomology of the Grassmannian ", "type": "text"}, {"bbox": [460, 281, 502, 294], "score": 0.93, "content": "G(k,N)", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [502, 281, 541, 295], "score": 1.0, "content": ", to the", "type": "text"}], "index": 14}, {"bbox": [70, 294, 541, 309], "spans": [{"bbox": [70, 294, 204, 309], "score": 1.0, "content": "fusion ring of the algebra ", "type": "text"}, {"bbox": [205, 296, 306, 308], "score": 0.92, "content": "\\mathrm{u}(k)\\cong\\mathrm{su}(k)\\oplus\\mathrm{u}(1)", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [306, 294, 350, 309], "score": 1.0, "content": " at level ", "type": "text"}, {"bbox": [351, 296, 407, 308], "score": 0.93, "content": "(N-k,N)", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [407, 294, 541, 309], "score": 1.0, "content": ". Witten used the duality", "type": "text"}], "index": 15}, {"bbox": [70, 309, 540, 323], "spans": [{"bbox": [70, 309, 117, 323], "score": 1.0, "content": "between ", "type": "text"}, {"bbox": [118, 310, 159, 322], "score": 0.94, "content": "G(k,N)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [160, 309, 186, 323], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [186, 310, 254, 322], "score": 0.94, "content": "G(N-k,N)", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [254, 309, 426, 323], "score": 1.0, "content": " to show that the fusion rings of", "type": "text"}, {"bbox": [427, 309, 450, 322], "score": 0.87, "content": "\\operatorname{u}(k)", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [451, 309, 481, 323], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [482, 309, 540, 322], "score": 0.92, "content": "(N-k,N)", "type": "inline_equation", "height": 13, "width": 58}], "index": 16}, {"bbox": [70, 322, 541, 338], "spans": [{"bbox": [70, 322, 94, 338], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [94, 324, 142, 336], "score": 0.94, "content": "\\mathrm{u}(N-k)", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [143, 322, 173, 338], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [173, 324, 205, 337], "score": 0.94, "content": "(k,N)", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [206, 322, 541, 338], "score": 1.0, "content": " should coincide. A considerable generalisation, applying to any", "type": "text"}], "index": 17}, {"bbox": [70, 337, 542, 353], "spans": [{"bbox": [70, 337, 542, 353], "score": 1.0, "content": "VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution", "type": "text"}], "index": 18}, {"bbox": [71, 351, 388, 368], "spans": [{"bbox": [71, 353, 179, 366], "score": 0.93, "content": "\\textstyle\\sum_{i}[x_{i}]\\leftrightarrow\\sum_{i}[1-x_{i}]", "type": "inline_equation", "height": 13, "width": 108}, {"bbox": [179, 351, 388, 368], "score": 1.0, "content": " of torsion elements of the Bloch group.", "type": "text"}], "index": 19}], "index": 14}, {"type": "text", "bbox": [71, 366, 541, 414], "lines": [{"bbox": [94, 367, 542, 385], "spans": [{"bbox": [94, 369, 253, 385], "score": 1.0, "content": "The Kac-Peterson matrices of", "type": "text"}, {"bbox": [253, 367, 276, 383], "score": 0.92, "content": "\\widehat{\\mathrm{sl}(\\ell)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [276, 369, 306, 385], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [306, 372, 313, 380], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [313, 369, 339, 385], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 367, 363, 384], "score": 0.92, "content": "\\widehat{\\mathrm{sl}(k)}", "type": "inline_equation", "height": 17, "width": 24}, {"bbox": [364, 369, 393, 385], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [393, 371, 399, 380], "score": 0.77, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [399, 369, 542, 385], "score": 1.0, "content": "are related, as are those of", "type": "text"}], "index": 20}, {"bbox": [71, 384, 541, 404], "spans": [{"bbox": [71, 389, 93, 402], "score": 0.92, "content": "C_{r,k}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [93, 387, 120, 404], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [120, 389, 143, 402], "score": 0.92, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [143, 387, 173, 404], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [174, 385, 199, 401], "score": 0.89, "content": "\\widehat{\\mathrm{so}(\\ell)}", "type": "inline_equation", "height": 16, "width": 25}, {"bbox": [200, 387, 231, 404], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [231, 389, 238, 398], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [238, 387, 265, 404], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [266, 384, 293, 401], "score": 0.86, "content": "\\widehat{\\mathrm{so}(k)}", "type": "inline_equation", "height": 17, "width": 27}, {"bbox": [293, 387, 324, 404], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [324, 389, 330, 398], "score": 0.86, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [330, 387, 541, 404], "score": 1.0, "content": ". We will need only the symplectic one;", "type": "text"}], "index": 21}, {"bbox": [70, 402, 239, 416], "spans": [{"bbox": [70, 402, 239, 416], "score": 1.0, "content": "the details will be given in §3.3.", "type": "text"}], "index": 22}], "index": 21}, {"type": "title", "bbox": [71, 426, 270, 441], "lines": [{"bbox": [72, 429, 269, 442], "spans": [{"bbox": [72, 429, 269, 442], "score": 1.0, "content": "2.2. Symmetries of fusion coefficients", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [71, 446, 541, 493], "lines": [{"bbox": [93, 447, 540, 465], "spans": [{"bbox": [93, 447, 423, 465], "score": 1.0, "content": "Definition 2.1. By an isomorphism between fusion rings ", "type": "text"}, {"bbox": [423, 449, 466, 463], "score": 0.92, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [466, 447, 496, 465], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [496, 449, 540, 462], "score": 0.9, "content": "\\mathcal{R}(Y_{s,m})", "type": "inline_equation", "height": 13, "width": 44}], "index": 24}, {"bbox": [69, 463, 541, 481], "spans": [{"bbox": [69, 464, 203, 481], "score": 1.0, "content": "(with fusion coefficients ", "type": "text"}, {"bbox": [204, 466, 216, 477], "score": 0.79, "content": "N", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [216, 464, 245, 481], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [245, 465, 259, 477], "score": 0.64, "content": "M", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [260, 464, 448, 481], "score": 1.0, "content": " respectively) we mean a bijection ", "type": "text"}, {"bbox": [448, 463, 541, 480], "score": 0.92, "content": "\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to", "type": "inline_equation", "height": 17, "width": 93}], "index": 25}, {"bbox": [71, 478, 177, 498], "spans": [{"bbox": [71, 480, 122, 498], "score": 0.94, "content": "{\\cal P}_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 18, "width": 51}, {"bbox": [122, 478, 177, 498], "score": 1.0, "content": " such that", "type": "text"}], "index": 26}], "index": 25}, {"type": "interline_equation", "bbox": [198, 496, 413, 513], "lines": [{"bbox": [198, 496, 413, 513], "spans": [{"bbox": [198, 496, 413, 513], "score": 0.9, "content": "N_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [71, 516, 542, 545], "lines": [{"bbox": [71, 516, 544, 535], "spans": [{"bbox": [71, 516, 107, 535], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [107, 518, 176, 532], "score": 0.91, "content": "X_{r,k}~=~Y_{s,m}", "type": "inline_equation", "height": 14, "width": 69}, {"bbox": [176, 516, 224, 535], "score": 1.0, "content": " we call ", "type": "text"}, {"bbox": [225, 523, 232, 529], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [233, 516, 544, 535], "score": 1.0, "content": " an automorphism or fusion-symmetry. Call the pair of", "type": "text"}], "index": 28}, {"bbox": [71, 533, 267, 547], "spans": [{"bbox": [71, 533, 142, 547], "score": 1.0, "content": "permutations ", "type": "text"}, {"bbox": [143, 533, 167, 546], "score": 0.87, "content": "\\pi,\\pi^{\\prime}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [167, 533, 187, 547], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [187, 534, 196, 543], "score": 0.84, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [196, 533, 267, 547], "score": 1.0, "content": "-symmetry if", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "interline_equation", "bbox": [225, 556, 387, 570], "lines": [{"bbox": [225, 556, 387, 570], "spans": [{"bbox": [225, 556, 387, 570], "score": 0.9, "content": "S_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "text", "bbox": [70, 577, 541, 700], "lines": [{"bbox": [94, 578, 541, 595], "spans": [{"bbox": [94, 578, 317, 595], "score": 1.0, "content": "The lemma below tells us that fusion- and ", "type": "text"}, {"bbox": [317, 582, 325, 591], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [326, 578, 541, 595], "score": 1.0, "content": "-symmetries form two isomorphic groups;", "type": "text"}], "index": 31}, {"bbox": [71, 594, 541, 610], "spans": [{"bbox": [71, 594, 196, 610], "score": 1.0, "content": "the former we will label ", "type": "text"}, {"bbox": [196, 595, 238, 608], "score": 0.94, "content": "\\boldsymbol{A}(\\boldsymbol{X}_{r,k})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [239, 594, 504, 610], "score": 1.0, "content": ". Equation (2.5a) says that the charge-conjugation ", "type": "text"}, {"bbox": [504, 596, 514, 605], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [514, 594, 541, 610], "score": 1.0, "content": ", and", "type": "text"}], "index": 32}, {"bbox": [70, 609, 541, 623], "spans": [{"bbox": [70, 609, 459, 623], "score": 1.0, "content": "more generally any conjugation, is a fusion-symmetry, while (2.2a) says ", "type": "text"}, {"bbox": [459, 610, 492, 622], "score": 0.94, "content": "(C,C)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [492, 609, 527, 623], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [527, 610, 536, 619], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [536, 609, 541, 623], "score": 1.0, "content": "-", "type": "text"}], "index": 33}, {"bbox": [67, 622, 542, 644], "spans": [{"bbox": [67, 622, 177, 644], "score": 1.0, "content": "symmetry. Because ", "type": "text"}, {"bbox": [177, 626, 248, 639], "score": 0.92, "content": "N_{0}=I=M_{\\tilde{0}}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [248, 622, 254, 644], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [255, 625, 313, 640], "score": 0.94, "content": "N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}", "type": "inline_equation", "height": 15, "width": 58}, {"bbox": [314, 622, 341, 644], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [341, 623, 407, 642], "score": 0.94, "content": "M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}", "type": "inline_equation", "height": 19, "width": 66}, {"bbox": [408, 622, 542, 644], "score": 1.0, "content": " (we use tilde’s to denote", "type": "text"}], "index": 34}, {"bbox": [69, 641, 542, 662], "spans": [{"bbox": [69, 641, 139, 662], "score": 1.0, "content": "quantities in ", "type": "text"}, {"bbox": [140, 643, 162, 657], "score": 0.92, "content": "Y_{s}^{(1)}", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [162, 641, 192, 662], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [192, 650, 203, 656], "score": 0.85, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [204, 641, 306, 662], "score": 1.0, "content": "), any isomorphism ", "type": "text"}, {"bbox": [306, 650, 313, 656], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [314, 641, 374, 662], "score": 1.0, "content": " must obey ", "type": "text"}, {"bbox": [374, 645, 410, 656], "score": 0.92, "content": "\\pi0=\\tilde{0}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [410, 641, 435, 662], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [435, 644, 504, 656], "score": 0.92, "content": "\\widetilde{C}\\circ\\pi=\\pi\\circ C", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [505, 641, 542, 662], "score": 1.0, "content": ". More", "type": "text"}], "index": 35}, {"bbox": [70, 659, 541, 675], "spans": [{"bbox": [70, 659, 153, 675], "score": 1.0, "content": "generally, since ", "type": "text"}, {"bbox": [154, 661, 169, 672], "score": 0.93, "content": "N_{\\lambda}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [169, 659, 347, 675], "score": 1.0, "content": " is a permutation matrix of order ", "type": "text"}, {"bbox": [347, 664, 354, 670], "score": 0.87, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 659, 372, 675], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [372, 661, 380, 670], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [380, 659, 528, 675], "score": 1.0, "content": " is a simpl e- current of order ", "type": "text"}, {"bbox": [529, 664, 536, 670], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [537, 659, 541, 675], "score": 1.0, "content": ",", "type": "text"}], "index": 36}, {"bbox": [71, 675, 541, 689], "spans": [{"bbox": [71, 675, 541, 689], "score": 1.0, "content": "we see that an isomorphism sends simple-currents to simple-currents of equal order. We", "type": "text"}], "index": 37}, {"bbox": [70, 690, 90, 703], "spans": [{"bbox": [70, 690, 90, 703], "score": 1.0, "content": "get", "type": "text"}], "index": 38}], "index": 34.5}, {"type": "interline_equation", "bbox": [254, 702, 358, 716], "lines": [{"bbox": [254, 702, 358, 716], "spans": [{"bbox": [254, 702, 358, 716], "score": 0.93, "content": "\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.", "type": "interline_equation"}], "index": 39}], "index": 39}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [198, 496, 413, 513], "lines": [{"bbox": [198, 496, 413, 513], "spans": [{"bbox": [198, 496, 413, 513], "score": 0.9, "content": "N_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "interline_equation", "bbox": [225, 556, 387, 570], "lines": [{"bbox": [225, 556, 387, 570], "spans": [{"bbox": [225, 556, 387, 570], "score": 0.9, "content": "S_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [254, 702, 358, 716], "lines": [{"bbox": [254, 702, 358, 716], "spans": [{"bbox": [254, 702, 358, 716], "score": 0.93, "content": "\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.", "type": "interline_equation"}], "index": 39}], "index": 39}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 543, 114], "lines": [], "index": 1, "bbox_fs": [70, 73, 541, 118], "lines_deleted": true}, {"type": "text", "bbox": [71, 115, 545, 200], "lines": [{"bbox": [94, 116, 548, 133], "spans": [{"bbox": [94, 116, 548, 133], "score": 1.0, "content": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions", "type": "text"}], "index": 3}, {"bbox": [70, 131, 540, 146], "spans": [{"bbox": [70, 131, 523, 146], "score": 1.0, "content": "and norms. Q-dimensions vary by at most a sign while norms are constant mod ", "type": "text"}, {"bbox": [523, 133, 536, 142], "score": 0.81, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [537, 131, 540, 146], "score": 1.0, "content": ":", "type": "text"}], "index": 4}, {"bbox": [71, 144, 541, 160], "spans": [{"bbox": [71, 146, 195, 159], "score": 0.92, "content": "{\\cal D}(w.\\lambda)\\;=\\;\\operatorname*{det}\\left(w\\right){\\cal D}(\\lambda)", "type": "inline_equation", "height": 13, "width": 124}, {"bbox": [196, 144, 225, 160], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 146, 317, 159], "score": 0.92, "content": "(w\\lambda|w\\lambda)\\,\\equiv\\,(\\lambda|\\lambda)", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [318, 144, 356, 160], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [356, 147, 369, 156], "score": 0.67, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [370, 144, 541, 160], "score": 1.0, "content": "). The point is that for excep-", "type": "text"}], "index": 5}, {"bbox": [71, 160, 540, 174], "spans": [{"bbox": [71, 160, 540, 174], "score": 1.0, "content": "tional algebras at small levels, the highest weights can often be distinguished by the pair", "type": "text"}], "index": 6}, {"bbox": [71, 173, 541, 190], "spans": [{"bbox": [71, 175, 172, 188], "score": 0.85, "content": "(\\mathcal{D}(\\lambda),(\\lambda+\\rho|\\lambda+\\rho)", "type": "inline_equation", "height": 13, "width": 101}, {"bbox": [172, 173, 207, 190], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [207, 176, 220, 185], "score": 0.65, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 173, 377, 190], "score": 1.0, "content": ")). For example this is true of ", "type": "text"}, {"bbox": [378, 176, 453, 188], "score": 0.93, "content": "E_{8,5},E_{8,6},F_{4,4}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [453, 173, 541, 190], "score": 1.0, "content": ". This is a useful", "type": "text"}], "index": 7}, {"bbox": [70, 189, 393, 203], "spans": [{"bbox": [70, 189, 393, 203], "score": 1.0, "content": "way in practise to use both (2.4) and the Galois action (2.3).", "type": "text"}], "index": 8}], "index": 5.5, "bbox_fs": [70, 116, 548, 203]}, {"type": "text", "bbox": [70, 201, 542, 365], "lines": [{"bbox": [94, 202, 541, 218], "spans": [{"bbox": [94, 202, 338, 218], "score": 1.0, "content": "An important property obeyed by the matrix ", "type": "text"}, {"bbox": [338, 205, 347, 214], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 202, 480, 218], "score": 1.0, "content": " for any classical algebra ", "type": "text"}, {"bbox": [480, 205, 496, 215], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [496, 202, 541, 218], "score": 1.0, "content": " is rank-", "type": "text"}], "index": 9}, {"bbox": [70, 216, 541, 233], "spans": [{"bbox": [70, 216, 541, 233], "score": 1.0, "content": "level duality. The first appearance of this curious duality seems to be by Frenkel [9],", "type": "text"}], "index": 10}, {"bbox": [70, 232, 542, 246], "spans": [{"bbox": [70, 232, 542, 246], "score": 1.0, "content": "but by now many aspects and generalisations have been explored in the literature. For", "type": "text"}], "index": 11}, {"bbox": [71, 244, 542, 265], "spans": [{"bbox": [71, 246, 93, 261], "score": 0.92, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 22}, {"bbox": [93, 244, 489, 265], "score": 1.0, "content": ", it is related to the existence of mutually commutative affine subalgbras", "type": "text"}, {"bbox": [490, 245, 515, 262], "score": 0.91, "content": "\\widehat{\\mathrm{sl}(n)}", "type": "inline_equation", "height": 17, "width": 25}, {"bbox": [515, 244, 542, 265], "score": 1.0, "content": " and", "type": "text"}], "index": 12}, {"bbox": [71, 263, 541, 281], "spans": [{"bbox": [71, 263, 95, 280], "score": 0.9, "content": "\\widehat{\\mathrm{sl}}(\\widehat{k})", "type": "inline_equation", "height": 17, "width": 24}, {"bbox": [96, 265, 114, 281], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [114, 263, 147, 279], "score": 0.91, "content": "\\widehat{\\mathrm{gl}}(n\\widehat{k})", "type": "inline_equation", "height": 16, "width": 33}, {"bbox": [147, 265, 541, 281], "score": 1.0, "content": ". Witten has another interpretation of it [37]: he found a natural map (a", "type": "text"}], "index": 13}, {"bbox": [71, 281, 541, 295], "spans": [{"bbox": [71, 281, 460, 295], "score": 1.0, "content": "ring homomorphism) from the quantum cohomology of the Grassmannian ", "type": "text"}, {"bbox": [460, 281, 502, 294], "score": 0.93, "content": "G(k,N)", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [502, 281, 541, 295], "score": 1.0, "content": ", to the", "type": "text"}], "index": 14}, {"bbox": [70, 294, 541, 309], "spans": [{"bbox": [70, 294, 204, 309], "score": 1.0, "content": "fusion ring of the algebra ", "type": "text"}, {"bbox": [205, 296, 306, 308], "score": 0.92, "content": "\\mathrm{u}(k)\\cong\\mathrm{su}(k)\\oplus\\mathrm{u}(1)", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [306, 294, 350, 309], "score": 1.0, "content": " at level ", "type": "text"}, {"bbox": [351, 296, 407, 308], "score": 0.93, "content": "(N-k,N)", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [407, 294, 541, 309], "score": 1.0, "content": ". Witten used the duality", "type": "text"}], "index": 15}, {"bbox": [70, 309, 540, 323], "spans": [{"bbox": [70, 309, 117, 323], "score": 1.0, "content": "between ", "type": "text"}, {"bbox": [118, 310, 159, 322], "score": 0.94, "content": "G(k,N)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [160, 309, 186, 323], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [186, 310, 254, 322], "score": 0.94, "content": "G(N-k,N)", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [254, 309, 426, 323], "score": 1.0, "content": " to show that the fusion rings of", "type": "text"}, {"bbox": [427, 309, 450, 322], "score": 0.87, "content": "\\operatorname{u}(k)", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [451, 309, 481, 323], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [482, 309, 540, 322], "score": 0.92, "content": "(N-k,N)", "type": "inline_equation", "height": 13, "width": 58}], "index": 16}, {"bbox": [70, 322, 541, 338], "spans": [{"bbox": [70, 322, 94, 338], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [94, 324, 142, 336], "score": 0.94, "content": "\\mathrm{u}(N-k)", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [143, 322, 173, 338], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [173, 324, 205, 337], "score": 0.94, "content": "(k,N)", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [206, 322, 541, 338], "score": 1.0, "content": " should coincide. A considerable generalisation, applying to any", "type": "text"}], "index": 17}, {"bbox": [70, 337, 542, 353], "spans": [{"bbox": [70, 337, 542, 353], "score": 1.0, "content": "VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution", "type": "text"}], "index": 18}, {"bbox": [71, 351, 388, 368], "spans": [{"bbox": [71, 353, 179, 366], "score": 0.93, "content": "\\textstyle\\sum_{i}[x_{i}]\\leftrightarrow\\sum_{i}[1-x_{i}]", "type": "inline_equation", "height": 13, "width": 108}, {"bbox": [179, 351, 388, 368], "score": 1.0, "content": " of torsion elements of the Bloch group.", "type": "text"}], "index": 19}], "index": 14, "bbox_fs": [70, 202, 542, 368]}, {"type": "text", "bbox": [71, 366, 541, 414], "lines": [{"bbox": [94, 367, 542, 385], "spans": [{"bbox": [94, 369, 253, 385], "score": 1.0, "content": "The Kac-Peterson matrices of", "type": "text"}, {"bbox": [253, 367, 276, 383], "score": 0.92, "content": "\\widehat{\\mathrm{sl}(\\ell)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [276, 369, 306, 385], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [306, 372, 313, 380], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [313, 369, 339, 385], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 367, 363, 384], "score": 0.92, "content": "\\widehat{\\mathrm{sl}(k)}", "type": "inline_equation", "height": 17, "width": 24}, {"bbox": [364, 369, 393, 385], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [393, 371, 399, 380], "score": 0.77, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [399, 369, 542, 385], "score": 1.0, "content": "are related, as are those of", "type": "text"}], "index": 20}, {"bbox": [71, 384, 541, 404], "spans": [{"bbox": [71, 389, 93, 402], "score": 0.92, "content": "C_{r,k}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [93, 387, 120, 404], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [120, 389, 143, 402], "score": 0.92, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [143, 387, 173, 404], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [174, 385, 199, 401], "score": 0.89, "content": "\\widehat{\\mathrm{so}(\\ell)}", "type": "inline_equation", "height": 16, "width": 25}, {"bbox": [200, 387, 231, 404], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [231, 389, 238, 398], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [238, 387, 265, 404], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [266, 384, 293, 401], "score": 0.86, "content": "\\widehat{\\mathrm{so}(k)}", "type": "inline_equation", "height": 17, "width": 27}, {"bbox": [293, 387, 324, 404], "score": 1.0, "content": " level ", "type": "text"}, {"bbox": [324, 389, 330, 398], "score": 0.86, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [330, 387, 541, 404], "score": 1.0, "content": ". We will need only the symplectic one;", "type": "text"}], "index": 21}, {"bbox": [70, 402, 239, 416], "spans": [{"bbox": [70, 402, 239, 416], "score": 1.0, "content": "the details will be given in §3.3.", "type": "text"}], "index": 22}], "index": 21, "bbox_fs": [70, 367, 542, 416]}, {"type": "title", "bbox": [71, 426, 270, 441], "lines": [{"bbox": [72, 429, 269, 442], "spans": [{"bbox": [72, 429, 269, 442], "score": 1.0, "content": "2.2. Symmetries of fusion coefficients", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [71, 446, 541, 493], "lines": [{"bbox": [93, 447, 540, 465], "spans": [{"bbox": [93, 447, 423, 465], "score": 1.0, "content": "Definition 2.1. By an isomorphism between fusion rings ", "type": "text"}, {"bbox": [423, 449, 466, 463], "score": 0.92, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [466, 447, 496, 465], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [496, 449, 540, 462], "score": 0.9, "content": "\\mathcal{R}(Y_{s,m})", "type": "inline_equation", "height": 13, "width": 44}], "index": 24}, {"bbox": [69, 463, 541, 481], "spans": [{"bbox": [69, 464, 203, 481], "score": 1.0, "content": "(with fusion coefficients ", "type": "text"}, {"bbox": [204, 466, 216, 477], "score": 0.79, "content": "N", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [216, 464, 245, 481], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [245, 465, 259, 477], "score": 0.64, "content": "M", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [260, 464, 448, 481], "score": 1.0, "content": " respectively) we mean a bijection ", "type": "text"}, {"bbox": [448, 463, 541, 480], "score": 0.92, "content": "\\pi\\ :\\ P_{+}^{k}(X_{r}^{(1)})\\ \\to", "type": "inline_equation", "height": 17, "width": 93}], "index": 25}, {"bbox": [71, 478, 177, 498], "spans": [{"bbox": [71, 480, 122, 498], "score": 0.94, "content": "{\\cal P}_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 18, "width": 51}, {"bbox": [122, 478, 177, 498], "score": 1.0, "content": " such that", "type": "text"}], "index": 26}], "index": 25, "bbox_fs": [69, 447, 541, 498]}, {"type": "interline_equation", "bbox": [198, 496, 413, 513], "lines": [{"bbox": [198, 496, 413, 513], "spans": [{"bbox": [198, 496, 413, 513], "score": 0.9, "content": "N_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [71, 516, 542, 545], "lines": [{"bbox": [71, 516, 544, 535], "spans": [{"bbox": [71, 516, 107, 535], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [107, 518, 176, 532], "score": 0.91, "content": "X_{r,k}~=~Y_{s,m}", "type": "inline_equation", "height": 14, "width": 69}, {"bbox": [176, 516, 224, 535], "score": 1.0, "content": " we call ", "type": "text"}, {"bbox": [225, 523, 232, 529], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [233, 516, 544, 535], "score": 1.0, "content": " an automorphism or fusion-symmetry. Call the pair of", "type": "text"}], "index": 28}, {"bbox": [71, 533, 267, 547], "spans": [{"bbox": [71, 533, 142, 547], "score": 1.0, "content": "permutations ", "type": "text"}, {"bbox": [143, 533, 167, 546], "score": 0.87, "content": "\\pi,\\pi^{\\prime}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [167, 533, 187, 547], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [187, 534, 196, 543], "score": 0.84, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [196, 533, 267, 547], "score": 1.0, "content": "-symmetry if", "type": "text"}], "index": 29}], "index": 28.5, "bbox_fs": [71, 516, 544, 547]}, {"type": "interline_equation", "bbox": [225, 556, 387, 570], "lines": [{"bbox": [225, 556, 387, 570], "spans": [{"bbox": [225, 556, 387, 570], "score": 0.9, "content": "S_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "text", "bbox": [70, 577, 541, 700], "lines": [{"bbox": [94, 578, 541, 595], "spans": [{"bbox": [94, 578, 317, 595], "score": 1.0, "content": "The lemma below tells us that fusion- and ", "type": "text"}, {"bbox": [317, 582, 325, 591], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [326, 578, 541, 595], "score": 1.0, "content": "-symmetries form two isomorphic groups;", "type": "text"}], "index": 31}, {"bbox": [71, 594, 541, 610], "spans": [{"bbox": [71, 594, 196, 610], "score": 1.0, "content": "the former we will label ", "type": "text"}, {"bbox": [196, 595, 238, 608], "score": 0.94, "content": "\\boldsymbol{A}(\\boldsymbol{X}_{r,k})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [239, 594, 504, 610], "score": 1.0, "content": ". Equation (2.5a) says that the charge-conjugation ", "type": "text"}, {"bbox": [504, 596, 514, 605], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [514, 594, 541, 610], "score": 1.0, "content": ", and", "type": "text"}], "index": 32}, {"bbox": [70, 609, 541, 623], "spans": [{"bbox": [70, 609, 459, 623], "score": 1.0, "content": "more generally any conjugation, is a fusion-symmetry, while (2.2a) says ", "type": "text"}, {"bbox": [459, 610, 492, 622], "score": 0.94, "content": "(C,C)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [492, 609, 527, 623], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [527, 610, 536, 619], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [536, 609, 541, 623], "score": 1.0, "content": "-", "type": "text"}], "index": 33}, {"bbox": [67, 622, 542, 644], "spans": [{"bbox": [67, 622, 177, 644], "score": 1.0, "content": "symmetry. Because ", "type": "text"}, {"bbox": [177, 626, 248, 639], "score": 0.92, "content": "N_{0}=I=M_{\\tilde{0}}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [248, 622, 254, 644], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [255, 625, 313, 640], "score": 0.94, "content": "N_{\\lambda\\mu}^{0}=C_{\\lambda\\mu}", "type": "inline_equation", "height": 15, "width": 58}, {"bbox": [314, 622, 341, 644], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [341, 623, 407, 642], "score": 0.94, "content": "M_{\\tilde{\\lambda},\\tilde{\\mu}}^{\\tilde{0}}=\\widetilde{C}_{\\tilde{\\lambda},\\tilde{\\mu}}", "type": "inline_equation", "height": 19, "width": 66}, {"bbox": [408, 622, 542, 644], "score": 1.0, "content": " (we use tilde’s to denote", "type": "text"}], "index": 34}, {"bbox": [69, 641, 542, 662], "spans": [{"bbox": [69, 641, 139, 662], "score": 1.0, "content": "quantities in ", "type": "text"}, {"bbox": [140, 643, 162, 657], "score": 0.92, "content": "Y_{s}^{(1)}", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [162, 641, 192, 662], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [192, 650, 203, 656], "score": 0.85, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [204, 641, 306, 662], "score": 1.0, "content": "), any isomorphism ", "type": "text"}, {"bbox": [306, 650, 313, 656], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [314, 641, 374, 662], "score": 1.0, "content": " must obey ", "type": "text"}, {"bbox": [374, 645, 410, 656], "score": 0.92, "content": "\\pi0=\\tilde{0}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [410, 641, 435, 662], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [435, 644, 504, 656], "score": 0.92, "content": "\\widetilde{C}\\circ\\pi=\\pi\\circ C", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [505, 641, 542, 662], "score": 1.0, "content": ". More", "type": "text"}], "index": 35}, {"bbox": [70, 659, 541, 675], "spans": [{"bbox": [70, 659, 153, 675], "score": 1.0, "content": "generally, since ", "type": "text"}, {"bbox": [154, 661, 169, 672], "score": 0.93, "content": "N_{\\lambda}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [169, 659, 347, 675], "score": 1.0, "content": " is a permutation matrix of order ", "type": "text"}, {"bbox": [347, 664, 354, 670], "score": 0.87, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 659, 372, 675], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [372, 661, 380, 670], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [380, 659, 528, 675], "score": 1.0, "content": " is a simpl e- current of order ", "type": "text"}, {"bbox": [529, 664, 536, 670], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [537, 659, 541, 675], "score": 1.0, "content": ",", "type": "text"}], "index": 36}, {"bbox": [71, 675, 541, 689], "spans": [{"bbox": [71, 675, 541, 689], "score": 1.0, "content": "we see that an isomorphism sends simple-currents to simple-currents of equal order. We", "type": "text"}], "index": 37}, {"bbox": [70, 690, 90, 703], "spans": [{"bbox": [70, 690, 90, 703], "score": 1.0, "content": "get", "type": "text"}], "index": 38}], "index": 34.5, "bbox_fs": [67, 578, 542, 703]}, {"type": "interline_equation", "bbox": [254, 702, 358, 716], "lines": [{"bbox": [254, 702, 358, 716], "spans": [{"bbox": [254, 702, 358, 716], "score": 0.93, "content": "\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~.", "type": "interline_equation"}], "index": 39}], "index": 39}]}
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[{"type": "text", "bbox": [70, 70, 543, 114], "content": "", "index": 0}, {"type": "text", "bbox": [71, 115, 545, 200], "content": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions and norms. Q-dimensions vary by at most a sign while norms are constant mod : and (mod ). The point is that for excep- tional algebras at small levels, the highest weights can often be distinguished by the pair (mod )). For example this is true of . This is a useful way in practise to use both (2.4) and the Galois action (2.3).", "index": 1}, {"type": "text", "bbox": [70, 201, 542, 365], "content": "An important property obeyed by the matrix for any classical algebra is rank- level duality. The first appearance of this curious duality seems to be by Frenkel [9], but by now many aspects and generalisations have been explored in the literature. For , it is related to the existence of mutually commutative affine subalgbras and in . Witten has another interpretation of it [37]: he found a natural map (a ring homomorphism) from the quantum cohomology of the Grassmannian , to the fusion ring of the algebra at level . Witten used the duality between and to show that the fusion rings of level and level should coincide. A considerable generalisation, applying to any VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution of torsion elements of the Bloch group.", "index": 2}, {"type": "text", "bbox": [71, 366, 541, 414], "content": "The Kac-Peterson matrices of level and level are related, as are those of and , and level and level . We will need only the symplectic one; the details will be given in §3.3.", "index": 3}, {"type": "title", "bbox": [71, 426, 270, 441], "content": "2.2. Symmetries of fusion coefficients", "index": 4}, {"type": "text", "bbox": [71, 446, 541, 493], "content": "Definition 2.1. By an isomorphism between fusion rings and (with fusion coefficients and respectively) we mean a bijection such that", "index": 5}, {"type": "interline_equation", "bbox": [198, 496, 413, 513], "content": "", "index": 6}, {"type": "text", "bbox": [71, 516, 542, 545], "content": "When we call an automorphism or fusion-symmetry. Call the pair of permutations an -symmetry if", "index": 7}, {"type": "interline_equation", "bbox": [225, 556, 387, 570], "content": "", "index": 8}, {"type": "text", "bbox": [70, 577, 541, 700], "content": "The lemma below tells us that fusion- and -symmetries form two isomorphic groups; the former we will label . Equation (2.5a) says that the charge-conjugation , and more generally any conjugation, is a fusion-symmetry, while (2.2a) says is an - symmetry. Because , and (we use tilde’s to denote quantities in level ), any isomorphism must obey and . More generally, since is a permutation matrix of order iff is a simpl e- current of order , we see that an isomorphism sends simple-currents to simple-currents of equal order. We get", "index": 9}, {"type": "interline_equation", "bbox": [254, 702, 358, 716], "content": "", "index": 10}]
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[{"bbox": [94, 116, 548, 133], "content": "A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions", "parent_index": 1, "line_index": 0}, {"bbox": [70, 131, 540, 146], "content": "and norms. Q-dimensions vary by at most a sign while norms are constant mod :", "parent_index": 1, "line_index": 1}, {"bbox": [71, 144, 541, 160], "content": "and (mod ). The point is that for excep-", "parent_index": 1, "line_index": 2}, {"bbox": [71, 160, 540, 174], "content": "tional algebras at small levels, the highest weights can often be distinguished by the pair", "parent_index": 1, "line_index": 3}, {"bbox": [71, 173, 541, 190], "content": "(mod )). For example this is true of . This is a useful", "parent_index": 1, "line_index": 4}, {"bbox": [70, 189, 393, 203], "content": "way in practise to use both (2.4) and the Galois action (2.3).", "parent_index": 1, "line_index": 5}, {"bbox": [94, 202, 541, 218], "content": "An important property obeyed by the matrix for any classical algebra is rank-", "parent_index": 2, "line_index": 0}, {"bbox": [70, 216, 541, 233], "content": "level duality. The first appearance of this curious duality seems to be by Frenkel [9],", "parent_index": 2, "line_index": 1}, {"bbox": [70, 232, 542, 246], "content": "but by now many aspects and generalisations have been explored in the literature. For", "parent_index": 2, "line_index": 2}, {"bbox": [71, 244, 542, 265], "content": ", it is related to the existence of mutually commutative affine subalgbras and", "parent_index": 2, "line_index": 3}, {"bbox": [71, 263, 541, 281], "content": "in . Witten has another interpretation of it [37]: he found a natural map (a", "parent_index": 2, "line_index": 4}, {"bbox": [71, 281, 541, 295], "content": "ring homomorphism) from the quantum cohomology of the Grassmannian , to the", "parent_index": 2, "line_index": 5}, {"bbox": [70, 294, 541, 309], "content": "fusion ring of the algebra at level . Witten used the duality", "parent_index": 2, "line_index": 6}, {"bbox": [70, 309, 540, 323], "content": "between and to show that the fusion rings of level", "parent_index": 2, "line_index": 7}, {"bbox": [70, 322, 541, 338], "content": "and level should coincide. A considerable generalisation, applying to any", "parent_index": 2, "line_index": 8}, {"bbox": [70, 337, 542, 353], "content": "VOA (or RCFT), has been conjectured by Nahm [30], and relates to the natural involution", "parent_index": 2, "line_index": 9}, {"bbox": [71, 351, 388, 368], "content": "of torsion elements of the Bloch group.", "parent_index": 2, "line_index": 10}, {"bbox": [94, 367, 542, 385], "content": "The Kac-Peterson matrices of level and level are related, as are those of", "parent_index": 3, "line_index": 0}, {"bbox": [71, 384, 541, 404], "content": "and , and level and level . We will need only the symplectic one;", "parent_index": 3, "line_index": 1}, {"bbox": [70, 402, 239, 416], "content": "the details will be given in §3.3.", "parent_index": 3, "line_index": 2}, {"bbox": [72, 429, 269, 442], "content": "2.2. Symmetries of fusion coefficients", "parent_index": 4, "line_index": 0}, {"bbox": [93, 447, 540, 465], "content": "Definition 2.1. By an isomorphism between fusion rings and", "parent_index": 5, "line_index": 0}, {"bbox": [69, 463, 541, 481], "content": "(with fusion coefficients and respectively) we mean a bijection", "parent_index": 5, "line_index": 1}, {"bbox": [71, 478, 177, 498], "content": "such that", "parent_index": 5, "line_index": 2}, {"bbox": [71, 516, 544, 535], "content": "When we call an automorphism or fusion-symmetry. Call the pair of", "parent_index": 7, "line_index": 0}, {"bbox": [71, 533, 267, 547], "content": "permutations an -symmetry if", "parent_index": 7, "line_index": 1}, {"bbox": [94, 578, 541, 595], "content": "The lemma below tells us that fusion- and -symmetries form two isomorphic groups;", "parent_index": 9, "line_index": 0}, {"bbox": [71, 594, 541, 610], "content": "the former we will label . Equation (2.5a) says that the charge-conjugation , and", "parent_index": 9, "line_index": 1}, {"bbox": [70, 609, 541, 623], "content": "more generally any conjugation, is a fusion-symmetry, while (2.2a) says is an -", "parent_index": 9, "line_index": 2}, {"bbox": [67, 622, 542, 644], "content": "symmetry. Because , and (we use tilde’s to denote", "parent_index": 9, "line_index": 3}, {"bbox": [69, 641, 542, 662], "content": "quantities in level ), any isomorphism must obey and . More", "parent_index": 9, "line_index": 4}, {"bbox": [70, 659, 541, 675], "content": "generally, since is a permutation matrix of order iff is a simpl e- current of order ,", "parent_index": 9, "line_index": 5}, {"bbox": [71, 675, 541, 689], "content": "we see that an isomorphism sends simple-currents to simple-currents of equal order. We", "parent_index": 9, "line_index": 6}, {"bbox": [70, 690, 90, 703], "content": "get", "parent_index": 9, "line_index": 7}]
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For instance $\pi$ must send $J$ -fixed-points to $\pi(J)$ -fixed-points.
More generally, a fusion-homomorphism $\pi$ is defined in the obvious algebraic way. It turns out that for such a $\pi$ , $\pi\lambda\,=\,\pi\mu$ iff $\mu\,=\,J\lambda$ for some simple-current $J$ for which $\pi(J0)=\tilde{0}$ . Moreover, $\pi(J0)=\tilde{0}$ is possible only if there are no $J$ -fixed-points. When $\pi$ is one-to-one (e.g. when there are no nontrivial simple-currents in $P_{+}^{k}(X_{r}^{(1)}))$ , then $\pi$ obeys (2.6). Fusion-homomorphisms will be studied elsewhere.
The key to finding fusion-symmetries is the following Lemma.
Lemma 2.2. Let $\widetilde{S}$ be the Kac-Peterson matrix for $Y_{s}^{(1)}$ level $m$ . Then a bijection $\pi\,:\,P_{+}^{k}(X_{r}^{(1)})\,\to\,P_{+}^{m}(Y_{s}^{(1)})$ defines an isomorphism of fusion rings iff there exists some bijection $\pi^{\prime}:P_{+}^{k}(X_{r}^{(1)})\to P_{+}^{m}(Y_{s}^{(1)})$ such that $S_{\lambda\mu}=\widetilde{S}_{\pi\lambda,\pi^{\prime}\mu}$ for all $\lambda,\mu\in P_{+}^{k}(X_{r}^{(1)})$ . In particular, a permutation $\pi$ is a fusion-symmetry iff $(\pi,\pi^{\prime})$ is an $S$ -symmetry for some $\pi^{\prime}$ .
Proof. The equality $N_{\lambda\mu}^{\nu}=M_{\pi\lambda,\pi\mu}^{\pi\nu}$ means that, for each $\mu$ , the column vectors $(\underline{{x}}_{\mu})_{\nu}=$ $\widetilde{S}_{\pi\nu,\pi\mu}$ are simultaneous eigenvectors for the fusion matrices $N_{\lambda}$ , with eigenvalues $\widetilde{S}_{\pi\lambda,\pi\mu}/\widetilde{S}_{0,\pi\mu}$ . I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll fusion matrices must be a scalar multiple of some column of $S$ . Thus there must be a permutation $\pi^{\prime\prime}$ of $P_{+}^{k}(X_{r}^{(1)})$ and scalars $\alpha(\mu)$ such that $\widetilde{S}_{\pi\nu,\pi\mu}\,=\,\alpha(\mu)\,S_{\nu,\pi^{\prime\prime}\mu}$ . Taking $\nu=0$ forces $\alpha(\mu)>0$ , and then unitarity forces $\alpha(\mu)=1$ . ■
Let $\pi$ be any isomorphism, and let $\pi^{\prime}$ be as in the Lemma. Then $\pi^{\prime}$ is also an isomorphism, with $(\pi^{\prime})^{\prime}\;=\;\pi$ . Equation (2.2b) implies for all $\lambda\:\in\:P_{+}$ and all simplecurrents $j$ , that
$$
Q_{j}(\lambda)\equiv\widetilde{Q}_{\pi^{\prime}j}(\pi\lambda)\equiv\widetilde{Q}_{\pi j}(\pi^{\prime}\lambda)\qquad(\mathrm{mod~1})~.
$$
Another quick consequence of the Lemma is that for any Galois automorphism $\sigma_{\ell}$ and isomorphism $\pi$ , we have $\tilde{\epsilon}_{\ell}(\pi\lambda)=\epsilon_{\ell}(\lambda)$ and $\pi(\lambda^{(\ell)})=(\pi\lambda)^{(\ell)}$ . To see this, apply the invertibility of $S$ to the equation
$$
\left\varepsilon(\lambda\right)S_{\lambda^{(\ell)},\mu}=\sigma_{\ell}S_{\lambda\mu}=\sigma_{\ell}\widetilde{S}_{\pi\lambda,\pi^{\prime}\mu}=\widetilde{\epsilon}_{\ell}(\pi\lambda)\,\widetilde{S}_{(\pi\lambda)^{(\ell)},\pi^{\prime}\mu}=\widetilde{\epsilon}_{\ell}(\pi\lambda)\,S_{\pi^{-1}(\pi\lambda)^{(\ell)},\mu}\ .
$$
A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a subset $\Gamma=\{\gamma_{1},...,\gamma_{m}\}$ of $P_{+}$ which generates $\mathcal{R}(X_{r,k})$ as a ring. Diagonalising, this is equivalent to requiring that there are $m$ -variable polynomials $P_{\lambda}(x_{1},\ldots,x_{m})$ such that
$$
\frac{S_{\lambda\mu}}{S_{0\mu}}=P_{\lambda}(\frac{S_{\gamma_{1}\mu}}{S_{0\mu}},\ldots,\frac{S_{\gamma_{m}\mu}}{S_{0\mu}})\qquad\forall\lambda,\mu\in{\cal P}_{+}\ .
$$
Let $(\pi,\pi^{\prime})$ be an $S$ -symmetry, and suppose we know that $\pi\gamma=\gamma$ for all $\gamma$ in the fusiongenerator $\Gamma$ . Then for any $\lambda\in P_{+}$ ,
$$
{\frac{S_{\lambda\mu}}{S_{0\mu}}}={\frac{S_{\pi\lambda,\pi^{\prime}\mu}}{S_{0,\pi^{\prime}\mu}}}=P_{\pi\lambda}({\frac{S_{\gamma_{1},\pi^{\prime}\mu}}{S_{0,\pi^{\prime}\mu}}},\dots)=P_{\pi\lambda}({\frac{S_{\gamma_{1}\mu}}{S_{0\mu}}},\dots)={\frac{S_{\pi\lambda,\mu}}{S_{0\mu}}}
$$
for all $\mu\in P_{+}$ , so $\pi\lambda=\lambda$ .
One of the reasons fusion-symmetries for the affine algebras are so tractible is the existence of small fusion-generators. In particular, because we know that any Lie character $\mathrm{ch}_{\overline{{{\mu}}}}$ for $X_{r}$ can be written as a polynomial in the fundamental characters $\mathrm{ch}_{\Lambda_{1}},\ldots,\mathrm{ch}_{\Lambda_{r}}$ , we know from (2.1b) that $\Gamma=\{\Lambda_{1},\dots,\Lambda_{r}\}$ is a fusion-generator for $X_{r}^{(1)}$ at any level $k$ sufficiently large that $P_{+}$ contains all $\Lambda_{i}$ (in other words, for any $k\geq\operatorname*{max}_{i}a_{i}^{\vee}$ ). In fact, it is easy to show [18] that a fusion-generator valid for any $X_{r,k}$ is $\{\Lambda_{1},\ldots,\Lambda_{r}\}\cap{\cal P}_{+}$ . Smaller fusion-generators usually exist — for example $\{\Lambda_{1}\}$ is a fusion-generator for $A_{8,k}$ whenever $k$ is even and coprime to 3.
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<html><body>
<p data-bbox="70 70 392 85">For instance $\pi$ must send $J$ -fixed-points to $\pi(J)$ -fixed-points. </p>
<p data-bbox="70 86 542 160">More generally, a fusion-homomorphism $\pi$ is defined in the obvious algebraic way. It turns out that for such a $\pi$ , $\pi\lambda\,=\,\pi\mu$ iff $\mu\,=\,J\lambda$ for some simple-current $J$ for which $\pi(J0)=\tilde{0}$ . Moreover, $\pi(J0)=\tilde{0}$ is possible only if there are no $J$ -fixed-points. When $\pi$ is one-to-one (e.g. when there are no nontrivial simple-currents in $P_{+}^{k}(X_{r}^{(1)}))$ , then $\pi$ obeys (2.6). Fusion-homomorphisms will be studied elsewhere. </p>
<p data-bbox="95 160 420 174">The key to finding fusion-symmetries is the following Lemma. </p>
<p data-bbox="70 180 543 246">Lemma 2.2. Let $\widetilde{S}$ be the Kac-Peterson matrix for $Y_{s}^{(1)}$ level $m$ . Then a bijection $\pi\,:\,P_{+}^{k}(X_{r}^{(1)})\,\to\,P_{+}^{m}(Y_{s}^{(1)})$ defines an isomorphism of fusion rings iff there exists some bijection $\pi^{\prime}:P_{+}^{k}(X_{r}^{(1)})\to P_{+}^{m}(Y_{s}^{(1)})$ such that $S_{\lambda\mu}=\widetilde{S}_{\pi\lambda,\pi^{\prime}\mu}$ for all $\lambda,\mu\in P_{+}^{k}(X_{r}^{(1)})$ . In particular, a permutation $\pi$ is a fusion-symmetry iff $(\pi,\pi^{\prime})$ is an $S$ -symmetry for some $\pi^{\prime}$ . </p>
<p data-bbox="70 250 558 344">Proof. The equality $N_{\lambda\mu}^{\nu}=M_{\pi\lambda,\pi\mu}^{\pi\nu}$ means that, for each $\mu$ , the column vectors $(\underline{{x}}_{\mu})_{\nu}=$ $\widetilde{S}_{\pi\nu,\pi\mu}$ are simultaneous eigenvectors for the fusion matrices $N_{\lambda}$ , with eigenvalues $\widetilde{S}_{\pi\lambda,\pi\mu}/\widetilde{S}_{0,\pi\mu}$ . I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll fusion matrices must be a scalar multiple of some column of $S$ . Thus there must be a permutation $\pi^{\prime\prime}$ of $P_{+}^{k}(X_{r}^{(1)})$ and scalars $\alpha(\mu)$ such that $\widetilde{S}_{\pi\nu,\pi\mu}\,=\,\alpha(\mu)\,S_{\nu,\pi^{\prime\prime}\mu}$ . Taking $\nu=0$ forces $\alpha(\mu)>0$ , and then unitarity forces $\alpha(\mu)=1$ . ■ </p>
<p data-bbox="70 349 541 391">Let $\pi$ be any isomorphism, and let $\pi^{\prime}$ be as in the Lemma. Then $\pi^{\prime}$ is also an isomorphism, with $(\pi^{\prime})^{\prime}\;=\;\pi$ . Equation (2.2b) implies for all $\lambda\:\in\:P_{+}$ and all simplecurrents $j$ , that </p>
<div class="equation" data-bbox="190 391 422 409">$$
Q_{j}(\lambda)\equiv\widetilde{Q}_{\pi^{\prime}j}(\pi\lambda)\equiv\widetilde{Q}_{\pi j}(\pi^{\prime}\lambda)\qquad(\mathrm{mod~1})~.
$$</div>
<p data-bbox="70 414 541 458">Another quick consequence of the Lemma is that for any Galois automorphism $\sigma_{\ell}$ and isomorphism $\pi$ , we have $\tilde{\epsilon}_{\ell}(\pi\lambda)=\epsilon_{\ell}(\lambda)$ and $\pi(\lambda^{(\ell)})=(\pi\lambda)^{(\ell)}$ . To see this, apply the invertibility of $S$ to the equation </p>
<div class="equation" data-bbox="111 468 507 488">$$
\left\varepsilon(\lambda\right)S_{\lambda^{(\ell)},\mu}=\sigma_{\ell}S_{\lambda\mu}=\sigma_{\ell}\widetilde{S}_{\pi\lambda,\pi^{\prime}\mu}=\widetilde{\epsilon}_{\ell}(\pi\lambda)\,\widetilde{S}_{(\pi\lambda)^{(\ell)},\pi^{\prime}\mu}=\widetilde{\epsilon}_{\ell}(\pi\lambda)\,S_{\pi^{-1}(\pi\lambda)^{(\ell)},\mu}\ .
$$</div>
<p data-bbox="69 496 542 540">A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a subset $\Gamma=\{\gamma_{1},...,\gamma_{m}\}$ of $P_{+}$ which generates $\mathcal{R}(X_{r,k})$ as a ring. Diagonalising, this is equivalent to requiring that there are $m$ -variable polynomials $P_{\lambda}(x_{1},\ldots,x_{m})$ such that </p>
<div class="equation" data-bbox="189 552 423 582">$$
\frac{S_{\lambda\mu}}{S_{0\mu}}=P_{\lambda}(\frac{S_{\gamma_{1}\mu}}{S_{0\mu}},\ldots,\frac{S_{\gamma_{m}\mu}}{S_{0\mu}})\qquad\forall\lambda,\mu\in{\cal P}_{+}\ .
$$</div>
<p data-bbox="69 592 541 621">Let $(\pi,\pi^{\prime})$ be an $S$ -symmetry, and suppose we know that $\pi\gamma=\gamma$ for all $\gamma$ in the fusiongenerator $\Gamma$ . Then for any $\lambda\in P_{+}$ , </p>
<div class="equation" data-bbox="146 632 465 663">$$
{\frac{S_{\lambda\mu}}{S_{0\mu}}}={\frac{S_{\pi\lambda,\pi^{\prime}\mu}}{S_{0,\pi^{\prime}\mu}}}=P_{\pi\lambda}({\frac{S_{\gamma_{1},\pi^{\prime}\mu}}{S_{0,\pi^{\prime}\mu}}},\dots)=P_{\pi\lambda}({\frac{S_{\gamma_{1}\mu}}{S_{0\mu}}},\dots)={\frac{S_{\pi\lambda,\mu}}{S_{0\mu}}}
$$</div>
<p data-bbox="69 672 207 686">for all $\mu\in P_{+}$ , so $\pi\lambda=\lambda$ . </p>
<p data-bbox="70 687 541 715">One of the reasons fusion-symmetries for the affine algebras are so tractible is the existence of small fusion-generators. In particular, because we know that any Lie character $\mathrm{ch}_{\overline{{{\mu}}}}$ for $X_{r}$ can be written as a polynomial in the fundamental characters $\mathrm{ch}_{\Lambda_{1}},\ldots,\mathrm{ch}_{\Lambda_{r}}$ , we know from (2.1b) that $\Gamma=\{\Lambda_{1},\dots,\Lambda_{r}\}$ is a fusion-generator for $X_{r}^{(1)}$ at any level $k$ sufficiently large that $P_{+}$ contains all $\Lambda_{i}$ (in other words, for any $k\geq\operatorname*{max}_{i}a_{i}^{\vee}$ ). In fact, it is easy to show [18] that a fusion-generator valid for any $X_{r,k}$ is $\{\Lambda_{1},\ldots,\Lambda_{r}\}\cap{\cal P}_{+}$ . Smaller fusion-generators usually exist — for example $\{\Lambda_{1}\}$ is a fusion-generator for $A_{8,k}$ whenever $k$ is even and coprime to 3. </p>
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0002044v1
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[{"type": "text", "text": "For instance $\\pi$ must send $J$ -fixed-points to $\\pi(J)$ -fixed-points. ", "page_idx": 6}, {"type": "text", "text": "More generally, a fusion-homomorphism $\\pi$ is defined in the obvious algebraic way. It turns out that for such a $\\pi$ , $\\pi\\lambda\\,=\\,\\pi\\mu$ iff $\\mu\\,=\\,J\\lambda$ for some simple-current $J$ for which $\\pi(J0)=\\tilde{0}$ . Moreover, $\\pi(J0)=\\tilde{0}$ is possible only if there are no $J$ -fixed-points. When $\\pi$ is one-to-one (e.g. when there are no nontrivial simple-currents in $P_{+}^{k}(X_{r}^{(1)}))$ , then $\\pi$ obeys (2.6). Fusion-homomorphisms will be studied elsewhere. ", "page_idx": 6}, {"type": "text", "text": "The key to finding fusion-symmetries is the following Lemma. ", "page_idx": 6}, {"type": "text", "text": "Lemma 2.2. Let $\\widetilde{S}$ be the Kac-Peterson matrix for $Y_{s}^{(1)}$ level $m$ . Then a bijection $\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})$ defines an isomorphism of fusion rings iff there exists some bijection $\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})$ such that $S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}$ for all $\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})$ . In particular, a permutation $\\pi$ is a fusion-symmetry iff $(\\pi,\\pi^{\\prime})$ is an $S$ -symmetry for some $\\pi^{\\prime}$ . ", "page_idx": 6}, {"type": "text", "text": "Proof. The equality $N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}$ means that, for each $\\mu$ , the column vectors $(\\underline{{x}}_{\\mu})_{\\nu}=$ $\\widetilde{S}_{\\pi\\nu,\\pi\\mu}$ are simultaneous eigenvectors for the fusion matrices $N_{\\lambda}$ , with eigenvalues $\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}$ . I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll fusion matrices must be a scalar multiple of some column of $S$ . Thus there must be a permutation $\\pi^{\\prime\\prime}$ of $P_{+}^{k}(X_{r}^{(1)})$ and scalars $\\alpha(\\mu)$ such that $\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}$ . Taking $\\nu=0$ forces $\\alpha(\\mu)>0$ , and then unitarity forces $\\alpha(\\mu)=1$ . ■ ", "page_idx": 6}, {"type": "text", "text": "Let $\\pi$ be any isomorphism, and let $\\pi^{\\prime}$ be as in the Lemma. Then $\\pi^{\\prime}$ is also an isomorphism, with $(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi$ . Equation (2.2b) implies for all $\\lambda\\:\\in\\:P_{+}$ and all simplecurrents $j$ , that ", "page_idx": 6}, {"type": "equation", "text": "$$\nQ_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "Another quick consequence of the Lemma is that for any Galois automorphism $\\sigma_{\\ell}$ and isomorphism $\\pi$ , we have $\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)$ and $\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}$ . To see this, apply the invertibility of $S$ to the equation ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a subset $\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}$ of $P_{+}$ which generates $\\mathcal{R}(X_{r,k})$ as a ring. Diagonalising, this is equivalent to requiring that there are $m$ -variable polynomials $P_{\\lambda}(x_{1},\\ldots,x_{m})$ such that ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "Let $(\\pi,\\pi^{\\prime})$ be an $S$ -symmetry, and suppose we know that $\\pi\\gamma=\\gamma$ for all $\\gamma$ in the fusiongenerator $\\Gamma$ . Then for any $\\lambda\\in P_{+}$ , ", "page_idx": 6}, {"type": "equation", "text": "$$\n{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "for all $\\mu\\in P_{+}$ , so $\\pi\\lambda=\\lambda$ . ", "page_idx": 6}, {"type": "text", "text": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the existence of small fusion-generators. In particular, because we know that any Lie character $\\mathrm{ch}_{\\overline{{{\\mu}}}}$ for $X_{r}$ can be written as a polynomial in the fundamental characters $\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}$ , we know from (2.1b) that $\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}$ is a fusion-generator for $X_{r}^{(1)}$ at any level $k$ sufficiently large that $P_{+}$ contains all $\\Lambda_{i}$ (in other words, for any $k\\geq\\operatorname*{max}_{i}a_{i}^{\\vee}$ ). In fact, it is easy to show [18] that a fusion-generator valid for any $X_{r,k}$ is $\\{\\Lambda_{1},\\ldots,\\Lambda_{r}\\}\\cap{\\cal P}_{+}$ . Smaller fusion-generators usually exist — for example $\\{\\Lambda_{1}\\}$ is a fusion-generator for $A_{8,k}$ whenever $k$ is even and coprime to 3. ", "page_idx": 6}]
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{"preproc_blocks": [{"type": "text", "bbox": [70, 70, 392, 85], "lines": [{"bbox": [70, 74, 390, 87], "spans": [{"bbox": [70, 74, 139, 87], "score": 1.0, "content": "For instance ", "type": "text"}, {"bbox": [139, 79, 146, 84], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [147, 74, 207, 87], "score": 1.0, "content": " must send ", "type": "text"}, {"bbox": [208, 75, 216, 84], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [216, 74, 298, 87], "score": 1.0, "content": "-fixed-points to ", "type": "text"}, {"bbox": [298, 75, 323, 87], "score": 0.94, "content": "\\pi(J)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [324, 74, 390, 87], "score": 1.0, "content": "-fixed-points.", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [70, 86, 542, 160], "lines": [{"bbox": [95, 88, 541, 102], "spans": [{"bbox": [95, 88, 309, 102], "score": 1.0, "content": "More generally, a fusion-homomorphism ", "type": "text"}, {"bbox": [309, 93, 316, 99], "score": 0.85, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [317, 88, 541, 102], "score": 1.0, "content": " is defined in the obvious algebraic way. It", "type": "text"}], "index": 1}, {"bbox": [70, 102, 541, 117], "spans": [{"bbox": [70, 102, 212, 117], "score": 1.0, "content": "turns out that for such a ", "type": "text"}, {"bbox": [213, 108, 220, 113], "score": 0.82, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [220, 102, 228, 117], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [228, 104, 278, 115], "score": 0.91, "content": "\\pi\\lambda\\,=\\,\\pi\\mu", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [278, 102, 298, 117], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [298, 104, 341, 115], "score": 0.93, "content": "\\mu\\,=\\,J\\lambda", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [341, 102, 476, 117], "score": 1.0, "content": " for some simple-current ", "type": "text"}, {"bbox": [477, 105, 484, 113], "score": 0.9, "content": "J", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [485, 102, 541, 117], "score": 1.0, "content": " for which", "type": "text"}], "index": 2}, {"bbox": [71, 116, 542, 132], "spans": [{"bbox": [71, 117, 124, 131], "score": 0.94, "content": "\\pi(J0)=\\tilde{0}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [124, 116, 187, 132], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [187, 117, 240, 131], "score": 0.95, "content": "\\pi(J0)=\\tilde{0}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [241, 116, 404, 132], "score": 1.0, "content": " is possible only if there are no ", "type": "text"}, {"bbox": [404, 119, 412, 127], "score": 0.9, "content": "J", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [413, 116, 520, 132], "score": 1.0, "content": "-fixed-points. When ", "type": "text"}, {"bbox": [520, 122, 528, 127], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [528, 116, 542, 132], "score": 1.0, "content": " is", "type": "text"}], "index": 3}, {"bbox": [69, 131, 542, 149], "spans": [{"bbox": [69, 131, 409, 149], "score": 1.0, "content": "one-to-one (e.g. when there are no nontrivial simple-currents in ", "type": "text"}, {"bbox": [410, 131, 463, 148], "score": 0.93, "content": "P_{+}^{k}(X_{r}^{(1)}))", "type": "inline_equation", "height": 17, "width": 53}, {"bbox": [463, 131, 498, 149], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [498, 138, 506, 144], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [506, 131, 542, 149], "score": 1.0, "content": " obeys", "type": "text"}], "index": 4}, {"bbox": [72, 148, 366, 162], "spans": [{"bbox": [72, 148, 366, 162], "score": 1.0, "content": "(2.6). Fusion-homomorphisms will be studied elsewhere.", "type": "text"}], "index": 5}], "index": 3}, {"type": "text", "bbox": [95, 160, 420, 174], "lines": [{"bbox": [95, 162, 419, 176], "spans": [{"bbox": [95, 162, 419, 176], "score": 1.0, "content": "The key to finding fusion-symmetries is the following Lemma.", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [70, 180, 543, 246], "lines": [{"bbox": [91, 178, 543, 201], "spans": [{"bbox": [91, 178, 194, 201], "score": 1.0, "content": "Lemma 2.2. Let", "type": "text"}, {"bbox": [194, 183, 203, 196], "score": 0.87, "content": "\\widetilde{S}", "type": "inline_equation", "height": 13, "width": 9}, {"bbox": [203, 178, 376, 201], "score": 1.0, "content": " be the Kac-Peterson matrix for ", "type": "text"}, {"bbox": [376, 181, 400, 197], "score": 0.9, "content": "Y_{s}^{(1)}", "type": "inline_equation", "height": 16, "width": 24}, {"bbox": [400, 178, 431, 201], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [431, 186, 443, 196], "score": 0.46, "content": "m", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [443, 178, 543, 201], "score": 1.0, "content": ". Then a bijection", "type": "text"}], "index": 7}, {"bbox": [71, 195, 543, 218], "spans": [{"bbox": [71, 198, 214, 216], "score": 0.92, "content": "\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 18, "width": 143}, {"bbox": [214, 195, 543, 218], "score": 1.0, "content": " defines an isomorphism of fusion rings iff there exists some", "type": "text"}], "index": 8}, {"bbox": [68, 212, 544, 236], "spans": [{"bbox": [68, 212, 119, 236], "score": 1.0, "content": "bijection ", "type": "text"}, {"bbox": [120, 216, 261, 233], "score": 0.92, "content": "\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 17, "width": 141}, {"bbox": [261, 212, 317, 236], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [317, 216, 392, 232], "score": 0.94, "content": "S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [392, 212, 433, 236], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [433, 215, 518, 233], "score": 0.91, "content": "\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 18, "width": 85}, {"bbox": [518, 212, 544, 236], "score": 1.0, "content": ". In", "type": "text"}], "index": 9}, {"bbox": [70, 232, 541, 247], "spans": [{"bbox": [70, 232, 205, 247], "score": 1.0, "content": "particular, a permutation ", "type": "text"}, {"bbox": [206, 238, 213, 243], "score": 0.67, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 232, 344, 247], "score": 1.0, "content": " is a fusion-symmetry iff", "type": "text"}, {"bbox": [344, 233, 377, 246], "score": 0.92, "content": "(\\pi,\\pi^{\\prime})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [377, 232, 409, 247], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [409, 234, 418, 244], "score": 0.83, "content": "S", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [418, 232, 524, 247], "score": 1.0, "content": "-symmetry for some ", "type": "text"}, {"bbox": [525, 234, 536, 243], "score": 0.87, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [536, 232, 541, 247], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 8.5}, {"type": "text", "bbox": [70, 250, 558, 344], "lines": [{"bbox": [68, 250, 542, 273], "spans": [{"bbox": [68, 250, 185, 273], "score": 1.0, "content": "Proof. The equality ", "type": "text"}, {"bbox": [185, 256, 263, 270], "score": 0.93, "content": "N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [263, 250, 378, 273], "score": 1.0, "content": " means that, for each ", "type": "text"}, {"bbox": [378, 256, 387, 267], "score": 0.75, "content": "\\mu", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [387, 250, 497, 273], "score": 1.0, "content": ", the column vectors ", "type": "text"}, {"bbox": [498, 254, 542, 269], "score": 0.9, "content": "(\\underline{{x}}_{\\mu})_{\\nu}=", "type": "inline_equation", "height": 15, "width": 44}], "index": 11}, {"bbox": [71, 268, 559, 290], "spans": [{"bbox": [71, 270, 105, 286], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}", "type": "inline_equation", "height": 16, "width": 34}, {"bbox": [105, 268, 377, 290], "score": 1.0, "content": " are simultaneous eigenvectors for the fusion matrices", "type": "text"}, {"bbox": [378, 272, 394, 284], "score": 0.89, "content": "N_{\\lambda}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [394, 268, 486, 290], "score": 1.0, "content": ", with eigenvalues", "type": "text"}, {"bbox": [487, 269, 555, 286], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}", "type": "inline_equation", "height": 17, "width": 68}, {"bbox": [555, 268, 559, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 12}, {"bbox": [70, 285, 542, 301], "spans": [{"bbox": [70, 285, 542, 301], "score": 1.0, "content": "I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll", "type": "text"}], "index": 13}, {"bbox": [71, 300, 542, 315], "spans": [{"bbox": [71, 300, 402, 315], "score": 1.0, "content": "fusion matrices must be a scalar multiple of some column of ", "type": "text"}, {"bbox": [402, 302, 410, 311], "score": 0.85, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [411, 300, 542, 315], "score": 1.0, "content": ". Thus there must be a", "type": "text"}], "index": 14}, {"bbox": [68, 312, 543, 335], "spans": [{"bbox": [68, 312, 140, 335], "score": 1.0, "content": "permutation ", "type": "text"}, {"bbox": [141, 317, 155, 327], "score": 0.9, "content": "\\pi^{\\prime\\prime}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [155, 312, 174, 335], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [174, 314, 223, 331], "score": 0.95, "content": "P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 49}, {"bbox": [224, 312, 293, 335], "score": 1.0, "content": " and scalars ", "type": "text"}, {"bbox": [293, 317, 317, 330], "score": 0.94, "content": "\\alpha(\\mu)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [318, 312, 378, 335], "score": 1.0, "content": " such that", "type": "text"}, {"bbox": [378, 315, 491, 330], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}", "type": "inline_equation", "height": 15, "width": 113}, {"bbox": [492, 312, 543, 335], "score": 1.0, "content": ". Taking", "type": "text"}], "index": 15}, {"bbox": [71, 331, 412, 345], "spans": [{"bbox": [71, 333, 100, 342], "score": 0.91, "content": "\\nu=0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [100, 331, 137, 345], "score": 1.0, "content": " forces ", "type": "text"}, {"bbox": [137, 332, 184, 344], "score": 0.94, "content": "\\alpha(\\mu)>0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [184, 331, 325, 345], "score": 1.0, "content": ", and then unitarity forces ", "type": "text"}, {"bbox": [325, 332, 371, 344], "score": 0.94, "content": "\\alpha(\\mu)=1", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [372, 331, 377, 345], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [401, 332, 412, 344], "score": 0.9251790046691895, "content": "■", "type": "text"}], "index": 16}], "index": 13.5}, {"type": "text", "bbox": [70, 349, 541, 391], "lines": [{"bbox": [92, 351, 541, 366], "spans": [{"bbox": [92, 351, 118, 366], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [119, 357, 126, 362], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [126, 351, 294, 366], "score": 1.0, "content": " be any isomorphism, and let ", "type": "text"}, {"bbox": [294, 353, 305, 362], "score": 0.9, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [306, 351, 470, 366], "score": 1.0, "content": " be as in the Lemma. Then ", "type": "text"}, {"bbox": [470, 353, 481, 362], "score": 0.89, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 351, 541, 366], "score": 1.0, "content": " is also an", "type": "text"}], "index": 17}, {"bbox": [71, 367, 538, 381], "spans": [{"bbox": [71, 367, 175, 381], "score": 1.0, "content": "isomorphism, with ", "type": "text"}, {"bbox": [175, 367, 227, 380], "score": 0.94, "content": "(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [228, 367, 410, 381], "score": 1.0, "content": ". Equation (2.2b) implies for all ", "type": "text"}, {"bbox": [411, 367, 453, 380], "score": 0.91, "content": "\\lambda\\:\\in\\:P_{+}", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [454, 367, 538, 381], "score": 1.0, "content": " and all simple-", "type": "text"}], "index": 18}, {"bbox": [71, 382, 154, 393], "spans": [{"bbox": [71, 382, 117, 393], "score": 1.0, "content": "currents ", "type": "text"}, {"bbox": [117, 383, 123, 393], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [124, 382, 154, 393], "score": 1.0, "content": ", that", "type": "text"}], "index": 19}], "index": 18}, {"type": "interline_equation", "bbox": [190, 391, 422, 409], "lines": [{"bbox": [190, 391, 422, 409], "spans": [{"bbox": [190, 391, 422, 409], "score": 0.92, "content": "Q_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 414, 541, 458], "lines": [{"bbox": [94, 416, 539, 432], "spans": [{"bbox": [94, 416, 528, 432], "score": 1.0, "content": "Another quick consequence of the Lemma is that for any Galois automorphism ", "type": "text"}, {"bbox": [528, 422, 539, 429], "score": 0.86, "content": "\\sigma_{\\ell}", "type": "inline_equation", "height": 7, "width": 11}], "index": 21}, {"bbox": [69, 430, 542, 447], "spans": [{"bbox": [69, 430, 164, 447], "score": 1.0, "content": "and isomorphism ", "type": "text"}, {"bbox": [165, 436, 172, 442], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [173, 430, 225, 447], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [225, 432, 300, 444], "score": 0.94, "content": "\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [301, 430, 327, 447], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 430, 416, 445], "score": 0.94, "content": "\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}", "type": "inline_equation", "height": 15, "width": 88}, {"bbox": [416, 430, 542, 447], "score": 1.0, "content": ". To see this, apply the", "type": "text"}], "index": 22}, {"bbox": [71, 445, 243, 460], "spans": [{"bbox": [71, 445, 149, 460], "score": 1.0, "content": "invertibility of ", "type": "text"}, {"bbox": [150, 447, 158, 456], "score": 0.91, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [158, 445, 243, 460], "score": 1.0, "content": " to the equation", "type": "text"}], "index": 23}], "index": 22}, {"type": "interline_equation", "bbox": [111, 468, 507, 488], "lines": [{"bbox": [111, 468, 507, 488], "spans": [{"bbox": [111, 468, 507, 488], "score": 0.9, "content": "\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [69, 496, 542, 540], "lines": [{"bbox": [95, 498, 542, 514], "spans": [{"bbox": [95, 498, 542, 514], "score": 1.0, "content": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a", "type": "text"}], "index": 25}, {"bbox": [70, 511, 542, 529], "spans": [{"bbox": [70, 511, 108, 529], "score": 1.0, "content": "subset ", "type": "text"}, {"bbox": [108, 514, 199, 527], "score": 0.93, "content": "\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [200, 511, 217, 529], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [218, 515, 233, 526], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [234, 511, 325, 529], "score": 1.0, "content": " which generates ", "type": "text"}, {"bbox": [325, 513, 369, 527], "score": 0.93, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [369, 511, 542, 529], "score": 1.0, "content": " as a ring. Diagonalising, this is", "type": "text"}], "index": 26}, {"bbox": [72, 527, 529, 543], "spans": [{"bbox": [72, 527, 270, 543], "score": 1.0, "content": "equivalent to requiring that there are ", "type": "text"}, {"bbox": [270, 532, 281, 538], "score": 0.78, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [281, 527, 396, 543], "score": 1.0, "content": "-variable polynomials ", "type": "text"}, {"bbox": [396, 528, 474, 541], "score": 0.93, "content": "P_{\\lambda}(x_{1},\\ldots,x_{m})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [474, 527, 529, 543], "score": 1.0, "content": " such that", "type": "text"}], "index": 27}], "index": 26}, {"type": "interline_equation", "bbox": [189, 552, 423, 582], "lines": [{"bbox": [189, 552, 423, 582], "spans": [{"bbox": [189, 552, 423, 582], "score": 0.93, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [69, 592, 541, 621], "lines": [{"bbox": [70, 594, 540, 609], "spans": [{"bbox": [70, 594, 93, 609], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [93, 595, 126, 608], "score": 0.94, "content": "(\\pi,\\pi^{\\prime})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [126, 594, 163, 609], "score": 1.0, "content": " be an ", "type": "text"}, {"bbox": [164, 596, 172, 605], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [172, 594, 378, 609], "score": 1.0, "content": "-symmetry, and suppose we know that ", "type": "text"}, {"bbox": [379, 596, 418, 607], "score": 0.84, "content": "\\pi\\gamma=\\gamma", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [419, 594, 457, 609], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [457, 599, 465, 607], "score": 0.89, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [465, 594, 540, 609], "score": 1.0, "content": " in the fusion-", "type": "text"}], "index": 29}, {"bbox": [70, 608, 254, 624], "spans": [{"bbox": [70, 608, 124, 624], "score": 1.0, "content": "generator ", "type": "text"}, {"bbox": [124, 610, 132, 619], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [132, 608, 212, 624], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [213, 610, 250, 622], "score": 0.93, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [250, 608, 254, 624], "score": 1.0, "content": ",", "type": "text"}], "index": 30}], "index": 29.5}, {"type": "interline_equation", "bbox": [146, 632, 465, 663], "lines": [{"bbox": [146, 632, 465, 663], "spans": [{"bbox": [146, 632, 465, 663], "score": 0.93, "content": "{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [69, 672, 207, 686], "lines": [{"bbox": [71, 674, 207, 687], "spans": [{"bbox": [71, 674, 106, 687], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [106, 676, 144, 687], "score": 0.93, "content": "\\mu\\in P_{+}", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [144, 674, 165, 687], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [165, 676, 203, 685], "score": 0.93, "content": "\\pi\\lambda=\\lambda", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [203, 674, 207, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [70, 687, 541, 715], "lines": [{"bbox": [95, 689, 541, 703], "spans": [{"bbox": [95, 689, 541, 703], "score": 1.0, "content": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the", "type": "text"}], "index": 33}, {"bbox": [72, 703, 540, 717], "spans": [{"bbox": [72, 703, 540, 717], "score": 1.0, "content": "existence of small fusion-generators. In particular, because we know that any Lie character", "type": "text"}], "index": 34}], "index": 33.5}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [190, 391, 422, 409], "lines": [{"bbox": [190, 391, 422, 409], "spans": [{"bbox": [190, 391, 422, 409], "score": 0.92, "content": "Q_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [111, 468, 507, 488], "lines": [{"bbox": [111, 468, 507, 488], "spans": [{"bbox": [111, 468, 507, 488], "score": 0.9, "content": "\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "interline_equation", "bbox": [189, 552, 423, 582], "lines": [{"bbox": [189, 552, 423, 582], "spans": [{"bbox": [189, 552, 423, 582], "score": 0.93, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "interline_equation", "bbox": [146, 632, 465, 663], "lines": [{"bbox": [146, 632, 465, 663], "spans": [{"bbox": [146, 632, 465, 663], "score": 0.93, "content": "{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}", "type": "interline_equation"}], "index": 31}], "index": 31}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 392, 85], "lines": [{"bbox": [70, 74, 390, 87], "spans": [{"bbox": [70, 74, 139, 87], "score": 1.0, "content": "For instance ", "type": "text"}, {"bbox": [139, 79, 146, 84], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [147, 74, 207, 87], "score": 1.0, "content": " must send ", "type": "text"}, {"bbox": [208, 75, 216, 84], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [216, 74, 298, 87], "score": 1.0, "content": "-fixed-points to ", "type": "text"}, {"bbox": [298, 75, 323, 87], "score": 0.94, "content": "\\pi(J)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [324, 74, 390, 87], "score": 1.0, "content": "-fixed-points.", "type": "text"}], "index": 0}], "index": 0, "bbox_fs": [70, 74, 390, 87]}, {"type": "text", "bbox": [70, 86, 542, 160], "lines": [{"bbox": [95, 88, 541, 102], "spans": [{"bbox": [95, 88, 309, 102], "score": 1.0, "content": "More generally, a fusion-homomorphism ", "type": "text"}, {"bbox": [309, 93, 316, 99], "score": 0.85, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [317, 88, 541, 102], "score": 1.0, "content": " is defined in the obvious algebraic way. It", "type": "text"}], "index": 1}, {"bbox": [70, 102, 541, 117], "spans": [{"bbox": [70, 102, 212, 117], "score": 1.0, "content": "turns out that for such a ", "type": "text"}, {"bbox": [213, 108, 220, 113], "score": 0.82, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [220, 102, 228, 117], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [228, 104, 278, 115], "score": 0.91, "content": "\\pi\\lambda\\,=\\,\\pi\\mu", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [278, 102, 298, 117], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [298, 104, 341, 115], "score": 0.93, "content": "\\mu\\,=\\,J\\lambda", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [341, 102, 476, 117], "score": 1.0, "content": " for some simple-current ", "type": "text"}, {"bbox": [477, 105, 484, 113], "score": 0.9, "content": "J", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [485, 102, 541, 117], "score": 1.0, "content": " for which", "type": "text"}], "index": 2}, {"bbox": [71, 116, 542, 132], "spans": [{"bbox": [71, 117, 124, 131], "score": 0.94, "content": "\\pi(J0)=\\tilde{0}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [124, 116, 187, 132], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [187, 117, 240, 131], "score": 0.95, "content": "\\pi(J0)=\\tilde{0}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [241, 116, 404, 132], "score": 1.0, "content": " is possible only if there are no ", "type": "text"}, {"bbox": [404, 119, 412, 127], "score": 0.9, "content": "J", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [413, 116, 520, 132], "score": 1.0, "content": "-fixed-points. When ", "type": "text"}, {"bbox": [520, 122, 528, 127], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [528, 116, 542, 132], "score": 1.0, "content": " is", "type": "text"}], "index": 3}, {"bbox": [69, 131, 542, 149], "spans": [{"bbox": [69, 131, 409, 149], "score": 1.0, "content": "one-to-one (e.g. when there are no nontrivial simple-currents in ", "type": "text"}, {"bbox": [410, 131, 463, 148], "score": 0.93, "content": "P_{+}^{k}(X_{r}^{(1)}))", "type": "inline_equation", "height": 17, "width": 53}, {"bbox": [463, 131, 498, 149], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [498, 138, 506, 144], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [506, 131, 542, 149], "score": 1.0, "content": " obeys", "type": "text"}], "index": 4}, {"bbox": [72, 148, 366, 162], "spans": [{"bbox": [72, 148, 366, 162], "score": 1.0, "content": "(2.6). Fusion-homomorphisms will be studied elsewhere.", "type": "text"}], "index": 5}], "index": 3, "bbox_fs": [69, 88, 542, 162]}, {"type": "text", "bbox": [95, 160, 420, 174], "lines": [{"bbox": [95, 162, 419, 176], "spans": [{"bbox": [95, 162, 419, 176], "score": 1.0, "content": "The key to finding fusion-symmetries is the following Lemma.", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [95, 162, 419, 176]}, {"type": "text", "bbox": [70, 180, 543, 246], "lines": [{"bbox": [91, 178, 543, 201], "spans": [{"bbox": [91, 178, 194, 201], "score": 1.0, "content": "Lemma 2.2. Let", "type": "text"}, {"bbox": [194, 183, 203, 196], "score": 0.87, "content": "\\widetilde{S}", "type": "inline_equation", "height": 13, "width": 9}, {"bbox": [203, 178, 376, 201], "score": 1.0, "content": " be the Kac-Peterson matrix for ", "type": "text"}, {"bbox": [376, 181, 400, 197], "score": 0.9, "content": "Y_{s}^{(1)}", "type": "inline_equation", "height": 16, "width": 24}, {"bbox": [400, 178, 431, 201], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [431, 186, 443, 196], "score": 0.46, "content": "m", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [443, 178, 543, 201], "score": 1.0, "content": ". Then a bijection", "type": "text"}], "index": 7}, {"bbox": [71, 195, 543, 218], "spans": [{"bbox": [71, 198, 214, 216], "score": 0.92, "content": "\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 18, "width": 143}, {"bbox": [214, 195, 543, 218], "score": 1.0, "content": " defines an isomorphism of fusion rings iff there exists some", "type": "text"}], "index": 8}, {"bbox": [68, 212, 544, 236], "spans": [{"bbox": [68, 212, 119, 236], "score": 1.0, "content": "bijection ", "type": "text"}, {"bbox": [120, 216, 261, 233], "score": 0.92, "content": "\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})", "type": "inline_equation", "height": 17, "width": 141}, {"bbox": [261, 212, 317, 236], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [317, 216, 392, 232], "score": 0.94, "content": "S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [392, 212, 433, 236], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [433, 215, 518, 233], "score": 0.91, "content": "\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 18, "width": 85}, {"bbox": [518, 212, 544, 236], "score": 1.0, "content": ". In", "type": "text"}], "index": 9}, {"bbox": [70, 232, 541, 247], "spans": [{"bbox": [70, 232, 205, 247], "score": 1.0, "content": "particular, a permutation ", "type": "text"}, {"bbox": [206, 238, 213, 243], "score": 0.67, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 232, 344, 247], "score": 1.0, "content": " is a fusion-symmetry iff", "type": "text"}, {"bbox": [344, 233, 377, 246], "score": 0.92, "content": "(\\pi,\\pi^{\\prime})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [377, 232, 409, 247], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [409, 234, 418, 244], "score": 0.83, "content": "S", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [418, 232, 524, 247], "score": 1.0, "content": "-symmetry for some ", "type": "text"}, {"bbox": [525, 234, 536, 243], "score": 0.87, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [536, 232, 541, 247], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 8.5, "bbox_fs": [68, 178, 544, 247]}, {"type": "text", "bbox": [70, 250, 558, 344], "lines": [{"bbox": [68, 250, 542, 273], "spans": [{"bbox": [68, 250, 185, 273], "score": 1.0, "content": "Proof. The equality ", "type": "text"}, {"bbox": [185, 256, 263, 270], "score": 0.93, "content": "N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [263, 250, 378, 273], "score": 1.0, "content": " means that, for each ", "type": "text"}, {"bbox": [378, 256, 387, 267], "score": 0.75, "content": "\\mu", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [387, 250, 497, 273], "score": 1.0, "content": ", the column vectors ", "type": "text"}, {"bbox": [498, 254, 542, 269], "score": 0.9, "content": "(\\underline{{x}}_{\\mu})_{\\nu}=", "type": "inline_equation", "height": 15, "width": 44}], "index": 11}, {"bbox": [71, 268, 559, 290], "spans": [{"bbox": [71, 270, 105, 286], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}", "type": "inline_equation", "height": 16, "width": 34}, {"bbox": [105, 268, 377, 290], "score": 1.0, "content": " are simultaneous eigenvectors for the fusion matrices", "type": "text"}, {"bbox": [378, 272, 394, 284], "score": 0.89, "content": "N_{\\lambda}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [394, 268, 486, 290], "score": 1.0, "content": ", with eigenvalues", "type": "text"}, {"bbox": [487, 269, 555, 286], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}", "type": "inline_equation", "height": 17, "width": 68}, {"bbox": [555, 268, 559, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 12}, {"bbox": [70, 285, 542, 301], "spans": [{"bbox": [70, 285, 542, 301], "score": 1.0, "content": "I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll", "type": "text"}], "index": 13}, {"bbox": [71, 300, 542, 315], "spans": [{"bbox": [71, 300, 402, 315], "score": 1.0, "content": "fusion matrices must be a scalar multiple of some column of ", "type": "text"}, {"bbox": [402, 302, 410, 311], "score": 0.85, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [411, 300, 542, 315], "score": 1.0, "content": ". Thus there must be a", "type": "text"}], "index": 14}, {"bbox": [68, 312, 543, 335], "spans": [{"bbox": [68, 312, 140, 335], "score": 1.0, "content": "permutation ", "type": "text"}, {"bbox": [141, 317, 155, 327], "score": 0.9, "content": "\\pi^{\\prime\\prime}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [155, 312, 174, 335], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [174, 314, 223, 331], "score": 0.95, "content": "P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 49}, {"bbox": [224, 312, 293, 335], "score": 1.0, "content": " and scalars ", "type": "text"}, {"bbox": [293, 317, 317, 330], "score": 0.94, "content": "\\alpha(\\mu)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [318, 312, 378, 335], "score": 1.0, "content": " such that", "type": "text"}, {"bbox": [378, 315, 491, 330], "score": 0.92, "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}", "type": "inline_equation", "height": 15, "width": 113}, {"bbox": [492, 312, 543, 335], "score": 1.0, "content": ". Taking", "type": "text"}], "index": 15}, {"bbox": [71, 331, 412, 345], "spans": [{"bbox": [71, 333, 100, 342], "score": 0.91, "content": "\\nu=0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [100, 331, 137, 345], "score": 1.0, "content": " forces ", "type": "text"}, {"bbox": [137, 332, 184, 344], "score": 0.94, "content": "\\alpha(\\mu)>0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [184, 331, 325, 345], "score": 1.0, "content": ", and then unitarity forces ", "type": "text"}, {"bbox": [325, 332, 371, 344], "score": 0.94, "content": "\\alpha(\\mu)=1", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [372, 331, 377, 345], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [401, 332, 412, 344], "score": 0.9251790046691895, "content": "■", "type": "text"}], "index": 16}], "index": 13.5, "bbox_fs": [68, 250, 559, 345]}, {"type": "text", "bbox": [70, 349, 541, 391], "lines": [{"bbox": [92, 351, 541, 366], "spans": [{"bbox": [92, 351, 118, 366], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [119, 357, 126, 362], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [126, 351, 294, 366], "score": 1.0, "content": " be any isomorphism, and let ", "type": "text"}, {"bbox": [294, 353, 305, 362], "score": 0.9, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [306, 351, 470, 366], "score": 1.0, "content": " be as in the Lemma. Then ", "type": "text"}, {"bbox": [470, 353, 481, 362], "score": 0.89, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 351, 541, 366], "score": 1.0, "content": " is also an", "type": "text"}], "index": 17}, {"bbox": [71, 367, 538, 381], "spans": [{"bbox": [71, 367, 175, 381], "score": 1.0, "content": "isomorphism, with ", "type": "text"}, {"bbox": [175, 367, 227, 380], "score": 0.94, "content": "(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [228, 367, 410, 381], "score": 1.0, "content": ". Equation (2.2b) implies for all ", "type": "text"}, {"bbox": [411, 367, 453, 380], "score": 0.91, "content": "\\lambda\\:\\in\\:P_{+}", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [454, 367, 538, 381], "score": 1.0, "content": " and all simple-", "type": "text"}], "index": 18}, {"bbox": [71, 382, 154, 393], "spans": [{"bbox": [71, 382, 117, 393], "score": 1.0, "content": "currents ", "type": "text"}, {"bbox": [117, 383, 123, 393], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [124, 382, 154, 393], "score": 1.0, "content": ", that", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [71, 351, 541, 393]}, {"type": "interline_equation", "bbox": [190, 391, 422, 409], "lines": [{"bbox": [190, 391, 422, 409], "spans": [{"bbox": [190, 391, 422, 409], "score": 0.92, "content": "Q_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 414, 541, 458], "lines": [{"bbox": [94, 416, 539, 432], "spans": [{"bbox": [94, 416, 528, 432], "score": 1.0, "content": "Another quick consequence of the Lemma is that for any Galois automorphism ", "type": "text"}, {"bbox": [528, 422, 539, 429], "score": 0.86, "content": "\\sigma_{\\ell}", "type": "inline_equation", "height": 7, "width": 11}], "index": 21}, {"bbox": [69, 430, 542, 447], "spans": [{"bbox": [69, 430, 164, 447], "score": 1.0, "content": "and isomorphism ", "type": "text"}, {"bbox": [165, 436, 172, 442], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [173, 430, 225, 447], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [225, 432, 300, 444], "score": 0.94, "content": "\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [301, 430, 327, 447], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 430, 416, 445], "score": 0.94, "content": "\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}", "type": "inline_equation", "height": 15, "width": 88}, {"bbox": [416, 430, 542, 447], "score": 1.0, "content": ". To see this, apply the", "type": "text"}], "index": 22}, {"bbox": [71, 445, 243, 460], "spans": [{"bbox": [71, 445, 149, 460], "score": 1.0, "content": "invertibility of ", "type": "text"}, {"bbox": [150, 447, 158, 456], "score": 0.91, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [158, 445, 243, 460], "score": 1.0, "content": " to the equation", "type": "text"}], "index": 23}], "index": 22, "bbox_fs": [69, 416, 542, 460]}, {"type": "interline_equation", "bbox": [111, 468, 507, 488], "lines": [{"bbox": [111, 468, 507, 488], "spans": [{"bbox": [111, 468, 507, 488], "score": 0.9, "content": "\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [69, 496, 542, 540], "lines": [{"bbox": [95, 498, 542, 514], "spans": [{"bbox": [95, 498, 542, 514], "score": 1.0, "content": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a", "type": "text"}], "index": 25}, {"bbox": [70, 511, 542, 529], "spans": [{"bbox": [70, 511, 108, 529], "score": 1.0, "content": "subset ", "type": "text"}, {"bbox": [108, 514, 199, 527], "score": 0.93, "content": "\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [200, 511, 217, 529], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [218, 515, 233, 526], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [234, 511, 325, 529], "score": 1.0, "content": " which generates ", "type": "text"}, {"bbox": [325, 513, 369, 527], "score": 0.93, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [369, 511, 542, 529], "score": 1.0, "content": " as a ring. Diagonalising, this is", "type": "text"}], "index": 26}, {"bbox": [72, 527, 529, 543], "spans": [{"bbox": [72, 527, 270, 543], "score": 1.0, "content": "equivalent to requiring that there are ", "type": "text"}, {"bbox": [270, 532, 281, 538], "score": 0.78, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [281, 527, 396, 543], "score": 1.0, "content": "-variable polynomials ", "type": "text"}, {"bbox": [396, 528, 474, 541], "score": 0.93, "content": "P_{\\lambda}(x_{1},\\ldots,x_{m})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [474, 527, 529, 543], "score": 1.0, "content": " such that", "type": "text"}], "index": 27}], "index": 26, "bbox_fs": [70, 498, 542, 543]}, {"type": "interline_equation", "bbox": [189, 552, 423, 582], "lines": [{"bbox": [189, 552, 423, 582], "spans": [{"bbox": [189, 552, 423, 582], "score": 0.93, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [69, 592, 541, 621], "lines": [{"bbox": [70, 594, 540, 609], "spans": [{"bbox": [70, 594, 93, 609], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [93, 595, 126, 608], "score": 0.94, "content": "(\\pi,\\pi^{\\prime})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [126, 594, 163, 609], "score": 1.0, "content": " be an ", "type": "text"}, {"bbox": [164, 596, 172, 605], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [172, 594, 378, 609], "score": 1.0, "content": "-symmetry, and suppose we know that ", "type": "text"}, {"bbox": [379, 596, 418, 607], "score": 0.84, "content": "\\pi\\gamma=\\gamma", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [419, 594, 457, 609], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [457, 599, 465, 607], "score": 0.89, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [465, 594, 540, 609], "score": 1.0, "content": " in the fusion-", "type": "text"}], "index": 29}, {"bbox": [70, 608, 254, 624], "spans": [{"bbox": [70, 608, 124, 624], "score": 1.0, "content": "generator ", "type": "text"}, {"bbox": [124, 610, 132, 619], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [132, 608, 212, 624], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [213, 610, 250, 622], "score": 0.93, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [250, 608, 254, 624], "score": 1.0, "content": ",", "type": "text"}], "index": 30}], "index": 29.5, "bbox_fs": [70, 594, 540, 624]}, {"type": "interline_equation", "bbox": [146, 632, 465, 663], "lines": [{"bbox": [146, 632, 465, 663], "spans": [{"bbox": [146, 632, 465, 663], "score": 0.93, "content": "{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [69, 672, 207, 686], "lines": [{"bbox": [71, 674, 207, 687], "spans": [{"bbox": [71, 674, 106, 687], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [106, 676, 144, 687], "score": 0.93, "content": "\\mu\\in P_{+}", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [144, 674, 165, 687], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [165, 676, 203, 685], "score": 0.93, "content": "\\pi\\lambda=\\lambda", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [203, 674, 207, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 32, "bbox_fs": [71, 674, 207, 687]}, {"type": "text", "bbox": [70, 687, 541, 715], "lines": [{"bbox": [95, 689, 541, 703], "spans": [{"bbox": [95, 689, 541, 703], "score": 1.0, "content": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the", "type": "text"}], "index": 33}, {"bbox": [72, 703, 540, 717], "spans": [{"bbox": [72, 703, 540, 717], "score": 1.0, "content": "existence of small fusion-generators. In particular, because we know that any Lie character", "type": "text"}], "index": 34}, {"bbox": [71, 71, 542, 91], "spans": [{"bbox": [71, 75, 90, 88], "score": 0.84, "content": "\\mathrm{ch}_{\\overline{{{\\mu}}}}", "type": "inline_equation", "height": 13, "width": 19, "cross_page": true}, {"bbox": [90, 71, 111, 91], "score": 1.0, "content": " for ", "type": "text", "cross_page": true}, {"bbox": [112, 75, 127, 86], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [128, 71, 462, 91], "score": 1.0, "content": " can be written as a polynomial in the fundamental characters ", "type": "text", "cross_page": true}, {"bbox": [462, 74, 537, 87], "score": 0.87, "content": "\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}", "type": "inline_equation", "height": 13, "width": 75, "cross_page": true}, {"bbox": [537, 71, 542, 91], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [69, 88, 540, 106], "spans": [{"bbox": [69, 88, 212, 106], "score": 1.0, "content": "we know from (2.1b) that ", "type": "text", "cross_page": true}, {"bbox": [212, 92, 303, 105], "score": 0.93, "content": "\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 91, "cross_page": true}, {"bbox": [304, 88, 438, 106], "score": 1.0, "content": " is a fusion-generator for ", "type": "text", "cross_page": true}, {"bbox": [438, 88, 462, 103], "score": 0.92, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24, "cross_page": true}, {"bbox": [463, 88, 532, 106], "score": 1.0, "content": "at any level ", "type": "text", "cross_page": true}, {"bbox": [532, 93, 540, 102], "score": 0.84, "content": "k", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}], "index": 1}, {"bbox": [70, 105, 541, 120], "spans": [{"bbox": [70, 105, 185, 120], "score": 1.0, "content": "sufficiently large that ", "type": "text", "cross_page": true}, {"bbox": [186, 108, 201, 119], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [202, 105, 268, 120], "score": 1.0, "content": " contains all ", "type": "text", "cross_page": true}, {"bbox": [268, 107, 281, 118], "score": 0.9, "content": "\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [281, 105, 412, 120], "score": 1.0, "content": " (in other words, for any ", "type": "text", "cross_page": true}, {"bbox": [412, 105, 478, 119], "score": 0.91, "content": "k\\geq\\operatorname*{max}_{i}a_{i}^{\\vee}", "type": "inline_equation", "height": 14, "width": 66, "cross_page": true}, {"bbox": [479, 105, 541, 120], "score": 1.0, "content": "). In fact, it", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [70, 119, 541, 135], "spans": [{"bbox": [70, 119, 362, 135], "score": 1.0, "content": "is easy to show [18] that a fusion-generator valid for any ", "type": "text", "cross_page": true}, {"bbox": [362, 122, 385, 134], "score": 0.93, "content": "X_{r,k}", "type": "inline_equation", "height": 12, "width": 23, "cross_page": true}, {"bbox": [385, 119, 399, 135], "score": 1.0, "content": " is ", "type": "text", "cross_page": true}, {"bbox": [400, 120, 492, 133], "score": 0.92, "content": "\\{\\Lambda_{1},\\ldots,\\Lambda_{r}\\}\\cap{\\cal P}_{+}", "type": "inline_equation", "height": 13, "width": 92, "cross_page": true}, {"bbox": [492, 119, 541, 135], "score": 1.0, "content": ". Smaller", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [70, 133, 541, 149], "spans": [{"bbox": [70, 133, 310, 149], "score": 1.0, "content": "fusion-generators usually exist — for example ", "type": "text", "cross_page": true}, {"bbox": [310, 135, 336, 147], "score": 0.94, "content": "\\{\\Lambda_{1}\\}", "type": "inline_equation", "height": 12, "width": 26, "cross_page": true}, {"bbox": [336, 133, 465, 149], "score": 1.0, "content": " is a fusion-generator for ", "type": "text", "cross_page": true}, {"bbox": [465, 135, 487, 148], "score": 0.91, "content": "A_{8,k}", "type": "inline_equation", "height": 13, "width": 22, "cross_page": true}, {"bbox": [488, 133, 541, 149], "score": 1.0, "content": " whenever", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [71, 149, 215, 163], "spans": [{"bbox": [71, 150, 78, 159], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 7, "cross_page": true}, {"bbox": [78, 149, 215, 163], "score": 1.0, "content": " is even and coprime to 3.", "type": "text", "cross_page": true}], "index": 5}], "index": 33.5, "bbox_fs": [72, 689, 541, 717]}]}
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[{"type": "text", "bbox": [70, 70, 392, 85], "content": "For instance must send -fixed-points to -fixed-points.", "index": 0}, {"type": "text", "bbox": [70, 86, 542, 160], "content": "More generally, a fusion-homomorphism is defined in the obvious algebraic way. It turns out that for such a , iff for some simple-current for which . Moreover, is possible only if there are no -fixed-points. When is one-to-one (e.g. when there are no nontrivial simple-currents in , then obeys (2.6). Fusion-homomorphisms will be studied elsewhere.", "index": 1}, {"type": "text", "bbox": [95, 160, 420, 174], "content": "The key to finding fusion-symmetries is the following Lemma.", "index": 2}, {"type": "text", "bbox": [70, 180, 543, 246], "content": "Lemma 2.2. Let be the Kac-Peterson matrix for level . Then a bijection defines an isomorphism of fusion rings iff there exists some bijection such that for all . In particular, a permutation is a fusion-symmetry iff is an -symmetry for some .", "index": 3}, {"type": "text", "bbox": [70, 250, 558, 344], "content": "Proof. The equality means that, for each , the column vectors are simultaneous eigenvectors for the fusion matrices , with eigenvalues . I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll fusion matrices must be a scalar multiple of some column of . Thus there must be a permutation of and scalars such that . Taking forces , and then unitarity forces . ■", "index": 4}, {"type": "text", "bbox": [70, 349, 541, 391], "content": "Let be any isomorphism, and let be as in the Lemma. Then is also an isomorphism, with . Equation (2.2b) implies for all and all simple- currents , that", "index": 5}, {"type": "interline_equation", "bbox": [190, 391, 422, 409], "content": "", "index": 6}, {"type": "text", "bbox": [70, 414, 541, 458], "content": "Another quick consequence of the Lemma is that for any Galois automorphism and isomorphism , we have and . To see this, apply the invertibility of to the equation", "index": 7}, {"type": "interline_equation", "bbox": [111, 468, 507, 488], "content": "", "index": 8}, {"type": "text", "bbox": [69, 496, 542, 540], "content": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a subset of which generates as a ring. Diagonalising, this is equivalent to requiring that there are -variable polynomials such that", "index": 9}, {"type": "interline_equation", "bbox": [189, 552, 423, 582], "content": "", "index": 10}, {"type": "text", "bbox": [69, 592, 541, 621], "content": "Let be an -symmetry, and suppose we know that for all in the fusion- generator . Then for any ,", "index": 11}, {"type": "interline_equation", "bbox": [146, 632, 465, 663], "content": "", "index": 12}, {"type": "text", "bbox": [69, 672, 207, 686], "content": "for all , so .", "index": 13}, {"type": "text", "bbox": [70, 687, 541, 715], "content": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the existence of small fusion-generators. In particular, because we know that any Lie character for can be written as a polynomial in the fundamental characters , we know from (2.1b) that is a fusion-generator for at any level sufficiently large that contains all (in other words, for any ). In fact, it is easy to show [18] that a fusion-generator valid for any is . Smaller fusion-generators usually exist — for example is a fusion-generator for whenever is even and coprime to 3.", "index": 14}]
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[{"bbox": [70, 74, 390, 87], "content": "For instance must send -fixed-points to -fixed-points.", "parent_index": 0, "line_index": 0}, {"bbox": [95, 88, 541, 102], "content": "More generally, a fusion-homomorphism is defined in the obvious algebraic way. It", "parent_index": 1, "line_index": 0}, {"bbox": [70, 102, 541, 117], "content": "turns out that for such a , iff for some simple-current for which", "parent_index": 1, "line_index": 1}, {"bbox": [71, 116, 542, 132], "content": ". Moreover, is possible only if there are no -fixed-points. When is", "parent_index": 1, "line_index": 2}, {"bbox": [69, 131, 542, 149], "content": "one-to-one (e.g. when there are no nontrivial simple-currents in , then obeys", "parent_index": 1, "line_index": 3}, {"bbox": [72, 148, 366, 162], "content": "(2.6). Fusion-homomorphisms will be studied elsewhere.", "parent_index": 1, "line_index": 4}, {"bbox": [95, 162, 419, 176], "content": "The key to finding fusion-symmetries is the following Lemma.", "parent_index": 2, "line_index": 0}, {"bbox": [91, 178, 543, 201], "content": "Lemma 2.2. Let be the Kac-Peterson matrix for level . Then a bijection", "parent_index": 3, "line_index": 0}, {"bbox": [71, 195, 543, 218], "content": "defines an isomorphism of fusion rings iff there exists some", "parent_index": 3, "line_index": 1}, {"bbox": [68, 212, 544, 236], "content": "bijection such that for all . In", "parent_index": 3, "line_index": 2}, {"bbox": [70, 232, 541, 247], "content": "particular, a permutation is a fusion-symmetry iff is an -symmetry for some .", "parent_index": 3, "line_index": 3}, {"bbox": [68, 250, 542, 273], "content": "Proof. The equality means that, for each , the column vectors", "parent_index": 4, "line_index": 0}, {"bbox": [71, 268, 559, 290], "content": "are simultaneous eigenvectors for the fusion matrices , with eigenvalues .", "parent_index": 4, "line_index": 1}, {"bbox": [70, 285, 542, 301], "content": "I t is easy to see from Verlinde’s formula (1.1b) that any simultaneous eigenvec t or for a ll", "parent_index": 4, "line_index": 2}, {"bbox": [71, 300, 542, 315], "content": "fusion matrices must be a scalar multiple of some column of . Thus there must be a", "parent_index": 4, "line_index": 3}, {"bbox": [68, 312, 543, 335], "content": "permutation of and scalars such that . Taking", "parent_index": 4, "line_index": 4}, {"bbox": [71, 331, 412, 345], "content": "forces , and then unitarity forces . ■", "parent_index": 4, "line_index": 5}, {"bbox": [92, 351, 541, 366], "content": "Let be any isomorphism, and let be as in the Lemma. Then is also an", "parent_index": 5, "line_index": 0}, {"bbox": [71, 367, 538, 381], "content": "isomorphism, with . Equation (2.2b) implies for all and all simple-", "parent_index": 5, "line_index": 1}, {"bbox": [71, 382, 154, 393], "content": "currents , that", "parent_index": 5, "line_index": 2}, {"bbox": [94, 416, 539, 432], "content": "Another quick consequence of the Lemma is that for any Galois automorphism", "parent_index": 7, "line_index": 0}, {"bbox": [69, 430, 542, 447], "content": "and isomorphism , we have and . To see this, apply the", "parent_index": 7, "line_index": 1}, {"bbox": [71, 445, 243, 460], "content": "invertibility of to the equation", "parent_index": 7, "line_index": 2}, {"bbox": [95, 498, 542, 514], "content": "A very useful notion for studying the fusion ring is that of fusion-generator, i.e. a", "parent_index": 9, "line_index": 0}, {"bbox": [70, 511, 542, 529], "content": "subset of which generates as a ring. Diagonalising, this is", "parent_index": 9, "line_index": 1}, {"bbox": [72, 527, 529, 543], "content": "equivalent to requiring that there are -variable polynomials such that", "parent_index": 9, "line_index": 2}, {"bbox": [70, 594, 540, 609], "content": "Let be an -symmetry, and suppose we know that for all in the fusion-", "parent_index": 11, "line_index": 0}, {"bbox": [70, 608, 254, 624], "content": "generator . Then for any ,", "parent_index": 11, "line_index": 1}, {"bbox": [71, 674, 207, 687], "content": "for all , so .", "parent_index": 13, "line_index": 0}, {"bbox": [95, 689, 541, 703], "content": "One of the reasons fusion-symmetries for the affine algebras are so tractible is the", "parent_index": 14, "line_index": 0}, {"bbox": [72, 703, 540, 717], "content": "existence of small fusion-generators. In particular, because we know that any Lie character", "parent_index": 14, "line_index": 1}, {"bbox": [71, 71, 542, 91], "content": "for can be written as a polynomial in the fundamental characters ,", "parent_index": 14, "line_index": 2}, {"bbox": [69, 88, 540, 106], "content": "we know from (2.1b) that is a fusion-generator for at any level", "parent_index": 14, "line_index": 3}, {"bbox": [70, 105, 541, 120], "content": "sufficiently large that contains all (in other words, for any ). In fact, it", "parent_index": 14, "line_index": 4}, {"bbox": [70, 119, 541, 135], "content": "is easy to show [18] that a fusion-generator valid for any is . Smaller", "parent_index": 14, "line_index": 5}, {"bbox": [70, 133, 541, 149], "content": "fusion-generators usually exist — for example is a fusion-generator for whenever", "parent_index": 14, "line_index": 6}, {"bbox": [71, 149, 215, 163], "content": "is even and coprime to 3.", "parent_index": 14, "line_index": 7}]
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[]
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[{"bbox": [139, 79, 146, 84], "content": "\\pi", "parent_index": 0, "subtype": "inline"}, {"bbox": [208, 75, 216, 84], "content": "J", "parent_index": 0, "subtype": "inline"}, {"bbox": [298, 75, 323, 87], "content": "\\pi(J)", "parent_index": 0, "subtype": "inline"}, {"bbox": [309, 93, 316, 99], "content": "\\pi", "parent_index": 1, "subtype": "inline"}, {"bbox": [213, 108, 220, 113], "content": "\\pi", "parent_index": 1, "subtype": "inline"}, {"bbox": [228, 104, 278, 115], "content": "\\pi\\lambda\\,=\\,\\pi\\mu", "parent_index": 1, "subtype": "inline"}, {"bbox": [298, 104, 341, 115], "content": "\\mu\\,=\\,J\\lambda", "parent_index": 1, "subtype": "inline"}, {"bbox": [477, 105, 484, 113], "content": "J", "parent_index": 1, "subtype": "inline"}, {"bbox": [71, 117, 124, 131], "content": "\\pi(J0)=\\tilde{0}", "parent_index": 1, "subtype": "inline"}, {"bbox": [187, 117, 240, 131], "content": "\\pi(J0)=\\tilde{0}", "parent_index": 1, "subtype": "inline"}, {"bbox": [404, 119, 412, 127], "content": "J", "parent_index": 1, "subtype": "inline"}, {"bbox": [520, 122, 528, 127], "content": "\\pi", "parent_index": 1, "subtype": "inline"}, {"bbox": [410, 131, 463, 148], "content": "P_{+}^{k}(X_{r}^{(1)}))", "parent_index": 1, "subtype": "inline"}, {"bbox": [498, 138, 506, 144], "content": "\\pi", "parent_index": 1, "subtype": "inline"}, {"bbox": [194, 183, 203, 196], "content": "\\widetilde{S}", "parent_index": 3, "subtype": "inline"}, {"bbox": [376, 181, 400, 197], "content": "Y_{s}^{(1)}", "parent_index": 3, "subtype": "inline"}, {"bbox": [431, 186, 443, 196], "content": "m", "parent_index": 3, "subtype": "inline"}, {"bbox": [71, 198, 214, 216], "content": "\\pi\\,:\\,P_{+}^{k}(X_{r}^{(1)})\\,\\to\\,P_{+}^{m}(Y_{s}^{(1)})", "parent_index": 3, "subtype": "inline"}, {"bbox": [120, 216, 261, 233], "content": "\\pi^{\\prime}:P_{+}^{k}(X_{r}^{(1)})\\to P_{+}^{m}(Y_{s}^{(1)})", "parent_index": 3, "subtype": "inline"}, {"bbox": [317, 216, 392, 232], "content": "S_{\\lambda\\mu}=\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}", "parent_index": 3, "subtype": "inline"}, {"bbox": [433, 215, 518, 233], "content": "\\lambda,\\mu\\in P_{+}^{k}(X_{r}^{(1)})", "parent_index": 3, "subtype": "inline"}, {"bbox": [206, 238, 213, 243], "content": "\\pi", "parent_index": 3, "subtype": "inline"}, {"bbox": [344, 233, 377, 246], "content": "(\\pi,\\pi^{\\prime})", "parent_index": 3, "subtype": "inline"}, {"bbox": [409, 234, 418, 244], "content": "S", "parent_index": 3, "subtype": "inline"}, {"bbox": [525, 234, 536, 243], "content": "\\pi^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [185, 256, 263, 270], "content": "N_{\\lambda\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}", "parent_index": 4, "subtype": "inline"}, {"bbox": [378, 256, 387, 267], "content": "\\mu", "parent_index": 4, "subtype": "inline"}, {"bbox": [498, 254, 542, 269], "content": "(\\underline{{x}}_{\\mu})_{\\nu}=", "parent_index": 4, "subtype": "inline"}, {"bbox": [71, 270, 105, 286], "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}", "parent_index": 4, "subtype": "inline"}, {"bbox": [378, 272, 394, 284], "content": "N_{\\lambda}", "parent_index": 4, "subtype": "inline"}, {"bbox": [487, 269, 555, 286], "content": "\\widetilde{S}_{\\pi\\lambda,\\pi\\mu}/\\widetilde{S}_{0,\\pi\\mu}", "parent_index": 4, "subtype": "inline"}, {"bbox": [402, 302, 410, 311], "content": "S", "parent_index": 4, "subtype": "inline"}, {"bbox": [141, 317, 155, 327], "content": "\\pi^{\\prime\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [174, 314, 223, 331], "content": "P_{+}^{k}(X_{r}^{(1)})", "parent_index": 4, "subtype": "inline"}, {"bbox": [293, 317, 317, 330], "content": "\\alpha(\\mu)", "parent_index": 4, "subtype": "inline"}, {"bbox": [378, 315, 491, 330], "content": "\\widetilde{S}_{\\pi\\nu,\\pi\\mu}\\,=\\,\\alpha(\\mu)\\,S_{\\nu,\\pi^{\\prime\\prime}\\mu}", "parent_index": 4, "subtype": "inline"}, {"bbox": [71, 333, 100, 342], "content": "\\nu=0", "parent_index": 4, "subtype": "inline"}, {"bbox": [137, 332, 184, 344], "content": "\\alpha(\\mu)>0", "parent_index": 4, "subtype": "inline"}, {"bbox": [325, 332, 371, 344], "content": "\\alpha(\\mu)=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [119, 357, 126, 362], "content": "\\pi", "parent_index": 5, "subtype": "inline"}, {"bbox": [294, 353, 305, 362], "content": "\\pi^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [470, 353, 481, 362], "content": "\\pi^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [175, 367, 227, 380], "content": "(\\pi^{\\prime})^{\\prime}\\;=\\;\\pi", "parent_index": 5, "subtype": "inline"}, {"bbox": [411, 367, 453, 380], "content": "\\lambda\\:\\in\\:P_{+}", "parent_index": 5, "subtype": "inline"}, {"bbox": [117, 383, 123, 393], "content": "j", "parent_index": 5, "subtype": "inline"}, {"bbox": [190, 391, 422, 409], "content": "Q_{j}(\\lambda)\\equiv\\widetilde{Q}_{\\pi^{\\prime}j}(\\pi\\lambda)\\equiv\\widetilde{Q}_{\\pi j}(\\pi^{\\prime}\\lambda)\\qquad(\\mathrm{mod~1})~.", "parent_index": 6, "subtype": "interline"}, {"bbox": [528, 422, 539, 429], "content": "\\sigma_{\\ell}", "parent_index": 7, "subtype": "inline"}, {"bbox": [165, 436, 172, 442], "content": "\\pi", "parent_index": 7, "subtype": "inline"}, {"bbox": [225, 432, 300, 444], "content": "\\tilde{\\epsilon}_{\\ell}(\\pi\\lambda)=\\epsilon_{\\ell}(\\lambda)", "parent_index": 7, "subtype": "inline"}, {"bbox": [328, 430, 416, 445], "content": "\\pi(\\lambda^{(\\ell)})=(\\pi\\lambda)^{(\\ell)}", "parent_index": 7, "subtype": "inline"}, {"bbox": [150, 447, 158, 456], "content": "S", "parent_index": 7, "subtype": "inline"}, {"bbox": [111, 468, 507, 488], "content": "\\left\\varepsilon(\\lambda\\right)S_{\\lambda^{(\\ell)},\\mu}=\\sigma_{\\ell}S_{\\lambda\\mu}=\\sigma_{\\ell}\\widetilde{S}_{\\pi\\lambda,\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,\\widetilde{S}_{(\\pi\\lambda)^{(\\ell)},\\pi^{\\prime}\\mu}=\\widetilde{\\epsilon}_{\\ell}(\\pi\\lambda)\\,S_{\\pi^{-1}(\\pi\\lambda)^{(\\ell)},\\mu}\\ .", "parent_index": 8, "subtype": "interline"}, {"bbox": [108, 514, 199, 527], "content": "\\Gamma=\\{\\gamma_{1},...,\\gamma_{m}\\}", "parent_index": 9, "subtype": "inline"}, {"bbox": [218, 515, 233, 526], "content": "P_{+}", "parent_index": 9, "subtype": "inline"}, {"bbox": [325, 513, 369, 527], "content": "\\mathcal{R}(X_{r,k})", "parent_index": 9, "subtype": "inline"}, {"bbox": [270, 532, 281, 538], "content": "m", "parent_index": 9, "subtype": "inline"}, {"bbox": [396, 528, 474, 541], "content": "P_{\\lambda}(x_{1},\\ldots,x_{m})", "parent_index": 9, "subtype": "inline"}, {"bbox": [189, 552, 423, 582], "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=P_{\\lambda}(\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}},\\ldots,\\frac{S_{\\gamma_{m}\\mu}}{S_{0\\mu}})\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .", "parent_index": 10, "subtype": "interline"}, {"bbox": [93, 595, 126, 608], "content": "(\\pi,\\pi^{\\prime})", "parent_index": 11, "subtype": "inline"}, {"bbox": [164, 596, 172, 605], "content": "S", "parent_index": 11, "subtype": "inline"}, {"bbox": [379, 596, 418, 607], "content": "\\pi\\gamma=\\gamma", "parent_index": 11, "subtype": "inline"}, {"bbox": [457, 599, 465, 607], "content": "\\gamma", "parent_index": 11, "subtype": "inline"}, {"bbox": [124, 610, 132, 619], "content": "\\Gamma", "parent_index": 11, "subtype": "inline"}, {"bbox": [213, 610, 250, 622], "content": "\\lambda\\in P_{+}", "parent_index": 11, "subtype": "inline"}, {"bbox": [146, 632, 465, 663], "content": "{\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}}={\\frac{S_{\\pi\\lambda,\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}}=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1},\\pi^{\\prime}\\mu}}{S_{0,\\pi^{\\prime}\\mu}}},\\dots)=P_{\\pi\\lambda}({\\frac{S_{\\gamma_{1}\\mu}}{S_{0\\mu}}},\\dots)={\\frac{S_{\\pi\\lambda,\\mu}}{S_{0\\mu}}}", "parent_index": 12, "subtype": "interline"}, {"bbox": [106, 676, 144, 687], "content": "\\mu\\in P_{+}", "parent_index": 13, "subtype": "inline"}, {"bbox": [165, 676, 203, 685], "content": "\\pi\\lambda=\\lambda", "parent_index": 13, "subtype": "inline"}, {"bbox": [71, 75, 90, 88], "content": "\\mathrm{ch}_{\\overline{{{\\mu}}}}", "parent_index": 14, "subtype": "inline"}, {"bbox": [112, 75, 127, 86], "content": "X_{r}", "parent_index": 14, "subtype": "inline"}, {"bbox": [462, 74, 537, 87], "content": "\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}", "parent_index": 14, "subtype": "inline"}, {"bbox": [212, 92, 303, 105], "content": "\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}", "parent_index": 14, "subtype": "inline"}, {"bbox": [438, 88, 462, 103], "content": "X_{r}^{(1)}", "parent_index": 14, "subtype": "inline"}, {"bbox": [532, 93, 540, 102], "content": "k", "parent_index": 14, "subtype": "inline"}, {"bbox": [186, 108, 201, 119], "content": "P_{+}", "parent_index": 14, "subtype": "inline"}, {"bbox": [268, 107, 281, 118], "content": "\\Lambda_{i}", "parent_index": 14, "subtype": "inline"}, {"bbox": [412, 105, 478, 119], "content": "k\\geq\\operatorname*{max}_{i}a_{i}^{\\vee}", "parent_index": 14, "subtype": "inline"}, {"bbox": [362, 122, 385, 134], "content": "X_{r,k}", "parent_index": 14, "subtype": "inline"}, {"bbox": [400, 120, 492, 133], "content": "\\{\\Lambda_{1},\\ldots,\\Lambda_{r}\\}\\cap{\\cal P}_{+}", "parent_index": 14, "subtype": "inline"}, {"bbox": [310, 135, 336, 147], "content": "\\{\\Lambda_{1}\\}", "parent_index": 14, "subtype": "inline"}, {"bbox": [465, 135, 487, 148], "content": "A_{8,k}", "parent_index": 14, "subtype": "inline"}, {"bbox": [71, 150, 78, 159], "content": "k", "parent_index": 14, "subtype": "inline"}]
|
[]
|
|
2.3. Standard constructions of fusion-symmetries
Simple-currents are a large source of fusion-symmetries. Let $j$ be any simple-current of order ${\boldsymbol{n}}$ . Choose any number $a\in\{0,1,\ldots,n-1\}$ such that
$$
\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\ .
$$
Any solution to this defines a fusion-symmetry $\lambda\mapsto J^{n a Q_{j}\,(\lambda)}\lambda$ , which we shall denote $\pi[a]$ or $\pi_{j}[a]$ . Note that from (2.2b), (2.5b) and (2.5c) that any $\pi\,=\,\pi[a]$ , $a\in\mathbb{Z}$ , obeys the relation $N_{\pi\lambda,\pi\mu}^{\pi\nu}=\ N_{\lambda\mu}^{\nu}$ when $N_{\lambda\mu}^{\nu}\neq0$ (it would in fact be a fusion-endomorphism — see §2.2); the ‘gcd’ condition forces $\pi[a]$ to be a permutation. Choosing $b\equiv-a\,(n a Q_{j}(j)\!+\!1)^{-1}$ (mod ${\boldsymbol{n}}$ ), we find that $(\pi[a],\pi[b])$ is an $S$ -symmetry.
When the group of simple-currents is not cyclic, this construction can be generalised in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix $\left(a_{i j}\right)$ . We will meet these in 3.4.
We will call these simple-current automorphisms. The first examples of these were found by Bernard [2], and were generalised further in [31].
For any affine algebra $X_{r}^{(1)}$ and any sufficiently high level, we will see in the next section that its fusion-symmetries consist entirely of simple-current automorphisms and conjugations. For this reason, any other fusion-symmetry is called exceptional.
There is another general construction of fusion-symmetries, generalising $C$ , although it yields few new examples for the affine fusion rings. If the Galois automorphism $\sigma_{\ell}$ is such that $0^{(\ell)}$ is a simple-current $j$ — equivalently, that $\sigma_{\ell}(S_{00}^{2})=S_{00}^{2}$ — then the permutation
$$
\pi\{\ell\}:\lambda\mapsto J(\lambda^{(\ell)})
$$
is a fusion-symmetry. The simplest example is $\pi\{-1\}=C$ . We call $\pi\{\ell\}$ a Galois fusionsymmetry. A special case of these was given in [13]. To see that $\pi\{\ell\}$ works, note from
$$
\epsilon_{\ell}(\lambda)\,S_{\lambda^{(\ell)},0}=\sigma_{\ell}S_{\lambda0}=\epsilon_{\ell}(0)\,e^{2\pi\mathrm{i}\,Q_{j}(\lambda)}S_{\lambda0}
$$
that $\epsilon_{\ell}(\lambda)\,\epsilon_{\ell}(0)=e^{2\pi\mathrm{i}\,Q_{j}(\lambda)}$ . Hence
$$
S_{J\lambda^{(\ell)},\mu}=e^{2\pi\mathrm{i}\,Q_{j}(\mu)}\epsilon_{\ell}(\lambda)\,\sigma_{\ell}(S_{\lambda\mu})=e^{2\pi\mathrm{i}\,Q_{j}(\mu)}\,\epsilon_{\ell}(\lambda)\,\epsilon_{\ell}(\mu)\,S_{\lambda,\mu^{(\ell)}}=S_{\lambda,J\mu^{(\ell)}}
$$
and so $(\pi\{\ell\},\pi\{\ell\}^{-1})$ is an $S$ -symmetry. Incidentally, $J$ will always be order 1 or 2 because $2\,Q_{j}(\lambda)\in\mathbb{Z}$ for all $\lambda\in P_{+}$ .
Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries have analogues in arbitrary (i.e. not necessarily affine) fusion rings.
|
<html><body>
<p data-bbox="72 173 331 188">2.3. Standard constructions of fusion-symmetries </p>
<p data-bbox="70 194 540 225">Simple-currents are a large source of fusion-symmetries. Let $j$ be any simple-current of order ${\boldsymbol{n}}$ . Choose any number $a\in\{0,1,\ldots,n-1\}$ such that </p>
<div class="equation" data-bbox="239 239 372 252">$$
\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\ .
$$</div>
<p data-bbox="70 261 541 336">Any solution to this defines a fusion-symmetry $\lambda\mapsto J^{n a Q_{j}\,(\lambda)}\lambda$ , which we shall denote $\pi[a]$ or $\pi_{j}[a]$ . Note that from (2.2b), (2.5b) and (2.5c) that any $\pi\,=\,\pi[a]$ , $a\in\mathbb{Z}$ , obeys the relation $N_{\pi\lambda,\pi\mu}^{\pi\nu}=\ N_{\lambda\mu}^{\nu}$ when $N_{\lambda\mu}^{\nu}\neq0$ (it would in fact be a fusion-endomorphism — see §2.2); the ‘gcd’ condition forces $\pi[a]$ to be a permutation. Choosing $b\equiv-a\,(n a Q_{j}(j)\!+\!1)^{-1}$ (mod ${\boldsymbol{n}}$ ), we find that $(\pi[a],\pi[b])$ is an $S$ -symmetry. </p>
<p data-bbox="71 336 541 378">When the group of simple-currents is not cyclic, this construction can be generalised in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix $\left(a_{i j}\right)$ . We will meet these in 3.4. </p>
<p data-bbox="70 380 541 408">We will call these simple-current automorphisms. The first examples of these were found by Bernard [2], and were generalised further in [31]. </p>
<p data-bbox="70 409 541 453">For any affine algebra $X_{r}^{(1)}$ and any sufficiently high level, we will see in the next section that its fusion-symmetries consist entirely of simple-current automorphisms and conjugations. For this reason, any other fusion-symmetry is called exceptional. </p>
<p data-bbox="70 453 541 497">There is another general construction of fusion-symmetries, generalising $C$ , although it yields few new examples for the affine fusion rings. If the Galois automorphism $\sigma_{\ell}$ is such that $0^{(\ell)}$ is a simple-current $j$ — equivalently, that $\sigma_{\ell}(S_{00}^{2})=S_{00}^{2}$ — then the permutation </p>
<div class="equation" data-bbox="257 509 354 524">$$
\pi\{\ell\}:\lambda\mapsto J(\lambda^{(\ell)})
$$</div>
<p data-bbox="69 534 540 564">is a fusion-symmetry. The simplest example is $\pi\{-1\}=C$ . We call $\pi\{\ell\}$ a Galois fusionsymmetry. A special case of these was given in [13]. To see that $\pi\{\ell\}$ works, note from </p>
<div class="equation" data-bbox="198 576 412 593">$$
\epsilon_{\ell}(\lambda)\,S_{\lambda^{(\ell)},0}=\sigma_{\ell}S_{\lambda0}=\epsilon_{\ell}(0)\,e^{2\pi\mathrm{i}\,Q_{j}(\lambda)}S_{\lambda0}
$$</div>
<p data-bbox="70 603 254 619">that $\epsilon_{\ell}(\lambda)\,\epsilon_{\ell}(0)=e^{2\pi\mathrm{i}\,Q_{j}(\lambda)}$ . Hence </p>
<div class="equation" data-bbox="116 631 493 648">$$
S_{J\lambda^{(\ell)},\mu}=e^{2\pi\mathrm{i}\,Q_{j}(\mu)}\epsilon_{\ell}(\lambda)\,\sigma_{\ell}(S_{\lambda\mu})=e^{2\pi\mathrm{i}\,Q_{j}(\mu)}\,\epsilon_{\ell}(\lambda)\,\epsilon_{\ell}(\mu)\,S_{\lambda,\mu^{(\ell)}}=S_{\lambda,J\mu^{(\ell)}}
$$</div>
<p data-bbox="70 657 542 686">and so $(\pi\{\ell\},\pi\{\ell\}^{-1})$ is an $S$ -symmetry. Incidentally, $J$ will always be order 1 or 2 because $2\,Q_{j}(\lambda)\in\mathbb{Z}$ for all $\lambda\in P_{+}$ . </p>
<p data-bbox="70 687 542 715">Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries have analogues in arbitrary (i.e. not necessarily affine) fusion rings. </p>
</body></html>
|
0002044v1
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[{"type": "text", "text": "", "page_idx": 7}, {"type": "text", "text": "2.3. Standard constructions of fusion-symmetries ", "page_idx": 7}, {"type": "text", "text": "Simple-currents are a large source of fusion-symmetries. Let $j$ be any simple-current of order ${\\boldsymbol{n}}$ . Choose any number $a\\in\\{0,1,\\ldots,n-1\\}$ such that ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\\ .\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Any solution to this defines a fusion-symmetry $\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda$ , which we shall denote $\\pi[a]$ or $\\pi_{j}[a]$ . Note that from (2.2b), (2.5b) and (2.5c) that any $\\pi\\,=\\,\\pi[a]$ , $a\\in\\mathbb{Z}$ , obeys the relation $N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}$ when $N_{\\lambda\\mu}^{\\nu}\\neq0$ (it would in fact be a fusion-endomorphism — see §2.2); the ‘gcd’ condition forces $\\pi[a]$ to be a permutation. Choosing $b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}$ (mod ${\\boldsymbol{n}}$ ), we find that $(\\pi[a],\\pi[b])$ is an $S$ -symmetry. ", "page_idx": 7}, {"type": "text", "text": "When the group of simple-currents is not cyclic, this construction can be generalised in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix $\\left(a_{i j}\\right)$ . We will meet these in 3.4. ", "page_idx": 7}, {"type": "text", "text": "We will call these simple-current automorphisms. The first examples of these were found by Bernard [2], and were generalised further in [31]. ", "page_idx": 7}, {"type": "text", "text": "For any affine algebra $X_{r}^{(1)}$ and any sufficiently high level, we will see in the next section that its fusion-symmetries consist entirely of simple-current automorphisms and conjugations. For this reason, any other fusion-symmetry is called exceptional. ", "page_idx": 7}, {"type": "text", "text": "There is another general construction of fusion-symmetries, generalising $C$ , although it yields few new examples for the affine fusion rings. If the Galois automorphism $\\sigma_{\\ell}$ is such that $0^{(\\ell)}$ is a simple-current $j$ — equivalently, that $\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}$ — then the permutation ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "is a fusion-symmetry. The simplest example is $\\pi\\{-1\\}=C$ . We call $\\pi\\{\\ell\\}$ a Galois fusionsymmetry. A special case of these was given in [13]. To see that $\\pi\\{\\ell\\}$ works, note from ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "that $\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}$ . Hence ", "page_idx": 7}, {"type": "equation", "text": "$$\nS_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and so $(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})$ is an $S$ -symmetry. Incidentally, $J$ will always be order 1 or 2 because $2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}$ for all $\\lambda\\in P_{+}$ . ", "page_idx": 7}, {"type": "text", "text": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries have analogues in arbitrary (i.e. not necessarily affine) fusion rings. ", "page_idx": 7}]
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{"preproc_blocks": [{"type": "text", "bbox": [70, 70, 541, 160], "lines": [{"bbox": [71, 71, 542, 91], "spans": [{"bbox": [71, 75, 90, 88], "score": 0.84, "content": "\\mathrm{ch}_{\\overline{{{\\mu}}}}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [90, 71, 111, 91], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [112, 75, 127, 86], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [128, 71, 462, 91], "score": 1.0, "content": " can be written as a polynomial in the fundamental characters ", "type": "text"}, {"bbox": [462, 74, 537, 87], "score": 0.87, "content": "\\mathrm{ch}_{\\Lambda_{1}},\\ldots,\\mathrm{ch}_{\\Lambda_{r}}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [537, 71, 542, 91], "score": 1.0, "content": ",", "type": "text"}], "index": 0}, {"bbox": [69, 88, 540, 106], "spans": [{"bbox": [69, 88, 212, 106], "score": 1.0, "content": "we know from (2.1b) that ", "type": "text"}, {"bbox": [212, 92, 303, 105], "score": 0.93, "content": "\\Gamma=\\{\\Lambda_{1},\\dots,\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [304, 88, 438, 106], "score": 1.0, "content": " is a fusion-generator for ", "type": "text"}, {"bbox": [438, 88, 462, 103], "score": 0.92, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [463, 88, 532, 106], "score": 1.0, "content": "at any level ", "type": "text"}, {"bbox": [532, 93, 540, 102], "score": 0.84, "content": "k", "type": "inline_equation", "height": 9, "width": 8}], "index": 1}, {"bbox": [70, 105, 541, 120], "spans": [{"bbox": [70, 105, 185, 120], "score": 1.0, "content": "sufficiently large that ", "type": "text"}, {"bbox": [186, 108, 201, 119], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [202, 105, 268, 120], "score": 1.0, "content": " contains all ", "type": "text"}, {"bbox": [268, 107, 281, 118], "score": 0.9, "content": "\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [281, 105, 412, 120], "score": 1.0, "content": " (in other words, for any ", "type": "text"}, {"bbox": [412, 105, 478, 119], "score": 0.91, "content": "k\\geq\\operatorname*{max}_{i}a_{i}^{\\vee}", "type": "inline_equation", "height": 14, "width": 66}, {"bbox": [479, 105, 541, 120], "score": 1.0, "content": "). In fact, it", "type": "text"}], "index": 2}, {"bbox": [70, 119, 541, 135], "spans": [{"bbox": [70, 119, 362, 135], "score": 1.0, "content": "is easy to show [18] that a fusion-generator valid for any ", "type": "text"}, {"bbox": [362, 122, 385, 134], "score": 0.93, "content": "X_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [385, 119, 399, 135], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [400, 120, 492, 133], "score": 0.92, "content": "\\{\\Lambda_{1},\\ldots,\\Lambda_{r}\\}\\cap{\\cal P}_{+}", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [492, 119, 541, 135], "score": 1.0, "content": ". Smaller", "type": "text"}], "index": 3}, {"bbox": [70, 133, 541, 149], "spans": [{"bbox": [70, 133, 310, 149], "score": 1.0, "content": "fusion-generators usually exist — for example ", "type": "text"}, {"bbox": [310, 135, 336, 147], "score": 0.94, "content": "\\{\\Lambda_{1}\\}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [336, 133, 465, 149], "score": 1.0, "content": " is a fusion-generator for ", "type": "text"}, {"bbox": [465, 135, 487, 148], "score": 0.91, "content": "A_{8,k}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [488, 133, 541, 149], "score": 1.0, "content": " whenever", "type": "text"}], "index": 4}, {"bbox": [71, 149, 215, 163], "spans": [{"bbox": [71, 150, 78, 159], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 149, 215, 163], "score": 1.0, "content": " is even and coprime to 3.", "type": "text"}], "index": 5}], "index": 2.5}, {"type": "text", "bbox": [72, 173, 331, 188], "lines": [{"bbox": [72, 177, 330, 189], "spans": [{"bbox": [72, 177, 330, 189], "score": 1.0, "content": "2.3. Standard constructions of fusion-symmetries", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [70, 194, 540, 225], "lines": [{"bbox": [95, 198, 540, 212], "spans": [{"bbox": [95, 198, 415, 212], "score": 1.0, "content": "Simple-currents are a large source of fusion-symmetries. Let ", "type": "text"}, {"bbox": [416, 199, 421, 210], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [421, 198, 540, 212], "score": 1.0, "content": " be any simple-current", "type": "text"}], "index": 7}, {"bbox": [70, 211, 399, 226], "spans": [{"bbox": [70, 211, 116, 226], "score": 1.0, "content": "of order ", "type": "text"}, {"bbox": [116, 217, 123, 222], "score": 0.89, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [124, 211, 239, 226], "score": 1.0, "content": ". Choose any number ", "type": "text"}, {"bbox": [240, 212, 344, 225], "score": 0.93, "content": "a\\in\\{0,1,\\ldots,n-1\\}", "type": "inline_equation", "height": 13, "width": 104}, {"bbox": [345, 211, 399, 226], "score": 1.0, "content": " such that", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "interline_equation", "bbox": [239, 239, 372, 252], "lines": [{"bbox": [239, 239, 372, 252], "spans": [{"bbox": [239, 239, 372, 252], "score": 0.88, "content": "\\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [70, 261, 541, 336], "lines": [{"bbox": [70, 262, 540, 280], "spans": [{"bbox": [70, 262, 317, 280], "score": 1.0, "content": "Any solution to this defines a fusion-symmetry ", "type": "text"}, {"bbox": [318, 264, 396, 276], "score": 0.93, "content": "\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [396, 262, 519, 280], "score": 1.0, "content": ", which we shall denote ", "type": "text"}, {"bbox": [519, 266, 540, 279], "score": 0.91, "content": "\\pi[a]", "type": "inline_equation", "height": 13, "width": 21}], "index": 10}, {"bbox": [70, 279, 541, 294], "spans": [{"bbox": [70, 279, 86, 294], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [87, 280, 112, 293], "score": 0.93, "content": "\\pi_{j}[a]", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [112, 279, 393, 294], "score": 1.0, "content": ". Note that from (2.2b), (2.5b) and (2.5c) that any ", "type": "text"}, {"bbox": [393, 280, 441, 293], "score": 0.92, "content": "\\pi\\,=\\,\\pi[a]", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [441, 279, 448, 294], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [448, 281, 480, 290], "score": 0.87, "content": "a\\in\\mathbb{Z}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [481, 279, 541, 294], "score": 1.0, "content": ", obeys the", "type": "text"}], "index": 11}, {"bbox": [69, 292, 543, 311], "spans": [{"bbox": [69, 292, 115, 311], "score": 1.0, "content": "relation ", "type": "text"}, {"bbox": [115, 295, 193, 309], "score": 0.94, "content": "N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [194, 292, 228, 311], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [228, 295, 273, 309], "score": 0.94, "content": "N_{\\lambda\\mu}^{\\nu}\\neq0", "type": "inline_equation", "height": 14, "width": 45}, {"bbox": [273, 292, 543, 311], "score": 1.0, "content": " (it would in fact be a fusion-endomorphism — see", "type": "text"}], "index": 12}, {"bbox": [69, 308, 539, 325], "spans": [{"bbox": [69, 308, 235, 325], "score": 1.0, "content": "§2.2); the ‘gcd’ condition forces ", "type": "text"}, {"bbox": [235, 311, 255, 323], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [255, 308, 418, 325], "score": 1.0, "content": " to be a permutation. Choosing ", "type": "text"}, {"bbox": [419, 309, 539, 324], "score": 0.92, "content": "b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}", "type": "inline_equation", "height": 15, "width": 120}], "index": 13}, {"bbox": [71, 322, 343, 339], "spans": [{"bbox": [71, 322, 102, 339], "score": 1.0, "content": "(mod ", "type": "text"}, {"bbox": [103, 329, 110, 334], "score": 0.82, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [111, 322, 189, 339], "score": 1.0, "content": "), we find that ", "type": "text"}, {"bbox": [189, 324, 244, 337], "score": 0.94, "content": "(\\pi[a],\\pi[b])", "type": "inline_equation", "height": 13, "width": 55}, {"bbox": [244, 322, 275, 339], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [276, 325, 284, 334], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [284, 322, 343, 339], "score": 1.0, "content": "-symmetry.", "type": "text"}], "index": 14}], "index": 12}, {"type": "text", "bbox": [71, 336, 541, 378], "lines": [{"bbox": [95, 338, 541, 353], "spans": [{"bbox": [95, 338, 541, 353], "score": 1.0, "content": "When the group of simple-currents is not cyclic, this construction can be generalised", "type": "text"}], "index": 15}, {"bbox": [69, 352, 540, 368], "spans": [{"bbox": [69, 352, 512, 368], "score": 1.0, "content": "in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix ", "type": "text"}, {"bbox": [512, 353, 536, 366], "score": 0.9, "content": "\\left(a_{i j}\\right)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [537, 352, 540, 368], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [71, 367, 213, 380], "spans": [{"bbox": [71, 367, 213, 380], "score": 1.0, "content": "We will meet these in 3.4.", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [70, 380, 541, 408], "lines": [{"bbox": [94, 380, 541, 396], "spans": [{"bbox": [94, 380, 541, 396], "score": 1.0, "content": "We will call these simple-current automorphisms. The first examples of these were", "type": "text"}], "index": 18}, {"bbox": [72, 396, 376, 410], "spans": [{"bbox": [72, 396, 376, 410], "score": 1.0, "content": "found by Bernard [2], and were generalised further in [31].", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "text", "bbox": [70, 409, 541, 453], "lines": [{"bbox": [92, 407, 544, 429], "spans": [{"bbox": [92, 407, 217, 429], "score": 1.0, "content": "For any affine algebra ", "type": "text"}, {"bbox": [218, 410, 242, 424], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [242, 407, 544, 429], "score": 1.0, "content": "and any sufficiently high level, we will see in the next", "type": "text"}], "index": 20}, {"bbox": [70, 426, 542, 442], "spans": [{"bbox": [70, 426, 542, 442], "score": 1.0, "content": "section that its fusion-symmetries consist entirely of simple-current automorphisms and", "type": "text"}], "index": 21}, {"bbox": [72, 442, 483, 455], "spans": [{"bbox": [72, 442, 483, 455], "score": 1.0, "content": "conjugations. For this reason, any other fusion-symmetry is called exceptional.", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [70, 453, 541, 497], "lines": [{"bbox": [94, 454, 541, 470], "spans": [{"bbox": [94, 454, 476, 470], "score": 1.0, "content": "There is another general construction of fusion-symmetries, generalising ", "type": "text"}, {"bbox": [477, 457, 486, 466], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [487, 454, 541, 470], "score": 1.0, "content": ", although", "type": "text"}], "index": 23}, {"bbox": [69, 469, 541, 484], "spans": [{"bbox": [69, 469, 490, 484], "score": 1.0, "content": "it yields few new examples for the affine fusion rings. If the Galois automorphism ", "type": "text"}, {"bbox": [491, 474, 502, 482], "score": 0.89, "content": "\\sigma_{\\ell}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [502, 469, 541, 484], "score": 1.0, "content": "is such", "type": "text"}], "index": 24}, {"bbox": [69, 481, 542, 501], "spans": [{"bbox": [69, 481, 96, 501], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [97, 483, 115, 495], "score": 0.91, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [115, 481, 218, 501], "score": 1.0, "content": " is a simple-current ", "type": "text"}, {"bbox": [218, 486, 224, 497], "score": 0.85, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [224, 481, 336, 501], "score": 1.0, "content": " — equivalently, that ", "type": "text"}, {"bbox": [337, 484, 409, 497], "score": 0.94, "content": "\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [409, 481, 542, 501], "score": 1.0, "content": " — then the permutation", "type": "text"}], "index": 25}], "index": 24}, {"type": "interline_equation", "bbox": [257, 509, 354, 524], "lines": [{"bbox": [257, 509, 354, 524], "spans": [{"bbox": [257, 509, 354, 524], "score": 0.93, "content": "\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [69, 534, 540, 564], "lines": [{"bbox": [68, 536, 541, 553], "spans": [{"bbox": [68, 536, 317, 553], "score": 1.0, "content": "is a fusion-symmetry. The simplest example is ", "type": "text"}, {"bbox": [317, 538, 377, 551], "score": 0.96, "content": "\\pi\\{-1\\}=C", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [378, 536, 428, 553], "score": 1.0, "content": ". We call ", "type": "text"}, {"bbox": [428, 538, 453, 551], "score": 0.93, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [453, 536, 541, 553], "score": 1.0, "content": " a Galois fusion-", "type": "text"}], "index": 27}, {"bbox": [71, 552, 528, 566], "spans": [{"bbox": [71, 552, 410, 566], "score": 1.0, "content": "symmetry. A special case of these was given in [13]. To see that ", "type": "text"}, {"bbox": [410, 552, 434, 565], "score": 0.93, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [435, 552, 528, 566], "score": 1.0, "content": " works, note from", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "interline_equation", "bbox": [198, 576, 412, 593], "lines": [{"bbox": [198, 576, 412, 593], "spans": [{"bbox": [198, 576, 412, 593], "score": 0.89, "content": "\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [70, 603, 254, 619], "lines": [{"bbox": [69, 603, 254, 620], "spans": [{"bbox": [69, 603, 97, 620], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [97, 606, 212, 620], "score": 0.93, "content": "\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}", "type": "inline_equation", "height": 14, "width": 115}, {"bbox": [212, 603, 254, 620], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [116, 631, 493, 648], "lines": [{"bbox": [116, 631, 493, 648], "spans": [{"bbox": [116, 631, 493, 648], "score": 0.91, "content": "S_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [70, 657, 542, 686], "lines": [{"bbox": [70, 658, 542, 675], "spans": [{"bbox": [70, 658, 107, 675], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [107, 660, 183, 673], "score": 0.93, "content": "(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [184, 658, 213, 675], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [213, 661, 222, 670], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [222, 658, 351, 675], "score": 1.0, "content": "-symmetry. Incidentally, ", "type": "text"}, {"bbox": [352, 661, 360, 670], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [360, 658, 542, 675], "score": 1.0, "content": " will always be order 1 or 2 because", "type": "text"}], "index": 32}, {"bbox": [70, 673, 214, 689], "spans": [{"bbox": [70, 675, 133, 688], "score": 0.92, "content": "2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [133, 673, 171, 689], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [172, 676, 209, 687], "score": 0.92, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [209, 673, 214, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32.5}, {"type": "text", "bbox": [70, 687, 542, 715], "lines": [{"bbox": [93, 687, 542, 704], "spans": [{"bbox": [93, 687, 542, 704], "score": 1.0, "content": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries", "type": "text"}], "index": 34}, {"bbox": [70, 702, 424, 719], "spans": [{"bbox": [70, 702, 424, 719], "score": 1.0, "content": "have analogues in arbitrary (i.e. not necessarily affine) fusion rings.", "type": "text"}], "index": 35}], "index": 34.5}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [239, 239, 372, 252], "lines": [{"bbox": [239, 239, 372, 252], "spans": [{"bbox": [239, 239, 372, 252], "score": 0.88, "content": "\\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [257, 509, 354, 524], "lines": [{"bbox": [257, 509, 354, 524], "spans": [{"bbox": [257, 509, 354, 524], "score": 0.93, "content": "\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "interline_equation", "bbox": [198, 576, 412, 593], "lines": [{"bbox": [198, 576, 412, 593], "spans": [{"bbox": [198, 576, 412, 593], "score": 0.89, "content": "\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "interline_equation", "bbox": [116, 631, 493, 648], "lines": [{"bbox": [116, 631, 493, 648], "spans": [{"bbox": [116, 631, 493, 648], "score": 0.91, "content": "S_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 31}], "index": 31}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 160], "lines": [], "index": 2.5, "bbox_fs": [69, 71, 542, 163], "lines_deleted": true}, {"type": "text", "bbox": [72, 173, 331, 188], "lines": [{"bbox": [72, 177, 330, 189], "spans": [{"bbox": [72, 177, 330, 189], "score": 1.0, "content": "2.3. Standard constructions of fusion-symmetries", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [72, 177, 330, 189]}, {"type": "text", "bbox": [70, 194, 540, 225], "lines": [{"bbox": [95, 198, 540, 212], "spans": [{"bbox": [95, 198, 415, 212], "score": 1.0, "content": "Simple-currents are a large source of fusion-symmetries. Let ", "type": "text"}, {"bbox": [416, 199, 421, 210], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [421, 198, 540, 212], "score": 1.0, "content": " be any simple-current", "type": "text"}], "index": 7}, {"bbox": [70, 211, 399, 226], "spans": [{"bbox": [70, 211, 116, 226], "score": 1.0, "content": "of order ", "type": "text"}, {"bbox": [116, 217, 123, 222], "score": 0.89, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [124, 211, 239, 226], "score": 1.0, "content": ". Choose any number ", "type": "text"}, {"bbox": [240, 212, 344, 225], "score": 0.93, "content": "a\\in\\{0,1,\\ldots,n-1\\}", "type": "inline_equation", "height": 13, "width": 104}, {"bbox": [345, 211, 399, 226], "score": 1.0, "content": " such that", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [70, 198, 540, 226]}, {"type": "interline_equation", "bbox": [239, 239, 372, 252], "lines": [{"bbox": [239, 239, 372, 252], "spans": [{"bbox": [239, 239, 372, 252], "score": 0.88, "content": "\\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [70, 261, 541, 336], "lines": [{"bbox": [70, 262, 540, 280], "spans": [{"bbox": [70, 262, 317, 280], "score": 1.0, "content": "Any solution to this defines a fusion-symmetry ", "type": "text"}, {"bbox": [318, 264, 396, 276], "score": 0.93, "content": "\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [396, 262, 519, 280], "score": 1.0, "content": ", which we shall denote ", "type": "text"}, {"bbox": [519, 266, 540, 279], "score": 0.91, "content": "\\pi[a]", "type": "inline_equation", "height": 13, "width": 21}], "index": 10}, {"bbox": [70, 279, 541, 294], "spans": [{"bbox": [70, 279, 86, 294], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [87, 280, 112, 293], "score": 0.93, "content": "\\pi_{j}[a]", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [112, 279, 393, 294], "score": 1.0, "content": ". Note that from (2.2b), (2.5b) and (2.5c) that any ", "type": "text"}, {"bbox": [393, 280, 441, 293], "score": 0.92, "content": "\\pi\\,=\\,\\pi[a]", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [441, 279, 448, 294], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [448, 281, 480, 290], "score": 0.87, "content": "a\\in\\mathbb{Z}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [481, 279, 541, 294], "score": 1.0, "content": ", obeys the", "type": "text"}], "index": 11}, {"bbox": [69, 292, 543, 311], "spans": [{"bbox": [69, 292, 115, 311], "score": 1.0, "content": "relation ", "type": "text"}, {"bbox": [115, 295, 193, 309], "score": 0.94, "content": "N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [194, 292, 228, 311], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [228, 295, 273, 309], "score": 0.94, "content": "N_{\\lambda\\mu}^{\\nu}\\neq0", "type": "inline_equation", "height": 14, "width": 45}, {"bbox": [273, 292, 543, 311], "score": 1.0, "content": " (it would in fact be a fusion-endomorphism — see", "type": "text"}], "index": 12}, {"bbox": [69, 308, 539, 325], "spans": [{"bbox": [69, 308, 235, 325], "score": 1.0, "content": "§2.2); the ‘gcd’ condition forces ", "type": "text"}, {"bbox": [235, 311, 255, 323], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [255, 308, 418, 325], "score": 1.0, "content": " to be a permutation. Choosing ", "type": "text"}, {"bbox": [419, 309, 539, 324], "score": 0.92, "content": "b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}", "type": "inline_equation", "height": 15, "width": 120}], "index": 13}, {"bbox": [71, 322, 343, 339], "spans": [{"bbox": [71, 322, 102, 339], "score": 1.0, "content": "(mod ", "type": "text"}, {"bbox": [103, 329, 110, 334], "score": 0.82, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [111, 322, 189, 339], "score": 1.0, "content": "), we find that ", "type": "text"}, {"bbox": [189, 324, 244, 337], "score": 0.94, "content": "(\\pi[a],\\pi[b])", "type": "inline_equation", "height": 13, "width": 55}, {"bbox": [244, 322, 275, 339], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [276, 325, 284, 334], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [284, 322, 343, 339], "score": 1.0, "content": "-symmetry.", "type": "text"}], "index": 14}], "index": 12, "bbox_fs": [69, 262, 543, 339]}, {"type": "text", "bbox": [71, 336, 541, 378], "lines": [{"bbox": [95, 338, 541, 353], "spans": [{"bbox": [95, 338, 541, 353], "score": 1.0, "content": "When the group of simple-currents is not cyclic, this construction can be generalised", "type": "text"}], "index": 15}, {"bbox": [69, 352, 540, 368], "spans": [{"bbox": [69, 352, 512, 368], "score": 1.0, "content": "in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix ", "type": "text"}, {"bbox": [512, 353, 536, 366], "score": 0.9, "content": "\\left(a_{i j}\\right)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [537, 352, 540, 368], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [71, 367, 213, 380], "spans": [{"bbox": [71, 367, 213, 380], "score": 1.0, "content": "We will meet these in 3.4.", "type": "text"}], "index": 17}], "index": 16, "bbox_fs": [69, 338, 541, 380]}, {"type": "text", "bbox": [70, 380, 541, 408], "lines": [{"bbox": [94, 380, 541, 396], "spans": [{"bbox": [94, 380, 541, 396], "score": 1.0, "content": "We will call these simple-current automorphisms. The first examples of these were", "type": "text"}], "index": 18}, {"bbox": [72, 396, 376, 410], "spans": [{"bbox": [72, 396, 376, 410], "score": 1.0, "content": "found by Bernard [2], and were generalised further in [31].", "type": "text"}], "index": 19}], "index": 18.5, "bbox_fs": [72, 380, 541, 410]}, {"type": "text", "bbox": [70, 409, 541, 453], "lines": [{"bbox": [92, 407, 544, 429], "spans": [{"bbox": [92, 407, 217, 429], "score": 1.0, "content": "For any affine algebra ", "type": "text"}, {"bbox": [218, 410, 242, 424], "score": 0.93, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [242, 407, 544, 429], "score": 1.0, "content": "and any sufficiently high level, we will see in the next", "type": "text"}], "index": 20}, {"bbox": [70, 426, 542, 442], "spans": [{"bbox": [70, 426, 542, 442], "score": 1.0, "content": "section that its fusion-symmetries consist entirely of simple-current automorphisms and", "type": "text"}], "index": 21}, {"bbox": [72, 442, 483, 455], "spans": [{"bbox": [72, 442, 483, 455], "score": 1.0, "content": "conjugations. For this reason, any other fusion-symmetry is called exceptional.", "type": "text"}], "index": 22}], "index": 21, "bbox_fs": [70, 407, 544, 455]}, {"type": "text", "bbox": [70, 453, 541, 497], "lines": [{"bbox": [94, 454, 541, 470], "spans": [{"bbox": [94, 454, 476, 470], "score": 1.0, "content": "There is another general construction of fusion-symmetries, generalising ", "type": "text"}, {"bbox": [477, 457, 486, 466], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [487, 454, 541, 470], "score": 1.0, "content": ", although", "type": "text"}], "index": 23}, {"bbox": [69, 469, 541, 484], "spans": [{"bbox": [69, 469, 490, 484], "score": 1.0, "content": "it yields few new examples for the affine fusion rings. If the Galois automorphism ", "type": "text"}, {"bbox": [491, 474, 502, 482], "score": 0.89, "content": "\\sigma_{\\ell}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [502, 469, 541, 484], "score": 1.0, "content": "is such", "type": "text"}], "index": 24}, {"bbox": [69, 481, 542, 501], "spans": [{"bbox": [69, 481, 96, 501], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [97, 483, 115, 495], "score": 0.91, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [115, 481, 218, 501], "score": 1.0, "content": " is a simple-current ", "type": "text"}, {"bbox": [218, 486, 224, 497], "score": 0.85, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [224, 481, 336, 501], "score": 1.0, "content": " — equivalently, that ", "type": "text"}, {"bbox": [337, 484, 409, 497], "score": 0.94, "content": "\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [409, 481, 542, 501], "score": 1.0, "content": " — then the permutation", "type": "text"}], "index": 25}], "index": 24, "bbox_fs": [69, 454, 542, 501]}, {"type": "interline_equation", "bbox": [257, 509, 354, 524], "lines": [{"bbox": [257, 509, 354, 524], "spans": [{"bbox": [257, 509, 354, 524], "score": 0.93, "content": "\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [69, 534, 540, 564], "lines": [{"bbox": [68, 536, 541, 553], "spans": [{"bbox": [68, 536, 317, 553], "score": 1.0, "content": "is a fusion-symmetry. The simplest example is ", "type": "text"}, {"bbox": [317, 538, 377, 551], "score": 0.96, "content": "\\pi\\{-1\\}=C", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [378, 536, 428, 553], "score": 1.0, "content": ". We call ", "type": "text"}, {"bbox": [428, 538, 453, 551], "score": 0.93, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [453, 536, 541, 553], "score": 1.0, "content": " a Galois fusion-", "type": "text"}], "index": 27}, {"bbox": [71, 552, 528, 566], "spans": [{"bbox": [71, 552, 410, 566], "score": 1.0, "content": "symmetry. A special case of these was given in [13]. To see that ", "type": "text"}, {"bbox": [410, 552, 434, 565], "score": 0.93, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [435, 552, 528, 566], "score": 1.0, "content": " works, note from", "type": "text"}], "index": 28}], "index": 27.5, "bbox_fs": [68, 536, 541, 566]}, {"type": "interline_equation", "bbox": [198, 576, 412, 593], "lines": [{"bbox": [198, 576, 412, 593], "spans": [{"bbox": [198, 576, 412, 593], "score": 0.89, "content": "\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [70, 603, 254, 619], "lines": [{"bbox": [69, 603, 254, 620], "spans": [{"bbox": [69, 603, 97, 620], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [97, 606, 212, 620], "score": 0.93, "content": "\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}", "type": "inline_equation", "height": 14, "width": 115}, {"bbox": [212, 603, 254, 620], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 30}], "index": 30, "bbox_fs": [69, 603, 254, 620]}, {"type": "interline_equation", "bbox": [116, 631, 493, 648], "lines": [{"bbox": [116, 631, 493, 648], "spans": [{"bbox": [116, 631, 493, 648], "score": 0.91, "content": "S_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [70, 657, 542, 686], "lines": [{"bbox": [70, 658, 542, 675], "spans": [{"bbox": [70, 658, 107, 675], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [107, 660, 183, 673], "score": 0.93, "content": "(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [184, 658, 213, 675], "score": 1.0, "content": " is an ", "type": "text"}, {"bbox": [213, 661, 222, 670], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [222, 658, 351, 675], "score": 1.0, "content": "-symmetry. Incidentally, ", "type": "text"}, {"bbox": [352, 661, 360, 670], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [360, 658, 542, 675], "score": 1.0, "content": " will always be order 1 or 2 because", "type": "text"}], "index": 32}, {"bbox": [70, 673, 214, 689], "spans": [{"bbox": [70, 675, 133, 688], "score": 0.92, "content": "2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [133, 673, 171, 689], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [172, 676, 209, 687], "score": 0.92, "content": "\\lambda\\in P_{+}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [209, 673, 214, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32.5, "bbox_fs": [70, 658, 542, 689]}, {"type": "text", "bbox": [70, 687, 542, 715], "lines": [{"bbox": [93, 687, 542, 704], "spans": [{"bbox": [93, 687, 542, 704], "score": 1.0, "content": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries", "type": "text"}], "index": 34}, {"bbox": [70, 702, 424, 719], "spans": [{"bbox": [70, 702, 424, 719], "score": 1.0, "content": "have analogues in arbitrary (i.e. not necessarily affine) fusion rings.", "type": "text"}], "index": 35}], "index": 34.5, "bbox_fs": [70, 687, 542, 719]}]}
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[{"type": "text", "bbox": [70, 70, 541, 160], "content": "", "index": 0}, {"type": "text", "bbox": [72, 173, 331, 188], "content": "2.3. Standard constructions of fusion-symmetries", "index": 1}, {"type": "text", "bbox": [70, 194, 540, 225], "content": "Simple-currents are a large source of fusion-symmetries. Let be any simple-current of order . Choose any number such that", "index": 2}, {"type": "interline_equation", "bbox": [239, 239, 372, 252], "content": "", "index": 3}, {"type": "text", "bbox": [70, 261, 541, 336], "content": "Any solution to this defines a fusion-symmetry , which we shall denote or . Note that from (2.2b), (2.5b) and (2.5c) that any , , obeys the relation when (it would in fact be a fusion-endomorphism — see §2.2); the ‘gcd’ condition forces to be a permutation. Choosing (mod ), we find that is an -symmetry.", "index": 4}, {"type": "text", "bbox": [71, 336, 541, 378], "content": "When the group of simple-currents is not cyclic, this construction can be generalised in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix . We will meet these in 3.4.", "index": 5}, {"type": "text", "bbox": [70, 380, 541, 408], "content": "We will call these simple-current automorphisms. The first examples of these were found by Bernard [2], and were generalised further in [31].", "index": 6}, {"type": "text", "bbox": [70, 409, 541, 453], "content": "For any affine algebra and any sufficiently high level, we will see in the next section that its fusion-symmetries consist entirely of simple-current automorphisms and conjugations. For this reason, any other fusion-symmetry is called exceptional.", "index": 7}, {"type": "text", "bbox": [70, 453, 541, 497], "content": "There is another general construction of fusion-symmetries, generalising , although it yields few new examples for the affine fusion rings. If the Galois automorphism is such that is a simple-current — equivalently, that — then the permutation", "index": 8}, {"type": "interline_equation", "bbox": [257, 509, 354, 524], "content": "", "index": 9}, {"type": "text", "bbox": [69, 534, 540, 564], "content": "is a fusion-symmetry. The simplest example is . We call a Galois fusion- symmetry. A special case of these was given in [13]. To see that works, note from", "index": 10}, {"type": "interline_equation", "bbox": [198, 576, 412, 593], "content": "", "index": 11}, {"type": "text", "bbox": [70, 603, 254, 619], "content": "that . Hence", "index": 12}, {"type": "interline_equation", "bbox": [116, 631, 493, 648], "content": "", "index": 13}, {"type": "text", "bbox": [70, 657, 542, 686], "content": "and so is an -symmetry. Incidentally, will always be order 1 or 2 because for all .", "index": 14}, {"type": "text", "bbox": [70, 687, 542, 715], "content": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries have analogues in arbitrary (i.e. not necessarily affine) fusion rings.", "index": 15}]
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[{"bbox": [72, 177, 330, 189], "content": "2.3. Standard constructions of fusion-symmetries", "parent_index": 1, "line_index": 0}, {"bbox": [95, 198, 540, 212], "content": "Simple-currents are a large source of fusion-symmetries. Let be any simple-current", "parent_index": 2, "line_index": 0}, {"bbox": [70, 211, 399, 226], "content": "of order . Choose any number such that", "parent_index": 2, "line_index": 1}, {"bbox": [70, 262, 540, 280], "content": "Any solution to this defines a fusion-symmetry , which we shall denote", "parent_index": 4, "line_index": 0}, {"bbox": [70, 279, 541, 294], "content": "or . Note that from (2.2b), (2.5b) and (2.5c) that any , , obeys the", "parent_index": 4, "line_index": 1}, {"bbox": [69, 292, 543, 311], "content": "relation when (it would in fact be a fusion-endomorphism — see", "parent_index": 4, "line_index": 2}, {"bbox": [69, 308, 539, 325], "content": "§2.2); the ‘gcd’ condition forces to be a permutation. Choosing", "parent_index": 4, "line_index": 3}, {"bbox": [71, 322, 343, 339], "content": "(mod ), we find that is an -symmetry.", "parent_index": 4, "line_index": 4}, {"bbox": [95, 338, 541, 353], "content": "When the group of simple-currents is not cyclic, this construction can be generalised", "parent_index": 5, "line_index": 0}, {"bbox": [69, 352, 540, 368], "content": "in a natural way, and the resulting fusion-symmetry will be parametrised by a matrix .", "parent_index": 5, "line_index": 1}, {"bbox": [71, 367, 213, 380], "content": "We will meet these in 3.4.", "parent_index": 5, "line_index": 2}, {"bbox": [94, 380, 541, 396], "content": "We will call these simple-current automorphisms. The first examples of these were", "parent_index": 6, "line_index": 0}, {"bbox": [72, 396, 376, 410], "content": "found by Bernard [2], and were generalised further in [31].", "parent_index": 6, "line_index": 1}, {"bbox": [92, 407, 544, 429], "content": "For any affine algebra and any sufficiently high level, we will see in the next", "parent_index": 7, "line_index": 0}, {"bbox": [70, 426, 542, 442], "content": "section that its fusion-symmetries consist entirely of simple-current automorphisms and", "parent_index": 7, "line_index": 1}, {"bbox": [72, 442, 483, 455], "content": "conjugations. For this reason, any other fusion-symmetry is called exceptional.", "parent_index": 7, "line_index": 2}, {"bbox": [94, 454, 541, 470], "content": "There is another general construction of fusion-symmetries, generalising , although", "parent_index": 8, "line_index": 0}, {"bbox": [69, 469, 541, 484], "content": "it yields few new examples for the affine fusion rings. If the Galois automorphism is such", "parent_index": 8, "line_index": 1}, {"bbox": [69, 481, 542, 501], "content": "that is a simple-current — equivalently, that — then the permutation", "parent_index": 8, "line_index": 2}, {"bbox": [68, 536, 541, 553], "content": "is a fusion-symmetry. The simplest example is . We call a Galois fusion-", "parent_index": 10, "line_index": 0}, {"bbox": [71, 552, 528, 566], "content": "symmetry. A special case of these was given in [13]. To see that works, note from", "parent_index": 10, "line_index": 1}, {"bbox": [69, 603, 254, 620], "content": "that . Hence", "parent_index": 12, "line_index": 0}, {"bbox": [70, 658, 542, 675], "content": "and so is an -symmetry. Incidentally, will always be order 1 or 2 because", "parent_index": 14, "line_index": 0}, {"bbox": [70, 673, 214, 689], "content": "for all .", "parent_index": 14, "line_index": 1}, {"bbox": [93, 687, 542, 704], "content": "Simple-currents (2.2), the Galois action (2.3), and the corresponding fusion-symmetries", "parent_index": 15, "line_index": 0}, {"bbox": [70, 702, 424, 719], "content": "have analogues in arbitrary (i.e. not necessarily affine) fusion rings.", "parent_index": 15, "line_index": 1}]
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[{"bbox": [416, 199, 421, 210], "content": "j", "parent_index": 2, "subtype": "inline"}, {"bbox": [116, 217, 123, 222], "content": "{\\boldsymbol{n}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [240, 212, 344, 225], "content": "a\\in\\{0,1,\\ldots,n-1\\}", "parent_index": 2, "subtype": "inline"}, {"bbox": [239, 239, 372, 252], "content": "\\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\\ .", "parent_index": 3, "subtype": "interline"}, {"bbox": [318, 264, 396, 276], "content": "\\lambda\\mapsto J^{n a Q_{j}\\,(\\lambda)}\\lambda", "parent_index": 4, "subtype": "inline"}, {"bbox": [519, 266, 540, 279], "content": "\\pi[a]", "parent_index": 4, "subtype": "inline"}, {"bbox": [87, 280, 112, 293], "content": "\\pi_{j}[a]", "parent_index": 4, "subtype": "inline"}, {"bbox": [393, 280, 441, 293], "content": "\\pi\\,=\\,\\pi[a]", "parent_index": 4, "subtype": "inline"}, {"bbox": [448, 281, 480, 290], "content": "a\\in\\mathbb{Z}", "parent_index": 4, "subtype": "inline"}, {"bbox": [115, 295, 193, 309], "content": "N_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}=\\ N_{\\lambda\\mu}^{\\nu}", "parent_index": 4, "subtype": "inline"}, {"bbox": [228, 295, 273, 309], "content": "N_{\\lambda\\mu}^{\\nu}\\neq0", "parent_index": 4, "subtype": "inline"}, {"bbox": [235, 311, 255, 323], "content": "\\pi[a]", "parent_index": 4, "subtype": "inline"}, {"bbox": [419, 309, 539, 324], "content": "b\\equiv-a\\,(n a Q_{j}(j)\\!+\\!1)^{-1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [103, 329, 110, 334], "content": "{\\boldsymbol{n}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [189, 324, 244, 337], "content": "(\\pi[a],\\pi[b])", "parent_index": 4, "subtype": "inline"}, {"bbox": [276, 325, 284, 334], "content": "S", "parent_index": 4, "subtype": "inline"}, {"bbox": [512, 353, 536, 366], "content": "\\left(a_{i j}\\right)", "parent_index": 5, "subtype": "inline"}, {"bbox": [218, 410, 242, 424], "content": "X_{r}^{(1)}", "parent_index": 7, "subtype": "inline"}, {"bbox": [477, 457, 486, 466], "content": "C", "parent_index": 8, "subtype": "inline"}, {"bbox": [491, 474, 502, 482], "content": "\\sigma_{\\ell}", "parent_index": 8, "subtype": "inline"}, {"bbox": [97, 483, 115, 495], "content": "0^{(\\ell)}", "parent_index": 8, "subtype": "inline"}, {"bbox": [218, 486, 224, 497], "content": "j", "parent_index": 8, "subtype": "inline"}, {"bbox": [337, 484, 409, 497], "content": "\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [257, 509, 354, 524], "content": "\\pi\\{\\ell\\}:\\lambda\\mapsto J(\\lambda^{(\\ell)})", "parent_index": 9, "subtype": "interline"}, {"bbox": [317, 538, 377, 551], "content": "\\pi\\{-1\\}=C", "parent_index": 10, "subtype": "inline"}, {"bbox": [428, 538, 453, 551], "content": "\\pi\\{\\ell\\}", "parent_index": 10, "subtype": "inline"}, {"bbox": [410, 552, 434, 565], "content": "\\pi\\{\\ell\\}", "parent_index": 10, "subtype": "inline"}, {"bbox": [198, 576, 412, 593], "content": "\\epsilon_{\\ell}(\\lambda)\\,S_{\\lambda^{(\\ell)},0}=\\sigma_{\\ell}S_{\\lambda0}=\\epsilon_{\\ell}(0)\\,e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}S_{\\lambda0}", "parent_index": 11, "subtype": "interline"}, {"bbox": [97, 606, 212, 620], "content": "\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(0)=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\lambda)}", "parent_index": 12, "subtype": "inline"}, {"bbox": [116, 631, 493, 648], "content": "S_{J\\lambda^{(\\ell)},\\mu}=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\epsilon_{\\ell}(\\lambda)\\,\\sigma_{\\ell}(S_{\\lambda\\mu})=e^{2\\pi\\mathrm{i}\\,Q_{j}(\\mu)}\\,\\epsilon_{\\ell}(\\lambda)\\,\\epsilon_{\\ell}(\\mu)\\,S_{\\lambda,\\mu^{(\\ell)}}=S_{\\lambda,J\\mu^{(\\ell)}}", "parent_index": 13, "subtype": "interline"}, {"bbox": [107, 660, 183, 673], "content": "(\\pi\\{\\ell\\},\\pi\\{\\ell\\}^{-1})", "parent_index": 14, "subtype": "inline"}, {"bbox": [213, 661, 222, 670], "content": "S", "parent_index": 14, "subtype": "inline"}, {"bbox": [352, 661, 360, 670], "content": "J", "parent_index": 14, "subtype": "inline"}, {"bbox": [70, 675, 133, 688], "content": "2\\,Q_{j}(\\lambda)\\in\\mathbb{Z}", "parent_index": 14, "subtype": "inline"}, {"bbox": [172, 676, 209, 687], "content": "\\lambda\\in P_{+}", "parent_index": 14, "subtype": "inline"}]
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Our main task in this paper is to find and construct all fusion-symmetries for the affine algebras $X_{r}^{(1)}$ , for simple $X_{r}$ . In this section we state the results, and in the next section we prove the completeness of our lists. Recall the simple-current automorphism $\pi[a]$ and Galois automorphism $\pi\{\ell\}$ defined in §2.3, and the notation $\kappa=k\!+\!h^{\vee}$ . It will be convenient to write $X{_{r,k}}^{,}$ for $\cdot X_{r}^{(1)}$ and level $k'$ . We write $_S$ for the group of symmetries of the extended Dynkin diagram.
# 3.1. The algebra $A_{r}^{(1)}$ , $r\geq1$
Define $\overline{r}\,=\,r\,+\,1$ and $\;n\;=\;k\,+\,{\overline{{r}}}$ . The level $k$ highest weights of $A_{r}^{(1)}$ constitute the set $P_{+}$ of $\overline{r}$ -tuples $\lambda\,=\,(\lambda_{0},.\,.\,.\,,\lambda_{r})$ of non-negative integers obeying $\textstyle\sum_{i=0}^{r}\lambda_{i}\;=\;k$ . The Dynkin diagram symmetries form the dihedral group $\boldsymbol{S}\;=\;\mathfrak{D}_{r+1}$ ; it is generated by the charge-conjugation $C$ and simple-current $J$ given by $C\lambda\,=\,(\lambda_{0},\lambda_{r},\lambda_{r-1},\ldots,\lambda_{1})$ and $J\lambda=\left(\lambda_{r},\lambda_{0},\lambda_{1},.\dots,\lambda_{r-1}\right)$ , with $Q_{J^{a}}(\lambda)=a\,t(\lambda)/\overline{{r}}$ for $\begin{array}{r}{t(\lambda)\overset{\mathrm{def}}{=}\sum_{j=1}^{r}j\lambda_{j}}\end{array}$ . Note that $C=i d$ . for ${A}_{1}^{(1)}$ .
The Kac-Peterson relation (2.1b) for $A_{r,k}$ takes the form
$$
\frac{S_{\lambda\mu}}{S_{0\mu}}=\exp[-2\pi\mathrm{i}\frac{t(\lambda)\,t(\mu)}{\kappa\,\overline{{r}}}]\,\,s_{(\lambda)}(\exp[-2\pi\mathrm{i}\frac{(\mu+\rho)(1)}{\kappa},\ldots,\exp[-2\pi\mathrm{i}\frac{(\mu+\rho)(\overline{{r}})}{\kappa}]\,\ \
$$
where $s_{(\lambda)}\big(x_{1},\ldots,x_{r+1}\big)$ is the Schur polynomial (see e.g. [27]) corresponding to the partition $(\lambda(1),\ldots,\lambda(\overline{{r}}))$ , and where $\textstyle\nu(\ell)=\sum_{i=\ell}^{r}\nu_{i}$ for any weight $\nu$ . In other words, $S_{\lambda\mu}/S_{0\mu}$ is the Schur polynomial corresponding to $\lambda$ , evaluated at roots of 1 determined by $\mu$ .
The fusion (derived from the Pieri rule and (2.4))
$$
\Lambda_{1}\mathinner{\left[\boxtimes\right]}\Lambda_{\ell}=\Lambda_{\ell+1}\mathinner{\left[\textstyle{\ H}\right.\left(\Lambda_{1}+\Lambda_{\ell}\right)\,,}
$$
valid for $k\geq2$ and $1\leq\ell<r$ , will be useful.
There are no exceptional fusion-symmetries for $A_{r}^{(1)}$ :
Theorem 3.A. The fusion-symmetries for $A_{r}^{(1)}$ level $k$ are $C^{i}\pi[a]$ , for $i\in\{0,1\}$ and any integer $0\leq a\leq r$ for which $1+k a$ is coprime to $r+1$ .
To avoid redundancies in the Theorem, for $r\,=\,1$ or $k\,=\,1$ take $i\,=\,0$ only. If we write ${\overline{{r}}}\,=\,r^{\prime}r^{\prime\prime}$ , where $r^{\prime}$ is coprime to $k$ and $r^{\prime\prime}|k^{\infty}$ , then the number of simple-current automorphisms will exactly equal $r^{\prime\prime}{\cdot}\varphi(r^{\prime})$ , where $\varphi$ is the Euler totient. The $\pi[a]$ commute with each other, and with $C$ .
For example, for $A_{1,k}$ when $k$ is odd, there is no nontrivial fusion-symmetry. When $k$ is even, there is exactly one, sending $\lambda=\lambda_{1}\Lambda_{1}$ to $\lambda$ (for $\lambda_{1}$ even) or $J\lambda=(k-\lambda_{1})\Lambda_{1}$ (for $\lambda_{1}$ odd). For $A_{2,k}$ , there are either six or four fusion-symmetries, depending on whether or not 3 divides $k$ .
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<html><body>
<p data-bbox="70 99 542 191">Our main task in this paper is to find and construct all fusion-symmetries for the affine algebras $X_{r}^{(1)}$ , for simple $X_{r}$ . In this section we state the results, and in the next section we prove the completeness of our lists. Recall the simple-current automorphism $\pi[a]$ and Galois automorphism $\pi\{\ell\}$ defined in §2.3, and the notation $\kappa=k\!+\!h^{\vee}$ . It will be convenient to write $X{_{r,k}}^{,}$ for $\cdot X_{r}^{(1)}$ and level $k'$ . We write $_S$ for the group of symmetries of the extended Dynkin diagram. </p>
<h1 data-bbox="71 203 218 221">3.1. The algebra $A_{r}^{(1)}$ , $r\geq1$ </h1>
<p data-bbox="69 228 541 323">Define $\overline{r}\,=\,r\,+\,1$ and $\;n\;=\;k\,+\,{\overline{{r}}}$ . The level $k$ highest weights of $A_{r}^{(1)}$ constitute the set $P_{+}$ of $\overline{r}$ -tuples $\lambda\,=\,(\lambda_{0},.\,.\,.\,,\lambda_{r})$ of non-negative integers obeying $\textstyle\sum_{i=0}^{r}\lambda_{i}\;=\;k$ . The Dynkin diagram symmetries form the dihedral group $\boldsymbol{S}\;=\;\mathfrak{D}_{r+1}$ ; it is generated by the charge-conjugation $C$ and simple-current $J$ given by $C\lambda\,=\,(\lambda_{0},\lambda_{r},\lambda_{r-1},\ldots,\lambda_{1})$ and $J\lambda=\left(\lambda_{r},\lambda_{0},\lambda_{1},.\dots,\lambda_{r-1}\right)$ , with $Q_{J^{a}}(\lambda)=a\,t(\lambda)/\overline{{r}}$ for $\begin{array}{r}{t(\lambda)\overset{\mathrm{def}}{=}\sum_{j=1}^{r}j\lambda_{j}}\end{array}$ . Note that $C=i d$ . for ${A}_{1}^{(1)}$ . </p>
<p data-bbox="95 324 394 339">The Kac-Peterson relation (2.1b) for $A_{r,k}$ takes the form </p>
<div class="equation" data-bbox="87 352 487 383">$$
\frac{S_{\lambda\mu}}{S_{0\mu}}=\exp[-2\pi\mathrm{i}\frac{t(\lambda)\,t(\mu)}{\kappa\,\overline{{r}}}]\,\,s_{(\lambda)}(\exp[-2\pi\mathrm{i}\frac{(\mu+\rho)(1)}{\kappa},\ldots,\exp[-2\pi\mathrm{i}\frac{(\mu+\rho)(\overline{{r}})}{\kappa}]\,\ \
$$</div>
<p data-bbox="70 395 540 439">where $s_{(\lambda)}\big(x_{1},\ldots,x_{r+1}\big)$ is the Schur polynomial (see e.g. [27]) corresponding to the partition $(\lambda(1),\ldots,\lambda(\overline{{r}}))$ , and where $\textstyle\nu(\ell)=\sum_{i=\ell}^{r}\nu_{i}$ for any weight $\nu$ . In other words, $S_{\lambda\mu}/S_{0\mu}$ is the Schur polynomial corresponding to $\lambda$ , evaluated at roots of 1 determined by $\mu$ . </p>
<p data-bbox="94 439 357 453">The fusion (derived from the Pieri rule and (2.4)) </p>
<div class="equation" data-bbox="223 467 386 484">$$
\Lambda_{1}\mathinner{\left[\boxtimes\right]}\Lambda_{\ell}=\Lambda_{\ell+1}\mathinner{\left[\textstyle{\ H}\right.\left(\Lambda_{1}+\Lambda_{\ell}\right)\,,}
$$</div>
<p data-bbox="71 496 304 510">valid for $k\geq2$ and $1\leq\ell<r$ , will be useful. </p>
<p data-bbox="95 512 371 527">There are no exceptional fusion-symmetries for $A_{r}^{(1)}$ : </p>
<p data-bbox="70 533 542 565">Theorem 3.A. The fusion-symmetries for $A_{r}^{(1)}$ level $k$ are $C^{i}\pi[a]$ , for $i\in\{0,1\}$ and any integer $0\leq a\leq r$ for which $1+k a$ is coprime to $r+1$ . </p>
<p data-bbox="70 571 541 628">To avoid redundancies in the Theorem, for $r\,=\,1$ or $k\,=\,1$ take $i\,=\,0$ only. If we write ${\overline{{r}}}\,=\,r^{\prime}r^{\prime\prime}$ , where $r^{\prime}$ is coprime to $k$ and $r^{\prime\prime}|k^{\infty}$ , then the number of simple-current automorphisms will exactly equal $r^{\prime\prime}{\cdot}\varphi(r^{\prime})$ , where $\varphi$ is the Euler totient. The $\pi[a]$ commute with each other, and with $C$ . </p>
<p data-bbox="70 630 541 686">For example, for $A_{1,k}$ when $k$ is odd, there is no nontrivial fusion-symmetry. When $k$ is even, there is exactly one, sending $\lambda=\lambda_{1}\Lambda_{1}$ to $\lambda$ (for $\lambda_{1}$ even) or $J\lambda=(k-\lambda_{1})\Lambda_{1}$ (for $\lambda_{1}$ odd). For $A_{2,k}$ , there are either six or four fusion-symmetries, depending on whether or not 3 divides $k$ . </p>
</body></html>
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0002044v1
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[{"type": "text", "text": "Our main task in this paper is to find and construct all fusion-symmetries for the affine algebras $X_{r}^{(1)}$ , for simple $X_{r}$ . In this section we state the results, and in the next section we prove the completeness of our lists. Recall the simple-current automorphism $\\pi[a]$ and Galois automorphism $\\pi\\{\\ell\\}$ defined in §2.3, and the notation $\\kappa=k\\!+\\!h^{\\vee}$ . It will be convenient to write $X{_{r,k}}^{,}$ for $\\cdot X_{r}^{(1)}$ and level $k'$ . We write $_S$ for the group of symmetries of the extended Dynkin diagram. ", "page_idx": 8}, {"type": "text", "text": "3.1. The algebra $A_{r}^{(1)}$ , $r\\geq1$ ", "text_level": 1, "page_idx": 8}, {"type": "text", "text": "Define $\\overline{r}\\,=\\,r\\,+\\,1$ and $\\;n\\;=\\;k\\,+\\,{\\overline{{r}}}$ . The level $k$ highest weights of $A_{r}^{(1)}$ constitute the set $P_{+}$ of $\\overline{r}$ -tuples $\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})$ of non-negative integers obeying $\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k$ . The Dynkin diagram symmetries form the dihedral group $\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}$ ; it is generated by the charge-conjugation $C$ and simple-current $J$ given by $C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})$ and $J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)$ , with $Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}$ for $\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}$ . Note that $C=i d$ . for ${A}_{1}^{(1)}$ . ", "page_idx": 8}, {"type": "text", "text": "The Kac-Peterson relation (2.1b) for $A_{r,k}$ takes the form ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "where $s_{(\\lambda)}\\big(x_{1},\\ldots,x_{r+1}\\big)$ is the Schur polynomial (see e.g. [27]) corresponding to the partition $(\\lambda(1),\\ldots,\\lambda(\\overline{{r}}))$ , and where $\\textstyle\\nu(\\ell)=\\sum_{i=\\ell}^{r}\\nu_{i}$ for any weight $\\nu$ . In other words, $S_{\\lambda\\mu}/S_{0\\mu}$ is the Schur polynomial corresponding to $\\lambda$ , evaluated at roots of 1 determined by $\\mu$ . ", "page_idx": 8}, {"type": "text", "text": "The fusion (derived from the Pieri rule and (2.4)) ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "valid for $k\\geq2$ and $1\\leq\\ell<r$ , will be useful. ", "page_idx": 8}, {"type": "text", "text": "There are no exceptional fusion-symmetries for $A_{r}^{(1)}$ : ", "page_idx": 8}, {"type": "text", "text": "Theorem 3.A. The fusion-symmetries for $A_{r}^{(1)}$ level $k$ are $C^{i}\\pi[a]$ , for $i\\in\\{0,1\\}$ and any integer $0\\leq a\\leq r$ for which $1+k a$ is coprime to $r+1$ . ", "page_idx": 8}, {"type": "text", "text": "To avoid redundancies in the Theorem, for $r\\,=\\,1$ or $k\\,=\\,1$ take $i\\,=\\,0$ only. If we write ${\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}$ , where $r^{\\prime}$ is coprime to $k$ and $r^{\\prime\\prime}|k^{\\infty}$ , then the number of simple-current automorphisms will exactly equal $r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})$ , where $\\varphi$ is the Euler totient. The $\\pi[a]$ commute with each other, and with $C$ . ", "page_idx": 8}, {"type": "text", "text": "For example, for $A_{1,k}$ when $k$ is odd, there is no nontrivial fusion-symmetry. When $k$ is even, there is exactly one, sending $\\lambda=\\lambda_{1}\\Lambda_{1}$ to $\\lambda$ (for $\\lambda_{1}$ even) or $J\\lambda=(k-\\lambda_{1})\\Lambda_{1}$ (for $\\lambda_{1}$ odd). For $A_{2,k}$ , there are either six or four fusion-symmetries, depending on whether or not 3 divides $k$ . ", "page_idx": 8}]
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The algebra ", "type": "text"}, {"bbox": [161, 205, 183, 221], "score": 0.48, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [183, 202, 189, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 208, 218, 221], "score": 0.48, "content": "r\\geq1", "type": "inline_equation", "height": 13, "width": 28}], "index": 6}], "index": 6}, {"type": "text", "bbox": [69, 228, 541, 323], "lines": [{"bbox": [93, 228, 542, 247], "spans": [{"bbox": [93, 228, 133, 247], "score": 1.0, "content": "Define ", "type": "text"}, {"bbox": [133, 232, 190, 243], "score": 0.89, "content": "\\overline{r}\\,=\\,r\\,+\\,1", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [190, 228, 219, 247], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [219, 233, 277, 243], "score": 0.92, "content": "\\;n\\;=\\;k\\,+\\,{\\overline{{r}}}", "type": "inline_equation", "height": 10, "width": 58}, {"bbox": [278, 228, 345, 247], "score": 1.0, "content": ". The level ", "type": "text"}, {"bbox": [345, 234, 352, 243], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [353, 228, 459, 247], "score": 1.0, "content": " highest weights of ", "type": "text"}, {"bbox": [460, 228, 482, 244], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [483, 228, 542, 247], "score": 1.0, "content": "constitute", "type": "text"}], "index": 7}, {"bbox": [70, 245, 541, 262], "spans": [{"bbox": [70, 245, 112, 262], "score": 1.0, "content": "the set ", "type": "text"}, {"bbox": [113, 246, 129, 259], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [129, 245, 148, 262], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [148, 247, 155, 257], "score": 0.75, "content": "\\overline{r}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [156, 245, 195, 262], "score": 1.0, "content": "-tuples ", "type": "text"}, {"bbox": [195, 246, 284, 259], "score": 0.91, "content": "\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})", "type": "inline_equation", "height": 13, "width": 89}, {"bbox": [284, 245, 466, 262], "score": 1.0, "content": " of non-negative integers obeying", "type": "text"}, {"bbox": [467, 245, 537, 261], "score": 0.91, "content": "\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k", "type": "inline_equation", "height": 16, "width": 70}, {"bbox": [537, 245, 541, 262], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [70, 259, 542, 276], "spans": [{"bbox": [70, 259, 392, 276], "score": 1.0, "content": "The Dynkin diagram symmetries form the dihedral group ", "type": "text"}, {"bbox": [392, 261, 451, 274], "score": 0.91, "content": "\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [451, 259, 542, 276], "score": 1.0, "content": "; it is generated", "type": "text"}], "index": 9}, {"bbox": [70, 274, 540, 290], "spans": [{"bbox": [70, 274, 213, 290], "score": 1.0, "content": "by the charge-conjugation ", "type": "text"}, {"bbox": [214, 275, 224, 285], "score": 0.83, "content": "C", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [225, 274, 332, 290], "score": 1.0, "content": " and simple-current ", "type": "text"}, {"bbox": [333, 276, 341, 285], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [341, 274, 395, 290], "score": 1.0, "content": " given by ", "type": "text"}, {"bbox": [395, 274, 540, 288], "score": 0.89, "content": "C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})", "type": "inline_equation", "height": 14, "width": 145}], "index": 10}, {"bbox": [69, 290, 542, 309], "spans": [{"bbox": [69, 291, 94, 309], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 292, 235, 306], "score": 0.92, "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)", "type": "inline_equation", "height": 14, "width": 140}, {"bbox": [235, 291, 270, 309], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [270, 293, 366, 306], "score": 0.92, "content": "Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [367, 291, 388, 309], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [389, 290, 478, 308], "score": 0.92, "content": "\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}", "type": "inline_equation", "height": 18, "width": 89}, {"bbox": [479, 291, 542, 309], "score": 1.0, "content": ". Note that", "type": "text"}], "index": 11}, {"bbox": [71, 309, 159, 325], "spans": [{"bbox": [71, 313, 108, 322], "score": 0.88, "content": "C=i d", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [108, 310, 132, 325], "score": 1.0, "content": ". for ", "type": "text"}, {"bbox": [132, 309, 154, 325], "score": 0.93, "content": "{A}_{1}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [155, 310, 159, 325], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5}, {"type": "text", "bbox": [95, 324, 394, 339], "lines": [{"bbox": [94, 325, 393, 341], "spans": [{"bbox": [94, 325, 290, 341], "score": 1.0, "content": "The Kac-Peterson relation (2.1b) for ", "type": "text"}, {"bbox": [290, 328, 313, 340], "score": 0.92, "content": "A_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [313, 325, 393, 341], "score": 1.0, "content": " takes the form", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [87, 352, 487, 383], "lines": [{"bbox": [87, 352, 487, 383], "spans": [{"bbox": [87, 352, 487, 383], "score": 0.85, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 395, 540, 439], "lines": [{"bbox": [71, 397, 540, 414], "spans": [{"bbox": [71, 397, 105, 414], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 399, 197, 412], "score": 0.92, "content": "s_{(\\lambda)}\\big(x_{1},\\ldots,x_{r+1}\\big)", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [198, 397, 540, 414], "score": 1.0, "content": " is the Schur polynomial (see e.g. [27]) corresponding to the parti-", "type": "text"}], "index": 15}, {"bbox": [69, 411, 539, 430], "spans": [{"bbox": [69, 411, 96, 430], "score": 1.0, "content": "tion ", "type": "text"}, {"bbox": [96, 413, 176, 426], "score": 0.9, "content": "(\\lambda(1),\\ldots,\\lambda(\\overline{{r}}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [176, 411, 239, 430], "score": 1.0, "content": ", and where ", "type": "text"}, {"bbox": [239, 411, 317, 427], "score": 0.94, "content": "\\textstyle\\nu(\\ell)=\\sum_{i=\\ell}^{r}\\nu_{i}", "type": "inline_equation", "height": 16, "width": 78}, {"bbox": [317, 411, 397, 430], "score": 1.0, "content": " for any weight ", "type": "text"}, {"bbox": [397, 415, 405, 423], "score": 0.72, "content": "\\nu", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [405, 411, 495, 430], "score": 1.0, "content": ". In other words, ", "type": "text"}, {"bbox": [495, 414, 539, 426], "score": 0.94, "content": "S_{\\lambda\\mu}/S_{0\\mu}", "type": "inline_equation", "height": 12, "width": 44}], "index": 16}, {"bbox": [69, 425, 519, 443], "spans": [{"bbox": [69, 425, 289, 443], "score": 1.0, "content": "is the Schur polynomial corresponding to", "type": "text"}, {"bbox": [290, 428, 298, 438], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [298, 425, 505, 443], "score": 1.0, "content": ", evaluated at roots of 1 determined by ", "type": "text"}, {"bbox": [506, 432, 513, 440], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [514, 425, 519, 443], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [94, 439, 357, 453], "lines": [{"bbox": [96, 441, 355, 455], "spans": [{"bbox": [96, 441, 355, 455], "score": 1.0, "content": "The fusion (derived from the Pieri rule and (2.4))", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [223, 467, 386, 484], "lines": [{"bbox": [223, 467, 386, 484], "spans": [{"bbox": [223, 467, 386, 484], "score": 0.55, "content": "\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [71, 496, 304, 510], "lines": [{"bbox": [71, 498, 302, 511], "spans": [{"bbox": [71, 498, 118, 511], "score": 1.0, "content": "valid for ", "type": "text"}, {"bbox": [118, 500, 147, 511], "score": 0.92, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [147, 498, 173, 511], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 499, 223, 511], "score": 0.9, "content": "1\\leq\\ell<r", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [224, 498, 302, 511], "score": 1.0, "content": ", will be useful.", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [95, 512, 371, 527], "lines": [{"bbox": [93, 510, 372, 529], "spans": [{"bbox": [93, 512, 344, 529], "score": 1.0, "content": "There are no exceptional fusion-symmetries for ", "type": "text"}, {"bbox": [345, 510, 367, 527], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 17, "width": 22}, {"bbox": [367, 512, 372, 529], "score": 1.0, "content": ":", "type": "text"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [70, 533, 542, 565], "lines": [{"bbox": [93, 533, 543, 554], "spans": [{"bbox": [93, 534, 323, 554], "score": 1.0, "content": "Theorem 3.A. The fusion-symmetries for ", "type": "text"}, {"bbox": [323, 533, 346, 550], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [347, 534, 375, 554], "score": 1.0, "content": "level", "type": "text"}, {"bbox": [375, 537, 384, 549], "score": 0.72, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [384, 534, 407, 554], "score": 1.0, "content": " are ", "type": "text"}, {"bbox": [407, 536, 442, 551], "score": 0.92, "content": "C^{i}\\pi[a]", "type": "inline_equation", "height": 15, "width": 35}, {"bbox": [442, 534, 467, 554], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [467, 537, 517, 551], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [517, 534, 543, 554], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [72, 551, 383, 568], "spans": [{"bbox": [72, 551, 133, 568], "score": 1.0, "content": "any integer ", "type": "text"}, {"bbox": [134, 553, 184, 565], "score": 0.86, "content": "0\\leq a\\leq r", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [185, 551, 240, 568], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [240, 552, 275, 564], "score": 0.88, "content": "1+k a", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [275, 551, 350, 568], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [350, 552, 378, 564], "score": 0.9, "content": "r+1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [378, 551, 383, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [70, 571, 541, 628], "lines": [{"bbox": [94, 573, 541, 587], "spans": [{"bbox": [94, 574, 329, 587], "score": 1.0, "content": "To avoid redundancies in the Theorem, for ", "type": "text"}, {"bbox": [330, 574, 362, 584], "score": 0.89, "content": "r\\,=\\,1", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [362, 574, 381, 587], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [381, 573, 414, 585], "score": 0.89, "content": "k\\,=\\,1", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [415, 574, 444, 587], "score": 1.0, "content": " take ", "type": "text"}, {"bbox": [445, 574, 475, 585], "score": 0.88, "content": "i\\,=\\,0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [476, 574, 541, 587], "score": 1.0, "content": " only. If we", "type": "text"}], "index": 24}, {"bbox": [72, 588, 541, 601], "spans": [{"bbox": [72, 588, 102, 601], "score": 1.0, "content": "write ", "type": "text"}, {"bbox": [102, 589, 149, 599], "score": 0.91, "content": "{\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [149, 588, 191, 601], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [192, 589, 201, 598], "score": 0.9, "content": "r^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [201, 588, 280, 601], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [280, 590, 287, 599], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [288, 588, 315, 601], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [316, 588, 348, 601], "score": 0.92, "content": "r^{\\prime\\prime}|k^{\\infty}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [348, 588, 541, 601], "score": 1.0, "content": ", then the number of simple-current", "type": "text"}], "index": 25}, {"bbox": [70, 601, 542, 617], "spans": [{"bbox": [70, 602, 246, 617], "score": 1.0, "content": "automorphisms will exactly equal ", "type": "text"}, {"bbox": [246, 603, 289, 616], "score": 0.93, "content": "r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [290, 602, 329, 617], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [329, 605, 338, 615], "score": 0.81, "content": "\\varphi", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [338, 602, 468, 617], "score": 1.0, "content": " is the Euler totient. The ", "type": "text"}, {"bbox": [468, 601, 489, 616], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [489, 602, 542, 617], "score": 1.0, "content": " commute", "type": "text"}], "index": 26}, {"bbox": [72, 617, 224, 630], "spans": [{"bbox": [72, 617, 210, 630], "score": 1.0, "content": "with each other, and with ", "type": "text"}, {"bbox": [210, 618, 220, 627], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [220, 617, 224, 630], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "text", "bbox": [70, 630, 541, 686], "lines": [{"bbox": [95, 631, 540, 646], "spans": [{"bbox": [95, 631, 184, 646], "score": 1.0, "content": "For example, for ", "type": "text"}, {"bbox": [185, 633, 207, 645], "score": 0.93, "content": "A_{1,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [207, 631, 241, 646], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [242, 633, 248, 641], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [249, 631, 532, 646], "score": 1.0, "content": " is odd, there is no nontrivial fusion-symmetry. When ", "type": "text"}, {"bbox": [532, 633, 540, 642], "score": 0.83, "content": "k", "type": "inline_equation", "height": 9, "width": 8}], "index": 28}, {"bbox": [69, 645, 541, 661], "spans": [{"bbox": [69, 645, 265, 661], "score": 1.0, "content": "is even, there is exactly one, sending ", "type": "text"}, {"bbox": [266, 647, 315, 658], "score": 0.93, "content": "\\lambda=\\lambda_{1}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [315, 645, 333, 661], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [333, 647, 340, 656], "score": 0.87, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [340, 645, 366, 661], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [367, 647, 379, 658], "score": 0.9, "content": "\\lambda_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [379, 645, 429, 661], "score": 1.0, "content": " even) or ", "type": "text"}, {"bbox": [429, 646, 516, 659], "score": 0.93, "content": "J\\lambda=(k-\\lambda_{1})\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [517, 645, 541, 661], "score": 1.0, "content": " (for", "type": "text"}], "index": 29}, {"bbox": [71, 659, 541, 675], "spans": [{"bbox": [71, 662, 83, 672], "score": 0.89, "content": "\\lambda_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [84, 659, 144, 675], "score": 1.0, "content": " odd). For ", "type": "text"}, {"bbox": [145, 662, 167, 674], "score": 0.93, "content": "A_{2,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [167, 659, 541, 675], "score": 1.0, "content": ", there are either six or four fusion-symmetries, depending on whether", "type": "text"}], "index": 30}, {"bbox": [70, 674, 169, 689], "spans": [{"bbox": [70, 674, 157, 689], "score": 1.0, "content": "or not 3 divides ", "type": "text"}, {"bbox": [158, 676, 164, 685], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [165, 674, 169, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 29.5}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [87, 352, 487, 383], "lines": [{"bbox": [87, 352, 487, 383], "spans": [{"bbox": [87, 352, 487, 383], "score": 0.85, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [223, 467, 386, 484], "lines": [{"bbox": [223, 467, 386, 484], "spans": [{"bbox": [223, 467, 386, 484], "score": 0.55, "content": "\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}", "type": "interline_equation"}], "index": 19}], "index": 19}], "discarded_blocks": [{"type": "discarded", "bbox": [200, 71, 410, 85], "lines": [{"bbox": [201, 73, 410, 86], "spans": [{"bbox": [201, 73, 410, 86], "score": 1.0, "content": "3. Data for the Affine Algebras.", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 99, 542, 191], "lines": [{"bbox": [95, 102, 541, 117], "spans": [{"bbox": [95, 102, 541, 117], "score": 1.0, "content": "Our main task in this paper is to find and construct all fusion-symmetries for the", "type": "text"}], "index": 0}, {"bbox": [69, 113, 544, 135], "spans": [{"bbox": [69, 113, 151, 135], "score": 1.0, "content": "affine algebras ", "type": "text"}, {"bbox": [151, 116, 174, 130], "score": 0.94, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [175, 113, 239, 135], "score": 1.0, "content": ", for simple ", "type": "text"}, {"bbox": [239, 120, 255, 131], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [255, 113, 544, 135], "score": 1.0, "content": ". In this section we state the results, and in the next", "type": "text"}], "index": 1}, {"bbox": [70, 132, 542, 148], "spans": [{"bbox": [70, 132, 542, 148], "score": 1.0, "content": "section we prove the completeness of our lists. 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We write ", "type": "text"}, {"bbox": [381, 166, 389, 174], "score": 0.91, "content": "_S", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [389, 163, 541, 178], "score": 1.0, "content": " for the group of symmetries", "type": "text"}], "index": 4}, {"bbox": [70, 177, 245, 192], "spans": [{"bbox": [70, 177, 245, 192], "score": 1.0, "content": "of the extended Dynkin diagram.", "type": "text"}], "index": 5}], "index": 2.5, "bbox_fs": [68, 102, 544, 192]}, {"type": "title", "bbox": [71, 203, 218, 221], "lines": [{"bbox": [69, 202, 218, 225], "spans": [{"bbox": [69, 202, 160, 225], "score": 1.0, "content": "3.1. 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The level ", "type": "text"}, {"bbox": [345, 234, 352, 243], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [353, 228, 459, 247], "score": 1.0, "content": " highest weights of ", "type": "text"}, {"bbox": [460, 228, 482, 244], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [483, 228, 542, 247], "score": 1.0, "content": "constitute", "type": "text"}], "index": 7}, {"bbox": [70, 245, 541, 262], "spans": [{"bbox": [70, 245, 112, 262], "score": 1.0, "content": "the set ", "type": "text"}, {"bbox": [113, 246, 129, 259], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [129, 245, 148, 262], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [148, 247, 155, 257], "score": 0.75, "content": "\\overline{r}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [156, 245, 195, 262], "score": 1.0, "content": "-tuples ", "type": "text"}, {"bbox": [195, 246, 284, 259], "score": 0.91, "content": "\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})", "type": "inline_equation", "height": 13, "width": 89}, {"bbox": [284, 245, 466, 262], "score": 1.0, "content": " of non-negative integers obeying", "type": "text"}, {"bbox": [467, 245, 537, 261], "score": 0.91, "content": "\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k", "type": "inline_equation", "height": 16, "width": 70}, {"bbox": [537, 245, 541, 262], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [70, 259, 542, 276], "spans": [{"bbox": [70, 259, 392, 276], "score": 1.0, "content": "The Dynkin diagram symmetries form the dihedral group ", "type": "text"}, {"bbox": [392, 261, 451, 274], "score": 0.91, "content": "\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [451, 259, 542, 276], "score": 1.0, "content": "; it is generated", "type": "text"}], "index": 9}, {"bbox": [70, 274, 540, 290], "spans": [{"bbox": [70, 274, 213, 290], "score": 1.0, "content": "by the charge-conjugation ", "type": "text"}, {"bbox": [214, 275, 224, 285], "score": 0.83, "content": "C", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [225, 274, 332, 290], "score": 1.0, "content": " and simple-current ", "type": "text"}, {"bbox": [333, 276, 341, 285], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [341, 274, 395, 290], "score": 1.0, "content": " given by ", "type": "text"}, {"bbox": [395, 274, 540, 288], "score": 0.89, "content": "C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})", "type": "inline_equation", "height": 14, "width": 145}], "index": 10}, {"bbox": [69, 290, 542, 309], "spans": [{"bbox": [69, 291, 94, 309], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 292, 235, 306], "score": 0.92, "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)", "type": "inline_equation", "height": 14, "width": 140}, {"bbox": [235, 291, 270, 309], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [270, 293, 366, 306], "score": 0.92, "content": "Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [367, 291, 388, 309], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [389, 290, 478, 308], "score": 0.92, "content": "\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}", "type": "inline_equation", "height": 18, "width": 89}, {"bbox": [479, 291, 542, 309], "score": 1.0, "content": ". Note that", "type": "text"}], "index": 11}, {"bbox": [71, 309, 159, 325], "spans": [{"bbox": [71, 313, 108, 322], "score": 0.88, "content": "C=i d", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [108, 310, 132, 325], "score": 1.0, "content": ". for ", "type": "text"}, {"bbox": [132, 309, 154, 325], "score": 0.93, "content": "{A}_{1}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [155, 310, 159, 325], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5, "bbox_fs": [69, 228, 542, 325]}, {"type": "text", "bbox": [95, 324, 394, 339], "lines": [{"bbox": [94, 325, 393, 341], "spans": [{"bbox": [94, 325, 290, 341], "score": 1.0, "content": "The Kac-Peterson relation (2.1b) for ", "type": "text"}, {"bbox": [290, 328, 313, 340], "score": 0.92, "content": "A_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [313, 325, 393, 341], "score": 1.0, "content": " takes the form", "type": "text"}], "index": 13}], "index": 13, "bbox_fs": [94, 325, 393, 341]}, {"type": "interline_equation", "bbox": [87, 352, 487, 383], "lines": [{"bbox": [87, 352, 487, 383], "spans": [{"bbox": [87, 352, 487, 383], "score": 0.85, "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 395, 540, 439], "lines": [{"bbox": [71, 397, 540, 414], "spans": [{"bbox": [71, 397, 105, 414], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 399, 197, 412], "score": 0.92, "content": "s_{(\\lambda)}\\big(x_{1},\\ldots,x_{r+1}\\big)", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [198, 397, 540, 414], "score": 1.0, "content": " is the Schur polynomial (see e.g. [27]) corresponding to the parti-", "type": "text"}], "index": 15}, {"bbox": [69, 411, 539, 430], "spans": [{"bbox": [69, 411, 96, 430], "score": 1.0, "content": "tion ", "type": "text"}, {"bbox": [96, 413, 176, 426], "score": 0.9, "content": "(\\lambda(1),\\ldots,\\lambda(\\overline{{r}}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [176, 411, 239, 430], "score": 1.0, "content": ", and where ", "type": "text"}, {"bbox": [239, 411, 317, 427], "score": 0.94, "content": "\\textstyle\\nu(\\ell)=\\sum_{i=\\ell}^{r}\\nu_{i}", "type": "inline_equation", "height": 16, "width": 78}, {"bbox": [317, 411, 397, 430], "score": 1.0, "content": " for any weight ", "type": "text"}, {"bbox": [397, 415, 405, 423], "score": 0.72, "content": "\\nu", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [405, 411, 495, 430], "score": 1.0, "content": ". In other words, ", "type": "text"}, {"bbox": [495, 414, 539, 426], "score": 0.94, "content": "S_{\\lambda\\mu}/S_{0\\mu}", "type": "inline_equation", "height": 12, "width": 44}], "index": 16}, {"bbox": [69, 425, 519, 443], "spans": [{"bbox": [69, 425, 289, 443], "score": 1.0, "content": "is the Schur polynomial corresponding to", "type": "text"}, {"bbox": [290, 428, 298, 438], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [298, 425, 505, 443], "score": 1.0, "content": ", evaluated at roots of 1 determined by ", "type": "text"}, {"bbox": [506, 432, 513, 440], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [514, 425, 519, 443], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16, "bbox_fs": [69, 397, 540, 443]}, {"type": "text", "bbox": [94, 439, 357, 453], "lines": [{"bbox": [96, 441, 355, 455], "spans": [{"bbox": [96, 441, 355, 455], "score": 1.0, "content": "The fusion (derived from the Pieri rule and (2.4))", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [96, 441, 355, 455]}, {"type": "interline_equation", "bbox": [223, 467, 386, 484], "lines": [{"bbox": [223, 467, 386, 484], "spans": [{"bbox": [223, 467, 386, 484], "score": 0.55, "content": "\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [71, 496, 304, 510], "lines": [{"bbox": [71, 498, 302, 511], "spans": [{"bbox": [71, 498, 118, 511], "score": 1.0, "content": "valid for ", "type": "text"}, {"bbox": [118, 500, 147, 511], "score": 0.92, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [147, 498, 173, 511], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 499, 223, 511], "score": 0.9, "content": "1\\leq\\ell<r", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [224, 498, 302, 511], "score": 1.0, "content": ", will be useful.", "type": "text"}], "index": 20}], "index": 20, "bbox_fs": [71, 498, 302, 511]}, {"type": "text", "bbox": [95, 512, 371, 527], "lines": [{"bbox": [93, 510, 372, 529], "spans": [{"bbox": [93, 512, 344, 529], "score": 1.0, "content": "There are no exceptional fusion-symmetries for ", "type": "text"}, {"bbox": [345, 510, 367, 527], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 17, "width": 22}, {"bbox": [367, 512, 372, 529], "score": 1.0, "content": ":", "type": "text"}], "index": 21}], "index": 21, "bbox_fs": [93, 510, 372, 529]}, {"type": "text", "bbox": [70, 533, 542, 565], "lines": [{"bbox": [93, 533, 543, 554], "spans": [{"bbox": [93, 534, 323, 554], "score": 1.0, "content": "Theorem 3.A. The fusion-symmetries for ", "type": "text"}, {"bbox": [323, 533, 346, 550], "score": 0.91, "content": "A_{r}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [347, 534, 375, 554], "score": 1.0, "content": "level", "type": "text"}, {"bbox": [375, 537, 384, 549], "score": 0.72, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [384, 534, 407, 554], "score": 1.0, "content": " are ", "type": "text"}, {"bbox": [407, 536, 442, 551], "score": 0.92, "content": "C^{i}\\pi[a]", "type": "inline_equation", "height": 15, "width": 35}, {"bbox": [442, 534, 467, 554], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [467, 537, 517, 551], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [517, 534, 543, 554], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [72, 551, 383, 568], "spans": [{"bbox": [72, 551, 133, 568], "score": 1.0, "content": "any integer ", "type": "text"}, {"bbox": [134, 553, 184, 565], "score": 0.86, "content": "0\\leq a\\leq r", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [185, 551, 240, 568], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [240, 552, 275, 564], "score": 0.88, "content": "1+k a", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [275, 551, 350, 568], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [350, 552, 378, 564], "score": 0.9, "content": "r+1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [378, 551, 383, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "bbox_fs": [72, 533, 543, 568]}, {"type": "text", "bbox": [70, 571, 541, 628], "lines": [{"bbox": [94, 573, 541, 587], "spans": [{"bbox": [94, 574, 329, 587], "score": 1.0, "content": "To avoid redundancies in the Theorem, for ", "type": "text"}, {"bbox": [330, 574, 362, 584], "score": 0.89, "content": "r\\,=\\,1", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [362, 574, 381, 587], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [381, 573, 414, 585], "score": 0.89, "content": "k\\,=\\,1", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [415, 574, 444, 587], "score": 1.0, "content": " take ", "type": "text"}, {"bbox": [445, 574, 475, 585], "score": 0.88, "content": "i\\,=\\,0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [476, 574, 541, 587], "score": 1.0, "content": " only. If we", "type": "text"}], "index": 24}, {"bbox": [72, 588, 541, 601], "spans": [{"bbox": [72, 588, 102, 601], "score": 1.0, "content": "write ", "type": "text"}, {"bbox": [102, 589, 149, 599], "score": 0.91, "content": "{\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [149, 588, 191, 601], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [192, 589, 201, 598], "score": 0.9, "content": "r^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [201, 588, 280, 601], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [280, 590, 287, 599], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [288, 588, 315, 601], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [316, 588, 348, 601], "score": 0.92, "content": "r^{\\prime\\prime}|k^{\\infty}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [348, 588, 541, 601], "score": 1.0, "content": ", then the number of simple-current", "type": "text"}], "index": 25}, {"bbox": [70, 601, 542, 617], "spans": [{"bbox": [70, 602, 246, 617], "score": 1.0, "content": "automorphisms will exactly equal ", "type": "text"}, {"bbox": [246, 603, 289, 616], "score": 0.93, "content": "r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [290, 602, 329, 617], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [329, 605, 338, 615], "score": 0.81, "content": "\\varphi", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [338, 602, 468, 617], "score": 1.0, "content": " is the Euler totient. The ", "type": "text"}, {"bbox": [468, 601, 489, 616], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [489, 602, 542, 617], "score": 1.0, "content": " commute", "type": "text"}], "index": 26}, {"bbox": [72, 617, 224, 630], "spans": [{"bbox": [72, 617, 210, 630], "score": 1.0, "content": "with each other, and with ", "type": "text"}, {"bbox": [210, 618, 220, 627], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [220, 617, 224, 630], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5, "bbox_fs": [70, 573, 542, 630]}, {"type": "text", "bbox": [70, 630, 541, 686], "lines": [{"bbox": [95, 631, 540, 646], "spans": [{"bbox": [95, 631, 184, 646], "score": 1.0, "content": "For example, for ", "type": "text"}, {"bbox": [185, 633, 207, 645], "score": 0.93, "content": "A_{1,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [207, 631, 241, 646], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [242, 633, 248, 641], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [249, 631, 532, 646], "score": 1.0, "content": " is odd, there is no nontrivial fusion-symmetry. When ", "type": "text"}, {"bbox": [532, 633, 540, 642], "score": 0.83, "content": "k", "type": "inline_equation", "height": 9, "width": 8}], "index": 28}, {"bbox": [69, 645, 541, 661], "spans": [{"bbox": [69, 645, 265, 661], "score": 1.0, "content": "is even, there is exactly one, sending ", "type": "text"}, {"bbox": [266, 647, 315, 658], "score": 0.93, "content": "\\lambda=\\lambda_{1}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [315, 645, 333, 661], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [333, 647, 340, 656], "score": 0.87, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [340, 645, 366, 661], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [367, 647, 379, 658], "score": 0.9, "content": "\\lambda_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [379, 645, 429, 661], "score": 1.0, "content": " even) or ", "type": "text"}, {"bbox": [429, 646, 516, 659], "score": 0.93, "content": "J\\lambda=(k-\\lambda_{1})\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [517, 645, 541, 661], "score": 1.0, "content": " (for", "type": "text"}], "index": 29}, {"bbox": [71, 659, 541, 675], "spans": [{"bbox": [71, 662, 83, 672], "score": 0.89, "content": "\\lambda_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [84, 659, 144, 675], "score": 1.0, "content": " odd). For ", "type": "text"}, {"bbox": [145, 662, 167, 674], "score": 0.93, "content": "A_{2,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [167, 659, 541, 675], "score": 1.0, "content": ", there are either six or four fusion-symmetries, depending on whether", "type": "text"}], "index": 30}, {"bbox": [70, 674, 169, 689], "spans": [{"bbox": [70, 674, 157, 689], "score": 1.0, "content": "or not 3 divides ", "type": "text"}, {"bbox": [158, 676, 164, 685], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [165, 674, 169, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 29.5, "bbox_fs": [69, 631, 541, 689]}]}
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[{"type": "text", "bbox": [70, 99, 542, 191], "content": "Our main task in this paper is to find and construct all fusion-symmetries for the affine algebras , for simple . In this section we state the results, and in the next section we prove the completeness of our lists. Recall the simple-current automorphism and Galois automorphism defined in §2.3, and the notation . It will be convenient to write for and level . We write for the group of symmetries of the extended Dynkin diagram.", "index": 0}, {"type": "title", "bbox": [71, 203, 218, 221], "content": "3.1. The algebra ,", "index": 1}, {"type": "text", "bbox": [69, 228, 541, 323], "content": "Define and . The level highest weights of constitute the set of -tuples of non-negative integers obeying . The Dynkin diagram symmetries form the dihedral group ; it is generated by the charge-conjugation and simple-current given by and , with for . Note that . for .", "index": 2}, {"type": "text", "bbox": [95, 324, 394, 339], "content": "The Kac-Peterson relation (2.1b) for takes the form", "index": 3}, {"type": "interline_equation", "bbox": [87, 352, 487, 383], "content": "", "index": 4}, {"type": "text", "bbox": [70, 395, 540, 439], "content": "where is the Schur polynomial (see e.g. [27]) corresponding to the parti- tion , and where for any weight . In other words, is the Schur polynomial corresponding to , evaluated at roots of 1 determined by .", "index": 5}, {"type": "text", "bbox": [94, 439, 357, 453], "content": "The fusion (derived from the Pieri rule and (2.4))", "index": 6}, {"type": "interline_equation", "bbox": [223, 467, 386, 484], "content": "", "index": 7}, {"type": "text", "bbox": [71, 496, 304, 510], "content": "valid for and , will be useful.", "index": 8}, {"type": "text", "bbox": [95, 512, 371, 527], "content": "There are no exceptional fusion-symmetries for :", "index": 9}, {"type": "text", "bbox": [70, 533, 542, 565], "content": "Theorem 3.A. The fusion-symmetries for level are , for and any integer for which is coprime to .", "index": 10}, {"type": "text", "bbox": [70, 571, 541, 628], "content": "To avoid redundancies in the Theorem, for or take only. If we write , where is coprime to and , then the number of simple-current automorphisms will exactly equal , where is the Euler totient. The commute with each other, and with .", "index": 11}, {"type": "text", "bbox": [70, 630, 541, 686], "content": "For example, for when is odd, there is no nontrivial fusion-symmetry. When is even, there is exactly one, sending to (for even) or (for odd). For , there are either six or four fusion-symmetries, depending on whether or not 3 divides .", "index": 12}]
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[{"bbox": [95, 102, 541, 117], "content": "Our main task in this paper is to find and construct all fusion-symmetries for the", "parent_index": 0, "line_index": 0}, {"bbox": [69, 113, 544, 135], "content": "affine algebras , for simple . In this section we state the results, and in the next", "parent_index": 0, "line_index": 1}, {"bbox": [70, 132, 542, 148], "content": "section we prove the completeness of our lists. Recall the simple-current automorphism", "parent_index": 0, "line_index": 2}, {"bbox": [71, 146, 541, 162], "content": "and Galois automorphism defined in §2.3, and the notation . It will be", "parent_index": 0, "line_index": 3}, {"bbox": [68, 161, 541, 178], "content": "convenient to write for and level . We write for the group of symmetries", "parent_index": 0, "line_index": 4}, {"bbox": [70, 177, 245, 192], "content": "of the extended Dynkin diagram.", "parent_index": 0, "line_index": 5}, {"bbox": [69, 202, 218, 225], "content": "3.1. The algebra ,", "parent_index": 1, "line_index": 0}, {"bbox": [93, 228, 542, 247], "content": "Define and . The level highest weights of constitute", "parent_index": 2, "line_index": 0}, {"bbox": [70, 245, 541, 262], "content": "the set of -tuples of non-negative integers obeying .", "parent_index": 2, "line_index": 1}, {"bbox": [70, 259, 542, 276], "content": "The Dynkin diagram symmetries form the dihedral group ; it is generated", "parent_index": 2, "line_index": 2}, {"bbox": [70, 274, 540, 290], "content": "by the charge-conjugation and simple-current given by", "parent_index": 2, "line_index": 3}, {"bbox": [69, 290, 542, 309], "content": "and , with for . Note that", "parent_index": 2, "line_index": 4}, {"bbox": [71, 309, 159, 325], "content": ". for .", "parent_index": 2, "line_index": 5}, {"bbox": [94, 325, 393, 341], "content": "The Kac-Peterson relation (2.1b) for takes the form", "parent_index": 3, "line_index": 0}, {"bbox": [71, 397, 540, 414], "content": "where is the Schur polynomial (see e.g. [27]) corresponding to the parti-", "parent_index": 5, "line_index": 0}, {"bbox": [69, 411, 539, 430], "content": "tion , and where for any weight . In other words,", "parent_index": 5, "line_index": 1}, {"bbox": [69, 425, 519, 443], "content": "is the Schur polynomial corresponding to , evaluated at roots of 1 determined by .", "parent_index": 5, "line_index": 2}, {"bbox": [96, 441, 355, 455], "content": "The fusion (derived from the Pieri rule and (2.4))", "parent_index": 6, "line_index": 0}, {"bbox": [71, 498, 302, 511], "content": "valid for and , will be useful.", "parent_index": 8, "line_index": 0}, {"bbox": [93, 510, 372, 529], "content": "There are no exceptional fusion-symmetries for :", "parent_index": 9, "line_index": 0}, {"bbox": [93, 533, 543, 554], "content": "Theorem 3.A. The fusion-symmetries for level are , for and", "parent_index": 10, "line_index": 0}, {"bbox": [72, 551, 383, 568], "content": "any integer for which is coprime to .", "parent_index": 10, "line_index": 1}, {"bbox": [94, 573, 541, 587], "content": "To avoid redundancies in the Theorem, for or take only. If we", "parent_index": 11, "line_index": 0}, {"bbox": [72, 588, 541, 601], "content": "write , where is coprime to and , then the number of simple-current", "parent_index": 11, "line_index": 1}, {"bbox": [70, 601, 542, 617], "content": "automorphisms will exactly equal , where is the Euler totient. The commute", "parent_index": 11, "line_index": 2}, {"bbox": [72, 617, 224, 630], "content": "with each other, and with .", "parent_index": 11, "line_index": 3}, {"bbox": [95, 631, 540, 646], "content": "For example, for when is odd, there is no nontrivial fusion-symmetry. When", "parent_index": 12, "line_index": 0}, {"bbox": [69, 645, 541, 661], "content": "is even, there is exactly one, sending to (for even) or (for", "parent_index": 12, "line_index": 1}, {"bbox": [71, 659, 541, 675], "content": "odd). For , there are either six or four fusion-symmetries, depending on whether", "parent_index": 12, "line_index": 2}, {"bbox": [70, 674, 169, 689], "content": "or not 3 divides .", "parent_index": 12, "line_index": 3}]
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[]
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[{"bbox": [151, 116, 174, 130], "content": "X_{r}^{(1)}", "parent_index": 0, "subtype": "inline"}, {"bbox": [239, 120, 255, 131], "content": "X_{r}", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 148, 92, 161], "content": "\\pi[a]", "parent_index": 0, "subtype": "inline"}, {"bbox": [231, 148, 256, 161], "content": "\\pi\\{\\ell\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [429, 148, 484, 159], "content": "\\kappa=k\\!+\\!h^{\\vee}", "parent_index": 0, "subtype": "inline"}, {"bbox": [179, 164, 206, 178], "content": "X{_{r,k}}^{,}", "parent_index": 0, "subtype": "inline"}, {"bbox": [231, 161, 255, 176], "content": "\\cdot X_{r}^{(1)}", "parent_index": 0, "subtype": "inline"}, {"bbox": [310, 165, 320, 174], "content": "k'", "parent_index": 0, "subtype": "inline"}, {"bbox": [381, 166, 389, 174], "content": "_S", "parent_index": 0, "subtype": "inline"}, {"bbox": [161, 205, 183, 221], "content": "A_{r}^{(1)}", "parent_index": 1, "subtype": "inline"}, {"bbox": [190, 208, 218, 221], "content": "r\\geq1", "parent_index": 1, "subtype": "inline"}, {"bbox": [133, 232, 190, 243], "content": "\\overline{r}\\,=\\,r\\,+\\,1", "parent_index": 2, "subtype": "inline"}, {"bbox": [219, 233, 277, 243], "content": "\\;n\\;=\\;k\\,+\\,{\\overline{{r}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [345, 234, 352, 243], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [460, 228, 482, 244], "content": "A_{r}^{(1)}", "parent_index": 2, "subtype": "inline"}, {"bbox": [113, 246, 129, 259], "content": "P_{+}", "parent_index": 2, "subtype": "inline"}, {"bbox": [148, 247, 155, 257], "content": "\\overline{r}", "parent_index": 2, "subtype": "inline"}, {"bbox": [195, 246, 284, 259], "content": "\\lambda\\,=\\,(\\lambda_{0},.\\,.\\,.\\,,\\lambda_{r})", "parent_index": 2, "subtype": "inline"}, {"bbox": [467, 245, 537, 261], "content": "\\textstyle\\sum_{i=0}^{r}\\lambda_{i}\\;=\\;k", "parent_index": 2, "subtype": "inline"}, {"bbox": [392, 261, 451, 274], "content": "\\boldsymbol{S}\\;=\\;\\mathfrak{D}_{r+1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [214, 275, 224, 285], "content": "C", "parent_index": 2, "subtype": "inline"}, {"bbox": [333, 276, 341, 285], "content": "J", "parent_index": 2, "subtype": "inline"}, {"bbox": [395, 274, 540, 288], "content": "C\\lambda\\,=\\,(\\lambda_{0},\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [95, 292, 235, 306], "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-1}\\right)", "parent_index": 2, "subtype": "inline"}, {"bbox": [270, 293, 366, 306], "content": "Q_{J^{a}}(\\lambda)=a\\,t(\\lambda)/\\overline{{r}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [389, 290, 478, 308], "content": "\\begin{array}{r}{t(\\lambda)\\overset{\\mathrm{def}}{=}\\sum_{j=1}^{r}j\\lambda_{j}}\\end{array}", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 313, 108, 322], "content": "C=i d", "parent_index": 2, "subtype": "inline"}, {"bbox": [132, 309, 154, 325], "content": "{A}_{1}^{(1)}", "parent_index": 2, "subtype": "inline"}, {"bbox": [290, 328, 313, 340], "content": "A_{r,k}", "parent_index": 3, "subtype": "inline"}, {"bbox": [87, 352, 487, 383], "content": "\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\exp[-2\\pi\\mathrm{i}\\frac{t(\\lambda)\\,t(\\mu)}{\\kappa\\,\\overline{{r}}}]\\,\\,s_{(\\lambda)}(\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(1)}{\\kappa},\\ldots,\\exp[-2\\pi\\mathrm{i}\\frac{(\\mu+\\rho)(\\overline{{r}})}{\\kappa}]\\,\\ \\", "parent_index": 4, "subtype": "interline"}, {"bbox": [105, 399, 197, 412], "content": "s_{(\\lambda)}\\big(x_{1},\\ldots,x_{r+1}\\big)", "parent_index": 5, "subtype": "inline"}, {"bbox": [96, 413, 176, 426], "content": "(\\lambda(1),\\ldots,\\lambda(\\overline{{r}}))", "parent_index": 5, "subtype": "inline"}, {"bbox": [239, 411, 317, 427], "content": "\\textstyle\\nu(\\ell)=\\sum_{i=\\ell}^{r}\\nu_{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [397, 415, 405, 423], "content": "\\nu", "parent_index": 5, "subtype": "inline"}, {"bbox": [495, 414, 539, 426], "content": "S_{\\lambda\\mu}/S_{0\\mu}", "parent_index": 5, "subtype": "inline"}, {"bbox": [290, 428, 298, 438], "content": "\\lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [506, 432, 513, 440], "content": "\\mu", "parent_index": 5, "subtype": "inline"}, {"bbox": [223, 467, 386, 484], "content": "\\Lambda_{1}\\mathinner{\\left[\\boxtimes\\right]}\\Lambda_{\\ell}=\\Lambda_{\\ell+1}\\mathinner{\\left[\\textstyle{\\ H}\\right.\\left(\\Lambda_{1}+\\Lambda_{\\ell}\\right)\\,,}", "parent_index": 7, "subtype": "interline"}, {"bbox": [118, 500, 147, 511], "content": "k\\geq2", "parent_index": 8, "subtype": "inline"}, {"bbox": [174, 499, 223, 511], "content": "1\\leq\\ell<r", "parent_index": 8, "subtype": "inline"}, {"bbox": [345, 510, 367, 527], "content": "A_{r}^{(1)}", "parent_index": 9, "subtype": "inline"}, {"bbox": [323, 533, 346, 550], "content": "A_{r}^{(1)}", "parent_index": 10, "subtype": "inline"}, {"bbox": [375, 537, 384, 549], "content": "k", "parent_index": 10, "subtype": "inline"}, {"bbox": [407, 536, 442, 551], "content": "C^{i}\\pi[a]", "parent_index": 10, "subtype": "inline"}, {"bbox": [467, 537, 517, 551], "content": "i\\in\\{0,1\\}", "parent_index": 10, "subtype": "inline"}, {"bbox": [134, 553, 184, 565], "content": "0\\leq a\\leq r", "parent_index": 10, "subtype": "inline"}, {"bbox": [240, 552, 275, 564], "content": "1+k a", "parent_index": 10, "subtype": "inline"}, {"bbox": [350, 552, 378, 564], "content": "r+1", "parent_index": 10, "subtype": "inline"}, {"bbox": [330, 574, 362, 584], "content": "r\\,=\\,1", "parent_index": 11, "subtype": "inline"}, {"bbox": [381, 573, 414, 585], "content": "k\\,=\\,1", "parent_index": 11, "subtype": "inline"}, {"bbox": [445, 574, 475, 585], "content": "i\\,=\\,0", "parent_index": 11, "subtype": "inline"}, {"bbox": [102, 589, 149, 599], "content": "{\\overline{{r}}}\\,=\\,r^{\\prime}r^{\\prime\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [192, 589, 201, 598], "content": "r^{\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [280, 590, 287, 599], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [316, 588, 348, 601], "content": "r^{\\prime\\prime}|k^{\\infty}", "parent_index": 11, "subtype": "inline"}, {"bbox": [246, 603, 289, 616], "content": "r^{\\prime\\prime}{\\cdot}\\varphi(r^{\\prime})", "parent_index": 11, "subtype": "inline"}, {"bbox": [329, 605, 338, 615], "content": "\\varphi", "parent_index": 11, "subtype": "inline"}, {"bbox": [468, 601, 489, 616], "content": "\\pi[a]", "parent_index": 11, "subtype": "inline"}, {"bbox": [210, 618, 220, 627], "content": "C", "parent_index": 11, "subtype": "inline"}, {"bbox": [185, 633, 207, 645], "content": "A_{1,k}", "parent_index": 12, "subtype": "inline"}, {"bbox": [242, 633, 248, 641], "content": "k", "parent_index": 12, "subtype": "inline"}, {"bbox": [532, 633, 540, 642], "content": "k", "parent_index": 12, "subtype": "inline"}, {"bbox": [266, 647, 315, 658], "content": "\\lambda=\\lambda_{1}\\Lambda_{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [333, 647, 340, 656], "content": "\\lambda", "parent_index": 12, "subtype": "inline"}, {"bbox": [367, 647, 379, 658], "content": "\\lambda_{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [429, 646, 516, 659], "content": "J\\lambda=(k-\\lambda_{1})\\Lambda_{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [71, 662, 83, 672], "content": "\\lambda_{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [145, 662, 167, 674], "content": "A_{2,k}", "parent_index": 12, "subtype": "inline"}, {"bbox": [158, 676, 164, 685], "content": "k", "parent_index": 12, "subtype": "inline"}]
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[]
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3.2. The algebra $B_{r}^{(1)}$ , $r\geq3$
A weight $\lambda$ in $P_{+}$ satisfies $k=\lambda_{0}+\lambda_{1}+2\lambda_{2}+\cdot\cdot\cdot+2\lambda_{r-1}+\lambda_{r}$ , and $\kappa=k+2r-1$ . The charge-conjugation is trivial, but there is a simple-current: $J\lambda=\left(\lambda_{1},\lambda_{0},\lambda_{2},.\ldots,\lambda_{r}\right)$ . It has $Q(\lambda)=\lambda_{r}/2$ .
The only fusion products we need are
for all $1\leq i<r-1$ , $k>2$ , and $0<\ell<k$ , where we drop $\mathrm{\Delta}^{\prime}\Lambda_{r-1}+(\ell-2)\Lambda_{r}{}^{\prime}$ if $\ell=1$ . We will also use the character formula (2.1b)
$$
\chi_{\Lambda_{1}}[\lambda]={\frac{S_{\Lambda_{1}\lambda}}{S_{0\lambda}}}=2\sum_{\ell=1}^{r}\cos(2\pi{\frac{\lambda^{+}(\ell)}{\kappa}})+1~,
$$
where $\lambda^{+}(\ell)=(\lambda+\rho)(\ell)$ and
$$
\lambda(\ell)=\sum_{i=\ell}^{r-1}\lambda_{i}+\frac{1}{2}\lambda_{r}\ .
$$
For $k\,=\,2$ ( $\kappa\,=\,2r+1$ ) there are several Galois fusion-symmetries — one for each Galois automorphism, since S020 = 41κ is rational. In particular, define $\gamma^{i}=\gamma^{\kappa-i}=\Lambda_{i}$ for $i\,=\,1,2,\dots,r\,-\,1$ , and $\gamma^{r}\,=\,\gamma^{r+1}\,=\,2\Lambda_{r}$ . Then for any ${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$ coprime to $\kappa$ , $\pi\{m\}$ fixes $0$ and $J$ , sends $\gamma^{a}$ to $\gamma^{m a}$ (where the superscript is taken mod $\kappa$ ), and stabilises $\{\Lambda_{r},J\Lambda_{r}\}$ $(\pi\{m\}\Lambda_{r}=\Lambda_{r}$ iff the Jacobi symbol $\Bigl(\frac{\kappa}{m}\Bigr)$ equals $+1$ ).
Why is $k=2$ so special here? One reason is that rank-level duality associates $B_{r,2}$ with $\mathrm{u}(1)_{2r+1}$ , and it is easy to confirm that $\widehat{\mathrm{u(1)}}$ has a rich variety of fusion-symmetries (and modular invariants) coming from its si mple-currents. Also, the $B_{r,2}$ matrix $S$ formally looks like the character table of the dihedral group and for some $r$ actually equals the Kac-Peterson matrix $S$ associated to the dihedral group ${\mathfrak{D}}_{{\sqrt{\kappa}}}$ twisted by an appropriate 3- cocycle [5] — finite group modular data tends to have significantly more modular invariants and fusion-symmetries than e.g. affine modular data.
Theorem 3.B. The fusion-symmetries of $B_{r}^{(1)}$ level $k$ for $k\ \neq\ 2$ are $\pi[1]^{i}$ where $i\in\{0,1\}$ . For $k=2$ a fusion-symmetry will equal $\pi[1]^{i}\,\pi\{m\}$ for $i\in\{0,1\}$ and $m\in\mathbb{Z}_{\kappa}^{\times}$ , $1\leq m\leq r$ .
When $k=1$ , $\pi[1]$ is trivial. We have $\mathcal{F}(B_{r,2})\cong\mathbb{Z}_{2}\times(\mathbb{Z}_{2r+1}^{\times}/\{\pm1\}).$
3.3. The algebra $C_{r}^{(1)}$ , $r\geq2$
A weight $\lambda$ of $P_{+}$ satisfies $k=\lambda_{0}+\lambda_{1}+\cdot\cdot\cdot+\lambda_{r}$ and $\kappa=k+r+1$ . Charge-conjugation $C$ again is trivial, and there is a simple-current $J$ defined by $J\lambda=\left(\lambda_{r},\lambda_{r-1},\ldots,\lambda_{1},\lambda_{0}\right)$ , with $\begin{array}{r}{Q(\lambda)=(\sum_{j=1}^{r}j\lambda_{j})/2}\end{array}$ .
Choose any $\lambda\:\in\:P_{+}$ . The Young diagram for $\lambda$ is defined in the usual way: for $1\leq\ell\leq r$ , the $\ell$ th row consists of $\begin{array}{r}{\lambda(\ell)\overset{\mathrm{def}}{=}\sum_{i=\ell}^{r}\lambda_{i}}\end{array}$ boxes. Let $\tau\lambda$ denote the $C_{k,r}$ weight whose diagram is the transpose of that for $\lambda$ . (For this purpose the algebra $C_{1}$ may be identified with $A_{1}$ .) For example, $\tau\Lambda_{a}=a\tilde{\Lambda}_{1}$ , where we use tilde’s to denote the quantities of $C_{k,r}$ . In fact, $\tau:P_{+}(C_{r,k})\rightarrow P_{+}(C_{k,r})$ is a bijection. Then
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<html><body>
<p data-bbox="70 69 221 86">3.2. The algebra $B_{r}^{(1)}$ , $r\geq3$ </p>
<p data-bbox="70 92 542 136">A weight $\lambda$ in $P_{+}$ satisfies $k=\lambda_{0}+\lambda_{1}+2\lambda_{2}+\cdot\cdot\cdot+2\lambda_{r-1}+\lambda_{r}$ , and $\kappa=k+2r-1$ . The charge-conjugation is trivial, but there is a simple-current: $J\lambda=\left(\lambda_{1},\lambda_{0},\lambda_{2},.\ldots,\lambda_{r}\right)$ . It has $Q(\lambda)=\lambda_{r}/2$ . </p>
<p data-bbox="94 137 294 151">The only fusion products we need are </p>
<p data-bbox="69 205 541 235">for all $1\leq i<r-1$ , $k>2$ , and $0<\ell<k$ , where we drop $\mathrm{\Delta}^{\prime}\Lambda_{r-1}+(\ell-2)\Lambda_{r}{}^{\prime}$ if $\ell=1$ . We will also use the character formula (2.1b) </p>
<div class="equation" data-bbox="195 246 415 285">$$
\chi_{\Lambda_{1}}[\lambda]={\frac{S_{\Lambda_{1}\lambda}}{S_{0\lambda}}}=2\sum_{\ell=1}^{r}\cos(2\pi{\frac{\lambda^{+}(\ell)}{\kappa}})+1~,
$$</div>
<p data-bbox="70 295 227 310">where $\lambda^{+}(\ell)=(\lambda+\rho)(\ell)$ and </p>
<div class="equation" data-bbox="249 309 362 349">$$
\lambda(\ell)=\sum_{i=\ell}^{r-1}\lambda_{i}+\frac{1}{2}\lambda_{r}\ .
$$</div>
<p data-bbox="70 353 541 426">For $k\,=\,2$ ( $\kappa\,=\,2r+1$ ) there are several Galois fusion-symmetries — one for each Galois automorphism, since S020 = 41κ is rational. In particular, define $\gamma^{i}=\gamma^{\kappa-i}=\Lambda_{i}$ for $i\,=\,1,2,\dots,r\,-\,1$ , and $\gamma^{r}\,=\,\gamma^{r+1}\,=\,2\Lambda_{r}$ . Then for any ${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$ coprime to $\kappa$ , $\pi\{m\}$ fixes $0$ and $J$ , sends $\gamma^{a}$ to $\gamma^{m a}$ (where the superscript is taken mod $\kappa$ ), and stabilises $\{\Lambda_{r},J\Lambda_{r}\}$ $(\pi\{m\}\Lambda_{r}=\Lambda_{r}$ iff the Jacobi symbol $\Bigl(\frac{\kappa}{m}\Bigr)$ equals $+1$ ). </p>
<p data-bbox="70 426 541 530">Why is $k=2$ so special here? One reason is that rank-level duality associates $B_{r,2}$ with $\mathrm{u}(1)_{2r+1}$ , and it is easy to confirm that $\widehat{\mathrm{u(1)}}$ has a rich variety of fusion-symmetries (and modular invariants) coming from its si mple-currents. Also, the $B_{r,2}$ matrix $S$ formally looks like the character table of the dihedral group and for some $r$ actually equals the Kac-Peterson matrix $S$ associated to the dihedral group ${\mathfrak{D}}_{{\sqrt{\kappa}}}$ twisted by an appropriate 3- cocycle [5] — finite group modular data tends to have significantly more modular invariants and fusion-symmetries than e.g. affine modular data. </p>
<p data-bbox="70 535 542 581">Theorem 3.B. The fusion-symmetries of $B_{r}^{(1)}$ level $k$ for $k\ \neq\ 2$ are $\pi[1]^{i}$ where $i\in\{0,1\}$ . For $k=2$ a fusion-symmetry will equal $\pi[1]^{i}\,\pi\{m\}$ for $i\in\{0,1\}$ and $m\in\mathbb{Z}_{\kappa}^{\times}$ , $1\leq m\leq r$ . </p>
<p data-bbox="93 586 451 604">When $k=1$ , $\pi[1]$ is trivial. We have $\mathcal{F}(B_{r,2})\cong\mathbb{Z}_{2}\times(\mathbb{Z}_{2r+1}^{\times}/\{\pm1\}).$ </p>
<p data-bbox="71 616 220 633">3.3. The algebra $C_{r}^{(1)}$ , $r\geq2$ </p>
<p data-bbox="71 639 541 683">A weight $\lambda$ of $P_{+}$ satisfies $k=\lambda_{0}+\lambda_{1}+\cdot\cdot\cdot+\lambda_{r}$ and $\kappa=k+r+1$ . Charge-conjugation $C$ again is trivial, and there is a simple-current $J$ defined by $J\lambda=\left(\lambda_{r},\lambda_{r-1},\ldots,\lambda_{1},\lambda_{0}\right)$ , with $\begin{array}{r}{Q(\lambda)=(\sum_{j=1}^{r}j\lambda_{j})/2}\end{array}$ . </p>
<p data-bbox="70 683 540 716">Choose any $\lambda\:\in\:P_{+}$ . The Young diagram for $\lambda$ is defined in the usual way: for $1\leq\ell\leq r$ , the $\ell$ th row consists of $\begin{array}{r}{\lambda(\ell)\overset{\mathrm{def}}{=}\sum_{i=\ell}^{r}\lambda_{i}}\end{array}$ boxes. Let $\tau\lambda$ denote the $C_{k,r}$ weight whose diagram is the transpose of that for $\lambda$ . (For this purpose the algebra $C_{1}$ may be identified with $A_{1}$ .) For example, $\tau\Lambda_{a}=a\tilde{\Lambda}_{1}$ , where we use tilde’s to denote the quantities of $C_{k,r}$ . In fact, $\tau:P_{+}(C_{r,k})\rightarrow P_{+}(C_{k,r})$ is a bijection. Then </p>
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0002044v1
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[{"type": "text", "text": "3.2. The algebra $B_{r}^{(1)}$ , $r\\geq3$ ", "page_idx": 9}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}$ , and $\\kappa=k+2r-1$ . The charge-conjugation is trivial, but there is a simple-current: $J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)$ . It has $Q(\\lambda)=\\lambda_{r}/2$ . ", "page_idx": 9}, {"type": "text", "text": "The only fusion products we need are ", "page_idx": 9}, {"type": "text", "text": "for all $1\\leq i<r-1$ , $k>2$ , and $0<\\ell<k$ , where we drop $\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}$ if $\\ell=1$ . We will also use the character formula (2.1b) ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "where $\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)$ and ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "For $k\\,=\\,2$ ( $\\kappa\\,=\\,2r+1$ ) there are several Galois fusion-symmetries — one for each Galois automorphism, since S020 = 41κ is rational. In particular, define $\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}$ for $i\\,=\\,1,2,\\dots,r\\,-\\,1$ , and $\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}$ . Then for any ${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$ coprime to $\\kappa$ , $\\pi\\{m\\}$ fixes $0$ and $J$ , sends $\\gamma^{a}$ to $\\gamma^{m a}$ (where the superscript is taken mod $\\kappa$ ), and stabilises $\\{\\Lambda_{r},J\\Lambda_{r}\\}$ $(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}$ iff the Jacobi symbol $\\Bigl(\\frac{\\kappa}{m}\\Bigr)$ equals $+1$ ). ", "page_idx": 9}, {"type": "text", "text": "Why is $k=2$ so special here? One reason is that rank-level duality associates $B_{r,2}$ with $\\mathrm{u}(1)_{2r+1}$ , and it is easy to confirm that $\\widehat{\\mathrm{u(1)}}$ has a rich variety of fusion-symmetries (and modular invariants) coming from its si mple-currents. Also, the $B_{r,2}$ matrix $S$ formally looks like the character table of the dihedral group and for some $r$ actually equals the Kac-Peterson matrix $S$ associated to the dihedral group ${\\mathfrak{D}}_{{\\sqrt{\\kappa}}}$ twisted by an appropriate 3- cocycle [5] — finite group modular data tends to have significantly more modular invariants and fusion-symmetries than e.g. affine modular data. ", "page_idx": 9}, {"type": "text", "text": "Theorem 3.B. The fusion-symmetries of $B_{r}^{(1)}$ level $k$ for $k\\ \\neq\\ 2$ are $\\pi[1]^{i}$ where $i\\in\\{0,1\\}$ . For $k=2$ a fusion-symmetry will equal $\\pi[1]^{i}\\,\\pi\\{m\\}$ for $i\\in\\{0,1\\}$ and $m\\in\\mathbb{Z}_{\\kappa}^{\\times}$ , $1\\leq m\\leq r$ . ", "page_idx": 9}, {"type": "text", "text": "When $k=1$ , $\\pi[1]$ is trivial. We have $\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).$ ", "page_idx": 9}, {"type": "text", "text": "3.3. The algebra $C_{r}^{(1)}$ , $r\\geq2$ ", "page_idx": 9}, {"type": "text", "text": "A weight $\\lambda$ of $P_{+}$ satisfies $k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}$ and $\\kappa=k+r+1$ . Charge-conjugation $C$ again is trivial, and there is a simple-current $J$ defined by $J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)$ , with $\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}$ . ", "page_idx": 9}, {"type": "text", "text": "Choose any $\\lambda\\:\\in\\:P_{+}$ . The Young diagram for $\\lambda$ is defined in the usual way: for $1\\leq\\ell\\leq r$ , the $\\ell$ th row consists of $\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}$ boxes. Let $\\tau\\lambda$ denote the $C_{k,r}$ weight whose diagram is the transpose of that for $\\lambda$ . (For this purpose the algebra $C_{1}$ may be identified with $A_{1}$ .) For example, $\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}$ , where we use tilde’s to denote the quantities of $C_{k,r}$ . In fact, $\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})$ is a bijection. Then ", "page_idx": 9}]
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The algebra ", "type": "text"}, {"bbox": [161, 72, 183, 86], "score": 0.73, "content": "B_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [184, 67, 190, 92], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 77, 219, 87], "score": 0.74, "content": "r\\geq3", "type": "inline_equation", "height": 10, "width": 29}], "index": 0}], "index": 0}, {"type": "text", "bbox": [70, 92, 542, 136], "lines": [{"bbox": [95, 95, 541, 110], "spans": [{"bbox": [95, 95, 145, 110], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [145, 96, 154, 106], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [154, 95, 170, 110], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [171, 96, 187, 109], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [187, 95, 233, 110], "score": 1.0, "content": " satisfies", "type": "text"}, {"bbox": [234, 95, 429, 108], "score": 0.91, "content": "k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}", "type": "inline_equation", "height": 13, "width": 195}, {"bbox": [430, 95, 459, 110], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [459, 96, 537, 107], "score": 0.9, "content": "\\kappa=k+2r-1", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [537, 95, 541, 110], "score": 1.0, "content": ".", "type": "text"}], "index": 1}, {"bbox": [70, 109, 541, 125], "spans": [{"bbox": [70, 109, 408, 125], "score": 1.0, "content": "The charge-conjugation is trivial, but there is a simple-current: ", "type": "text"}, {"bbox": [408, 109, 536, 123], "score": 0.91, "content": "J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)", "type": "inline_equation", "height": 14, "width": 128}, {"bbox": [536, 109, 541, 125], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [69, 123, 176, 140], "spans": [{"bbox": [69, 123, 105, 140], "score": 1.0, "content": "It has ", "type": "text"}, {"bbox": [105, 123, 172, 138], "score": 0.92, "content": "Q(\\lambda)=\\lambda_{r}/2", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [172, 123, 176, 140], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2}, {"type": "text", "bbox": [94, 137, 294, 151], "lines": [{"bbox": [95, 138, 294, 152], "spans": [{"bbox": [95, 138, 294, 152], "score": 1.0, "content": "The only fusion products we need are", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [69, 205, 541, 235], "lines": [{"bbox": [70, 208, 541, 223], "spans": [{"bbox": [70, 208, 105, 223], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [106, 209, 175, 221], "score": 0.83, "content": "1\\leq i<r-1", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [175, 208, 180, 223], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [181, 208, 210, 221], "score": 0.68, "content": "k>2", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [211, 208, 240, 223], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [240, 208, 290, 221], "score": 0.91, "content": "0<\\ell<k", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [291, 208, 378, 223], "score": 1.0, "content": ", where we drop ", "type": "text"}, {"bbox": [379, 208, 473, 222], "score": 0.91, "content": "\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}", "type": "inline_equation", "height": 14, "width": 94}, {"bbox": [473, 208, 487, 223], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [487, 209, 514, 219], "score": 0.87, "content": "\\ell=1", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [515, 208, 541, 223], "score": 1.0, "content": ". We", "type": "text"}], "index": 5}, {"bbox": [71, 223, 288, 238], "spans": [{"bbox": [71, 223, 288, 238], "score": 1.0, "content": "will also use the character formula (2.1b)", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "interline_equation", "bbox": [195, 246, 415, 285], "lines": [{"bbox": [195, 246, 415, 285], "spans": [{"bbox": [195, 246, 415, 285], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 295, 227, 310], "lines": [{"bbox": [72, 297, 227, 312], "spans": [{"bbox": [72, 297, 105, 312], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 298, 203, 311], "score": 0.94, "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [203, 297, 227, 312], "score": 1.0, "content": " and", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [249, 309, 362, 349], "lines": [{"bbox": [249, 309, 362, 349], "spans": [{"bbox": [249, 309, 362, 349], "score": 0.94, "content": "\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [70, 353, 541, 426], "lines": [{"bbox": [93, 354, 541, 371], "spans": [{"bbox": [93, 354, 117, 371], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 358, 150, 367], "score": 0.89, "content": "k\\,=\\,2", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [150, 354, 158, 371], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [158, 357, 219, 368], "score": 0.69, "content": "\\kappa\\,=\\,2r+1", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [220, 354, 541, 371], "score": 1.0, "content": ") there are several Galois fusion-symmetries — one for each", "type": "text"}], "index": 10}, {"bbox": [69, 368, 541, 388], "spans": [{"bbox": [69, 368, 272, 388], "score": 1.0, "content": "Galois automorphism, since S020 = 41κ ", "type": "text"}, {"bbox": [267, 369, 441, 387], "score": 1.0, "content": "is rational. In particular, define ", "type": "text"}, {"bbox": [441, 371, 521, 384], "score": 0.94, "content": "\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [521, 369, 541, 387], "score": 1.0, "content": " for", "type": "text"}], "index": 11}, {"bbox": [71, 383, 540, 402], "spans": [{"bbox": [71, 387, 166, 398], "score": 0.9, "content": "i\\,=\\,1,2,\\dots,r\\,-\\,1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [166, 383, 198, 402], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [198, 385, 293, 398], "score": 0.92, "content": "\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [293, 383, 378, 402], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [379, 390, 389, 396], "score": 0.85, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [390, 383, 455, 402], "score": 1.0, "content": " coprime to ", "type": "text"}, {"bbox": [456, 390, 463, 396], "score": 0.69, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [463, 383, 470, 402], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [471, 386, 501, 398], "score": 0.93, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [501, 383, 533, 402], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [533, 387, 540, 396], "score": 0.39, "content": "0", "type": "inline_equation", "height": 9, "width": 7}], "index": 12}, {"bbox": [71, 399, 540, 414], "spans": [{"bbox": [71, 399, 95, 414], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 401, 103, 410], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [103, 399, 142, 414], "score": 1.0, "content": ", sends ", "type": "text"}, {"bbox": [142, 401, 155, 412], "score": 0.91, "content": "\\gamma^{a}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [155, 399, 173, 414], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [174, 401, 195, 412], "score": 0.91, "content": "\\gamma^{m a}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [196, 399, 393, 414], "score": 1.0, "content": " (where the superscript is taken mod ", "type": "text"}, {"bbox": [393, 404, 401, 410], "score": 0.84, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [401, 399, 487, 414], "score": 1.0, "content": "), and stabilises ", "type": "text"}, {"bbox": [487, 400, 540, 413], "score": 0.93, "content": "\\{\\Lambda_{r},J\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 53}], "index": 13}, {"bbox": [72, 412, 351, 429], "spans": [{"bbox": [72, 415, 149, 427], "score": 0.89, "content": "(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 77}, {"bbox": [149, 412, 266, 429], "score": 1.0, "content": " iff the Jacobi symbol ", "type": "text"}, {"bbox": [266, 414, 286, 428], "score": 0.88, "content": "\\Bigl(\\frac{\\kappa}{m}\\Bigr)", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [287, 412, 326, 429], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [327, 416, 342, 425], "score": 0.54, "content": "+1", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [343, 412, 351, 429], "score": 1.0, "content": ").", "type": "text"}], "index": 14}], "index": 12}, {"type": "text", "bbox": [70, 426, 541, 530], "lines": [{"bbox": [94, 427, 541, 443], "spans": [{"bbox": [94, 427, 133, 443], "score": 1.0, "content": "Why is", "type": "text"}, {"bbox": [134, 430, 162, 438], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [163, 427, 491, 443], "score": 1.0, "content": " so special here? One reason is that rank-level duality associates", "type": "text"}, {"bbox": [492, 430, 513, 442], "score": 0.95, "content": "B_{r,2}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [514, 427, 541, 443], "score": 1.0, "content": " with", "type": "text"}], "index": 15}, {"bbox": [71, 442, 541, 460], "spans": [{"bbox": [71, 447, 115, 459], "score": 0.93, "content": "\\mathrm{u}(1)_{2r+1}", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [116, 445, 281, 460], "score": 1.0, "content": ", and it is easy to confirm that", "type": "text"}, {"bbox": [281, 442, 304, 459], "score": 0.9, "content": "\\widehat{\\mathrm{u(1)}}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [304, 445, 541, 460], "score": 1.0, "content": " has a rich variety of fusion-symmetries (and", "type": "text"}], "index": 16}, {"bbox": [72, 460, 540, 474], "spans": [{"bbox": [72, 460, 415, 474], "score": 1.0, "content": "modular invariants) coming from its si mple-currents. Also, the ", "type": "text"}, {"bbox": [416, 461, 438, 474], "score": 0.92, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [438, 460, 483, 474], "score": 1.0, "content": " matrix ", "type": "text"}, {"bbox": [483, 461, 491, 470], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [492, 460, 540, 474], "score": 1.0, "content": " formally", "type": "text"}], "index": 17}, {"bbox": [70, 473, 541, 488], "spans": [{"bbox": [70, 473, 427, 488], "score": 1.0, "content": "looks like the character table of the dihedral group and for some ", "type": "text"}, {"bbox": [427, 479, 434, 484], "score": 0.8, "content": "r", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [434, 473, 541, 488], "score": 1.0, "content": " actually equals the", "type": "text"}], "index": 18}, {"bbox": [69, 487, 541, 504], "spans": [{"bbox": [69, 487, 182, 504], "score": 1.0, "content": "Kac-Peterson matrix ", "type": "text"}, {"bbox": [183, 490, 191, 499], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [191, 487, 363, 504], "score": 1.0, "content": " associated to the dihedral group", "type": "text"}, {"bbox": [364, 488, 389, 504], "score": 0.91, "content": "{\\mathfrak{D}}_{{\\sqrt{\\kappa}}}", "type": "inline_equation", "height": 16, "width": 25}, {"bbox": [390, 487, 541, 504], "score": 1.0, "content": " twisted by an appropriate 3-", "type": "text"}], "index": 19}, {"bbox": [69, 503, 541, 518], "spans": [{"bbox": [69, 503, 541, 518], "score": 1.0, "content": "cocycle [5] — finite group modular data tends to have significantly more modular invariants", "type": "text"}], "index": 20}, {"bbox": [70, 517, 349, 532], "spans": [{"bbox": [70, 517, 349, 532], "score": 1.0, "content": "and fusion-symmetries than e.g. affine modular data.", "type": "text"}], "index": 21}], "index": 18}, {"type": "text", "bbox": [70, 535, 542, 581], "lines": [{"bbox": [93, 536, 542, 556], "spans": [{"bbox": [93, 537, 326, 556], "score": 1.0, "content": "Theorem 3.B. The fusion-symmetries of ", "type": "text"}, {"bbox": [327, 536, 350, 552], "score": 0.9, "content": "B_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [351, 537, 383, 556], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [383, 539, 392, 551], "score": 0.71, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [392, 537, 416, 556], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [416, 539, 453, 553], "score": 0.9, "content": "k\\ \\neq\\ 2", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [453, 537, 479, 556], "score": 1.0, "content": " are ", "type": "text"}, {"bbox": [479, 539, 504, 553], "score": 0.91, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [505, 537, 542, 556], "score": 1.0, "content": " where", "type": "text"}], "index": 22}, {"bbox": [71, 553, 539, 570], "spans": [{"bbox": [71, 555, 120, 568], "score": 0.93, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [120, 553, 151, 570], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [151, 556, 181, 565], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [181, 553, 338, 570], "score": 1.0, "content": " a fusion-symmetry will equal ", "type": "text"}, {"bbox": [339, 553, 395, 568], "score": 0.93, "content": "\\pi[1]^{i}\\,\\pi\\{m\\}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [396, 553, 417, 570], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [417, 554, 467, 568], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [468, 553, 493, 570], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [493, 554, 536, 568], "score": 0.91, "content": "m\\in\\mathbb{Z}_{\\kappa}^{\\times}", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [536, 553, 539, 570], "score": 1.0, "content": ",", "type": "text"}], "index": 23}, {"bbox": [71, 568, 131, 585], "spans": [{"bbox": [71, 570, 126, 581], "score": 0.88, "content": "1\\leq m\\leq r", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [126, 568, 131, 585], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23}, {"type": "text", "bbox": [93, 586, 451, 604], "lines": [{"bbox": [95, 587, 449, 606], "spans": [{"bbox": [95, 587, 129, 606], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 591, 159, 600], "score": 0.59, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [159, 587, 165, 606], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [165, 589, 186, 603], "score": 0.41, "content": "\\pi[1]", "type": "inline_equation", "height": 14, "width": 21}, {"bbox": [187, 587, 290, 606], "score": 1.0, "content": " is trivial. We have ", "type": "text"}, {"bbox": [290, 588, 449, 604], "score": 0.92, "content": "\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).", "type": "inline_equation", "height": 16, "width": 159}], "index": 25}], "index": 25}, {"type": "text", "bbox": [71, 616, 220, 633], "lines": [{"bbox": [68, 613, 218, 638], "spans": [{"bbox": [68, 613, 160, 638], "score": 1.0, "content": "3.3. The algebra ", "type": "text"}, {"bbox": [161, 618, 183, 633], "score": 0.75, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 22}, {"bbox": [183, 613, 190, 638], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 623, 218, 633], "score": 0.66, "content": "r\\geq2", "type": "inline_equation", "height": 10, "width": 28}], "index": 26}], "index": 26}, {"type": "text", "bbox": [71, 639, 541, 683], "lines": [{"bbox": [96, 642, 540, 656], "spans": [{"bbox": [96, 642, 144, 656], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 643, 152, 653], "score": 0.82, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [153, 642, 167, 656], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [167, 644, 183, 655], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [183, 642, 230, 656], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [230, 642, 339, 654], "score": 0.92, "content": "k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}", "type": "inline_equation", "height": 12, "width": 109}, {"bbox": [339, 642, 364, 656], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [364, 642, 429, 653], "score": 0.9, "content": "\\kappa=k+r+1", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [430, 642, 540, 656], "score": 1.0, "content": ". Charge-conjugation", "type": "text"}], "index": 27}, {"bbox": [71, 656, 540, 671], "spans": [{"bbox": [71, 658, 81, 667], "score": 0.83, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [81, 656, 325, 671], "score": 1.0, "content": " again is trivial, and there is a simple-current ", "type": "text"}, {"bbox": [325, 658, 334, 667], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [334, 656, 396, 671], "score": 1.0, "content": " defined by ", "type": "text"}, {"bbox": [396, 657, 536, 670], "score": 0.9, "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)", "type": "inline_equation", "height": 13, "width": 140}, {"bbox": [536, 656, 540, 671], "score": 1.0, "content": ",", "type": "text"}], "index": 28}, {"bbox": [70, 669, 217, 688], "spans": [{"bbox": [70, 669, 98, 688], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [99, 671, 213, 686], "score": 0.92, "content": "\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}", "type": "inline_equation", "height": 15, "width": 114}, {"bbox": [214, 669, 217, 688], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 28}, {"type": "text", "bbox": [70, 683, 540, 716], "lines": [{"bbox": [94, 683, 541, 702], "spans": [{"bbox": [94, 683, 162, 702], "score": 1.0, "content": "Choose any ", "type": "text"}, {"bbox": [162, 686, 205, 698], "score": 0.88, "content": "\\lambda\\:\\in\\:P_{+}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [206, 683, 352, 702], "score": 1.0, "content": ". The Young diagram for ", "type": "text"}, {"bbox": [352, 686, 360, 695], "score": 0.83, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [360, 683, 541, 702], "score": 1.0, "content": " is defined in the usual way: for", "type": "text"}], "index": 30}, {"bbox": [71, 699, 541, 719], "spans": [{"bbox": [71, 704, 123, 715], "score": 0.89, "content": "1\\leq\\ell\\leq r", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [124, 699, 151, 719], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [151, 704, 157, 713], "score": 0.54, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [157, 699, 254, 719], "score": 1.0, "content": "th row consists of ", "type": "text"}, {"bbox": [254, 700, 335, 717], "score": 0.93, "content": "\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}", "type": "inline_equation", "height": 17, "width": 81}, {"bbox": [336, 699, 400, 719], "score": 1.0, "content": " boxes. Let ", "type": "text"}, {"bbox": [401, 704, 415, 713], "score": 0.9, "content": "\\tau\\lambda", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [415, 699, 479, 719], "score": 1.0, "content": " denote the ", "type": "text"}, {"bbox": [479, 704, 501, 717], "score": 0.93, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [501, 699, 541, 719], "score": 1.0, "content": " weight", "type": "text"}], "index": 31}], "index": 30.5}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [195, 246, 415, 285], "lines": [{"bbox": [195, 246, 415, 285], "spans": [{"bbox": [195, 246, 415, 285], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [249, 309, 362, 349], "lines": [{"bbox": [249, 309, 362, 349], "spans": [{"bbox": [249, 309, 362, 349], "score": 0.94, "content": "\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 313, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 69, 221, 86], "lines": [{"bbox": [68, 67, 219, 92], "spans": [{"bbox": [68, 67, 160, 92], "score": 1.0, "content": "3.2. The algebra ", "type": "text"}, {"bbox": [161, 72, 183, 86], "score": 0.73, "content": "B_{r}^{(1)}", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [184, 67, 190, 92], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 77, 219, 87], "score": 0.74, "content": "r\\geq3", "type": "inline_equation", "height": 10, "width": 29}], "index": 0}], "index": 0, "bbox_fs": [68, 67, 219, 92]}, {"type": "text", "bbox": [70, 92, 542, 136], "lines": [{"bbox": [95, 95, 541, 110], "spans": [{"bbox": [95, 95, 145, 110], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [145, 96, 154, 106], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [154, 95, 170, 110], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [171, 96, 187, 109], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [187, 95, 233, 110], "score": 1.0, "content": " satisfies", "type": "text"}, {"bbox": [234, 95, 429, 108], "score": 0.91, "content": "k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}", "type": "inline_equation", "height": 13, "width": 195}, {"bbox": [430, 95, 459, 110], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [459, 96, 537, 107], "score": 0.9, "content": "\\kappa=k+2r-1", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [537, 95, 541, 110], "score": 1.0, "content": ".", "type": "text"}], "index": 1}, {"bbox": [70, 109, 541, 125], "spans": [{"bbox": [70, 109, 408, 125], "score": 1.0, "content": "The charge-conjugation is trivial, but there is a simple-current: ", "type": "text"}, {"bbox": [408, 109, 536, 123], "score": 0.91, "content": "J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)", "type": "inline_equation", "height": 14, "width": 128}, {"bbox": [536, 109, 541, 125], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [69, 123, 176, 140], "spans": [{"bbox": [69, 123, 105, 140], "score": 1.0, "content": "It has ", "type": "text"}, {"bbox": [105, 123, 172, 138], "score": 0.92, "content": "Q(\\lambda)=\\lambda_{r}/2", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [172, 123, 176, 140], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2, "bbox_fs": [69, 95, 541, 140]}, {"type": "text", "bbox": [94, 137, 294, 151], "lines": [{"bbox": [95, 138, 294, 152], "spans": [{"bbox": [95, 138, 294, 152], "score": 1.0, "content": "The only fusion products we need are", "type": "text"}], "index": 4}], "index": 4, "bbox_fs": [95, 138, 294, 152]}, {"type": "text", "bbox": [69, 205, 541, 235], "lines": [{"bbox": [70, 208, 541, 223], "spans": [{"bbox": [70, 208, 105, 223], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [106, 209, 175, 221], "score": 0.83, "content": "1\\leq i<r-1", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [175, 208, 180, 223], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [181, 208, 210, 221], "score": 0.68, "content": "k>2", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [211, 208, 240, 223], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [240, 208, 290, 221], "score": 0.91, "content": "0<\\ell<k", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [291, 208, 378, 223], "score": 1.0, "content": ", where we drop ", "type": "text"}, {"bbox": [379, 208, 473, 222], "score": 0.91, "content": "\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}", "type": "inline_equation", "height": 14, "width": 94}, {"bbox": [473, 208, 487, 223], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [487, 209, 514, 219], "score": 0.87, "content": "\\ell=1", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [515, 208, 541, 223], "score": 1.0, "content": ". We", "type": "text"}], "index": 5}, {"bbox": [71, 223, 288, 238], "spans": [{"bbox": [71, 223, 288, 238], "score": 1.0, "content": "will also use the character formula (2.1b)", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [70, 208, 541, 238]}, {"type": "interline_equation", "bbox": [195, 246, 415, 285], "lines": [{"bbox": [195, 246, 415, 285], "spans": [{"bbox": [195, 246, 415, 285], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 295, 227, 310], "lines": [{"bbox": [72, 297, 227, 312], "spans": [{"bbox": [72, 297, 105, 312], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 298, 203, 311], "score": 0.94, "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [203, 297, 227, 312], "score": 1.0, "content": " and", "type": "text"}], "index": 8}], "index": 8, "bbox_fs": [72, 297, 227, 312]}, {"type": "interline_equation", "bbox": [249, 309, 362, 349], "lines": [{"bbox": [249, 309, 362, 349], "spans": [{"bbox": [249, 309, 362, 349], "score": 0.94, "content": "\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [70, 353, 541, 426], "lines": [{"bbox": [93, 354, 541, 371], "spans": [{"bbox": [93, 354, 117, 371], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 358, 150, 367], "score": 0.89, "content": "k\\,=\\,2", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [150, 354, 158, 371], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [158, 357, 219, 368], "score": 0.69, "content": "\\kappa\\,=\\,2r+1", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [220, 354, 541, 371], "score": 1.0, "content": ") there are several Galois fusion-symmetries — one for each", "type": "text"}], "index": 10}, {"bbox": [69, 368, 541, 388], "spans": [{"bbox": [69, 368, 272, 388], "score": 1.0, "content": "Galois automorphism, since S020 = 41κ ", "type": "text"}, {"bbox": [267, 369, 441, 387], "score": 1.0, "content": "is rational. In particular, define ", "type": "text"}, {"bbox": [441, 371, 521, 384], "score": 0.94, "content": "\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [521, 369, 541, 387], "score": 1.0, "content": " for", "type": "text"}], "index": 11}, {"bbox": [71, 383, 540, 402], "spans": [{"bbox": [71, 387, 166, 398], "score": 0.9, "content": "i\\,=\\,1,2,\\dots,r\\,-\\,1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [166, 383, 198, 402], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [198, 385, 293, 398], "score": 0.92, "content": "\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [293, 383, 378, 402], "score": 1.0, "content": ". Then for any ", "type": "text"}, {"bbox": [379, 390, 389, 396], "score": 0.85, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [390, 383, 455, 402], "score": 1.0, "content": " coprime to ", "type": "text"}, {"bbox": [456, 390, 463, 396], "score": 0.69, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [463, 383, 470, 402], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [471, 386, 501, 398], "score": 0.93, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [501, 383, 533, 402], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [533, 387, 540, 396], "score": 0.39, "content": "0", "type": "inline_equation", "height": 9, "width": 7}], "index": 12}, {"bbox": [71, 399, 540, 414], "spans": [{"bbox": [71, 399, 95, 414], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 401, 103, 410], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [103, 399, 142, 414], "score": 1.0, "content": ", sends ", "type": "text"}, {"bbox": [142, 401, 155, 412], "score": 0.91, "content": "\\gamma^{a}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [155, 399, 173, 414], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [174, 401, 195, 412], "score": 0.91, "content": "\\gamma^{m a}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [196, 399, 393, 414], "score": 1.0, "content": " (where the superscript is taken mod ", "type": "text"}, {"bbox": [393, 404, 401, 410], "score": 0.84, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [401, 399, 487, 414], "score": 1.0, "content": "), and stabilises ", "type": "text"}, {"bbox": [487, 400, 540, 413], "score": 0.93, "content": "\\{\\Lambda_{r},J\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 53}], "index": 13}, {"bbox": [72, 412, 351, 429], "spans": [{"bbox": [72, 415, 149, 427], "score": 0.89, "content": "(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 77}, {"bbox": [149, 412, 266, 429], "score": 1.0, "content": " iff the Jacobi symbol ", "type": "text"}, {"bbox": [266, 414, 286, 428], "score": 0.88, "content": "\\Bigl(\\frac{\\kappa}{m}\\Bigr)", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [287, 412, 326, 429], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [327, 416, 342, 425], "score": 0.54, "content": "+1", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [343, 412, 351, 429], "score": 1.0, "content": ").", "type": "text"}], "index": 14}], "index": 12, "bbox_fs": [69, 354, 541, 429]}, {"type": "text", "bbox": [70, 426, 541, 530], "lines": [{"bbox": [94, 427, 541, 443], "spans": [{"bbox": [94, 427, 133, 443], "score": 1.0, "content": "Why is", "type": "text"}, {"bbox": [134, 430, 162, 438], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [163, 427, 491, 443], "score": 1.0, "content": " so special here? One reason is that rank-level duality associates", "type": "text"}, {"bbox": [492, 430, 513, 442], "score": 0.95, "content": "B_{r,2}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [514, 427, 541, 443], "score": 1.0, "content": " with", "type": "text"}], "index": 15}, {"bbox": [71, 442, 541, 460], "spans": [{"bbox": [71, 447, 115, 459], "score": 0.93, "content": "\\mathrm{u}(1)_{2r+1}", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [116, 445, 281, 460], "score": 1.0, "content": ", and it is easy to confirm that", "type": "text"}, {"bbox": [281, 442, 304, 459], "score": 0.9, "content": "\\widehat{\\mathrm{u(1)}}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [304, 445, 541, 460], "score": 1.0, "content": " has a rich variety of fusion-symmetries (and", "type": "text"}], "index": 16}, {"bbox": [72, 460, 540, 474], "spans": [{"bbox": [72, 460, 415, 474], "score": 1.0, "content": "modular invariants) coming from its si mple-currents. Also, the ", "type": "text"}, {"bbox": [416, 461, 438, 474], "score": 0.92, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [438, 460, 483, 474], "score": 1.0, "content": " matrix ", "type": "text"}, {"bbox": [483, 461, 491, 470], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [492, 460, 540, 474], "score": 1.0, "content": " formally", "type": "text"}], "index": 17}, {"bbox": [70, 473, 541, 488], "spans": [{"bbox": [70, 473, 427, 488], "score": 1.0, "content": "looks like the character table of the dihedral group and for some ", "type": "text"}, {"bbox": [427, 479, 434, 484], "score": 0.8, "content": "r", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [434, 473, 541, 488], "score": 1.0, "content": " actually equals the", "type": "text"}], "index": 18}, {"bbox": [69, 487, 541, 504], "spans": [{"bbox": [69, 487, 182, 504], "score": 1.0, "content": "Kac-Peterson matrix ", "type": "text"}, {"bbox": [183, 490, 191, 499], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [191, 487, 363, 504], "score": 1.0, "content": " associated to the dihedral group", "type": "text"}, {"bbox": [364, 488, 389, 504], "score": 0.91, "content": "{\\mathfrak{D}}_{{\\sqrt{\\kappa}}}", "type": "inline_equation", "height": 16, "width": 25}, {"bbox": [390, 487, 541, 504], "score": 1.0, "content": " twisted by an appropriate 3-", "type": "text"}], "index": 19}, {"bbox": [69, 503, 541, 518], "spans": [{"bbox": [69, 503, 541, 518], "score": 1.0, "content": "cocycle [5] — finite group modular data tends to have significantly more modular invariants", "type": "text"}], "index": 20}, {"bbox": [70, 517, 349, 532], "spans": [{"bbox": [70, 517, 349, 532], "score": 1.0, "content": "and fusion-symmetries than e.g. affine modular data.", "type": "text"}], "index": 21}], "index": 18, "bbox_fs": [69, 427, 541, 532]}, {"type": "text", "bbox": [70, 535, 542, 581], "lines": [{"bbox": [93, 536, 542, 556], "spans": [{"bbox": [93, 537, 326, 556], "score": 1.0, "content": "Theorem 3.B. The fusion-symmetries of ", "type": "text"}, {"bbox": [327, 536, 350, 552], "score": 0.9, "content": "B_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [351, 537, 383, 556], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [383, 539, 392, 551], "score": 0.71, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [392, 537, 416, 556], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [416, 539, 453, 553], "score": 0.9, "content": "k\\ \\neq\\ 2", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [453, 537, 479, 556], "score": 1.0, "content": " are ", "type": "text"}, {"bbox": [479, 539, 504, 553], "score": 0.91, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [505, 537, 542, 556], "score": 1.0, "content": " where", "type": "text"}], "index": 22}, {"bbox": [71, 553, 539, 570], "spans": [{"bbox": [71, 555, 120, 568], "score": 0.93, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [120, 553, 151, 570], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [151, 556, 181, 565], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [181, 553, 338, 570], "score": 1.0, "content": " a fusion-symmetry will equal ", "type": "text"}, {"bbox": [339, 553, 395, 568], "score": 0.93, "content": "\\pi[1]^{i}\\,\\pi\\{m\\}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [396, 553, 417, 570], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [417, 554, 467, 568], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [468, 553, 493, 570], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [493, 554, 536, 568], "score": 0.91, "content": "m\\in\\mathbb{Z}_{\\kappa}^{\\times}", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [536, 553, 539, 570], "score": 1.0, "content": ",", "type": "text"}], "index": 23}, {"bbox": [71, 568, 131, 585], "spans": [{"bbox": [71, 570, 126, 581], "score": 0.88, "content": "1\\leq m\\leq r", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [126, 568, 131, 585], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23, "bbox_fs": [71, 536, 542, 585]}, {"type": "text", "bbox": [93, 586, 451, 604], "lines": [{"bbox": [95, 587, 449, 606], "spans": [{"bbox": [95, 587, 129, 606], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 591, 159, 600], "score": 0.59, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [159, 587, 165, 606], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [165, 589, 186, 603], "score": 0.41, "content": "\\pi[1]", "type": "inline_equation", "height": 14, "width": 21}, {"bbox": [187, 587, 290, 606], "score": 1.0, "content": " is trivial. We have ", "type": "text"}, {"bbox": [290, 588, 449, 604], "score": 0.92, "content": "\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).", "type": "inline_equation", "height": 16, "width": 159}], "index": 25}], "index": 25, "bbox_fs": [95, 587, 449, 606]}, {"type": "text", "bbox": [71, 616, 220, 633], "lines": [{"bbox": [68, 613, 218, 638], "spans": [{"bbox": [68, 613, 160, 638], "score": 1.0, "content": "3.3. The algebra ", "type": "text"}, {"bbox": [161, 618, 183, 633], "score": 0.75, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 22}, {"bbox": [183, 613, 190, 638], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [190, 623, 218, 633], "score": 0.66, "content": "r\\geq2", "type": "inline_equation", "height": 10, "width": 28}], "index": 26}], "index": 26, "bbox_fs": [68, 613, 218, 638]}, {"type": "text", "bbox": [71, 639, 541, 683], "lines": [{"bbox": [96, 642, 540, 656], "spans": [{"bbox": [96, 642, 144, 656], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 643, 152, 653], "score": 0.82, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [153, 642, 167, 656], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [167, 644, 183, 655], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [183, 642, 230, 656], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [230, 642, 339, 654], "score": 0.92, "content": "k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}", "type": "inline_equation", "height": 12, "width": 109}, {"bbox": [339, 642, 364, 656], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [364, 642, 429, 653], "score": 0.9, "content": "\\kappa=k+r+1", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [430, 642, 540, 656], "score": 1.0, "content": ". Charge-conjugation", "type": "text"}], "index": 27}, {"bbox": [71, 656, 540, 671], "spans": [{"bbox": [71, 658, 81, 667], "score": 0.83, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [81, 656, 325, 671], "score": 1.0, "content": " again is trivial, and there is a simple-current ", "type": "text"}, {"bbox": [325, 658, 334, 667], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [334, 656, 396, 671], "score": 1.0, "content": " defined by ", "type": "text"}, {"bbox": [396, 657, 536, 670], "score": 0.9, "content": "J\\lambda=\\left(\\lambda_{r},\\lambda_{r-1},\\ldots,\\lambda_{1},\\lambda_{0}\\right)", "type": "inline_equation", "height": 13, "width": 140}, {"bbox": [536, 656, 540, 671], "score": 1.0, "content": ",", "type": "text"}], "index": 28}, {"bbox": [70, 669, 217, 688], "spans": [{"bbox": [70, 669, 98, 688], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [99, 671, 213, 686], "score": 0.92, "content": "\\begin{array}{r}{Q(\\lambda)=(\\sum_{j=1}^{r}j\\lambda_{j})/2}\\end{array}", "type": "inline_equation", "height": 15, "width": 114}, {"bbox": [214, 669, 217, 688], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 28, "bbox_fs": [70, 642, 540, 688]}, {"type": "text", "bbox": [70, 683, 540, 716], "lines": [{"bbox": [94, 683, 541, 702], "spans": [{"bbox": [94, 683, 162, 702], "score": 1.0, "content": "Choose any ", "type": "text"}, {"bbox": [162, 686, 205, 698], "score": 0.88, "content": "\\lambda\\:\\in\\:P_{+}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [206, 683, 352, 702], "score": 1.0, "content": ". The Young diagram for ", "type": "text"}, {"bbox": [352, 686, 360, 695], "score": 0.83, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [360, 683, 541, 702], "score": 1.0, "content": " is defined in the usual way: for", "type": "text"}], "index": 30}, {"bbox": [71, 699, 541, 719], "spans": [{"bbox": [71, 704, 123, 715], "score": 0.89, "content": "1\\leq\\ell\\leq r", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [124, 699, 151, 719], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [151, 704, 157, 713], "score": 0.54, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [157, 699, 254, 719], "score": 1.0, "content": "th row consists of ", "type": "text"}, {"bbox": [254, 700, 335, 717], "score": 0.93, "content": "\\begin{array}{r}{\\lambda(\\ell)\\overset{\\mathrm{def}}{=}\\sum_{i=\\ell}^{r}\\lambda_{i}}\\end{array}", "type": "inline_equation", "height": 17, "width": 81}, {"bbox": [336, 699, 400, 719], "score": 1.0, "content": " boxes. Let ", "type": "text"}, {"bbox": [401, 704, 415, 713], "score": 0.9, "content": "\\tau\\lambda", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [415, 699, 479, 719], "score": 1.0, "content": " denote the ", "type": "text"}, {"bbox": [479, 704, 501, 717], "score": 0.93, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [501, 699, 541, 719], "score": 1.0, "content": " weight", "type": "text"}], "index": 31}, {"bbox": [72, 74, 541, 88], "spans": [{"bbox": [72, 74, 303, 88], "score": 1.0, "content": "whose diagram is the transpose of that for ", "type": "text", "cross_page": true}, {"bbox": [303, 75, 311, 84], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [311, 74, 481, 88], "score": 1.0, "content": ". (For this purpose the algebra ", "type": "text", "cross_page": true}, {"bbox": [482, 75, 495, 86], "score": 0.92, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [496, 74, 541, 88], "score": 1.0, "content": " may be", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [70, 87, 541, 103], "spans": [{"bbox": [70, 87, 149, 103], "score": 1.0, "content": "identified with ", "type": "text", "cross_page": true}, {"bbox": [149, 90, 164, 101], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 11, "width": 15, "cross_page": true}, {"bbox": [164, 87, 246, 103], "score": 1.0, "content": ".) For example,", "type": "text", "cross_page": true}, {"bbox": [247, 87, 304, 101], "score": 0.94, "content": "\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 57, "cross_page": true}, {"bbox": [304, 87, 541, 103], "score": 1.0, "content": ", where we use tilde’s to denote the quantities", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [70, 102, 394, 117], "spans": [{"bbox": [70, 102, 84, 117], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [85, 104, 107, 117], "score": 0.93, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 22, "cross_page": true}, {"bbox": [107, 102, 157, 117], "score": 1.0, "content": ". In fact, ", "type": "text", "cross_page": true}, {"bbox": [157, 103, 286, 117], "score": 0.94, "content": "\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 129, "cross_page": true}, {"bbox": [286, 102, 394, 117], "score": 1.0, "content": " is a bijection. Then", "type": "text", "cross_page": true}], "index": 2}], "index": 30.5, "bbox_fs": [71, 683, 541, 719]}]}
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[{"type": "text", "bbox": [70, 69, 221, 86], "content": "3.2. The algebra ,", "index": 0}, {"type": "text", "bbox": [70, 92, 542, 136], "content": "A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current: . It has .", "index": 1}, {"type": "text", "bbox": [94, 137, 294, 151], "content": "The only fusion products we need are", "index": 2}, {"type": "text", "bbox": [69, 205, 541, 235], "content": "for all , , and , where we drop if . We will also use the character formula (2.1b)", "index": 3}, {"type": "interline_equation", "bbox": [195, 246, 415, 285], "content": "", "index": 4}, {"type": "text", "bbox": [70, 295, 227, 310], "content": "where and", "index": 5}, {"type": "interline_equation", "bbox": [249, 309, 362, 349], "content": "", "index": 6}, {"type": "text", "bbox": [70, 353, 541, 426], "content": "For ( ) there are several Galois fusion-symmetries — one for each Galois automorphism, since S020 = 41κ is rational. In particular, define for , and . Then for any coprime to , fixes and , sends to (where the superscript is taken mod ), and stabilises iff the Jacobi symbol equals ).", "index": 7}, {"type": "text", "bbox": [70, 426, 541, 530], "content": "Why is so special here? One reason is that rank-level duality associates with , and it is easy to confirm that has a rich variety of fusion-symmetries (and modular invariants) coming from its si mple-currents. Also, the matrix formally looks like the character table of the dihedral group and for some actually equals the Kac-Peterson matrix associated to the dihedral group twisted by an appropriate 3- cocycle [5] — finite group modular data tends to have significantly more modular invariants and fusion-symmetries than e.g. affine modular data.", "index": 8}, {"type": "text", "bbox": [70, 535, 542, 581], "content": "Theorem 3.B. The fusion-symmetries of level for are where . For a fusion-symmetry will equal for and , .", "index": 9}, {"type": "text", "bbox": [93, 586, 451, 604], "content": "When , is trivial. We have", "index": 10}, {"type": "text", "bbox": [71, 616, 220, 633], "content": "3.3. The algebra ,", "index": 11}, {"type": "text", "bbox": [71, 639, 541, 683], "content": "A weight of satisfies and . Charge-conjugation again is trivial, and there is a simple-current defined by , with .", "index": 12}, {"type": "text", "bbox": [70, 683, 540, 716], "content": "Choose any . The Young diagram for is defined in the usual way: for , the th row consists of boxes. Let denote the weight whose diagram is the transpose of that for . (For this purpose the algebra may be identified with .) For example, , where we use tilde’s to denote the quantities of . In fact, is a bijection. Then", "index": 13}]
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[{"bbox": [68, 67, 219, 92], "content": "3.2. The algebra ,", "parent_index": 0, "line_index": 0}, {"bbox": [95, 95, 541, 110], "content": "A weight in satisfies , and .", "parent_index": 1, "line_index": 0}, {"bbox": [70, 109, 541, 125], "content": "The charge-conjugation is trivial, but there is a simple-current: .", "parent_index": 1, "line_index": 1}, {"bbox": [69, 123, 176, 140], "content": "It has .", "parent_index": 1, "line_index": 2}, {"bbox": [95, 138, 294, 152], "content": "The only fusion products we need are", "parent_index": 2, "line_index": 0}, {"bbox": [70, 208, 541, 223], "content": "for all , , and , where we drop if . We", "parent_index": 3, "line_index": 0}, {"bbox": [71, 223, 288, 238], "content": "will also use the character formula (2.1b)", "parent_index": 3, "line_index": 1}, {"bbox": [72, 297, 227, 312], "content": "where and", "parent_index": 5, "line_index": 0}, {"bbox": [93, 354, 541, 371], "content": "For ( ) there are several Galois fusion-symmetries — one for each", "parent_index": 7, "line_index": 0}, {"bbox": [69, 368, 541, 388], "content": "Galois automorphism, since S020 = 41κ is rational. In particular, define for", "parent_index": 7, "line_index": 1}, {"bbox": [71, 383, 540, 402], "content": ", and . Then for any coprime to , fixes", "parent_index": 7, "line_index": 2}, {"bbox": [71, 399, 540, 414], "content": "and , sends to (where the superscript is taken mod ), and stabilises", "parent_index": 7, "line_index": 3}, {"bbox": [72, 412, 351, 429], "content": "iff the Jacobi symbol equals ).", "parent_index": 7, "line_index": 4}, {"bbox": [94, 427, 541, 443], "content": "Why is so special here? One reason is that rank-level duality associates with", "parent_index": 8, "line_index": 0}, {"bbox": [71, 442, 541, 460], "content": ", and it is easy to confirm that has a rich variety of fusion-symmetries (and", "parent_index": 8, "line_index": 1}, {"bbox": [72, 460, 540, 474], "content": "modular invariants) coming from its si mple-currents. Also, the matrix formally", "parent_index": 8, "line_index": 2}, {"bbox": [70, 473, 541, 488], "content": "looks like the character table of the dihedral group and for some actually equals the", "parent_index": 8, "line_index": 3}, {"bbox": [69, 487, 541, 504], "content": "Kac-Peterson matrix associated to the dihedral group twisted by an appropriate 3-", "parent_index": 8, "line_index": 4}, {"bbox": [69, 503, 541, 518], "content": "cocycle [5] — finite group modular data tends to have significantly more modular invariants", "parent_index": 8, "line_index": 5}, {"bbox": [70, 517, 349, 532], "content": "and fusion-symmetries than e.g. affine modular data.", "parent_index": 8, "line_index": 6}, {"bbox": [93, 536, 542, 556], "content": "Theorem 3.B. The fusion-symmetries of level for are where", "parent_index": 9, "line_index": 0}, {"bbox": [71, 553, 539, 570], "content": ". For a fusion-symmetry will equal for and ,", "parent_index": 9, "line_index": 1}, {"bbox": [71, 568, 131, 585], "content": ".", "parent_index": 9, "line_index": 2}, {"bbox": [95, 587, 449, 606], "content": "When , is trivial. We have", "parent_index": 10, "line_index": 0}, {"bbox": [68, 613, 218, 638], "content": "3.3. The algebra ,", "parent_index": 11, "line_index": 0}, {"bbox": [96, 642, 540, 656], "content": "A weight of satisfies and . Charge-conjugation", "parent_index": 12, "line_index": 0}, {"bbox": [71, 656, 540, 671], "content": "again is trivial, and there is a simple-current defined by ,", "parent_index": 12, "line_index": 1}, {"bbox": [70, 669, 217, 688], "content": "with .", "parent_index": 12, "line_index": 2}, {"bbox": [94, 683, 541, 702], "content": "Choose any . The Young diagram for is defined in the usual way: for", "parent_index": 13, "line_index": 0}, {"bbox": [71, 699, 541, 719], "content": ", the th row consists of boxes. Let denote the weight", "parent_index": 13, "line_index": 1}, {"bbox": [72, 74, 541, 88], "content": "whose diagram is the transpose of that for . (For this purpose the algebra may be", "parent_index": 13, "line_index": 2}, {"bbox": [70, 87, 541, 103], "content": "identified with .) For example, , where we use tilde’s to denote the quantities", "parent_index": 13, "line_index": 3}, {"bbox": [70, 102, 394, 117], "content": "of . In fact, is a bijection. Then", "parent_index": 13, "line_index": 4}]
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[]
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[{"bbox": [161, 72, 183, 86], "content": "B_{r}^{(1)}", "parent_index": 0, "subtype": "inline"}, {"bbox": [190, 77, 219, 87], "content": "r\\geq3", "parent_index": 0, "subtype": "inline"}, {"bbox": [145, 96, 154, 106], "content": "\\lambda", "parent_index": 1, "subtype": "inline"}, {"bbox": [171, 96, 187, 109], "content": "P_{+}", "parent_index": 1, "subtype": "inline"}, {"bbox": [234, 95, 429, 108], "content": "k=\\lambda_{0}+\\lambda_{1}+2\\lambda_{2}+\\cdot\\cdot\\cdot+2\\lambda_{r-1}+\\lambda_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [459, 96, 537, 107], "content": "\\kappa=k+2r-1", "parent_index": 1, "subtype": "inline"}, {"bbox": [408, 109, 536, 123], "content": "J\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},.\\ldots,\\lambda_{r}\\right)", "parent_index": 1, "subtype": "inline"}, {"bbox": [105, 123, 172, 138], "content": "Q(\\lambda)=\\lambda_{r}/2", "parent_index": 1, "subtype": "inline"}, {"bbox": [106, 209, 175, 221], "content": "1\\leq i<r-1", "parent_index": 3, "subtype": "inline"}, {"bbox": [181, 208, 210, 221], "content": "k>2", "parent_index": 3, "subtype": "inline"}, {"bbox": [240, 208, 290, 221], "content": "0<\\ell<k", "parent_index": 3, "subtype": "inline"}, {"bbox": [379, 208, 473, 222], "content": "\\mathrm{\\Delta}^{\\prime}\\Lambda_{r-1}+(\\ell-2)\\Lambda_{r}{}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [487, 209, 514, 219], "content": "\\ell=1", "parent_index": 3, "subtype": "inline"}, {"bbox": [195, 246, 415, 285], "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})+1~,", "parent_index": 4, "subtype": "interline"}, {"bbox": [106, 298, 203, 311], "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "parent_index": 5, "subtype": "inline"}, {"bbox": [249, 309, 362, 349], "content": "\\lambda(\\ell)=\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{1}{2}\\lambda_{r}\\ .", "parent_index": 6, "subtype": "interline"}, {"bbox": [117, 358, 150, 367], "content": "k\\,=\\,2", "parent_index": 7, "subtype": "inline"}, {"bbox": [158, 357, 219, 368], "content": "\\kappa\\,=\\,2r+1", "parent_index": 7, "subtype": "inline"}, {"bbox": [441, 371, 521, 384], "content": "\\gamma^{i}=\\gamma^{\\kappa-i}=\\Lambda_{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [71, 387, 166, 398], "content": "i\\,=\\,1,2,\\dots,r\\,-\\,1", "parent_index": 7, "subtype": "inline"}, {"bbox": [198, 385, 293, 398], "content": "\\gamma^{r}\\,=\\,\\gamma^{r+1}\\,=\\,2\\Lambda_{r}", "parent_index": 7, "subtype": "inline"}, {"bbox": [379, 390, 389, 396], "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [456, 390, 463, 396], "content": "\\kappa", "parent_index": 7, "subtype": "inline"}, {"bbox": [471, 386, 501, 398], "content": "\\pi\\{m\\}", "parent_index": 7, "subtype": "inline"}, {"bbox": [533, 387, 540, 396], "content": "0", "parent_index": 7, "subtype": "inline"}, {"bbox": [95, 401, 103, 410], "content": "J", "parent_index": 7, "subtype": "inline"}, {"bbox": [142, 401, 155, 412], "content": "\\gamma^{a}", "parent_index": 7, "subtype": "inline"}, {"bbox": [174, 401, 195, 412], "content": "\\gamma^{m a}", "parent_index": 7, "subtype": "inline"}, {"bbox": [393, 404, 401, 410], "content": "\\kappa", "parent_index": 7, "subtype": "inline"}, {"bbox": [487, 400, 540, 413], "content": "\\{\\Lambda_{r},J\\Lambda_{r}\\}", "parent_index": 7, "subtype": "inline"}, {"bbox": [72, 415, 149, 427], "content": "(\\pi\\{m\\}\\Lambda_{r}=\\Lambda_{r}", "parent_index": 7, "subtype": "inline"}, {"bbox": [266, 414, 286, 428], "content": "\\Bigl(\\frac{\\kappa}{m}\\Bigr)", "parent_index": 7, "subtype": "inline"}, {"bbox": [327, 416, 342, 425], "content": "+1", "parent_index": 7, "subtype": "inline"}, {"bbox": [134, 430, 162, 438], "content": "k=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [492, 430, 513, 442], "content": "B_{r,2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [71, 447, 115, 459], "content": "\\mathrm{u}(1)_{2r+1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [281, 442, 304, 459], "content": "\\widehat{\\mathrm{u(1)}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [416, 461, 438, 474], "content": "B_{r,2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [483, 461, 491, 470], "content": "S", "parent_index": 8, "subtype": "inline"}, {"bbox": [427, 479, 434, 484], "content": "r", "parent_index": 8, "subtype": "inline"}, {"bbox": [183, 490, 191, 499], "content": "S", "parent_index": 8, "subtype": "inline"}, {"bbox": [364, 488, 389, 504], "content": "{\\mathfrak{D}}_{{\\sqrt{\\kappa}}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [327, 536, 350, 552], "content": "B_{r}^{(1)}", "parent_index": 9, "subtype": "inline"}, {"bbox": [383, 539, 392, 551], "content": "k", "parent_index": 9, "subtype": "inline"}, {"bbox": [416, 539, 453, 553], "content": "k\\ \\neq\\ 2", "parent_index": 9, "subtype": "inline"}, {"bbox": [479, 539, 504, 553], "content": "\\pi[1]^{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [71, 555, 120, 568], "content": "i\\in\\{0,1\\}", "parent_index": 9, "subtype": "inline"}, {"bbox": [151, 556, 181, 565], "content": "k=2", "parent_index": 9, "subtype": "inline"}, {"bbox": [339, 553, 395, 568], "content": "\\pi[1]^{i}\\,\\pi\\{m\\}", "parent_index": 9, "subtype": "inline"}, {"bbox": [417, 554, 467, 568], "content": "i\\in\\{0,1\\}", "parent_index": 9, "subtype": "inline"}, {"bbox": [493, 554, 536, 568], "content": "m\\in\\mathbb{Z}_{\\kappa}^{\\times}", "parent_index": 9, "subtype": "inline"}, {"bbox": [71, 570, 126, 581], "content": "1\\leq m\\leq r", "parent_index": 9, "subtype": "inline"}, {"bbox": [130, 591, 159, 600], "content": "k=1", "parent_index": 10, "subtype": "inline"}, {"bbox": [165, 589, 186, 603], "content": "\\pi[1]", "parent_index": 10, "subtype": "inline"}, {"bbox": [290, 588, 449, 604], "content": "\\mathcal{F}(B_{r,2})\\cong\\mathbb{Z}_{2}\\times(\\mathbb{Z}_{2r+1}^{\\times}/\\{\\pm1\\}).", "parent_index": 10, "subtype": "inline"}, {"bbox": [161, 618, 183, 633], "content": "C_{r}^{(1)}", "parent_index": 11, "subtype": "inline"}, {"bbox": [190, 623, 218, 633], "content": "r\\geq2", "parent_index": 11, "subtype": "inline"}, {"bbox": [144, 643, 152, 653], "content": "\\lambda", "parent_index": 12, "subtype": "inline"}, {"bbox": [167, 644, 183, 655], "content": "P_{+}", "parent_index": 12, "subtype": "inline"}, {"bbox": [230, 642, 339, 654], "content": "k=\\lambda_{0}+\\lambda_{1}+\\cdot\\cdot\\cdot+\\lambda_{r}", "parent_index": 12, "subtype": "inline"}, {"bbox": [364, 642, 429, 653], "content": "\\kappa=k+r+1", "parent_index": 12, "subtype": "inline"}, {"bbox": [71, 658, 81, 667], "content": "C", "parent_index": 12, "subtype": "inline"}, {"bbox": [325, 658, 334, 667], "content": "J", "parent_index": 12, "subtype": "inline"}, {"bbox": [396, 657, 536, 670], "content": 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[479, 704, 501, 717], "content": "C_{k,r}", "parent_index": 13, "subtype": "inline"}, {"bbox": [303, 75, 311, 84], "content": "\\lambda", "parent_index": 13, "subtype": "inline"}, {"bbox": [482, 75, 495, 86], "content": "C_{1}", "parent_index": 13, "subtype": "inline"}, {"bbox": [149, 90, 164, 101], "content": "A_{1}", "parent_index": 13, "subtype": "inline"}, {"bbox": [247, 87, 304, 101], "content": "\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}", "parent_index": 13, "subtype": "inline"}, {"bbox": [85, 104, 107, 117], "content": "C_{k,r}", "parent_index": 13, "subtype": "inline"}, {"bbox": [157, 103, 286, 117], "content": "\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})", "parent_index": 13, "subtype": "inline"}]
|
[]
|
|
$$
\tilde{S}_{\tau\lambda,\tau\mu}=S_{\lambda\mu}\ .
$$
This rank-level duality for $C_{r}^{(1)}$ is especially interesting, as it defines a fusion ring isomorphism $\mathcal{R}(C_{r,k})\cong\mathcal{R}(C_{k,r})$ (see §5). When $k=r$ , we get a nontrivial fusion-symmetry: $\pi_{\mathrm{rld}}\lambda\,{\overset{\mathrm{def}}{=}}\,\tau\lambda$ .
The only fusion product we need is
$$
\Lambda_{1}\sqcup\Lambda_{i}=\Lambda_{i-1}\sqcup\Lambda_{i+1}\sqcup\left(\Lambda_{1}+\Lambda_{i}\right)\,,
$$
valid for $i<r$ and $k\geq2$ . The following character formula (2.1b) will also be used:
$$
\chi_{\Lambda_{1}}[\lambda]={\frac{S_{\Lambda_{1}\lambda}}{S_{0\lambda}}}=2\sum_{\ell=1}^{r}\cos(\pi{\frac{\lambda^{+}(\ell)}{\kappa}})~,
$$
where $\lambda^{+}(\ell)=(\lambda+\rho)(\ell)$ as before.
Theorem 3.C. The fusion-symmetries for $C_{r}^{(1)}$ level $k$ , when $k\neq r$ and either $k$ or $r$ is even, are $\pi[1]^{i}$ for $i\in\{0,1\}$ . When $k\neq r$ but both $k$ and $r$ are odd, then there is no nontrivial fusion-symmetry. When $k=r$ , they are $\pi[1]^{i}\,\pi_{\mathrm{rld}}^{j}$ $\mathit{\Pi}_{k}$ even) or $\pi[1]^{i}$ ( $\mathit{k}$ odd), for $i,j\in\{0,1\}$ .
When $r=k$ is even, $A(C_{r,k})\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ .
3.4. The algebra $D_{r}^{(1)},\,r\geq4$
A weight $\lambda$ of $P_{+}$ satisfies $k=\lambda_{0}\!+\!\lambda_{1}\!+\!2\lambda_{2}\!+\!\cdot\!\cdot\!+\!2\lambda_{r-2}\!+\!\lambda_{r-1}\!+\!\lambda_{r}$ , and $\kappa=k{+}2r{-}2$ . For any $r$ , there are the conjugations $C_{0}=i d$ . and $C_{1}\lambda=(\lambda_{0},\lambda_{1},.\dots,\lambda_{r-2},\lambda_{r},\lambda_{r-1})$ . The charge-conjugation $C$ equals $C_{1}$ for odd $r$ , and $C_{0}$ for even $r$ . When $r=4$ there are four additional conjugations; these six $C_{i}$ correspond to all permutations of the ${D}_{4}^{(1)}$ Dynkin labels $\lambda_{1},\lambda_{3},\lambda_{4}$ .
There are three non-trivial simple-currents, $J_{v}$ , $J_{s}$ and $J_{v}J_{s}$ . Explicitly, we have $J_{v}\lambda=\left(\lambda_{1},\lambda_{0},\lambda_{2},...\,,\lambda_{r-2},\lambda_{r},\lambda_{r-1}\right)$ with $Q_{v}(\lambda)=(\lambda_{r-1}+\lambda_{r})/2$ , and
$$
J_{s}\lambda={\left\{\begin{array}{l l}{(\lambda_{r},\lambda_{r-1},\lambda_{r-2},\ldots,\lambda_{1},\lambda_{0})}&{{\mathrm{if~}}r{\mathrm{~is~even}},}\\ {(\lambda_{r-1},\lambda_{r},\lambda_{r-2},\ldots,\lambda_{1},\lambda_{0})}&{{\mathrm{if~}}r{\mathrm{~is~odd}},}\end{array}\right.}
$$
with $\begin{array}{r}{Q_{s}(\lambda)=(2\sum_{j=1}^{r-2}j\lambda_{j}\!-\!(r\!-\!2)\lambda_{r-1}\!-\!r\lambda_{r})/4}\end{array}$ . From this we compute $Q_{s}(J_{s}0)=-r k/4$ . The fusion products we need are
|
<html><body>
<div class="equation" data-bbox="267 129 344 146">$$
\tilde{S}_{\tau\lambda,\tau\mu}=S_{\lambda\mu}\ .
$$</div>
<p data-bbox="70 158 541 205">This rank-level duality for $C_{r}^{(1)}$ is especially interesting, as it defines a fusion ring isomorphism $\mathcal{R}(C_{r,k})\cong\mathcal{R}(C_{k,r})$ (see §5). When $k=r$ , we get a nontrivial fusion-symmetry: $\pi_{\mathrm{rld}}\lambda\,{\overset{\mathrm{def}}{=}}\,\tau\lambda$ . </p>
<p data-bbox="95 207 282 221">The only fusion product we need is </p>
<div class="equation" data-bbox="201 234 406 251">$$
\Lambda_{1}\sqcup\Lambda_{i}=\Lambda_{i-1}\sqcup\Lambda_{i+1}\sqcup\left(\Lambda_{1}+\Lambda_{i}\right)\,,
$$</div>
<p data-bbox="71 262 505 277">valid for $i<r$ and $k\geq2$ . The following character formula (2.1b) will also be used: </p>
<div class="equation" data-bbox="209 289 401 329">$$
\chi_{\Lambda_{1}}[\lambda]={\frac{S_{\Lambda_{1}\lambda}}{S_{0\lambda}}}=2\sum_{\ell=1}^{r}\cos(\pi{\frac{\lambda^{+}(\ell)}{\kappa}})~,
$$</div>
<p data-bbox="70 339 258 355">where $\lambda^{+}(\ell)=(\lambda+\rho)(\ell)$ as before. </p>
<p data-bbox="70 361 542 421">Theorem 3.C. The fusion-symmetries for $C_{r}^{(1)}$ level $k$ , when $k\neq r$ and either $k$ or $r$ is even, are $\pi[1]^{i}$ for $i\in\{0,1\}$ . When $k\neq r$ but both $k$ and $r$ are odd, then there is no nontrivial fusion-symmetry. When $k=r$ , they are $\pi[1]^{i}\,\pi_{\mathrm{rld}}^{j}$ $\mathit{\Pi}_{k}$ even) or $\pi[1]^{i}$ ( $\mathit{k}$ odd), for $i,j\in\{0,1\}$ . </p>
<p data-bbox="94 427 306 443">When $r=k$ is even, $A(C_{r,k})\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ . </p>
<p data-bbox="71 456 221 474">3.4. The algebra $D_{r}^{(1)},\,r\geq4$ </p>
<p data-bbox="70 479 541 554">A weight $\lambda$ of $P_{+}$ satisfies $k=\lambda_{0}\!+\!\lambda_{1}\!+\!2\lambda_{2}\!+\!\cdot\!\cdot\!+\!2\lambda_{r-2}\!+\!\lambda_{r-1}\!+\!\lambda_{r}$ , and $\kappa=k{+}2r{-}2$ . For any $r$ , there are the conjugations $C_{0}=i d$ . and $C_{1}\lambda=(\lambda_{0},\lambda_{1},.\dots,\lambda_{r-2},\lambda_{r},\lambda_{r-1})$ . The charge-conjugation $C$ equals $C_{1}$ for odd $r$ , and $C_{0}$ for even $r$ . When $r=4$ there are four additional conjugations; these six $C_{i}$ correspond to all permutations of the ${D}_{4}^{(1)}$ Dynkin labels $\lambda_{1},\lambda_{3},\lambda_{4}$ . </p>
<p data-bbox="70 554 541 584">There are three non-trivial simple-currents, $J_{v}$ , $J_{s}$ and $J_{v}J_{s}$ . Explicitly, we have $J_{v}\lambda=\left(\lambda_{1},\lambda_{0},\lambda_{2},...\,,\lambda_{r-2},\lambda_{r},\lambda_{r-1}\right)$ with $Q_{v}(\lambda)=(\lambda_{r-1}+\lambda_{r})/2$ , and </p>
<div class="equation" data-bbox="174 596 430 627">$$
J_{s}\lambda={\left\{\begin{array}{l l}{(\lambda_{r},\lambda_{r-1},\lambda_{r-2},\ldots,\lambda_{1},\lambda_{0})}&{{\mathrm{if~}}r{\mathrm{~is~even}},}\\ {(\lambda_{r-1},\lambda_{r},\lambda_{r-2},\ldots,\lambda_{1},\lambda_{0})}&{{\mathrm{if~}}r{\mathrm{~is~odd}},}\end{array}\right.}
$$</div>
<p data-bbox="69 639 541 669">with $\begin{array}{r}{Q_{s}(\lambda)=(2\sum_{j=1}^{r-2}j\lambda_{j}\!-\!(r\!-\!2)\lambda_{r-1}\!-\!r\lambda_{r})/4}\end{array}$ . From this we compute $Q_{s}(J_{s}0)=-r k/4$ . The fusion products we need are </p>
</body></html>
|
0002044v1
| 10 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "", "page_idx": 10}, {"type": "equation", "text": "$$\n\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "This rank-level duality for $C_{r}^{(1)}$ is especially interesting, as it defines a fusion ring isomorphism $\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})$ (see §5). When $k=r$ , we get a nontrivial fusion-symmetry: $\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda$ . ", "page_idx": 10}, {"type": "text", "text": "The only fusion product we need is ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "valid for $i<r$ and $k\\geq2$ . The following character formula (2.1b) will also be used: ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "where $\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)$ as before. ", "page_idx": 10}, {"type": "text", "text": "Theorem 3.C. The fusion-symmetries for $C_{r}^{(1)}$ level $k$ , when $k\\neq r$ and either $k$ or $r$ is even, are $\\pi[1]^{i}$ for $i\\in\\{0,1\\}$ . When $k\\neq r$ but both $k$ and $r$ are odd, then there is no nontrivial fusion-symmetry. When $k=r$ , they are $\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}$ $\\mathit{\\Pi}_{k}$ even) or $\\pi[1]^{i}$ ( $\\mathit{k}$ odd), for $i,j\\in\\{0,1\\}$ . ", "page_idx": 10}, {"type": "text", "text": "When $r=k$ is even, $A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}$ . ", "page_idx": 10}, {"type": "text", "text": "3.4. The algebra $D_{r}^{(1)},\\,r\\geq4$ ", "page_idx": 10}, {"type": "text", "text": "A weight $\\lambda$ of $P_{+}$ satisfies $k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}$ , and $\\kappa=k{+}2r{-}2$ . For any $r$ , there are the conjugations $C_{0}=i d$ . and $C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})$ . The charge-conjugation $C$ equals $C_{1}$ for odd $r$ , and $C_{0}$ for even $r$ . When $r=4$ there are four additional conjugations; these six $C_{i}$ correspond to all permutations of the ${D}_{4}^{(1)}$ Dynkin labels $\\lambda_{1},\\lambda_{3},\\lambda_{4}$ . ", "page_idx": 10}, {"type": "text", "text": "There are three non-trivial simple-currents, $J_{v}$ , $J_{s}$ and $J_{v}J_{s}$ . Explicitly, we have $J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)$ with $Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2$ , and ", "page_idx": 10}, {"type": "equation", "text": "$$\nJ_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "with $\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}$ . From this we compute $Q_{s}(J_{s}0)=-r k/4$ . The fusion products we need are ", "page_idx": 10}]
|
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For example,", "type": "text"}, {"bbox": [247, 87, 304, 101], "score": 0.94, "content": "\\tau\\Lambda_{a}=a\\tilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 57}, {"bbox": [304, 87, 541, 103], "score": 1.0, "content": ", where we use tilde’s to denote the quantities", "type": "text"}], "index": 1}, {"bbox": [70, 102, 394, 117], "spans": [{"bbox": [70, 102, 84, 117], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [85, 104, 107, 117], "score": 0.93, "content": "C_{k,r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [107, 102, 157, 117], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [157, 103, 286, 117], "score": 0.94, "content": "\\tau:P_{+}(C_{r,k})\\rightarrow P_{+}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 129}, {"bbox": [286, 102, 394, 117], "score": 1.0, "content": " is a bijection. Then", "type": "text"}], "index": 2}], "index": 1}, {"type": "interline_equation", "bbox": [267, 129, 344, 146], "lines": [{"bbox": [267, 129, 344, 146], "spans": [{"bbox": [267, 129, 344, 146], "score": 0.91, "content": "\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [70, 158, 541, 205], "lines": [{"bbox": [93, 158, 540, 178], "spans": [{"bbox": [93, 159, 234, 178], "score": 1.0, "content": "This rank-level duality for ", "type": "text"}, {"bbox": [234, 158, 257, 174], "score": 0.91, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [258, 159, 540, 178], "score": 1.0, "content": "is especially interesting, as it defines a fusion ring iso-", "type": "text"}], "index": 4}, {"bbox": [71, 176, 540, 191], "spans": [{"bbox": [71, 176, 127, 191], "score": 1.0, "content": "morphism ", "type": "text"}, {"bbox": [127, 176, 225, 190], "score": 0.91, "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [226, 176, 311, 191], "score": 1.0, "content": " (see §5). When ", "type": "text"}, {"bbox": [312, 176, 341, 188], "score": 0.89, "content": "k=r", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [342, 176, 540, 191], "score": 1.0, "content": ", we get a nontrivial fusion-symmetry:", "type": "text"}], "index": 5}, {"bbox": [71, 189, 134, 210], "spans": [{"bbox": [71, 192, 128, 208], "score": 0.92, "content": "\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda", "type": "inline_equation", "height": 16, "width": 57}, {"bbox": [129, 189, 134, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [95, 207, 282, 221], "lines": [{"bbox": [95, 209, 281, 223], "spans": [{"bbox": [95, 209, 281, 223], "score": 1.0, "content": "The only fusion product we need is", "type": "text"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [201, 234, 406, 251], "lines": [{"bbox": [201, 234, 406, 251], "spans": [{"bbox": [201, 234, 406, 251], "score": 0.72, "content": "\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [71, 262, 505, 277], "lines": [{"bbox": [70, 263, 506, 281], "spans": [{"bbox": [70, 263, 118, 281], "score": 1.0, "content": "valid for ", "type": "text"}, {"bbox": [118, 267, 144, 276], "score": 0.9, "content": "i<r", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [145, 263, 171, 281], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [171, 264, 200, 277], "score": 0.9, "content": "k\\geq2", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [201, 263, 506, 281], "score": 1.0, "content": ". The following character formula (2.1b) will also be used:", "type": "text"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [209, 289, 401, 329], "lines": [{"bbox": [209, 289, 401, 329], "spans": [{"bbox": [209, 289, 401, 329], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 339, 258, 355], "lines": [{"bbox": [71, 341, 257, 357], "spans": [{"bbox": [71, 341, 105, 357], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 342, 203, 356], "score": 0.92, "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [203, 341, 257, 357], "score": 1.0, "content": " as before.", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [70, 361, 542, 421], "lines": [{"bbox": [93, 362, 542, 381], "spans": [{"bbox": [93, 362, 324, 381], "score": 1.0, "content": "Theorem 3.C. The fusion-symmetries for ", "type": "text"}, {"bbox": [325, 362, 348, 378], "score": 0.9, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [348, 362, 378, 381], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [378, 365, 387, 377], "score": 0.75, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [387, 362, 424, 381], "score": 1.0, "content": ", when ", "type": "text"}, {"bbox": [424, 365, 455, 379], "score": 0.9, "content": "k\\neq r", "type": "inline_equation", "height": 14, "width": 31}, {"bbox": [456, 362, 516, 381], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [516, 366, 524, 377], "score": 0.69, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [525, 362, 542, 381], "score": 1.0, "content": " or", "type": "text"}], "index": 12}, {"bbox": [71, 379, 541, 394], "spans": [{"bbox": [71, 385, 78, 391], "score": 0.66, "content": "r", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [78, 380, 146, 394], "score": 1.0, "content": " is even, are ", "type": "text"}, {"bbox": [146, 379, 171, 393], "score": 0.9, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [171, 380, 193, 394], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [193, 379, 243, 394], "score": 0.93, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 15, "width": 50}, {"bbox": [243, 380, 286, 394], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [286, 380, 317, 393], "score": 0.91, "content": "k\\neq r", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [317, 380, 365, 394], "score": 1.0, "content": " but both ", "type": "text"}, {"bbox": [365, 380, 373, 391], "score": 0.77, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [374, 380, 399, 394], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [399, 382, 407, 391], "score": 0.69, "content": "r", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [407, 380, 541, 394], "score": 1.0, "content": " are odd, then there is no", "type": "text"}], "index": 13}, {"bbox": [70, 392, 542, 410], "spans": [{"bbox": [70, 394, 255, 410], "score": 1.0, "content": "nontrivial fusion-symmetry. When ", "type": "text"}, {"bbox": [255, 394, 285, 406], "score": 0.88, "content": "k=r", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [285, 394, 337, 410], "score": 1.0, "content": ", they are ", "type": "text"}, {"bbox": [337, 392, 384, 409], "score": 0.91, "content": "\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}", "type": "inline_equation", "height": 17, "width": 47}, {"bbox": [384, 394, 390, 410], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [391, 393, 399, 406], "score": 0.6, "content": "\\mathit{\\Pi}_{k}", "type": "inline_equation", "height": 13, "width": 8}, {"bbox": [400, 394, 449, 410], "score": 1.0, "content": " even) or ", "type": "text"}, {"bbox": [450, 393, 475, 408], "score": 0.79, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [475, 394, 482, 410], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [483, 394, 491, 406], "score": 0.43, "content": "\\mathit{k}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [491, 394, 542, 410], "score": 1.0, "content": " odd), for", "type": "text"}], "index": 14}, {"bbox": [71, 407, 136, 423], "spans": [{"bbox": [71, 409, 131, 422], "score": 0.91, "content": "i,j\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [131, 407, 136, 423], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13.5}, {"type": "text", "bbox": [94, 427, 306, 443], "lines": [{"bbox": [95, 429, 305, 444], "spans": [{"bbox": [95, 429, 129, 444], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 431, 159, 441], "score": 0.88, "content": "r=k", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [159, 429, 204, 444], "score": 1.0, "content": " is even, ", "type": "text"}, {"bbox": [205, 429, 303, 444], "score": 0.91, "content": "A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [303, 429, 305, 444], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [71, 456, 221, 474], "lines": [{"bbox": [70, 456, 219, 476], "spans": [{"bbox": [70, 456, 160, 476], "score": 1.0, "content": "3.4. The algebra ", "type": "text"}, {"bbox": [161, 458, 219, 474], "score": 0.3, "content": "D_{r}^{(1)},\\,r\\geq4", "type": "inline_equation", "height": 16, "width": 58}], "index": 17}], "index": 17}, {"type": "text", "bbox": [70, 479, 541, 554], "lines": [{"bbox": [94, 481, 540, 497], "spans": [{"bbox": [94, 481, 144, 497], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 483, 151, 493], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [151, 481, 166, 497], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [166, 484, 182, 496], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [182, 481, 227, 497], "score": 1.0, "content": " satisfies", "type": "text"}, {"bbox": [228, 483, 438, 495], "score": 0.88, "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}", "type": "inline_equation", "height": 12, "width": 210}, {"bbox": [439, 481, 467, 497], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [468, 484, 536, 494], "score": 0.91, "content": "\\kappa=k{+}2r{-}2", "type": "inline_equation", "height": 10, "width": 68}, {"bbox": [537, 481, 540, 497], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [69, 495, 541, 513], "spans": [{"bbox": [69, 495, 114, 513], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [114, 501, 120, 507], "score": 0.82, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [121, 495, 265, 513], "score": 1.0, "content": ", there are the conjugations ", "type": "text"}, {"bbox": [266, 498, 307, 509], "score": 0.92, "content": "C_{0}=i d", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [307, 495, 335, 513], "score": 1.0, "content": ". and ", "type": "text"}, {"bbox": [335, 498, 510, 510], "score": 0.91, "content": "C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})", "type": "inline_equation", "height": 12, "width": 175}, {"bbox": [511, 495, 541, 513], "score": 1.0, "content": ". The", "type": "text"}], "index": 19}, {"bbox": [70, 511, 541, 527], "spans": [{"bbox": [70, 511, 173, 527], "score": 1.0, "content": "charge-conjugation ", "type": "text"}, {"bbox": [174, 513, 183, 522], "score": 0.87, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 511, 223, 527], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [223, 513, 237, 524], "score": 0.91, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [238, 511, 284, 527], "score": 1.0, "content": " for odd ", "type": "text"}, {"bbox": [284, 516, 290, 522], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [291, 511, 321, 527], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [321, 513, 335, 524], "score": 0.92, "content": "C_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [335, 511, 385, 527], "score": 1.0, "content": " for even ", "type": "text"}, {"bbox": [385, 516, 391, 522], "score": 0.88, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [391, 511, 434, 527], "score": 1.0, "content": ". 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Explicitly, we have", "type": "text"}], "index": 23}, {"bbox": [71, 570, 442, 586], "spans": [{"bbox": [71, 571, 262, 584], "score": 0.91, "content": "J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)", "type": "inline_equation", "height": 13, "width": 191}, {"bbox": [263, 570, 293, 586], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [293, 571, 414, 584], "score": 0.9, "content": "Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2", "type": "inline_equation", "height": 13, "width": 121}, {"bbox": [414, 570, 442, 586], "score": 1.0, "content": ", and", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "interline_equation", "bbox": [174, 596, 430, 627], "lines": [{"bbox": [174, 596, 430, 627], "spans": [{"bbox": [174, 596, 430, 627], "score": 0.87, "content": "J_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [69, 639, 541, 669], "lines": [{"bbox": [69, 639, 541, 660], "spans": [{"bbox": [69, 641, 97, 660], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [97, 639, 320, 658], "score": 0.91, "content": "\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}", "type": "inline_equation", "height": 19, "width": 223}, {"bbox": [321, 641, 444, 660], "score": 1.0, "content": ". From this we compute", "type": "text"}, {"bbox": [444, 642, 536, 656], "score": 0.91, "content": "Q_{s}(J_{s}0)=-r k/4", "type": "inline_equation", "height": 14, "width": 92}, {"bbox": [536, 641, 541, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [94, 657, 268, 672], "spans": [{"bbox": [94, 657, 268, 672], "score": 1.0, "content": "The fusion products we need are", "type": "text"}], "index": 27}], "index": 26.5}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [267, 129, 344, 146], "lines": [{"bbox": [267, 129, 344, 146], "spans": [{"bbox": [267, 129, 344, 146], "score": 0.91, "content": "\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [201, 234, 406, 251], "lines": [{"bbox": [201, 234, 406, 251], "spans": [{"bbox": [201, 234, 406, 251], "score": 0.72, "content": "\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [209, 289, 401, 329], "lines": [{"bbox": [209, 289, 401, 329], "spans": [{"bbox": [209, 289, 401, 329], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "interline_equation", "bbox": [174, 596, 430, 627], "lines": [{"bbox": [174, 596, 430, 627], "spans": [{"bbox": [174, 596, 430, 627], "score": 0.87, "content": "J_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}", "type": "interline_equation"}], "index": 25}], "index": 25}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 542, 115], "lines": [], "index": 1, "bbox_fs": [70, 74, 541, 117], "lines_deleted": true}, {"type": "interline_equation", "bbox": [267, 129, 344, 146], "lines": [{"bbox": [267, 129, 344, 146], "spans": [{"bbox": [267, 129, 344, 146], "score": 0.91, "content": "\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [70, 158, 541, 205], "lines": [{"bbox": [93, 158, 540, 178], "spans": [{"bbox": [93, 159, 234, 178], "score": 1.0, "content": "This rank-level duality for ", "type": "text"}, {"bbox": [234, 158, 257, 174], "score": 0.91, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [258, 159, 540, 178], "score": 1.0, "content": "is especially interesting, as it defines a fusion ring iso-", "type": "text"}], "index": 4}, {"bbox": [71, 176, 540, 191], "spans": [{"bbox": [71, 176, 127, 191], "score": 1.0, "content": "morphism ", "type": "text"}, {"bbox": [127, 176, 225, 190], "score": 0.91, "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [226, 176, 311, 191], "score": 1.0, "content": " (see §5). When ", "type": "text"}, {"bbox": [312, 176, 341, 188], "score": 0.89, "content": "k=r", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [342, 176, 540, 191], "score": 1.0, "content": ", we get a nontrivial fusion-symmetry:", "type": "text"}], "index": 5}, {"bbox": [71, 189, 134, 210], "spans": [{"bbox": [71, 192, 128, 208], "score": 0.92, "content": "\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda", "type": "inline_equation", "height": 16, "width": 57}, {"bbox": [129, 189, 134, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5, "bbox_fs": [71, 158, 540, 210]}, {"type": "text", "bbox": [95, 207, 282, 221], "lines": [{"bbox": [95, 209, 281, 223], "spans": [{"bbox": [95, 209, 281, 223], "score": 1.0, "content": "The only fusion product we need is", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [95, 209, 281, 223]}, {"type": "interline_equation", "bbox": [201, 234, 406, 251], "lines": [{"bbox": [201, 234, 406, 251], "spans": [{"bbox": [201, 234, 406, 251], "score": 0.72, "content": "\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [71, 262, 505, 277], "lines": [{"bbox": [70, 263, 506, 281], "spans": [{"bbox": [70, 263, 118, 281], "score": 1.0, "content": "valid for ", "type": "text"}, {"bbox": [118, 267, 144, 276], "score": 0.9, "content": "i<r", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [145, 263, 171, 281], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [171, 264, 200, 277], "score": 0.9, "content": "k\\geq2", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [201, 263, 506, 281], "score": 1.0, "content": ". The following character formula (2.1b) will also be used:", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [70, 263, 506, 281]}, {"type": "interline_equation", "bbox": [209, 289, 401, 329], "lines": [{"bbox": [209, 289, 401, 329], "spans": [{"bbox": [209, 289, 401, 329], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 339, 258, 355], "lines": [{"bbox": [71, 341, 257, 357], "spans": [{"bbox": [71, 341, 105, 357], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 342, 203, 356], "score": 0.92, "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [203, 341, 257, 357], "score": 1.0, "content": " as before.", "type": "text"}], "index": 11}], "index": 11, "bbox_fs": [71, 341, 257, 357]}, {"type": "text", "bbox": [70, 361, 542, 421], "lines": [{"bbox": [93, 362, 542, 381], "spans": [{"bbox": [93, 362, 324, 381], "score": 1.0, "content": "Theorem 3.C. The fusion-symmetries for ", "type": "text"}, {"bbox": [325, 362, 348, 378], "score": 0.9, "content": "C_{r}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [348, 362, 378, 381], "score": 1.0, "content": "level ", "type": "text"}, {"bbox": [378, 365, 387, 377], "score": 0.75, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [387, 362, 424, 381], "score": 1.0, "content": ", when ", "type": "text"}, {"bbox": [424, 365, 455, 379], "score": 0.9, "content": "k\\neq r", "type": "inline_equation", "height": 14, "width": 31}, {"bbox": [456, 362, 516, 381], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [516, 366, 524, 377], "score": 0.69, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [525, 362, 542, 381], "score": 1.0, "content": " or", "type": "text"}], "index": 12}, {"bbox": [71, 379, 541, 394], "spans": [{"bbox": [71, 385, 78, 391], "score": 0.66, "content": "r", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [78, 380, 146, 394], "score": 1.0, "content": " is even, are ", "type": "text"}, {"bbox": [146, 379, 171, 393], "score": 0.9, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [171, 380, 193, 394], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [193, 379, 243, 394], "score": 0.93, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 15, "width": 50}, {"bbox": [243, 380, 286, 394], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [286, 380, 317, 393], "score": 0.91, "content": "k\\neq r", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [317, 380, 365, 394], "score": 1.0, "content": " but both ", "type": "text"}, {"bbox": [365, 380, 373, 391], "score": 0.77, "content": "k", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [374, 380, 399, 394], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [399, 382, 407, 391], "score": 0.69, "content": "r", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [407, 380, 541, 394], "score": 1.0, "content": " are odd, then there is no", "type": "text"}], "index": 13}, {"bbox": [70, 392, 542, 410], "spans": [{"bbox": [70, 394, 255, 410], "score": 1.0, "content": "nontrivial fusion-symmetry. When ", "type": "text"}, {"bbox": [255, 394, 285, 406], "score": 0.88, "content": "k=r", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [285, 394, 337, 410], "score": 1.0, "content": ", they are ", "type": "text"}, {"bbox": [337, 392, 384, 409], "score": 0.91, "content": "\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}", "type": "inline_equation", "height": 17, "width": 47}, {"bbox": [384, 394, 390, 410], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [391, 393, 399, 406], "score": 0.6, "content": "\\mathit{\\Pi}_{k}", "type": "inline_equation", "height": 13, "width": 8}, {"bbox": [400, 394, 449, 410], "score": 1.0, "content": " even) or ", "type": "text"}, {"bbox": [450, 393, 475, 408], "score": 0.79, "content": "\\pi[1]^{i}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [475, 394, 482, 410], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [483, 394, 491, 406], "score": 0.43, "content": "\\mathit{k}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [491, 394, 542, 410], "score": 1.0, "content": " odd), for", "type": "text"}], "index": 14}, {"bbox": [71, 407, 136, 423], "spans": [{"bbox": [71, 409, 131, 422], "score": 0.91, "content": "i,j\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [131, 407, 136, 423], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13.5, "bbox_fs": [70, 362, 542, 423]}, {"type": "text", "bbox": [94, 427, 306, 443], "lines": [{"bbox": [95, 429, 305, 444], "spans": [{"bbox": [95, 429, 129, 444], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 431, 159, 441], "score": 0.88, "content": "r=k", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [159, 429, 204, 444], "score": 1.0, "content": " is even, ", "type": "text"}, {"bbox": [205, 429, 303, 444], "score": 0.91, "content": "A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [303, 429, 305, 444], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 16, "bbox_fs": [95, 429, 305, 444]}, {"type": "text", "bbox": [71, 456, 221, 474], "lines": [{"bbox": [70, 456, 219, 476], "spans": [{"bbox": [70, 456, 160, 476], "score": 1.0, "content": "3.4. The algebra ", "type": "text"}, {"bbox": [161, 458, 219, 474], "score": 0.3, "content": "D_{r}^{(1)},\\,r\\geq4", "type": "inline_equation", "height": 16, "width": 58}], "index": 17}], "index": 17, "bbox_fs": [70, 456, 219, 476]}, {"type": "text", "bbox": [70, 479, 541, 554], "lines": [{"bbox": [94, 481, 540, 497], "spans": [{"bbox": [94, 481, 144, 497], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 483, 151, 493], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [151, 481, 166, 497], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [166, 484, 182, 496], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [182, 481, 227, 497], "score": 1.0, "content": " satisfies", "type": "text"}, {"bbox": [228, 483, 438, 495], "score": 0.88, "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}", "type": "inline_equation", "height": 12, "width": 210}, {"bbox": [439, 481, 467, 497], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [468, 484, 536, 494], "score": 0.91, "content": "\\kappa=k{+}2r{-}2", "type": "inline_equation", "height": 10, "width": 68}, {"bbox": [537, 481, 540, 497], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [69, 495, 541, 513], "spans": [{"bbox": [69, 495, 114, 513], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [114, 501, 120, 507], "score": 0.82, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [121, 495, 265, 513], "score": 1.0, "content": ", there are the conjugations ", "type": "text"}, {"bbox": [266, 498, 307, 509], "score": 0.92, "content": "C_{0}=i d", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [307, 495, 335, 513], "score": 1.0, "content": ". and ", "type": "text"}, {"bbox": [335, 498, 510, 510], "score": 0.91, "content": "C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})", "type": "inline_equation", "height": 12, "width": 175}, {"bbox": [511, 495, 541, 513], "score": 1.0, "content": ". The", "type": "text"}], "index": 19}, {"bbox": [70, 511, 541, 527], "spans": [{"bbox": [70, 511, 173, 527], "score": 1.0, "content": "charge-conjugation ", "type": "text"}, {"bbox": [174, 513, 183, 522], "score": 0.87, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 511, 223, 527], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [223, 513, 237, 524], "score": 0.91, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [238, 511, 284, 527], "score": 1.0, "content": " for odd ", "type": "text"}, {"bbox": [284, 516, 290, 522], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [291, 511, 321, 527], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [321, 513, 335, 524], "score": 0.92, "content": "C_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [335, 511, 385, 527], "score": 1.0, "content": " for even ", "type": "text"}, {"bbox": [385, 516, 391, 522], "score": 0.88, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [391, 511, 434, 527], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [435, 513, 464, 522], "score": 0.9, "content": "r=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [464, 511, 541, 527], "score": 1.0, "content": " there are four", "type": "text"}], "index": 20}, {"bbox": [69, 524, 541, 542], "spans": [{"bbox": [69, 526, 252, 542], "score": 1.0, "content": "additional conjugations; these six ", "type": "text"}, {"bbox": [253, 529, 265, 540], "score": 0.91, "content": "C_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [265, 526, 473, 542], "score": 1.0, "content": " correspond to all permutations of the ", "type": "text"}, {"bbox": [473, 524, 496, 541], "score": 0.94, "content": "{D}_{4}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [497, 526, 541, 542], "score": 1.0, "content": "Dynkin", "type": "text"}], "index": 21}, {"bbox": [71, 542, 156, 557], "spans": [{"bbox": [71, 542, 104, 557], "score": 1.0, "content": "labels ", "type": "text"}, {"bbox": [105, 543, 153, 554], "score": 0.92, "content": "\\lambda_{1},\\lambda_{3},\\lambda_{4}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [153, 542, 156, 557], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 20, "bbox_fs": [69, 481, 541, 557]}, {"type": "text", "bbox": [70, 554, 541, 584], "lines": [{"bbox": [93, 554, 541, 572], "spans": [{"bbox": [93, 554, 335, 572], "score": 1.0, "content": "There are three non-trivial simple-currents, ", "type": "text"}, {"bbox": [335, 557, 347, 568], "score": 0.84, "content": "J_{v}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [348, 554, 356, 572], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [357, 558, 369, 568], "score": 0.81, "content": "J_{s}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [369, 554, 399, 572], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [399, 558, 423, 568], "score": 0.91, "content": "J_{v}J_{s}", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [424, 554, 541, 572], "score": 1.0, "content": ". Explicitly, we have", "type": "text"}], "index": 23}, {"bbox": [71, 570, 442, 586], "spans": [{"bbox": [71, 571, 262, 584], "score": 0.91, "content": "J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)", "type": "inline_equation", "height": 13, "width": 191}, {"bbox": [263, 570, 293, 586], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [293, 571, 414, 584], "score": 0.9, "content": "Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2", "type": "inline_equation", "height": 13, "width": 121}, {"bbox": [414, 570, 442, 586], "score": 1.0, "content": ", and", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [71, 554, 541, 586]}, {"type": "interline_equation", "bbox": [174, 596, 430, 627], "lines": [{"bbox": [174, 596, 430, 627], "spans": [{"bbox": [174, 596, 430, 627], "score": 0.87, "content": "J_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [69, 639, 541, 669], "lines": [{"bbox": [69, 639, 541, 660], "spans": [{"bbox": [69, 641, 97, 660], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [97, 639, 320, 658], "score": 0.91, "content": "\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}", "type": "inline_equation", "height": 19, "width": 223}, {"bbox": [321, 641, 444, 660], "score": 1.0, "content": ". From this we compute", "type": "text"}, {"bbox": [444, 642, 536, 656], "score": 0.91, "content": "Q_{s}(J_{s}0)=-r k/4", "type": "inline_equation", "height": 14, "width": 92}, {"bbox": [536, 641, 541, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [94, 657, 268, 672], "spans": [{"bbox": [94, 657, 268, 672], "score": 1.0, "content": "The fusion products we need are", "type": "text"}], "index": 27}], "index": 26.5, "bbox_fs": [69, 639, 541, 672]}]}
|
[{"type": "text", "bbox": [70, 70, 542, 115], "content": "", "index": 0}, {"type": "interline_equation", "bbox": [267, 129, 344, 146], "content": "", "index": 1}, {"type": "text", "bbox": [70, 158, 541, 205], "content": "This rank-level duality for is especially interesting, as it defines a fusion ring iso- morphism (see §5). When , we get a nontrivial fusion-symmetry: .", "index": 2}, {"type": "text", "bbox": [95, 207, 282, 221], "content": "The only fusion product we need is", "index": 3}, {"type": "interline_equation", "bbox": [201, 234, 406, 251], "content": "", "index": 4}, {"type": "text", "bbox": [71, 262, 505, 277], "content": "valid for and . The following character formula (2.1b) will also be used:", "index": 5}, {"type": "interline_equation", "bbox": [209, 289, 401, 329], "content": "", "index": 6}, {"type": "text", "bbox": [70, 339, 258, 355], "content": "where as before.", "index": 7}, {"type": "text", "bbox": [70, 361, 542, 421], "content": "Theorem 3.C. The fusion-symmetries for level , when and either or is even, are for . When but both and are odd, then there is no nontrivial fusion-symmetry. When , they are even) or ( odd), for .", "index": 8}, {"type": "text", "bbox": [94, 427, 306, 443], "content": "When is even, .", "index": 9}, {"type": "text", "bbox": [71, 456, 221, 474], "content": "3.4. The algebra", "index": 10}, {"type": "text", "bbox": [70, 479, 541, 554], "content": "A weight of satisfies , and . For any , there are the conjugations . and . The charge-conjugation equals for odd , and for even . When there are four additional conjugations; these six correspond to all permutations of the Dynkin labels .", "index": 11}, {"type": "text", "bbox": [70, 554, 541, 584], "content": "There are three non-trivial simple-currents, , and . Explicitly, we have with , and", "index": 12}, {"type": "interline_equation", "bbox": [174, 596, 430, 627], "content": "", "index": 13}, {"type": "text", "bbox": [69, 639, 541, 669], "content": "with . From this we compute . The fusion products we need are", "index": 14}]
|
[{"bbox": [93, 158, 540, 178], "content": "This rank-level duality for is especially interesting, as it defines a fusion ring iso-", "parent_index": 2, "line_index": 0}, {"bbox": [71, 176, 540, 191], "content": "morphism (see §5). When , we get a nontrivial fusion-symmetry:", "parent_index": 2, "line_index": 1}, {"bbox": [71, 189, 134, 210], "content": ".", "parent_index": 2, "line_index": 2}, {"bbox": [95, 209, 281, 223], "content": "The only fusion product we need is", "parent_index": 3, "line_index": 0}, {"bbox": [70, 263, 506, 281], "content": "valid for and . The following character formula (2.1b) will also be used:", "parent_index": 5, "line_index": 0}, {"bbox": [71, 341, 257, 357], "content": "where as before.", "parent_index": 7, "line_index": 0}, {"bbox": [93, 362, 542, 381], "content": "Theorem 3.C. The fusion-symmetries for level , when and either or", "parent_index": 8, "line_index": 0}, {"bbox": [71, 379, 541, 394], "content": "is even, are for . When but both and are odd, then there is no", "parent_index": 8, "line_index": 1}, {"bbox": [70, 392, 542, 410], "content": "nontrivial fusion-symmetry. When , they are even) or ( odd), for", "parent_index": 8, "line_index": 2}, {"bbox": [71, 407, 136, 423], "content": ".", "parent_index": 8, "line_index": 3}, {"bbox": [95, 429, 305, 444], "content": "When is even, .", "parent_index": 9, "line_index": 0}, {"bbox": [70, 456, 219, 476], "content": "3.4. The algebra", "parent_index": 10, "line_index": 0}, {"bbox": [94, 481, 540, 497], "content": "A weight of satisfies , and .", "parent_index": 11, "line_index": 0}, {"bbox": [69, 495, 541, 513], "content": "For any , there are the conjugations . and . The", "parent_index": 11, "line_index": 1}, {"bbox": [70, 511, 541, 527], "content": "charge-conjugation equals for odd , and for even . When there are four", "parent_index": 11, "line_index": 2}, {"bbox": [69, 524, 541, 542], "content": "additional conjugations; these six correspond to all permutations of the Dynkin", "parent_index": 11, "line_index": 3}, {"bbox": [71, 542, 156, 557], "content": "labels .", "parent_index": 11, "line_index": 4}, {"bbox": [93, 554, 541, 572], "content": "There are three non-trivial simple-currents, , and . Explicitly, we have", "parent_index": 12, "line_index": 0}, {"bbox": [71, 570, 442, 586], "content": "with , and", "parent_index": 12, "line_index": 1}, {"bbox": [69, 639, 541, 660], "content": "with . From this we compute .", "parent_index": 14, "line_index": 0}, {"bbox": [94, 657, 268, 672], "content": "The fusion products we need are", "parent_index": 14, "line_index": 1}]
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[]
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[{"bbox": [267, 129, 344, 146], "content": "\\tilde{S}_{\\tau\\lambda,\\tau\\mu}=S_{\\lambda\\mu}\\ .", "parent_index": 1, "subtype": "interline"}, {"bbox": [234, 158, 257, 174], "content": "C_{r}^{(1)}", "parent_index": 2, "subtype": "inline"}, {"bbox": [127, 176, 225, 190], "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "parent_index": 2, "subtype": "inline"}, {"bbox": [312, 176, 341, 188], "content": "k=r", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 192, 128, 208], "content": "\\pi_{\\mathrm{rld}}\\lambda\\,{\\overset{\\mathrm{def}}{=}}\\,\\tau\\lambda", "parent_index": 2, "subtype": "inline"}, {"bbox": [201, 234, 406, 251], "content": "\\Lambda_{1}\\sqcup\\Lambda_{i}=\\Lambda_{i-1}\\sqcup\\Lambda_{i+1}\\sqcup\\left(\\Lambda_{1}+\\Lambda_{i}\\right)\\,,", "parent_index": 4, "subtype": "interline"}, {"bbox": [118, 267, 144, 276], "content": "i<r", "parent_index": 5, "subtype": "inline"}, {"bbox": [171, 264, 200, 277], "content": "k\\geq2", "parent_index": 5, "subtype": "inline"}, {"bbox": [209, 289, 401, 329], "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "parent_index": 6, "subtype": "interline"}, {"bbox": [105, 342, 203, 356], "content": "\\lambda^{+}(\\ell)=(\\lambda+\\rho)(\\ell)", "parent_index": 7, "subtype": "inline"}, {"bbox": [325, 362, 348, 378], "content": "C_{r}^{(1)}", "parent_index": 8, "subtype": "inline"}, {"bbox": [378, 365, 387, 377], "content": "k", "parent_index": 8, "subtype": "inline"}, {"bbox": [424, 365, 455, 379], "content": "k\\neq r", "parent_index": 8, "subtype": "inline"}, {"bbox": [516, 366, 524, 377], "content": "k", "parent_index": 8, "subtype": "inline"}, {"bbox": [71, 385, 78, 391], "content": "r", "parent_index": 8, "subtype": "inline"}, {"bbox": [146, 379, 171, 393], "content": "\\pi[1]^{i}", "parent_index": 8, "subtype": "inline"}, {"bbox": [193, 379, 243, 394], "content": "i\\in\\{0,1\\}", "parent_index": 8, "subtype": "inline"}, {"bbox": [286, 380, 317, 393], "content": "k\\neq r", "parent_index": 8, "subtype": "inline"}, {"bbox": [365, 380, 373, 391], "content": "k", "parent_index": 8, "subtype": "inline"}, {"bbox": [399, 382, 407, 391], "content": "r", "parent_index": 8, "subtype": "inline"}, {"bbox": [255, 394, 285, 406], "content": "k=r", "parent_index": 8, "subtype": "inline"}, {"bbox": [337, 392, 384, 409], "content": "\\pi[1]^{i}\\,\\pi_{\\mathrm{rld}}^{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [391, 393, 399, 406], "content": "\\mathit{\\Pi}_{k}", "parent_index": 8, "subtype": "inline"}, {"bbox": [450, 393, 475, 408], "content": "\\pi[1]^{i}", "parent_index": 8, "subtype": "inline"}, {"bbox": [483, 394, 491, 406], "content": "\\mathit{k}", "parent_index": 8, "subtype": "inline"}, {"bbox": [71, 409, 131, 422], "content": "i,j\\in\\{0,1\\}", "parent_index": 8, "subtype": "inline"}, {"bbox": [130, 431, 159, 441], "content": "r=k", "parent_index": 9, "subtype": "inline"}, {"bbox": [205, 429, 303, 444], "content": "A(C_{r,k})\\cong\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [161, 458, 219, 474], "content": "D_{r}^{(1)},\\,r\\geq4", "parent_index": 10, "subtype": "inline"}, {"bbox": [144, 483, 151, 493], "content": "\\lambda", "parent_index": 11, "subtype": "inline"}, {"bbox": [166, 484, 182, 496], "content": "P_{+}", "parent_index": 11, "subtype": "inline"}, {"bbox": [228, 483, 438, 495], "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!\\cdot\\!\\cdot\\!+\\!2\\lambda_{r-2}\\!+\\!\\lambda_{r-1}\\!+\\!\\lambda_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [468, 484, 536, 494], "content": "\\kappa=k{+}2r{-}2", "parent_index": 11, "subtype": "inline"}, {"bbox": [114, 501, 120, 507], "content": "r", "parent_index": 11, "subtype": "inline"}, {"bbox": [266, 498, 307, 509], "content": "C_{0}=i d", "parent_index": 11, "subtype": "inline"}, {"bbox": [335, 498, 510, 510], "content": "C_{1}\\lambda=(\\lambda_{0},\\lambda_{1},.\\dots,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1})", "parent_index": 11, "subtype": "inline"}, {"bbox": [174, 513, 183, 522], "content": "C", "parent_index": 11, "subtype": "inline"}, {"bbox": [223, 513, 237, 524], "content": "C_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [284, 516, 290, 522], "content": "r", "parent_index": 11, "subtype": "inline"}, {"bbox": [321, 513, 335, 524], "content": "C_{0}", "parent_index": 11, "subtype": "inline"}, {"bbox": [385, 516, 391, 522], "content": "r", "parent_index": 11, "subtype": "inline"}, {"bbox": [435, 513, 464, 522], "content": "r=4", "parent_index": 11, "subtype": "inline"}, {"bbox": [253, 529, 265, 540], "content": "C_{i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [473, 524, 496, 541], "content": "{D}_{4}^{(1)}", "parent_index": 11, "subtype": "inline"}, {"bbox": [105, 543, 153, 554], "content": "\\lambda_{1},\\lambda_{3},\\lambda_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [335, 557, 347, 568], "content": "J_{v}", "parent_index": 12, "subtype": "inline"}, {"bbox": [357, 558, 369, 568], "content": "J_{s}", "parent_index": 12, "subtype": "inline"}, {"bbox": [399, 558, 423, 568], "content": "J_{v}J_{s}", "parent_index": 12, "subtype": "inline"}, {"bbox": [71, 571, 262, 584], "content": "J_{v}\\lambda=\\left(\\lambda_{1},\\lambda_{0},\\lambda_{2},...\\,,\\lambda_{r-2},\\lambda_{r},\\lambda_{r-1}\\right)", "parent_index": 12, "subtype": "inline"}, {"bbox": [293, 571, 414, 584], "content": "Q_{v}(\\lambda)=(\\lambda_{r-1}+\\lambda_{r})/2", "parent_index": 12, "subtype": "inline"}, {"bbox": [174, 596, 430, 627], "content": "J_{s}\\lambda={\\left\\{\\begin{array}{l l}{(\\lambda_{r},\\lambda_{r-1},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~even}},}\\\\ {(\\lambda_{r-1},\\lambda_{r},\\lambda_{r-2},\\ldots,\\lambda_{1},\\lambda_{0})}&{{\\mathrm{if~}}r{\\mathrm{~is~odd}},}\\end{array}\\right.}", "parent_index": 13, "subtype": "interline"}, {"bbox": [97, 639, 320, 658], "content": "\\begin{array}{r}{Q_{s}(\\lambda)=(2\\sum_{j=1}^{r-2}j\\lambda_{j}\\!-\\!(r\\!-\\!2)\\lambda_{r-1}\\!-\\!r\\lambda_{r})/4}\\end{array}", "parent_index": 14, "subtype": "inline"}, {"bbox": [444, 642, 536, 656], "content": "Q_{s}(J_{s}0)=-r k/4", "parent_index": 14, "subtype": "inline"}]
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[]
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valid for all $1\leq i<r-2$ and $k>2$ . We also will use the character formula (2.1b)
$$
\chi_{\Lambda_{1}}[\lambda]={\frac{S_{\Lambda_{1}\lambda}}{S_{0\lambda}}}=2\sum_{\ell=1}^{r}\cos(2\pi{\frac{\lambda^{+}(\ell)}{\kappa}})~,
$$
where $\lambda^{+}(\ell)\,=\,(\lambda+\rho)(\ell)$ and the orthonormal components $\lambda(\ell)$ are defined by $\lambda(\ell)\,=$ $\begin{array}{r}{\sum_{i=\ell}^{r-1}\lambda_{i}+\frac{\lambda_{r}-\lambda_{r-1}}{2}}\end{array}$ .
The simple-current automorphisms are as follows, and depend on whether $r$ and $k$ are even or odd. When $r$ is odd, the group of simple-currents is generated by $J_{s}$ . If in addition $k$ is odd, there will be only two simple-current automorphisms: $\pi=\pi^{\prime}=\pi[a]=J_{s}^{4a\cup_{s}}$ for $a\in\{0,2\}$ . If instead $k$ is even, there will be four simple-current automorphisms: $\pi=\pi[a]$ and $\pi^{\prime}=\pi[a k-a]$ for $0\leq a\leq3$ . When $k\equiv2$ (mod 4), these form the group $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ , otherwise when $4|k$ the group is $\mathbb{Z}_{4}$ .
When $r$ is even, the simple-currents are generated by both $J_{v}$ and $J_{s}$ . If in addition $k$ is even, we have 16 simple-current automorphisms:
$$
{\boldsymbol{\pi}}={\boldsymbol{\pi}}\left[{\boldsymbol{a}}\quad b\,\right]\qquad{\mathrm{and}}\qquad{\boldsymbol{\pi}}^{\prime}={\boldsymbol{\pi}}\left[{\boldsymbol{a}}\quad c\right]
$$
for any $a,b,c,d\in\{0,1\}$ , forming a group isomorphic to $\mathbb{Z}_{2}^{4}$ . This notation means
$$
\pi\left[\!\!\begin{array}{r c}{{a}}&{{b}}\\ {{c}}&{{d}}\end{array}\!\!\right](\lambda)\ensuremath{\stackrel{\mathrm{def}}{=}}\ J_{v}^{2a\,Q_{v}(\lambda)+2b\,Q_{s}(\lambda)}J_{s}^{2c\,Q_{v}(\lambda)+2d\,Q_{s}(\lambda)}\lambda\ .
$$
When $k$ is odd, we will have six simple-current automorphisms:
$$
{\begin{array}{r l}&{\pi\,=\pi\left[{\begin{array}{c c}{a}&{0}\\ {0}&{d}\end{array}}\right]\qquad{\mathrm{with}}\qquad\pi^{\prime}=\pi\left[{\begin{array}{c c}{a\left(d+1\right)}&{{\frac{d r}{2}}}\\ {{\frac{d r}{2}}}&{d}\end{array}}\right]}\\ {{\mathrm{rr}}\quad\pi\,=\pi\left[{\begin{array}{c c}{{\frac{r}{2}}+1}&{b}\\ {c}&{1}\end{array}}\right]\qquad{\mathrm{with}}\qquad\pi^{\prime}=\pi\left[{\begin{array}{c c}{{\frac{r}{2}}+1+b c{\frac{r}{2}}}&{b+{\frac{r}{2}}}\\ {{\frac{r}{2}}+1+b c+b}&{1}\end{array}}\right]}\end{array}},
$$
where $a={\frac{r}{2}}$ or $d=0$ , and where $b=1$ or $d=1$ . The corresponding permutation of $P_{+}$ is still given by (3.5). Again, for these $r,k$ , these are the values of $a,b,c,d$ for which (3.5) is invertible. For $k$ odd, the group of simple-current automorphisms is isomorphic to the symmetric group $\mathfrak{S}_{3}$ when 4 divides $r$ , and to $\mathbb{Z}_{6}$ when $r\equiv2$ (mod 4).
For $k=2$ (so $\kappa=2r$ ), there are several Galois fusion-symmetries. In particular, write $\lambda^{i}\,=\,\lambda^{2r-i}\,=\,\Lambda_{i}$ for $1\,\leq\,i\,\leq\,r\,-\,2$ , and $\lambda^{r\pm1}=\Lambda_{r-1}+\Lambda_{r}$ . As with $B_{r,2}$ , $\begin{array}{r}{S_{00}^{2}\,=\,\frac{1}{4\kappa}}\end{array}$ 1 is rational so for any ${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$ coprime to $2r$ , we get a Galois fusion-symmetry $\pi\{m\}$ . It takes $\lambda^{a}$ to $\lambda^{m a}$ , where the superscript is taken mod $2r$ , and will fix $J_{v}0$ . Also, $\pi\{m\}$ will send $J_{s}0$ to $J_{s}^{m}0$ , as well as stabilise the set $\left\{\Lambda_{r},\Lambda_{r-1},J_{v}\Lambda_{r},J_{v}\Lambda_{r-1}\right\}$ . (In particular, put $t=r$ when $r$ is even or when $m\equiv1$ (mod 4), otherwise put $t=r-1$ ; then for any $i,j,\,\pi\{m\}\,C_{1}^{j}J_{v}^{i}\Lambda_{r}$ is $C_{1}^{j}J_{v}^{i}\Lambda_{t}$ or $C_{1}^{j}J_{v}^{i+1}\Lambda_{t}$ , when the Jacobi symbol $\textstyle\left({\frac{\kappa}{m}}\right)$ is $\pm1$ , respectively.)
Theorem 3.D. The fusion-symmetries of $D_{r}^{(1)}$ for $k\neq2$ are all of the form $C_{i}\,\pi$ , where $C_{i}$ is a conjugation, and where $\pi$ is a simple-current automorphism. Similarly for ${D}_{4}^{(1)}$ at $k=2$ . Finally, when both $k=2$ and $r>4$ , any fusion-symmetry $\pi$ can be written as $\pi=C_{1}^{a}\,\pi_{v}^{b}\,\pi\{m\}$ for $a,b\in\{0,1\}$ and any $m\in\mathbb{Z}_{2r}^{\times}$ , $1\leq m<r$ .
|
<html><body>
<p data-bbox="70 70 505 86">valid for all $1\leq i<r-2$ and $k>2$ . We also will use the character formula (2.1b) </p>
<div class="equation" data-bbox="205 99 405 137">$$
\chi_{\Lambda_{1}}[\lambda]={\frac{S_{\Lambda_{1}\lambda}}{S_{0\lambda}}}=2\sum_{\ell=1}^{r}\cos(2\pi{\frac{\lambda^{+}(\ell)}{\kappa}})~,
$$</div>
<p data-bbox="70 147 541 178">where $\lambda^{+}(\ell)\,=\,(\lambda+\rho)(\ell)$ and the orthonormal components $\lambda(\ell)$ are defined by $\lambda(\ell)\,=$ $\begin{array}{r}{\sum_{i=\ell}^{r-1}\lambda_{i}+\frac{\lambda_{r}-\lambda_{r-1}}{2}}\end{array}$ . </p>
<p data-bbox="70 178 542 264">The simple-current automorphisms are as follows, and depend on whether $r$ and $k$ are even or odd. When $r$ is odd, the group of simple-currents is generated by $J_{s}$ . If in addition $k$ is odd, there will be only two simple-current automorphisms: $\pi=\pi^{\prime}=\pi[a]=J_{s}^{4a\cup_{s}}$ for $a\in\{0,2\}$ . If instead $k$ is even, there will be four simple-current automorphisms: $\pi=\pi[a]$ and $\pi^{\prime}=\pi[a k-a]$ for $0\leq a\leq3$ . When $k\equiv2$ (mod 4), these form the group $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ , otherwise when $4|k$ the group is $\mathbb{Z}_{4}$ . </p>
<p data-bbox="70 264 541 293">When $r$ is even, the simple-currents are generated by both $J_{v}$ and $J_{s}$ . If in addition $k$ is even, we have 16 simple-current automorphisms: </p>
<div class="equation" data-bbox="195 305 415 337">$$
{\boldsymbol{\pi}}={\boldsymbol{\pi}}\left[{\boldsymbol{a}}\quad b\,\right]\qquad{\mathrm{and}}\qquad{\boldsymbol{\pi}}^{\prime}={\boldsymbol{\pi}}\left[{\boldsymbol{a}}\quad c\right]
$$</div>
<p data-bbox="69 346 495 362">for any $a,b,c,d\in\{0,1\}$ , forming a group isomorphic to $\mathbb{Z}_{2}^{4}$ . This notation means </p>
<div class="equation" data-bbox="164 375 448 406">$$
\pi\left[\!\!\begin{array}{r c}{{a}}&{{b}}\\ {{c}}&{{d}}\end{array}\!\!\right](\lambda)\ensuremath{\stackrel{\mathrm{def}}{=}}\ J_{v}^{2a\,Q_{v}(\lambda)+2b\,Q_{s}(\lambda)}J_{s}^{2c\,Q_{v}(\lambda)+2d\,Q_{s}(\lambda)}\lambda\ .
$$</div>
<p data-bbox="70 415 407 430">When $k$ is odd, we will have six simple-current automorphisms: </p>
<div class="equation" data-bbox="126 441 490 510">$$
{\begin{array}{r l}&{\pi\,=\pi\left[{\begin{array}{c c}{a}&{0}\\ {0}&{d}\end{array}}\right]\qquad{\mathrm{with}}\qquad\pi^{\prime}=\pi\left[{\begin{array}{c c}{a\left(d+1\right)}&{{\frac{d r}{2}}}\\ {{\frac{d r}{2}}}&{d}\end{array}}\right]}\\ {{\mathrm{rr}}\quad\pi\,=\pi\left[{\begin{array}{c c}{{\frac{r}{2}}+1}&{b}\\ {c}&{1}\end{array}}\right]\qquad{\mathrm{with}}\qquad\pi^{\prime}=\pi\left[{\begin{array}{c c}{{\frac{r}{2}}+1+b c{\frac{r}{2}}}&{b+{\frac{r}{2}}}\\ {{\frac{r}{2}}+1+b c+b}&{1}\end{array}}\right]}\end{array}},
$$</div>
<p data-bbox="69 516 541 574">where $a={\frac{r}{2}}$ or $d=0$ , and where $b=1$ or $d=1$ . The corresponding permutation of $P_{+}$ is still given by (3.5). Again, for these $r,k$ , these are the values of $a,b,c,d$ for which (3.5) is invertible. For $k$ odd, the group of simple-current automorphisms is isomorphic to the symmetric group $\mathfrak{S}_{3}$ when 4 divides $r$ , and to $\mathbb{Z}_{6}$ when $r\equiv2$ (mod 4). </p>
<p data-bbox="70 574 542 678">For $k=2$ (so $\kappa=2r$ ), there are several Galois fusion-symmetries. In particular, write $\lambda^{i}\,=\,\lambda^{2r-i}\,=\,\Lambda_{i}$ for $1\,\leq\,i\,\leq\,r\,-\,2$ , and $\lambda^{r\pm1}=\Lambda_{r-1}+\Lambda_{r}$ . As with $B_{r,2}$ , $\begin{array}{r}{S_{00}^{2}\,=\,\frac{1}{4\kappa}}\end{array}$ 1 is rational so for any ${\boldsymbol{\mathit{\varepsilon}}}^{\prime}{\boldsymbol{\mathit{m}}}$ coprime to $2r$ , we get a Galois fusion-symmetry $\pi\{m\}$ . It takes $\lambda^{a}$ to $\lambda^{m a}$ , where the superscript is taken mod $2r$ , and will fix $J_{v}0$ . Also, $\pi\{m\}$ will send $J_{s}0$ to $J_{s}^{m}0$ , as well as stabilise the set $\left\{\Lambda_{r},\Lambda_{r-1},J_{v}\Lambda_{r},J_{v}\Lambda_{r-1}\right\}$ . (In particular, put $t=r$ when $r$ is even or when $m\equiv1$ (mod 4), otherwise put $t=r-1$ ; then for any $i,j,\,\pi\{m\}\,C_{1}^{j}J_{v}^{i}\Lambda_{r}$ is $C_{1}^{j}J_{v}^{i}\Lambda_{t}$ or $C_{1}^{j}J_{v}^{i+1}\Lambda_{t}$ , when the Jacobi symbol $\textstyle\left({\frac{\kappa}{m}}\right)$ is $\pm1$ , respectively.) </p>
<p data-bbox="70 684 541 716">Theorem 3.D. The fusion-symmetries of $D_{r}^{(1)}$ for $k\neq2$ are all of the form $C_{i}\,\pi$ , where $C_{i}$ is a conjugation, and where $\pi$ is a simple-current automorphism. Similarly for ${D}_{4}^{(1)}$ at $k=2$ . Finally, when both $k=2$ and $r>4$ , any fusion-symmetry $\pi$ can be written as $\pi=C_{1}^{a}\,\pi_{v}^{b}\,\pi\{m\}$ for $a,b\in\{0,1\}$ and any $m\in\mathbb{Z}_{2r}^{\times}$ , $1\leq m<r$ . </p>
</body></html>
|
0002044v1
| 11 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "valid for all $1\\leq i<r-2$ and $k>2$ . We also will use the character formula (2.1b) ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "where $\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)$ and the orthonormal components $\\lambda(\\ell)$ are defined by $\\lambda(\\ell)\\,=$ $\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}$ . ", "page_idx": 11}, {"type": "text", "text": "The simple-current automorphisms are as follows, and depend on whether $r$ and $k$ are even or odd. When $r$ is odd, the group of simple-currents is generated by $J_{s}$ . If in addition $k$ is odd, there will be only two simple-current automorphisms: $\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}$ for $a\\in\\{0,2\\}$ . If instead $k$ is even, there will be four simple-current automorphisms: $\\pi=\\pi[a]$ and $\\pi^{\\prime}=\\pi[a k-a]$ for $0\\leq a\\leq3$ . When $k\\equiv2$ (mod 4), these form the group $\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}$ , otherwise when $4|k$ the group is $\\mathbb{Z}_{4}$ . ", "page_idx": 11}, {"type": "text", "text": "When $r$ is even, the simple-currents are generated by both $J_{v}$ and $J_{s}$ . If in addition $k$ is even, we have 16 simple-current automorphisms: ", "page_idx": 11}, {"type": "equation", "text": "$$\n{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "for any $a,b,c,d\\in\\{0,1\\}$ , forming a group isomorphic to $\\mathbb{Z}_{2}^{4}$ . This notation means ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "When $k$ is odd, we will have six simple-current automorphisms: ", "page_idx": 11}, {"type": "equation", "text": "$$\n{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "where $a={\\frac{r}{2}}$ or $d=0$ , and where $b=1$ or $d=1$ . The corresponding permutation of $P_{+}$ is still given by (3.5). Again, for these $r,k$ , these are the values of $a,b,c,d$ for which (3.5) is invertible. For $k$ odd, the group of simple-current automorphisms is isomorphic to the symmetric group $\\mathfrak{S}_{3}$ when 4 divides $r$ , and to $\\mathbb{Z}_{6}$ when $r\\equiv2$ (mod 4). ", "page_idx": 11}, {"type": "text", "text": "For $k=2$ (so $\\kappa=2r$ ), there are several Galois fusion-symmetries. In particular, write $\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}$ for $1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2$ , and $\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}$ . As with $B_{r,2}$ , $\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}$ 1 is rational so for any ${\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}$ coprime to $2r$ , we get a Galois fusion-symmetry $\\pi\\{m\\}$ . It takes $\\lambda^{a}$ to $\\lambda^{m a}$ , where the superscript is taken mod $2r$ , and will fix $J_{v}0$ . Also, $\\pi\\{m\\}$ will send $J_{s}0$ to $J_{s}^{m}0$ , as well as stabilise the set $\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}$ . (In particular, put $t=r$ when $r$ is even or when $m\\equiv1$ (mod 4), otherwise put $t=r-1$ ; then for any $i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}$ is $C_{1}^{j}J_{v}^{i}\\Lambda_{t}$ or $C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}$ , when the Jacobi symbol $\\textstyle\\left({\\frac{\\kappa}{m}}\\right)$ is $\\pm1$ , respectively.) ", "page_idx": 11}, {"type": "text", "text": "Theorem 3.D. The fusion-symmetries of $D_{r}^{(1)}$ for $k\\neq2$ are all of the form $C_{i}\\,\\pi$ , where $C_{i}$ is a conjugation, and where $\\pi$ is a simple-current automorphism. Similarly for ${D}_{4}^{(1)}$ at $k=2$ . Finally, when both $k=2$ and $r>4$ , any fusion-symmetry $\\pi$ can be written as $\\pi=C_{1}^{a}\\,\\pi_{v}^{b}\\,\\pi\\{m\\}$ for $a,b\\in\\{0,1\\}$ and any $m\\in\\mathbb{Z}_{2r}^{\\times}$ , $1\\leq m<r$ . ", "page_idx": 11}]
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{"preproc_blocks": [{"type": "text", "bbox": [70, 70, 505, 86], "lines": [{"bbox": [71, 73, 504, 88], "spans": [{"bbox": [71, 73, 135, 88], "score": 1.0, "content": "valid for all ", "type": "text"}, {"bbox": [135, 75, 204, 86], "score": 0.88, "content": "1\\leq i<r-2", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [204, 73, 230, 88], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [230, 75, 259, 85], "score": 0.88, "content": "k>2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [259, 73, 504, 88], "score": 1.0, "content": ". We also will use the character formula (2.1b)", "type": "text"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [205, 99, 405, 137], "lines": [{"bbox": [205, 99, 405, 137], "spans": [{"bbox": [205, 99, 405, 137], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [70, 147, 541, 178], "lines": [{"bbox": [70, 149, 541, 166], "spans": [{"bbox": [70, 149, 106, 166], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 150, 209, 163], "score": 0.92, "content": "\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [209, 149, 396, 166], "score": 1.0, "content": " and the orthonormal components ", "type": "text"}, {"bbox": [396, 151, 418, 163], "score": 0.93, "content": "\\lambda(\\ell)", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [418, 149, 503, 166], "score": 1.0, "content": " are defined by ", "type": "text"}, {"bbox": [503, 150, 541, 164], "score": 0.91, "content": "\\lambda(\\ell)\\,=", "type": "inline_equation", "height": 14, "width": 38}], "index": 2}, {"bbox": [71, 162, 175, 180], "spans": [{"bbox": [71, 163, 171, 180], "score": 0.93, "content": "\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}", "type": "inline_equation", "height": 17, "width": 100}, {"bbox": [171, 162, 175, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [70, 178, 542, 264], "lines": [{"bbox": [76, 178, 542, 196], "spans": [{"bbox": [76, 178, 481, 196], "score": 1.0, "content": "The simple-current automorphisms are as follows, and depend on whether ", "type": "text"}, {"bbox": [482, 185, 487, 191], "score": 0.86, "content": "r", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [488, 178, 513, 196], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [513, 182, 520, 191], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [520, 178, 542, 196], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [69, 194, 541, 209], "spans": [{"bbox": [69, 194, 173, 209], "score": 1.0, "content": "even or odd. When", "type": "text"}, {"bbox": [174, 199, 180, 205], "score": 0.67, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [180, 194, 451, 209], "score": 1.0, "content": " is odd, the group of simple-currents is generated by ", "type": "text"}, {"bbox": [451, 196, 463, 207], "score": 0.9, "content": "J_{s}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [464, 194, 541, 209], "score": 1.0, "content": ". If in addition", "type": "text"}], "index": 5}, {"bbox": [71, 207, 543, 224], "spans": [{"bbox": [71, 210, 78, 219], "score": 0.86, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 207, 404, 224], "score": 1.0, "content": " is odd, there will be only two simple-current automorphisms: ", "type": "text"}, {"bbox": [405, 209, 521, 222], "score": 0.93, "content": "\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [521, 207, 543, 224], "score": 1.0, "content": "for", "type": "text"}], "index": 6}, {"bbox": [71, 222, 540, 239], "spans": [{"bbox": [71, 223, 122, 237], "score": 0.92, "content": "a\\in\\{0,2\\}", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [122, 222, 182, 239], "score": 1.0, "content": ". If instead ", "type": "text"}, {"bbox": [182, 224, 190, 234], "score": 0.84, "content": "k", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [190, 222, 495, 239], "score": 1.0, "content": " is even, there will be four simple-current automorphisms: ", "type": "text"}, {"bbox": [496, 224, 540, 236], "score": 0.93, "content": "\\pi=\\pi[a]", "type": "inline_equation", "height": 12, "width": 44}], "index": 7}, {"bbox": [70, 236, 541, 254], "spans": [{"bbox": [70, 236, 95, 254], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 238, 172, 251], "score": 0.93, "content": "\\pi^{\\prime}=\\pi[a k-a]", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [172, 236, 194, 254], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [195, 238, 248, 250], "score": 0.85, "content": "0\\leq a\\leq3", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [249, 236, 293, 254], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [294, 239, 324, 248], "score": 0.92, "content": "k\\equiv2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [324, 236, 493, 254], "score": 1.0, "content": " (mod 4), these form the group ", "type": "text"}, {"bbox": [494, 239, 536, 250], "score": 0.92, "content": "\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [536, 236, 541, 254], "score": 1.0, "content": ",", "type": "text"}], "index": 8}, {"bbox": [70, 250, 260, 268], "spans": [{"bbox": [70, 250, 155, 268], "score": 1.0, "content": "otherwise when ", "type": "text"}, {"bbox": [155, 253, 172, 265], "score": 0.88, "content": "4|k", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [172, 250, 241, 268], "score": 1.0, "content": " the group is ", "type": "text"}, {"bbox": [242, 254, 255, 264], "score": 0.9, "content": "\\mathbb{Z}_{4}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [256, 250, 260, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 6.5}, {"type": "text", "bbox": [70, 264, 541, 293], "lines": [{"bbox": [95, 266, 541, 281], "spans": [{"bbox": [95, 266, 130, 281], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 271, 136, 277], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [136, 266, 408, 281], "score": 1.0, "content": " is even, the simple-currents are generated by both ", "type": "text"}, {"bbox": [408, 268, 421, 279], "score": 0.92, "content": "J_{v}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [421, 266, 448, 281], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [448, 268, 460, 279], "score": 0.91, "content": "J_{s}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [461, 266, 541, 281], "score": 1.0, "content": ". If in addition", "type": "text"}], "index": 10}, {"bbox": [71, 280, 349, 297], "spans": [{"bbox": [71, 282, 78, 291], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 280, 349, 297], "score": 1.0, "content": " is even, we have 16 simple-current automorphisms:", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "interline_equation", "bbox": [195, 305, 415, 337], "lines": [{"bbox": [195, 305, 415, 337], "spans": [{"bbox": [195, 305, 415, 337], "score": 0.92, "content": "{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 346, 495, 362], "lines": [{"bbox": [70, 348, 497, 366], "spans": [{"bbox": [70, 348, 112, 366], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [112, 351, 195, 363], "score": 0.93, "content": "a,b,c,d\\in\\{0,1\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [195, 348, 365, 366], "score": 1.0, "content": ", forming a group isomorphic to ", "type": "text"}, {"bbox": [365, 350, 379, 363], "score": 0.92, "content": "\\mathbb{Z}_{2}^{4}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [379, 348, 497, 366], "score": 1.0, "content": ". This notation means", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [164, 375, 448, 406], "lines": [{"bbox": [164, 375, 448, 406], "spans": [{"bbox": [164, 375, 448, 406], "score": 0.92, "content": "\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 415, 407, 430], "lines": [{"bbox": [71, 417, 405, 432], "spans": [{"bbox": [71, 417, 106, 432], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [106, 420, 113, 429], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [113, 417, 405, 432], "score": 1.0, "content": " is odd, we will have six simple-current automorphisms:", "type": "text"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [126, 441, 490, 510], "lines": [{"bbox": [126, 441, 490, 510], "spans": [{"bbox": [126, 441, 490, 510], "score": 0.93, "content": "{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [69, 516, 541, 574], "lines": [{"bbox": [70, 517, 539, 536], "spans": [{"bbox": [70, 517, 106, 536], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 520, 137, 533], "score": 0.93, "content": "a={\\frac{r}{2}}", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [138, 517, 156, 536], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [156, 520, 186, 529], "score": 0.89, "content": "d=0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [187, 517, 252, 536], "score": 1.0, "content": ", and where ", "type": "text"}, {"bbox": [252, 519, 281, 529], "score": 0.88, "content": "b=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [281, 517, 299, 536], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [299, 520, 329, 529], "score": 0.89, "content": "d=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [330, 517, 523, 536], "score": 1.0, "content": ". The corresponding permutation of ", "type": "text"}, {"bbox": [523, 520, 539, 532], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}], "index": 17}, {"bbox": [70, 533, 540, 547], "spans": [{"bbox": [70, 533, 274, 547], "score": 1.0, "content": "is still given by (3.5). Again, for these ", "type": "text"}, {"bbox": [275, 533, 293, 546], "score": 0.87, "content": "r,k", "type": "inline_equation", "height": 13, "width": 18}, {"bbox": [293, 533, 419, 547], "score": 1.0, "content": ", these are the values of ", "type": "text"}, {"bbox": [419, 534, 459, 546], "score": 0.91, "content": "a,b,c,d", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [459, 533, 540, 547], "score": 1.0, "content": " for which (3.5)", "type": "text"}], "index": 18}, {"bbox": [69, 547, 542, 562], "spans": [{"bbox": [69, 547, 163, 562], "score": 1.0, "content": "is invertible. For ", "type": "text"}, {"bbox": [164, 549, 171, 558], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [171, 547, 542, 562], "score": 1.0, "content": " odd, the group of simple-current automorphisms is isomorphic to the", "type": "text"}], "index": 19}, {"bbox": [70, 562, 442, 576], "spans": [{"bbox": [70, 562, 163, 576], "score": 1.0, "content": "symmetric group ", "type": "text"}, {"bbox": [163, 564, 178, 574], "score": 0.91, "content": "\\mathfrak{S}_{3}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [179, 562, 264, 576], "score": 1.0, "content": " when 4 divides ", "type": "text"}, {"bbox": [264, 567, 270, 573], "score": 0.85, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [271, 562, 314, 576], "score": 1.0, "content": ", and to ", "type": "text"}, {"bbox": [315, 563, 328, 574], "score": 0.89, "content": "\\mathbb{Z}_{6}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [329, 562, 363, 576], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [363, 564, 392, 573], "score": 0.85, "content": "r\\equiv2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [392, 562, 442, 576], "score": 1.0, "content": " (mod 4).", "type": "text"}], "index": 20}], "index": 18.5}, {"type": "text", "bbox": [70, 574, 542, 678], "lines": [{"bbox": [93, 575, 541, 592], "spans": [{"bbox": [93, 575, 115, 592], "score": 1.0, "content": "For", "type": "text"}, {"bbox": [116, 578, 145, 587], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [145, 575, 167, 592], "score": 1.0, "content": " (so ", "type": "text"}, {"bbox": [167, 578, 201, 587], "score": 0.86, "content": "\\kappa=2r", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [202, 575, 541, 592], "score": 1.0, "content": "), there are several Galois fusion-symmetries. In particular, write", "type": "text"}], "index": 21}, {"bbox": [71, 587, 540, 609], "spans": [{"bbox": [71, 591, 160, 602], "score": 0.9, "content": "\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [160, 587, 183, 609], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [184, 593, 259, 603], "score": 0.91, "content": "1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [259, 587, 290, 609], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [291, 591, 390, 604], "score": 0.92, "content": "\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [390, 587, 447, 609], "score": 1.0, "content": ". As with ", "type": "text"}, {"bbox": [447, 592, 469, 605], "score": 0.92, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [469, 587, 476, 609], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 590, 527, 605], "score": 0.93, "content": "\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}", "type": "inline_equation", "height": 15, "width": 50}, {"bbox": [511, 588, 540, 604], "score": 1.0, "content": "1 is", "type": "text"}], "index": 22}, {"bbox": [71, 605, 542, 620], "spans": [{"bbox": [71, 605, 169, 620], "score": 1.0, "content": "rational so for any ", "type": "text"}, {"bbox": [169, 610, 180, 615], "score": 0.86, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [180, 605, 241, 620], "score": 1.0, "content": " coprime to ", "type": "text"}, {"bbox": [241, 607, 253, 616], "score": 0.86, "content": "2r", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [254, 605, 432, 620], "score": 1.0, "content": ", we get a Galois fusion-symmetry ", "type": "text"}, {"bbox": [432, 606, 462, 619], "score": 0.94, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [462, 605, 512, 620], "score": 1.0, "content": ". It takes ", "type": "text"}, {"bbox": [513, 607, 525, 616], "score": 0.9, "content": "\\lambda^{a}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [525, 605, 542, 620], "score": 1.0, "content": " to", "type": "text"}], "index": 23}, {"bbox": [71, 617, 542, 635], "spans": [{"bbox": [71, 621, 93, 630], "score": 0.89, "content": "\\lambda^{m a}", "type": "inline_equation", "height": 9, "width": 22}, {"bbox": [93, 617, 286, 635], "score": 1.0, "content": ", where the superscript is taken mod ", "type": "text"}, {"bbox": [286, 620, 299, 630], "score": 0.8, "content": "2r", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [299, 617, 367, 635], "score": 1.0, "content": ", and will fix ", "type": "text"}, {"bbox": [367, 619, 386, 631], "score": 0.88, "content": "J_{v}0", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [387, 617, 423, 635], "score": 1.0, "content": ". Also,", "type": "text"}, {"bbox": [424, 620, 454, 632], "score": 0.94, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [455, 617, 507, 635], "score": 1.0, "content": " will send ", "type": "text"}, {"bbox": [507, 621, 525, 631], "score": 0.92, "content": "J_{s}0", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [526, 617, 542, 635], "score": 1.0, "content": " to", "type": "text"}], "index": 24}, {"bbox": [71, 633, 541, 650], "spans": [{"bbox": [71, 635, 95, 647], "score": 0.92, "content": "J_{s}^{m}0", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [95, 633, 241, 650], "score": 1.0, "content": ", as well as stabilise the set ", "type": "text"}, {"bbox": [241, 634, 372, 647], "score": 0.91, "content": "\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}", "type": "inline_equation", "height": 13, "width": 131}, {"bbox": [373, 633, 481, 650], "score": 1.0, "content": ". (In particular, put ", "type": "text"}, {"bbox": [482, 636, 508, 644], "score": 0.9, "content": "t=r", "type": "inline_equation", "height": 8, "width": 26}, {"bbox": [508, 633, 541, 650], "score": 1.0, "content": " when", "type": "text"}], "index": 25}, {"bbox": [71, 646, 540, 663], "spans": [{"bbox": [71, 653, 77, 658], "score": 0.87, "content": "r", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [77, 648, 164, 663], "score": 1.0, "content": " is even or when ", "type": "text"}, {"bbox": [165, 650, 198, 659], "score": 0.88, "content": "m\\equiv1", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [198, 648, 324, 663], "score": 1.0, "content": " (mod 4), otherwise put ", "type": "text"}, {"bbox": [325, 649, 371, 660], "score": 0.87, "content": "t=r-1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [371, 648, 445, 663], "score": 1.0, "content": "; then for any ", "type": "text"}, {"bbox": [445, 646, 540, 662], "score": 0.84, "content": "i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}", "type": "inline_equation", "height": 16, "width": 95}], "index": 26}, {"bbox": [68, 661, 459, 681], "spans": [{"bbox": [68, 661, 83, 681], "score": 1.0, "content": "is ", "type": "text"}, {"bbox": [83, 663, 123, 677], "score": 0.94, "content": "C_{1}^{j}J_{v}^{i}\\Lambda_{t}", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [123, 661, 141, 681], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [141, 663, 192, 677], "score": 0.93, "content": "C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [193, 661, 329, 681], "score": 1.0, "content": ", when the Jacobi symbol ", "type": "text"}, {"bbox": [329, 663, 350, 678], "score": 0.88, "content": "\\textstyle\\left({\\frac{\\kappa}{m}}\\right)", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [351, 661, 365, 681], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [365, 664, 382, 676], "score": 0.82, "content": "\\pm1", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [382, 661, 459, 681], "score": 1.0, "content": ", respectively.)", "type": "text"}], "index": 27}], "index": 24}, {"type": "text", "bbox": [70, 684, 541, 716], "lines": [{"bbox": [93, 684, 541, 705], "spans": [{"bbox": [93, 684, 323, 705], "score": 1.0, "content": "Theorem 3.D. The fusion-symmetries of ", "type": "text"}, {"bbox": [324, 685, 347, 700], "score": 0.93, "content": "D_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 23}, {"bbox": [348, 684, 371, 705], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [371, 687, 405, 701], "score": 0.89, "content": "k\\neq2", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [405, 684, 513, 705], "score": 1.0, "content": " are all of the form ", "type": "text"}, {"bbox": [513, 688, 536, 701], "score": 0.88, "content": "C_{i}\\,\\pi", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [537, 684, 541, 705], "score": 1.0, "content": ",", "type": "text"}], "index": 28}, {"bbox": [72, 702, 541, 718], "spans": [{"bbox": [72, 702, 105, 718], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 704, 118, 715], "score": 0.9, "content": "C_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [118, 702, 272, 718], "score": 1.0, "content": " is a conjugation, and where ", "type": "text"}, {"bbox": [273, 708, 280, 713], "score": 0.79, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [281, 702, 541, 718], "score": 1.0, "content": " is a simple-current automorphism. Similarly for", "type": "text"}], "index": 29}], "index": 28.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [205, 99, 405, 137], "lines": [{"bbox": [205, 99, 405, 137], "spans": [{"bbox": [205, 99, 405, 137], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [195, 305, 415, 337], "lines": [{"bbox": [195, 305, 415, 337], "spans": [{"bbox": [195, 305, 415, 337], "score": 0.92, "content": "{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [164, 375, 448, 406], "lines": [{"bbox": [164, 375, 448, 406], "spans": [{"bbox": [164, 375, 448, 406], "score": 0.92, "content": "\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [126, 441, 490, 510], "lines": [{"bbox": [126, 441, 490, 510], "spans": [{"bbox": [126, 441, 490, 510], "score": 0.93, "content": "{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},", "type": "interline_equation"}], "index": 16}], "index": 16}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 505, 86], "lines": [{"bbox": [71, 73, 504, 88], "spans": [{"bbox": [71, 73, 135, 88], "score": 1.0, "content": "valid for all ", "type": "text"}, {"bbox": [135, 75, 204, 86], "score": 0.88, "content": "1\\leq i<r-2", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [204, 73, 230, 88], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [230, 75, 259, 85], "score": 0.88, "content": "k>2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [259, 73, 504, 88], "score": 1.0, "content": ". We also will use the character formula (2.1b)", "type": "text"}], "index": 0}], "index": 0, "bbox_fs": [71, 73, 504, 88]}, {"type": "interline_equation", "bbox": [205, 99, 405, 137], "lines": [{"bbox": [205, 99, 405, 137], "spans": [{"bbox": [205, 99, 405, 137], "score": 0.94, "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [70, 147, 541, 178], "lines": [{"bbox": [70, 149, 541, 166], "spans": [{"bbox": [70, 149, 106, 166], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 150, 209, 163], "score": 0.92, "content": "\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [209, 149, 396, 166], "score": 1.0, "content": " and the orthonormal components ", "type": "text"}, {"bbox": [396, 151, 418, 163], "score": 0.93, "content": "\\lambda(\\ell)", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [418, 149, 503, 166], "score": 1.0, "content": " are defined by ", "type": "text"}, {"bbox": [503, 150, 541, 164], "score": 0.91, "content": "\\lambda(\\ell)\\,=", "type": "inline_equation", "height": 14, "width": 38}], "index": 2}, {"bbox": [71, 162, 175, 180], "spans": [{"bbox": [71, 163, 171, 180], "score": 0.93, "content": "\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}", "type": "inline_equation", "height": 17, "width": 100}, {"bbox": [171, 162, 175, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2.5, "bbox_fs": [70, 149, 541, 180]}, {"type": "text", "bbox": [70, 178, 542, 264], "lines": [{"bbox": [76, 178, 542, 196], "spans": [{"bbox": [76, 178, 481, 196], "score": 1.0, "content": "The simple-current automorphisms are as follows, and depend on whether ", "type": "text"}, {"bbox": [482, 185, 487, 191], "score": 0.86, "content": "r", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [488, 178, 513, 196], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [513, 182, 520, 191], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [520, 178, 542, 196], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [69, 194, 541, 209], "spans": [{"bbox": [69, 194, 173, 209], "score": 1.0, "content": "even or odd. When", "type": "text"}, {"bbox": [174, 199, 180, 205], "score": 0.67, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [180, 194, 451, 209], "score": 1.0, "content": " is odd, the group of simple-currents is generated by ", "type": "text"}, {"bbox": [451, 196, 463, 207], "score": 0.9, "content": "J_{s}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [464, 194, 541, 209], "score": 1.0, "content": ". If in addition", "type": "text"}], "index": 5}, {"bbox": [71, 207, 543, 224], "spans": [{"bbox": [71, 210, 78, 219], "score": 0.86, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 207, 404, 224], "score": 1.0, "content": " is odd, there will be only two simple-current automorphisms: ", "type": "text"}, {"bbox": [405, 209, 521, 222], "score": 0.93, "content": "\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [521, 207, 543, 224], "score": 1.0, "content": "for", "type": "text"}], "index": 6}, {"bbox": [71, 222, 540, 239], "spans": [{"bbox": [71, 223, 122, 237], "score": 0.92, "content": "a\\in\\{0,2\\}", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [122, 222, 182, 239], "score": 1.0, "content": ". If instead ", "type": "text"}, {"bbox": [182, 224, 190, 234], "score": 0.84, "content": "k", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [190, 222, 495, 239], "score": 1.0, "content": " is even, there will be four simple-current automorphisms: ", "type": "text"}, {"bbox": [496, 224, 540, 236], "score": 0.93, "content": "\\pi=\\pi[a]", "type": "inline_equation", "height": 12, "width": 44}], "index": 7}, {"bbox": [70, 236, 541, 254], "spans": [{"bbox": [70, 236, 95, 254], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 238, 172, 251], "score": 0.93, "content": "\\pi^{\\prime}=\\pi[a k-a]", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [172, 236, 194, 254], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [195, 238, 248, 250], "score": 0.85, "content": "0\\leq a\\leq3", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [249, 236, 293, 254], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [294, 239, 324, 248], "score": 0.92, "content": "k\\equiv2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [324, 236, 493, 254], "score": 1.0, "content": " (mod 4), these form the group ", "type": "text"}, {"bbox": [494, 239, 536, 250], "score": 0.92, "content": "\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [536, 236, 541, 254], "score": 1.0, "content": ",", "type": "text"}], "index": 8}, {"bbox": [70, 250, 260, 268], "spans": [{"bbox": [70, 250, 155, 268], "score": 1.0, "content": "otherwise when ", "type": "text"}, {"bbox": [155, 253, 172, 265], "score": 0.88, "content": "4|k", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [172, 250, 241, 268], "score": 1.0, "content": " the group is ", "type": "text"}, {"bbox": [242, 254, 255, 264], "score": 0.9, "content": "\\mathbb{Z}_{4}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [256, 250, 260, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 6.5, "bbox_fs": [69, 178, 543, 268]}, {"type": "text", "bbox": [70, 264, 541, 293], "lines": [{"bbox": [95, 266, 541, 281], "spans": [{"bbox": [95, 266, 130, 281], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [130, 271, 136, 277], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [136, 266, 408, 281], "score": 1.0, "content": " is even, the simple-currents are generated by both ", "type": "text"}, {"bbox": [408, 268, 421, 279], "score": 0.92, "content": "J_{v}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [421, 266, 448, 281], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [448, 268, 460, 279], "score": 0.91, "content": "J_{s}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [461, 266, 541, 281], "score": 1.0, "content": ". If in addition", "type": "text"}], "index": 10}, {"bbox": [71, 280, 349, 297], "spans": [{"bbox": [71, 282, 78, 291], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 280, 349, 297], "score": 1.0, "content": " is even, we have 16 simple-current automorphisms:", "type": "text"}], "index": 11}], "index": 10.5, "bbox_fs": [71, 266, 541, 297]}, {"type": "interline_equation", "bbox": [195, 305, 415, 337], "lines": [{"bbox": [195, 305, 415, 337], "spans": [{"bbox": [195, 305, 415, 337], "score": 0.92, "content": "{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 346, 495, 362], "lines": [{"bbox": [70, 348, 497, 366], "spans": [{"bbox": [70, 348, 112, 366], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [112, 351, 195, 363], "score": 0.93, "content": "a,b,c,d\\in\\{0,1\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [195, 348, 365, 366], "score": 1.0, "content": ", forming a group isomorphic to ", "type": "text"}, {"bbox": [365, 350, 379, 363], "score": 0.92, "content": "\\mathbb{Z}_{2}^{4}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [379, 348, 497, 366], "score": 1.0, "content": ". This notation means", "type": "text"}], "index": 13}], "index": 13, "bbox_fs": [70, 348, 497, 366]}, {"type": "interline_equation", "bbox": [164, 375, 448, 406], "lines": [{"bbox": [164, 375, 448, 406], "spans": [{"bbox": [164, 375, 448, 406], "score": 0.92, "content": "\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 415, 407, 430], "lines": [{"bbox": [71, 417, 405, 432], "spans": [{"bbox": [71, 417, 106, 432], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [106, 420, 113, 429], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [113, 417, 405, 432], "score": 1.0, "content": " is odd, we will have six simple-current automorphisms:", "type": "text"}], "index": 15}], "index": 15, "bbox_fs": [71, 417, 405, 432]}, {"type": "interline_equation", "bbox": [126, 441, 490, 510], "lines": [{"bbox": [126, 441, 490, 510], "spans": [{"bbox": [126, 441, 490, 510], "score": 0.93, "content": "{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [69, 516, 541, 574], "lines": [{"bbox": [70, 517, 539, 536], "spans": [{"bbox": [70, 517, 106, 536], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 520, 137, 533], "score": 0.93, "content": "a={\\frac{r}{2}}", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [138, 517, 156, 536], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [156, 520, 186, 529], "score": 0.89, "content": "d=0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [187, 517, 252, 536], "score": 1.0, "content": ", and where ", "type": "text"}, {"bbox": [252, 519, 281, 529], "score": 0.88, "content": "b=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [281, 517, 299, 536], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [299, 520, 329, 529], "score": 0.89, "content": "d=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [330, 517, 523, 536], "score": 1.0, "content": ". The corresponding permutation of ", "type": "text"}, {"bbox": [523, 520, 539, 532], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}], "index": 17}, {"bbox": [70, 533, 540, 547], "spans": [{"bbox": [70, 533, 274, 547], "score": 1.0, "content": "is still given by (3.5). Again, for these ", "type": "text"}, {"bbox": [275, 533, 293, 546], "score": 0.87, "content": "r,k", "type": "inline_equation", "height": 13, "width": 18}, {"bbox": [293, 533, 419, 547], "score": 1.0, "content": ", these are the values of ", "type": "text"}, {"bbox": [419, 534, 459, 546], "score": 0.91, "content": "a,b,c,d", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [459, 533, 540, 547], "score": 1.0, "content": " for which (3.5)", "type": "text"}], "index": 18}, {"bbox": [69, 547, 542, 562], "spans": [{"bbox": [69, 547, 163, 562], "score": 1.0, "content": "is invertible. For ", "type": "text"}, {"bbox": [164, 549, 171, 558], "score": 0.87, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [171, 547, 542, 562], "score": 1.0, "content": " odd, the group of simple-current automorphisms is isomorphic to the", "type": "text"}], "index": 19}, {"bbox": [70, 562, 442, 576], "spans": [{"bbox": [70, 562, 163, 576], "score": 1.0, "content": "symmetric group ", "type": "text"}, {"bbox": [163, 564, 178, 574], "score": 0.91, "content": "\\mathfrak{S}_{3}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [179, 562, 264, 576], "score": 1.0, "content": " when 4 divides ", "type": "text"}, {"bbox": [264, 567, 270, 573], "score": 0.85, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [271, 562, 314, 576], "score": 1.0, "content": ", and to ", "type": "text"}, {"bbox": [315, 563, 328, 574], "score": 0.89, "content": "\\mathbb{Z}_{6}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [329, 562, 363, 576], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [363, 564, 392, 573], "score": 0.85, "content": "r\\equiv2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [392, 562, 442, 576], "score": 1.0, "content": " (mod 4).", "type": "text"}], "index": 20}], "index": 18.5, "bbox_fs": [69, 517, 542, 576]}, {"type": "text", "bbox": [70, 574, 542, 678], "lines": [{"bbox": [93, 575, 541, 592], "spans": [{"bbox": [93, 575, 115, 592], "score": 1.0, "content": "For", "type": "text"}, {"bbox": [116, 578, 145, 587], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [145, 575, 167, 592], "score": 1.0, "content": " (so ", "type": "text"}, {"bbox": [167, 578, 201, 587], "score": 0.86, "content": "\\kappa=2r", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [202, 575, 541, 592], "score": 1.0, "content": "), there are several Galois fusion-symmetries. In particular, write", "type": "text"}], "index": 21}, {"bbox": [71, 587, 540, 609], "spans": [{"bbox": [71, 591, 160, 602], "score": 0.9, "content": "\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [160, 587, 183, 609], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [184, 593, 259, 603], "score": 0.91, "content": "1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [259, 587, 290, 609], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [291, 591, 390, 604], "score": 0.92, "content": "\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [390, 587, 447, 609], "score": 1.0, "content": ". As with ", "type": "text"}, {"bbox": [447, 592, 469, 605], "score": 0.92, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [469, 587, 476, 609], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 590, 527, 605], "score": 0.93, "content": "\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}", "type": "inline_equation", "height": 15, "width": 50}, {"bbox": [511, 588, 540, 604], "score": 1.0, "content": "1 is", "type": "text"}], "index": 22}, {"bbox": [71, 605, 542, 620], "spans": [{"bbox": [71, 605, 169, 620], "score": 1.0, "content": "rational so for any ", "type": "text"}, {"bbox": [169, 610, 180, 615], "score": 0.86, "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [180, 605, 241, 620], "score": 1.0, "content": " coprime to ", "type": "text"}, {"bbox": [241, 607, 253, 616], "score": 0.86, "content": "2r", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [254, 605, 432, 620], "score": 1.0, "content": ", we get a Galois fusion-symmetry ", "type": "text"}, {"bbox": [432, 606, 462, 619], "score": 0.94, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [462, 605, 512, 620], "score": 1.0, "content": ". It takes ", "type": "text"}, {"bbox": [513, 607, 525, 616], "score": 0.9, "content": "\\lambda^{a}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [525, 605, 542, 620], "score": 1.0, "content": " to", "type": "text"}], "index": 23}, {"bbox": [71, 617, 542, 635], "spans": [{"bbox": [71, 621, 93, 630], "score": 0.89, "content": "\\lambda^{m a}", "type": "inline_equation", "height": 9, "width": 22}, {"bbox": [93, 617, 286, 635], "score": 1.0, "content": ", where the superscript is taken mod ", "type": "text"}, {"bbox": [286, 620, 299, 630], "score": 0.8, "content": "2r", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [299, 617, 367, 635], "score": 1.0, "content": ", and will fix ", "type": "text"}, {"bbox": [367, 619, 386, 631], "score": 0.88, "content": "J_{v}0", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [387, 617, 423, 635], "score": 1.0, "content": ". Also,", "type": "text"}, {"bbox": [424, 620, 454, 632], "score": 0.94, "content": "\\pi\\{m\\}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [455, 617, 507, 635], "score": 1.0, "content": " will send ", "type": "text"}, {"bbox": [507, 621, 525, 631], "score": 0.92, "content": "J_{s}0", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [526, 617, 542, 635], "score": 1.0, "content": " to", "type": "text"}], "index": 24}, {"bbox": [71, 633, 541, 650], "spans": [{"bbox": [71, 635, 95, 647], "score": 0.92, "content": "J_{s}^{m}0", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [95, 633, 241, 650], "score": 1.0, "content": ", as well as stabilise the set ", "type": "text"}, {"bbox": [241, 634, 372, 647], "score": 0.91, "content": "\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}", "type": "inline_equation", "height": 13, "width": 131}, {"bbox": [373, 633, 481, 650], "score": 1.0, "content": ". (In particular, put ", "type": "text"}, {"bbox": [482, 636, 508, 644], "score": 0.9, "content": "t=r", "type": "inline_equation", "height": 8, "width": 26}, {"bbox": [508, 633, 541, 650], "score": 1.0, "content": " when", "type": "text"}], "index": 25}, {"bbox": [71, 646, 540, 663], "spans": [{"bbox": [71, 653, 77, 658], "score": 0.87, "content": "r", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [77, 648, 164, 663], "score": 1.0, "content": " is even or when ", "type": "text"}, {"bbox": [165, 650, 198, 659], "score": 0.88, "content": "m\\equiv1", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [198, 648, 324, 663], "score": 1.0, "content": " (mod 4), otherwise put ", "type": "text"}, {"bbox": [325, 649, 371, 660], "score": 0.87, "content": "t=r-1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [371, 648, 445, 663], "score": 1.0, "content": "; then for any ", "type": "text"}, {"bbox": [445, 646, 540, 662], "score": 0.84, "content": "i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}", "type": "inline_equation", "height": 16, "width": 95}], "index": 26}, {"bbox": [68, 661, 459, 681], "spans": [{"bbox": [68, 661, 83, 681], "score": 1.0, "content": "is ", "type": "text"}, {"bbox": [83, 663, 123, 677], "score": 0.94, "content": "C_{1}^{j}J_{v}^{i}\\Lambda_{t}", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [123, 661, 141, 681], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [141, 663, 192, 677], "score": 0.93, "content": "C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [193, 661, 329, 681], "score": 1.0, "content": ", when the Jacobi symbol ", "type": "text"}, {"bbox": [329, 663, 350, 678], "score": 0.88, "content": "\\textstyle\\left({\\frac{\\kappa}{m}}\\right)", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [351, 661, 365, 681], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [365, 664, 382, 676], "score": 0.82, "content": "\\pm1", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [382, 661, 459, 681], "score": 1.0, "content": ", respectively.)", "type": "text"}], "index": 27}], "index": 24, "bbox_fs": [68, 575, 542, 681]}, {"type": "text", "bbox": [70, 684, 541, 716], "lines": [{"bbox": [93, 684, 541, 705], "spans": [{"bbox": [93, 684, 323, 705], "score": 1.0, "content": "Theorem 3.D. The fusion-symmetries of ", "type": "text"}, {"bbox": [324, 685, 347, 700], "score": 0.93, "content": "D_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 23}, {"bbox": [348, 684, 371, 705], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [371, 687, 405, 701], "score": 0.89, "content": "k\\neq2", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [405, 684, 513, 705], "score": 1.0, "content": " are all of the form ", "type": "text"}, {"bbox": [513, 688, 536, 701], "score": 0.88, "content": "C_{i}\\,\\pi", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [537, 684, 541, 705], "score": 1.0, "content": ",", "type": "text"}], "index": 28}, {"bbox": [72, 702, 541, 718], "spans": [{"bbox": [72, 702, 105, 718], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 704, 118, 715], "score": 0.9, "content": "C_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [118, 702, 272, 718], "score": 1.0, "content": " is a conjugation, and where ", "type": "text"}, {"bbox": [273, 708, 280, 713], "score": 0.79, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [281, 702, 541, 718], "score": 1.0, "content": " is a simple-current automorphism. Similarly for", "type": "text"}], "index": 29}, {"bbox": [71, 67, 544, 92], "spans": [{"bbox": [71, 70, 95, 88], "score": 0.9, "content": "{D}_{4}^{(1)}", "type": "inline_equation", "height": 18, "width": 24, "cross_page": true}, {"bbox": [95, 67, 112, 92], "score": 1.0, "content": "at ", "type": "text", "cross_page": true}, {"bbox": [112, 74, 142, 86], "score": 0.86, "content": "k=2", "type": "inline_equation", "height": 12, "width": 30, "cross_page": true}, {"bbox": [142, 67, 249, 92], "score": 1.0, "content": ". Finally, when both", "type": "text", "cross_page": true}, {"bbox": [250, 74, 280, 86], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 12, "width": 30, "cross_page": true}, {"bbox": [280, 67, 306, 92], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [306, 74, 335, 86], "score": 0.87, "content": "r>4", "type": "inline_equation", "height": 12, "width": 29, "cross_page": true}, {"bbox": [335, 67, 454, 92], "score": 1.0, "content": ", any fusion-symmetry ", "type": "text", "cross_page": true}, {"bbox": [454, 77, 463, 85], "score": 0.63, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9, "cross_page": true}, {"bbox": [463, 67, 544, 92], "score": 1.0, "content": " can be written", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [70, 86, 418, 106], "spans": [{"bbox": [70, 86, 86, 106], "score": 1.0, "content": "as ", "type": "text", "cross_page": true}, {"bbox": [86, 87, 171, 102], "score": 0.89, "content": "\\pi=C_{1}^{a}\\,\\pi_{v}^{b}\\,\\pi\\{m\\}", "type": "inline_equation", "height": 15, "width": 85, "cross_page": true}, {"bbox": [172, 86, 194, 106], "score": 1.0, "content": " for ", "type": "text", "cross_page": true}, {"bbox": [194, 88, 255, 102], "score": 0.91, "content": "a,b\\in\\{0,1\\}", "type": "inline_equation", "height": 14, "width": 61, "cross_page": true}, {"bbox": [256, 86, 305, 106], "score": 1.0, "content": " and any ", "type": "text", "cross_page": true}, {"bbox": [305, 87, 349, 102], "score": 0.9, "content": "m\\in\\mathbb{Z}_{2r}^{\\times}", "type": "inline_equation", "height": 15, "width": 44, "cross_page": true}, {"bbox": [349, 86, 356, 106], "score": 1.0, "content": ", ", "type": "text", "cross_page": true}, {"bbox": [356, 88, 411, 102], "score": 0.9, "content": "1\\leq m<r", "type": "inline_equation", "height": 14, "width": 55, "cross_page": true}, {"bbox": [412, 86, 418, 106], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 1}], "index": 28.5, "bbox_fs": [72, 684, 541, 718]}]}
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[{"type": "text", "bbox": [70, 70, 505, 86], "content": "valid for all and . We also will use the character formula (2.1b)", "index": 0}, {"type": "interline_equation", "bbox": [205, 99, 405, 137], "content": "", "index": 1}, {"type": "text", "bbox": [70, 147, 541, 178], "content": "where and the orthonormal components are defined by .", "index": 2}, {"type": "text", "bbox": [70, 178, 542, 264], "content": "The simple-current automorphisms are as follows, and depend on whether and are even or odd. When is odd, the group of simple-currents is generated by . If in addition is odd, there will be only two simple-current automorphisms: for . If instead is even, there will be four simple-current automorphisms: and for . When (mod 4), these form the group , otherwise when the group is .", "index": 3}, {"type": "text", "bbox": [70, 264, 541, 293], "content": "When is even, the simple-currents are generated by both and . If in addition is even, we have 16 simple-current automorphisms:", "index": 4}, {"type": "interline_equation", "bbox": [195, 305, 415, 337], "content": "", "index": 5}, {"type": "text", "bbox": [69, 346, 495, 362], "content": "for any , forming a group isomorphic to . This notation means", "index": 6}, {"type": "interline_equation", "bbox": [164, 375, 448, 406], "content": "", "index": 7}, {"type": "text", "bbox": [70, 415, 407, 430], "content": "When is odd, we will have six simple-current automorphisms:", "index": 8}, {"type": "interline_equation", "bbox": [126, 441, 490, 510], "content": "", "index": 9}, {"type": "text", "bbox": [69, 516, 541, 574], "content": "where or , and where or . The corresponding permutation of is still given by (3.5). Again, for these , these are the values of for which (3.5) is invertible. For odd, the group of simple-current automorphisms is isomorphic to the symmetric group when 4 divides , and to when (mod 4).", "index": 10}, {"type": "text", "bbox": [70, 574, 542, 678], "content": "For (so ), there are several Galois fusion-symmetries. In particular, write for , and . As with , 1 is rational so for any coprime to , we get a Galois fusion-symmetry . It takes to , where the superscript is taken mod , and will fix . Also, will send to , as well as stabilise the set . (In particular, put when is even or when (mod 4), otherwise put ; then for any is or , when the Jacobi symbol is , respectively.)", "index": 11}, {"type": "text", "bbox": [70, 684, 541, 716], "content": "Theorem 3.D. The fusion-symmetries of for are all of the form , where is a conjugation, and where is a simple-current automorphism. Similarly for at . Finally, when both and , any fusion-symmetry can be written as for and any , .", "index": 12}]
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[{"bbox": [71, 73, 504, 88], "content": "valid for all and . We also will use the character formula (2.1b)", "parent_index": 0, "line_index": 0}, {"bbox": [70, 149, 541, 166], "content": "where and the orthonormal components are defined by", "parent_index": 2, "line_index": 0}, {"bbox": [71, 162, 175, 180], "content": ".", "parent_index": 2, "line_index": 1}, {"bbox": [76, 178, 542, 196], "content": "The simple-current automorphisms are as follows, and depend on whether and are", "parent_index": 3, "line_index": 0}, {"bbox": [69, 194, 541, 209], "content": "even or odd. When is odd, the group of simple-currents is generated by . If in addition", "parent_index": 3, "line_index": 1}, {"bbox": [71, 207, 543, 224], "content": "is odd, there will be only two simple-current automorphisms: for", "parent_index": 3, "line_index": 2}, {"bbox": [71, 222, 540, 239], "content": ". If instead is even, there will be four simple-current automorphisms:", "parent_index": 3, "line_index": 3}, {"bbox": [70, 236, 541, 254], "content": "and for . When (mod 4), these form the group ,", "parent_index": 3, "line_index": 4}, {"bbox": [70, 250, 260, 268], "content": "otherwise when the group is .", "parent_index": 3, "line_index": 5}, {"bbox": [95, 266, 541, 281], "content": "When is even, the simple-currents are generated by both and . If in addition", "parent_index": 4, "line_index": 0}, {"bbox": [71, 280, 349, 297], "content": "is even, we have 16 simple-current automorphisms:", "parent_index": 4, "line_index": 1}, {"bbox": [70, 348, 497, 366], "content": "for any , forming a group isomorphic to . This notation means", "parent_index": 6, "line_index": 0}, {"bbox": [71, 417, 405, 432], "content": "When is odd, we will have six simple-current automorphisms:", "parent_index": 8, "line_index": 0}, {"bbox": [70, 517, 539, 536], "content": "where or , and where or . The corresponding permutation of", "parent_index": 10, "line_index": 0}, {"bbox": [70, 533, 540, 547], "content": "is still given by (3.5). Again, for these , these are the values of for which (3.5)", "parent_index": 10, "line_index": 1}, {"bbox": [69, 547, 542, 562], "content": "is invertible. For odd, the group of simple-current automorphisms is isomorphic to the", "parent_index": 10, "line_index": 2}, {"bbox": [70, 562, 442, 576], "content": "symmetric group when 4 divides , and to when (mod 4).", "parent_index": 10, "line_index": 3}, {"bbox": [93, 575, 541, 592], "content": "For (so ), there are several Galois fusion-symmetries. In particular, write", "parent_index": 11, "line_index": 0}, {"bbox": [71, 587, 540, 609], "content": "for , and . As with , 1 is", "parent_index": 11, "line_index": 1}, {"bbox": [71, 605, 542, 620], "content": "rational so for any coprime to , we get a Galois fusion-symmetry . It takes to", "parent_index": 11, "line_index": 2}, {"bbox": [71, 617, 542, 635], "content": ", where the superscript is taken mod , and will fix . Also, will send to", "parent_index": 11, "line_index": 3}, {"bbox": [71, 633, 541, 650], "content": ", as well as stabilise the set . (In particular, put when", "parent_index": 11, "line_index": 4}, {"bbox": [71, 646, 540, 663], "content": "is even or when (mod 4), otherwise put ; then for any", "parent_index": 11, "line_index": 5}, {"bbox": [68, 661, 459, 681], "content": "is or , when the Jacobi symbol is , respectively.)", "parent_index": 11, "line_index": 6}, {"bbox": [93, 684, 541, 705], "content": "Theorem 3.D. The fusion-symmetries of for are all of the form ,", "parent_index": 12, "line_index": 0}, {"bbox": [72, 702, 541, 718], "content": "where is a conjugation, and where is a simple-current automorphism. Similarly for", "parent_index": 12, "line_index": 1}, {"bbox": [71, 67, 544, 92], "content": "at . Finally, when both and , any fusion-symmetry can be written", "parent_index": 12, "line_index": 2}, {"bbox": [70, 86, 418, 106], "content": "as for and any , .", "parent_index": 12, "line_index": 3}]
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[]
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[{"bbox": [135, 75, 204, 86], "content": "1\\leq i<r-2", "parent_index": 0, "subtype": "inline"}, {"bbox": [230, 75, 259, 85], "content": "k>2", "parent_index": 0, "subtype": "inline"}, {"bbox": [205, 99, 405, 137], "content": "\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,", "parent_index": 1, "subtype": "interline"}, {"bbox": [106, 150, 209, 163], "content": "\\lambda^{+}(\\ell)\\,=\\,(\\lambda+\\rho)(\\ell)", "parent_index": 2, "subtype": "inline"}, {"bbox": [396, 151, 418, 163], "content": "\\lambda(\\ell)", "parent_index": 2, "subtype": "inline"}, {"bbox": [503, 150, 541, 164], "content": "\\lambda(\\ell)\\,=", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 163, 171, 180], "content": "\\begin{array}{r}{\\sum_{i=\\ell}^{r-1}\\lambda_{i}+\\frac{\\lambda_{r}-\\lambda_{r-1}}{2}}\\end{array}", "parent_index": 2, "subtype": "inline"}, {"bbox": [482, 185, 487, 191], "content": "r", "parent_index": 3, "subtype": "inline"}, {"bbox": [513, 182, 520, 191], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [174, 199, 180, 205], "content": "r", "parent_index": 3, "subtype": "inline"}, {"bbox": [451, 196, 463, 207], "content": "J_{s}", "parent_index": 3, "subtype": "inline"}, {"bbox": [71, 210, 78, 219], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [405, 209, 521, 222], "content": "\\pi=\\pi^{\\prime}=\\pi[a]=J_{s}^{4a\\cup_{s}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [71, 223, 122, 237], "content": "a\\in\\{0,2\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [182, 224, 190, 234], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [496, 224, 540, 236], "content": "\\pi=\\pi[a]", "parent_index": 3, "subtype": "inline"}, {"bbox": [95, 238, 172, 251], "content": "\\pi^{\\prime}=\\pi[a k-a]", "parent_index": 3, "subtype": "inline"}, {"bbox": [195, 238, 248, 250], "content": "0\\leq a\\leq3", "parent_index": 3, "subtype": "inline"}, {"bbox": [294, 239, 324, 248], "content": "k\\equiv2", "parent_index": 3, "subtype": "inline"}, {"bbox": [494, 239, 536, 250], "content": "\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [155, 253, 172, 265], "content": "4|k", "parent_index": 3, "subtype": "inline"}, {"bbox": [242, 254, 255, 264], "content": "\\mathbb{Z}_{4}", "parent_index": 3, "subtype": "inline"}, {"bbox": [130, 271, 136, 277], "content": "r", "parent_index": 4, "subtype": "inline"}, {"bbox": [408, 268, 421, 279], "content": "J_{v}", "parent_index": 4, "subtype": "inline"}, {"bbox": [448, 268, 460, 279], "content": "J_{s}", "parent_index": 4, "subtype": "inline"}, {"bbox": [71, 282, 78, 291], "content": "k", "parent_index": 4, "subtype": "inline"}, {"bbox": [195, 305, 415, 337], "content": "{\\boldsymbol{\\pi}}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad b\\,\\right]\\qquad{\\mathrm{and}}\\qquad{\\boldsymbol{\\pi}}^{\\prime}={\\boldsymbol{\\pi}}\\left[{\\boldsymbol{a}}\\quad c\\right]", "parent_index": 5, "subtype": "interline"}, {"bbox": [112, 351, 195, 363], "content": "a,b,c,d\\in\\{0,1\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [365, 350, 379, 363], "content": "\\mathbb{Z}_{2}^{4}", "parent_index": 6, "subtype": "inline"}, {"bbox": [164, 375, 448, 406], "content": "\\pi\\left[\\!\\!\\begin{array}{r c}{{a}}&{{b}}\\\\ {{c}}&{{d}}\\end{array}\\!\\!\\right](\\lambda)\\ensuremath{\\stackrel{\\mathrm{def}}{=}}\\ J_{v}^{2a\\,Q_{v}(\\lambda)+2b\\,Q_{s}(\\lambda)}J_{s}^{2c\\,Q_{v}(\\lambda)+2d\\,Q_{s}(\\lambda)}\\lambda\\ .", "parent_index": 7, "subtype": "interline"}, {"bbox": [106, 420, 113, 429], "content": "k", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 441, 490, 510], "content": "{\\begin{array}{r l}&{\\pi\\,=\\pi\\left[{\\begin{array}{c c}{a}&{0}\\\\ {0}&{d}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{a\\left(d+1\\right)}&{{\\frac{d r}{2}}}\\\\ {{\\frac{d r}{2}}}&{d}\\end{array}}\\right]}\\\\ {{\\mathrm{rr}}\\quad\\pi\\,=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1}&{b}\\\\ {c}&{1}\\end{array}}\\right]\\qquad{\\mathrm{with}}\\qquad\\pi^{\\prime}=\\pi\\left[{\\begin{array}{c c}{{\\frac{r}{2}}+1+b c{\\frac{r}{2}}}&{b+{\\frac{r}{2}}}\\\\ {{\\frac{r}{2}}+1+b c+b}&{1}\\end{array}}\\right]}\\end{array}},", "parent_index": 9, "subtype": "interline"}, {"bbox": [106, 520, 137, 533], "content": "a={\\frac{r}{2}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [156, 520, 186, 529], "content": "d=0", "parent_index": 10, "subtype": "inline"}, {"bbox": [252, 519, 281, 529], "content": "b=1", "parent_index": 10, "subtype": "inline"}, {"bbox": [299, 520, 329, 529], "content": "d=1", "parent_index": 10, "subtype": "inline"}, {"bbox": [523, 520, 539, 532], "content": "P_{+}", "parent_index": 10, "subtype": "inline"}, {"bbox": [275, 533, 293, 546], "content": "r,k", "parent_index": 10, "subtype": "inline"}, {"bbox": [419, 534, 459, 546], "content": "a,b,c,d", "parent_index": 10, "subtype": "inline"}, {"bbox": [164, 549, 171, 558], "content": "k", "parent_index": 10, "subtype": "inline"}, {"bbox": [163, 564, 178, 574], "content": "\\mathfrak{S}_{3}", "parent_index": 10, "subtype": "inline"}, {"bbox": [264, 567, 270, 573], "content": "r", "parent_index": 10, "subtype": "inline"}, {"bbox": [315, 563, 328, 574], "content": "\\mathbb{Z}_{6}", "parent_index": 10, "subtype": "inline"}, {"bbox": [363, 564, 392, 573], "content": "r\\equiv2", "parent_index": 10, "subtype": "inline"}, {"bbox": [116, 578, 145, 587], "content": "k=2", "parent_index": 11, "subtype": "inline"}, {"bbox": [167, 578, 201, 587], "content": "\\kappa=2r", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 591, 160, 602], "content": "\\lambda^{i}\\,=\\,\\lambda^{2r-i}\\,=\\,\\Lambda_{i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [184, 593, 259, 603], "content": "1\\,\\leq\\,i\\,\\leq\\,r\\,-\\,2", "parent_index": 11, "subtype": "inline"}, {"bbox": [291, 591, 390, 604], "content": "\\lambda^{r\\pm1}=\\Lambda_{r-1}+\\Lambda_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [447, 592, 469, 605], "content": "B_{r,2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [477, 590, 527, 605], "content": "\\begin{array}{r}{S_{00}^{2}\\,=\\,\\frac{1}{4\\kappa}}\\end{array}", "parent_index": 11, "subtype": "inline"}, {"bbox": [169, 610, 180, 615], "content": "{\\boldsymbol{\\mathit{\\varepsilon}}}^{\\prime}{\\boldsymbol{\\mathit{m}}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [241, 607, 253, 616], "content": "2r", "parent_index": 11, "subtype": "inline"}, {"bbox": [432, 606, 462, 619], "content": "\\pi\\{m\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [513, 607, 525, 616], "content": "\\lambda^{a}", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 621, 93, 630], "content": "\\lambda^{m a}", "parent_index": 11, "subtype": "inline"}, {"bbox": [286, 620, 299, 630], "content": "2r", "parent_index": 11, "subtype": "inline"}, {"bbox": [367, 619, 386, 631], "content": "J_{v}0", "parent_index": 11, "subtype": "inline"}, {"bbox": [424, 620, 454, 632], "content": "\\pi\\{m\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [507, 621, 525, 631], "content": "J_{s}0", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 635, 95, 647], "content": "J_{s}^{m}0", "parent_index": 11, "subtype": "inline"}, {"bbox": [241, 634, 372, 647], "content": "\\left\\{\\Lambda_{r},\\Lambda_{r-1},J_{v}\\Lambda_{r},J_{v}\\Lambda_{r-1}\\right\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [482, 636, 508, 644], "content": "t=r", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 653, 77, 658], "content": "r", "parent_index": 11, "subtype": "inline"}, {"bbox": [165, 650, 198, 659], "content": "m\\equiv1", "parent_index": 11, "subtype": "inline"}, {"bbox": [325, 649, 371, 660], "content": "t=r-1", "parent_index": 11, "subtype": "inline"}, {"bbox": [445, 646, 540, 662], "content": "i,j,\\,\\pi\\{m\\}\\,C_{1}^{j}J_{v}^{i}\\Lambda_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [83, 663, 123, 677], "content": "C_{1}^{j}J_{v}^{i}\\Lambda_{t}", "parent_index": 11, "subtype": "inline"}, {"bbox": [141, 663, 192, 677], "content": "C_{1}^{j}J_{v}^{i+1}\\Lambda_{t}", "parent_index": 11, "subtype": "inline"}, {"bbox": [329, 663, 350, 678], "content": "\\textstyle\\left({\\frac{\\kappa}{m}}\\right)", "parent_index": 11, "subtype": "inline"}, {"bbox": [365, 664, 382, 676], "content": "\\pm1", "parent_index": 11, "subtype": "inline"}, {"bbox": [324, 685, 347, 700], "content": "D_{r}^{(1)}", "parent_index": 12, "subtype": "inline"}, {"bbox": [371, 687, 405, 701], "content": "k\\neq2", "parent_index": 12, "subtype": "inline"}, {"bbox": [513, 688, 536, 701], "content": "C_{i}\\,\\pi", "parent_index": 12, "subtype": "inline"}, {"bbox": [105, 704, 118, 715], "content": "C_{i}", "parent_index": 12, "subtype": "inline"}, {"bbox": [273, 708, 280, 713], "content": "\\pi", "parent_index": 12, "subtype": "inline"}, {"bbox": [71, 70, 95, 88], "content": "{D}_{4}^{(1)}", "parent_index": 12, "subtype": "inline"}, {"bbox": [112, 74, 142, 86], "content": "k=2", "parent_index": 12, "subtype": "inline"}, {"bbox": [250, 74, 280, 86], "content": "k=2", "parent_index": 12, "subtype": "inline"}, {"bbox": [306, 74, 335, 86], "content": "r>4", "parent_index": 12, "subtype": "inline"}, {"bbox": [454, 77, 463, 85], "content": "\\pi", "parent_index": 12, "subtype": "inline"}, {"bbox": [86, 87, 171, 102], "content": "\\pi=C_{1}^{a}\\,\\pi_{v}^{b}\\,\\pi\\{m\\}", "parent_index": 12, "subtype": "inline"}, {"bbox": [194, 88, 255, 102], "content": "a,b\\in\\{0,1\\}", "parent_index": 12, "subtype": "inline"}, {"bbox": [305, 87, 349, 102], "content": "m\\in\\mathbb{Z}_{2r}^{\\times}", "parent_index": 12, "subtype": "inline"}, {"bbox": [356, 88, 411, 102], "content": "1\\leq m<r", "parent_index": 12, "subtype": "inline"}]
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$\pi_{v}$ here refers to the simple-current automorphism $\pi[2]$ or 10 00 ], for r odd/even. When $k\,=\,1$ , $A(D_{e v e n,1})\cong{\mathfrak{S}}_{3}$ , corresponding to any permutation of $\Lambda_{1},\Lambda_{r-1},\Lambda_{r}$ , and $A(D_{o d d,1})\:=\:\langle{C_{1}}\rangle\:\cong\:\mathbb{Z}_{2}$ . When $r\,>\,4$ , ${\cal A}(D_{r,2})\,\cong\,(\mathbb{Z}_{2r}^{\times}/\{\pm1\})\,\times\,\mathbb{Z}_{2}\,\times\,\mathbb{Z}_{2}$ or $\mathbb{Z}_{r}^{\times}\times\mathbb{Z}_{2}$ for $r$ even/odd. $A(D_{4,2})$ has 24 elements, and any element can be written uniquely as $C_{i}\,\pi\,\left[\begin{array}{l l}{a}&{0}\\ {0}&{d}\end{array}\right]$
# 3.5. The algebra $E_{6}^{(1)}$
A weight $\lambda$ of $P_{+}$ satisfies $k=\lambda_{0}\!+\!\lambda_{1}\!+\!2\lambda_{2}\!+\!3\lambda_{3}\!+\!2\lambda_{4}\!+\!\lambda_{5}\!+\!2\lambda_{6}$ and $\kappa=k\!+\!12$ . The charge-conjugation acts as $C\lambda=(\lambda_{0},\lambda_{5},\lambda_{4},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{6})$ . The order 3 simple-current $J$ is given by $J\lambda=(\lambda_{5},\lambda_{0},\lambda_{6},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{4})$ with $Q(\lambda)=(-\lambda_{1}+\lambda_{2}-\lambda_{4}+\lambda_{5})/3$ .
The fusion products we need can be derived from [29] using (2.4):
$$
(\Lambda_{1}+\Lambda_{2})_{3}\sqcup(\Lambda_{1}+\Lambda_{5})_{2}
$$
where the outer subscript on any summand denotes the smallest level where that summand appears (it will also appear at all larger levels). So for example $\Lambda_{1}\boxtimes\Lambda_{1}$ equals $\Lambda_{2}$ + $\Lambda_{5}$ + $(2\Lambda_{1})$ for any $k\geq2$ , but equals $\Lambda_{5}$ at $k\,=\,1$ . A similar convention is used in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will equal the numerical value of the fusion coefficient).
Theorem 3.E6. The fusion-symmetries of $E_{6}^{(1)}$ are $C^{i}\,\pi[a]$ , for any $i\in\{0,1\}$ and any $a\in\{0,1,2\}$ for which $a k\not\equiv1$ (mod 3).
# 3.6. The algebra E7(1)
A weight $\lambda$ in $P_{+}$ satisfies $k\,=\,\lambda_{0}+2\lambda_{1}+3\lambda_{2}+4\lambda_{3}+3\lambda_{4}+2\lambda_{5}+\lambda_{6}+2\lambda_{7}$ , and $\kappa\,=\,k\,+\,18$ . The charge-conjugation is trivial, but there is a simple-current $J$ given by $J\lambda=(\lambda_{6},\lambda_{5},...\,,\lambda_{1},\lambda_{0},\lambda_{7})$ . It has $Q(\lambda)=(\lambda_{4}+\lambda_{6}+\lambda_{7})/2$ .
The only fusion products we need can be obtained from [29] and (2.4):
$\Lambda_{6}$ × $\Lambda_{6}=(0)_{1}$ + $(\Lambda_{1})_{2}$ + $(\Lambda_{5})_{2}$ + $(2\Lambda_{6})_{2}$ $\Lambda_{1}$ × $\Lambda_{6}=(\Lambda_{6})_{2}$ + $(\Lambda_{7})_{2}$ + $(\Lambda_{1}+\Lambda_{6})_{3}$ $\Lambda_{5}$ × $\Lambda_{6}=(\Lambda_{4})_{3}$ + $(\Lambda_{6})_{2}$ + $(\Lambda_{7})_{2}$ + $(\Lambda_{1}+\Lambda_{6})_{3}$ + $(\Lambda_{5}+\Lambda_{6})_{3}$ $\Lambda_{6}$ × $(2\Lambda_{6})=(\Lambda_{6})_{2}$ + $(\Lambda_{1}+\Lambda_{6})_{3}$ + $(3\Lambda_{6})_{3}$ + $(\Lambda_{5}+\Lambda_{6})_{3}$ $\Lambda_{4}$ × $\Lambda_{6}=(\Lambda_{2})_{3}$ + $(\Lambda_{3})_{4}$ + $(\Lambda_{5})_{3}$ + $(\Lambda_{1}+\Lambda_{5})_{4}$ + $(\Lambda_{4}+\Lambda_{6})_{4}$ + $(\Lambda_{6}+\Lambda_{7})_{3}$ $\Lambda_{6}$ × $\Lambda_{7}=(\Lambda_{1})_{2}$ + $(\Lambda_{2})_{3}$ + $(\Lambda_{5})_{2}$ + $(\Lambda_{6}+\Lambda_{7})_{3}$ $\Lambda_{6}$ × $(\Lambda_{5}+\Lambda_{6})=(\Lambda_{5})_{3}$ + $(2\Lambda_{5})_{4}$ + $(2\Lambda_{6})_{3}$ + $(\Lambda_{6}+\Lambda_{7})_{3}$ + $(\Lambda_{1}+\Lambda_{5})_{4}$ + $(\Lambda_{4}+\Lambda_{6})_{4}$ + $(\Lambda_{1}+2\Lambda_{6})_{4}$ + $(\Lambda_{5}+2\Lambda_{6})_{4}$
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<p data-bbox="70 106 541 204">$\pi_{v}$ here refers to the simple-current automorphism $\pi[2]$ or 10 00 ], for r odd/even. When $k\,=\,1$ , $A(D_{e v e n,1})\cong{\mathfrak{S}}_{3}$ , corresponding to any permutation of $\Lambda_{1},\Lambda_{r-1},\Lambda_{r}$ , and $A(D_{o d d,1})\:=\:\langle{C_{1}}\rangle\:\cong\:\mathbb{Z}_{2}$ . When $r\,>\,4$ , ${\cal A}(D_{r,2})\,\cong\,(\mathbb{Z}_{2r}^{\times}/\{\pm1\})\,\times\,\mathbb{Z}_{2}\,\times\,\mathbb{Z}_{2}$ or $\mathbb{Z}_{r}^{\times}\times\mathbb{Z}_{2}$ for $r$ even/odd. $A(D_{4,2})$ has 24 elements, and any element can be written uniquely as $C_{i}\,\pi\,\left[\begin{array}{l l}{a}&{0}\\ {0}&{d}\end{array}\right]$ </p>
<h1 data-bbox="70 216 183 232">3.5. The algebra $E_{6}^{(1)}$ </h1>
<p data-bbox="70 237 541 280">A weight $\lambda$ of $P_{+}$ satisfies $k=\lambda_{0}\!+\!\lambda_{1}\!+\!2\lambda_{2}\!+\!3\lambda_{3}\!+\!2\lambda_{4}\!+\!\lambda_{5}\!+\!2\lambda_{6}$ and $\kappa=k\!+\!12$ . The charge-conjugation acts as $C\lambda=(\lambda_{0},\lambda_{5},\lambda_{4},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{6})$ . The order 3 simple-current $J$ is given by $J\lambda=(\lambda_{5},\lambda_{0},\lambda_{6},\lambda_{3},\lambda_{2},\lambda_{1},\lambda_{4})$ with $Q(\lambda)=(-\lambda_{1}+\lambda_{2}-\lambda_{4}+\lambda_{5})/3$ . </p>
<p data-bbox="96 281 443 295">The fusion products we need can be derived from [29] using (2.4): </p>
<div class="equation" data-bbox="282 354 416 370">$$
(\Lambda_{1}+\Lambda_{2})_{3}\sqcup(\Lambda_{1}+\Lambda_{5})_{2}
$$</div>
<p data-bbox="70 372 541 445">where the outer subscript on any summand denotes the smallest level where that summand appears (it will also appear at all larger levels). So for example $\Lambda_{1}\boxtimes\Lambda_{1}$ equals $\Lambda_{2}$ + $\Lambda_{5}$ + $(2\Lambda_{1})$ for any $k\geq2$ , but equals $\Lambda_{5}$ at $k\,=\,1$ . A similar convention is used in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will equal the numerical value of the fusion coefficient). </p>
<p data-bbox="71 450 542 482">Theorem 3.E6. The fusion-symmetries of $E_{6}^{(1)}$ are $C^{i}\,\pi[a]$ , for any $i\in\{0,1\}$ and any $a\in\{0,1,2\}$ for which $a k\not\equiv1$ (mod 3). </p>
<h1 data-bbox="71 491 183 507">3.6. The algebra E7(1) </h1>
<p data-bbox="70 512 542 556">A weight $\lambda$ in $P_{+}$ satisfies $k\,=\,\lambda_{0}+2\lambda_{1}+3\lambda_{2}+4\lambda_{3}+3\lambda_{4}+2\lambda_{5}+\lambda_{6}+2\lambda_{7}$ , and $\kappa\,=\,k\,+\,18$ . The charge-conjugation is trivial, but there is a simple-current $J$ given by $J\lambda=(\lambda_{6},\lambda_{5},...\,,\lambda_{1},\lambda_{0},\lambda_{7})$ . It has $Q(\lambda)=(\lambda_{4}+\lambda_{6}+\lambda_{7})/2$ . </p>
<p data-bbox="94 556 468 570">The only fusion products we need can be obtained from [29] and (2.4): </p>
<p data-bbox="82 573 530 716">$\Lambda_{6}$ × $\Lambda_{6}=(0)_{1}$ + $(\Lambda_{1})_{2}$ + $(\Lambda_{5})_{2}$ + $(2\Lambda_{6})_{2}$ $\Lambda_{1}$ × $\Lambda_{6}=(\Lambda_{6})_{2}$ + $(\Lambda_{7})_{2}$ + $(\Lambda_{1}+\Lambda_{6})_{3}$ $\Lambda_{5}$ × $\Lambda_{6}=(\Lambda_{4})_{3}$ + $(\Lambda_{6})_{2}$ + $(\Lambda_{7})_{2}$ + $(\Lambda_{1}+\Lambda_{6})_{3}$ + $(\Lambda_{5}+\Lambda_{6})_{3}$ $\Lambda_{6}$ × $(2\Lambda_{6})=(\Lambda_{6})_{2}$ + $(\Lambda_{1}+\Lambda_{6})_{3}$ + $(3\Lambda_{6})_{3}$ + $(\Lambda_{5}+\Lambda_{6})_{3}$ $\Lambda_{4}$ × $\Lambda_{6}=(\Lambda_{2})_{3}$ + $(\Lambda_{3})_{4}$ + $(\Lambda_{5})_{3}$ + $(\Lambda_{1}+\Lambda_{5})_{4}$ + $(\Lambda_{4}+\Lambda_{6})_{4}$ + $(\Lambda_{6}+\Lambda_{7})_{3}$ $\Lambda_{6}$ × $\Lambda_{7}=(\Lambda_{1})_{2}$ + $(\Lambda_{2})_{3}$ + $(\Lambda_{5})_{2}$ + $(\Lambda_{6}+\Lambda_{7})_{3}$ $\Lambda_{6}$ × $(\Lambda_{5}+\Lambda_{6})=(\Lambda_{5})_{3}$ + $(2\Lambda_{5})_{4}$ + $(2\Lambda_{6})_{3}$ + $(\Lambda_{6}+\Lambda_{7})_{3}$ + $(\Lambda_{1}+\Lambda_{5})_{4}$ + $(\Lambda_{4}+\Lambda_{6})_{4}$ + $(\Lambda_{1}+2\Lambda_{6})_{4}$ + $(\Lambda_{5}+2\Lambda_{6})_{4}$ </p>
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0002044v1
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[{"type": "text", "text": "", "page_idx": 12}, {"type": "text", "text": "$\\pi_{v}$ here refers to the simple-current automorphism $\\pi[2]$ or 10 00 ], for r odd/even. When $k\\,=\\,1$ , $A(D_{e v e n,1})\\cong{\\mathfrak{S}}_{3}$ , corresponding to any permutation of $\\Lambda_{1},\\Lambda_{r-1},\\Lambda_{r}$ , and $A(D_{o d d,1})\\:=\\:\\langle{C_{1}}\\rangle\\:\\cong\\:\\mathbb{Z}_{2}$ . When $r\\,>\\,4$ , ${\\cal A}(D_{r,2})\\,\\cong\\,(\\mathbb{Z}_{2r}^{\\times}/\\{\\pm1\\})\\,\\times\\,\\mathbb{Z}_{2}\\,\\times\\,\\mathbb{Z}_{2}$ or $\\mathbb{Z}_{r}^{\\times}\\times\\mathbb{Z}_{2}$ for $r$ even/odd. $A(D_{4,2})$ has 24 elements, and any element can be written uniquely as $C_{i}\\,\\pi\\,\\left[\\begin{array}{l l}{a}&{0}\\\\ {0}&{d}\\end{array}\\right]$ ", "page_idx": 12}, {"type": "text", "text": "3.5. The algebra $E_{6}^{(1)}$ ", "text_level": 1, "page_idx": 12}, {"type": "text", "text": "A weight $\\lambda$ of $P_{+}$ satisfies $k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!3\\lambda_{3}\\!+\\!2\\lambda_{4}\\!+\\!\\lambda_{5}\\!+\\!2\\lambda_{6}$ and $\\kappa=k\\!+\\!12$ . The charge-conjugation acts as $C\\lambda=(\\lambda_{0},\\lambda_{5},\\lambda_{4},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{6})$ . The order 3 simple-current $J$ is given by $J\\lambda=(\\lambda_{5},\\lambda_{0},\\lambda_{6},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{4})$ with $Q(\\lambda)=(-\\lambda_{1}+\\lambda_{2}-\\lambda_{4}+\\lambda_{5})/3$ . ", "page_idx": 12}, {"type": "text", "text": "The fusion products we need can be derived from [29] using (2.4): ", "page_idx": 12}, {"type": "equation", "text": "$$\n(\\Lambda_{1}+\\Lambda_{2})_{3}\\sqcup(\\Lambda_{1}+\\Lambda_{5})_{2}\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "where the outer subscript on any summand denotes the smallest level where that summand appears (it will also appear at all larger levels). So for example $\\Lambda_{1}\\boxtimes\\Lambda_{1}$ equals $\\Lambda_{2}$ + $\\Lambda_{5}$ + $(2\\Lambda_{1})$ for any $k\\geq2$ , but equals $\\Lambda_{5}$ at $k\\,=\\,1$ . A similar convention is used in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will equal the numerical value of the fusion coefficient). ", "page_idx": 12}, {"type": "text", "text": "Theorem 3.E6. The fusion-symmetries of $E_{6}^{(1)}$ are $C^{i}\\,\\pi[a]$ , for any $i\\in\\{0,1\\}$ and any $a\\in\\{0,1,2\\}$ for which $a k\\not\\equiv1$ (mod 3). ", "page_idx": 12}, {"type": "text", "text": "3.6. The algebra E7(1) ", "text_level": 1, "page_idx": 12}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}$ , and $\\kappa\\,=\\,k\\,+\\,18$ . The charge-conjugation is trivial, but there is a simple-current $J$ given by $J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})$ . It has $Q(\\lambda)=(\\lambda_{4}+\\lambda_{6}+\\lambda_{7})/2$ . ", "page_idx": 12}, {"type": "text", "text": "The only fusion products we need can be obtained from [29] and (2.4): ", "page_idx": 12}, {"type": "text", "text": "$\\Lambda_{6}$ × $\\Lambda_{6}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{5})_{2}$ + $(2\\Lambda_{6})_{2}$ $\\Lambda_{1}$ × $\\Lambda_{6}=(\\Lambda_{6})_{2}$ + $(\\Lambda_{7})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ $\\Lambda_{5}$ × $\\Lambda_{6}=(\\Lambda_{4})_{3}$ + $(\\Lambda_{6})_{2}$ + $(\\Lambda_{7})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ + $(\\Lambda_{5}+\\Lambda_{6})_{3}$ $\\Lambda_{6}$ × $(2\\Lambda_{6})=(\\Lambda_{6})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ + $(3\\Lambda_{6})_{3}$ + $(\\Lambda_{5}+\\Lambda_{6})_{3}$ $\\Lambda_{4}$ × $\\Lambda_{6}=(\\Lambda_{2})_{3}$ + $(\\Lambda_{3})_{4}$ + $(\\Lambda_{5})_{3}$ + $(\\Lambda_{1}+\\Lambda_{5})_{4}$ + $(\\Lambda_{4}+\\Lambda_{6})_{4}$ + $(\\Lambda_{6}+\\Lambda_{7})_{3}$ $\\Lambda_{6}$ × $\\Lambda_{7}=(\\Lambda_{1})_{2}$ + $(\\Lambda_{2})_{3}$ + $(\\Lambda_{5})_{2}$ + $(\\Lambda_{6}+\\Lambda_{7})_{3}$ $\\Lambda_{6}$ × $(\\Lambda_{5}+\\Lambda_{6})=(\\Lambda_{5})_{3}$ + $(2\\Lambda_{5})_{4}$ + $(2\\Lambda_{6})_{3}$ + $(\\Lambda_{6}+\\Lambda_{7})_{3}$ + $(\\Lambda_{1}+\\Lambda_{5})_{4}$ + $(\\Lambda_{4}+\\Lambda_{6})_{4}$ + $(\\Lambda_{1}+2\\Lambda_{6})_{4}$ + $(\\Lambda_{5}+2\\Lambda_{6})_{4}$ ", "page_idx": 12}]
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The order 3 simple-current ", "type": "text"}, {"bbox": [531, 256, 540, 265], "score": 0.87, "content": "J", "type": "inline_equation", "height": 9, "width": 9}], "index": 9}, {"bbox": [69, 267, 490, 284], "spans": [{"bbox": [69, 268, 131, 284], "score": 1.0, "content": "is given by ", "type": "text"}, {"bbox": [131, 268, 290, 282], "score": 0.92, "content": "J\\lambda=(\\lambda_{5},\\lambda_{0},\\lambda_{6},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{4})", "type": "inline_equation", "height": 14, "width": 159}, {"bbox": [290, 268, 319, 284], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [320, 267, 487, 282], "score": 0.91, "content": "Q(\\lambda)=(-\\lambda_{1}+\\lambda_{2}-\\lambda_{4}+\\lambda_{5})/3", "type": "inline_equation", "height": 15, "width": 167}, {"bbox": [487, 268, 490, 284], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9}, {"type": "text", "bbox": [96, 281, 443, 295], "lines": [{"bbox": [95, 282, 440, 297], "spans": [{"bbox": [95, 282, 440, 297], "score": 1.0, "content": "The fusion products we need can be derived from [29] using (2.4):", "type": "text"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [282, 354, 416, 370], "lines": [{"bbox": [282, 354, 416, 370], "spans": [{"bbox": [282, 354, 416, 370], "score": 0.38, "content": "(\\Lambda_{1}+\\Lambda_{2})_{3}\\sqcup(\\Lambda_{1}+\\Lambda_{5})_{2}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [70, 372, 541, 445], "lines": [{"bbox": [70, 374, 540, 390], "spans": [{"bbox": [70, 374, 540, 390], "score": 1.0, "content": "where the outer subscript on any summand denotes the smallest level where that sum-", "type": "text"}], "index": 13}, {"bbox": [70, 388, 541, 403], "spans": [{"bbox": [70, 389, 455, 403], "score": 1.0, "content": "mand appears (it will also appear at all larger levels). So for example ", "type": "text"}, {"bbox": [456, 388, 502, 402], "score": 0.36, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [503, 389, 541, 403], "score": 1.0, "content": " equals", "type": "text"}], "index": 14}, {"bbox": [71, 402, 542, 420], "spans": [{"bbox": [71, 404, 85, 416], "score": 0.86, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [86, 402, 102, 420], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [102, 403, 117, 416], "score": 0.86, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [118, 402, 135, 420], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [135, 403, 164, 417], "score": 0.87, "content": "(2\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [165, 402, 210, 420], "score": 1.0, "content": " for any ", "type": "text"}, {"bbox": [211, 403, 244, 416], "score": 0.91, "content": "k\\geq2", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [244, 402, 311, 420], "score": 1.0, "content": ", but equals ", "type": "text"}, {"bbox": [311, 403, 326, 416], "score": 0.88, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [326, 402, 344, 420], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [344, 403, 378, 415], "score": 0.91, "content": "k\\,=\\,1", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [378, 402, 542, 420], "score": 1.0, "content": ". A similar convention is used", "type": "text"}], "index": 15}, {"bbox": [70, 418, 541, 433], "spans": [{"bbox": [70, 418, 541, 433], "score": 1.0, "content": "in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will", "type": "text"}], "index": 16}, {"bbox": [71, 433, 339, 447], "spans": [{"bbox": [71, 433, 339, 447], "score": 1.0, "content": "equal the numerical value of the fusion coefficient).", "type": "text"}], "index": 17}], "index": 15}, {"type": "text", "bbox": [71, 450, 542, 482], "lines": [{"bbox": [90, 450, 542, 475], "spans": [{"bbox": [90, 450, 329, 475], "score": 1.0, "content": "Theorem 3.E6. The fusion-symmetries of ", "type": "text"}, {"bbox": [330, 450, 353, 468], "score": 0.92, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 18, "width": 23}, {"bbox": [353, 450, 377, 475], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [378, 452, 414, 468], "score": 0.9, "content": "C^{i}\\,\\pi[a]", "type": "inline_equation", "height": 16, "width": 36}, {"bbox": [415, 450, 465, 475], "score": 1.0, "content": ", for any ", "type": "text"}, {"bbox": [465, 453, 516, 468], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 15, "width": 51}, {"bbox": [516, 450, 542, 475], "score": 1.0, "content": " and", "type": "text"}], "index": 18}, {"bbox": [71, 467, 299, 485], "spans": [{"bbox": [71, 467, 93, 485], "score": 1.0, "content": "any ", "type": "text"}, {"bbox": [94, 468, 156, 482], "score": 0.93, "content": "a\\in\\{0,1,2\\}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [156, 467, 212, 485], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [212, 468, 248, 482], "score": 0.74, "content": "a k\\not\\equiv1", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [249, 467, 299, 485], "score": 1.0, "content": " (mod 3).", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "title", "bbox": [71, 491, 183, 507], "lines": [{"bbox": [68, 491, 186, 512], "spans": [{"bbox": [68, 491, 186, 512], "score": 1.0, "content": "3.6. The algebra E7(1)", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 512, 542, 556], "lines": [{"bbox": [94, 514, 542, 529], "spans": [{"bbox": [94, 514, 146, 529], "score": 1.0, "content": "A weight", "type": "text"}, {"bbox": [147, 514, 156, 526], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [156, 514, 174, 529], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [174, 514, 191, 528], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 14, "width": 17}, {"bbox": [191, 514, 239, 529], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [239, 514, 511, 527], "score": 0.88, "content": "k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}", "type": "inline_equation", "height": 13, "width": 272}, {"bbox": [512, 514, 542, 529], "score": 1.0, "content": ", and", "type": "text"}], "index": 21}, {"bbox": [71, 528, 541, 545], "spans": [{"bbox": [71, 528, 132, 541], "score": 0.91, "content": "\\kappa\\,=\\,k\\,+\\,18", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [132, 529, 481, 545], "score": 1.0, "content": ". The charge-conjugation is trivial, but there is a simple-current ", "type": "text"}, {"bbox": [482, 531, 490, 540], "score": 0.84, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [491, 529, 541, 545], "score": 1.0, "content": " given by", "type": "text"}], "index": 22}, {"bbox": [71, 542, 392, 559], "spans": [{"bbox": [71, 542, 216, 557], "score": 0.91, "content": "J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})", "type": "inline_equation", "height": 15, "width": 145}, {"bbox": [216, 543, 258, 559], "score": 1.0, "content": ". It has ", "type": "text"}, {"bbox": [258, 542, 388, 557], "score": 0.92, "content": "Q(\\lambda)=(\\lambda_{4}+\\lambda_{6}+\\lambda_{7})/2", "type": "inline_equation", "height": 15, "width": 130}, {"bbox": [388, 543, 392, 559], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22}, {"type": "text", "bbox": [94, 556, 468, 570], "lines": [{"bbox": [96, 558, 465, 572], "spans": [{"bbox": [96, 558, 465, 572], "score": 1.0, "content": "The only fusion products we need can be obtained from [29] and (2.4):", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [82, 573, 530, 716], "lines": [{"bbox": [120, 575, 348, 592], "spans": [{"bbox": [120, 576, 135, 590], "score": 0.87, "content": "\\Lambda_{6}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [136, 575, 151, 592], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [152, 575, 202, 591], "score": 0.93, "content": "\\Lambda_{6}=(0)_{1}", "type": "inline_equation", "height": 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", "type": "text"}, {"bbox": [162, 163, 204, 177], "score": 0.92, "content": "A(D_{4,2})", "type": "inline_equation", "height": 14, "width": 42}, {"bbox": [205, 163, 541, 178], "score": 1.0, "content": " has 24 elements, and any element can be written uniquely as", "type": "text"}], "index": 5}, {"bbox": [71, 176, 137, 208], "spans": [{"bbox": [71, 176, 137, 208], "score": 0.89, "content": "C_{i}\\,\\pi\\,\\left[\\begin{array}{l l}{a}&{0}\\\\ {0}&{d}\\end{array}\\right]", "type": "inline_equation", "height": 32, "width": 66}], "index": 6}], "index": 4, "bbox_fs": [70, 105, 545, 208]}, {"type": "title", "bbox": [70, 216, 183, 232], "lines": [{"bbox": [68, 217, 183, 237], "spans": [{"bbox": [68, 217, 160, 237], "score": 1.0, "content": "3.5. The algebra ", "type": "text"}, {"bbox": [161, 217, 183, 234], "score": 0.9, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 17, "width": 22}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 237, 541, 280], "lines": [{"bbox": [95, 239, 541, 255], "spans": [{"bbox": [95, 239, 144, 255], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 240, 152, 250], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [153, 239, 167, 255], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [167, 240, 183, 253], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [184, 239, 228, 255], "score": 1.0, "content": " satisfies", "type": "text"}, {"bbox": [228, 240, 432, 252], "score": 0.89, "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!3\\lambda_{3}\\!+\\!2\\lambda_{4}\\!+\\!\\lambda_{5}\\!+\\!2\\lambda_{6}", "type": "inline_equation", "height": 12, "width": 204}, {"bbox": [432, 239, 457, 255], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [457, 241, 510, 251], "score": 0.92, "content": "\\kappa=k\\!+\\!12", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [511, 239, 541, 255], "score": 1.0, "content": ". The", "type": "text"}], "index": 8}, {"bbox": [72, 253, 540, 269], "spans": [{"bbox": [72, 253, 213, 269], "score": 1.0, "content": "charge-conjugation acts as ", "type": "text"}, {"bbox": [213, 253, 375, 267], "score": 0.89, "content": "C\\lambda=(\\lambda_{0},\\lambda_{5},\\lambda_{4},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{6})", "type": "inline_equation", "height": 14, "width": 162}, {"bbox": [376, 253, 531, 269], "score": 1.0, "content": ". The order 3 simple-current ", "type": "text"}, {"bbox": [531, 256, 540, 265], "score": 0.87, "content": "J", "type": "inline_equation", "height": 9, "width": 9}], "index": 9}, {"bbox": [69, 267, 490, 284], "spans": [{"bbox": [69, 268, 131, 284], "score": 1.0, "content": "is given by ", "type": "text"}, {"bbox": [131, 268, 290, 282], "score": 0.92, "content": "J\\lambda=(\\lambda_{5},\\lambda_{0},\\lambda_{6},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{4})", "type": "inline_equation", "height": 14, "width": 159}, {"bbox": [290, 268, 319, 284], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [320, 267, 487, 282], "score": 0.91, "content": "Q(\\lambda)=(-\\lambda_{1}+\\lambda_{2}-\\lambda_{4}+\\lambda_{5})/3", "type": "inline_equation", "height": 15, "width": 167}, {"bbox": [487, 268, 490, 284], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9, "bbox_fs": [69, 239, 541, 284]}, {"type": "text", "bbox": [96, 281, 443, 295], "lines": [{"bbox": [95, 282, 440, 297], "spans": [{"bbox": [95, 282, 440, 297], "score": 1.0, "content": "The fusion products we need can be derived from [29] using (2.4):", "type": "text"}], "index": 11}], "index": 11, "bbox_fs": [95, 282, 440, 297]}, {"type": "interline_equation", "bbox": [282, 354, 416, 370], "lines": [{"bbox": [282, 354, 416, 370], "spans": [{"bbox": [282, 354, 416, 370], "score": 0.38, "content": "(\\Lambda_{1}+\\Lambda_{2})_{3}\\sqcup(\\Lambda_{1}+\\Lambda_{5})_{2}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [70, 372, 541, 445], "lines": [{"bbox": [70, 374, 540, 390], "spans": [{"bbox": [70, 374, 540, 390], "score": 1.0, "content": "where the outer subscript on any summand denotes the smallest level where that sum-", "type": "text"}], "index": 13}, {"bbox": [70, 388, 541, 403], "spans": [{"bbox": [70, 389, 455, 403], "score": 1.0, "content": "mand appears (it will also appear at all larger levels). So for example ", "type": "text"}, {"bbox": [456, 388, 502, 402], "score": 0.36, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [503, 389, 541, 403], "score": 1.0, "content": " equals", "type": "text"}], "index": 14}, {"bbox": [71, 402, 542, 420], "spans": [{"bbox": [71, 404, 85, 416], "score": 0.86, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [86, 402, 102, 420], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [102, 403, 117, 416], "score": 0.86, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [118, 402, 135, 420], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [135, 403, 164, 417], "score": 0.87, "content": "(2\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [165, 402, 210, 420], "score": 1.0, "content": " for any ", "type": "text"}, {"bbox": [211, 403, 244, 416], "score": 0.91, "content": "k\\geq2", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [244, 402, 311, 420], "score": 1.0, "content": ", but equals ", "type": "text"}, {"bbox": [311, 403, 326, 416], "score": 0.88, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [326, 402, 344, 420], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [344, 403, 378, 415], "score": 0.91, "content": "k\\,=\\,1", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [378, 402, 542, 420], "score": 1.0, "content": ". A similar convention is used", "type": "text"}], "index": 15}, {"bbox": [70, 418, 541, 433], "spans": [{"bbox": [70, 418, 541, 433], "score": 1.0, "content": "in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will", "type": "text"}], "index": 16}, {"bbox": [71, 433, 339, 447], "spans": [{"bbox": [71, 433, 339, 447], "score": 1.0, "content": "equal the numerical value of the fusion coefficient).", "type": "text"}], "index": 17}], "index": 15, "bbox_fs": [70, 374, 542, 447]}, {"type": "text", "bbox": [71, 450, 542, 482], "lines": [{"bbox": [90, 450, 542, 475], "spans": [{"bbox": [90, 450, 329, 475], "score": 1.0, "content": "Theorem 3.E6. The fusion-symmetries of ", "type": "text"}, {"bbox": [330, 450, 353, 468], "score": 0.92, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 18, "width": 23}, {"bbox": [353, 450, 377, 475], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [378, 452, 414, 468], "score": 0.9, "content": "C^{i}\\,\\pi[a]", "type": "inline_equation", "height": 16, "width": 36}, {"bbox": [415, 450, 465, 475], "score": 1.0, "content": ", for any ", "type": "text"}, {"bbox": [465, 453, 516, 468], "score": 0.92, "content": "i\\in\\{0,1\\}", "type": "inline_equation", "height": 15, "width": 51}, {"bbox": [516, 450, 542, 475], "score": 1.0, "content": " and", "type": "text"}], "index": 18}, {"bbox": [71, 467, 299, 485], "spans": [{"bbox": [71, 467, 93, 485], "score": 1.0, "content": "any ", "type": "text"}, {"bbox": [94, 468, 156, 482], "score": 0.93, "content": "a\\in\\{0,1,2\\}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [156, 467, 212, 485], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [212, 468, 248, 482], "score": 0.74, "content": "a k\\not\\equiv1", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [249, 467, 299, 485], "score": 1.0, "content": " (mod 3).", "type": "text"}], "index": 19}], "index": 18.5, "bbox_fs": [71, 450, 542, 485]}, {"type": "title", "bbox": [71, 491, 183, 507], "lines": [{"bbox": [68, 491, 186, 512], "spans": [{"bbox": [68, 491, 186, 512], "score": 1.0, "content": "3.6. The algebra E7(1)", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 512, 542, 556], "lines": [{"bbox": [94, 514, 542, 529], "spans": [{"bbox": [94, 514, 146, 529], "score": 1.0, "content": "A weight", "type": "text"}, {"bbox": [147, 514, 156, 526], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [156, 514, 174, 529], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [174, 514, 191, 528], "score": 0.89, "content": "P_{+}", "type": "inline_equation", "height": 14, "width": 17}, {"bbox": [191, 514, 239, 529], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [239, 514, 511, 527], "score": 0.88, "content": "k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}", "type": "inline_equation", "height": 13, "width": 272}, {"bbox": [512, 514, 542, 529], "score": 1.0, "content": ", and", "type": "text"}], "index": 21}, {"bbox": [71, 528, 541, 545], "spans": [{"bbox": [71, 528, 132, 541], "score": 0.91, "content": "\\kappa\\,=\\,k\\,+\\,18", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [132, 529, 481, 545], "score": 1.0, "content": ". The charge-conjugation is trivial, but there is a simple-current ", "type": "text"}, {"bbox": [482, 531, 490, 540], "score": 0.84, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [491, 529, 541, 545], "score": 1.0, "content": " given by", "type": "text"}], "index": 22}, {"bbox": [71, 542, 392, 559], "spans": [{"bbox": [71, 542, 216, 557], "score": 0.91, "content": "J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})", "type": "inline_equation", "height": 15, "width": 145}, {"bbox": [216, 543, 258, 559], "score": 1.0, "content": ". It has ", "type": "text"}, {"bbox": [258, 542, 388, 557], "score": 0.92, "content": "Q(\\lambda)=(\\lambda_{4}+\\lambda_{6}+\\lambda_{7})/2", "type": "inline_equation", "height": 15, "width": 130}, {"bbox": [388, 543, 392, 559], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22, "bbox_fs": [71, 514, 542, 559]}, {"type": "text", "bbox": [94, 556, 468, 570], "lines": [{"bbox": [96, 558, 465, 572], "spans": [{"bbox": [96, 558, 465, 572], "score": 1.0, "content": "The only fusion products we need can be obtained from [29] and (2.4):", "type": "text"}], "index": 24}], "index": 24, "bbox_fs": [96, 558, 465, 572]}, {"type": "text", "bbox": [82, 573, 530, 716], "lines": [{"bbox": [120, 575, 348, 592], "spans": [{"bbox": [120, 576, 135, 590], "score": 0.87, "content": "\\Lambda_{6}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [136, 575, 151, 592], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [152, 575, 202, 591], "score": 0.93, 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"text"}, {"bbox": [199, 701, 257, 716], "score": 0.93, "content": "(\\Lambda_{4}+\\Lambda_{6})_{4}", "type": "inline_equation", "height": 15, "width": 58}, {"bbox": [257, 700, 273, 718], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [274, 701, 338, 716], "score": 0.93, "content": "(\\Lambda_{1}+2\\Lambda_{6})_{4}", "type": "inline_equation", "height": 15, "width": 64}, {"bbox": [338, 700, 354, 718], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [355, 701, 419, 716], "score": 0.89, "content": "(\\Lambda_{5}+2\\Lambda_{6})_{4}", "type": "inline_equation", "height": 15, "width": 64}], "index": 32}], "index": 28.5, "bbox_fs": [82, 575, 528, 718]}]}
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[{"type": "text", "bbox": [69, 69, 542, 102], "content": "", "index": 0}, {"type": "text", "bbox": [70, 106, 541, 204], "content": "here refers to the simple-current automorphism or 10 00 ], for r odd/even. When , , corresponding to any permutation of , and . When , or for even/odd. has 24 elements, and any element can be written uniquely as", "index": 1}, {"type": "title", "bbox": [70, 216, 183, 232], "content": "3.5. The algebra", "index": 2}, {"type": "text", "bbox": [70, 237, 541, 280], "content": "A weight of satisfies and . The charge-conjugation acts as . The order 3 simple-current is given by with .", "index": 3}, {"type": "text", "bbox": [96, 281, 443, 295], "content": "The fusion products we need can be derived from [29] using (2.4):", "index": 4}, {"type": "interline_equation", "bbox": [282, 354, 416, 370], "content": "", "index": 5}, {"type": "text", "bbox": [70, 372, 541, 445], "content": "where the outer subscript on any summand denotes the smallest level where that sum- mand appears (it will also appear at all larger levels). So for example equals + + for any , but equals at . A similar convention is used in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will equal the numerical value of the fusion coefficient).", "index": 6}, {"type": "text", "bbox": [71, 450, 542, 482], "content": "Theorem 3.E6. The fusion-symmetries of are , for any and any for which (mod 3).", "index": 7}, {"type": "title", "bbox": [71, 491, 183, 507], "content": "3.6. The algebra E7(1)", "index": 8}, {"type": "text", "bbox": [70, 512, 542, 556], "content": "A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current given by . It has .", "index": 9}, {"type": "text", "bbox": [94, 556, 468, 570], "content": "The only fusion products we need can be obtained from [29] and (2.4):", "index": 10}, {"type": "text", "bbox": [82, 573, 530, 716], "content": "× + + + × + + × + + + + × + + + × + + + + + × + + + × + + + + + + +", "index": 11}]
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[{"bbox": [95, 105, 545, 141], "content": "here refers to the simple-current automorphism or 10 00 ], for r odd/even.", "parent_index": 1, "line_index": 0}, {"bbox": [70, 133, 542, 150], "content": "When , , corresponding to any permutation of , and", "parent_index": 1, "line_index": 1}, {"bbox": [71, 147, 540, 166], "content": ". When , or", "parent_index": 1, "line_index": 2}, {"bbox": [70, 163, 541, 178], "content": "for even/odd. has 24 elements, and any element can be written uniquely as", "parent_index": 1, "line_index": 3}, {"bbox": [71, 176, 137, 208], "content": "", "parent_index": 1, "line_index": 4}, {"bbox": [68, 217, 183, 237], "content": "3.5. The algebra", "parent_index": 2, "line_index": 0}, {"bbox": [95, 239, 541, 255], "content": "A weight of satisfies and . The", "parent_index": 3, "line_index": 0}, {"bbox": [72, 253, 540, 269], "content": "charge-conjugation acts as . The order 3 simple-current", "parent_index": 3, "line_index": 1}, {"bbox": [69, 267, 490, 284], "content": "is given by with .", "parent_index": 3, "line_index": 2}, {"bbox": [95, 282, 440, 297], "content": "The fusion products we need can be derived from [29] using (2.4):", "parent_index": 4, "line_index": 0}, {"bbox": [70, 374, 540, 390], "content": "where the outer subscript on any summand denotes the smallest level where that sum-", "parent_index": 6, "line_index": 0}, {"bbox": [70, 388, 541, 403], "content": "mand appears (it will also appear at all larger levels). So for example equals", "parent_index": 6, "line_index": 1}, {"bbox": [71, 402, 542, 420], "content": "+ + for any , but equals at . A similar convention is used", "parent_index": 6, "line_index": 2}, {"bbox": [70, 418, 541, 433], "content": "in (3.7) and elsewhere for higher fusion multiplicities (the number of subscripts used will", "parent_index": 6, "line_index": 3}, {"bbox": [71, 433, 339, 447], "content": "equal the numerical value of the fusion coefficient).", "parent_index": 6, "line_index": 4}, {"bbox": [90, 450, 542, 475], "content": "Theorem 3.E6. The fusion-symmetries of are , for any and", "parent_index": 7, "line_index": 0}, {"bbox": [71, 467, 299, 485], "content": "any for which (mod 3).", "parent_index": 7, "line_index": 1}, {"bbox": [68, 491, 186, 512], "content": "3.6. The algebra E7(1)", "parent_index": 8, "line_index": 0}, {"bbox": [94, 514, 542, 529], "content": "A weight in satisfies , and", "parent_index": 9, "line_index": 0}, {"bbox": [71, 528, 541, 545], "content": ". The charge-conjugation is trivial, but there is a simple-current given by", "parent_index": 9, "line_index": 1}, {"bbox": [71, 542, 392, 559], "content": ". It has .", "parent_index": 9, "line_index": 2}, {"bbox": [96, 558, 465, 572], "content": "The only fusion products we need can be obtained from [29] and (2.4):", "parent_index": 10, "line_index": 0}, {"bbox": [120, 575, 348, 592], "content": "× + + +", "parent_index": 11, "line_index": 0}, {"bbox": [120, 594, 332, 610], "content": "× + +", "parent_index": 11, "line_index": 1}, {"bbox": [120, 611, 453, 627], "content": "× + + + +", "parent_index": 11, "line_index": 2}, {"bbox": [105, 630, 412, 646], "content": "× + + +", "parent_index": 11, "line_index": 3}, {"bbox": [120, 647, 528, 664], "content": "× + + + + +", "parent_index": 11, "line_index": 4}, {"bbox": [120, 665, 378, 682], "content": "× + + +", "parent_index": 11, "line_index": 5}, {"bbox": [82, 683, 465, 699], "content": "× + + + +", "parent_index": 11, "line_index": 6}, {"bbox": [183, 700, 419, 718], "content": "+ + +", "parent_index": 11, "line_index": 7}]
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[{"bbox": [95, 116, 108, 127], "content": "\\pi_{v}", "parent_index": 1, "subtype": "inline"}, {"bbox": [367, 113, 389, 128], "content": "\\pi[2]", "parent_index": 1, "subtype": "inline"}, {"bbox": [106, 134, 139, 146], "content": "k\\,=\\,1", "parent_index": 1, "subtype": "inline"}, {"bbox": [147, 133, 239, 148], "content": "A(D_{e v e n,1})\\cong{\\mathfrak{S}}_{3}", "parent_index": 1, "subtype": "inline"}, {"bbox": [447, 134, 512, 147], "content": "\\Lambda_{1},\\Lambda_{r-1},\\Lambda_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [71, 148, 200, 163], "content": "A(D_{o d d,1})\\:=\\:\\langle{C_{1}}\\rangle\\:\\cong\\:\\mathbb{Z}_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [248, 149, 282, 161], "content": "r\\,>\\,4", "parent_index": 1, "subtype": "inline"}, {"bbox": [290, 147, 473, 163], "content": "{\\cal A}(D_{r,2})\\,\\cong\\,(\\mathbb{Z}_{2r}^{\\times}/\\{\\pm1\\})\\,\\times\\,\\mathbb{Z}_{2}\\,\\times\\,\\mathbb{Z}_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [493, 148, 540, 162], "content": "\\mathbb{Z}_{r}^{\\times}\\times\\mathbb{Z}_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [90, 166, 97, 174], "content": "r", "parent_index": 1, "subtype": "inline"}, {"bbox": [162, 163, 204, 177], "content": "A(D_{4,2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [71, 176, 137, 208], "content": "C_{i}\\,\\pi\\,\\left[\\begin{array}{l l}{a}&{0}\\\\ {0}&{d}\\end{array}\\right]", "parent_index": 1, "subtype": "inline"}, {"bbox": [161, 217, 183, 234], "content": "E_{6}^{(1)}", "parent_index": 2, "subtype": "inline"}, {"bbox": [144, 240, 152, 250], "content": "\\lambda", "parent_index": 3, "subtype": "inline"}, {"bbox": [167, 240, 183, 253], "content": "P_{+}", "parent_index": 3, "subtype": "inline"}, {"bbox": [228, 240, 432, 252], "content": "k=\\lambda_{0}\\!+\\!\\lambda_{1}\\!+\\!2\\lambda_{2}\\!+\\!3\\lambda_{3}\\!+\\!2\\lambda_{4}\\!+\\!\\lambda_{5}\\!+\\!2\\lambda_{6}", "parent_index": 3, "subtype": "inline"}, {"bbox": [457, 241, 510, 251], "content": "\\kappa=k\\!+\\!12", "parent_index": 3, "subtype": "inline"}, {"bbox": [213, 253, 375, 267], "content": "C\\lambda=(\\lambda_{0},\\lambda_{5},\\lambda_{4},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{6})", "parent_index": 3, "subtype": "inline"}, {"bbox": [531, 256, 540, 265], "content": "J", "parent_index": 3, "subtype": "inline"}, {"bbox": [131, 268, 290, 282], "content": "J\\lambda=(\\lambda_{5},\\lambda_{0},\\lambda_{6},\\lambda_{3},\\lambda_{2},\\lambda_{1},\\lambda_{4})", "parent_index": 3, "subtype": "inline"}, {"bbox": [320, 267, 487, 282], "content": "Q(\\lambda)=(-\\lambda_{1}+\\lambda_{2}-\\lambda_{4}+\\lambda_{5})/3", "parent_index": 3, "subtype": "inline"}, {"bbox": [282, 354, 416, 370], "content": "(\\Lambda_{1}+\\Lambda_{2})_{3}\\sqcup(\\Lambda_{1}+\\Lambda_{5})_{2}", "parent_index": 5, "subtype": "interline"}, {"bbox": [456, 388, 502, 402], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [71, 404, 85, 416], "content": "\\Lambda_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [102, 403, 117, 416], "content": "\\Lambda_{5}", "parent_index": 6, "subtype": "inline"}, {"bbox": [135, 403, 164, 417], "content": "(2\\Lambda_{1})", "parent_index": 6, "subtype": "inline"}, {"bbox": [211, 403, 244, 416], "content": "k\\geq2", "parent_index": 6, "subtype": "inline"}, {"bbox": [311, 403, 326, 416], "content": "\\Lambda_{5}", "parent_index": 6, "subtype": "inline"}, {"bbox": [344, 403, 378, 415], "content": "k\\,=\\,1", "parent_index": 6, "subtype": "inline"}, {"bbox": [330, 450, 353, 468], "content": "E_{6}^{(1)}", "parent_index": 7, "subtype": "inline"}, {"bbox": [378, 452, 414, 468], "content": "C^{i}\\,\\pi[a]", "parent_index": 7, "subtype": "inline"}, {"bbox": [465, 453, 516, 468], "content": "i\\in\\{0,1\\}", "parent_index": 7, "subtype": "inline"}, {"bbox": [94, 468, 156, 482], "content": "a\\in\\{0,1,2\\}", "parent_index": 7, "subtype": "inline"}, {"bbox": [212, 468, 248, 482], "content": "a k\\not\\equiv1", "parent_index": 7, "subtype": "inline"}, {"bbox": [147, 514, 156, 526], "content": "\\lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [174, 514, 191, 528], "content": "P_{+}", "parent_index": 9, "subtype": "inline"}, {"bbox": [239, 514, 511, 527], "content": "k\\,=\\,\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+3\\lambda_{4}+2\\lambda_{5}+\\lambda_{6}+2\\lambda_{7}", "parent_index": 9, "subtype": "inline"}, {"bbox": [71, 528, 132, 541], "content": "\\kappa\\,=\\,k\\,+\\,18", "parent_index": 9, "subtype": "inline"}, {"bbox": [482, 531, 490, 540], "content": "J", "parent_index": 9, "subtype": "inline"}, {"bbox": [71, 542, 216, 557], "content": "J\\lambda=(\\lambda_{6},\\lambda_{5},...\\,,\\lambda_{1},\\lambda_{0},\\lambda_{7})", "parent_index": 9, "subtype": "inline"}, {"bbox": [258, 542, 388, 557], "content": "Q(\\lambda)=(\\lambda_{4}+\\lambda_{6}+\\lambda_{7})/2", "parent_index": 9, "subtype": "inline"}, {"bbox": [120, 576, 135, 590], "content": "\\Lambda_{6}", "parent_index": 11, "subtype": "inline"}, {"bbox": [152, 575, 202, 591], "content": "\\Lambda_{6}=(0)_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [220, 576, 249, 591], "content": "(\\Lambda_{1})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [266, 575, 295, 591], "content": "(\\Lambda_{5})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [313, 575, 348, 591], "content": "(2\\Lambda_{6})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [120, 594, 135, 608], "content": "\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [152, 594, 210, 609], "content": "\\Lambda_{6}=(\\Lambda_{6})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [227, 594, 257, 609], "content": "(\\Lambda_{7})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [274, 594, 332, 609], "content": "(\\Lambda_{1}+\\Lambda_{6})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [120, 612, 135, 626], "content": "\\Lambda_{5}", "parent_index": 11, "subtype": "inline"}, {"bbox": [152, 612, 210, 627], "content": "\\Lambda_{6}=(\\Lambda_{4})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [227, 612, 257, 627], "content": "(\\Lambda_{6})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [274, 612, 303, 627], "content": "(\\Lambda_{7})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [321, 612, 378, 627], "content": "(\\Lambda_{1}+\\Lambda_{6})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [395, 612, 453, 627], "content": "(\\Lambda_{5}+\\Lambda_{6})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [105, 630, 120, 644], "content": "\\Lambda_{6}", "parent_index": 11, "subtype": "inline"}, {"bbox": [137, 630, 210, 645], "content": "(2\\Lambda_{6})=(\\Lambda_{6})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [227, 630, 285, 645], "content": "(\\Lambda_{1}+\\Lambda_{6})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [302, 630, 338, 645], "content": "(3\\Lambda_{6})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [355, 630, 412, 645], "content": "(\\Lambda_{5}+\\Lambda_{6})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [120, 648, 136, 662], "content": "\\Lambda_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [152, 648, 210, 663], "content": "\\Lambda_{6}=(\\Lambda_{2})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [227, 647, 257, 663], "content": "(\\Lambda_{3})_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [274, 647, 303, 663], "content": "(\\Lambda_{5})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [320, 647, 378, 663], "content": "(\\Lambda_{1}+\\Lambda_{5})_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [395, 647, 453, 663], "content": "(\\Lambda_{4}+\\Lambda_{6})_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [470, 647, 528, 663], "content": "(\\Lambda_{6}+\\Lambda_{7})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [120, 666, 136, 680], "content": "\\Lambda_{6}", "parent_index": 11, "subtype": "inline"}, {"bbox": [152, 666, 210, 680], "content": "\\Lambda_{7}=(\\Lambda_{1})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [227, 666, 257, 680], "content": "(\\Lambda_{2})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [274, 666, 303, 680], "content": "(\\Lambda_{5})_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [320, 666, 378, 680], "content": "(\\Lambda_{6}+\\Lambda_{7})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [82, 683, 98, 698], "content": "\\Lambda_{6}", "parent_index": 11, "subtype": "inline"}, {"bbox": [115, 683, 210, 698], "content": "(\\Lambda_{5}+\\Lambda_{6})=(\\Lambda_{5})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [227, 683, 263, 698], "content": "(2\\Lambda_{5})_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [280, 683, 315, 698], "content": "(2\\Lambda_{6})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [332, 683, 390, 698], "content": "(\\Lambda_{6}+\\Lambda_{7})_{3}", "parent_index": 11, "subtype": "inline"}, {"bbox": [407, 683, 465, 698], "content": "(\\Lambda_{1}+\\Lambda_{5})_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [199, 701, 257, 716], "content": "(\\Lambda_{4}+\\Lambda_{6})_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [274, 701, 338, 716], "content": "(\\Lambda_{1}+2\\Lambda_{6})_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [355, 701, 419, 716], "content": "(\\Lambda_{5}+2\\Lambda_{6})_{4}", "parent_index": 11, "subtype": "inline"}]
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At $k=3$ there is an order 3 Galois fusion-symmetry $\pi_{3}=\pi\{5\}$ , which sends $J^{i}\Lambda_{1}\mapsto$ $J^{i}(2\Lambda_{6})\mapsto J^{i}\Lambda_{2}\mapsto J^{i}\Lambda_{1}$ and fixes the other six weights.
Theorem 3.E7. The only nontrivial fusion-symmetries for ${E}_{7}^{(1)}$ are $\pi[1]$ at even $k$ , as well as $\pi_{3}$ and its inverse at $k=3$ .
# 3.7. The algebra E8(1)
A weight $\lambda$ in $P_{+}$ satisfies $k=\lambda_{0}+2\lambda_{1}+3\lambda_{2}+4\lambda_{3}+5\lambda_{4}+6\lambda_{5}+4\lambda_{6}+2\lambda_{7}+3\lambda_{8}$ , and $\kappa=k+30$ . The conjugations and simple-currents are all trivial, except for an anomolous simple-current at $k=2$ , sending $P_{+}=(0,\Lambda_{1},\Lambda_{7})$ to $(\Lambda_{7},\Lambda_{1},0)$ , which plays no role in this paper (except in Theorem 5.1).
The only fusion products we need can be derived from [28] and (2.4):
$(2\Lambda_{1})\vert\mathrm{\bf\sfXI}(2\Lambda_{1})=(0)_{4}$ + $(\Lambda_{1})_{5}$ + $(\Lambda_{2})_{5}$ + $(\Lambda_{3})_{4}$ + $(\Lambda_{7})_{4}$ + $2\,\mathbf{E}\left(2\Lambda_{1}\right)_{46}$ + $(2\Lambda_{2})_{6}$ $\Lambda_{1}$ × $\Lambda_{4}=(\Lambda_{3})_{5}$ + $(\Lambda_{4})_{6}$ + $(\Lambda_{5})_{6}$ + $(\Lambda_{6})_{5}$ + $(\Lambda_{1}+\Lambda_{3})_{6}$ + $(\Lambda_{1}+\Lambda_{4})_{7}$ + $(\Lambda_{1}+\Lambda_{6})_{6}$ $\Lambda_{1}$ × $(\Lambda_{1}\!+\!\Lambda_{3})=(\Lambda_{3})_{6}$ + $(\Lambda_{4})_{6}$ + $(\Lambda_{1}+\Lambda_{2})_{6}$ + $2\,\mathtt{H}\,(\Lambda_{1}+\Lambda_{3})_{67}$ + $(\Lambda_{1}+\Lambda_{4})_{7}$ $\Lambda_{1}$ × $(2\Lambda_{7})=\!(\Lambda_{6})_{4}$ + $(\Lambda_{1}+\Lambda_{7})_{4}$ + $(2\Lambda_{7})_{5}$ + $(\Lambda_{2}+\Lambda_{7})_{5}$ + $(\Lambda_{7}+\Lambda_{8})_{5}$ + $(\Lambda_{1}+2\Lambda_{7})_{6}$ $(3.7g)$
A fusion-symmetry at $k=4$ , called $\pi_{4}$ , was first found in [15]. It interchanges $\Lambda_{1}\leftrightarrow\Lambda_{6}$ and fixes the other eight weights in $P_{+}$ . There also is a fusion-symmetry, called $\pi_{5}$ , at $k=5$ which interchanges $\Lambda_{7}\leftrightarrow2\Lambda_{1}$ , $\Lambda_{8}\leftrightarrow\Lambda_{1}+\Lambda_{2}$ , and $\Lambda_{6}\leftrightarrow\Lambda_{2}+\Lambda_{7}$ , and fixes the nine other weights. The exceptional $\pi_{5}$ is closely related to the Galois permutation $\lambda\mapsto\lambda^{(13)}$ .
Theorem 3.E8. The only nontrivial fusion-symmetries for ${E}_{8}^{(1)}$ are $\pi_{4}$ and $\pi_{5}$ , occurring at $k=4$ and 5 respectively.
# 3.8. The algebra F 4(1)
A weight $\lambda$ in $P_{+}$ satisfies $k=\lambda_{0}+2\lambda_{1}+3\lambda_{2}+2\lambda_{3}+\lambda_{4}$ , and $\kappa=k+9$ . Again, the conjugations and simple-currents are trivial.
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<p data-bbox="71 70 541 100">At $k=3$ there is an order 3 Galois fusion-symmetry $\pi_{3}=\pi\{5\}$ , which sends $J^{i}\Lambda_{1}\mapsto$ $J^{i}(2\Lambda_{6})\mapsto J^{i}\Lambda_{2}\mapsto J^{i}\Lambda_{1}$ and fixes the other six weights. </p>
<p data-bbox="71 107 541 137">Theorem 3.E7. The only nontrivial fusion-symmetries for ${E}_{7}^{(1)}$ are $\pi[1]$ at even $k$ , as well as $\pi_{3}$ and its inverse at $k=3$ . </p>
<h1 data-bbox="70 150 183 167">3.7. The algebra E8(1) </h1>
<p data-bbox="70 173 542 230">A weight $\lambda$ in $P_{+}$ satisfies $k=\lambda_{0}+2\lambda_{1}+3\lambda_{2}+4\lambda_{3}+5\lambda_{4}+6\lambda_{5}+4\lambda_{6}+2\lambda_{7}+3\lambda_{8}$ , and $\kappa=k+30$ . The conjugations and simple-currents are all trivial, except for an anomolous simple-current at $k=2$ , sending $P_{+}=(0,\Lambda_{1},\Lambda_{7})$ to $(\Lambda_{7},\Lambda_{1},0)$ , which plays no role in this paper (except in Theorem 5.1). </p>
<p data-bbox="94 230 460 245">The only fusion products we need can be derived from [28] and (2.4): </p>
<p data-bbox="47 364 469 381">$(2\Lambda_{1})\vert\mathrm{\bf\sfXI}(2\Lambda_{1})=(0)_{4}$ + $(\Lambda_{1})_{5}$ + $(\Lambda_{2})_{5}$ + $(\Lambda_{3})_{4}$ + $(\Lambda_{7})_{4}$ + $2\,\mathbf{E}\left(2\Lambda_{1}\right)_{46}$ + $(2\Lambda_{2})_{6}$ $\Lambda_{1}$ × $\Lambda_{4}=(\Lambda_{3})_{5}$ + $(\Lambda_{4})_{6}$ + $(\Lambda_{5})_{6}$ + $(\Lambda_{6})_{5}$ + $(\Lambda_{1}+\Lambda_{3})_{6}$ + $(\Lambda_{1}+\Lambda_{4})_{7}$ + $(\Lambda_{1}+\Lambda_{6})_{6}$ $\Lambda_{1}$ × $(\Lambda_{1}\!+\!\Lambda_{3})=(\Lambda_{3})_{6}$ + $(\Lambda_{4})_{6}$ + $(\Lambda_{1}+\Lambda_{2})_{6}$ + $2\,\mathtt{H}\,(\Lambda_{1}+\Lambda_{3})_{67}$ + $(\Lambda_{1}+\Lambda_{4})_{7}$ $\Lambda_{1}$ × $(2\Lambda_{7})=\!(\Lambda_{6})_{4}$ + $(\Lambda_{1}+\Lambda_{7})_{4}$ + $(2\Lambda_{7})_{5}$ + $(\Lambda_{2}+\Lambda_{7})_{5}$ + $(\Lambda_{7}+\Lambda_{8})_{5}$ + $(\Lambda_{1}+2\Lambda_{7})_{6}$ $(3.7g)$ </p>
<p data-bbox="70 554 542 613">A fusion-symmetry at $k=4$ , called $\pi_{4}$ , was first found in [15]. It interchanges $\Lambda_{1}\leftrightarrow\Lambda_{6}$ and fixes the other eight weights in $P_{+}$ . There also is a fusion-symmetry, called $\pi_{5}$ , at $k=5$ which interchanges $\Lambda_{7}\leftrightarrow2\Lambda_{1}$ , $\Lambda_{8}\leftrightarrow\Lambda_{1}+\Lambda_{2}$ , and $\Lambda_{6}\leftrightarrow\Lambda_{2}+\Lambda_{7}$ , and fixes the nine other weights. The exceptional $\pi_{5}$ is closely related to the Galois permutation $\lambda\mapsto\lambda^{(13)}$ . </p>
<p data-bbox="70 618 541 650">Theorem 3.E8. The only nontrivial fusion-symmetries for ${E}_{8}^{(1)}$ are $\pi_{4}$ and $\pi_{5}$ , occurring at $k=4$ and 5 respectively. </p>
<h1 data-bbox="71 663 183 680">3.8. The algebra F 4(1) </h1>
<p data-bbox="70 686 541 715">A weight $\lambda$ in $P_{+}$ satisfies $k=\lambda_{0}+2\lambda_{1}+3\lambda_{2}+2\lambda_{3}+\lambda_{4}$ , and $\kappa=k+9$ . Again, the conjugations and simple-currents are trivial. </p>
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0002044v1
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[{"type": "text", "text": "At $k=3$ there is an order 3 Galois fusion-symmetry $\\pi_{3}=\\pi\\{5\\}$ , which sends $J^{i}\\Lambda_{1}\\mapsto$ $J^{i}(2\\Lambda_{6})\\mapsto J^{i}\\Lambda_{2}\\mapsto J^{i}\\Lambda_{1}$ and fixes the other six weights. ", "page_idx": 13}, {"type": "text", "text": "Theorem 3.E7. The only nontrivial fusion-symmetries for ${E}_{7}^{(1)}$ are $\\pi[1]$ at even $k$ , as well as $\\pi_{3}$ and its inverse at $k=3$ . ", "page_idx": 13}, {"type": "text", "text": "3.7. The algebra E8(1) ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+5\\lambda_{4}+6\\lambda_{5}+4\\lambda_{6}+2\\lambda_{7}+3\\lambda_{8}$ , and $\\kappa=k+30$ . The conjugations and simple-currents are all trivial, except for an anomolous simple-current at $k=2$ , sending $P_{+}=(0,\\Lambda_{1},\\Lambda_{7})$ to $(\\Lambda_{7},\\Lambda_{1},0)$ , which plays no role in this paper (except in Theorem 5.1). ", "page_idx": 13}, {"type": "text", "text": "The only fusion products we need can be derived from [28] and (2.4): ", "page_idx": 13}, {"type": "text", "text": "$(2\\Lambda_{1})\\vert\\mathrm{\\bf\\sfXI}(2\\Lambda_{1})=(0)_{4}$ + $(\\Lambda_{1})_{5}$ + $(\\Lambda_{2})_{5}$ + $(\\Lambda_{3})_{4}$ + $(\\Lambda_{7})_{4}$ + $2\\,\\mathbf{E}\\left(2\\Lambda_{1}\\right)_{46}$ + $(2\\Lambda_{2})_{6}$ $\\Lambda_{1}$ × $\\Lambda_{4}=(\\Lambda_{3})_{5}$ + $(\\Lambda_{4})_{6}$ + $(\\Lambda_{5})_{6}$ + $(\\Lambda_{6})_{5}$ + $(\\Lambda_{1}+\\Lambda_{3})_{6}$ + $(\\Lambda_{1}+\\Lambda_{4})_{7}$ + $(\\Lambda_{1}+\\Lambda_{6})_{6}$ $\\Lambda_{1}$ × $(\\Lambda_{1}\\!+\\!\\Lambda_{3})=(\\Lambda_{3})_{6}$ + $(\\Lambda_{4})_{6}$ + $(\\Lambda_{1}+\\Lambda_{2})_{6}$ + $2\\,\\mathtt{H}\\,(\\Lambda_{1}+\\Lambda_{3})_{67}$ + $(\\Lambda_{1}+\\Lambda_{4})_{7}$ $\\Lambda_{1}$ × $(2\\Lambda_{7})=\\!(\\Lambda_{6})_{4}$ + $(\\Lambda_{1}+\\Lambda_{7})_{4}$ + $(2\\Lambda_{7})_{5}$ + $(\\Lambda_{2}+\\Lambda_{7})_{5}$ + $(\\Lambda_{7}+\\Lambda_{8})_{5}$ + $(\\Lambda_{1}+2\\Lambda_{7})_{6}$ $(3.7g)$ ", "page_idx": 13}, {"type": "text", "text": "", "page_idx": 13}, {"type": "text", "text": "", "page_idx": 13}, {"type": "text", "text": "", "page_idx": 13}, {"type": "text", "text": "A fusion-symmetry at $k=4$ , called $\\pi_{4}$ , was first found in [15]. It interchanges $\\Lambda_{1}\\leftrightarrow\\Lambda_{6}$ and fixes the other eight weights in $P_{+}$ . There also is a fusion-symmetry, called $\\pi_{5}$ , at $k=5$ which interchanges $\\Lambda_{7}\\leftrightarrow2\\Lambda_{1}$ , $\\Lambda_{8}\\leftrightarrow\\Lambda_{1}+\\Lambda_{2}$ , and $\\Lambda_{6}\\leftrightarrow\\Lambda_{2}+\\Lambda_{7}$ , and fixes the nine other weights. The exceptional $\\pi_{5}$ is closely related to the Galois permutation $\\lambda\\mapsto\\lambda^{(13)}$ . ", "page_idx": 13}, {"type": "text", "text": "Theorem 3.E8. The only nontrivial fusion-symmetries for ${E}_{8}^{(1)}$ are $\\pi_{4}$ and $\\pi_{5}$ , occurring at $k=4$ and 5 respectively. ", "page_idx": 13}, {"type": "text", "text": "3.8. The algebra F 4(1) ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+2\\lambda_{3}+\\lambda_{4}$ , and $\\kappa=k+9$ . Again, the conjugations and simple-currents are trivial. ", "page_idx": 13}]
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{"preproc_blocks": [{"type": "text", "bbox": [71, 70, 541, 100], "lines": [{"bbox": [94, 73, 541, 90], "spans": [{"bbox": [94, 73, 111, 90], "score": 1.0, "content": "At", "type": "text"}, {"bbox": [112, 75, 141, 84], "score": 0.9, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [141, 73, 371, 90], "score": 1.0, "content": " there is an order 3 Galois fusion-symmetry ", "type": "text"}, {"bbox": [372, 73, 426, 87], "score": 0.93, "content": "\\pi_{3}=\\pi\\{5\\}", "type": "inline_equation", "height": 14, "width": 54}, {"bbox": [426, 73, 498, 90], "score": 1.0, "content": ", which sends ", "type": "text"}, {"bbox": [498, 73, 541, 86], "score": 0.89, "content": "J^{i}\\Lambda_{1}\\mapsto", "type": "inline_equation", "height": 13, "width": 43}], "index": 0}, {"bbox": [71, 87, 367, 103], "spans": [{"bbox": [71, 88, 200, 101], "score": 0.92, "content": "J^{i}(2\\Lambda_{6})\\mapsto J^{i}\\Lambda_{2}\\mapsto J^{i}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 129}, {"bbox": [200, 87, 367, 103], "score": 1.0, "content": " and fixes the other six weights.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [71, 107, 541, 137], "lines": [{"bbox": [92, 108, 540, 128], "spans": [{"bbox": [92, 109, 414, 128], "score": 1.0, "content": "Theorem 3.E7. The only nontrivial fusion-symmetries for ", "type": "text"}, {"bbox": [414, 108, 438, 126], "score": 0.9, "content": "{E}_{7}^{(1)}", "type": "inline_equation", "height": 18, "width": 24}, {"bbox": [438, 109, 461, 128], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [461, 110, 483, 126], "score": 0.71, "content": "\\pi[1]", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [483, 109, 528, 128], "score": 1.0, "content": " at even ", "type": "text"}, {"bbox": [529, 112, 536, 123], "score": 0.73, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [536, 109, 540, 128], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [72, 126, 271, 139], "spans": [{"bbox": [72, 126, 126, 139], "score": 1.0, "content": "as well as ", "type": "text"}, {"bbox": [126, 129, 139, 138], "score": 0.75, "content": "\\pi_{3}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 126, 237, 139], "score": 1.0, "content": " and its inverse at ", "type": "text"}, {"bbox": [238, 128, 267, 137], "score": 0.91, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [268, 126, 271, 139], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "title", "bbox": [70, 150, 183, 167], "lines": [{"bbox": [67, 148, 189, 174], "spans": [{"bbox": [67, 148, 189, 174], "score": 1.0, "content": "3.7. 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The conjugations and simple-currents are all trivial, except for an anomolous", "type": "text"}], "index": 6}, {"bbox": [70, 203, 541, 218], "spans": [{"bbox": [70, 205, 163, 218], "score": 1.0, "content": "simple-current at ", "type": "text"}, {"bbox": [163, 204, 193, 215], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [193, 205, 241, 218], "score": 1.0, "content": ", sending ", "type": "text"}, {"bbox": [241, 203, 327, 218], "score": 0.92, "content": "P_{+}=(0,\\Lambda_{1},\\Lambda_{7})", "type": "inline_equation", "height": 15, "width": 86}, {"bbox": [327, 205, 344, 218], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [344, 203, 397, 218], "score": 0.92, "content": "(\\Lambda_{7},\\Lambda_{1},0)", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [398, 205, 541, 218], "score": 1.0, "content": ", which plays no role in this", "type": "text"}], "index": 7}, {"bbox": [70, 219, 236, 232], "spans": [{"bbox": [70, 219, 236, 232], "score": 1.0, "content": "paper (except in Theorem 5.1).", "type": "text"}], "index": 8}], "index": 6.5}, {"type": "text", "bbox": [94, 230, 460, 245], "lines": [{"bbox": [95, 232, 458, 247], "spans": [{"bbox": [95, 232, 458, 247], "score": 1.0, "content": "The only fusion products we need can be derived from [28] and (2.4):", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [47, 364, 469, 381], "lines": [{"bbox": [48, 366, 468, 385], "spans": [{"bbox": [48, 366, 158, 382], "score": 0.73, "content": "(2\\Lambda_{1})\\vert\\mathrm{\\bf\\sfXI}(2\\Lambda_{1})=(0)_{4}", "type": "inline_equation", "height": 16, "width": 110}, {"bbox": [158, 368, 174, 385], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [174, 366, 204, 381], "score": 0.92, "content": "(\\Lambda_{1})_{5}", "type": "inline_equation", "height": 15, "width": 30}, {"bbox": [204, 368, 221, 385], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [221, 366, 250, 381], 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397, 474], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [397, 456, 455, 471], "score": 0.85, "content": "(\\Lambda_{1}+\\Lambda_{4})_{7}", "type": "inline_equation", "height": 15, "width": 58}], "index": 12}], "index": 12}, {"type": "text", "bbox": [59, 526, 530, 543], "lines": [{"bbox": [59, 528, 540, 545], "spans": [{"bbox": [59, 528, 74, 542], "score": 0.86, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [75, 530, 92, 545], "score": 1.0, "content": " × ", "type": "text"}, {"bbox": [92, 528, 162, 543], "score": 0.91, "content": "(2\\Lambda_{7})=\\!(\\Lambda_{6})_{4}", "type": "inline_equation", "height": 15, "width": 70}, {"bbox": [163, 530, 176, 545], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [177, 528, 233, 543], "score": 0.91, "content": "(\\Lambda_{1}+\\Lambda_{7})_{4}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [234, 530, 247, 545], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [248, 528, 282, 543], "score": 0.9, "content": "(2\\Lambda_{7})_{5}", "type": "inline_equation", "height": 15, "width": 34}, {"bbox": [282, 530, 296, 545], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [296, 528, 353, 543], "score": 0.88, "content": "(\\Lambda_{2}+\\Lambda_{7})_{5}", "type": "inline_equation", "height": 15, "width": 57}, {"bbox": [353, 530, 367, 545], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [367, 528, 424, 543], "score": 0.25, "content": "(\\Lambda_{7}+\\Lambda_{8})_{5}", "type": "inline_equation", "height": 15, "width": 57}, {"bbox": [424, 530, 437, 545], "score": 1.0, "content": " +", "type": "text"}, {"bbox": [438, 528, 500, 543], "score": 0.35, "content": "(\\Lambda_{1}+2\\Lambda_{7})_{6}", "type": "inline_equation", "height": 15, "width": 62}, {"bbox": [501, 530, 508, 545], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [509, 529, 540, 543], "score": 0.29, "content": "(3.7g)", "type": "inline_equation", "height": 14, "width": 31}], "index": 13}], "index": 13}, {"type": "text", "bbox": [70, 554, 542, 613], "lines": [{"bbox": [95, 556, 540, 571], "spans": [{"bbox": [95, 556, 209, 571], "score": 1.0, "content": "A fusion-symmetry at", "type": "text"}, {"bbox": [209, 556, 239, 568], "score": 0.9, "content": "k=4", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [240, 556, 277, 571], "score": 1.0, "content": ", called", "type": "text"}, {"bbox": [277, 558, 290, 569], "score": 0.85, "content": "\\pi_{4}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [291, 556, 493, 571], "score": 1.0, "content": ", was first found in [15]. It interchanges ", "type": "text"}, {"bbox": [493, 556, 540, 569], "score": 0.92, "content": "\\Lambda_{1}\\leftrightarrow\\Lambda_{6}", "type": "inline_equation", "height": 13, "width": 47}], "index": 14}, {"bbox": [71, 571, 540, 586], "spans": [{"bbox": [71, 571, 252, 586], "score": 1.0, "content": "and fixes the other eight weights in ", "type": "text"}, {"bbox": [252, 571, 268, 585], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [269, 571, 477, 586], "score": 1.0, "content": ". There also is a fusion-symmetry, called", "type": "text"}, {"bbox": [478, 572, 491, 584], "score": 0.85, "content": "\\pi_{5}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [491, 571, 510, 586], "score": 1.0, "content": ", at ", "type": "text"}, {"bbox": [510, 571, 540, 583], "score": 0.88, "content": "k=5", "type": "inline_equation", "height": 12, "width": 30}], "index": 15}, {"bbox": [72, 585, 541, 600], "spans": [{"bbox": [72, 586, 173, 600], "score": 1.0, "content": "which interchanges ", "type": "text"}, {"bbox": [173, 585, 226, 598], "score": 0.93, "content": "\\Lambda_{7}\\leftrightarrow2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [226, 586, 232, 600], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 585, 306, 598], "score": 0.91, "content": "\\Lambda_{8}\\leftrightarrow\\Lambda_{1}+\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 586, 335, 600], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [335, 585, 410, 598], "score": 0.93, "content": "\\Lambda_{6}\\leftrightarrow\\Lambda_{2}+\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [410, 586, 541, 600], "score": 1.0, "content": ", and fixes the nine other", "type": "text"}], "index": 16}, {"bbox": [69, 598, 508, 615], "spans": [{"bbox": [69, 599, 205, 615], "score": 1.0, "content": "weights. The exceptional ", "type": "text"}, {"bbox": [205, 602, 218, 613], "score": 0.88, "content": "\\pi_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [219, 599, 452, 615], "score": 1.0, "content": " is closely related to the Galois permutation ", "type": "text"}, {"bbox": [452, 598, 503, 611], "score": 0.91, "content": "\\lambda\\mapsto\\lambda^{(13)}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [503, 599, 508, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 15.5}, {"type": "text", "bbox": [70, 618, 541, 650], "lines": [{"bbox": [90, 619, 540, 641], "spans": [{"bbox": [90, 619, 415, 641], "score": 1.0, "content": "Theorem 3.E8. The only nontrivial fusion-symmetries for ", "type": "text"}, {"bbox": [415, 619, 438, 636], "score": 0.91, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 17, "width": 23}, {"bbox": [438, 619, 462, 641], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [462, 624, 475, 635], "score": 0.78, "content": "\\pi_{4}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [476, 619, 502, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [503, 624, 516, 635], "score": 0.79, "content": "\\pi_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [516, 619, 540, 641], "score": 1.0, "content": ", oc-", "type": "text"}], "index": 18}, {"bbox": [72, 637, 257, 653], "spans": [{"bbox": [72, 637, 127, 653], "score": 1.0, "content": "curring at ", "type": "text"}, {"bbox": [128, 639, 157, 648], "score": 0.85, "content": "k=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [157, 637, 257, 653], "score": 1.0, "content": " and 5 respectively.", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "title", "bbox": [71, 663, 183, 680], "lines": [{"bbox": [69, 663, 186, 683], "spans": [{"bbox": [69, 663, 186, 683], "score": 1.0, "content": "3.8. The algebra F 4(1)", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 686, 541, 715], "lines": [{"bbox": [94, 688, 541, 703], "spans": [{"bbox": [94, 688, 145, 703], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [146, 690, 153, 699], "score": 0.86, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [154, 688, 171, 703], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [171, 690, 186, 702], "score": 0.93, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [187, 688, 234, 703], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [234, 689, 396, 701], "score": 0.93, "content": "k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+2\\lambda_{3}+\\lambda_{4}", "type": "inline_equation", "height": 12, "width": 162}, {"bbox": [396, 688, 425, 703], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [425, 690, 476, 700], "score": 0.89, "content": "\\kappa=k+9", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [477, 688, 541, 703], "score": 1.0, "content": ". Again, the", "type": "text"}], "index": 21}, {"bbox": [72, 703, 303, 717], "spans": [{"bbox": [72, 703, 303, 717], "score": 1.0, "content": "conjugations and simple-currents are trivial.", "type": "text"}], "index": 22}], "index": 21.5}], "layout_bboxes": [], "page_idx": 13, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 742], "lines": [{"bbox": [298, 731, 313, 744], "spans": [{"bbox": [298, 731, 313, 744], "score": 1.0, "content": "14", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [71, 70, 541, 100], "lines": [{"bbox": [94, 73, 541, 90], "spans": [{"bbox": [94, 73, 111, 90], "score": 1.0, "content": "At", "type": "text"}, {"bbox": [112, 75, 141, 84], "score": 0.9, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [141, 73, 371, 90], "score": 1.0, "content": " there is an order 3 Galois fusion-symmetry ", "type": "text"}, {"bbox": [372, 73, 426, 87], "score": 0.93, "content": "\\pi_{3}=\\pi\\{5\\}", "type": "inline_equation", "height": 14, "width": 54}, {"bbox": [426, 73, 498, 90], "score": 1.0, "content": ", which sends ", "type": "text"}, {"bbox": [498, 73, 541, 86], "score": 0.89, "content": "J^{i}\\Lambda_{1}\\mapsto", "type": "inline_equation", "height": 13, "width": 43}], "index": 0}, {"bbox": [71, 87, 367, 103], "spans": [{"bbox": [71, 88, 200, 101], "score": 0.92, "content": "J^{i}(2\\Lambda_{6})\\mapsto J^{i}\\Lambda_{2}\\mapsto J^{i}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 129}, {"bbox": [200, 87, 367, 103], "score": 1.0, "content": " and fixes the other six weights.", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [71, 73, 541, 103]}, {"type": "text", "bbox": [71, 107, 541, 137], "lines": [{"bbox": [92, 108, 540, 128], "spans": [{"bbox": [92, 109, 414, 128], "score": 1.0, "content": "Theorem 3.E7. 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Again, the", "type": "text"}], "index": 21}, {"bbox": [72, 703, 303, 717], "spans": [{"bbox": [72, 703, 303, 717], "score": 1.0, "content": "conjugations and simple-currents are trivial.", "type": "text"}], "index": 22}], "index": 21.5, "bbox_fs": [72, 688, 541, 717]}]}
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[{"type": "text", "bbox": [71, 70, 541, 100], "content": "At there is an order 3 Galois fusion-symmetry , which sends and fixes the other six weights.", "index": 0}, {"type": "text", "bbox": [71, 107, 541, 137], "content": "Theorem 3.E7. The only nontrivial fusion-symmetries for are at even , as well as and its inverse at .", "index": 1}, {"type": "title", "bbox": [70, 150, 183, 167], "content": "3.7. The algebra E8(1)", "index": 2}, {"type": "text", "bbox": [70, 173, 542, 230], "content": "A weight in satisfies , and . The conjugations and simple-currents are all trivial, except for an anomolous simple-current at , sending to , which plays no role in this paper (except in Theorem 5.1).", "index": 3}, {"type": "text", "bbox": [94, 230, 460, 245], "content": "The only fusion products we need can be derived from [28] and (2.4):", "index": 4}, {"type": "text", "bbox": [47, 364, 469, 381], "content": "+ + + + + + × + + + + + + × + + + + × + + + + +", "index": 5}, {"type": "text", "bbox": [73, 418, 520, 435], "content": "", "index": 6}, {"type": "text", "bbox": [45, 454, 460, 470], "content": "", "index": 7}, {"type": "text", "bbox": [59, 526, 530, 543], "content": "", "index": 8}, {"type": "text", "bbox": [70, 554, 542, 613], "content": "A fusion-symmetry at , called , was first found in [15]. It interchanges and fixes the other eight weights in . There also is a fusion-symmetry, called , at which interchanges , , and , and fixes the nine other weights. The exceptional is closely related to the Galois permutation .", "index": 9}, {"type": "text", "bbox": [70, 618, 541, 650], "content": "Theorem 3.E8. The only nontrivial fusion-symmetries for are and , oc- curring at and 5 respectively.", "index": 10}, {"type": "title", "bbox": [71, 663, 183, 680], "content": "3.8. The algebra F 4(1)", "index": 11}, {"type": "text", "bbox": [70, 686, 541, 715], "content": "A weight in satisfies , and . Again, the conjugations and simple-currents are trivial.", "index": 12}]
|
[{"bbox": [94, 73, 541, 90], "content": "At there is an order 3 Galois fusion-symmetry , which sends", "parent_index": 0, "line_index": 0}, {"bbox": [71, 87, 367, 103], "content": "and fixes the other six weights.", "parent_index": 0, "line_index": 1}, {"bbox": [92, 108, 540, 128], "content": "Theorem 3.E7. The only nontrivial fusion-symmetries for are at even ,", "parent_index": 1, "line_index": 0}, {"bbox": [72, 126, 271, 139], "content": "as well as and its inverse at .", "parent_index": 1, "line_index": 1}, {"bbox": [67, 148, 189, 174], "content": "3.7. The algebra E8(1)", "parent_index": 2, "line_index": 0}, {"bbox": [95, 176, 541, 190], "content": "A weight in satisfies , and", "parent_index": 3, "line_index": 0}, {"bbox": [71, 190, 541, 205], "content": ". The conjugations and simple-currents are all trivial, except for an anomolous", "parent_index": 3, "line_index": 1}, {"bbox": [70, 203, 541, 218], "content": "simple-current at , sending to , which plays no role in this", "parent_index": 3, "line_index": 2}, {"bbox": [70, 219, 236, 232], "content": "paper (except in Theorem 5.1).", "parent_index": 3, "line_index": 3}, {"bbox": [95, 232, 458, 247], "content": "The only fusion products we need can be derived from [28] and (2.4):", "parent_index": 4, "line_index": 0}, {"bbox": [48, 366, 468, 385], "content": "+ + + + + +", "parent_index": 5, "line_index": 0}, {"bbox": [75, 420, 523, 438], "content": "× + + + + + +", "parent_index": 5, "line_index": 1}, {"bbox": [45, 456, 455, 474], "content": "× + + + +", "parent_index": 5, "line_index": 2}, {"bbox": [59, 528, 540, 545], "content": "× + + + + +", "parent_index": 5, "line_index": 3}, {"bbox": [95, 556, 540, 571], "content": "A fusion-symmetry at , called , was first found in [15]. It interchanges", "parent_index": 9, "line_index": 0}, {"bbox": [71, 571, 540, 586], "content": "and fixes the other eight weights in . There also is a fusion-symmetry, called , at", "parent_index": 9, "line_index": 1}, {"bbox": [72, 585, 541, 600], "content": "which interchanges , , and , and fixes the nine other", "parent_index": 9, "line_index": 2}, {"bbox": [69, 598, 508, 615], "content": "weights. The exceptional is closely related to the Galois permutation .", "parent_index": 9, "line_index": 3}, {"bbox": [90, 619, 540, 641], "content": "Theorem 3.E8. The only nontrivial fusion-symmetries for are and , oc-", "parent_index": 10, "line_index": 0}, {"bbox": [72, 637, 257, 653], "content": "curring at and 5 respectively.", "parent_index": 10, "line_index": 1}, {"bbox": [69, 663, 186, 683], "content": "3.8. The algebra F 4(1)", "parent_index": 11, "line_index": 0}, {"bbox": [94, 688, 541, 703], "content": "A weight in satisfies , and . Again, the", "parent_index": 12, "line_index": 0}, {"bbox": [72, 703, 303, 717], "content": "conjugations and simple-currents are trivial.", "parent_index": 12, "line_index": 1}]
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[]
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[{"bbox": [112, 75, 141, 84], "content": "k=3", "parent_index": 0, "subtype": "inline"}, {"bbox": [372, 73, 426, 87], "content": "\\pi_{3}=\\pi\\{5\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [498, 73, 541, 86], "content": "J^{i}\\Lambda_{1}\\mapsto", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 88, 200, 101], "content": "J^{i}(2\\Lambda_{6})\\mapsto J^{i}\\Lambda_{2}\\mapsto J^{i}\\Lambda_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [414, 108, 438, 126], "content": "{E}_{7}^{(1)}", "parent_index": 1, "subtype": "inline"}, {"bbox": [461, 110, 483, 126], "content": "\\pi[1]", "parent_index": 1, "subtype": "inline"}, {"bbox": [529, 112, 536, 123], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 129, 139, 138], "content": "\\pi_{3}", "parent_index": 1, "subtype": "inline"}, {"bbox": [238, 128, 267, 137], "content": "k=3", "parent_index": 1, "subtype": "inline"}, {"bbox": [144, 177, 153, 186], "content": "\\lambda", "parent_index": 3, "subtype": "inline"}, {"bbox": [168, 176, 185, 189], "content": "P_{+}", "parent_index": 3, "subtype": "inline"}, {"bbox": [231, 176, 514, 188], "content": "k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+4\\lambda_{3}+5\\lambda_{4}+6\\lambda_{5}+4\\lambda_{6}+2\\lambda_{7}+3\\lambda_{8}", "parent_index": 3, "subtype": "inline"}, {"bbox": [71, 190, 128, 202], "content": "\\kappa=k+30", "parent_index": 3, "subtype": "inline"}, {"bbox": [163, 204, 193, 215], "content": "k=2", "parent_index": 3, "subtype": "inline"}, {"bbox": [241, 203, 327, 218], "content": "P_{+}=(0,\\Lambda_{1},\\Lambda_{7})", "parent_index": 3, "subtype": "inline"}, {"bbox": [344, 203, 397, 218], "content": "(\\Lambda_{7},\\Lambda_{1},0)", "parent_index": 3, "subtype": "inline"}, {"bbox": [48, 366, 158, 382], "content": "(2\\Lambda_{1})\\vert\\mathrm{\\bf\\sfXI}(2\\Lambda_{1})=(0)_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [174, 366, 204, 381], "content": "(\\Lambda_{1})_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [221, 366, 250, 381], "content": "(\\Lambda_{2})_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [267, 366, 297, 381], "content": "(\\Lambda_{3})_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [314, 366, 343, 381], "content": "(\\Lambda_{7})_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [360, 366, 415, 381], "content": "2\\,\\mathbf{E}\\left(2\\Lambda_{1}\\right)_{46}", "parent_index": 5, "subtype": "inline"}, {"bbox": [433, 366, 468, 381], "content": "(2\\Lambda_{2})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [75, 420, 90, 434], "content": "\\Lambda_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [106, 420, 165, 435], "content": "\\Lambda_{4}=(\\Lambda_{3})_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [182, 420, 211, 435], "content": "(\\Lambda_{4})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [227, 420, 255, 435], "content": "(\\Lambda_{5})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [271, 420, 300, 435], "content": "(\\Lambda_{6})_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [316, 420, 373, 435], "content": "(\\Lambda_{1}+\\Lambda_{3})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [390, 420, 448, 435], "content": "(\\Lambda_{1}+\\Lambda_{4})_{7}", "parent_index": 5, "subtype": "inline"}, {"bbox": [465, 420, 523, 436], "content": "(\\Lambda_{1}+\\Lambda_{6})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [45, 456, 60, 470], "content": "\\Lambda_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [74, 456, 165, 471], "content": "(\\Lambda_{1}\\!+\\!\\Lambda_{3})=(\\Lambda_{3})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [182, 456, 212, 471], "content": "(\\Lambda_{4})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [228, 456, 286, 471], "content": "(\\Lambda_{1}+\\Lambda_{2})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [303, 456, 380, 471], "content": "2\\,\\mathtt{H}\\,(\\Lambda_{1}+\\Lambda_{3})_{67}", "parent_index": 5, "subtype": "inline"}, {"bbox": [397, 456, 455, 471], "content": "(\\Lambda_{1}+\\Lambda_{4})_{7}", "parent_index": 5, "subtype": "inline"}, {"bbox": [59, 528, 74, 542], "content": "\\Lambda_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [92, 528, 162, 543], "content": "(2\\Lambda_{7})=\\!(\\Lambda_{6})_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [177, 528, 233, 543], "content": "(\\Lambda_{1}+\\Lambda_{7})_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [248, 528, 282, 543], "content": "(2\\Lambda_{7})_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [296, 528, 353, 543], "content": "(\\Lambda_{2}+\\Lambda_{7})_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [367, 528, 424, 543], "content": "(\\Lambda_{7}+\\Lambda_{8})_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [438, 528, 500, 543], "content": "(\\Lambda_{1}+2\\Lambda_{7})_{6}", "parent_index": 5, "subtype": "inline"}, {"bbox": [509, 529, 540, 543], "content": "(3.7g)", "parent_index": 5, "subtype": "inline"}, {"bbox": [209, 556, 239, 568], "content": "k=4", "parent_index": 9, "subtype": "inline"}, {"bbox": [277, 558, 290, 569], "content": "\\pi_{4}", "parent_index": 9, "subtype": "inline"}, {"bbox": [493, 556, 540, 569], "content": "\\Lambda_{1}\\leftrightarrow\\Lambda_{6}", "parent_index": 9, "subtype": "inline"}, {"bbox": [252, 571, 268, 585], "content": "P_{+}", "parent_index": 9, "subtype": "inline"}, {"bbox": [478, 572, 491, 584], "content": "\\pi_{5}", "parent_index": 9, "subtype": "inline"}, {"bbox": [510, 571, 540, 583], "content": "k=5", "parent_index": 9, "subtype": "inline"}, {"bbox": [173, 585, 226, 598], "content": "\\Lambda_{7}\\leftrightarrow2\\Lambda_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [232, 585, 306, 598], "content": "\\Lambda_{8}\\leftrightarrow\\Lambda_{1}+\\Lambda_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [335, 585, 410, 598], "content": "\\Lambda_{6}\\leftrightarrow\\Lambda_{2}+\\Lambda_{7}", "parent_index": 9, "subtype": "inline"}, {"bbox": [205, 602, 218, 613], "content": "\\pi_{5}", "parent_index": 9, "subtype": "inline"}, {"bbox": [452, 598, 503, 611], "content": "\\lambda\\mapsto\\lambda^{(13)}", "parent_index": 9, "subtype": "inline"}, {"bbox": [415, 619, 438, 636], "content": "{E}_{8}^{(1)}", "parent_index": 10, "subtype": "inline"}, {"bbox": [462, 624, 475, 635], "content": "\\pi_{4}", "parent_index": 10, "subtype": "inline"}, {"bbox": [503, 624, 516, 635], "content": "\\pi_{5}", "parent_index": 10, "subtype": "inline"}, {"bbox": [128, 639, 157, 648], "content": "k=4", "parent_index": 10, "subtype": "inline"}, {"bbox": [146, 690, 153, 699], "content": "\\lambda", "parent_index": 12, "subtype": "inline"}, {"bbox": [171, 690, 186, 702], "content": "P_{+}", "parent_index": 12, "subtype": "inline"}, {"bbox": [234, 689, 396, 701], "content": "k=\\lambda_{0}+2\\lambda_{1}+3\\lambda_{2}+2\\lambda_{3}+\\lambda_{4}", "parent_index": 12, "subtype": "inline"}, {"bbox": [425, 690, 476, 700], "content": "\\kappa=k+9", "parent_index": 12, "subtype": "inline"}]
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There are Galois fusion-symmetries at levels $k=3$ and 4. In particular, for $k=3$ we have the fusion-symmetry $\pi_{3}=\pi\{5\}$ which interchanges both $\Lambda_{2}\leftrightarrow\Lambda_{4}$ and $\Lambda_{1}\leftrightarrow3\Lambda_{4}$ , and fixes the other five weights in $P_{+}$ . The exceptional $\pi_{3}$ was found independently in [34,14]. For $k\,=\,4$ we get a fusion-symmetry of order 4, which we will call $\pi_{4}$ . It fixes 0, $\Lambda_{2}+\Lambda_{4}$ , $\Lambda_{3}+\Lambda_{4}$ , and $2\Lambda_{4}$ , and permutes $\Lambda_{4}\;\mapsto\;\Lambda_{1}\;\mapsto\;2\Lambda_{1}\;\mapsto\;4\Lambda_{4}\;\mapsto\;\Lambda_{4}$ , $\Lambda_{2}\mapsto2\Lambda_{3}\mapsto3\Lambda_{4}\mapsto\Lambda_{3}\mapsto\Lambda_{2}$ , and $\Lambda_{1}\!+\!\Lambda_{3}\mapsto\Lambda_{3}\!+\!2\Lambda_{4}\mapsto\Lambda_{1}\!+\!\Lambda_{4}\mapsto\Lambda_{1}+2\Lambda_{4}\mapsto\Lambda_{1}\!+\!\Lambda_{3}$ . Its square $\pi_{4}^{2}$ equals the fusion-symmetry $\pi\{5\}$ .
The only fusion products we need can be obtained from [29] and (2.4):
$\Lambda_{4}$ × $\Lambda_{4}=(0)_{1}$ + $(\Lambda_{1})_{2}$ + $(\Lambda_{3})_{2}$ + $(\Lambda_{4})_{1}$ + $(2\Lambda_{4})_{2}$ $\Lambda_{1}$ × $\Lambda_{4}=(\Lambda_{3})_{2}$ + $(\Lambda_{4})_{2}$ + $(\Lambda_{1}+\Lambda_{4})_{3}$ $\Lambda_{3}$ × $\Lambda_{4}=(\Lambda_{1})_{2}$ + $(\Lambda_{2})_{3}$ + $(\Lambda_{3})_{2}$ + $(\Lambda_{4})_{2}$ + $(\Lambda_{1}+\Lambda_{4})_{3}$ + $(\Lambda_{3}+\Lambda_{4})_{3}$ + $(2\Lambda_{4})_{2}$ $(2\Lambda_{4})$ × $\Lambda_{4}=(\Lambda_{3})_{2}$ + $(\Lambda_{4})_{2}$ + $(2\Lambda_{4})_{2}$ + $(3\Lambda_{4})_{3}$ + $(\Lambda_{1}+\Lambda_{4})_{3}$ + $(\Lambda_{3}+\Lambda_{4})_{3}$
Theorem 3.F4. The only nontrivial fusion-symmetries of ${F}_{4}^{(1)}$ are $\pi_{3}$ at level 3, and $\pi_{4}^{i}$ for $1\leq i\leq3$ , which occur at level 4.
# 3.9. The algebra ${G_{2}^{(1)}}$
A weight $\lambda$ in $P_{+}$ satisfies $k=\lambda_{0}+2\lambda_{1}+\lambda_{2}$ , and $\kappa=k+4$ . The conjugations and simple-currents are all trivial.
Again there are nontrivial Galois fusion-symmetries. At $k=3$ , we have the order 3 fusion-symmetry $\pi_{3}=\pi\{4\}$ sending $\Lambda_{1}\mapsto3\Lambda_{2}\mapsto\Lambda_{2}\mapsto\Lambda_{1}$ , and fixing the remaining three weights. It was found in [14]. At $k=4$ , we have $\pi_{4}=\pi\{5\}$ permuting both $\Lambda_{1}\leftrightarrow4\Lambda_{2}$ and $2\Lambda_{1}\leftrightarrow\Lambda_{2}$ , and fixing the other five weights. It was found independently in [34,14], and in $\S5$ we will see that it is closely related to the $\pi_{3}$ of $F_{4,3}$ .
The only fusion products we will need can be obtained from [29] and (2.4):
$\Lambda_{2}$ × $\Lambda_{2}=(0)_{1}$ + $(\Lambda_{1})_{2}$ + $(\Lambda_{2})_{1}$ + $(2\Lambda_{2})_{2}$ $\Lambda_{2}$ × $\Lambda_{2}$ × $\Lambda_{2}=(0)_{1}$ + $2\,\pmb{\nabla}\,(\Lambda_{1})_{22}$ + $4\,\pmb{\mathrm{{E}}}\left(\Lambda_{2}\right)_{1122}$ + $3\,\pm\,(2\Lambda_{2})_{222}$ + $2\,\Xi\,(\Lambda_{1}+\Lambda_{2})_{33}$ + $(3\Lambda_{2})_{3}$
Theorem 3.G2. The only nontrivial fusion-symmetries for ${G_{2}^{(1)}}$ are $(\pi_{3})^{\pm1}$ at $k=3$ , and $\pi_{4}$ at $k=4$ .
# 4. The Arguments
The fundamental reason the classification of fusion-symmetries for the affine algebras is so accessible is (2.1b), which reduces the problem to studying Lie group characters at elements of finite order. These values have been studied by a number of people — see e.g. [22,28] — and the resulting combinatorics is often quite pretty.
Lemma 2.2 implies that a fusion-symmetry $\pi$ preserves q-dimensions: ${\mathcal{D}}(\lambda)={\mathcal{D}}(\pi\lambda)$ $\forall\lambda\in P_{+}$ . In this subsection we use that to find a weight $\Lambda_{\star}$ for each algebra which must be essentially fixed by $\pi$ .
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<p data-bbox="70 70 541 172">There are Galois fusion-symmetries at levels $k=3$ and 4. In particular, for $k=3$ we have the fusion-symmetry $\pi_{3}=\pi\{5\}$ which interchanges both $\Lambda_{2}\leftrightarrow\Lambda_{4}$ and $\Lambda_{1}\leftrightarrow3\Lambda_{4}$ , and fixes the other five weights in $P_{+}$ . The exceptional $\pi_{3}$ was found independently in [34,14]. For $k\,=\,4$ we get a fusion-symmetry of order 4, which we will call $\pi_{4}$ . It fixes 0, $\Lambda_{2}+\Lambda_{4}$ , $\Lambda_{3}+\Lambda_{4}$ , and $2\Lambda_{4}$ , and permutes $\Lambda_{4}\;\mapsto\;\Lambda_{1}\;\mapsto\;2\Lambda_{1}\;\mapsto\;4\Lambda_{4}\;\mapsto\;\Lambda_{4}$ , $\Lambda_{2}\mapsto2\Lambda_{3}\mapsto3\Lambda_{4}\mapsto\Lambda_{3}\mapsto\Lambda_{2}$ , and $\Lambda_{1}\!+\!\Lambda_{3}\mapsto\Lambda_{3}\!+\!2\Lambda_{4}\mapsto\Lambda_{1}\!+\!\Lambda_{4}\mapsto\Lambda_{1}+2\Lambda_{4}\mapsto\Lambda_{1}\!+\!\Lambda_{3}$ . Its square $\pi_{4}^{2}$ equals the fusion-symmetry $\pi\{5\}$ . </p>
<p data-bbox="95 172 467 186">The only fusion products we need can be obtained from [29] and (2.4): </p>
<p data-bbox="86 197 531 270">$\Lambda_{4}$ × $\Lambda_{4}=(0)_{1}$ + $(\Lambda_{1})_{2}$ + $(\Lambda_{3})_{2}$ + $(\Lambda_{4})_{1}$ + $(2\Lambda_{4})_{2}$ $\Lambda_{1}$ × $\Lambda_{4}=(\Lambda_{3})_{2}$ + $(\Lambda_{4})_{2}$ + $(\Lambda_{1}+\Lambda_{4})_{3}$ $\Lambda_{3}$ × $\Lambda_{4}=(\Lambda_{1})_{2}$ + $(\Lambda_{2})_{3}$ + $(\Lambda_{3})_{2}$ + $(\Lambda_{4})_{2}$ + $(\Lambda_{1}+\Lambda_{4})_{3}$ + $(\Lambda_{3}+\Lambda_{4})_{3}$ + $(2\Lambda_{4})_{2}$ $(2\Lambda_{4})$ × $\Lambda_{4}=(\Lambda_{3})_{2}$ + $(\Lambda_{4})_{2}$ + $(2\Lambda_{4})_{2}$ + $(3\Lambda_{4})_{3}$ + $(\Lambda_{1}+\Lambda_{4})_{3}$ + $(\Lambda_{3}+\Lambda_{4})_{3}$ </p>
<p data-bbox="71 282 542 312">Theorem 3.F4. The only nontrivial fusion-symmetries of ${F}_{4}^{(1)}$ are $\pi_{3}$ at level 3, and $\pi_{4}^{i}$ for $1\leq i\leq3$ , which occur at level 4. </p>
<h1 data-bbox="71 326 183 343">3.9. The algebra ${G_{2}^{(1)}}$ </h1>
<p data-bbox="70 350 541 378">A weight $\lambda$ in $P_{+}$ satisfies $k=\lambda_{0}+2\lambda_{1}+\lambda_{2}$ , and $\kappa=k+4$ . The conjugations and simple-currents are all trivial. </p>
<p data-bbox="70 379 541 451">Again there are nontrivial Galois fusion-symmetries. At $k=3$ , we have the order 3 fusion-symmetry $\pi_{3}=\pi\{4\}$ sending $\Lambda_{1}\mapsto3\Lambda_{2}\mapsto\Lambda_{2}\mapsto\Lambda_{1}$ , and fixing the remaining three weights. It was found in [14]. At $k=4$ , we have $\pi_{4}=\pi\{5\}$ permuting both $\Lambda_{1}\leftrightarrow4\Lambda_{2}$ and $2\Lambda_{1}\leftrightarrow\Lambda_{2}$ , and fixing the other five weights. It was found independently in [34,14], and in $\S5$ we will see that it is closely related to the $\pi_{3}$ of $F_{4,3}$ . </p>
<p data-bbox="95 451 490 466">The only fusion products we will need can be obtained from [29] and (2.4): </p>
<p data-bbox="66 477 545 513">$\Lambda_{2}$ × $\Lambda_{2}=(0)_{1}$ + $(\Lambda_{1})_{2}$ + $(\Lambda_{2})_{1}$ + $(2\Lambda_{2})_{2}$ $\Lambda_{2}$ × $\Lambda_{2}$ × $\Lambda_{2}=(0)_{1}$ + $2\,\pmb{\nabla}\,(\Lambda_{1})_{22}$ + $4\,\pmb{\mathrm{{E}}}\left(\Lambda_{2}\right)_{1122}$ + $3\,\pm\,(2\Lambda_{2})_{222}$ + $2\,\Xi\,(\Lambda_{1}+\Lambda_{2})_{33}$ + $(3\Lambda_{2})_{3}$ </p>
<p data-bbox="70 525 542 556">Theorem 3.G2. The only nontrivial fusion-symmetries for ${G_{2}^{(1)}}$ are $(\pi_{3})^{\pm1}$ at $k=3$ , and $\pi_{4}$ at $k=4$ . </p>
<h1 data-bbox="249 583 362 598">4. The Arguments </h1>
<p data-bbox="70 612 541 669">The fundamental reason the classification of fusion-symmetries for the affine algebras is so accessible is (2.1b), which reduces the problem to studying Lie group characters at elements of finite order. These values have been studied by a number of people — see e.g. [22,28] — and the resulting combinatorics is often quite pretty. </p>
<p data-bbox="69 670 540 713">Lemma 2.2 implies that a fusion-symmetry $\pi$ preserves q-dimensions: ${\mathcal{D}}(\lambda)={\mathcal{D}}(\pi\lambda)$ $\forall\lambda\in P_{+}$ . In this subsection we use that to find a weight $\Lambda_{\star}$ for each algebra which must be essentially fixed by $\pi$ . </p>
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0002044v1
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[{"type": "text", "text": "There are Galois fusion-symmetries at levels $k=3$ and 4. In particular, for $k=3$ we have the fusion-symmetry $\\pi_{3}=\\pi\\{5\\}$ which interchanges both $\\Lambda_{2}\\leftrightarrow\\Lambda_{4}$ and $\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}$ , and fixes the other five weights in $P_{+}$ . The exceptional $\\pi_{3}$ was found independently in [34,14]. For $k\\,=\\,4$ we get a fusion-symmetry of order 4, which we will call $\\pi_{4}$ . It fixes 0, $\\Lambda_{2}+\\Lambda_{4}$ , $\\Lambda_{3}+\\Lambda_{4}$ , and $2\\Lambda_{4}$ , and permutes $\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}$ , $\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}$ , and $\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}$ . Its square $\\pi_{4}^{2}$ equals the fusion-symmetry $\\pi\\{5\\}$ . ", "page_idx": 14}, {"type": "text", "text": "The only fusion products we need can be obtained from [29] and (2.4): ", "page_idx": 14}, {"type": "text", "text": "$\\Lambda_{4}$ × $\\Lambda_{4}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{1}$ + $(2\\Lambda_{4})_{2}$ $\\Lambda_{1}$ × $\\Lambda_{4}=(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ $\\Lambda_{3}$ × $\\Lambda_{4}=(\\Lambda_{1})_{2}$ + $(\\Lambda_{2})_{3}$ + $(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ + $(\\Lambda_{3}+\\Lambda_{4})_{3}$ + $(2\\Lambda_{4})_{2}$ $(2\\Lambda_{4})$ × $\\Lambda_{4}=(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(2\\Lambda_{4})_{2}$ + $(3\\Lambda_{4})_{3}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ + $(\\Lambda_{3}+\\Lambda_{4})_{3}$ ", "page_idx": 14}, {"type": "text", "text": "Theorem 3.F4. The only nontrivial fusion-symmetries of ${F}_{4}^{(1)}$ are $\\pi_{3}$ at level 3, and $\\pi_{4}^{i}$ for $1\\leq i\\leq3$ , which occur at level 4. ", "page_idx": 14}, {"type": "text", "text": "3.9. The algebra ${G_{2}^{(1)}}$ ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "A weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}$ , and $\\kappa=k+4$ . The conjugations and simple-currents are all trivial. ", "page_idx": 14}, {"type": "text", "text": "Again there are nontrivial Galois fusion-symmetries. At $k=3$ , we have the order 3 fusion-symmetry $\\pi_{3}=\\pi\\{4\\}$ sending $\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}$ , and fixing the remaining three weights. It was found in [14]. At $k=4$ , we have $\\pi_{4}=\\pi\\{5\\}$ permuting both $\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}$ and $2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}$ , and fixing the other five weights. It was found independently in [34,14], and in $\\S5$ we will see that it is closely related to the $\\pi_{3}$ of $F_{4,3}$ . ", "page_idx": 14}, {"type": "text", "text": "The only fusion products we will need can be obtained from [29] and (2.4): ", "page_idx": 14}, {"type": "text", "text": "$\\Lambda_{2}$ × $\\Lambda_{2}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{2})_{1}$ + $(2\\Lambda_{2})_{2}$ $\\Lambda_{2}$ × $\\Lambda_{2}$ × $\\Lambda_{2}=(0)_{1}$ + $2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}$ + $4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}$ + $3\\,\\pm\\,(2\\Lambda_{2})_{222}$ + $2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}$ + $(3\\Lambda_{2})_{3}$ ", "page_idx": 14}, {"type": "text", "text": "Theorem 3.G2. The only nontrivial fusion-symmetries for ${G_{2}^{(1)}}$ are $(\\pi_{3})^{\\pm1}$ at $k=3$ , and $\\pi_{4}$ at $k=4$ . ", "page_idx": 14}, {"type": "text", "text": "4. The Arguments ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "The fundamental reason the classification of fusion-symmetries for the affine algebras is so accessible is (2.1b), which reduces the problem to studying Lie group characters at elements of finite order. These values have been studied by a number of people — see e.g. [22,28] — and the resulting combinatorics is often quite pretty. ", "page_idx": 14}, {"type": "text", "text": "Lemma 2.2 implies that a fusion-symmetry $\\pi$ preserves q-dimensions: ${\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)$ $\\forall\\lambda\\in P_{+}$ . In this subsection we use that to find a weight $\\Lambda_{\\star}$ for each algebra which must be essentially fixed by $\\pi$ . ", "page_idx": 14}]
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In particular, for ", "type": "text"}, {"bbox": [493, 75, 522, 84], "score": 0.91, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [522, 73, 541, 88], "score": 1.0, "content": " we", "type": "text"}], "index": 0}, {"bbox": [70, 87, 540, 104], "spans": [{"bbox": [70, 87, 212, 104], "score": 1.0, "content": "have the fusion-symmetry ", "type": "text"}, {"bbox": [212, 89, 268, 101], "score": 0.94, "content": "\\pi_{3}=\\pi\\{5\\}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [268, 87, 405, 104], "score": 1.0, "content": " which interchanges both ", "type": "text"}, {"bbox": [405, 90, 453, 100], "score": 0.92, "content": "\\Lambda_{2}\\leftrightarrow\\Lambda_{4}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [454, 87, 481, 104], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [482, 90, 536, 101], "score": 0.91, "content": "\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [537, 87, 540, 104], "score": 1.0, "content": ",", "type": "text"}], "index": 1}, {"bbox": [70, 101, 540, 118], "spans": [{"bbox": [70, 101, 265, 118], "score": 1.0, "content": "and fixes the other five weights in ", "type": "text"}, {"bbox": [265, 104, 281, 116], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [282, 101, 387, 118], "score": 1.0, "content": ". The exceptional ", "type": "text"}, {"bbox": [387, 108, 399, 115], "score": 0.88, "content": "\\pi_{3}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [399, 101, 540, 118], "score": 1.0, "content": " was found independently", "type": "text"}], "index": 2}, {"bbox": [69, 116, 542, 132], "spans": [{"bbox": [69, 116, 156, 132], "score": 1.0, "content": "in [34,14]. For ", "type": "text"}, {"bbox": [156, 118, 191, 128], "score": 0.87, "content": "k\\,=\\,4", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [191, 116, 505, 132], "score": 1.0, "content": " we get a fusion-symmetry of order 4, which we will call ", "type": "text"}, {"bbox": [505, 122, 517, 129], "score": 0.79, "content": "\\pi_{4}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [518, 116, 542, 132], "score": 1.0, "content": ". It", "type": "text"}], "index": 3}, {"bbox": [69, 130, 541, 146], "spans": [{"bbox": [69, 130, 114, 146], "score": 1.0, "content": "fixes 0, ", "type": "text"}, {"bbox": [115, 131, 160, 144], "score": 0.87, "content": "\\Lambda_{2}+\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [161, 130, 168, 146], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [169, 131, 214, 144], "score": 0.87, "content": "\\Lambda_{3}+\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [215, 130, 248, 146], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [248, 131, 269, 144], "score": 0.88, "content": "2\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [269, 130, 356, 146], "score": 1.0, "content": ", and permutes ", "type": "text"}, {"bbox": [356, 132, 536, 144], "score": 0.92, "content": "\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}", "type": "inline_equation", "height": 12, "width": 180}, {"bbox": [537, 130, 541, 146], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [71, 145, 540, 160], "spans": [{"bbox": [71, 145, 226, 158], "score": 0.92, "content": "\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 155}, {"bbox": [227, 145, 256, 160], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [256, 146, 536, 158], "score": 0.81, "content": "\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}", "type": "inline_equation", "height": 12, "width": 280}, {"bbox": [537, 145, 540, 160], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [70, 159, 321, 174], "spans": [{"bbox": [70, 159, 126, 174], "score": 1.0, "content": "Its square ", "type": "text"}, {"bbox": [126, 159, 140, 173], "score": 0.89, "content": "\\pi_{4}^{2}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [140, 159, 290, 174], "score": 1.0, "content": " equals the fusion-symmetry ", "type": "text"}, {"bbox": [290, 159, 317, 173], "score": 0.92, "content": "\\pi\\{5\\}", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [317, 159, 321, 174], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [95, 172, 467, 186], "lines": [{"bbox": [95, 173, 466, 189], "spans": [{"bbox": [95, 173, 466, 189], "score": 1.0, "content": "The only fusion products we need can be obtained from [29] and (2.4):", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [86, 197, 531, 270], "lines": [{"bbox": [97, 200, 371, 217], "spans": [{"bbox": [97, 201, 112, 215], "score": 0.85, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [113, 200, 128, 217], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [128, 200, 179, 216], "score": 0.92, "content": "\\Lambda_{4}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [180, 200, 196, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [196, 201, 226, 216], "score": 0.9, "content": "(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 15, "width": 30}, {"bbox": [226, 200, 243, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [243, 200, 272, 216], "score": 0.9, "content": "(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [273, 200, 289, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [290, 200, 318, 216], "score": 0.85, "content": "(\\Lambda_{4})_{1}", "type": "inline_equation", "height": 16, "width": 28}, {"bbox": [319, 200, 336, 217], "score": 1.0, "content": " 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59}], "index": 9}, {"bbox": [97, 236, 529, 253], "spans": [{"bbox": [97, 237, 112, 251], "score": 0.88, "content": "\\Lambda_{3}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [113, 237, 128, 253], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [128, 236, 187, 252], "score": 0.93, "content": "\\Lambda_{4}=(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 16, "width": 59}, {"bbox": [187, 237, 204, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [204, 236, 234, 252], "score": 0.92, "content": "(\\Lambda_{2})_{3}", "type": "inline_equation", "height": 16, "width": 30}, {"bbox": [234, 237, 250, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [251, 236, 280, 252], "score": 0.92, "content": "(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [280, 237, 297, 253], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [297, 236, 326, 252], "score": 0.92, "content": "(\\Lambda_{4})_{2}", 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"width": 58}, {"bbox": [414, 254, 430, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [431, 254, 488, 270], "score": 0.85, "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 57}], "index": 11}], "index": 9.5}, {"type": "text", "bbox": [71, 282, 542, 312], "lines": [{"bbox": [90, 281, 542, 306], "spans": [{"bbox": [90, 281, 403, 306], "score": 1.0, "content": "Theorem 3.F4. The only nontrivial fusion-symmetries of ", "type": "text"}, {"bbox": [403, 282, 427, 300], "score": 0.92, "content": "{F}_{4}^{(1)}", "type": "inline_equation", "height": 18, "width": 24}, {"bbox": [427, 281, 449, 306], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [449, 287, 463, 299], "score": 0.83, "content": "\\pi_{3}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [464, 281, 542, 306], "score": 1.0, "content": " at level 3, and", "type": "text"}], "index": 12}, {"bbox": [70, 299, 281, 315], "spans": [{"bbox": [70, 299, 84, 314], "score": 0.9, "content": "\\pi_{4}^{i}", "type": "inline_equation", "height": 15, "width": 14}, {"bbox": [84, 300, 105, 315], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [106, 300, 156, 313], "score": 0.88, "content": "1\\leq i\\leq3", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [156, 300, 281, 315], "score": 1.0, "content": ", which occur at level 4.", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "title", "bbox": [71, 326, 183, 343], "lines": [{"bbox": [67, 327, 183, 348], "spans": [{"bbox": [67, 328, 160, 348], "score": 1.0, "content": "3.9. The algebra ", "type": "text"}, {"bbox": [161, 327, 183, 345], "score": 0.91, "content": "{G_{2}^{(1)}}", "type": "inline_equation", "height": 18, "width": 22}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 350, 541, 378], "lines": [{"bbox": [93, 352, 542, 368], "spans": [{"bbox": [93, 352, 146, 368], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [146, 353, 154, 363], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [155, 352, 172, 368], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [172, 353, 188, 366], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [188, 352, 235, 368], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [236, 352, 333, 365], "score": 0.92, "content": "k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [334, 352, 364, 368], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [364, 354, 416, 364], "score": 0.9, "content": "\\kappa=k+4", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [416, 352, 542, 368], "score": 1.0, "content": ". The conjugations and", "type": "text"}], "index": 15}, {"bbox": [71, 367, 228, 381], "spans": [{"bbox": [71, 367, 228, 381], "score": 1.0, "content": "simple-currents are all trivial.", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "text", "bbox": [70, 379, 541, 451], "lines": [{"bbox": [95, 381, 541, 396], "spans": [{"bbox": [95, 381, 396, 396], "score": 1.0, "content": "Again there are nontrivial Galois fusion-symmetries. At ", "type": "text"}, {"bbox": [396, 383, 427, 392], "score": 0.89, "content": "k=3", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [427, 381, 541, 396], "score": 1.0, "content": ", we have the order 3", "type": "text"}], "index": 17}, {"bbox": [69, 394, 541, 411], "spans": [{"bbox": [69, 394, 160, 411], "score": 1.0, "content": "fusion-symmetry", "type": "text"}, {"bbox": [161, 396, 215, 409], "score": 0.92, "content": "\\pi_{3}=\\pi\\{4\\}", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [216, 394, 259, 411], "score": 1.0, "content": " sending", "type": "text"}, {"bbox": [260, 397, 377, 408], "score": 0.92, "content": "\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 117}, {"bbox": [378, 394, 541, 411], "score": 1.0, "content": ", and fixing the remaining three", "type": "text"}], "index": 18}, {"bbox": [71, 409, 540, 425], "spans": [{"bbox": [71, 409, 252, 425], "score": 1.0, "content": "weights. It was found in [14]. At ", "type": "text"}, {"bbox": [252, 410, 284, 421], "score": 0.87, "content": "k=4", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 409, 338, 425], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [338, 411, 394, 423], "score": 0.93, "content": "\\pi_{4}=\\pi\\{5\\}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [394, 409, 485, 425], "score": 1.0, "content": " permuting both ", "type": "text"}, {"bbox": [485, 412, 540, 423], "score": 0.91, "content": "\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 55}], "index": 19}, {"bbox": [70, 423, 541, 439], "spans": [{"bbox": [70, 423, 94, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [94, 424, 149, 437], "score": 0.92, "content": "2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 55}, {"bbox": [150, 423, 541, 439], "score": 1.0, "content": ", and fixing the other five weights. It was found independently in [34,14],", "type": "text"}], "index": 20}, {"bbox": [70, 437, 401, 455], "spans": [{"bbox": [70, 437, 108, 455], "score": 1.0, "content": "and in ", "type": "text"}, {"bbox": [108, 438, 120, 452], "score": 0.39, "content": "\\S5", "type": "inline_equation", "height": 14, "width": 12}, {"bbox": [121, 437, 345, 455], "score": 1.0, "content": " we will see that it is closely related to the", "type": "text"}, {"bbox": [345, 440, 359, 451], "score": 0.84, "content": "\\pi_{3}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [360, 437, 375, 455], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [376, 438, 397, 453], "score": 0.91, "content": "F_{4,3}", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [398, 437, 401, 455], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [95, 451, 490, 466], "lines": [{"bbox": [95, 452, 489, 468], "spans": [{"bbox": [95, 452, 489, 468], "score": 1.0, "content": "The only fusion products we will need can be obtained from [29] and (2.4):", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [66, 477, 545, 513], "lines": [{"bbox": [98, 479, 326, 497], "spans": [{"bbox": [98, 479, 114, 493], "score": 0.87, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [114, 479, 130, 497], "score": 1.0, "content": " × ", "type": "text"}, {"bbox": [130, 479, 181, 495], "score": 0.93, "content": "\\Lambda_{2}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [181, 479, 198, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [198, 479, 227, 495], "score": 0.91, "content": "(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [228, 479, 244, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [244, 479, 273, 495], "score": 0.91, "content": "(\\Lambda_{2})_{1}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [274, 479, 291, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [291, 479, 326, 495], "score": 0.89, "content": "(2\\Lambda_{2})_{2}", "type": "inline_equation", "height": 16, "width": 35}], "index": 23}, {"bbox": [66, 497, 544, 516], "spans": [{"bbox": [66, 497, 82, 511], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [82, 497, 97, 516], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [98, 498, 114, 511], "score": 0.9, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [114, 497, 129, 516], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [130, 497, 180, 513], "score": 0.94, "content": "\\Lambda_{2}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 50}, {"bbox": [181, 497, 197, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [197, 497, 246, 513], "score": 0.91, "content": "2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}", "type": "inline_equation", "height": 16, "width": 49}, {"bbox": [246, 497, 262, 516], "score": 1.0, "content": " +", "type": "text"}, {"bbox": [263, 497, 321, 513], "score": 0.87, "content": "4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}", "type": "inline_equation", "height": 16, "width": 58}, {"bbox": [322, 497, 338, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [338, 497, 398, 513], "score": 0.89, "content": "3\\,\\pm\\,(2\\Lambda_{2})_{222}", "type": "inline_equation", "height": 16, "width": 60}, {"bbox": [398, 497, 414, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [414, 497, 491, 513], "score": 0.89, "content": "2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}", "type": "inline_equation", "height": 16, "width": 77}, {"bbox": [492, 497, 509, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [509, 497, 544, 513], "score": 0.89, "content": "(3\\Lambda_{2})_{3}", "type": "inline_equation", "height": 16, "width": 35}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [70, 525, 542, 556], "lines": [{"bbox": [91, 522, 543, 548], "spans": [{"bbox": [91, 522, 408, 548], "score": 1.0, "content": "Theorem 3.G2. The only nontrivial fusion-symmetries for", "type": "text"}, {"bbox": [409, 525, 432, 543], "score": 0.92, "content": "{G_{2}^{(1)}}", "type": "inline_equation", "height": 18, "width": 23}, {"bbox": [433, 522, 455, 548], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [455, 527, 490, 543], "score": 0.91, "content": "(\\pi_{3})^{\\pm1}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [491, 522, 506, 548], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [506, 528, 537, 541], "score": 0.87, "content": "k=3", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [537, 522, 543, 548], "score": 1.0, "content": ",", "type": "text"}], "index": 25}, {"bbox": [72, 543, 160, 557], "spans": [{"bbox": [72, 544, 93, 557], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [93, 544, 107, 556], "score": 0.84, "content": "\\pi_{4}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [108, 544, 124, 557], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [124, 543, 155, 555], "score": 0.89, "content": "k=4", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [155, 544, 160, 557], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "title", "bbox": [249, 583, 362, 598], "lines": [{"bbox": [249, 585, 362, 599], "spans": [{"bbox": [249, 585, 362, 599], "score": 1.0, "content": "4. The Arguments", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [70, 612, 541, 669], "lines": [{"bbox": [95, 614, 540, 628], "spans": [{"bbox": [95, 614, 540, 628], "score": 1.0, "content": "The fundamental reason the classification of fusion-symmetries for the affine algebras", "type": "text"}], "index": 28}, {"bbox": [69, 628, 541, 642], "spans": [{"bbox": [69, 628, 541, 642], "score": 1.0, "content": "is so accessible is (2.1b), which reduces the problem to studying Lie group characters at", "type": "text"}], "index": 29}, {"bbox": [70, 641, 541, 659], "spans": [{"bbox": [70, 641, 541, 659], "score": 1.0, "content": "elements of finite order. These values have been studied by a number of people — see e.g.", "type": "text"}], "index": 30}, {"bbox": [71, 656, 401, 673], "spans": [{"bbox": [71, 656, 401, 673], "score": 1.0, "content": "[22,28] — and the resulting combinatorics is often quite pretty.", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [69, 670, 540, 713], "lines": [{"bbox": [93, 671, 540, 686], "spans": [{"bbox": [93, 671, 325, 686], "score": 1.0, "content": "Lemma 2.2 implies that a fusion-symmetry ", "type": "text"}, {"bbox": [325, 677, 332, 682], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [333, 671, 464, 686], "score": 1.0, "content": " preserves q-dimensions: ", "type": "text"}, {"bbox": [464, 673, 540, 685], "score": 0.94, "content": "{\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)", "type": "inline_equation", "height": 12, "width": 76}], "index": 32}, {"bbox": [70, 686, 540, 701], "spans": [{"bbox": [70, 688, 116, 699], "score": 0.92, "content": "\\forall\\lambda\\in P_{+}", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [116, 686, 373, 701], "score": 1.0, "content": ". In this subsection we use that to find a weight ", "type": "text"}, {"bbox": [374, 688, 387, 698], "score": 0.92, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [388, 686, 540, 701], "score": 1.0, "content": "for each algebra which must", "type": "text"}], "index": 33}, {"bbox": [70, 700, 203, 715], "spans": [{"bbox": [70, 700, 190, 715], "score": 1.0, "content": "be essentially fixed by ", "type": "text"}, {"bbox": [190, 705, 198, 711], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [198, 700, 203, 715], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 33}], "layout_bboxes": [], "page_idx": 14, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "15", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 172], "lines": [{"bbox": [94, 73, 541, 88], "spans": [{"bbox": [94, 73, 330, 88], "score": 1.0, "content": "There are Galois fusion-symmetries at levels ", "type": "text"}, {"bbox": [331, 75, 360, 84], "score": 0.91, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [360, 73, 492, 88], "score": 1.0, "content": " and 4. In particular, for ", "type": "text"}, {"bbox": [493, 75, 522, 84], "score": 0.91, "content": "k=3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [522, 73, 541, 88], "score": 1.0, "content": " we", "type": "text"}], "index": 0}, {"bbox": [70, 87, 540, 104], "spans": [{"bbox": [70, 87, 212, 104], "score": 1.0, "content": "have the fusion-symmetry ", "type": "text"}, {"bbox": [212, 89, 268, 101], "score": 0.94, "content": "\\pi_{3}=\\pi\\{5\\}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [268, 87, 405, 104], "score": 1.0, "content": " which interchanges both ", "type": "text"}, {"bbox": [405, 90, 453, 100], "score": 0.92, "content": "\\Lambda_{2}\\leftrightarrow\\Lambda_{4}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [454, 87, 481, 104], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [482, 90, 536, 101], "score": 0.91, "content": "\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [537, 87, 540, 104], "score": 1.0, "content": ",", "type": "text"}], "index": 1}, {"bbox": [70, 101, 540, 118], "spans": [{"bbox": [70, 101, 265, 118], "score": 1.0, "content": "and fixes the other five weights in ", "type": "text"}, {"bbox": [265, 104, 281, 116], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [282, 101, 387, 118], "score": 1.0, "content": ". The exceptional ", "type": "text"}, {"bbox": [387, 108, 399, 115], "score": 0.88, "content": "\\pi_{3}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [399, 101, 540, 118], "score": 1.0, "content": " was found independently", "type": "text"}], "index": 2}, {"bbox": [69, 116, 542, 132], "spans": [{"bbox": [69, 116, 156, 132], "score": 1.0, "content": "in [34,14]. For ", "type": "text"}, {"bbox": [156, 118, 191, 128], "score": 0.87, "content": "k\\,=\\,4", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [191, 116, 505, 132], "score": 1.0, "content": " we get a fusion-symmetry of order 4, which we will call ", "type": "text"}, {"bbox": [505, 122, 517, 129], "score": 0.79, "content": "\\pi_{4}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [518, 116, 542, 132], "score": 1.0, "content": ". It", "type": "text"}], "index": 3}, {"bbox": [69, 130, 541, 146], "spans": [{"bbox": [69, 130, 114, 146], "score": 1.0, "content": "fixes 0, ", "type": "text"}, {"bbox": [115, 131, 160, 144], "score": 0.87, "content": "\\Lambda_{2}+\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [161, 130, 168, 146], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [169, 131, 214, 144], "score": 0.87, "content": "\\Lambda_{3}+\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [215, 130, 248, 146], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [248, 131, 269, 144], "score": 0.88, "content": "2\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [269, 130, 356, 146], "score": 1.0, "content": ", and permutes ", "type": "text"}, {"bbox": [356, 132, 536, 144], "score": 0.92, "content": "\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}", "type": "inline_equation", "height": 12, "width": 180}, {"bbox": [537, 130, 541, 146], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [71, 145, 540, 160], "spans": [{"bbox": [71, 145, 226, 158], "score": 0.92, "content": "\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 155}, {"bbox": [227, 145, 256, 160], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [256, 146, 536, 158], "score": 0.81, "content": "\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}", "type": "inline_equation", "height": 12, "width": 280}, {"bbox": [537, 145, 540, 160], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [70, 159, 321, 174], "spans": [{"bbox": [70, 159, 126, 174], "score": 1.0, "content": "Its square ", "type": "text"}, {"bbox": [126, 159, 140, 173], "score": 0.89, "content": "\\pi_{4}^{2}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [140, 159, 290, 174], "score": 1.0, "content": " equals the fusion-symmetry ", "type": "text"}, {"bbox": [290, 159, 317, 173], "score": 0.92, "content": "\\pi\\{5\\}", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [317, 159, 321, 174], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 3, "bbox_fs": [69, 73, 542, 174]}, {"type": "text", "bbox": [95, 172, 467, 186], "lines": [{"bbox": [95, 173, 466, 189], "spans": [{"bbox": [95, 173, 466, 189], "score": 1.0, "content": "The only fusion products we need can be obtained from [29] and (2.4):", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [95, 173, 466, 189]}, {"type": "text", "bbox": [86, 197, 531, 270], "lines": [{"bbox": [97, 200, 371, 217], "spans": [{"bbox": [97, 201, 112, 215], "score": 0.85, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [113, 200, 128, 217], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [128, 200, 179, 216], "score": 0.92, "content": "\\Lambda_{4}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [180, 200, 196, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [196, 201, 226, 216], "score": 0.9, "content": "(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 15, "width": 30}, {"bbox": [226, 200, 243, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [243, 200, 272, 216], "score": 0.9, "content": "(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [273, 200, 289, 217], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [290, 200, 318, 216], "score": 0.85, "content": "(\\Lambda_{4})_{1}", "type": "inline_equation", "height": 16, "width": 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"content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 58}, {"bbox": [414, 254, 430, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [431, 254, 488, 270], "score": 0.85, "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 57}], "index": 11}], "index": 9.5, "bbox_fs": [82, 200, 529, 271]}, {"type": "text", "bbox": [71, 282, 542, 312], "lines": [{"bbox": [90, 281, 542, 306], "spans": [{"bbox": [90, 281, 403, 306], "score": 1.0, "content": "Theorem 3.F4. The only nontrivial fusion-symmetries of ", "type": "text"}, {"bbox": [403, 282, 427, 300], "score": 0.92, "content": "{F}_{4}^{(1)}", "type": "inline_equation", "height": 18, "width": 24}, {"bbox": [427, 281, 449, 306], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [449, 287, 463, 299], "score": 0.83, "content": "\\pi_{3}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [464, 281, 542, 306], "score": 1.0, "content": " at level 3, and", "type": "text"}], "index": 12}, {"bbox": [70, 299, 281, 315], "spans": [{"bbox": [70, 299, 84, 314], "score": 0.9, "content": "\\pi_{4}^{i}", "type": "inline_equation", "height": 15, "width": 14}, {"bbox": [84, 300, 105, 315], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [106, 300, 156, 313], "score": 0.88, "content": "1\\leq i\\leq3", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [156, 300, 281, 315], "score": 1.0, "content": ", which occur at level 4.", "type": "text"}], "index": 13}], "index": 12.5, "bbox_fs": [70, 281, 542, 315]}, {"type": "title", "bbox": [71, 326, 183, 343], "lines": [{"bbox": [67, 327, 183, 348], "spans": [{"bbox": [67, 328, 160, 348], "score": 1.0, "content": "3.9. The algebra ", "type": "text"}, {"bbox": [161, 327, 183, 345], "score": 0.91, "content": "{G_{2}^{(1)}}", "type": "inline_equation", "height": 18, "width": 22}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 350, 541, 378], "lines": [{"bbox": [93, 352, 542, 368], "spans": [{"bbox": [93, 352, 146, 368], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [146, 353, 154, 363], "score": 0.8, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [155, 352, 172, 368], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [172, 353, 188, 366], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [188, 352, 235, 368], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [236, 352, 333, 365], "score": 0.92, "content": "k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [334, 352, 364, 368], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [364, 354, 416, 364], "score": 0.9, "content": "\\kappa=k+4", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [416, 352, 542, 368], "score": 1.0, "content": ". The conjugations and", "type": "text"}], "index": 15}, {"bbox": [71, 367, 228, 381], "spans": [{"bbox": [71, 367, 228, 381], "score": 1.0, "content": "simple-currents are all trivial.", "type": "text"}], "index": 16}], "index": 15.5, "bbox_fs": [71, 352, 542, 381]}, {"type": "text", "bbox": [70, 379, 541, 451], "lines": [{"bbox": [95, 381, 541, 396], "spans": [{"bbox": [95, 381, 396, 396], "score": 1.0, "content": "Again there are nontrivial Galois fusion-symmetries. At ", "type": "text"}, {"bbox": [396, 383, 427, 392], "score": 0.89, "content": "k=3", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [427, 381, 541, 396], "score": 1.0, "content": ", we have the order 3", "type": "text"}], "index": 17}, {"bbox": [69, 394, 541, 411], "spans": [{"bbox": [69, 394, 160, 411], "score": 1.0, "content": "fusion-symmetry", "type": "text"}, {"bbox": [161, 396, 215, 409], "score": 0.92, "content": "\\pi_{3}=\\pi\\{4\\}", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [216, 394, 259, 411], "score": 1.0, "content": " sending", "type": "text"}, {"bbox": [260, 397, 377, 408], "score": 0.92, "content": "\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 117}, {"bbox": [378, 394, 541, 411], "score": 1.0, "content": ", and fixing the remaining three", "type": "text"}], "index": 18}, {"bbox": [71, 409, 540, 425], "spans": [{"bbox": [71, 409, 252, 425], "score": 1.0, "content": "weights. It was found in [14]. At ", "type": "text"}, {"bbox": [252, 410, 284, 421], "score": 0.87, "content": "k=4", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 409, 338, 425], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [338, 411, 394, 423], "score": 0.93, "content": "\\pi_{4}=\\pi\\{5\\}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [394, 409, 485, 425], "score": 1.0, "content": " permuting both ", "type": "text"}, {"bbox": [485, 412, 540, 423], "score": 0.91, "content": "\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 55}], "index": 19}, {"bbox": [70, 423, 541, 439], "spans": [{"bbox": [70, 423, 94, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [94, 424, 149, 437], "score": 0.92, "content": "2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 55}, {"bbox": [150, 423, 541, 439], "score": 1.0, "content": ", and fixing the other five weights. It was found independently in [34,14],", "type": "text"}], "index": 20}, {"bbox": [70, 437, 401, 455], "spans": [{"bbox": [70, 437, 108, 455], "score": 1.0, "content": "and in ", "type": "text"}, {"bbox": [108, 438, 120, 452], "score": 0.39, "content": "\\S5", "type": "inline_equation", "height": 14, "width": 12}, {"bbox": [121, 437, 345, 455], "score": 1.0, "content": " we will see that it is closely related to the", "type": "text"}, {"bbox": [345, 440, 359, 451], "score": 0.84, "content": "\\pi_{3}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [360, 437, 375, 455], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [376, 438, 397, 453], "score": 0.91, "content": "F_{4,3}", "type": "inline_equation", "height": 15, "width": 21}, {"bbox": [398, 437, 401, 455], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19, "bbox_fs": [69, 381, 541, 455]}, {"type": "text", "bbox": [95, 451, 490, 466], "lines": [{"bbox": [95, 452, 489, 468], "spans": [{"bbox": [95, 452, 489, 468], "score": 1.0, "content": "The only fusion products we will need can be obtained from [29] and (2.4):", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [95, 452, 489, 468]}, {"type": "text", "bbox": [66, 477, 545, 513], "lines": [{"bbox": [98, 479, 326, 497], "spans": [{"bbox": [98, 479, 114, 493], "score": 0.87, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [114, 479, 130, 497], "score": 1.0, "content": " × ", "type": "text"}, {"bbox": [130, 479, 181, 495], "score": 0.93, "content": "\\Lambda_{2}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [181, 479, 198, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [198, 479, 227, 495], "score": 0.91, "content": "(\\Lambda_{1})_{2}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [228, 479, 244, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [244, 479, 273, 495], "score": 0.91, "content": "(\\Lambda_{2})_{1}", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [274, 479, 291, 497], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [291, 479, 326, 495], "score": 0.89, "content": "(2\\Lambda_{2})_{2}", "type": "inline_equation", "height": 16, "width": 35}], "index": 23}, {"bbox": [66, 497, 544, 516], "spans": [{"bbox": [66, 497, 82, 511], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [82, 497, 97, 516], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [98, 498, 114, 511], "score": 0.9, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [114, 497, 129, 516], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [130, 497, 180, 513], "score": 0.94, "content": "\\Lambda_{2}=(0)_{1}", "type": "inline_equation", "height": 16, "width": 50}, {"bbox": [181, 497, 197, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [197, 497, 246, 513], "score": 0.91, "content": "2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}", "type": "inline_equation", "height": 16, "width": 49}, {"bbox": [246, 497, 262, 516], "score": 1.0, "content": " +", "type": "text"}, {"bbox": [263, 497, 321, 513], "score": 0.87, "content": "4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}", "type": "inline_equation", "height": 16, "width": 58}, {"bbox": [322, 497, 338, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [338, 497, 398, 513], "score": 0.89, "content": "3\\,\\pm\\,(2\\Lambda_{2})_{222}", "type": "inline_equation", "height": 16, "width": 60}, {"bbox": [398, 497, 414, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [414, 497, 491, 513], "score": 0.89, "content": "2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}", "type": "inline_equation", "height": 16, "width": 77}, {"bbox": [492, 497, 509, 516], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [509, 497, 544, 513], "score": 0.89, "content": "(3\\Lambda_{2})_{3}", "type": "inline_equation", "height": 16, "width": 35}], "index": 24}], "index": 23.5, "bbox_fs": [66, 479, 544, 516]}, {"type": "text", "bbox": [70, 525, 542, 556], "lines": [{"bbox": [91, 522, 543, 548], "spans": [{"bbox": [91, 522, 408, 548], "score": 1.0, "content": "Theorem 3.G2. The only nontrivial fusion-symmetries for", "type": "text"}, {"bbox": [409, 525, 432, 543], "score": 0.92, "content": "{G_{2}^{(1)}}", "type": "inline_equation", "height": 18, "width": 23}, {"bbox": [433, 522, 455, 548], "score": 1.0, "content": "are ", "type": "text"}, {"bbox": [455, 527, 490, 543], "score": 0.91, "content": "(\\pi_{3})^{\\pm1}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [491, 522, 506, 548], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [506, 528, 537, 541], "score": 0.87, "content": "k=3", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [537, 522, 543, 548], "score": 1.0, "content": ",", "type": "text"}], "index": 25}, {"bbox": [72, 543, 160, 557], "spans": [{"bbox": [72, 544, 93, 557], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [93, 544, 107, 556], "score": 0.84, "content": "\\pi_{4}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [108, 544, 124, 557], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [124, 543, 155, 555], "score": 0.89, "content": "k=4", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [155, 544, 160, 557], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [72, 522, 543, 557]}, {"type": "title", "bbox": [249, 583, 362, 598], "lines": [{"bbox": [249, 585, 362, 599], "spans": [{"bbox": [249, 585, 362, 599], "score": 1.0, "content": "4. The Arguments", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [70, 612, 541, 669], "lines": [{"bbox": [95, 614, 540, 628], "spans": [{"bbox": [95, 614, 540, 628], "score": 1.0, "content": "The fundamental reason the classification of fusion-symmetries for the affine algebras", "type": "text"}], "index": 28}, {"bbox": [69, 628, 541, 642], "spans": [{"bbox": [69, 628, 541, 642], "score": 1.0, "content": "is so accessible is (2.1b), which reduces the problem to studying Lie group characters at", "type": "text"}], "index": 29}, {"bbox": [70, 641, 541, 659], "spans": [{"bbox": [70, 641, 541, 659], "score": 1.0, "content": "elements of finite order. These values have been studied by a number of people — see e.g.", "type": "text"}], "index": 30}, {"bbox": [71, 656, 401, 673], "spans": [{"bbox": [71, 656, 401, 673], "score": 1.0, "content": "[22,28] — and the resulting combinatorics is often quite pretty.", "type": "text"}], "index": 31}], "index": 29.5, "bbox_fs": [69, 614, 541, 673]}, {"type": "text", "bbox": [69, 670, 540, 713], "lines": [{"bbox": [93, 671, 540, 686], "spans": [{"bbox": [93, 671, 325, 686], "score": 1.0, "content": "Lemma 2.2 implies that a fusion-symmetry ", "type": "text"}, {"bbox": [325, 677, 332, 682], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [333, 671, 464, 686], "score": 1.0, "content": " preserves q-dimensions: ", "type": "text"}, {"bbox": [464, 673, 540, 685], "score": 0.94, "content": "{\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)", "type": "inline_equation", "height": 12, "width": 76}], "index": 32}, {"bbox": [70, 686, 540, 701], "spans": [{"bbox": [70, 688, 116, 699], "score": 0.92, "content": "\\forall\\lambda\\in P_{+}", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [116, 686, 373, 701], "score": 1.0, "content": ". In this subsection we use that to find a weight ", "type": "text"}, {"bbox": [374, 688, 387, 698], "score": 0.92, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [388, 686, 540, 701], "score": 1.0, "content": "for each algebra which must", "type": "text"}], "index": 33}, {"bbox": [70, 700, 203, 715], "spans": [{"bbox": [70, 700, 190, 715], "score": 1.0, "content": "be essentially fixed by ", "type": "text"}, {"bbox": [190, 705, 198, 711], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [198, 700, 203, 715], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 33, "bbox_fs": [70, 671, 540, 715]}]}
|
[{"type": "text", "bbox": [70, 70, 541, 172], "content": "There are Galois fusion-symmetries at levels and 4. In particular, for we have the fusion-symmetry which interchanges both and , and fixes the other five weights in . The exceptional was found independently in [34,14]. For we get a fusion-symmetry of order 4, which we will call . It fixes 0, , , and , and permutes , , and . Its square equals the fusion-symmetry .", "index": 0}, {"type": "text", "bbox": [95, 172, 467, 186], "content": "The only fusion products we need can be obtained from [29] and (2.4):", "index": 1}, {"type": "text", "bbox": [86, 197, 531, 270], "content": "× + + + + × + + × + + + + + + × + + + + +", "index": 2}, {"type": "text", "bbox": [71, 282, 542, 312], "content": "Theorem 3.F4. The only nontrivial fusion-symmetries of are at level 3, and for , which occur at level 4.", "index": 3}, {"type": "title", "bbox": [71, 326, 183, 343], "content": "3.9. The algebra", "index": 4}, {"type": "text", "bbox": [70, 350, 541, 378], "content": "A weight in satisfies , and . The conjugations and simple-currents are all trivial.", "index": 5}, {"type": "text", "bbox": [70, 379, 541, 451], "content": "Again there are nontrivial Galois fusion-symmetries. At , we have the order 3 fusion-symmetry sending , and fixing the remaining three weights. It was found in [14]. At , we have permuting both and , and fixing the other five weights. It was found independently in [34,14], and in we will see that it is closely related to the of .", "index": 6}, {"type": "text", "bbox": [95, 451, 490, 466], "content": "The only fusion products we will need can be obtained from [29] and (2.4):", "index": 7}, {"type": "text", "bbox": [66, 477, 545, 513], "content": "× + + + × × + + + + +", "index": 8}, {"type": "text", "bbox": [70, 525, 542, 556], "content": "Theorem 3.G2. The only nontrivial fusion-symmetries for are at , and at .", "index": 9}, {"type": "title", "bbox": [249, 583, 362, 598], "content": "4. The Arguments", "index": 10}, {"type": "text", "bbox": [70, 612, 541, 669], "content": "The fundamental reason the classification of fusion-symmetries for the affine algebras is so accessible is (2.1b), which reduces the problem to studying Lie group characters at elements of finite order. These values have been studied by a number of people — see e.g. [22,28] — and the resulting combinatorics is often quite pretty.", "index": 11}, {"type": "text", "bbox": [69, 670, 540, 713], "content": "Lemma 2.2 implies that a fusion-symmetry preserves q-dimensions: . In this subsection we use that to find a weight for each algebra which must be essentially fixed by .", "index": 12}]
|
[{"bbox": [94, 73, 541, 88], "content": "There are Galois fusion-symmetries at levels and 4. In particular, for we", "parent_index": 0, "line_index": 0}, {"bbox": [70, 87, 540, 104], "content": "have the fusion-symmetry which interchanges both and ,", "parent_index": 0, "line_index": 1}, {"bbox": [70, 101, 540, 118], "content": "and fixes the other five weights in . The exceptional was found independently", "parent_index": 0, "line_index": 2}, {"bbox": [69, 116, 542, 132], "content": "in [34,14]. For we get a fusion-symmetry of order 4, which we will call . It", "parent_index": 0, "line_index": 3}, {"bbox": [69, 130, 541, 146], "content": "fixes 0, , , and , and permutes ,", "parent_index": 0, "line_index": 4}, {"bbox": [71, 145, 540, 160], "content": ", and .", "parent_index": 0, "line_index": 5}, {"bbox": [70, 159, 321, 174], "content": "Its square equals the fusion-symmetry .", "parent_index": 0, "line_index": 6}, {"bbox": [95, 173, 466, 189], "content": "The only fusion products we need can be obtained from [29] and (2.4):", "parent_index": 1, "line_index": 0}, {"bbox": [97, 200, 371, 217], "content": "× + + + +", "parent_index": 2, "line_index": 0}, {"bbox": [97, 219, 309, 235], "content": "× + +", "parent_index": 2, "line_index": 1}, {"bbox": [97, 236, 529, 253], "content": "× + + + + + +", "parent_index": 2, "line_index": 2}, {"bbox": [82, 254, 488, 271], "content": "× + + + + +", "parent_index": 2, "line_index": 3}, {"bbox": [90, 281, 542, 306], "content": "Theorem 3.F4. The only nontrivial fusion-symmetries of are at level 3, and", "parent_index": 3, "line_index": 0}, {"bbox": [70, 299, 281, 315], "content": "for , which occur at level 4.", "parent_index": 3, "line_index": 1}, {"bbox": [67, 327, 183, 348], "content": "3.9. The algebra", "parent_index": 4, "line_index": 0}, {"bbox": [93, 352, 542, 368], "content": "A weight in satisfies , and . The conjugations and", "parent_index": 5, "line_index": 0}, {"bbox": [71, 367, 228, 381], "content": "simple-currents are all trivial.", "parent_index": 5, "line_index": 1}, {"bbox": [95, 381, 541, 396], "content": "Again there are nontrivial Galois fusion-symmetries. At , we have the order 3", "parent_index": 6, "line_index": 0}, {"bbox": [69, 394, 541, 411], "content": "fusion-symmetry sending , and fixing the remaining three", "parent_index": 6, "line_index": 1}, {"bbox": [71, 409, 540, 425], "content": "weights. It was found in [14]. At , we have permuting both", "parent_index": 6, "line_index": 2}, {"bbox": [70, 423, 541, 439], "content": "and , and fixing the other five weights. It was found independently in [34,14],", "parent_index": 6, "line_index": 3}, {"bbox": [70, 437, 401, 455], "content": "and in we will see that it is closely related to the of .", "parent_index": 6, "line_index": 4}, {"bbox": [95, 452, 489, 468], "content": "The only fusion products we will need can be obtained from [29] and (2.4):", "parent_index": 7, "line_index": 0}, {"bbox": [98, 479, 326, 497], "content": "× + + +", "parent_index": 8, "line_index": 0}, {"bbox": [66, 497, 544, 516], "content": "× × + + + + +", "parent_index": 8, "line_index": 1}, {"bbox": [91, 522, 543, 548], "content": "Theorem 3.G2. The only nontrivial fusion-symmetries for are at ,", "parent_index": 9, "line_index": 0}, {"bbox": [72, 543, 160, 557], "content": "and at .", "parent_index": 9, "line_index": 1}, {"bbox": [249, 585, 362, 599], "content": "4. The Arguments", "parent_index": 10, "line_index": 0}, {"bbox": [95, 614, 540, 628], "content": "The fundamental reason the classification of fusion-symmetries for the affine algebras", "parent_index": 11, "line_index": 0}, {"bbox": [69, 628, 541, 642], "content": "is so accessible is (2.1b), which reduces the problem to studying Lie group characters at", "parent_index": 11, "line_index": 1}, {"bbox": [70, 641, 541, 659], "content": "elements of finite order. These values have been studied by a number of people — see e.g.", "parent_index": 11, "line_index": 2}, {"bbox": [71, 656, 401, 673], "content": "[22,28] — and the resulting combinatorics is often quite pretty.", "parent_index": 11, "line_index": 3}, {"bbox": [93, 671, 540, 686], "content": "Lemma 2.2 implies that a fusion-symmetry preserves q-dimensions:", "parent_index": 12, "line_index": 0}, {"bbox": [70, 686, 540, 701], "content": ". In this subsection we use that to find a weight for each algebra which must", "parent_index": 12, "line_index": 1}, {"bbox": [70, 700, 203, 715], "content": "be essentially fixed by .", "parent_index": 12, "line_index": 2}]
|
[]
|
[{"bbox": [331, 75, 360, 84], "content": "k=3", "parent_index": 0, "subtype": "inline"}, {"bbox": [493, 75, 522, 84], "content": "k=3", "parent_index": 0, "subtype": "inline"}, {"bbox": [212, 89, 268, 101], "content": "\\pi_{3}=\\pi\\{5\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [405, 90, 453, 100], "content": "\\Lambda_{2}\\leftrightarrow\\Lambda_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [482, 90, 536, 101], "content": "\\Lambda_{1}\\leftrightarrow3\\Lambda_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [265, 104, 281, 116], "content": "P_{+}", "parent_index": 0, "subtype": "inline"}, {"bbox": [387, 108, 399, 115], "content": "\\pi_{3}", "parent_index": 0, "subtype": "inline"}, {"bbox": [156, 118, 191, 128], "content": "k\\,=\\,4", "parent_index": 0, "subtype": "inline"}, {"bbox": [505, 122, 517, 129], "content": "\\pi_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [115, 131, 160, 144], "content": "\\Lambda_{2}+\\Lambda_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [169, 131, 214, 144], "content": "\\Lambda_{3}+\\Lambda_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [248, 131, 269, 144], "content": "2\\Lambda_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [356, 132, 536, 144], "content": "\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{1}\\;\\mapsto\\;2\\Lambda_{1}\\;\\mapsto\\;4\\Lambda_{4}\\;\\mapsto\\;\\Lambda_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 145, 226, 158], "content": "\\Lambda_{2}\\mapsto2\\Lambda_{3}\\mapsto3\\Lambda_{4}\\mapsto\\Lambda_{3}\\mapsto\\Lambda_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [256, 146, 536, 158], "content": "\\Lambda_{1}\\!+\\!\\Lambda_{3}\\mapsto\\Lambda_{3}\\!+\\!2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{4}\\mapsto\\Lambda_{1}+2\\Lambda_{4}\\mapsto\\Lambda_{1}\\!+\\!\\Lambda_{3}", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 159, 140, 173], "content": "\\pi_{4}^{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [290, 159, 317, 173], "content": "\\pi\\{5\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [97, 201, 112, 215], "content": "\\Lambda_{4}", "parent_index": 2, "subtype": "inline"}, {"bbox": [128, 200, 179, 216], "content": "\\Lambda_{4}=(0)_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [196, 201, 226, 216], "content": "(\\Lambda_{1})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [243, 200, 272, 216], "content": "(\\Lambda_{3})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [290, 200, 318, 216], "content": "(\\Lambda_{4})_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [336, 201, 371, 216], "content": "(2\\Lambda_{4})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [97, 219, 112, 233], "content": "\\Lambda_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [129, 219, 187, 234], "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [204, 219, 234, 234], "content": "(\\Lambda_{4})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [250, 219, 309, 234], "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [97, 237, 112, 251], "content": "\\Lambda_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [128, 236, 187, 252], "content": "\\Lambda_{4}=(\\Lambda_{1})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [204, 236, 234, 252], "content": "(\\Lambda_{2})_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [251, 236, 280, 252], "content": "(\\Lambda_{3})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [297, 236, 326, 252], "content": "(\\Lambda_{4})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [344, 236, 402, 252], "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [419, 236, 476, 252], "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [493, 236, 529, 252], "content": "(2\\Lambda_{4})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [82, 254, 111, 270], "content": "(2\\Lambda_{4})", "parent_index": 2, "subtype": "inline"}, {"bbox": [128, 254, 187, 270], "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [204, 254, 234, 270], "content": "(\\Lambda_{4})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [251, 254, 286, 270], "content": "(2\\Lambda_{4})_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [304, 254, 339, 270], "content": "(3\\Lambda_{4})_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [356, 254, 414, 270], "content": "(\\Lambda_{1}+\\Lambda_{4})_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [431, 254, 488, 270], "content": "(\\Lambda_{3}+\\Lambda_{4})_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [403, 282, 427, 300], "content": "{F}_{4}^{(1)}", "parent_index": 3, "subtype": "inline"}, {"bbox": [449, 287, 463, 299], "content": "\\pi_{3}", "parent_index": 3, "subtype": "inline"}, {"bbox": [70, 299, 84, 314], "content": "\\pi_{4}^{i}", "parent_index": 3, "subtype": "inline"}, {"bbox": [106, 300, 156, 313], "content": "1\\leq i\\leq3", "parent_index": 3, "subtype": "inline"}, {"bbox": [161, 327, 183, 345], "content": "{G_{2}^{(1)}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [146, 353, 154, 363], "content": "\\lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [172, 353, 188, 366], "content": "P_{+}", "parent_index": 5, "subtype": "inline"}, {"bbox": [236, 352, 333, 365], "content": "k=\\lambda_{0}+2\\lambda_{1}+\\lambda_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [364, 354, 416, 364], "content": "\\kappa=k+4", "parent_index": 5, "subtype": "inline"}, {"bbox": [396, 383, 427, 392], "content": "k=3", "parent_index": 6, "subtype": "inline"}, {"bbox": [161, 396, 215, 409], "content": "\\pi_{3}=\\pi\\{4\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [260, 397, 377, 408], "content": "\\Lambda_{1}\\mapsto3\\Lambda_{2}\\mapsto\\Lambda_{2}\\mapsto\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [252, 410, 284, 421], "content": "k=4", "parent_index": 6, "subtype": "inline"}, {"bbox": [338, 411, 394, 423], "content": "\\pi_{4}=\\pi\\{5\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [485, 412, 540, 423], "content": "\\Lambda_{1}\\leftrightarrow4\\Lambda_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [94, 424, 149, 437], "content": "2\\Lambda_{1}\\leftrightarrow\\Lambda_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [108, 438, 120, 452], "content": "\\S5", "parent_index": 6, "subtype": "inline"}, {"bbox": [345, 440, 359, 451], "content": "\\pi_{3}", "parent_index": 6, "subtype": "inline"}, {"bbox": [376, 438, 397, 453], "content": "F_{4,3}", "parent_index": 6, "subtype": "inline"}, {"bbox": [98, 479, 114, 493], "content": "\\Lambda_{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [130, 479, 181, 495], "content": 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"3\\,\\pm\\,(2\\Lambda_{2})_{222}", "parent_index": 8, "subtype": "inline"}, {"bbox": [414, 497, 491, 513], "content": "2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}", "parent_index": 8, "subtype": "inline"}, {"bbox": [509, 497, 544, 513], "content": "(3\\Lambda_{2})_{3}", "parent_index": 8, "subtype": "inline"}, {"bbox": [409, 525, 432, 543], "content": "{G_{2}^{(1)}}", "parent_index": 9, "subtype": "inline"}, {"bbox": [455, 527, 490, 543], "content": "(\\pi_{3})^{\\pm1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [506, 528, 537, 541], "content": "k=3", "parent_index": 9, "subtype": "inline"}, {"bbox": [93, 544, 107, 556], "content": "\\pi_{4}", "parent_index": 9, "subtype": "inline"}, {"bbox": [124, 543, 155, 555], "content": "k=4", "parent_index": 9, "subtype": "inline"}, {"bbox": [325, 677, 332, 682], "content": "\\pi", "parent_index": 12, "subtype": "inline"}, {"bbox": [464, 673, 540, 685], "content": "{\\mathcal{D}}(\\lambda)={\\mathcal{D}}(\\pi\\lambda)", "parent_index": 12, "subtype": "inline"}, {"bbox": [70, 688, 116, 699], "content": "\\forall\\lambda\\in P_{+}", "parent_index": 12, "subtype": "inline"}, {"bbox": [374, 688, 387, 698], "content": "\\Lambda_{\\star}", "parent_index": 12, "subtype": "inline"}, {"bbox": [190, 705, 198, 711], "content": "\\pi", "parent_index": 12, "subtype": "inline"}]
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[]
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4.1. $q$ -dimensions
The most basic properties obeyed by the q-dimensions $\begin{array}{r}{\mathcal{D}(\lambda)=\frac{S_{\lambda0}}{S_{00}}}\end{array}$ are that $\mathcal{D}(\lambda)\geq1$ , and $\mathcal{D}(s\lambda)=\mathcal{D}(\lambda)$ for any $s\in S$ . Recall that $\boldsymbol{S}$ is the symmetry group of the extended Dynkin diagram of $X_{r}^{(1)}$ , and that $s\in S$ acts on $P_{+}$ by permuting the Dynkin labels.
The argument yielding Proposition 4.1 below relies heavily on the following observation. Use (2.1c) to extend the domain of $\mathcal{D}$ from $P_{+}$ to the fundamental chamber $C_{+}$ :
$$
C_{+}\stackrel{\mathrm{def}}{=}\{\sum_{i=0}^{r}x_{i}\Lambda_{i}\left|\right.x_{i}\in\mathbb{R},\ x_{i}>-1,\ \sum_{i=0}^{r}x_{i}a_{i}^{\vee}=k\}\ .
$$
Choose any $a,b\in C_{+}$ . Then a straightforward calculation from (2.1c) gives
$$
\frac{d}{d t}\mathcal{D}(t a+(1-t)b)=0\quad\Longrightarrow\quad\frac{d^{2}}{d t^{2}}\mathcal{D}(t a+(1-t)b)<0
$$
for $0<t<1$ . This means that for all $0<t<1$ ,
$$
{\mathcal{D}}(t a+(1-t)b)>\operatorname*{min}\{{\mathcal{D}}(a),\,{\mathcal{D}}(b)\}~.
$$
Proposition 4.1 [17,18]. For the following algebras $X_{r}^{(1)}$ and levels $k$ , and choices of weight $\Lambda_{\star}$ , $\mathcal{D}(\lambda)=\mathcal{D}(\Lambda_{\star})$ implies $\lambda\in{\mathcal{S}}\Lambda_{\star}$ :
(a) For A(r1) any level $k$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (b) For Br(1) any level $k\neq2$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (c) For Cr(1) any level $k$ (except for $(r,k)=(2,3)$ or $(3,2).$ ), where $\Lambda_{\star}=\Lambda_{1}$ ; (d) For $D_{r}^{(1)}$ any level $k\neq2$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (e6) For E6(1) any level $k$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (e7) For ${E}_{7}^{(1)}$ any level $k\neq3$ , where $\Lambda_{\star}=\Lambda_{6}$ ; (e8) For E8(1) any level $k\neq1,4$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (f4) For ${F}_{4}^{(1)}$ any level $k\neq3,4$ , where $\Lambda_{\star}=\Lambda_{4}$ ; (g2) For G(21) level any $k\neq3,4$ , where $\Lambda_{\star}=\Lambda_{2}$ .
The missing cases are: $B_{r,2}$ where ${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{2})=\cdots={\mathcal{D}}(\Lambda_{r-1})={\mathcal{D}}(2\Lambda_{r});$ $D_{r,2}$ where ${\mathcal{D}}(\Lambda_{1})=\cdots={\mathcal{D}}(\Lambda_{r-2})$ ;
$C_{2,3}$ where ${\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(3\Lambda_{1})={\mathcal{D}}(\Lambda_{1})$ , and its rank-level dual $C_{3,2}$ ;
$E_{7,3}$ where ${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(\Lambda_{6})$ ;
$E_{8,1}$ where $\Lambda_{1}\notin P_{+}=\{0\}$ , and $E_{8,4}$ where ${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{6})$ ;
$F_{4,3}$ where ${\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(\Lambda_{4})$ , and $F_{4,4}$ where $\mathcal{D}(\Lambda_{1})=\mathcal{D}(2\Lambda_{1})=\mathcal{D}(4\Lambda_{4})=\mathcal{D}(\Lambda_{4})$ ;
$G_{2,3}$ where $\mathcal{D}(\Lambda_{1})=\mathcal{D}(\Lambda_{2})=\mathcal{D}(3\Lambda_{2})$ , and $G_{2,4}$ where $\mathcal{D}(\Lambda_{2})=\mathcal{D}(2\Lambda_{1})$ .
The weight $\Lambda_{\star}$ singled out by Proposition 4.1 (i.e. $\Lambda_{\star}=\Lambda_{1}$ for $A_{r}^{(1)}$ , ..., $\Lambda_{\star}=\Lambda_{2}$ for $G_{2}^{(1)})$ is the nonzero weight with smallest Weyl dimension. What we find is that, for all but the smallest levels (see [18, Table 3]), $\Lambda_{\star}$ will also have the smallest q-dimension after the simple-currents.
|
<html><body>
<p data-bbox="71 71 167 86">4.1. $q$ -dimensions </p>
<p data-bbox="70 92 541 138">The most basic properties obeyed by the q-dimensions $\begin{array}{r}{\mathcal{D}(\lambda)=\frac{S_{\lambda0}}{S_{00}}}\end{array}$ are that $\mathcal{D}(\lambda)\geq1$ , and $\mathcal{D}(s\lambda)=\mathcal{D}(\lambda)$ for any $s\in S$ . Recall that $\boldsymbol{S}$ is the symmetry group of the extended Dynkin diagram of $X_{r}^{(1)}$ , and that $s\in S$ acts on $P_{+}$ by permuting the Dynkin labels. </p>
<p data-bbox="70 138 541 167">The argument yielding Proposition 4.1 below relies heavily on the following observation. Use (2.1c) to extend the domain of $\mathcal{D}$ from $P_{+}$ to the fundamental chamber $C_{+}$ : </p>
<div class="equation" data-bbox="173 181 438 218">$$
C_{+}\stackrel{\mathrm{def}}{=}\{\sum_{i=0}^{r}x_{i}\Lambda_{i}\left|\right.x_{i}\in\mathbb{R},\ x_{i}>-1,\ \sum_{i=0}^{r}x_{i}a_{i}^{\vee}=k\}\ .
$$</div>
<p data-bbox="70 228 468 244">Choose any $a,b\in C_{+}$ . Then a straightforward calculation from (2.1c) gives </p>
<div class="equation" data-bbox="159 257 453 286">$$
\frac{d}{d t}\mathcal{D}(t a+(1-t)b)=0\quad\Longrightarrow\quad\frac{d^{2}}{d t^{2}}\mathcal{D}(t a+(1-t)b)<0
$$</div>
<p data-bbox="70 296 324 311">for $0<t<1$ . This means that for all $0<t<1$ , </p>
<div class="equation" data-bbox="207 325 404 340">$$
{\mathcal{D}}(t a+(1-t)b)>\operatorname*{min}\{{\mathcal{D}}(a),\,{\mathcal{D}}(b)\}~.
$$</div>
<p data-bbox="71 360 542 390">Proposition 4.1 [17,18]. For the following algebras $X_{r}^{(1)}$ and levels $k$ , and choices of weight $\Lambda_{\star}$ , $\mathcal{D}(\lambda)=\mathcal{D}(\Lambda_{\star})$ implies $\lambda\in{\mathcal{S}}\Lambda_{\star}$ : </p>
<p data-bbox="70 392 479 542">(a) For A(r1) any level $k$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (b) For Br(1) any level $k\neq2$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (c) For Cr(1) any level $k$ (except for $(r,k)=(2,3)$ or $(3,2).$ ), where $\Lambda_{\star}=\Lambda_{1}$ ; (d) For $D_{r}^{(1)}$ any level $k\neq2$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (e6) For E6(1) any level $k$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (e7) For ${E}_{7}^{(1)}$ any level $k\neq3$ , where $\Lambda_{\star}=\Lambda_{6}$ ; (e8) For E8(1) any level $k\neq1,4$ , where $\Lambda_{\star}=\Lambda_{1}$ ; (f4) For ${F}_{4}^{(1)}$ any level $k\neq3,4$ , where $\Lambda_{\star}=\Lambda_{4}$ ; (g2) For G(21) level any $k\neq3,4$ , where $\Lambda_{\star}=\Lambda_{2}$ . </p>
<p data-bbox="93 547 521 650">The missing cases are: $B_{r,2}$ where ${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{2})=\cdots={\mathcal{D}}(\Lambda_{r-1})={\mathcal{D}}(2\Lambda_{r});$ $D_{r,2}$ where ${\mathcal{D}}(\Lambda_{1})=\cdots={\mathcal{D}}(\Lambda_{r-2})$ ;
$C_{2,3}$ where ${\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(3\Lambda_{1})={\mathcal{D}}(\Lambda_{1})$ , and its rank-level dual $C_{3,2}$ ;
$E_{7,3}$ where ${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(\Lambda_{6})$ ;
$E_{8,1}$ where $\Lambda_{1}\notin P_{+}=\{0\}$ , and $E_{8,4}$ where ${\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{6})$ ;
$F_{4,3}$ where ${\mathcal{D}}(\Lambda_{2})={\mathcal{D}}(\Lambda_{4})$ , and $F_{4,4}$ where $\mathcal{D}(\Lambda_{1})=\mathcal{D}(2\Lambda_{1})=\mathcal{D}(4\Lambda_{4})=\mathcal{D}(\Lambda_{4})$ ;
$G_{2,3}$ where $\mathcal{D}(\Lambda_{1})=\mathcal{D}(\Lambda_{2})=\mathcal{D}(3\Lambda_{2})$ , and $G_{2,4}$ where $\mathcal{D}(\Lambda_{2})=\mathcal{D}(2\Lambda_{1})$ . </p>
<p data-bbox="70 654 541 715">The weight $\Lambda_{\star}$ singled out by Proposition 4.1 (i.e. $\Lambda_{\star}=\Lambda_{1}$ for $A_{r}^{(1)}$ , ..., $\Lambda_{\star}=\Lambda_{2}$ for $G_{2}^{(1)})$ is the nonzero weight with smallest Weyl dimension. What we find is that, for all but the smallest levels (see [18, Table 3]), $\Lambda_{\star}$ will also have the smallest q-dimension after the simple-currents. </p>
</body></html>
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0002044v1
| 15 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "4.1. $q$ -dimensions ", "page_idx": 15}, {"type": "text", "text": "The most basic properties obeyed by the q-dimensions $\\begin{array}{r}{\\mathcal{D}(\\lambda)=\\frac{S_{\\lambda0}}{S_{00}}}\\end{array}$ are that $\\mathcal{D}(\\lambda)\\geq1$ , and $\\mathcal{D}(s\\lambda)=\\mathcal{D}(\\lambda)$ for any $s\\in S$ . Recall that $\\boldsymbol{S}$ is the symmetry group of the extended Dynkin diagram of $X_{r}^{(1)}$ , and that $s\\in S$ acts on $P_{+}$ by permuting the Dynkin labels. ", "page_idx": 15}, {"type": "text", "text": "The argument yielding Proposition 4.1 below relies heavily on the following observation. Use (2.1c) to extend the domain of $\\mathcal{D}$ from $P_{+}$ to the fundamental chamber $C_{+}$ : ", "page_idx": 15}, {"type": "equation", "text": "$$\nC_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{i=0}^{r}x_{i}\\Lambda_{i}\\left|\\right.x_{i}\\in\\mathbb{R},\\ x_{i}>-1,\\ \\sum_{i=0}^{r}x_{i}a_{i}^{\\vee}=k\\}\\ .\n$$", "text_format": "latex", "page_idx": 15}, {"type": "text", "text": "Choose any $a,b\\in C_{+}$ . Then a straightforward calculation from (2.1c) gives ", "page_idx": 15}, {"type": "equation", "text": "$$\n\\frac{d}{d t}\\mathcal{D}(t a+(1-t)b)=0\\quad\\Longrightarrow\\quad\\frac{d^{2}}{d t^{2}}\\mathcal{D}(t a+(1-t)b)<0\n$$", "text_format": "latex", "page_idx": 15}, {"type": "text", "text": "for $0<t<1$ . This means that for all $0<t<1$ , ", "page_idx": 15}, {"type": "equation", "text": "$$\n{\\mathcal{D}}(t a+(1-t)b)>\\operatorname*{min}\\{{\\mathcal{D}}(a),\\,{\\mathcal{D}}(b)\\}~.\n$$", "text_format": "latex", "page_idx": 15}, {"type": "text", "text": "Proposition 4.1 [17,18]. For the following algebras $X_{r}^{(1)}$ and levels $k$ , and choices of weight $\\Lambda_{\\star}$ , $\\mathcal{D}(\\lambda)=\\mathcal{D}(\\Lambda_{\\star})$ implies $\\lambda\\in{\\mathcal{S}}\\Lambda_{\\star}$ : ", "page_idx": 15}, {"type": "text", "text": "(a) For A(r1) any level $k$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (b) For Br(1) any level $k\\neq2$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (c) For Cr(1) any level $k$ (except for $(r,k)=(2,3)$ or $(3,2).$ ), where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (d) For $D_{r}^{(1)}$ any level $k\\neq2$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (e6) For E6(1) any level $k$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (e7) For ${E}_{7}^{(1)}$ any level $k\\neq3$ , where $\\Lambda_{\\star}=\\Lambda_{6}$ ; (e8) For E8(1) any level $k\\neq1,4$ , where $\\Lambda_{\\star}=\\Lambda_{1}$ ; (f4) For ${F}_{4}^{(1)}$ any level $k\\neq3,4$ , where $\\Lambda_{\\star}=\\Lambda_{4}$ ; (g2) For G(21) level any $k\\neq3,4$ , where $\\Lambda_{\\star}=\\Lambda_{2}$ . ", "page_idx": 15}, {"type": "text", "text": "The missing cases are: $B_{r,2}$ where ${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})=\\cdots={\\mathcal{D}}(\\Lambda_{r-1})={\\mathcal{D}}(2\\Lambda_{r});$ $D_{r,2}$ where ${\\mathcal{D}}(\\Lambda_{1})=\\cdots={\\mathcal{D}}(\\Lambda_{r-2})$ ; \n$C_{2,3}$ where ${\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(3\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{1})$ , and its rank-level dual $C_{3,2}$ ; \n$E_{7,3}$ where ${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{6})$ ; \n$E_{8,1}$ where $\\Lambda_{1}\\notin P_{+}=\\{0\\}$ , and $E_{8,4}$ where ${\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{6})$ ; \n$F_{4,3}$ where ${\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{4})$ , and $F_{4,4}$ where $\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(2\\Lambda_{1})=\\mathcal{D}(4\\Lambda_{4})=\\mathcal{D}(\\Lambda_{4})$ ; \n$G_{2,3}$ where $\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(3\\Lambda_{2})$ , and $G_{2,4}$ where $\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(2\\Lambda_{1})$ . ", "page_idx": 15}, {"type": "text", "text": "The weight $\\Lambda_{\\star}$ singled out by Proposition 4.1 (i.e. $\\Lambda_{\\star}=\\Lambda_{1}$ for $A_{r}^{(1)}$ , ..., $\\Lambda_{\\star}=\\Lambda_{2}$ for $G_{2}^{(1)})$ is the nonzero weight with smallest Weyl dimension. What we find is that, for all but the smallest levels (see [18, Table 3]), $\\Lambda_{\\star}$ will also have the smallest q-dimension after the simple-currents. ", "page_idx": 15}]
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Use (2.1c) to extend the domain of ", "type": "text"}, {"bbox": [288, 156, 298, 165], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [298, 155, 329, 169], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [329, 156, 345, 168], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [345, 155, 501, 169], "score": 1.0, "content": " to the fundamental chamber ", "type": "text"}, {"bbox": [501, 156, 518, 168], "score": 0.92, "content": "C_{+}", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [518, 155, 522, 169], "score": 1.0, "content": ":", "type": "text"}], "index": 5}], "index": 4.5, "bbox_fs": [71, 140, 540, 169]}, {"type": "interline_equation", "bbox": [173, 181, 438, 218], "lines": [{"bbox": [173, 181, 438, 218], "spans": [{"bbox": [173, 181, 438, 218], "score": 0.94, "content": "C_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{i=0}^{r}x_{i}\\Lambda_{i}\\left|\\right.x_{i}\\in\\mathbb{R},\\ x_{i}>-1,\\ \\sum_{i=0}^{r}x_{i}a_{i}^{\\vee}=k\\}\\ .", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [70, 228, 468, 244], "lines": [{"bbox": [71, 231, 467, 246], "spans": [{"bbox": [71, 231, 135, 246], "score": 1.0, "content": "Choose any ", "type": "text"}, {"bbox": [136, 234, 183, 245], "score": 0.94, "content": "a,b\\in C_{+}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [184, 231, 467, 246], "score": 1.0, "content": ". 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This means that for all ", "type": "text"}, {"bbox": [271, 301, 320, 310], "score": 0.88, "content": "0<t<1", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [320, 298, 324, 313], "score": 1.0, "content": ",", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [70, 298, 324, 313]}, {"type": "interline_equation", "bbox": [207, 325, 404, 340], "lines": [{"bbox": [207, 325, 404, 340], "spans": [{"bbox": [207, 325, 404, 340], "score": 0.88, "content": "{\\mathcal{D}}(t a+(1-t)b)>\\operatorname*{min}\\{{\\mathcal{D}}(a),\\,{\\mathcal{D}}(b)\\}~.", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [71, 360, 542, 390], "lines": [{"bbox": [91, 360, 543, 380], "spans": [{"bbox": [91, 360, 380, 380], "score": 1.0, "content": "Proposition 4.1 [17,18]. 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"\\Lambda_{\\star}=\\Lambda_{2}", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [521, 654, 542, 673], "score": 1.0, "content": " for", "type": "text"}], "index": 29}, {"bbox": [71, 669, 543, 691], "spans": [{"bbox": [71, 671, 97, 687], "score": 0.89, "content": "G_{2}^{(1)})", "type": "inline_equation", "height": 16, "width": 26}, {"bbox": [97, 669, 543, 691], "score": 1.0, "content": " is the nonzero weight with smallest Weyl dimension. What we find is that, for all", "type": "text"}], "index": 30}, {"bbox": [69, 687, 541, 703], "spans": [{"bbox": [69, 687, 290, 703], "score": 1.0, "content": "but the smallest levels (see [18, Table 3]), ", "type": "text"}, {"bbox": [291, 690, 305, 700], "score": 0.91, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [305, 687, 541, 703], "score": 1.0, "content": "will also have the smallest q-dimension after", "type": "text"}], "index": 31}, {"bbox": [70, 702, 176, 718], "spans": [{"bbox": [70, 702, 176, 718], "score": 1.0, "content": "the simple-currents.", "type": "text"}], "index": 32}], "index": 30.5, "bbox_fs": [69, 654, 543, 718]}]}
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[{"type": "text", "bbox": [71, 71, 167, 86], "content": "4.1. -dimensions", "index": 0}, {"type": "text", "bbox": [70, 92, 541, 138], "content": "The most basic properties obeyed by the q-dimensions are that , and for any . Recall that is the symmetry group of the extended Dynkin diagram of , and that acts on by permuting the Dynkin labels.", "index": 1}, {"type": "text", "bbox": [70, 138, 541, 167], "content": "The argument yielding Proposition 4.1 below relies heavily on the following observa- tion. Use (2.1c) to extend the domain of from to the fundamental chamber :", "index": 2}, {"type": "interline_equation", "bbox": [173, 181, 438, 218], "content": "", "index": 3}, {"type": "text", "bbox": [70, 228, 468, 244], "content": "Choose any . Then a straightforward calculation from (2.1c) gives", "index": 4}, {"type": "interline_equation", "bbox": [159, 257, 453, 286], "content": "", "index": 5}, {"type": "text", "bbox": [70, 296, 324, 311], "content": "for . This means that for all ,", "index": 6}, {"type": "interline_equation", "bbox": [207, 325, 404, 340], "content": "", "index": 7}, {"type": "text", "bbox": [71, 360, 542, 390], "content": "Proposition 4.1 [17,18]. For the following algebras and levels , and choices of weight , implies :", "index": 8}, {"type": "text", "bbox": [70, 392, 479, 542], "content": "(a) For A(r1) any level , where ; (b) For Br(1) any level , where ; (c) For Cr(1) any level (except for or ), where ; (d) For any level , where ; (e6) For E6(1) any level , where ; (e7) For any level , where ; (e8) For E8(1) any level , where ; (f4) For any level , where ; (g2) For G(21) level any , where .", "index": 9}, {"type": "list", "bbox": [93, 547, 521, 650], "content": "", "index": 10}, {"type": "text", "bbox": [70, 654, 541, 715], "content": "The weight singled out by Proposition 4.1 (i.e. for , ..., for is the nonzero weight with smallest Weyl dimension. What we find is that, for all but the smallest levels (see [18, Table 3]), will also have the smallest q-dimension after the simple-currents.", "index": 11}]
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[{"bbox": [70, 74, 167, 87], "content": "4.1. -dimensions", "parent_index": 0, "line_index": 0}, {"bbox": [92, 91, 542, 112], "content": "The most basic properties obeyed by the q-dimensions are that ,", "parent_index": 1, "line_index": 0}, {"bbox": [70, 108, 541, 125], "content": "and for any . Recall that is the symmetry group of the extended", "parent_index": 1, "line_index": 1}, {"bbox": [70, 123, 519, 142], "content": "Dynkin diagram of , and that acts on by permuting the Dynkin labels.", "parent_index": 1, "line_index": 2}, {"bbox": [93, 140, 540, 156], "content": "The argument yielding Proposition 4.1 below relies heavily on the following observa-", "parent_index": 2, "line_index": 0}, {"bbox": [71, 155, 522, 169], "content": "tion. Use (2.1c) to extend the domain of from to the fundamental chamber :", "parent_index": 2, "line_index": 1}, {"bbox": [71, 231, 467, 246], "content": "Choose any . Then a straightforward calculation from (2.1c) gives", "parent_index": 4, "line_index": 0}, {"bbox": [70, 298, 324, 313], "content": "for . This means that for all ,", "parent_index": 6, "line_index": 0}, {"bbox": [91, 360, 543, 380], "content": "Proposition 4.1 [17,18]. For the following algebras and levels , and choices", "parent_index": 8, "line_index": 0}, {"bbox": [71, 378, 314, 394], "content": "of weight , implies :", "parent_index": 8, "line_index": 1}, {"bbox": [72, 392, 290, 409], "content": "(a) For A(r1) any level , where ;", "parent_index": 9, "line_index": 0}, {"bbox": [72, 409, 312, 425], "content": "(b) For Br(1) any level , where ;", "parent_index": 9, "line_index": 1}, {"bbox": [72, 424, 474, 443], "content": "(c) For Cr(1) any level (except for or ), where ;", "parent_index": 9, "line_index": 2}, {"bbox": [71, 442, 314, 459], "content": "(d) For any level , where ;", "parent_index": 9, "line_index": 3}, {"bbox": [65, 457, 291, 477], "content": "(e6) For E6(1) any level , where ;", "parent_index": 9, "line_index": 4}, {"bbox": [67, 475, 312, 493], "content": "(e7) For any level , where ;", "parent_index": 9, "line_index": 5}, {"bbox": [63, 489, 324, 516], "content": "(e8) For E8(1) any level , where ;", "parent_index": 9, "line_index": 6}, {"bbox": [70, 510, 324, 526], "content": "(f4) For any level , where ;", "parent_index": 9, "line_index": 7}, {"bbox": [64, 523, 324, 549], "content": "(g2) For G(21) level any , where .", "parent_index": 9, "line_index": 8}, {"bbox": [94, 550, 505, 566], "content": "The missing cases are: where", "parent_index": 10, "line_index": 0}, {"bbox": [95, 564, 285, 580], "content": "where ;", "parent_index": 10, "line_index": 1}, {"bbox": [95, 578, 446, 596], "content": "where , and its rank-level dual ;", "parent_index": 10, "line_index": 2}, {"bbox": [95, 593, 290, 608], "content": "where ;", "parent_index": 10, "line_index": 3}, {"bbox": [95, 607, 410, 622], "content": "where , and where ;", "parent_index": 10, "line_index": 4}, {"bbox": [95, 622, 519, 637], "content": "where , and where ;", "parent_index": 10, "line_index": 5}, {"bbox": [95, 636, 475, 653], "content": "where , and where .", "parent_index": 10, "line_index": 6}, {"bbox": [92, 654, 542, 673], "content": "The weight singled out by Proposition 4.1 (i.e. for , ..., for", "parent_index": 11, "line_index": 0}, {"bbox": [71, 669, 543, 691], "content": "is the nonzero weight with smallest Weyl dimension. What we find is that, for all", "parent_index": 11, "line_index": 1}, {"bbox": [69, 687, 541, 703], "content": "but the smallest levels (see [18, Table 3]), will also have the smallest q-dimension after", "parent_index": 11, "line_index": 2}, {"bbox": [70, 702, 176, 718], "content": "the simple-currents.", "parent_index": 11, "line_index": 3}]
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[{"bbox": [97, 79, 102, 87], "content": "q", "parent_index": 0, "subtype": "inline"}, {"bbox": [380, 95, 440, 111], "content": "\\begin{array}{r}{\\mathcal{D}(\\lambda)=\\frac{S_{\\lambda0}}{S_{00}}}\\end{array}", "parent_index": 1, "subtype": "inline"}, {"bbox": [488, 96, 536, 109], "content": "\\mathcal{D}(\\lambda)\\geq1", "parent_index": 1, "subtype": "inline"}, {"bbox": [95, 110, 170, 123], "content": "\\mathcal{D}(s\\lambda)=\\mathcal{D}(\\lambda)", "parent_index": 1, "subtype": "inline"}, {"bbox": [217, 111, 246, 120], "content": "s\\in S", "parent_index": 1, "subtype": "inline"}, {"bbox": [320, 111, 328, 120], "content": "\\boldsymbol{S}", "parent_index": 1, "subtype": "inline"}, {"bbox": [174, 123, 198, 138], "content": "X_{r}^{(1)}", "parent_index": 1, "subtype": "inline"}, {"bbox": [254, 128, 282, 137], "content": "s\\in S", "parent_index": 1, "subtype": "inline"}, {"bbox": [327, 128, 343, 140], "content": "P_{+}", "parent_index": 1, "subtype": "inline"}, {"bbox": [288, 156, 298, 165], "content": "\\mathcal{D}", "parent_index": 2, "subtype": "inline"}, {"bbox": [329, 156, 345, 168], "content": "P_{+}", "parent_index": 2, "subtype": "inline"}, {"bbox": [501, 156, 518, 168], "content": "C_{+}", "parent_index": 2, "subtype": "inline"}, {"bbox": [173, 181, 438, 218], "content": "C_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{i=0}^{r}x_{i}\\Lambda_{i}\\left|\\right.x_{i}\\in\\mathbb{R},\\ x_{i}>-1,\\ \\sum_{i=0}^{r}x_{i}a_{i}^{\\vee}=k\\}\\ .", "parent_index": 3, "subtype": "interline"}, {"bbox": [136, 234, 183, 245], "content": "a,b\\in C_{+}", "parent_index": 4, "subtype": "inline"}, {"bbox": [159, 257, 453, 286], "content": "\\frac{d}{d t}\\mathcal{D}(t a+(1-t)b)=0\\quad\\Longrightarrow\\quad\\frac{d^{2}}{d t^{2}}\\mathcal{D}(t a+(1-t)b)<0", "parent_index": 5, "subtype": "interline"}, {"bbox": [90, 301, 138, 309], "content": "0<t<1", "parent_index": 6, "subtype": "inline"}, {"bbox": [271, 301, 320, 310], "content": "0<t<1", "parent_index": 6, "subtype": "inline"}, {"bbox": [207, 325, 404, 340], "content": "{\\mathcal{D}}(t a+(1-t)b)>\\operatorname*{min}\\{{\\mathcal{D}}(a),\\,{\\mathcal{D}}(b)\\}~.", "parent_index": 7, "subtype": "interline"}, {"bbox": [380, 360, 406, 376], "content": "X_{r}^{(1)}", "parent_index": 8, "subtype": "inline"}, {"bbox": [464, 364, 472, 375], "content": "k", "parent_index": 8, "subtype": "inline"}, {"bbox": [122, 380, 136, 391], "content": "\\Lambda_{\\star}", "parent_index": 8, "subtype": "inline"}, {"bbox": [144, 379, 218, 392], "content": "\\mathcal{D}(\\lambda)=\\mathcal{D}(\\Lambda_{\\star})", "parent_index": 8, "subtype": "inline"}, {"bbox": [263, 378, 308, 390], "content": "\\lambda\\in{\\mathcal{S}}\\Lambda_{\\star}", "parent_index": 8, "subtype": "inline"}, {"bbox": [193, 396, 200, 406], "content": "k", "parent_index": 9, "subtype": "inline"}, {"bbox": [241, 394, 285, 408], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [194, 411, 223, 425], "content": "k\\neq2", "parent_index": 9, "subtype": "inline"}, {"bbox": [264, 411, 308, 424], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [193, 428, 201, 439], "content": "k", "parent_index": 9, "subtype": "inline"}, {"bbox": [264, 427, 334, 442], "content": "(r,k)=(2,3)", "parent_index": 9, "subtype": "inline"}, {"bbox": [353, 428, 380, 442], "content": "(3,2).", "parent_index": 9, "subtype": "inline"}, {"bbox": [426, 428, 470, 441], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [117, 442, 141, 457], "content": "D_{r}^{(1)}", "parent_index": 9, "subtype": "inline"}, {"bbox": [194, 444, 224, 458], "content": "k\\neq2", "parent_index": 9, "subtype": "inline"}, {"bbox": [264, 444, 308, 458], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [193, 462, 201, 473], "content": "k", "parent_index": 9, "subtype": "inline"}, {"bbox": [242, 461, 286, 474], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [117, 475, 140, 492], "content": "{E}_{7}^{(1)}", "parent_index": 9, "subtype": "inline"}, {"bbox": [193, 478, 223, 492], "content": "k\\neq3", "parent_index": 9, "subtype": "inline"}, {"bbox": [264, 478, 308, 491], "content": "\\Lambda_{\\star}=\\Lambda_{6}", "parent_index": 9, "subtype": "inline"}, {"bbox": [193, 495, 235, 509], "content": "k\\neq1,4", "parent_index": 9, "subtype": "inline"}, {"bbox": [275, 495, 319, 508], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [117, 510, 140, 526], "content": "{F}_{4}^{(1)}", "parent_index": 9, "subtype": "inline"}, {"bbox": [194, 513, 235, 525], "content": "k\\neq3,4", "parent_index": 9, "subtype": "inline"}, {"bbox": [275, 511, 319, 525], "content": "\\Lambda_{\\star}=\\Lambda_{4}", "parent_index": 9, "subtype": "inline"}, {"bbox": [194, 529, 235, 542], "content": "k\\neq3,4", "parent_index": 9, "subtype": "inline"}, {"bbox": [275, 528, 319, 542], "content": "\\Lambda_{\\star}=\\Lambda_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [217, 551, 239, 564], "content": "B_{r,2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [277, 551, 505, 564], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})=\\cdots={\\mathcal{D}}(\\Lambda_{r-1})={\\mathcal{D}}(2\\Lambda_{r});", "parent_index": 10, "subtype": "inline"}, {"bbox": [95, 567, 117, 579], "content": "D_{r,2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [156, 565, 279, 578], "content": "{\\mathcal{D}}(\\Lambda_{1})=\\cdots={\\mathcal{D}}(\\Lambda_{r-2})", "parent_index": 10, "subtype": "inline"}, {"bbox": [95, 581, 117, 593], "content": "C_{2,3}", "parent_index": 10, "subtype": "inline"}, {"bbox": [155, 580, 290, 593], "content": "{\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(3\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [419, 579, 441, 593], "content": "C_{3,2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [95, 595, 117, 608], "content": "E_{7,3}", "parent_index": 10, "subtype": "inline"}, {"bbox": [155, 594, 285, 607], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{6})", "parent_index": 10, "subtype": "inline"}, {"bbox": [95, 610, 117, 622], "content": "E_{8,1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [155, 609, 234, 622], "content": "\\Lambda_{1}\\notin P_{+}=\\{0\\}", "parent_index": 10, "subtype": "inline"}, {"bbox": [264, 610, 286, 622], "content": "E_{8,4}", "parent_index": 10, "subtype": "inline"}, {"bbox": [324, 609, 405, 621], "content": "{\\mathcal{D}}(\\Lambda_{1})={\\mathcal{D}}(\\Lambda_{6})", "parent_index": 10, "subtype": "inline"}, {"bbox": [95, 624, 115, 636], "content": "F_{4,3}", "parent_index": 10, "subtype": "inline"}, {"bbox": [155, 623, 236, 636], "content": "{\\mathcal{D}}(\\Lambda_{2})={\\mathcal{D}}(\\Lambda_{4})", "parent_index": 10, "subtype": "inline"}, {"bbox": [266, 624, 286, 636], "content": "F_{4,4}", "parent_index": 10, "subtype": "inline"}, {"bbox": [325, 623, 516, 636], "content": "\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(2\\Lambda_{1})=\\mathcal{D}(4\\Lambda_{4})=\\mathcal{D}(\\Lambda_{4})", "parent_index": 10, "subtype": "inline"}, {"bbox": [95, 639, 118, 651], "content": "G_{2,3}", "parent_index": 10, "subtype": "inline"}, {"bbox": [156, 638, 292, 650], "content": "\\mathcal{D}(\\Lambda_{1})=\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(3\\Lambda_{2})", "parent_index": 10, "subtype": "inline"}, {"bbox": [322, 638, 344, 651], "content": "G_{2,4}", "parent_index": 10, "subtype": "inline"}, {"bbox": [383, 637, 470, 650], "content": "\\mathcal{D}(\\Lambda_{2})=\\mathcal{D}(2\\Lambda_{1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [158, 659, 172, 669], "content": "\\Lambda_{\\star}", "parent_index": 11, "subtype": "inline"}, {"bbox": [362, 659, 407, 669], "content": "\\Lambda_{\\star}=\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [429, 655, 451, 669], "content": "A_{r}^{(1)}", "parent_index": 11, "subtype": "inline"}, {"bbox": [477, 659, 521, 669], "content": "\\Lambda_{\\star}=\\Lambda_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 671, 97, 687], "content": "G_{2}^{(1)})", "parent_index": 11, "subtype": "inline"}, {"bbox": [291, 690, 305, 700], "content": "\\Lambda_{\\star}", "parent_index": 11, "subtype": "inline"}]
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The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we will sketch here the most interesting $(A_{r}^{(1)})$ and the most difficult $(E_{8}^{(1)})$ cases.
Consider first $A_{r,k}$ . By choosing $a\!-\!b=\Lambda_{i}-\Lambda_{j}$ in (4.1), we get that either $\lambda=k\Lambda_{\ell}$ for some $\ell$ , in which case $\lambda$ is a simple-current and (for $k\neq1$ ) $D(\lambda)<\mathcal{D}(\Lambda_{1})$ , or $\mathcal{D}(\lambda)\ge\mathcal{D}(\Lambda_{\ell})$ for some $\ell$ , with equality iff $\lambda\in S\Lambda_{\ell}$ . But then rank-level duality $A_{r,k}\leftrightarrow A_{k-1,r+1}$ (defined as for $C_{r,k}$ , and which is exact for $A_{r,k}$ q-dimensions) and (4.1) with $a-b=\widetilde{\Lambda_{0}}-\widetilde{\Lambda_{1}}$ give us $\mathcal{D}(\Lambda_{\ell})\,=\,\widetilde{\mathcal{D}}(\ell\widetilde{\Lambda_{1}})\,\geq\,\widetilde{\mathcal{D}}(\widetilde{\Lambda_{1}})\,=\,\mathcal{D}(\Lambda_{1})$ , with equality iff $\ell\,=\,1$ or $r$ . Com bining these results yields Proposition 4.1(a).
For $E_{8,k}$ , run through each $a\mathrm{~-~}b\mathrm{~=~}a_{j}^{\vee}\Lambda_{i}\mathrm{~-~}a_{i}^{\vee}\Lambda_{j}$ to reduce the proof to comparing $\mathcal{D}(\Lambda_{1})$ with $\textstyle\mathcal{D}(\frac{k}{a_{i}^{\vee}}\Lambda_{i})$ for $i\neq0$ , or $\mathcal{D}(\Lambda_{i})$ for $i\neq0,1$ (the argument in [18] unnecessarily complicated things by restricting to integral weights). Standard arguments (see [18] for details) quickly show that the q-dimension $\textstyle\mathcal{D}(\frac{k}{a_{i}^{\vee}}\Lambda_{i})$ monotonically increases with $k$ to $\infty$ , while $\mathcal{D}(\Lambda_{i})$ monotonically increases with $k$ to the Weyl dimension of $\Lambda_{i}$ . The proof of Proposition 4.1(e8) then reduces to a short computation.
# 4.2. The $A$ -series argument
Recall that $\overline{r}\,=\,r\,+\,1$ . Proposition 4.1(a) tells us that $\pi\Lambda_{1}\,=\,C^{a}J^{b}\Lambda_{1}$ , for some $a,b$ . Hitting $\pi$ with $C^{a}$ , we can assume without loss of generality that $a\;=\;0$ . Write $\pi(J0)=J^{c}0$ ; then $\pi$ can be a permutation of $P_{+}$ only if $c$ is coprime to $\overline{r}$ .
If $k=1$ then $P_{+}=\{0,J0,\dots,J^{r}0\}$ so $\pi=\pi[c-1]$ . Thus we can assume $k\geq2$ .
Useful is the coefficient of $\lambda$ in the tensor product $\Lambda_{1}\otimes\cdot\cdot\otimes\Lambda_{1}$ ( $\ell\,\mathrm{times})$ : it is 0 unless $t(\lambda)=\ell$ , in which case the coefficient is $\frac{\ell!}{h(\lambda)}$ (to get this, compare (3.1) above with [27, p.114]) — we equate here the fundamental weights $\Lambda_{\overline{{r}}}$ and $\Lambda_{0}$ , so e.g. $^{\star}{\frac{k}{r}}\Lambda_{\overline{{r}}}^{\ \ \,}{}^{\ ,}$ equals $\mathrm{\Delta}^{\prime}0^{\circ}$ when $\overline{r}$ divides $k$ . Here, $h(\lambda)=\prod h(x)$ is the product of the hook-lengths of the Young diagram corresponding to $\lambda$ . Equation (2.4) tells us that as long as $t(\lambda)=\ell\leq k$ , the number $\frac{\ell!}{h(\lambda)}$ will also be the coefficient of $N_{\lambda}$ in the fusion power $(N_{\Lambda_{1}})^{\ell}$ . Note that $J0=k\Lambda_{1}$ is the only simple-current appearing in the fusion product $\Lambda_{1}$ × · · · × $\Lambda_{1}$ ( $k$ times). Thus the only nontrivial simple-current appearing in the fusion $\pi\Lambda_{1}$ × · · · × $\pi\Lambda_{1}$ will be $J^{b k}J0$ (0 will appear iff $\overline{r}$ divides $k$ ). Hence $b k+1\equiv c$ (mod $\overline{r}$ ) must be coprime to $\overline{r}$ . This is precisely the condition needed for $\pi[b]$ to be a simple-current automorphism.
In other words, it suffices to consider $\pi\Lambda_{1}=\Lambda_{1}$ and hence $\pi[J0]=J0$ . We are done if $r=1$ , so assume $r\geq2$ . From the $\Lambda_{1}$ × $\Lambda_{1}$ fusion, we get that $\pi\Lambda_{2}\in\{\Lambda_{2},2\Lambda_{1}\}$ . Note that $k\Lambda_{1}$ occurs (with multiplicity 1) in the tensor and fusion product of $2\Lambda_{1}$ with $k-2\;\Lambda_{1}\,{}^{\prime}\mathrm{s}$ s, but that it doesn’t in the tensor (hence fusion) product of $\Lambda_{2}$ with $k-2~\Lambda_{1}$ ’s (recall that $k\Lambda_{1}\succ(k-2)\Lambda_{1}+\Lambda_{2}$ in the usual partial order on weights). Since $\Lambda_{2}$ × $\Lambda_{1}$ × · · · × $\Lambda_{1}$ does not contain $J0$ , $(\pi\Lambda_{2})\boxtimes(\pi\Lambda_{1})\boxtimes\dots\boxtimes(\pi\Lambda_{1})$ should also avoid $\pi(J0)\,=\,J0$ , and thus $\pi\Lambda_{2}$ cannot equal $2\Lambda_{1}$ .
Thus we know $\pi\Lambda_{2}=\Lambda_{2}$ . The remaining $\pi\Lambda_{\ell}=\Lambda_{\ell}$ follow quickly from induction: if $\pi\Lambda_{\ell}=\Lambda_{\ell}$ for some $2\leq\ell<r$ , then the fusion $\Lambda_{1}\boxtimes\Lambda_{\ell}$ tells us $\pi\Lambda_{\ell+1}\in\{\Lambda_{\ell+1},\Lambda_{1}+\Lambda_{\ell}\}$ . But $h(\Lambda_{1}+\Lambda_{\ell})=(\ell+1)!/\ell$ and $h(\Lambda_{\ell+1})=(\ell+1)!$ , so $\pi\Lambda_{\ell+1}=\Lambda_{\ell+1}$ . Thus $\pi$ fixes all fundamental weights, and since these comprise a fusion-generator (see the discussion at the end of §2.2) we know that $\pi$ must fix everything in $P_{+}$ .
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<p data-bbox="71 70 542 102">The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we will sketch here the most interesting $(A_{r}^{(1)})$ and the most difficult $(E_{8}^{(1)})$ cases. </p>
<p data-bbox="70 103 541 192">Consider first $A_{r,k}$ . By choosing $a\!-\!b=\Lambda_{i}-\Lambda_{j}$ in (4.1), we get that either $\lambda=k\Lambda_{\ell}$ for some $\ell$ , in which case $\lambda$ is a simple-current and (for $k\neq1$ ) $D(\lambda)<\mathcal{D}(\Lambda_{1})$ , or $\mathcal{D}(\lambda)\ge\mathcal{D}(\Lambda_{\ell})$ for some $\ell$ , with equality iff $\lambda\in S\Lambda_{\ell}$ . But then rank-level duality $A_{r,k}\leftrightarrow A_{k-1,r+1}$ (defined as for $C_{r,k}$ , and which is exact for $A_{r,k}$ q-dimensions) and (4.1) with $a-b=\widetilde{\Lambda_{0}}-\widetilde{\Lambda_{1}}$ give us $\mathcal{D}(\Lambda_{\ell})\,=\,\widetilde{\mathcal{D}}(\ell\widetilde{\Lambda_{1}})\,\geq\,\widetilde{\mathcal{D}}(\widetilde{\Lambda_{1}})\,=\,\mathcal{D}(\Lambda_{1})$ , with equality iff $\ell\,=\,1$ or $r$ . Com bining these results yields Proposition 4.1(a). </p>
<p data-bbox="70 193 541 288">For $E_{8,k}$ , run through each $a\mathrm{~-~}b\mathrm{~=~}a_{j}^{\vee}\Lambda_{i}\mathrm{~-~}a_{i}^{\vee}\Lambda_{j}$ to reduce the proof to comparing $\mathcal{D}(\Lambda_{1})$ with $\textstyle\mathcal{D}(\frac{k}{a_{i}^{\vee}}\Lambda_{i})$ for $i\neq0$ , or $\mathcal{D}(\Lambda_{i})$ for $i\neq0,1$ (the argument in [18] unnecessarily complicated things by restricting to integral weights). Standard arguments (see [18] for details) quickly show that the q-dimension $\textstyle\mathcal{D}(\frac{k}{a_{i}^{\vee}}\Lambda_{i})$ monotonically increases with $k$ to $\infty$ , while $\mathcal{D}(\Lambda_{i})$ monotonically increases with $k$ to the Weyl dimension of $\Lambda_{i}$ . The proof of Proposition 4.1(e8) then reduces to a short computation. </p>
<h1 data-bbox="72 301 218 316">4.2. The $A$ -series argument </h1>
<p data-bbox="70 323 541 366">Recall that $\overline{r}\,=\,r\,+\,1$ . Proposition 4.1(a) tells us that $\pi\Lambda_{1}\,=\,C^{a}J^{b}\Lambda_{1}$ , for some $a,b$ . Hitting $\pi$ with $C^{a}$ , we can assume without loss of generality that $a\;=\;0$ . Write $\pi(J0)=J^{c}0$ ; then $\pi$ can be a permutation of $P_{+}$ only if $c$ is coprime to $\overline{r}$ . </p>
<p data-bbox="88 365 518 380">If $k=1$ then $P_{+}=\{0,J0,\dots,J^{r}0\}$ so $\pi=\pi[c-1]$ . Thus we can assume $k\geq2$ . </p>
<p data-bbox="69 381 541 530">Useful is the coefficient of $\lambda$ in the tensor product $\Lambda_{1}\otimes\cdot\cdot\otimes\Lambda_{1}$ ( $\ell\,\mathrm{times})$ : it is 0 unless $t(\lambda)=\ell$ , in which case the coefficient is $\frac{\ell!}{h(\lambda)}$ (to get this, compare (3.1) above with [27, p.114]) — we equate here the fundamental weights $\Lambda_{\overline{{r}}}$ and $\Lambda_{0}$ , so e.g. $^{\star}{\frac{k}{r}}\Lambda_{\overline{{r}}}^{\ \ \,}{}^{\ ,}$ equals $\mathrm{\Delta}^{\prime}0^{\circ}$ when $\overline{r}$ divides $k$ . Here, $h(\lambda)=\prod h(x)$ is the product of the hook-lengths of the Young diagram corresponding to $\lambda$ . Equation (2.4) tells us that as long as $t(\lambda)=\ell\leq k$ , the number $\frac{\ell!}{h(\lambda)}$ will also be the coefficient of $N_{\lambda}$ in the fusion power $(N_{\Lambda_{1}})^{\ell}$ . Note that $J0=k\Lambda_{1}$ is the only simple-current appearing in the fusion product $\Lambda_{1}$ × · · · × $\Lambda_{1}$ ( $k$ times). Thus the only nontrivial simple-current appearing in the fusion $\pi\Lambda_{1}$ × · · · × $\pi\Lambda_{1}$ will be $J^{b k}J0$ (0 will appear iff $\overline{r}$ divides $k$ ). Hence $b k+1\equiv c$ (mod $\overline{r}$ ) must be coprime to $\overline{r}$ . This is precisely the condition needed for $\pi[b]$ to be a simple-current automorphism. </p>
<p data-bbox="70 531 541 631">In other words, it suffices to consider $\pi\Lambda_{1}=\Lambda_{1}$ and hence $\pi[J0]=J0$ . We are done if $r=1$ , so assume $r\geq2$ . From the $\Lambda_{1}$ × $\Lambda_{1}$ fusion, we get that $\pi\Lambda_{2}\in\{\Lambda_{2},2\Lambda_{1}\}$ . Note that $k\Lambda_{1}$ occurs (with multiplicity 1) in the tensor and fusion product of $2\Lambda_{1}$ with $k-2\;\Lambda_{1}\,{}^{\prime}\mathrm{s}$ s, but that it doesn’t in the tensor (hence fusion) product of $\Lambda_{2}$ with $k-2~\Lambda_{1}$ ’s (recall that $k\Lambda_{1}\succ(k-2)\Lambda_{1}+\Lambda_{2}$ in the usual partial order on weights). Since $\Lambda_{2}$ × $\Lambda_{1}$ × · · · × $\Lambda_{1}$ does not contain $J0$ , $(\pi\Lambda_{2})\boxtimes(\pi\Lambda_{1})\boxtimes\dots\boxtimes(\pi\Lambda_{1})$ should also avoid $\pi(J0)\,=\,J0$ , and thus $\pi\Lambda_{2}$ cannot equal $2\Lambda_{1}$ . </p>
<p data-bbox="70 632 541 703">Thus we know $\pi\Lambda_{2}=\Lambda_{2}$ . The remaining $\pi\Lambda_{\ell}=\Lambda_{\ell}$ follow quickly from induction: if $\pi\Lambda_{\ell}=\Lambda_{\ell}$ for some $2\leq\ell<r$ , then the fusion $\Lambda_{1}\boxtimes\Lambda_{\ell}$ tells us $\pi\Lambda_{\ell+1}\in\{\Lambda_{\ell+1},\Lambda_{1}+\Lambda_{\ell}\}$ . But $h(\Lambda_{1}+\Lambda_{\ell})=(\ell+1)!/\ell$ and $h(\Lambda_{\ell+1})=(\ell+1)!$ , so $\pi\Lambda_{\ell+1}=\Lambda_{\ell+1}$ . Thus $\pi$ fixes all fundamental weights, and since these comprise a fusion-generator (see the discussion at the end of §2.2) we know that $\pi$ must fix everything in $P_{+}$ . </p>
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0002044v1
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[{"type": "text", "text": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we will sketch here the most interesting $(A_{r}^{(1)})$ and the most difficult $(E_{8}^{(1)})$ cases. ", "page_idx": 16}, {"type": "text", "text": "Consider first $A_{r,k}$ . By choosing $a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}$ in (4.1), we get that either $\\lambda=k\\Lambda_{\\ell}$ for some $\\ell$ , in which case $\\lambda$ is a simple-current and (for $k\\neq1$ ) $D(\\lambda)<\\mathcal{D}(\\Lambda_{1})$ , or $\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})$ for some $\\ell$ , with equality iff $\\lambda\\in S\\Lambda_{\\ell}$ . But then rank-level duality $A_{r,k}\\leftrightarrow A_{k-1,r+1}$ (defined as for $C_{r,k}$ , and which is exact for $A_{r,k}$ q-dimensions) and (4.1) with $a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}$ give us $\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})$ , with equality iff $\\ell\\,=\\,1$ or $r$ . Com bining these results yields Proposition 4.1(a). ", "page_idx": 16}, {"type": "text", "text": "For $E_{8,k}$ , run through each $a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}$ to reduce the proof to comparing $\\mathcal{D}(\\Lambda_{1})$ with $\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})$ for $i\\neq0$ , or $\\mathcal{D}(\\Lambda_{i})$ for $i\\neq0,1$ (the argument in [18] unnecessarily complicated things by restricting to integral weights). Standard arguments (see [18] for details) quickly show that the q-dimension $\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})$ monotonically increases with $k$ to $\\infty$ , while $\\mathcal{D}(\\Lambda_{i})$ monotonically increases with $k$ to the Weyl dimension of $\\Lambda_{i}$ . The proof of Proposition 4.1(e8) then reduces to a short computation. ", "page_idx": 16}, {"type": "text", "text": "4.2. The $A$ -series argument ", "text_level": 1, "page_idx": 16}, {"type": "text", "text": "Recall that $\\overline{r}\\,=\\,r\\,+\\,1$ . Proposition 4.1(a) tells us that $\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}$ , for some $a,b$ . Hitting $\\pi$ with $C^{a}$ , we can assume without loss of generality that $a\\;=\\;0$ . Write $\\pi(J0)=J^{c}0$ ; then $\\pi$ can be a permutation of $P_{+}$ only if $c$ is coprime to $\\overline{r}$ . ", "page_idx": 16}, {"type": "text", "text": "If $k=1$ then $P_{+}=\\{0,J0,\\dots,J^{r}0\\}$ so $\\pi=\\pi[c-1]$ . Thus we can assume $k\\geq2$ . ", "page_idx": 16}, {"type": "text", "text": "Useful is the coefficient of $\\lambda$ in the tensor product $\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}$ ( $\\ell\\,\\mathrm{times})$ : it is 0 unless $t(\\lambda)=\\ell$ , in which case the coefficient is $\\frac{\\ell!}{h(\\lambda)}$ (to get this, compare (3.1) above with [27, p.114]) — we equate here the fundamental weights $\\Lambda_{\\overline{{r}}}$ and $\\Lambda_{0}$ , so e.g. $^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}$ equals $\\mathrm{\\Delta}^{\\prime}0^{\\circ}$ when $\\overline{r}$ divides $k$ . Here, $h(\\lambda)=\\prod h(x)$ is the product of the hook-lengths of the Young diagram corresponding to $\\lambda$ . Equation (2.4) tells us that as long as $t(\\lambda)=\\ell\\leq k$ , the number $\\frac{\\ell!}{h(\\lambda)}$ will also be the coefficient of $N_{\\lambda}$ in the fusion power $(N_{\\Lambda_{1}})^{\\ell}$ . Note that $J0=k\\Lambda_{1}$ is the only simple-current appearing in the fusion product $\\Lambda_{1}$ × · · · × $\\Lambda_{1}$ ( $k$ times). Thus the only nontrivial simple-current appearing in the fusion $\\pi\\Lambda_{1}$ × · · · × $\\pi\\Lambda_{1}$ will be $J^{b k}J0$ (0 will appear iff $\\overline{r}$ divides $k$ ). Hence $b k+1\\equiv c$ (mod $\\overline{r}$ ) must be coprime to $\\overline{r}$ . This is precisely the condition needed for $\\pi[b]$ to be a simple-current automorphism. ", "page_idx": 16}, {"type": "text", "text": "In other words, it suffices to consider $\\pi\\Lambda_{1}=\\Lambda_{1}$ and hence $\\pi[J0]=J0$ . We are done if $r=1$ , so assume $r\\geq2$ . From the $\\Lambda_{1}$ × $\\Lambda_{1}$ fusion, we get that $\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}$ . Note that $k\\Lambda_{1}$ occurs (with multiplicity 1) in the tensor and fusion product of $2\\Lambda_{1}$ with $k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}$ s, but that it doesn’t in the tensor (hence fusion) product of $\\Lambda_{2}$ with $k-2~\\Lambda_{1}$ ’s (recall that $k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}$ in the usual partial order on weights). Since $\\Lambda_{2}$ × $\\Lambda_{1}$ × · · · × $\\Lambda_{1}$ does not contain $J0$ , $(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})$ should also avoid $\\pi(J0)\\,=\\,J0$ , and thus $\\pi\\Lambda_{2}$ cannot equal $2\\Lambda_{1}$ . ", "page_idx": 16}, {"type": "text", "text": "Thus we know $\\pi\\Lambda_{2}=\\Lambda_{2}$ . The remaining $\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$ follow quickly from induction: if $\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$ for some $2\\leq\\ell<r$ , then the fusion $\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}$ tells us $\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}$ . But $h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell$ and $h(\\Lambda_{\\ell+1})=(\\ell+1)!$ , so $\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}$ . Thus $\\pi$ fixes all fundamental weights, and since these comprise a fusion-generator (see the discussion at the end of §2.2) we know that $\\pi$ must fix everything in $P_{+}$ . ", "page_idx": 16}]
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{"preproc_blocks": [{"type": "text", "bbox": [71, 70, 542, 102], "lines": [{"bbox": [95, 74, 541, 88], "spans": [{"bbox": [95, 74, 541, 88], "score": 1.0, "content": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we", "type": "text"}], "index": 0}, {"bbox": [68, 85, 484, 106], "spans": [{"bbox": [68, 85, 264, 106], "score": 1.0, "content": "will sketch here the most interesting ", "type": "text"}, {"bbox": [265, 88, 296, 104], "score": 0.88, "content": "(A_{r}^{(1)})", "type": "inline_equation", "height": 16, "width": 31}, {"bbox": [297, 85, 416, 106], "score": 1.0, "content": " and the most difficult ", "type": "text"}, {"bbox": [417, 88, 449, 104], "score": 0.92, "content": "(E_{8}^{(1)})", "type": "inline_equation", "height": 16, "width": 32}, {"bbox": [449, 85, 484, 106], "score": 1.0, "content": " cases.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [70, 103, 541, 192], "lines": [{"bbox": [94, 104, 542, 121], "spans": [{"bbox": [94, 104, 167, 121], "score": 1.0, "content": "Consider first", "type": "text"}, {"bbox": [168, 106, 190, 119], "score": 0.92, "content": "A_{r,k}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [190, 104, 263, 121], "score": 1.0, "content": ". By choosing ", "type": "text"}, {"bbox": [264, 106, 339, 119], "score": 0.91, "content": "a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [339, 104, 479, 121], "score": 1.0, "content": " in (4.1), we get that either ", "type": "text"}, {"bbox": [479, 106, 522, 117], "score": 0.93, "content": "\\lambda=k\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [522, 104, 542, 121], "score": 1.0, "content": "for", "type": "text"}], "index": 2}, {"bbox": [70, 119, 540, 135], "spans": [{"bbox": [70, 119, 100, 135], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [100, 121, 106, 129], "score": 0.86, "content": "\\ell", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [106, 119, 182, 135], "score": 1.0, "content": ", in which case ", "type": "text"}, {"bbox": [182, 120, 190, 130], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [190, 119, 335, 135], "score": 1.0, "content": " is a simple-current and (for ", "type": "text"}, {"bbox": [335, 120, 364, 132], "score": 0.9, "content": "k\\neq1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [364, 119, 371, 135], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [371, 120, 446, 133], "score": 0.93, "content": "D(\\lambda)<\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [446, 119, 465, 135], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [465, 120, 540, 133], "score": 0.93, "content": "\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})", "type": "inline_equation", "height": 13, "width": 75}], "index": 3}, {"bbox": [70, 132, 541, 150], "spans": [{"bbox": [70, 132, 117, 150], "score": 1.0, "content": "for some ", "type": "text"}, {"bbox": [117, 135, 123, 144], "score": 0.87, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [123, 132, 213, 150], "score": 1.0, "content": ", with equality iff", "type": "text"}, {"bbox": [214, 135, 256, 146], "score": 0.91, "content": "\\lambda\\in S\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [257, 132, 407, 150], "score": 1.0, "content": ". But then rank-level duality ", "type": "text"}, {"bbox": [407, 135, 494, 147], "score": 0.92, "content": "A_{r,k}\\leftrightarrow A_{k-1,r+1}", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [495, 132, 541, 150], "score": 1.0, "content": " (defined", "type": "text"}], "index": 4}, {"bbox": [71, 148, 540, 165], "spans": [{"bbox": [71, 151, 104, 165], "score": 1.0, "content": "as for ", "type": "text"}, {"bbox": [104, 152, 126, 164], "score": 0.92, "content": "C_{r,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [126, 151, 251, 165], "score": 1.0, "content": ", and which is exact for ", "type": "text"}, {"bbox": [252, 152, 273, 164], "score": 0.91, "content": "A_{r,k}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [273, 151, 432, 165], "score": 1.0, "content": " q-dimensions) and (4.1) with ", "type": "text"}, {"bbox": [432, 148, 515, 162], "score": 0.94, "content": "a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}", "type": "inline_equation", "height": 14, "width": 83}, {"bbox": [515, 151, 540, 165], "score": 1.0, "content": " give", "type": "text"}], "index": 5}, {"bbox": [70, 165, 541, 182], "spans": [{"bbox": [70, 165, 87, 182], "score": 1.0, "content": "us ", "type": "text"}, {"bbox": [87, 165, 282, 180], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 15, "width": 195}, {"bbox": [282, 165, 380, 182], "score": 1.0, "content": ", with equality iff", "type": "text"}, {"bbox": [380, 168, 412, 177], "score": 0.92, "content": "\\ell\\,=\\,1", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [412, 165, 432, 182], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [433, 171, 439, 177], "score": 0.86, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [439, 165, 541, 182], "score": 1.0, "content": ". Com bining these", "type": "text"}], "index": 6}, {"bbox": [71, 181, 243, 196], "spans": [{"bbox": [71, 181, 243, 196], "score": 1.0, "content": "results yields Proposition 4.1(a).", "type": "text"}], "index": 7}], "index": 4.5}, {"type": "text", "bbox": [70, 193, 541, 288], "lines": [{"bbox": [93, 194, 541, 212], "spans": [{"bbox": [93, 194, 117, 212], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 197, 140, 209], "score": 0.93, "content": "E_{8,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [140, 194, 244, 212], "score": 1.0, "content": ", run through each ", "type": "text"}, {"bbox": [244, 196, 357, 211], "score": 0.94, "content": "a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}", "type": "inline_equation", "height": 15, "width": 113}, {"bbox": [358, 194, 541, 212], "score": 1.0, "content": " to reduce the proof to comparing", "type": "text"}], "index": 8}, {"bbox": [71, 209, 543, 232], "spans": [{"bbox": [71, 213, 104, 226], "score": 0.93, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [104, 209, 135, 232], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [136, 212, 181, 230], "score": 0.95, "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "type": "inline_equation", "height": 18, "width": 45}, {"bbox": [182, 209, 204, 232], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [204, 214, 232, 225], "score": 0.91, "content": "i\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [233, 209, 255, 232], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [255, 213, 286, 226], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{i})", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [287, 209, 309, 232], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [309, 214, 349, 225], "score": 0.93, "content": "i\\neq0,1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [349, 209, 543, 232], "score": 1.0, "content": " (the argument in [18] unnecessarily", "type": "text"}], "index": 9}, {"bbox": [71, 228, 540, 244], "spans": [{"bbox": [71, 228, 540, 244], "score": 1.0, "content": "complicated things by restricting to integral weights). Standard arguments (see [18] for", "type": "text"}], "index": 10}, {"bbox": [69, 241, 543, 262], "spans": [{"bbox": [69, 241, 296, 262], "score": 1.0, "content": "details) quickly show that the q-dimension ", "type": "text"}, {"bbox": [297, 243, 342, 261], "score": 0.95, "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "type": "inline_equation", "height": 18, "width": 45}, {"bbox": [342, 241, 499, 262], "score": 1.0, "content": " monotonically increases with ", "type": "text"}, {"bbox": [499, 245, 506, 254], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [506, 241, 524, 262], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [524, 249, 536, 254], "score": 0.84, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 12}, {"bbox": [537, 241, 543, 262], "score": 1.0, "content": ",", "type": "text"}], "index": 11}, {"bbox": [72, 261, 542, 275], "spans": [{"bbox": [72, 261, 103, 275], "score": 1.0, "content": "while ", "type": "text"}, {"bbox": [104, 262, 135, 274], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{i})", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [135, 261, 296, 275], "score": 1.0, "content": " monotonically increases with ", "type": "text"}, {"bbox": [297, 263, 304, 271], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [304, 261, 448, 275], "score": 1.0, "content": " to the Weyl dimension of ", "type": "text"}, {"bbox": [448, 263, 461, 273], "score": 0.92, "content": "\\Lambda_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [461, 261, 542, 275], "score": 1.0, "content": ". The proof of", "type": "text"}], "index": 12}, {"bbox": [70, 276, 371, 290], "spans": [{"bbox": [70, 276, 371, 290], "score": 1.0, "content": "Proposition 4.1(e8) then reduces to a short computation.", "type": "text"}], "index": 13}], "index": 10.5}, {"type": "title", "bbox": [72, 301, 218, 316], "lines": [{"bbox": [71, 304, 219, 316], "spans": [{"bbox": [71, 304, 120, 316], "score": 1.0, "content": "4.2. The ", "type": "text"}, {"bbox": [121, 306, 130, 314], "score": 0.33, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [130, 304, 219, 316], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 323, 541, 366], "lines": [{"bbox": [93, 325, 541, 339], "spans": [{"bbox": [93, 325, 159, 339], "score": 1.0, "content": "Recall that ", "type": "text"}, {"bbox": [159, 327, 215, 337], "score": 0.93, "content": "\\overline{r}\\,=\\,r\\,+\\,1", "type": "inline_equation", "height": 10, "width": 56}, {"bbox": [216, 325, 401, 339], "score": 1.0, "content": ". Proposition 4.1(a) tells us that ", "type": "text"}, {"bbox": [401, 325, 484, 338], "score": 0.94, "content": "\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [484, 325, 541, 339], "score": 1.0, "content": ", for some", "type": "text"}], "index": 15}, {"bbox": [71, 339, 542, 354], "spans": [{"bbox": [71, 342, 88, 353], "score": 0.91, "content": "a,b", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [89, 339, 144, 354], "score": 1.0, "content": ". Hitting ", "type": "text"}, {"bbox": [144, 345, 151, 350], "score": 0.84, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [152, 339, 185, 354], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [185, 342, 200, 350], "score": 0.9, "content": "C^{a}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [201, 339, 464, 354], "score": 1.0, "content": ", we can assume without loss of generality that ", "type": "text"}, {"bbox": [464, 342, 497, 351], "score": 0.91, "content": "a\\;=\\;0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [497, 339, 542, 354], "score": 1.0, "content": ". Write", "type": "text"}], "index": 16}, {"bbox": [71, 353, 461, 369], "spans": [{"bbox": [71, 355, 137, 368], "score": 0.92, "content": "\\pi(J0)=J^{c}0", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [137, 353, 170, 369], "score": 1.0, "content": "; then ", "type": "text"}, {"bbox": [171, 359, 178, 365], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [178, 353, 312, 369], "score": 1.0, "content": " can be a permutation of ", "type": "text"}, {"bbox": [312, 356, 327, 367], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [328, 353, 369, 369], "score": 1.0, "content": " only if ", "type": "text"}, {"bbox": [369, 359, 375, 365], "score": 0.88, "content": "c", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [375, 353, 449, 369], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [450, 357, 456, 365], "score": 0.89, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [456, 353, 461, 369], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [88, 365, 518, 380], "lines": [{"bbox": [93, 366, 516, 384], "spans": [{"bbox": [93, 366, 106, 384], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [107, 370, 136, 379], "score": 0.92, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [136, 366, 166, 384], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [167, 370, 281, 382], "score": 0.93, "content": "P_{+}=\\{0,J0,\\dots,J^{r}0\\}", "type": "inline_equation", "height": 12, "width": 114}, {"bbox": [281, 366, 299, 384], "score": 1.0, "content": " so ", "type": "text"}, {"bbox": [299, 370, 363, 382], "score": 0.94, "content": "\\pi=\\pi[c-1]", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [363, 366, 482, 384], "score": 1.0, "content": ". Thus we can assume ", "type": "text"}, {"bbox": [482, 370, 511, 381], "score": 0.95, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [511, 366, 516, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [69, 381, 541, 530], "lines": [{"bbox": [93, 381, 541, 398], "spans": [{"bbox": [93, 381, 230, 398], "score": 1.0, "content": "Useful is the coefficient of ", "type": "text"}, {"bbox": [231, 384, 238, 393], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [238, 381, 353, 398], "score": 1.0, "content": " in the tensor product ", "type": "text"}, {"bbox": [353, 384, 419, 395], "score": 0.91, "content": "\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 66}, {"bbox": [419, 381, 426, 398], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [426, 384, 468, 396], "score": 0.29, "content": "\\ell\\,\\mathrm{times})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [468, 381, 541, 398], "score": 1.0, "content": ": it is 0 unless", "type": "text"}], "index": 19}, {"bbox": [71, 393, 543, 419], "spans": [{"bbox": [71, 398, 115, 411], "score": 0.93, "content": "t(\\lambda)=\\ell", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [115, 393, 288, 419], "score": 1.0, "content": ", in which case the coefficient is", "type": "text"}, {"bbox": [288, 396, 309, 414], "score": 0.94, "content": "\\frac{\\ell!}{h(\\lambda)}", "type": "inline_equation", "height": 18, "width": 21}, {"bbox": [309, 393, 543, 419], "score": 1.0, "content": " (to get this, compare (3.1) above with [27,", "type": "text"}], "index": 20}, {"bbox": [70, 414, 541, 430], "spans": [{"bbox": [70, 414, 334, 430], "score": 1.0, "content": "p.114]) — we equate here the fundamental weights ", "type": "text"}, {"bbox": [335, 416, 349, 427], "score": 0.91, "content": "\\Lambda_{\\overline{{r}}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [349, 414, 373, 430], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [374, 417, 388, 428], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [388, 414, 430, 430], "score": 1.0, "content": ", so e.g. ", "type": "text"}, {"bbox": [430, 414, 458, 430], "score": 0.91, "content": "^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}", "type": "inline_equation", "height": 16, "width": 28}, {"bbox": [458, 414, 498, 430], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [498, 416, 509, 426], "score": 0.39, "content": "\\mathrm{\\Delta}^{\\prime}0^{\\circ}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [509, 414, 541, 430], "score": 1.0, "content": " when", "type": "text"}], "index": 21}, {"bbox": [71, 429, 541, 445], "spans": [{"bbox": [71, 432, 77, 440], "score": 0.86, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [78, 429, 120, 445], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [120, 431, 127, 440], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [128, 429, 167, 445], "score": 1.0, "content": ". Here, ", "type": "text"}, {"bbox": [167, 430, 242, 443], "score": 0.95, "content": "h(\\lambda)=\\prod h(x)", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [243, 429, 541, 445], "score": 1.0, "content": " is the product of the hook-lengths of the Young diagram", "type": "text"}], "index": 22}, {"bbox": [70, 443, 539, 460], "spans": [{"bbox": [70, 443, 162, 459], "score": 1.0, "content": "corresponding to ", "type": "text"}, {"bbox": [163, 446, 170, 454], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [171, 443, 381, 459], "score": 1.0, "content": ". Equation (2.4) tells us that as long as ", "type": "text"}, {"bbox": [381, 444, 446, 457], "score": 0.91, "content": "t(\\lambda)=\\ell\\leq k", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [447, 443, 517, 459], "score": 1.0, "content": ", the number", "type": "text"}, {"bbox": [518, 443, 539, 460], "score": 0.93, "content": "\\frac{\\ell!}{h(\\lambda)}", "type": "inline_equation", "height": 17, "width": 21}], "index": 23}, {"bbox": [69, 460, 542, 476], "spans": [{"bbox": [69, 460, 226, 476], "score": 1.0, "content": "will also be the coefficient of ", "type": "text"}, {"bbox": [227, 463, 243, 474], "score": 0.93, "content": "N_{\\lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [243, 460, 353, 476], "score": 1.0, "content": " in the fusion power ", "type": "text"}, {"bbox": [353, 461, 389, 475], "score": 0.92, "content": "(N_{\\Lambda_{1}})^{\\ell}", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [389, 460, 454, 476], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [454, 463, 506, 474], "score": 0.92, "content": "J0=k\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [506, 460, 542, 476], "score": 1.0, "content": " is the", "type": "text"}], "index": 24}, {"bbox": [70, 475, 542, 491], "spans": [{"bbox": [70, 475, 348, 491], "score": 1.0, "content": "only simple-current appearing in the fusion product ", "type": "text"}, {"bbox": [348, 476, 362, 488], "score": 0.73, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [362, 475, 415, 491], "score": 1.0, "content": " × · · · × ", "type": "text"}, {"bbox": [415, 475, 430, 488], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [430, 475, 438, 491], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [439, 476, 446, 487], "score": 0.67, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [446, 475, 542, 491], "score": 1.0, "content": " times). Thus the", "type": "text"}], "index": 25}, {"bbox": [70, 489, 542, 505], "spans": [{"bbox": [70, 489, 354, 505], "score": 1.0, "content": "only nontrivial simple-current appearing in the fusion ", "type": "text"}, {"bbox": [354, 489, 376, 502], "score": 0.81, "content": "\\pi\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [376, 489, 429, 505], "score": 1.0, "content": " × · · · × ", "type": "text"}, {"bbox": [429, 489, 451, 502], "score": 0.87, "content": "\\pi\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [451, 489, 492, 505], "score": 1.0, "content": " will be", "type": "text"}, {"bbox": [493, 489, 525, 501], "score": 0.83, "content": "J^{b k}J0", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [526, 489, 542, 505], "score": 1.0, "content": " (0", "type": "text"}], "index": 26}, {"bbox": [72, 505, 541, 518], "spans": [{"bbox": [72, 505, 150, 518], "score": 1.0, "content": "will appear iff", "type": "text"}, {"bbox": [151, 507, 157, 515], "score": 0.86, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [158, 505, 203, 518], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [203, 506, 210, 515], "score": 0.82, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [210, 505, 262, 518], "score": 1.0, "content": "). Hence ", "type": "text"}, {"bbox": [263, 506, 322, 516], "score": 0.89, "content": "b k+1\\equiv c", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [323, 505, 358, 518], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [359, 506, 366, 515], "score": 0.59, "content": "\\overline{r}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [366, 505, 484, 518], "score": 1.0, "content": ") must be coprime to ", "type": "text"}, {"bbox": [484, 506, 491, 515], "score": 0.76, "content": "\\overline{r}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [492, 505, 541, 518], "score": 1.0, "content": ". This is", "type": "text"}], "index": 27}, {"bbox": [72, 519, 472, 533], "spans": [{"bbox": [72, 520, 251, 533], "score": 1.0, "content": "precisely the condition needed for ", "type": "text"}, {"bbox": [251, 519, 271, 532], "score": 0.92, "content": "\\pi[b]", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [271, 520, 472, 533], "score": 1.0, "content": " to be a simple-current automorphism.", "type": "text"}], "index": 28}], "index": 23.5}, {"type": "text", "bbox": [70, 531, 541, 631], "lines": [{"bbox": [94, 532, 543, 547], "spans": [{"bbox": [94, 533, 289, 547], "score": 1.0, "content": "In other words, it suffices to consider ", "type": "text"}, {"bbox": [289, 533, 340, 545], "score": 0.92, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [340, 533, 398, 547], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [398, 532, 457, 546], "score": 0.92, "content": "\\pi[J0]=J0", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [457, 533, 543, 547], "score": 1.0, "content": ". We are done if", "type": "text"}], "index": 29}, {"bbox": [71, 547, 541, 563], "spans": [{"bbox": [71, 550, 99, 558], "score": 0.89, "content": "r=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [100, 547, 160, 563], "score": 1.0, "content": ", so assume ", "type": "text"}, {"bbox": [160, 549, 188, 560], "score": 0.89, "content": "r\\geq2", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [189, 547, 246, 563], "score": 1.0, "content": ". From the ", "type": "text"}, {"bbox": [246, 547, 261, 560], "score": 0.78, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [261, 547, 277, 563], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [278, 547, 293, 560], "score": 0.77, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [293, 547, 394, 563], "score": 1.0, "content": " fusion, we get that ", "type": "text"}, {"bbox": [394, 547, 481, 561], "score": 0.91, "content": "\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}", "type": "inline_equation", "height": 14, "width": 87}, {"bbox": [482, 547, 541, 563], "score": 1.0, "content": ". Note that", "type": "text"}], "index": 30}, {"bbox": [71, 561, 540, 576], "spans": [{"bbox": [71, 563, 91, 574], "score": 0.91, "content": "k\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [92, 561, 431, 576], "score": 1.0, "content": " occurs (with multiplicity 1) in the tensor and fusion product of ", "type": "text"}, {"bbox": [431, 561, 452, 574], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [452, 561, 482, 576], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [482, 561, 534, 574], "score": 0.69, "content": "k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [534, 561, 540, 576], "score": 1.0, "content": "s,", "type": "text"}], "index": 31}, {"bbox": [70, 575, 541, 590], "spans": [{"bbox": [70, 576, 378, 590], "score": 1.0, "content": "but that it doesn’t in the tensor (hence fusion) product of ", "type": "text"}, {"bbox": [379, 576, 393, 588], "score": 0.89, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [393, 576, 423, 590], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [423, 575, 469, 588], "score": 0.79, "content": "k-2~\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [470, 576, 541, 590], "score": 1.0, "content": "’s (recall that", "type": "text"}], "index": 32}, {"bbox": [71, 589, 540, 606], "spans": [{"bbox": [71, 591, 186, 604], "score": 0.92, "content": "k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [187, 590, 425, 606], "score": 1.0, "content": " in the usual partial order on weights). Since ", "type": "text"}, {"bbox": [425, 590, 440, 603], "score": 0.72, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [440, 590, 456, 606], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [457, 590, 472, 603], "score": 0.74, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [472, 590, 524, 606], "score": 1.0, "content": " × · · · ×", "type": "text"}, {"bbox": [524, 589, 540, 603], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 16}], "index": 33}, {"bbox": [69, 603, 542, 621], "spans": [{"bbox": [69, 604, 163, 621], "score": 1.0, "content": "does not contain ", "type": "text"}, {"bbox": [164, 606, 178, 616], "score": 0.8, "content": "J0", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [178, 604, 185, 621], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [185, 603, 348, 618], "score": 0.25, "content": "(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})", "type": "inline_equation", "height": 15, "width": 163}, {"bbox": [349, 604, 449, 621], "score": 1.0, "content": " should also avoid ", "type": "text"}, {"bbox": [449, 604, 513, 618], "score": 0.93, "content": "\\pi(J0)\\,=\\,J0", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [513, 604, 542, 621], "score": 1.0, "content": ", and", "type": "text"}], "index": 34}, {"bbox": [70, 618, 218, 634], "spans": [{"bbox": [70, 618, 97, 634], "score": 1.0, "content": "thus ", "type": "text"}, {"bbox": [97, 621, 118, 631], "score": 0.92, "content": "\\pi\\Lambda_{2}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [119, 618, 192, 634], "score": 1.0, "content": " cannot equal ", "type": "text"}, {"bbox": [193, 619, 213, 632], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [213, 618, 218, 634], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 32}, {"type": "text", "bbox": [70, 632, 541, 703], "lines": [{"bbox": [94, 632, 543, 649], "spans": [{"bbox": [94, 632, 174, 649], "score": 1.0, "content": "Thus we know ", "type": "text"}, {"bbox": [174, 633, 226, 646], "score": 0.91, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [226, 632, 315, 649], "score": 1.0, "content": ". The remaining ", "type": "text"}, {"bbox": [316, 633, 366, 646], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [366, 632, 543, 649], "score": 1.0, "content": "follow quickly from induction: if", "type": "text"}], "index": 36}, {"bbox": [71, 647, 541, 663], "spans": [{"bbox": [71, 649, 121, 660], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [121, 647, 173, 663], "score": 1.0, "content": "for some ", "type": "text"}, {"bbox": [173, 648, 224, 660], "score": 0.87, "content": "2\\leq\\ell<r", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [225, 647, 314, 663], "score": 1.0, "content": ", then the fusion ", "type": "text"}, {"bbox": [314, 647, 360, 660], "score": 0.44, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [360, 647, 404, 663], "score": 1.0, "content": "tells us ", "type": "text"}, {"bbox": [405, 648, 537, 662], "score": 0.93, "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "type": "inline_equation", "height": 14, "width": 132}, {"bbox": [537, 647, 541, 663], "score": 1.0, "content": ".", "type": "text"}], "index": 37}, {"bbox": [70, 661, 542, 678], "spans": [{"bbox": [70, 661, 95, 678], "score": 1.0, "content": "But ", "type": "text"}, {"bbox": [96, 662, 222, 675], "score": 0.92, "content": "h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [222, 661, 249, 678], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [250, 662, 347, 676], "score": 0.92, "content": "h(\\Lambda_{\\ell+1})=(\\ell+1)!", "type": "inline_equation", "height": 14, "width": 97}, {"bbox": [348, 661, 370, 678], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 663, 446, 675], "score": 0.92, "content": "\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [446, 661, 487, 678], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [487, 664, 495, 673], "score": 0.72, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [495, 661, 542, 678], "score": 1.0, "content": " fixes all", "type": "text"}], "index": 38}, {"bbox": [70, 676, 542, 691], "spans": [{"bbox": [70, 676, 542, 691], "score": 1.0, "content": "fundamental weights, and since these comprise a fusion-generator (see the discussion at", "type": "text"}], "index": 39}, {"bbox": [70, 689, 384, 707], "spans": [{"bbox": [70, 689, 231, 707], "score": 1.0, "content": "the end of §2.2) we know that ", "type": "text"}, {"bbox": [232, 695, 240, 701], "score": 0.67, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [240, 689, 362, 707], "score": 1.0, "content": " must fix everything in ", "type": "text"}, {"bbox": [362, 692, 378, 704], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [379, 689, 384, 707], "score": 1.0, "content": ".", "type": "text"}], "index": 40}], "index": 38}], "layout_bboxes": [], "page_idx": 16, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 742], "lines": [{"bbox": [298, 731, 314, 744], "spans": [{"bbox": [298, 731, 314, 744], "score": 1.0, "content": "17", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [71, 70, 542, 102], "lines": [{"bbox": [95, 74, 541, 88], "spans": [{"bbox": [95, 74, 541, 88], "score": 1.0, "content": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we", "type": "text"}], "index": 0}, {"bbox": [68, 85, 484, 106], "spans": [{"bbox": [68, 85, 264, 106], "score": 1.0, "content": "will sketch here the most interesting ", "type": "text"}, {"bbox": [265, 88, 296, 104], "score": 0.88, "content": "(A_{r}^{(1)})", "type": "inline_equation", "height": 16, "width": 31}, {"bbox": [297, 85, 416, 106], "score": 1.0, "content": " and the most difficult ", "type": "text"}, {"bbox": [417, 88, 449, 104], "score": 0.92, "content": "(E_{8}^{(1)})", "type": "inline_equation", "height": 16, "width": 32}, {"bbox": [449, 85, 484, 106], "score": 1.0, "content": " cases.", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [68, 74, 541, 106]}, {"type": "text", "bbox": [70, 103, 541, 192], "lines": [{"bbox": [94, 104, 542, 121], "spans": [{"bbox": [94, 104, 167, 121], "score": 1.0, "content": "Consider first", "type": "text"}, {"bbox": [168, 106, 190, 119], "score": 0.92, "content": "A_{r,k}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [190, 104, 263, 121], "score": 1.0, "content": ". By choosing ", "type": "text"}, {"bbox": [264, 106, 339, 119], "score": 0.91, "content": "a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [339, 104, 479, 121], "score": 1.0, "content": " in (4.1), we get that either ", "type": "text"}, {"bbox": [479, 106, 522, 117], "score": 0.93, "content": "\\lambda=k\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [522, 104, 542, 121], "score": 1.0, "content": "for", "type": "text"}], "index": 2}, {"bbox": [70, 119, 540, 135], "spans": [{"bbox": [70, 119, 100, 135], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [100, 121, 106, 129], "score": 0.86, "content": "\\ell", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [106, 119, 182, 135], "score": 1.0, "content": ", in which case ", "type": "text"}, {"bbox": [182, 120, 190, 130], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [190, 119, 335, 135], "score": 1.0, "content": " is a simple-current and (for ", "type": "text"}, {"bbox": [335, 120, 364, 132], "score": 0.9, "content": "k\\neq1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [364, 119, 371, 135], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [371, 120, 446, 133], "score": 0.93, "content": "D(\\lambda)<\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [446, 119, 465, 135], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [465, 120, 540, 133], "score": 0.93, "content": "\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})", "type": "inline_equation", "height": 13, "width": 75}], "index": 3}, {"bbox": [70, 132, 541, 150], "spans": [{"bbox": [70, 132, 117, 150], "score": 1.0, "content": "for some ", "type": "text"}, {"bbox": [117, 135, 123, 144], "score": 0.87, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [123, 132, 213, 150], "score": 1.0, "content": ", with equality iff", "type": "text"}, {"bbox": [214, 135, 256, 146], "score": 0.91, "content": "\\lambda\\in S\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [257, 132, 407, 150], "score": 1.0, "content": ". But then rank-level duality ", "type": "text"}, {"bbox": [407, 135, 494, 147], "score": 0.92, "content": "A_{r,k}\\leftrightarrow A_{k-1,r+1}", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [495, 132, 541, 150], "score": 1.0, "content": " (defined", "type": "text"}], "index": 4}, {"bbox": [71, 148, 540, 165], "spans": [{"bbox": [71, 151, 104, 165], "score": 1.0, "content": "as for ", "type": "text"}, {"bbox": [104, 152, 126, 164], "score": 0.92, "content": "C_{r,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [126, 151, 251, 165], "score": 1.0, "content": ", and which is exact for ", "type": "text"}, {"bbox": [252, 152, 273, 164], "score": 0.91, "content": "A_{r,k}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [273, 151, 432, 165], "score": 1.0, "content": " q-dimensions) and (4.1) with ", "type": "text"}, {"bbox": [432, 148, 515, 162], "score": 0.94, "content": "a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}", "type": "inline_equation", "height": 14, "width": 83}, {"bbox": [515, 151, 540, 165], "score": 1.0, "content": " give", "type": "text"}], "index": 5}, {"bbox": [70, 165, 541, 182], "spans": [{"bbox": [70, 165, 87, 182], "score": 1.0, "content": "us ", "type": "text"}, {"bbox": [87, 165, 282, 180], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 15, "width": 195}, {"bbox": [282, 165, 380, 182], "score": 1.0, "content": ", with equality iff", "type": "text"}, {"bbox": [380, 168, 412, 177], "score": 0.92, "content": "\\ell\\,=\\,1", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [412, 165, 432, 182], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [433, 171, 439, 177], "score": 0.86, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [439, 165, 541, 182], "score": 1.0, "content": ". Com bining these", "type": "text"}], "index": 6}, {"bbox": [71, 181, 243, 196], "spans": [{"bbox": [71, 181, 243, 196], "score": 1.0, "content": "results yields Proposition 4.1(a).", "type": "text"}], "index": 7}], "index": 4.5, "bbox_fs": [70, 104, 542, 196]}, {"type": "text", "bbox": [70, 193, 541, 288], "lines": [{"bbox": [93, 194, 541, 212], "spans": [{"bbox": [93, 194, 117, 212], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 197, 140, 209], "score": 0.93, "content": "E_{8,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [140, 194, 244, 212], "score": 1.0, "content": ", run through each ", "type": "text"}, {"bbox": [244, 196, 357, 211], "score": 0.94, "content": "a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}", "type": "inline_equation", "height": 15, "width": 113}, {"bbox": [358, 194, 541, 212], "score": 1.0, "content": " to reduce the proof to comparing", "type": "text"}], "index": 8}, {"bbox": [71, 209, 543, 232], "spans": [{"bbox": [71, 213, 104, 226], "score": 0.93, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [104, 209, 135, 232], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [136, 212, 181, 230], "score": 0.95, "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "type": "inline_equation", "height": 18, "width": 45}, {"bbox": [182, 209, 204, 232], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [204, 214, 232, 225], "score": 0.91, "content": "i\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [233, 209, 255, 232], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [255, 213, 286, 226], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{i})", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [287, 209, 309, 232], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [309, 214, 349, 225], "score": 0.93, "content": "i\\neq0,1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [349, 209, 543, 232], "score": 1.0, "content": " (the argument in [18] unnecessarily", "type": "text"}], "index": 9}, {"bbox": [71, 228, 540, 244], "spans": [{"bbox": [71, 228, 540, 244], "score": 1.0, "content": "complicated things by restricting to integral weights). Standard arguments (see [18] for", "type": "text"}], "index": 10}, {"bbox": [69, 241, 543, 262], "spans": [{"bbox": [69, 241, 296, 262], "score": 1.0, "content": "details) quickly show that the q-dimension ", "type": "text"}, {"bbox": [297, 243, 342, 261], "score": 0.95, "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "type": "inline_equation", "height": 18, "width": 45}, {"bbox": [342, 241, 499, 262], "score": 1.0, "content": " monotonically increases with ", "type": "text"}, {"bbox": [499, 245, 506, 254], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [506, 241, 524, 262], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [524, 249, 536, 254], "score": 0.84, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 12}, {"bbox": [537, 241, 543, 262], "score": 1.0, "content": ",", "type": "text"}], "index": 11}, {"bbox": [72, 261, 542, 275], "spans": [{"bbox": [72, 261, 103, 275], "score": 1.0, "content": "while ", "type": "text"}, {"bbox": [104, 262, 135, 274], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{i})", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [135, 261, 296, 275], "score": 1.0, "content": " monotonically increases with ", "type": "text"}, {"bbox": [297, 263, 304, 271], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [304, 261, 448, 275], "score": 1.0, "content": " to the Weyl dimension of ", "type": "text"}, {"bbox": [448, 263, 461, 273], "score": 0.92, "content": "\\Lambda_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [461, 261, 542, 275], "score": 1.0, "content": ". The proof of", "type": "text"}], "index": 12}, {"bbox": [70, 276, 371, 290], "spans": [{"bbox": [70, 276, 371, 290], "score": 1.0, "content": "Proposition 4.1(e8) then reduces to a short computation.", "type": "text"}], "index": 13}], "index": 10.5, "bbox_fs": [69, 194, 543, 290]}, {"type": "title", "bbox": [72, 301, 218, 316], "lines": [{"bbox": [71, 304, 219, 316], "spans": [{"bbox": [71, 304, 120, 316], "score": 1.0, "content": "4.2. The ", "type": "text"}, {"bbox": [121, 306, 130, 314], "score": 0.33, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [130, 304, 219, 316], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 323, 541, 366], "lines": [{"bbox": [93, 325, 541, 339], "spans": [{"bbox": [93, 325, 159, 339], "score": 1.0, "content": "Recall that ", "type": "text"}, {"bbox": [159, 327, 215, 337], "score": 0.93, "content": "\\overline{r}\\,=\\,r\\,+\\,1", "type": "inline_equation", "height": 10, "width": 56}, {"bbox": [216, 325, 401, 339], "score": 1.0, "content": ". Proposition 4.1(a) tells us that ", "type": "text"}, {"bbox": [401, 325, 484, 338], "score": 0.94, "content": "\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [484, 325, 541, 339], "score": 1.0, "content": ", for some", "type": "text"}], "index": 15}, {"bbox": [71, 339, 542, 354], "spans": [{"bbox": [71, 342, 88, 353], "score": 0.91, "content": "a,b", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [89, 339, 144, 354], "score": 1.0, "content": ". Hitting ", "type": "text"}, {"bbox": [144, 345, 151, 350], "score": 0.84, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [152, 339, 185, 354], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [185, 342, 200, 350], "score": 0.9, "content": "C^{a}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [201, 339, 464, 354], "score": 1.0, "content": ", we can assume without loss of generality that ", "type": "text"}, {"bbox": [464, 342, 497, 351], "score": 0.91, "content": "a\\;=\\;0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [497, 339, 542, 354], "score": 1.0, "content": ". Write", "type": "text"}], "index": 16}, {"bbox": [71, 353, 461, 369], "spans": [{"bbox": [71, 355, 137, 368], "score": 0.92, "content": "\\pi(J0)=J^{c}0", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [137, 353, 170, 369], "score": 1.0, "content": "; then ", "type": "text"}, {"bbox": [171, 359, 178, 365], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [178, 353, 312, 369], "score": 1.0, "content": " can be a permutation of ", "type": "text"}, {"bbox": [312, 356, 327, 367], "score": 0.92, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [328, 353, 369, 369], "score": 1.0, "content": " only if ", "type": "text"}, {"bbox": [369, 359, 375, 365], "score": 0.88, "content": "c", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [375, 353, 449, 369], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [450, 357, 456, 365], "score": 0.89, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [456, 353, 461, 369], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16, "bbox_fs": [71, 325, 542, 369]}, {"type": "text", "bbox": [88, 365, 518, 380], "lines": [{"bbox": [93, 366, 516, 384], "spans": [{"bbox": [93, 366, 106, 384], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [107, 370, 136, 379], "score": 0.92, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [136, 366, 166, 384], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [167, 370, 281, 382], "score": 0.93, "content": "P_{+}=\\{0,J0,\\dots,J^{r}0\\}", "type": "inline_equation", "height": 12, "width": 114}, {"bbox": [281, 366, 299, 384], "score": 1.0, "content": " so ", "type": "text"}, {"bbox": [299, 370, 363, 382], "score": 0.94, "content": "\\pi=\\pi[c-1]", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [363, 366, 482, 384], "score": 1.0, "content": ". Thus we can assume ", "type": "text"}, {"bbox": [482, 370, 511, 381], "score": 0.95, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [511, 366, 516, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [93, 366, 516, 384]}, {"type": "text", "bbox": [69, 381, 541, 530], "lines": [{"bbox": [93, 381, 541, 398], "spans": [{"bbox": [93, 381, 230, 398], "score": 1.0, "content": "Useful is the coefficient of ", "type": "text"}, {"bbox": [231, 384, 238, 393], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [238, 381, 353, 398], "score": 1.0, "content": " in the tensor product ", "type": "text"}, {"bbox": [353, 384, 419, 395], "score": 0.91, "content": "\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 66}, {"bbox": [419, 381, 426, 398], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [426, 384, 468, 396], "score": 0.29, "content": "\\ell\\,\\mathrm{times})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [468, 381, 541, 398], "score": 1.0, "content": ": it is 0 unless", "type": "text"}], "index": 19}, {"bbox": [71, 393, 543, 419], "spans": [{"bbox": [71, 398, 115, 411], "score": 0.93, "content": "t(\\lambda)=\\ell", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [115, 393, 288, 419], "score": 1.0, "content": ", in which case the coefficient is", "type": "text"}, {"bbox": [288, 396, 309, 414], "score": 0.94, "content": "\\frac{\\ell!}{h(\\lambda)}", "type": "inline_equation", "height": 18, "width": 21}, {"bbox": [309, 393, 543, 419], "score": 1.0, "content": " (to get this, compare (3.1) above with [27,", "type": "text"}], "index": 20}, {"bbox": [70, 414, 541, 430], "spans": [{"bbox": [70, 414, 334, 430], "score": 1.0, "content": "p.114]) — we equate here the fundamental weights ", "type": "text"}, {"bbox": [335, 416, 349, 427], "score": 0.91, "content": "\\Lambda_{\\overline{{r}}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [349, 414, 373, 430], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [374, 417, 388, 428], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [388, 414, 430, 430], "score": 1.0, "content": ", so e.g. ", "type": "text"}, {"bbox": [430, 414, 458, 430], "score": 0.91, "content": "^{\\star}{\\frac{k}{r}}\\Lambda_{\\overline{{r}}}^{\\ \\ \\,}{}^{\\ ,}", "type": "inline_equation", "height": 16, "width": 28}, {"bbox": [458, 414, 498, 430], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [498, 416, 509, 426], "score": 0.39, "content": "\\mathrm{\\Delta}^{\\prime}0^{\\circ}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [509, 414, 541, 430], "score": 1.0, "content": " when", "type": "text"}], "index": 21}, {"bbox": [71, 429, 541, 445], "spans": [{"bbox": [71, 432, 77, 440], "score": 0.86, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [78, 429, 120, 445], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [120, 431, 127, 440], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [128, 429, 167, 445], "score": 1.0, "content": ". Here, ", "type": "text"}, {"bbox": [167, 430, 242, 443], "score": 0.95, "content": "h(\\lambda)=\\prod h(x)", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [243, 429, 541, 445], "score": 1.0, "content": " is the product of the hook-lengths of the Young diagram", "type": "text"}], "index": 22}, {"bbox": [70, 443, 539, 460], "spans": [{"bbox": [70, 443, 162, 459], "score": 1.0, "content": "corresponding to ", "type": "text"}, {"bbox": [163, 446, 170, 454], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [171, 443, 381, 459], "score": 1.0, "content": ". Equation (2.4) tells us that as long as ", "type": "text"}, {"bbox": [381, 444, 446, 457], "score": 0.91, "content": "t(\\lambda)=\\ell\\leq k", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [447, 443, 517, 459], "score": 1.0, "content": ", the number", "type": "text"}, {"bbox": [518, 443, 539, 460], "score": 0.93, "content": "\\frac{\\ell!}{h(\\lambda)}", "type": "inline_equation", "height": 17, "width": 21}], "index": 23}, {"bbox": [69, 460, 542, 476], "spans": [{"bbox": [69, 460, 226, 476], "score": 1.0, "content": "will also be the coefficient of ", "type": "text"}, {"bbox": [227, 463, 243, 474], "score": 0.93, "content": "N_{\\lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [243, 460, 353, 476], "score": 1.0, "content": " in the fusion power ", "type": "text"}, {"bbox": [353, 461, 389, 475], "score": 0.92, "content": "(N_{\\Lambda_{1}})^{\\ell}", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [389, 460, 454, 476], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [454, 463, 506, 474], "score": 0.92, "content": "J0=k\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [506, 460, 542, 476], "score": 1.0, "content": " is the", "type": "text"}], "index": 24}, {"bbox": [70, 475, 542, 491], "spans": [{"bbox": [70, 475, 348, 491], "score": 1.0, "content": "only simple-current appearing in the fusion product ", "type": "text"}, {"bbox": [348, 476, 362, 488], "score": 0.73, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [362, 475, 415, 491], "score": 1.0, "content": " × · · · × ", "type": "text"}, {"bbox": [415, 475, 430, 488], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [430, 475, 438, 491], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [439, 476, 446, 487], "score": 0.67, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [446, 475, 542, 491], "score": 1.0, "content": " times). Thus the", "type": "text"}], "index": 25}, {"bbox": [70, 489, 542, 505], "spans": [{"bbox": [70, 489, 354, 505], "score": 1.0, "content": "only nontrivial simple-current appearing in the fusion ", "type": "text"}, {"bbox": [354, 489, 376, 502], "score": 0.81, "content": "\\pi\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [376, 489, 429, 505], "score": 1.0, "content": " × · · · × ", "type": "text"}, {"bbox": [429, 489, 451, 502], "score": 0.87, "content": "\\pi\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [451, 489, 492, 505], "score": 1.0, "content": " will be", "type": "text"}, {"bbox": [493, 489, 525, 501], "score": 0.83, "content": "J^{b k}J0", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [526, 489, 542, 505], "score": 1.0, "content": " (0", "type": "text"}], "index": 26}, {"bbox": [72, 505, 541, 518], "spans": [{"bbox": [72, 505, 150, 518], "score": 1.0, "content": "will appear iff", "type": "text"}, {"bbox": [151, 507, 157, 515], "score": 0.86, "content": "\\overline{r}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [158, 505, 203, 518], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [203, 506, 210, 515], "score": 0.82, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [210, 505, 262, 518], "score": 1.0, "content": "). Hence ", "type": "text"}, {"bbox": [263, 506, 322, 516], "score": 0.89, "content": "b k+1\\equiv c", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [323, 505, 358, 518], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [359, 506, 366, 515], "score": 0.59, "content": "\\overline{r}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [366, 505, 484, 518], "score": 1.0, "content": ") must be coprime to ", "type": "text"}, {"bbox": [484, 506, 491, 515], "score": 0.76, "content": "\\overline{r}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [492, 505, 541, 518], "score": 1.0, "content": ". This is", "type": "text"}], "index": 27}, {"bbox": [72, 519, 472, 533], "spans": [{"bbox": [72, 520, 251, 533], "score": 1.0, "content": "precisely the condition needed for ", "type": "text"}, {"bbox": [251, 519, 271, 532], "score": 0.92, "content": "\\pi[b]", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [271, 520, 472, 533], "score": 1.0, "content": " to be a simple-current automorphism.", "type": "text"}], "index": 28}], "index": 23.5, "bbox_fs": [69, 381, 543, 533]}, {"type": "text", "bbox": [70, 531, 541, 631], "lines": [{"bbox": [94, 532, 543, 547], "spans": [{"bbox": [94, 533, 289, 547], "score": 1.0, "content": "In other words, it suffices to consider ", "type": "text"}, {"bbox": [289, 533, 340, 545], "score": 0.92, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [340, 533, 398, 547], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [398, 532, 457, 546], "score": 0.92, "content": "\\pi[J0]=J0", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [457, 533, 543, 547], "score": 1.0, "content": ". We are done if", "type": "text"}], "index": 29}, {"bbox": [71, 547, 541, 563], "spans": [{"bbox": [71, 550, 99, 558], "score": 0.89, "content": "r=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [100, 547, 160, 563], "score": 1.0, "content": ", so assume ", "type": "text"}, {"bbox": [160, 549, 188, 560], "score": 0.89, "content": "r\\geq2", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [189, 547, 246, 563], "score": 1.0, "content": ". From the ", "type": "text"}, {"bbox": [246, 547, 261, 560], "score": 0.78, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [261, 547, 277, 563], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [278, 547, 293, 560], "score": 0.77, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [293, 547, 394, 563], "score": 1.0, "content": " fusion, we get that ", "type": "text"}, {"bbox": [394, 547, 481, 561], "score": 0.91, "content": "\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}", "type": "inline_equation", "height": 14, "width": 87}, {"bbox": [482, 547, 541, 563], "score": 1.0, "content": ". Note that", "type": "text"}], "index": 30}, {"bbox": [71, 561, 540, 576], "spans": [{"bbox": [71, 563, 91, 574], "score": 0.91, "content": "k\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [92, 561, 431, 576], "score": 1.0, "content": " occurs (with multiplicity 1) in the tensor and fusion product of ", "type": "text"}, {"bbox": [431, 561, 452, 574], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [452, 561, 482, 576], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [482, 561, 534, 574], "score": 0.69, "content": "k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [534, 561, 540, 576], "score": 1.0, "content": "s,", "type": "text"}], "index": 31}, {"bbox": [70, 575, 541, 590], "spans": [{"bbox": [70, 576, 378, 590], "score": 1.0, "content": "but that it doesn’t in the tensor (hence fusion) product of ", "type": "text"}, {"bbox": [379, 576, 393, 588], "score": 0.89, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [393, 576, 423, 590], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [423, 575, 469, 588], "score": 0.79, "content": "k-2~\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [470, 576, 541, 590], "score": 1.0, "content": "’s (recall that", "type": "text"}], "index": 32}, {"bbox": [71, 589, 540, 606], "spans": [{"bbox": [71, 591, 186, 604], "score": 0.92, "content": "k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [187, 590, 425, 606], "score": 1.0, "content": " in the usual partial order on weights). Since ", "type": "text"}, {"bbox": [425, 590, 440, 603], "score": 0.72, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [440, 590, 456, 606], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [457, 590, 472, 603], "score": 0.74, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [472, 590, 524, 606], "score": 1.0, "content": " × · · · ×", "type": "text"}, {"bbox": [524, 589, 540, 603], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 16}], "index": 33}, {"bbox": [69, 603, 542, 621], "spans": [{"bbox": [69, 604, 163, 621], "score": 1.0, "content": "does not contain ", "type": "text"}, {"bbox": [164, 606, 178, 616], "score": 0.8, "content": "J0", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [178, 604, 185, 621], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [185, 603, 348, 618], "score": 0.25, "content": "(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})", "type": "inline_equation", "height": 15, "width": 163}, {"bbox": [349, 604, 449, 621], "score": 1.0, "content": " should also avoid ", "type": "text"}, {"bbox": [449, 604, 513, 618], "score": 0.93, "content": "\\pi(J0)\\,=\\,J0", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [513, 604, 542, 621], "score": 1.0, "content": ", and", "type": "text"}], "index": 34}, {"bbox": [70, 618, 218, 634], "spans": [{"bbox": [70, 618, 97, 634], "score": 1.0, "content": "thus ", "type": "text"}, {"bbox": [97, 621, 118, 631], "score": 0.92, "content": "\\pi\\Lambda_{2}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [119, 618, 192, 634], "score": 1.0, "content": " cannot equal ", "type": "text"}, {"bbox": [193, 619, 213, 632], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [213, 618, 218, 634], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 32, "bbox_fs": [69, 532, 543, 634]}, {"type": "text", "bbox": [70, 632, 541, 703], "lines": [{"bbox": [94, 632, 543, 649], "spans": [{"bbox": [94, 632, 174, 649], "score": 1.0, "content": "Thus we know ", "type": "text"}, {"bbox": [174, 633, 226, 646], "score": 0.91, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [226, 632, 315, 649], "score": 1.0, "content": ". The remaining ", "type": "text"}, {"bbox": [316, 633, 366, 646], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [366, 632, 543, 649], "score": 1.0, "content": "follow quickly from induction: if", "type": "text"}], "index": 36}, {"bbox": [71, 647, 541, 663], "spans": [{"bbox": [71, 649, 121, 660], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [121, 647, 173, 663], "score": 1.0, "content": "for some ", "type": "text"}, {"bbox": [173, 648, 224, 660], "score": 0.87, "content": "2\\leq\\ell<r", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [225, 647, 314, 663], "score": 1.0, "content": ", then the fusion ", "type": "text"}, {"bbox": [314, 647, 360, 660], "score": 0.44, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [360, 647, 404, 663], "score": 1.0, "content": "tells us ", "type": "text"}, {"bbox": [405, 648, 537, 662], "score": 0.93, "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "type": "inline_equation", "height": 14, "width": 132}, {"bbox": [537, 647, 541, 663], "score": 1.0, "content": ".", "type": "text"}], "index": 37}, {"bbox": [70, 661, 542, 678], "spans": [{"bbox": [70, 661, 95, 678], "score": 1.0, "content": "But ", "type": "text"}, {"bbox": [96, 662, 222, 675], "score": 0.92, "content": "h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [222, 661, 249, 678], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [250, 662, 347, 676], "score": 0.92, "content": "h(\\Lambda_{\\ell+1})=(\\ell+1)!", "type": "inline_equation", "height": 14, "width": 97}, {"bbox": [348, 661, 370, 678], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 663, 446, 675], "score": 0.92, "content": "\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [446, 661, 487, 678], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [487, 664, 495, 673], "score": 0.72, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [495, 661, 542, 678], "score": 1.0, "content": " fixes all", "type": "text"}], "index": 38}, {"bbox": [70, 676, 542, 691], "spans": [{"bbox": [70, 676, 542, 691], "score": 1.0, "content": "fundamental weights, and since these comprise a fusion-generator (see the discussion at", "type": "text"}], "index": 39}, {"bbox": [70, 689, 384, 707], "spans": [{"bbox": [70, 689, 231, 707], "score": 1.0, "content": "the end of §2.2) we know that ", "type": "text"}, {"bbox": [232, 695, 240, 701], "score": 0.67, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [240, 689, 362, 707], "score": 1.0, "content": " must fix everything in ", "type": "text"}, {"bbox": [362, 692, 378, 704], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [379, 689, 384, 707], "score": 1.0, "content": ".", "type": "text"}], "index": 40}], "index": 38, "bbox_fs": [70, 632, 543, 707]}]}
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[{"type": "text", "bbox": [71, 70, 542, 102], "content": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we will sketch here the most interesting and the most difficult cases.", "index": 0}, {"type": "text", "bbox": [70, 103, 541, 192], "content": "Consider first . By choosing in (4.1), we get that either for some , in which case is a simple-current and (for ) , or for some , with equality iff . But then rank-level duality (defined as for , and which is exact for q-dimensions) and (4.1) with give us , with equality iff or . Com bining these results yields Proposition 4.1(a).", "index": 1}, {"type": "text", "bbox": [70, 193, 541, 288], "content": "For , run through each to reduce the proof to comparing with for , or for (the argument in [18] unnecessarily complicated things by restricting to integral weights). Standard arguments (see [18] for details) quickly show that the q-dimension monotonically increases with to , while monotonically increases with to the Weyl dimension of . The proof of Proposition 4.1(e8) then reduces to a short computation.", "index": 2}, {"type": "title", "bbox": [72, 301, 218, 316], "content": "4.2. The -series argument", "index": 3}, {"type": "text", "bbox": [70, 323, 541, 366], "content": "Recall that . Proposition 4.1(a) tells us that , for some . Hitting with , we can assume without loss of generality that . Write ; then can be a permutation of only if is coprime to .", "index": 4}, {"type": "text", "bbox": [88, 365, 518, 380], "content": "If then so . Thus we can assume .", "index": 5}, {"type": "text", "bbox": [69, 381, 541, 530], "content": "Useful is the coefficient of in the tensor product ( : it is 0 unless , in which case the coefficient is (to get this, compare (3.1) above with [27, p.114]) — we equate here the fundamental weights and , so e.g. equals when divides . Here, is the product of the hook-lengths of the Young diagram corresponding to . Equation (2.4) tells us that as long as , the number will also be the coefficient of in the fusion power . Note that is the only simple-current appearing in the fusion product × · · · × ( times). Thus the only nontrivial simple-current appearing in the fusion × · · · × will be (0 will appear iff divides ). Hence (mod ) must be coprime to . This is precisely the condition needed for to be a simple-current automorphism.", "index": 6}, {"type": "text", "bbox": [70, 531, 541, 631], "content": "In other words, it suffices to consider and hence . We are done if , so assume . From the × fusion, we get that . Note that occurs (with multiplicity 1) in the tensor and fusion product of with s, but that it doesn’t in the tensor (hence fusion) product of with ’s (recall that in the usual partial order on weights). Since × × · · · × does not contain , should also avoid , and thus cannot equal .", "index": 7}, {"type": "text", "bbox": [70, 632, 541, 703], "content": "Thus we know . The remaining follow quickly from induction: if for some , then the fusion tells us . But and , so . Thus fixes all fundamental weights, and since these comprise a fusion-generator (see the discussion at the end of §2.2) we know that must fix everything in .", "index": 8}]
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[{"bbox": [95, 74, 541, 88], "content": "The complete proof of Proposition 4.1 is given in [18], but to illustrate the ideas we", "parent_index": 0, "line_index": 0}, {"bbox": [68, 85, 484, 106], "content": "will sketch here the most interesting and the most difficult cases.", "parent_index": 0, "line_index": 1}, {"bbox": [94, 104, 542, 121], "content": "Consider first . By choosing in (4.1), we get that either for", "parent_index": 1, "line_index": 0}, {"bbox": [70, 119, 540, 135], "content": "some , in which case is a simple-current and (for ) , or", "parent_index": 1, "line_index": 1}, {"bbox": [70, 132, 541, 150], "content": "for some , with equality iff . But then rank-level duality (defined", "parent_index": 1, "line_index": 2}, {"bbox": [71, 148, 540, 165], "content": "as for , and which is exact for q-dimensions) and (4.1) with give", "parent_index": 1, "line_index": 3}, {"bbox": [70, 165, 541, 182], "content": "us , with equality iff or . Com bining these", "parent_index": 1, "line_index": 4}, {"bbox": [71, 181, 243, 196], "content": "results yields Proposition 4.1(a).", "parent_index": 1, "line_index": 5}, {"bbox": [93, 194, 541, 212], "content": "For , run through each to reduce the proof to comparing", "parent_index": 2, "line_index": 0}, {"bbox": [71, 209, 543, 232], "content": "with for , or for (the argument in [18] unnecessarily", "parent_index": 2, "line_index": 1}, {"bbox": [71, 228, 540, 244], "content": "complicated things by restricting to integral weights). Standard arguments (see [18] for", "parent_index": 2, "line_index": 2}, {"bbox": [69, 241, 543, 262], "content": "details) quickly show that the q-dimension monotonically increases with to ,", "parent_index": 2, "line_index": 3}, {"bbox": [72, 261, 542, 275], "content": "while monotonically increases with to the Weyl dimension of . The proof of", "parent_index": 2, "line_index": 4}, {"bbox": [70, 276, 371, 290], "content": "Proposition 4.1(e8) then reduces to a short computation.", "parent_index": 2, "line_index": 5}, {"bbox": [71, 304, 219, 316], "content": "4.2. The -series argument", "parent_index": 3, "line_index": 0}, {"bbox": [93, 325, 541, 339], "content": "Recall that . Proposition 4.1(a) tells us that , for some", "parent_index": 4, "line_index": 0}, {"bbox": [71, 339, 542, 354], "content": ". Hitting with , we can assume without loss of generality that . Write", "parent_index": 4, "line_index": 1}, {"bbox": [71, 353, 461, 369], "content": "; then can be a permutation of only if is coprime to .", "parent_index": 4, "line_index": 2}, {"bbox": [93, 366, 516, 384], "content": "If then so . Thus we can assume .", "parent_index": 5, "line_index": 0}, {"bbox": [93, 381, 541, 398], "content": "Useful is the coefficient of in the tensor product ( : it is 0 unless", "parent_index": 6, "line_index": 0}, {"bbox": [71, 393, 543, 419], "content": ", in which case the coefficient is (to get this, compare (3.1) above with [27,", "parent_index": 6, "line_index": 1}, {"bbox": [70, 414, 541, 430], "content": "p.114]) — we equate here the fundamental weights and , so e.g. equals when", "parent_index": 6, "line_index": 2}, {"bbox": [71, 429, 541, 445], "content": "divides . Here, is the product of the hook-lengths of the Young diagram", "parent_index": 6, "line_index": 3}, {"bbox": [70, 443, 539, 460], "content": "corresponding to . Equation (2.4) tells us that as long as , the number", "parent_index": 6, "line_index": 4}, {"bbox": [69, 460, 542, 476], "content": "will also be the coefficient of in the fusion power . Note that is the", "parent_index": 6, "line_index": 5}, {"bbox": [70, 475, 542, 491], "content": "only simple-current appearing in the fusion product × · · · × ( times). Thus the", "parent_index": 6, "line_index": 6}, {"bbox": [70, 489, 542, 505], "content": "only nontrivial simple-current appearing in the fusion × · · · × will be (0", "parent_index": 6, "line_index": 7}, {"bbox": [72, 505, 541, 518], "content": "will appear iff divides ). Hence (mod ) must be coprime to . This is", "parent_index": 6, "line_index": 8}, {"bbox": [72, 519, 472, 533], "content": "precisely the condition needed for to be a simple-current automorphism.", "parent_index": 6, "line_index": 9}, {"bbox": [94, 532, 543, 547], "content": "In other words, it suffices to consider and hence . We are done if", "parent_index": 7, "line_index": 0}, {"bbox": [71, 547, 541, 563], "content": ", so assume . From the × fusion, we get that . Note that", "parent_index": 7, "line_index": 1}, {"bbox": [71, 561, 540, 576], "content": "occurs (with multiplicity 1) in the tensor and fusion product of with s,", "parent_index": 7, "line_index": 2}, {"bbox": [70, 575, 541, 590], "content": "but that it doesn’t in the tensor (hence fusion) product of with ’s (recall that", "parent_index": 7, "line_index": 3}, {"bbox": [71, 589, 540, 606], "content": "in the usual partial order on weights). Since × × · · · ×", "parent_index": 7, "line_index": 4}, {"bbox": [69, 603, 542, 621], "content": "does not contain , should also avoid , and", "parent_index": 7, "line_index": 5}, {"bbox": [70, 618, 218, 634], "content": "thus cannot equal .", "parent_index": 7, "line_index": 6}, {"bbox": [94, 632, 543, 649], "content": "Thus we know . The remaining follow quickly from induction: if", "parent_index": 8, "line_index": 0}, {"bbox": [71, 647, 541, 663], "content": "for some , then the fusion tells us .", "parent_index": 8, "line_index": 1}, {"bbox": [70, 661, 542, 678], "content": "But and , so . Thus fixes all", "parent_index": 8, "line_index": 2}, {"bbox": [70, 676, 542, 691], "content": "fundamental weights, and since these comprise a fusion-generator (see the discussion at", "parent_index": 8, "line_index": 3}, {"bbox": [70, 689, 384, 707], "content": "the end of §2.2) we know that must fix everything in .", "parent_index": 8, "line_index": 4}]
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[]
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[{"bbox": [265, 88, 296, 104], "content": "(A_{r}^{(1)})", "parent_index": 0, "subtype": "inline"}, {"bbox": [417, 88, 449, 104], "content": "(E_{8}^{(1)})", "parent_index": 0, "subtype": "inline"}, {"bbox": [168, 106, 190, 119], "content": "A_{r,k}", "parent_index": 1, "subtype": "inline"}, {"bbox": [264, 106, 339, 119], "content": "a\\!-\\!b=\\Lambda_{i}-\\Lambda_{j}", "parent_index": 1, "subtype": "inline"}, {"bbox": [479, 106, 522, 117], "content": "\\lambda=k\\Lambda_{\\ell}", "parent_index": 1, "subtype": "inline"}, {"bbox": [100, 121, 106, 129], "content": "\\ell", "parent_index": 1, "subtype": "inline"}, {"bbox": [182, 120, 190, 130], "content": "\\lambda", "parent_index": 1, "subtype": "inline"}, {"bbox": [335, 120, 364, 132], "content": "k\\neq1", "parent_index": 1, "subtype": "inline"}, {"bbox": [371, 120, 446, 133], "content": "D(\\lambda)<\\mathcal{D}(\\Lambda_{1})", "parent_index": 1, "subtype": "inline"}, {"bbox": [465, 120, 540, 133], "content": "\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})", "parent_index": 1, "subtype": "inline"}, {"bbox": [117, 135, 123, 144], "content": "\\ell", "parent_index": 1, "subtype": "inline"}, {"bbox": [214, 135, 256, 146], "content": "\\lambda\\in S\\Lambda_{\\ell}", "parent_index": 1, "subtype": "inline"}, {"bbox": [407, 135, 494, 147], "content": "A_{r,k}\\leftrightarrow A_{k-1,r+1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [104, 152, 126, 164], "content": "C_{r,k}", "parent_index": 1, "subtype": "inline"}, {"bbox": [252, 152, 273, 164], "content": "A_{r,k}", "parent_index": 1, "subtype": "inline"}, {"bbox": [432, 148, 515, 162], "content": "a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [87, 165, 282, 180], "content": "\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})", "parent_index": 1, "subtype": "inline"}, {"bbox": [380, 168, 412, 177], "content": "\\ell\\,=\\,1", "parent_index": 1, "subtype": "inline"}, {"bbox": [433, 171, 439, 177], "content": "r", "parent_index": 1, "subtype": "inline"}, {"bbox": [117, 197, 140, 209], "content": "E_{8,k}", "parent_index": 2, "subtype": "inline"}, {"bbox": [244, 196, 357, 211], "content": "a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 213, 104, 226], "content": "\\mathcal{D}(\\Lambda_{1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [136, 212, 181, 230], "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "parent_index": 2, "subtype": "inline"}, {"bbox": [204, 214, 232, 225], "content": "i\\neq0", "parent_index": 2, "subtype": "inline"}, {"bbox": [255, 213, 286, 226], "content": "\\mathcal{D}(\\Lambda_{i})", "parent_index": 2, "subtype": "inline"}, {"bbox": [309, 214, 349, 225], "content": "i\\neq0,1", "parent_index": 2, "subtype": "inline"}, {"bbox": [297, 243, 342, 261], "content": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})", "parent_index": 2, "subtype": "inline"}, {"bbox": [499, 245, 506, 254], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [524, 249, 536, 254], "content": "\\infty", "parent_index": 2, "subtype": "inline"}, {"bbox": [104, 262, 135, 274], "content": "\\mathcal{D}(\\Lambda_{i})", "parent_index": 2, "subtype": "inline"}, {"bbox": [297, 263, 304, 271], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [448, 263, 461, 273], "content": "\\Lambda_{i}", "parent_index": 2, "subtype": "inline"}, {"bbox": [121, 306, 130, 314], "content": "A", "parent_index": 3, "subtype": "inline"}, {"bbox": [159, 327, 215, 337], "content": "\\overline{r}\\,=\\,r\\,+\\,1", "parent_index": 4, "subtype": "inline"}, {"bbox": [401, 325, 484, 338], "content": "\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [71, 342, 88, 353], "content": "a,b", "parent_index": 4, "subtype": "inline"}, {"bbox": [144, 345, 151, 350], "content": "\\pi", "parent_index": 4, "subtype": "inline"}, {"bbox": [185, 342, 200, 350], "content": "C^{a}", "parent_index": 4, "subtype": "inline"}, {"bbox": [464, 342, 497, 351], "content": "a\\;=\\;0", "parent_index": 4, "subtype": "inline"}, {"bbox": [71, 355, 137, 368], "content": "\\pi(J0)=J^{c}0", "parent_index": 4, "subtype": "inline"}, {"bbox": [171, 359, 178, 365], "content": "\\pi", "parent_index": 4, "subtype": "inline"}, {"bbox": [312, 356, 327, 367], "content": "P_{+}", "parent_index": 4, "subtype": "inline"}, {"bbox": [369, 359, 375, 365], "content": "c", "parent_index": 4, "subtype": "inline"}, {"bbox": [450, 357, 456, 365], "content": "\\overline{r}", "parent_index": 4, "subtype": "inline"}, {"bbox": [107, 370, 136, 379], "content": "k=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [167, 370, 281, 382], "content": "P_{+}=\\{0,J0,\\dots,J^{r}0\\}", "parent_index": 5, "subtype": "inline"}, {"bbox": [299, 370, 363, 382], "content": "\\pi=\\pi[c-1]", "parent_index": 5, "subtype": "inline"}, {"bbox": [482, 370, 511, 381], "content": "k\\geq2", "parent_index": 5, "subtype": "inline"}, {"bbox": [231, 384, 238, 393], "content": "\\lambda", "parent_index": 6, "subtype": "inline"}, {"bbox": [353, 384, 419, 395], "content": "\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [426, 384, 468, 396], "content": "\\ell\\,\\mathrm{times})", "parent_index": 6, "subtype": "inline"}, {"bbox": [71, 398, 115, 411], "content": "t(\\lambda)=\\ell", "parent_index": 6, "subtype": "inline"}, {"bbox": [288, 396, 309, 414], "content": "\\frac{\\ell!}{h(\\lambda)}", "parent_index": 6, "subtype": "inline"}, {"bbox": [335, 416, 349, 427], "content": "\\Lambda_{\\overline{{r}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [374, 417, 388, 428], "content": 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"N_{\\lambda}", "parent_index": 6, "subtype": "inline"}, {"bbox": [353, 461, 389, 475], "content": "(N_{\\Lambda_{1}})^{\\ell}", "parent_index": 6, "subtype": "inline"}, {"bbox": [454, 463, 506, 474], "content": "J0=k\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [348, 476, 362, 488], "content": "\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [415, 475, 430, 488], "content": "\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [439, 476, 446, 487], "content": "k", "parent_index": 6, "subtype": "inline"}, {"bbox": [354, 489, 376, 502], "content": "\\pi\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [429, 489, 451, 502], "content": "\\pi\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [493, 489, 525, 501], "content": "J^{b k}J0", "parent_index": 6, "subtype": "inline"}, {"bbox": [151, 507, 157, 515], "content": "\\overline{r}", "parent_index": 6, "subtype": "inline"}, {"bbox": [203, 506, 210, 515], "content": "k", "parent_index": 6, "subtype": "inline"}, {"bbox": [263, 506, 322, 516], "content": "b k+1\\equiv c", "parent_index": 6, "subtype": "inline"}, {"bbox": [359, 506, 366, 515], "content": "\\overline{r}", "parent_index": 6, "subtype": "inline"}, {"bbox": [484, 506, 491, 515], "content": "\\overline{r}", "parent_index": 6, "subtype": "inline"}, {"bbox": [251, 519, 271, 532], "content": "\\pi[b]", "parent_index": 6, "subtype": "inline"}, {"bbox": [289, 533, 340, 545], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [398, 532, 457, 546], "content": "\\pi[J0]=J0", "parent_index": 7, "subtype": "inline"}, {"bbox": [71, 550, 99, 558], "content": "r=1", "parent_index": 7, "subtype": "inline"}, {"bbox": [160, 549, 188, 560], "content": "r\\geq2", "parent_index": 7, "subtype": "inline"}, {"bbox": [246, 547, 261, 560], "content": "\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [278, 547, 293, 560], "content": "\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [394, 547, 481, 561], "content": "\\pi\\Lambda_{2}\\in\\{\\Lambda_{2},2\\Lambda_{1}\\}", "parent_index": 7, "subtype": "inline"}, {"bbox": [71, 563, 91, 574], "content": "k\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [431, 561, 452, 574], "content": "2\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [482, 561, 534, 574], "content": "k-2\\;\\Lambda_{1}\\,{}^{\\prime}\\mathrm{s}", "parent_index": 7, "subtype": "inline"}, {"bbox": [379, 576, 393, 588], "content": "\\Lambda_{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [423, 575, 469, 588], "content": "k-2~\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [71, 591, 186, 604], "content": "k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [425, 590, 440, 603], "content": "\\Lambda_{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [457, 590, 472, 603], "content": "\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [524, 589, 540, 603], "content": "\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [164, 606, 178, 616], "content": "J0", "parent_index": 7, "subtype": "inline"}, {"bbox": [185, 603, 348, 618], "content": "(\\pi\\Lambda_{2})\\boxtimes(\\pi\\Lambda_{1})\\boxtimes\\dots\\boxtimes(\\pi\\Lambda_{1})", "parent_index": 7, "subtype": "inline"}, {"bbox": [449, 604, 513, 618], "content": "\\pi(J0)\\,=\\,J0", "parent_index": 7, "subtype": "inline"}, {"bbox": [97, 621, 118, 631], "content": "\\pi\\Lambda_{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [193, 619, 213, 632], "content": "2\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [174, 633, 226, 646], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [316, 633, 366, 646], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "parent_index": 8, "subtype": "inline"}, {"bbox": [71, 649, 121, 660], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "parent_index": 8, "subtype": "inline"}, {"bbox": [173, 648, 224, 660], "content": "2\\leq\\ell<r", "parent_index": 8, "subtype": "inline"}, {"bbox": [314, 647, 360, 660], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{\\ell}", "parent_index": 8, "subtype": "inline"}, {"bbox": [405, 648, 537, 662], "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "parent_index": 8, "subtype": "inline"}, {"bbox": [96, 662, 222, 675], "content": "h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell", "parent_index": 8, "subtype": "inline"}, {"bbox": [250, 662, 347, 676], "content": "h(\\Lambda_{\\ell+1})=(\\ell+1)!", "parent_index": 8, "subtype": "inline"}, {"bbox": [371, 663, 446, 675], "content": "\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [487, 664, 495, 673], "content": "\\pi", "parent_index": 8, "subtype": "inline"}, {"bbox": [232, 695, 240, 701], "content": "\\pi", "parent_index": 8, "subtype": "inline"}, {"bbox": [362, 692, 378, 704], "content": "P_{+}", "parent_index": 8, "subtype": "inline"}]
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[]
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$k=1$ is easy: $P_{+}=\{0,J0,\Lambda_{r}\}$ and $\pi=i d$ . is automatic. $k=2$ will be done later in this subsection. Assume now that $k\geq3$ .
From Proposition 4.1(b) we can write $\pi\Lambda_{1}\,=\,J^{a}\Lambda_{1}$ and $\pi^{\prime}\Lambda_{1}\,=\,J^{a^{\prime}}\Lambda_{1}$ . We know $\pi J0\,=\,J0$ , so (2.7b) says $\pi$ must take spinors to spinors, and nonspinors to nonspinors. Then we will have $\chi_{\Lambda_{1}}[\psi]\,=\,(-1)^{a^{\prime}}\chi_{\Lambda_{1}}[\pi\psi]$ for any spinor $\psi$ . Now if $a^{\prime}=1$ , then $\pi$ will take the spinors which maximize $\chi_{\Lambda_{1}}$ , to those which minimize it. Both these maxima and minima can be easily found from (3.2). Thus we get that $\pi(S\Lambda_{r})$ equals $k\Lambda_{r}$ (when $k$ odd) or $S((k-1)\Lambda_{r})$ (when $k$ even). But the sets $S\Lambda_{r}$ and $k\Lambda_{r}$ have different cardinalities ( $k\Lambda_{r}$ is a $J$ -fixed-point), and so can’t get mapped to each other. Also, the fusions $\Lambda_{1}$ × $\Lambda_{r}=\Lambda_{r}$ + $(\Lambda_{1}+\Lambda_{r})$ and $J^{a}\Lambda_{1}\boxtimes\left(J^{i}(k-1)\Lambda_{r}\right)=\left(J^{a+i}(k-1)\Lambda_{r}\right)$ + $(J^{a+i+1}(k-$ $1)\Lambda_{r})$ + $J^{a+i+1}(\Lambda_{r-1}+(k-3)\Lambda_{r})$ have different numbers of weights on their right sides, so also $\pi\Lambda_{r}\notin{\cal S}(k-1)\Lambda_{r}$ .
Thus $a^{\prime}=0$ and $\pi\Lambda_{r}=J^{b}\Lambda_{r}$ for some $b$ . Similarly, $a=0$ . Hitting $\pi$ with $\pi[1]^{b}$ , we may assume that $\pi$ fixes $\Lambda_{r}$ .
Now assume $\pi$ fixes $\Lambda_{\ell}$ , for $1\leq\ell<r-1$ . Then the fusion $\Lambda_{1}$ × $\Lambda_{\ell}$ says that $\pi\Lambda_{\ell+1}$ equals $\Lambda_{\ell+1}$ or $\Lambda_{1}+\Lambda_{\ell}$ . But from (3.2) we find
$$
\begin{array}{c}{{-\chi_{\Lambda_{1}}[\Lambda_{1}+\Lambda_{\ell}]=2\,\{\displaystyle\mathrm{cos}(\pi\frac{2r-2\ell+1}{\kappa})-\mathrm{cos}(\pi\frac{2r-2\ell-1}{\kappa})+\mathrm{cos}(\pi\frac{2r+1}{\kappa})\}}}\\ {{-\cos(\pi\frac{2r+3}{\kappa})\}=4\cos(\pi\frac{2r-\ell+1}{\kappa})\,\{\displaystyle\mathrm{cos}(2\pi\frac{\ell}{\kappa})-\mathrm{cos}(2\pi\frac{\ell+1}{\kappa})\},}}\end{array}
$$
Hence $\pi$ will fix $\Lambda_{\ell+1}$ if it fixes $\Lambda_{\ell}$ , concluding the argument.
Now consider the more interesting case: $k=2$ . Then $\kappa=2r+1$ ; recall the weights in $P_{+}(B_{r,2})$ are the simple-currents $0,J0$ , the $J$ -fixed-points $\gamma^{1},\ldots,\gamma^{r}$ (notation defined in §3.2), and the spinors $\Lambda_{r},J\Lambda_{r}$ . Because $\pi(J0)\,=\,\pi^{\prime}(J0)\,=\,J0$ , we know both $\pi$ and $\pi^{\prime}$ must take $J$ -fixed-points to $J$ -fixed-points, i.e. $\pi\Lambda_{1}\,=\,\gamma^{m}$ and $\pi^{\prime}\Lambda_{1}\,=\,\gamma^{m^{\prime}}$ for some $1\leq m,m^{\prime}\leq r$ . It is easy to compute [25]
$$
\frac{S_{\gamma^{a}\gamma^{b}}}{S_{0\gamma^{b}}}=2\cos(2\pi\frac{a b}{\kappa})\ .
$$
From this we see $m\,m^{\prime}\equiv\pm1$ (mod $\kappa$ ), so $^{\prime\prime}$ is coprime to $\kappa$ . Hitting it with the Galois fusion-symmetry $\pi\{m^{\prime}\}$ , we see that we may assume $\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ .
Now use (4.2) to get $\pi\gamma^{i}=\gamma^{i}$ for all $i$ . Then $\pi$ equals the identity or $\pi[1]$ , depending on what $\pi$ does to $\Lambda_{r}$ .
# 4.4. The $C$ -series argument
By rank-level duality, we may take $r\le k$ . For now assume $(r,k)\neq(2,3)$ . Then we know $\pi\Lambda_{1}=J^{a}\Lambda_{1}$ and $\pi\Lambda_{1}=J^{a^{\prime}}\Lambda_{1}$ for some $a,a^{\prime}$ . Since $\pi J0=\pi^{\prime}J0=J0$ , (2.7b) says $a=a^{\prime}=0$ if $k r$ is odd. Since $\chi_{\Lambda_{1}}[\Lambda_{1}]>0$ (using (3.3)), $S_{\Lambda_{1}\Lambda_{1}}=S_{J^{a}\Lambda_{1},J^{a^{\prime}}\Lambda_{1}}$ implies that $a=a^{\prime}$ also holds when $k r$ is even, and hence we may assume (hitting with $\pi[1]^{a}$ ) that also $a=a^{\prime}=0$ holds for $k r$ even. From the fusion $\Lambda_{1}$ × $\Lambda_{\ell}$ we get $\pi\Lambda_{\ell+1}\in\{\Lambda_{\ell+1},\Lambda_{1}+\Lambda_{\ell}\}$ if $\pi\Lambda_{\ell}=\Lambda_{\ell}$ ; for $r<k$ conclude the argument with the calculation
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<p data-bbox="70 93 541 122">$k=1$ is easy: $P_{+}=\{0,J0,\Lambda_{r}\}$ and $\pi=i d$ . is automatic. $k=2$ will be done later in this subsection. Assume now that $k\geq3$ . </p>
<p data-bbox="70 123 541 266">From Proposition 4.1(b) we can write $\pi\Lambda_{1}\,=\,J^{a}\Lambda_{1}$ and $\pi^{\prime}\Lambda_{1}\,=\,J^{a^{\prime}}\Lambda_{1}$ . We know $\pi J0\,=\,J0$ , so (2.7b) says $\pi$ must take spinors to spinors, and nonspinors to nonspinors. Then we will have $\chi_{\Lambda_{1}}[\psi]\,=\,(-1)^{a^{\prime}}\chi_{\Lambda_{1}}[\pi\psi]$ for any spinor $\psi$ . Now if $a^{\prime}=1$ , then $\pi$ will take the spinors which maximize $\chi_{\Lambda_{1}}$ , to those which minimize it. Both these maxima and minima can be easily found from (3.2). Thus we get that $\pi(S\Lambda_{r})$ equals $k\Lambda_{r}$ (when $k$ odd) or $S((k-1)\Lambda_{r})$ (when $k$ even). But the sets $S\Lambda_{r}$ and $k\Lambda_{r}$ have different cardinalities ( $k\Lambda_{r}$ is a $J$ -fixed-point), and so can’t get mapped to each other. Also, the fusions $\Lambda_{1}$ × $\Lambda_{r}=\Lambda_{r}$ + $(\Lambda_{1}+\Lambda_{r})$ and $J^{a}\Lambda_{1}\boxtimes\left(J^{i}(k-1)\Lambda_{r}\right)=\left(J^{a+i}(k-1)\Lambda_{r}\right)$ + $(J^{a+i+1}(k-$ $1)\Lambda_{r})$ + $J^{a+i+1}(\Lambda_{r-1}+(k-3)\Lambda_{r})$ have different numbers of weights on their right sides, so also $\pi\Lambda_{r}\notin{\cal S}(k-1)\Lambda_{r}$ . </p>
<p data-bbox="70 267 541 295">Thus $a^{\prime}=0$ and $\pi\Lambda_{r}=J^{b}\Lambda_{r}$ for some $b$ . Similarly, $a=0$ . Hitting $\pi$ with $\pi[1]^{b}$ , we may assume that $\pi$ fixes $\Lambda_{r}$ . </p>
<p data-bbox="70 296 540 326">Now assume $\pi$ fixes $\Lambda_{\ell}$ , for $1\leq\ell<r-1$ . Then the fusion $\Lambda_{1}$ × $\Lambda_{\ell}$ says that $\pi\Lambda_{\ell+1}$ equals $\Lambda_{\ell+1}$ or $\Lambda_{1}+\Lambda_{\ell}$ . But from (3.2) we find </p>
<div class="equation" data-bbox="124 340 527 401">$$
\begin{array}{c}{{-\chi_{\Lambda_{1}}[\Lambda_{1}+\Lambda_{\ell}]=2\,\{\displaystyle\mathrm{cos}(\pi\frac{2r-2\ell+1}{\kappa})-\mathrm{cos}(\pi\frac{2r-2\ell-1}{\kappa})+\mathrm{cos}(\pi\frac{2r+1}{\kappa})\}}}\\ {{-\cos(\pi\frac{2r+3}{\kappa})\}=4\cos(\pi\frac{2r-\ell+1}{\kappa})\,\{\displaystyle\mathrm{cos}(2\pi\frac{\ell}{\kappa})-\mathrm{cos}(2\pi\frac{\ell+1}{\kappa})\},}}\end{array}
$$</div>
<p data-bbox="70 409 391 425">Hence $\pi$ will fix $\Lambda_{\ell+1}$ if it fixes $\Lambda_{\ell}$ , concluding the argument. </p>
<p data-bbox="70 428 541 501">Now consider the more interesting case: $k=2$ . Then $\kappa=2r+1$ ; recall the weights in $P_{+}(B_{r,2})$ are the simple-currents $0,J0$ , the $J$ -fixed-points $\gamma^{1},\ldots,\gamma^{r}$ (notation defined in §3.2), and the spinors $\Lambda_{r},J\Lambda_{r}$ . Because $\pi(J0)\,=\,\pi^{\prime}(J0)\,=\,J0$ , we know both $\pi$ and $\pi^{\prime}$ must take $J$ -fixed-points to $J$ -fixed-points, i.e. $\pi\Lambda_{1}\,=\,\gamma^{m}$ and $\pi^{\prime}\Lambda_{1}\,=\,\gamma^{m^{\prime}}$ for some $1\leq m,m^{\prime}\leq r$ . It is easy to compute [25] </p>
<div class="equation" data-bbox="247 515 365 547">$$
\frac{S_{\gamma^{a}\gamma^{b}}}{S_{0\gamma^{b}}}=2\cos(2\pi\frac{a b}{\kappa})\ .
$$</div>
<p data-bbox="70 560 541 589">From this we see $m\,m^{\prime}\equiv\pm1$ (mod $\kappa$ ), so $^{\prime\prime}$ is coprime to $\kappa$ . Hitting it with the Galois fusion-symmetry $\pi\{m^{\prime}\}$ , we see that we may assume $\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ . </p>
<p data-bbox="71 590 541 619">Now use (4.2) to get $\pi\gamma^{i}=\gamma^{i}$ for all $i$ . Then $\pi$ equals the identity or $\pi[1]$ , depending on what $\pi$ does to $\Lambda_{r}$ . </p>
<h1 data-bbox="71 634 218 649">4.4. The $C$ -series argument </h1>
<p data-bbox="70 657 541 715">By rank-level duality, we may take $r\le k$ . For now assume $(r,k)\neq(2,3)$ . Then we know $\pi\Lambda_{1}=J^{a}\Lambda_{1}$ and $\pi\Lambda_{1}=J^{a^{\prime}}\Lambda_{1}$ for some $a,a^{\prime}$ . Since $\pi J0=\pi^{\prime}J0=J0$ , (2.7b) says $a=a^{\prime}=0$ if $k r$ is odd. Since $\chi_{\Lambda_{1}}[\Lambda_{1}]>0$ (using (3.3)), $S_{\Lambda_{1}\Lambda_{1}}=S_{J^{a}\Lambda_{1},J^{a^{\prime}}\Lambda_{1}}$ implies that $a=a^{\prime}$ also holds when $k r$ is even, and hence we may assume (hitting with $\pi[1]^{a}$ ) that also $a=a^{\prime}=0$ holds for $k r$ even. From the fusion $\Lambda_{1}$ × $\Lambda_{\ell}$ we get $\pi\Lambda_{\ell+1}\in\{\Lambda_{\ell+1},\Lambda_{1}+\Lambda_{\ell}\}$ if $\pi\Lambda_{\ell}=\Lambda_{\ell}$ ; for $r<k$ conclude the argument with the calculation </p>
</body></html>
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0002044v1
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[{"type": "text", "text": "$k=1$ is easy: $P_{+}=\\{0,J0,\\Lambda_{r}\\}$ and $\\pi=i d$ . is automatic. $k=2$ will be done later in this subsection. Assume now that $k\\geq3$ . ", "page_idx": 17}, {"type": "text", "text": "From Proposition 4.1(b) we can write $\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}$ and $\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}$ . We know $\\pi J0\\,=\\,J0$ , so (2.7b) says $\\pi$ must take spinors to spinors, and nonspinors to nonspinors. Then we will have $\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]$ for any spinor $\\psi$ . Now if $a^{\\prime}=1$ , then $\\pi$ will take the spinors which maximize $\\chi_{\\Lambda_{1}}$ , to those which minimize it. Both these maxima and minima can be easily found from (3.2). Thus we get that $\\pi(S\\Lambda_{r})$ equals $k\\Lambda_{r}$ (when $k$ odd) or $S((k-1)\\Lambda_{r})$ (when $k$ even). But the sets $S\\Lambda_{r}$ and $k\\Lambda_{r}$ have different cardinalities ( $k\\Lambda_{r}$ is a $J$ -fixed-point), and so can’t get mapped to each other. Also, the fusions $\\Lambda_{1}$ × $\\Lambda_{r}=\\Lambda_{r}$ + $(\\Lambda_{1}+\\Lambda_{r})$ and $J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)$ + $(J^{a+i+1}(k-$ $1)\\Lambda_{r})$ + $J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})$ have different numbers of weights on their right sides, so also $\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}$ . ", "page_idx": 17}, {"type": "text", "text": "Thus $a^{\\prime}=0$ and $\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}$ for some $b$ . Similarly, $a=0$ . Hitting $\\pi$ with $\\pi[1]^{b}$ , we may assume that $\\pi$ fixes $\\Lambda_{r}$ . ", "page_idx": 17}, {"type": "text", "text": "Now assume $\\pi$ fixes $\\Lambda_{\\ell}$ , for $1\\leq\\ell<r-1$ . Then the fusion $\\Lambda_{1}$ × $\\Lambda_{\\ell}$ says that $\\pi\\Lambda_{\\ell+1}$ equals $\\Lambda_{\\ell+1}$ or $\\Lambda_{1}+\\Lambda_{\\ell}$ . But from (3.2) we find ", "page_idx": 17}, {"type": "equation", "text": "$$\n\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}\n$$", "text_format": "latex", "page_idx": 17}, {"type": "text", "text": "Hence $\\pi$ will fix $\\Lambda_{\\ell+1}$ if it fixes $\\Lambda_{\\ell}$ , concluding the argument. ", "page_idx": 17}, {"type": "text", "text": "Now consider the more interesting case: $k=2$ . Then $\\kappa=2r+1$ ; recall the weights in $P_{+}(B_{r,2})$ are the simple-currents $0,J0$ , the $J$ -fixed-points $\\gamma^{1},\\ldots,\\gamma^{r}$ (notation defined in §3.2), and the spinors $\\Lambda_{r},J\\Lambda_{r}$ . Because $\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0$ , we know both $\\pi$ and $\\pi^{\\prime}$ must take $J$ -fixed-points to $J$ -fixed-points, i.e. $\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}$ and $\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}$ for some $1\\leq m,m^{\\prime}\\leq r$ . It is easy to compute [25] ", "page_idx": 17}, {"type": "equation", "text": "$$\n\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .\n$$", "text_format": "latex", "page_idx": 17}, {"type": "text", "text": "From this we see $m\\,m^{\\prime}\\equiv\\pm1$ (mod $\\kappa$ ), so $^{\\prime\\prime}$ is coprime to $\\kappa$ . Hitting it with the Galois fusion-symmetry $\\pi\\{m^{\\prime}\\}$ , we see that we may assume $\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ . ", "page_idx": 17}, {"type": "text", "text": "Now use (4.2) to get $\\pi\\gamma^{i}=\\gamma^{i}$ for all $i$ . Then $\\pi$ equals the identity or $\\pi[1]$ , depending on what $\\pi$ does to $\\Lambda_{r}$ . ", "page_idx": 17}, {"type": "text", "text": "4.4. The $C$ -series argument ", "text_level": 1, "page_idx": 17}, {"type": "text", "text": "By rank-level duality, we may take $r\\le k$ . For now assume $(r,k)\\neq(2,3)$ . Then we know $\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}$ and $\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}$ for some $a,a^{\\prime}$ . Since $\\pi J0=\\pi^{\\prime}J0=J0$ , (2.7b) says $a=a^{\\prime}=0$ if $k r$ is odd. Since $\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0$ (using (3.3)), $S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}$ implies that $a=a^{\\prime}$ also holds when $k r$ is even, and hence we may assume (hitting with $\\pi[1]^{a}$ ) that also $a=a^{\\prime}=0$ holds for $k r$ even. From the fusion $\\Lambda_{1}$ × $\\Lambda_{\\ell}$ we get $\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}$ if $\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$ ; for $r<k$ conclude the argument with the calculation ", "page_idx": 17}]
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{"preproc_blocks": [{"type": "text", "bbox": [70, 93, 541, 122], "lines": [{"bbox": [95, 95, 541, 111], "spans": [{"bbox": [95, 98, 124, 107], "score": 0.9, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [124, 95, 171, 111], "score": 1.0, "content": " is easy: ", "type": "text"}, {"bbox": [171, 97, 259, 110], "score": 0.95, "content": "P_{+}=\\{0,J0,\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [259, 95, 286, 111], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [286, 98, 321, 107], "score": 0.92, "content": "\\pi=i d", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [321, 95, 400, 111], "score": 1.0, "content": ". is automatic. ", "type": "text"}, {"bbox": [401, 98, 430, 107], "score": 0.92, "content": "k=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [430, 95, 541, 111], "score": 1.0, "content": " will be done later in", "type": "text"}], "index": 0}, {"bbox": [72, 111, 285, 124], "spans": [{"bbox": [72, 111, 252, 124], "score": 1.0, "content": "this subsection. Assume now that ", "type": "text"}, {"bbox": [252, 113, 281, 123], "score": 0.92, "content": "k\\geq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [281, 111, 285, 124], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [70, 123, 541, 266], "lines": [{"bbox": [93, 124, 541, 140], "spans": [{"bbox": [93, 125, 304, 140], "score": 1.0, "content": "From Proposition 4.1(b) we can write ", "type": "text"}, {"bbox": [304, 127, 372, 138], "score": 0.95, "content": "\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [373, 125, 402, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [402, 124, 478, 138], "score": 0.95, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 76}, {"bbox": [478, 125, 541, 140], "score": 1.0, "content": ". We know", "type": "text"}], "index": 2}, {"bbox": [71, 140, 540, 155], "spans": [{"bbox": [71, 142, 125, 151], "score": 0.88, "content": "\\pi J0\\,=\\,J0", "type": "inline_equation", "height": 9, "width": 54}, {"bbox": [125, 140, 209, 155], "score": 1.0, "content": ", so (2.7b) says ", "type": "text"}, {"bbox": [209, 145, 217, 150], "score": 0.81, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [217, 140, 540, 155], "score": 1.0, "content": " must take spinors to spinors, and nonspinors to nonspinors.", "type": "text"}], "index": 3}, {"bbox": [70, 153, 542, 169], "spans": [{"bbox": [70, 154, 172, 169], "score": 1.0, "content": "Then we will have ", "type": "text"}, {"bbox": [172, 153, 300, 168], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]", "type": "inline_equation", "height": 15, "width": 128}, {"bbox": [300, 154, 381, 169], "score": 1.0, "content": " for any spinor ", "type": "text"}, {"bbox": [382, 156, 390, 167], "score": 0.9, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [391, 154, 439, 169], "score": 1.0, "content": ". Now if ", "type": "text"}, {"bbox": [440, 155, 474, 165], "score": 0.88, "content": "a^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [474, 154, 508, 169], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [509, 158, 516, 165], "score": 0.61, "content": "\\pi", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [517, 154, 542, 169], "score": 1.0, "content": " will", "type": "text"}], "index": 4}, {"bbox": [70, 168, 542, 184], "spans": [{"bbox": [70, 168, 251, 184], "score": 1.0, "content": "take the spinors which maximize ", "type": "text"}, {"bbox": [251, 173, 271, 182], "score": 0.86, "content": "\\chi_{\\Lambda_{1}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [271, 168, 542, 184], "score": 1.0, "content": ", to those which minimize it. Both these maxima", "type": "text"}], "index": 5}, {"bbox": [70, 182, 541, 198], "spans": [{"bbox": [70, 182, 403, 198], "score": 1.0, "content": "and minima can be easily found from (3.2). Thus we get that ", "type": "text"}, {"bbox": [403, 183, 442, 196], "score": 0.92, "content": "\\pi(S\\Lambda_{r})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [442, 182, 482, 198], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [482, 182, 503, 195], "score": 0.89, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [504, 182, 541, 198], "score": 1.0, "content": " (when", "type": "text"}], "index": 6}, {"bbox": [70, 196, 541, 212], "spans": [{"bbox": [70, 197, 79, 209], "score": 0.74, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [79, 198, 126, 212], "score": 1.0, "content": " odd) or ", "type": "text"}, {"bbox": [127, 196, 196, 211], "score": 0.92, "content": "S((k-1)\\Lambda_{r})", "type": "inline_equation", "height": 15, "width": 69}, {"bbox": [196, 198, 236, 212], "score": 1.0, "content": " (when ", "type": "text"}, {"bbox": [236, 198, 244, 208], "score": 0.74, "content": "k", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [244, 198, 357, 212], "score": 1.0, "content": " even). But the sets ", "type": "text"}, {"bbox": [357, 199, 379, 210], "score": 0.92, "content": "S\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [379, 198, 407, 212], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [408, 198, 429, 210], "score": 0.89, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [429, 198, 541, 212], "score": 1.0, "content": " have different cardi-", "type": "text"}], "index": 7}, {"bbox": [71, 211, 541, 226], "spans": [{"bbox": [71, 212, 116, 226], "score": 1.0, "content": "nalities (", "type": "text"}, {"bbox": [116, 211, 138, 224], "score": 0.88, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [138, 212, 162, 226], "score": 1.0, "content": " is a ", "type": "text"}, {"bbox": [163, 212, 171, 223], "score": 0.83, "content": "J", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [172, 212, 541, 226], "score": 1.0, "content": "-fixed-point), and so can’t get mapped to each other. Also, the fusions", "type": "text"}], "index": 8}, {"bbox": [71, 224, 541, 241], "spans": [{"bbox": [71, 225, 85, 238], "score": 0.86, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [86, 225, 101, 241], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [102, 225, 147, 238], "score": 0.89, "content": "\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [147, 225, 164, 241], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [164, 225, 215, 239], "score": 0.89, "content": "(\\Lambda_{1}+\\Lambda_{r})", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [216, 225, 241, 241], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [241, 226, 457, 240], "score": 0.85, "content": "J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)", "type": "inline_equation", "height": 14, "width": 216}, {"bbox": [457, 225, 476, 241], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [477, 224, 541, 240], "score": 0.85, "content": "(J^{a+i+1}(k-", "type": "inline_equation", "height": 16, "width": 64}], "index": 9}, {"bbox": [70, 237, 541, 257], "spans": [{"bbox": [70, 239, 99, 254], "score": 0.85, "content": "1)\\Lambda_{r})", "type": "inline_equation", "height": 15, "width": 29}, {"bbox": [99, 237, 117, 257], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [118, 240, 254, 254], "score": 0.9, "content": "J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})", "type": "inline_equation", "height": 14, "width": 136}, {"bbox": [254, 237, 541, 257], "score": 1.0, "content": " have different numbers of weights on their right sides,", "type": "text"}], "index": 10}, {"bbox": [70, 254, 209, 269], "spans": [{"bbox": [70, 255, 109, 269], "score": 1.0, "content": "so also ", "type": "text"}, {"bbox": [110, 254, 204, 268], "score": 0.92, "content": "\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}", "type": "inline_equation", "height": 14, "width": 94}, {"bbox": [204, 255, 209, 269], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 6.5}, {"type": "text", "bbox": [70, 267, 541, 295], "lines": [{"bbox": [94, 268, 541, 285], "spans": [{"bbox": [94, 268, 125, 285], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [126, 269, 160, 281], "score": 0.9, "content": "a^{\\prime}=0", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 268, 186, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [186, 269, 251, 282], "score": 0.95, "content": "\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [252, 268, 304, 285], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [304, 271, 310, 280], "score": 0.85, "content": "b", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [310, 268, 372, 285], "score": 1.0, "content": ". Similarly, ", "type": "text"}, {"bbox": [372, 270, 403, 281], "score": 0.84, "content": "a=0", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [403, 268, 453, 285], "score": 1.0, "content": ". Hitting ", "type": "text"}, {"bbox": [453, 272, 462, 280], "score": 0.69, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [462, 268, 492, 285], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [493, 268, 518, 283], "score": 0.92, "content": "\\pi[1]^{b}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [519, 268, 541, 285], "score": 1.0, "content": ", we", "type": "text"}], "index": 12}, {"bbox": [71, 284, 220, 298], "spans": [{"bbox": [71, 284, 163, 298], "score": 1.0, "content": "may assume that", "type": "text"}, {"bbox": [164, 286, 172, 295], "score": 0.75, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [173, 284, 202, 298], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [202, 285, 216, 296], "score": 0.89, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [217, 284, 220, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [70, 296, 540, 326], "lines": [{"bbox": [93, 297, 539, 315], "spans": [{"bbox": [93, 297, 163, 315], "score": 1.0, "content": "Now assume ", "type": "text"}, {"bbox": [163, 301, 172, 309], "score": 0.76, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [172, 297, 201, 315], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [202, 300, 215, 311], "score": 0.88, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [216, 297, 240, 315], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [241, 300, 311, 311], "score": 0.91, "content": "1\\leq\\ell<r-1", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [311, 297, 406, 315], "score": 1.0, "content": ". Then the fusion ", "type": "text"}, {"bbox": [406, 298, 421, 311], "score": 0.79, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [421, 297, 437, 315], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [438, 298, 452, 311], "score": 0.73, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [452, 297, 506, 315], "score": 1.0, "content": "says that ", "type": "text"}, {"bbox": [506, 299, 539, 312], "score": 0.89, "content": "\\pi\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 13, "width": 33}], "index": 14}, {"bbox": [71, 313, 321, 327], "spans": [{"bbox": [71, 313, 106, 327], "score": 1.0, "content": "equals", "type": "text"}, {"bbox": [107, 313, 133, 326], "score": 0.91, "content": "\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [133, 313, 150, 327], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [151, 313, 192, 325], "score": 0.91, "content": "\\Lambda_{1}+\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [193, 313, 321, 327], "score": 1.0, "content": ". But from (3.2) we find", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [124, 340, 527, 401], "lines": [{"bbox": [124, 340, 527, 401], "spans": [{"bbox": [124, 340, 527, 401], "score": 0.86, "content": "\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [70, 409, 391, 425], "lines": [{"bbox": [70, 411, 390, 426], "spans": [{"bbox": [70, 411, 106, 426], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [107, 417, 114, 423], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [114, 411, 157, 426], "score": 1.0, "content": " will fix ", "type": "text"}, {"bbox": [158, 413, 182, 425], "score": 0.92, "content": "\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [182, 411, 236, 426], "score": 1.0, "content": " if it fixes ", "type": "text"}, {"bbox": [236, 414, 249, 424], "score": 0.9, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [250, 411, 390, 426], "score": 1.0, "content": ", concluding the argument.", "type": "text"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [70, 428, 541, 501], "lines": [{"bbox": [94, 430, 540, 446], "spans": [{"bbox": [94, 430, 311, 446], "score": 1.0, "content": "Now consider the more interesting case: ", "type": "text"}, {"bbox": [311, 432, 341, 441], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [342, 430, 382, 446], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [382, 433, 440, 442], "score": 0.91, "content": "\\kappa=2r+1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [440, 430, 540, 446], "score": 1.0, "content": "; recall the weights", "type": "text"}], "index": 18}, {"bbox": [69, 443, 542, 460], "spans": [{"bbox": [69, 443, 85, 460], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [86, 446, 132, 459], "score": 0.94, "content": "P_{+}(B_{r,2})", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [132, 443, 262, 460], "score": 1.0, "content": " are the simple-currents ", "type": "text"}, {"bbox": [262, 447, 288, 458], "score": 0.92, "content": "0,J0", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [288, 443, 316, 460], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [316, 447, 325, 456], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [325, 443, 393, 460], "score": 1.0, "content": "-fixed-points ", "type": "text"}, {"bbox": [393, 445, 444, 458], "score": 0.92, "content": "\\gamma^{1},\\ldots,\\gamma^{r}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [445, 443, 542, 460], "score": 1.0, "content": " (notation defined", "type": "text"}], "index": 19}, {"bbox": [69, 458, 541, 474], "spans": [{"bbox": [69, 458, 206, 474], "score": 1.0, "content": "in §3.2), and the spinors ", "type": "text"}, {"bbox": [207, 461, 247, 472], "score": 0.93, "content": "\\Lambda_{r},J\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [248, 458, 304, 474], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [304, 460, 421, 473], "score": 0.93, "content": "\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [421, 458, 507, 474], "score": 1.0, "content": ", we know both ", "type": "text"}, {"bbox": [508, 464, 515, 470], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [516, 458, 541, 474], "score": 1.0, "content": " and", "type": "text"}], "index": 20}, {"bbox": [71, 472, 542, 489], "spans": [{"bbox": [71, 475, 82, 484], "score": 0.9, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [82, 472, 144, 489], "score": 1.0, "content": " must take ", "type": "text"}, {"bbox": [145, 475, 153, 484], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [153, 472, 237, 489], "score": 1.0, "content": "-fixed-points to ", "type": "text"}, {"bbox": [238, 475, 246, 484], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [246, 472, 339, 489], "score": 1.0, "content": "-fixed-points, i.e. ", "type": "text"}, {"bbox": [340, 475, 397, 487], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [397, 472, 425, 489], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [425, 473, 489, 487], "score": 0.94, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [489, 472, 542, 489], "score": 1.0, "content": " for some", "type": "text"}], "index": 21}, {"bbox": [71, 487, 287, 504], "spans": [{"bbox": [71, 489, 145, 501], "score": 0.92, "content": "1\\leq m,m^{\\prime}\\leq r", "type": "inline_equation", "height": 12, "width": 74}, {"bbox": [145, 487, 287, 504], "score": 1.0, "content": ". It is easy to compute [25]", "type": "text"}], "index": 22}], "index": 20}, {"type": "interline_equation", "bbox": [247, 515, 365, 547], "lines": [{"bbox": [247, 515, 365, 547], "spans": [{"bbox": [247, 515, 365, 547], "score": 0.94, "content": "\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [70, 560, 541, 589], "lines": [{"bbox": [70, 561, 541, 578], "spans": [{"bbox": [70, 561, 165, 578], "score": 1.0, "content": "From this we see ", "type": "text"}, {"bbox": [165, 564, 225, 574], "score": 0.8, "content": "m\\,m^{\\prime}\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [226, 561, 261, 578], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [262, 567, 269, 573], "score": 0.58, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [269, 561, 296, 578], "score": 1.0, "content": "), so ", "type": "text"}, {"bbox": [297, 568, 307, 573], "score": 0.85, "content": "^{\\prime\\prime}", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [308, 561, 385, 578], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [385, 566, 393, 573], "score": 0.74, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [393, 561, 541, 578], "score": 1.0, "content": ". Hitting it with the Galois", "type": "text"}], "index": 24}, {"bbox": [70, 576, 445, 592], "spans": [{"bbox": [70, 577, 162, 592], "score": 1.0, "content": "fusion-symmetry ", "type": "text"}, {"bbox": [162, 578, 195, 591], "score": 0.93, "content": "\\pi\\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [196, 577, 349, 592], "score": 1.0, "content": ", we see that we may assume ", "type": "text"}, {"bbox": [350, 576, 441, 590], "score": 0.92, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [442, 577, 445, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "text", "bbox": [71, 590, 541, 619], "lines": [{"bbox": [94, 591, 540, 607], "spans": [{"bbox": [94, 591, 205, 607], "score": 1.0, "content": "Now use (4.2) to get ", "type": "text"}, {"bbox": [205, 591, 251, 605], "score": 0.93, "content": "\\pi\\gamma^{i}=\\gamma^{i}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [251, 591, 288, 607], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [289, 594, 293, 603], "score": 0.75, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [293, 591, 331, 607], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [332, 594, 340, 603], "score": 0.72, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [340, 591, 458, 607], "score": 1.0, "content": " equals the identity or ", "type": "text"}, {"bbox": [459, 592, 479, 605], "score": 0.74, "content": "\\pi[1]", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [479, 591, 540, 607], "score": 1.0, "content": ", depending", "type": "text"}], "index": 26}, {"bbox": [70, 605, 189, 621], "spans": [{"bbox": [70, 605, 117, 621], "score": 1.0, "content": "on what ", "type": "text"}, {"bbox": [118, 611, 125, 617], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [126, 605, 170, 621], "score": 1.0, "content": " does to ", "type": "text"}, {"bbox": [170, 608, 184, 619], "score": 0.91, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [185, 605, 189, 621], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "title", "bbox": [71, 634, 218, 649], "lines": [{"bbox": [71, 637, 218, 649], "spans": [{"bbox": [71, 637, 119, 649], "score": 1.0, "content": "4.4. The ", "type": "text"}, {"bbox": [119, 639, 129, 647], "score": 0.87, "content": "C", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [129, 637, 218, 649], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [70, 657, 541, 715], "lines": [{"bbox": [94, 659, 541, 674], "spans": [{"bbox": [94, 659, 284, 674], "score": 1.0, "content": "By rank-level duality, we may take ", "type": "text"}, {"bbox": [284, 661, 314, 672], "score": 0.93, "content": "r\\le k", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [315, 659, 413, 674], "score": 1.0, "content": ". For now assume ", "type": "text"}, {"bbox": [413, 659, 484, 673], "score": 0.91, "content": "(r,k)\\neq(2,3)", "type": "inline_equation", "height": 14, "width": 71}, {"bbox": [484, 659, 541, 674], "score": 1.0, "content": ". Then we", "type": "text"}], "index": 29}, {"bbox": [70, 672, 540, 688], "spans": [{"bbox": [70, 673, 102, 688], "score": 1.0, "content": "know ", "type": "text"}, {"bbox": [102, 676, 168, 686], "score": 0.92, "content": "\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 66}, {"bbox": [168, 673, 195, 688], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [196, 672, 265, 686], "score": 0.93, "content": "\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 69}, {"bbox": [265, 673, 318, 688], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [318, 675, 340, 687], "score": 0.91, "content": "a,a^{\\prime}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [340, 673, 380, 688], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [380, 674, 475, 686], "score": 0.86, "content": "\\pi J0=\\pi^{\\prime}J0=J0", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [475, 673, 540, 688], "score": 1.0, "content": ", (2.7b) says", "type": "text"}], "index": 30}, {"bbox": [71, 685, 542, 707], "spans": [{"bbox": [71, 689, 126, 699], "score": 0.92, "content": "a=a^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 55}, {"bbox": [126, 685, 140, 707], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [140, 690, 153, 699], "score": 0.89, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [154, 685, 228, 707], "score": 1.0, "content": " is odd. Since ", "type": "text"}, {"bbox": [228, 689, 291, 702], "score": 0.95, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [291, 685, 366, 707], "score": 1.0, "content": " (using (3.3)), ", "type": "text"}, {"bbox": [367, 689, 473, 703], "score": 0.9, "content": "S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}", "type": "inline_equation", "height": 14, "width": 106}, {"bbox": [473, 685, 542, 707], "score": 1.0, "content": " implies that", "type": "text"}], "index": 31}, {"bbox": [71, 702, 540, 717], "spans": [{"bbox": [71, 704, 103, 713], "score": 0.92, "content": "a=a^{\\prime}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [103, 702, 192, 717], "score": 1.0, "content": " also holds when ", "type": "text"}, {"bbox": [192, 704, 205, 713], "score": 0.88, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [205, 702, 459, 717], "score": 1.0, "content": " is even, and hence we may assume (hitting with ", "type": "text"}, {"bbox": [459, 703, 487, 716], "score": 0.84, "content": "\\pi[1]^{a}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [487, 702, 540, 717], "score": 1.0, "content": ") that also", "type": "text"}], "index": 32}], "index": 30.5}], "layout_bboxes": [], "page_idx": 17, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [124, 340, 527, 401], "lines": [{"bbox": [124, 340, 527, 401], "spans": [{"bbox": [124, 340, 527, 401], "score": 0.86, "content": "\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [247, 515, 365, 547], "lines": [{"bbox": [247, 515, 365, 547], "spans": [{"bbox": [247, 515, 365, 547], "score": 0.94, "content": "\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .", "type": "interline_equation"}], "index": 23}], "index": 23}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 741], "lines": [{"bbox": [298, 731, 313, 744], "spans": [{"bbox": [298, 731, 313, 744], "score": 1.0, "content": "18", "type": "text"}]}]}, {"type": "discarded", "bbox": [71, 71, 219, 85], "lines": [{"bbox": [72, 74, 219, 86], "spans": [{"bbox": [72, 74, 120, 86], "score": 1.0, "content": "4.3. The ", "type": "text"}, {"bbox": [120, 75, 131, 84], "score": 0.86, "content": "B", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [131, 74, 219, 86], "score": 1.0, "content": "-series argument", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 93, 541, 122], "lines": [{"bbox": [95, 95, 541, 111], "spans": [{"bbox": [95, 98, 124, 107], "score": 0.9, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [124, 95, 171, 111], "score": 1.0, "content": " is easy: ", "type": "text"}, {"bbox": [171, 97, 259, 110], "score": 0.95, "content": "P_{+}=\\{0,J0,\\Lambda_{r}\\}", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [259, 95, 286, 111], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [286, 98, 321, 107], "score": 0.92, "content": "\\pi=i d", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [321, 95, 400, 111], "score": 1.0, "content": ". is automatic. ", "type": "text"}, {"bbox": [401, 98, 430, 107], "score": 0.92, "content": "k=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [430, 95, 541, 111], "score": 1.0, "content": " will be done later in", "type": "text"}], "index": 0}, {"bbox": [72, 111, 285, 124], "spans": [{"bbox": [72, 111, 252, 124], "score": 1.0, "content": "this subsection. Assume now that ", "type": "text"}, {"bbox": [252, 113, 281, 123], "score": 0.92, "content": "k\\geq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [281, 111, 285, 124], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [72, 95, 541, 124]}, {"type": "text", "bbox": [70, 123, 541, 266], "lines": [{"bbox": [93, 124, 541, 140], "spans": [{"bbox": [93, 125, 304, 140], "score": 1.0, "content": "From Proposition 4.1(b) we can write ", "type": "text"}, {"bbox": [304, 127, 372, 138], "score": 0.95, "content": "\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [373, 125, 402, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [402, 124, 478, 138], "score": 0.95, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 76}, {"bbox": [478, 125, 541, 140], "score": 1.0, "content": ". We know", "type": "text"}], "index": 2}, {"bbox": [71, 140, 540, 155], "spans": [{"bbox": [71, 142, 125, 151], "score": 0.88, "content": "\\pi J0\\,=\\,J0", "type": "inline_equation", "height": 9, "width": 54}, {"bbox": [125, 140, 209, 155], "score": 1.0, "content": ", so (2.7b) says ", "type": "text"}, {"bbox": [209, 145, 217, 150], "score": 0.81, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [217, 140, 540, 155], "score": 1.0, "content": " must take spinors to spinors, and nonspinors to nonspinors.", "type": "text"}], "index": 3}, {"bbox": [70, 153, 542, 169], "spans": [{"bbox": [70, 154, 172, 169], "score": 1.0, "content": "Then we will have ", "type": "text"}, {"bbox": [172, 153, 300, 168], "score": 0.93, "content": "\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]", "type": "inline_equation", "height": 15, "width": 128}, {"bbox": [300, 154, 381, 169], "score": 1.0, "content": " for any spinor ", "type": "text"}, {"bbox": [382, 156, 390, 167], "score": 0.9, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [391, 154, 439, 169], "score": 1.0, "content": ". Now if ", "type": "text"}, {"bbox": [440, 155, 474, 165], "score": 0.88, "content": "a^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [474, 154, 508, 169], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [509, 158, 516, 165], "score": 0.61, "content": "\\pi", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [517, 154, 542, 169], "score": 1.0, "content": " will", "type": "text"}], "index": 4}, {"bbox": [70, 168, 542, 184], "spans": [{"bbox": [70, 168, 251, 184], "score": 1.0, "content": "take the spinors which maximize ", "type": "text"}, {"bbox": [251, 173, 271, 182], "score": 0.86, "content": "\\chi_{\\Lambda_{1}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [271, 168, 542, 184], "score": 1.0, "content": ", to those which minimize it. Both these maxima", "type": "text"}], "index": 5}, {"bbox": [70, 182, 541, 198], "spans": [{"bbox": [70, 182, 403, 198], "score": 1.0, "content": "and minima can be easily found from (3.2). Thus we get that ", "type": "text"}, {"bbox": [403, 183, 442, 196], "score": 0.92, "content": "\\pi(S\\Lambda_{r})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [442, 182, 482, 198], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [482, 182, 503, 195], "score": 0.89, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [504, 182, 541, 198], "score": 1.0, "content": " (when", "type": "text"}], "index": 6}, {"bbox": [70, 196, 541, 212], "spans": [{"bbox": [70, 197, 79, 209], "score": 0.74, "content": "k", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [79, 198, 126, 212], "score": 1.0, "content": " odd) or ", "type": "text"}, {"bbox": [127, 196, 196, 211], "score": 0.92, "content": "S((k-1)\\Lambda_{r})", "type": "inline_equation", "height": 15, "width": 69}, {"bbox": [196, 198, 236, 212], "score": 1.0, "content": " (when ", "type": "text"}, {"bbox": [236, 198, 244, 208], "score": 0.74, "content": "k", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [244, 198, 357, 212], "score": 1.0, "content": " even). But the sets ", "type": "text"}, {"bbox": [357, 199, 379, 210], "score": 0.92, "content": "S\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [379, 198, 407, 212], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [408, 198, 429, 210], "score": 0.89, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [429, 198, 541, 212], "score": 1.0, "content": " have different cardi-", "type": "text"}], "index": 7}, {"bbox": [71, 211, 541, 226], "spans": [{"bbox": [71, 212, 116, 226], "score": 1.0, "content": "nalities (", "type": "text"}, {"bbox": [116, 211, 138, 224], "score": 0.88, "content": "k\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [138, 212, 162, 226], "score": 1.0, "content": " is a ", "type": "text"}, {"bbox": [163, 212, 171, 223], "score": 0.83, "content": "J", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [172, 212, 541, 226], "score": 1.0, "content": "-fixed-point), and so can’t get mapped to each other. Also, the fusions", "type": "text"}], "index": 8}, {"bbox": [71, 224, 541, 241], "spans": [{"bbox": [71, 225, 85, 238], "score": 0.86, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [86, 225, 101, 241], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [102, 225, 147, 238], "score": 0.89, "content": "\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [147, 225, 164, 241], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [164, 225, 215, 239], "score": 0.89, "content": "(\\Lambda_{1}+\\Lambda_{r})", "type": "inline_equation", "height": 14, "width": 51}, {"bbox": [216, 225, 241, 241], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [241, 226, 457, 240], "score": 0.85, "content": "J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)", "type": "inline_equation", "height": 14, "width": 216}, {"bbox": [457, 225, 476, 241], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [477, 224, 541, 240], "score": 0.85, "content": "(J^{a+i+1}(k-", "type": "inline_equation", "height": 16, "width": 64}], "index": 9}, {"bbox": [70, 237, 541, 257], "spans": [{"bbox": [70, 239, 99, 254], "score": 0.85, "content": "1)\\Lambda_{r})", "type": "inline_equation", "height": 15, "width": 29}, {"bbox": [99, 237, 117, 257], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [118, 240, 254, 254], "score": 0.9, "content": "J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})", "type": "inline_equation", "height": 14, "width": 136}, {"bbox": [254, 237, 541, 257], "score": 1.0, "content": " have different numbers of weights on their right sides,", "type": "text"}], "index": 10}, {"bbox": [70, 254, 209, 269], "spans": [{"bbox": [70, 255, 109, 269], "score": 1.0, "content": "so also ", "type": "text"}, {"bbox": [110, 254, 204, 268], "score": 0.92, "content": "\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}", "type": "inline_equation", "height": 14, "width": 94}, {"bbox": [204, 255, 209, 269], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 6.5, "bbox_fs": [70, 124, 542, 269]}, {"type": "text", "bbox": [70, 267, 541, 295], "lines": [{"bbox": [94, 268, 541, 285], "spans": [{"bbox": [94, 268, 125, 285], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [126, 269, 160, 281], "score": 0.9, "content": "a^{\\prime}=0", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 268, 186, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [186, 269, 251, 282], "score": 0.95, "content": "\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [252, 268, 304, 285], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [304, 271, 310, 280], "score": 0.85, "content": "b", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [310, 268, 372, 285], "score": 1.0, "content": ". Similarly, ", "type": "text"}, {"bbox": [372, 270, 403, 281], "score": 0.84, "content": "a=0", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [403, 268, 453, 285], "score": 1.0, "content": ". Hitting ", "type": "text"}, {"bbox": [453, 272, 462, 280], "score": 0.69, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [462, 268, 492, 285], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [493, 268, 518, 283], "score": 0.92, "content": "\\pi[1]^{b}", "type": "inline_equation", "height": 15, "width": 25}, {"bbox": [519, 268, 541, 285], "score": 1.0, "content": ", we", "type": "text"}], "index": 12}, {"bbox": [71, 284, 220, 298], "spans": [{"bbox": [71, 284, 163, 298], "score": 1.0, "content": "may assume that", "type": "text"}, {"bbox": [164, 286, 172, 295], "score": 0.75, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [173, 284, 202, 298], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [202, 285, 216, 296], "score": 0.89, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [217, 284, 220, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5, "bbox_fs": [71, 268, 541, 298]}, {"type": "text", "bbox": [70, 296, 540, 326], "lines": [{"bbox": [93, 297, 539, 315], "spans": [{"bbox": [93, 297, 163, 315], "score": 1.0, "content": "Now assume ", "type": "text"}, {"bbox": [163, 301, 172, 309], "score": 0.76, "content": "\\pi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [172, 297, 201, 315], "score": 1.0, "content": " fixes ", "type": "text"}, {"bbox": [202, 300, 215, 311], "score": 0.88, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [216, 297, 240, 315], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [241, 300, 311, 311], "score": 0.91, "content": "1\\leq\\ell<r-1", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [311, 297, 406, 315], "score": 1.0, "content": ". Then the fusion ", "type": "text"}, {"bbox": [406, 298, 421, 311], "score": 0.79, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [421, 297, 437, 315], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [438, 298, 452, 311], "score": 0.73, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [452, 297, 506, 315], "score": 1.0, "content": "says that ", "type": "text"}, {"bbox": [506, 299, 539, 312], "score": 0.89, "content": "\\pi\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 13, "width": 33}], "index": 14}, {"bbox": [71, 313, 321, 327], "spans": [{"bbox": [71, 313, 106, 327], "score": 1.0, "content": "equals", "type": "text"}, {"bbox": [107, 313, 133, 326], "score": 0.91, "content": "\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [133, 313, 150, 327], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [151, 313, 192, 325], "score": 0.91, "content": "\\Lambda_{1}+\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [193, 313, 321, 327], "score": 1.0, "content": ". But from (3.2) we find", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [71, 297, 539, 327]}, {"type": "interline_equation", "bbox": [124, 340, 527, 401], "lines": [{"bbox": [124, 340, 527, 401], "spans": [{"bbox": [124, 340, 527, 401], "score": 0.86, "content": "\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [70, 409, 391, 425], "lines": [{"bbox": [70, 411, 390, 426], "spans": [{"bbox": [70, 411, 106, 426], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [107, 417, 114, 423], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [114, 411, 157, 426], "score": 1.0, "content": " will fix ", "type": "text"}, {"bbox": [158, 413, 182, 425], "score": 0.92, "content": "\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [182, 411, 236, 426], "score": 1.0, "content": " if it fixes ", "type": "text"}, {"bbox": [236, 414, 249, 424], "score": 0.9, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [250, 411, 390, 426], "score": 1.0, "content": ", concluding the argument.", "type": "text"}], "index": 17}], "index": 17, "bbox_fs": [70, 411, 390, 426]}, {"type": "text", "bbox": [70, 428, 541, 501], "lines": [{"bbox": [94, 430, 540, 446], "spans": [{"bbox": [94, 430, 311, 446], "score": 1.0, "content": "Now consider the more interesting case: ", "type": "text"}, {"bbox": [311, 432, 341, 441], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [342, 430, 382, 446], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [382, 433, 440, 442], "score": 0.91, "content": "\\kappa=2r+1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [440, 430, 540, 446], "score": 1.0, "content": "; recall the weights", "type": "text"}], "index": 18}, {"bbox": [69, 443, 542, 460], "spans": [{"bbox": [69, 443, 85, 460], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [86, 446, 132, 459], "score": 0.94, "content": "P_{+}(B_{r,2})", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [132, 443, 262, 460], "score": 1.0, "content": " are the simple-currents ", "type": "text"}, {"bbox": [262, 447, 288, 458], "score": 0.92, "content": "0,J0", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [288, 443, 316, 460], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [316, 447, 325, 456], "score": 0.88, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [325, 443, 393, 460], "score": 1.0, "content": "-fixed-points ", "type": "text"}, {"bbox": [393, 445, 444, 458], "score": 0.92, "content": "\\gamma^{1},\\ldots,\\gamma^{r}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [445, 443, 542, 460], "score": 1.0, "content": " (notation defined", "type": "text"}], "index": 19}, {"bbox": [69, 458, 541, 474], "spans": [{"bbox": [69, 458, 206, 474], "score": 1.0, "content": "in §3.2), and the spinors ", "type": "text"}, {"bbox": [207, 461, 247, 472], "score": 0.93, "content": "\\Lambda_{r},J\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [248, 458, 304, 474], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [304, 460, 421, 473], "score": 0.93, "content": "\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [421, 458, 507, 474], "score": 1.0, "content": ", we know both ", "type": "text"}, {"bbox": [508, 464, 515, 470], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [516, 458, 541, 474], "score": 1.0, "content": " and", "type": "text"}], "index": 20}, {"bbox": [71, 472, 542, 489], "spans": [{"bbox": [71, 475, 82, 484], "score": 0.9, "content": "\\pi^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [82, 472, 144, 489], "score": 1.0, "content": " must take ", "type": "text"}, {"bbox": [145, 475, 153, 484], "score": 0.9, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [153, 472, 237, 489], "score": 1.0, "content": "-fixed-points to ", "type": "text"}, {"bbox": [238, 475, 246, 484], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [246, 472, 339, 489], "score": 1.0, "content": "-fixed-points, i.e. ", "type": "text"}, {"bbox": [340, 475, 397, 487], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [397, 472, 425, 489], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [425, 473, 489, 487], "score": 0.94, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [489, 472, 542, 489], "score": 1.0, "content": " for some", "type": "text"}], "index": 21}, {"bbox": [71, 487, 287, 504], "spans": [{"bbox": [71, 489, 145, 501], "score": 0.92, "content": "1\\leq m,m^{\\prime}\\leq r", "type": "inline_equation", "height": 12, "width": 74}, {"bbox": [145, 487, 287, 504], "score": 1.0, "content": ". It is easy to compute [25]", "type": "text"}], "index": 22}], "index": 20, "bbox_fs": [69, 430, 542, 504]}, {"type": "interline_equation", "bbox": [247, 515, 365, 547], "lines": [{"bbox": [247, 515, 365, 547], "spans": [{"bbox": [247, 515, 365, 547], "score": 0.94, "content": "\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [70, 560, 541, 589], "lines": [{"bbox": [70, 561, 541, 578], "spans": [{"bbox": [70, 561, 165, 578], "score": 1.0, "content": "From this we see ", "type": "text"}, {"bbox": [165, 564, 225, 574], "score": 0.8, "content": "m\\,m^{\\prime}\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [226, 561, 261, 578], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [262, 567, 269, 573], "score": 0.58, "content": "\\kappa", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [269, 561, 296, 578], "score": 1.0, "content": "), so ", "type": "text"}, {"bbox": [297, 568, 307, 573], "score": 0.85, "content": "^{\\prime\\prime}", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [308, 561, 385, 578], "score": 1.0, "content": " is coprime to ", "type": "text"}, {"bbox": [385, 566, 393, 573], "score": 0.74, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [393, 561, 541, 578], "score": 1.0, "content": ". Hitting it with the Galois", "type": "text"}], "index": 24}, {"bbox": [70, 576, 445, 592], "spans": [{"bbox": [70, 577, 162, 592], "score": 1.0, "content": "fusion-symmetry ", "type": "text"}, {"bbox": [162, 578, 195, 591], "score": 0.93, "content": "\\pi\\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [196, 577, 349, 592], "score": 1.0, "content": ", we see that we may assume ", "type": "text"}, {"bbox": [350, 576, 441, 590], "score": 0.92, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [442, 577, 445, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [70, 561, 541, 592]}, {"type": "text", "bbox": [71, 590, 541, 619], "lines": [{"bbox": [94, 591, 540, 607], "spans": [{"bbox": [94, 591, 205, 607], "score": 1.0, "content": "Now use (4.2) to get ", "type": "text"}, {"bbox": [205, 591, 251, 605], "score": 0.93, "content": "\\pi\\gamma^{i}=\\gamma^{i}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [251, 591, 288, 607], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [289, 594, 293, 603], "score": 0.75, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [293, 591, 331, 607], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [332, 594, 340, 603], "score": 0.72, "content": "\\pi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [340, 591, 458, 607], "score": 1.0, "content": " equals the identity or ", "type": "text"}, {"bbox": [459, 592, 479, 605], "score": 0.74, "content": "\\pi[1]", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [479, 591, 540, 607], "score": 1.0, "content": ", depending", "type": "text"}], "index": 26}, {"bbox": [70, 605, 189, 621], "spans": [{"bbox": [70, 605, 117, 621], "score": 1.0, "content": "on what ", "type": "text"}, {"bbox": [118, 611, 125, 617], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [126, 605, 170, 621], "score": 1.0, "content": " does to ", "type": "text"}, {"bbox": [170, 608, 184, 619], "score": 0.91, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [185, 605, 189, 621], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26.5, "bbox_fs": [70, 591, 540, 621]}, {"type": "title", "bbox": [71, 634, 218, 649], "lines": [{"bbox": [71, 637, 218, 649], "spans": [{"bbox": [71, 637, 119, 649], "score": 1.0, "content": "4.4. The ", "type": "text"}, {"bbox": [119, 639, 129, 647], "score": 0.87, "content": "C", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [129, 637, 218, 649], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [70, 657, 541, 715], "lines": [{"bbox": [94, 659, 541, 674], "spans": [{"bbox": [94, 659, 284, 674], "score": 1.0, "content": "By rank-level duality, we may take ", "type": "text"}, {"bbox": [284, 661, 314, 672], "score": 0.93, "content": "r\\le k", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [315, 659, 413, 674], "score": 1.0, "content": ". For now assume ", "type": "text"}, {"bbox": [413, 659, 484, 673], "score": 0.91, "content": "(r,k)\\neq(2,3)", "type": "inline_equation", "height": 14, "width": 71}, {"bbox": [484, 659, 541, 674], "score": 1.0, "content": ". Then we", "type": "text"}], "index": 29}, {"bbox": [70, 672, 540, 688], "spans": [{"bbox": [70, 673, 102, 688], "score": 1.0, "content": "know ", "type": "text"}, {"bbox": [102, 676, 168, 686], "score": 0.92, "content": "\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 66}, {"bbox": [168, 673, 195, 688], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [196, 672, 265, 686], "score": 0.93, "content": "\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 69}, {"bbox": [265, 673, 318, 688], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [318, 675, 340, 687], "score": 0.91, "content": "a,a^{\\prime}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [340, 673, 380, 688], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [380, 674, 475, 686], "score": 0.86, "content": "\\pi J0=\\pi^{\\prime}J0=J0", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [475, 673, 540, 688], "score": 1.0, "content": ", (2.7b) says", "type": "text"}], "index": 30}, {"bbox": [71, 685, 542, 707], "spans": [{"bbox": [71, 689, 126, 699], "score": 0.92, "content": "a=a^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 55}, {"bbox": [126, 685, 140, 707], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [140, 690, 153, 699], "score": 0.89, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [154, 685, 228, 707], "score": 1.0, "content": " is odd. Since ", "type": "text"}, {"bbox": [228, 689, 291, 702], "score": 0.95, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [291, 685, 366, 707], "score": 1.0, "content": " (using (3.3)), ", "type": "text"}, {"bbox": [367, 689, 473, 703], "score": 0.9, "content": "S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}", "type": "inline_equation", "height": 14, "width": 106}, {"bbox": [473, 685, 542, 707], "score": 1.0, "content": " implies that", "type": "text"}], "index": 31}, {"bbox": [71, 702, 540, 717], "spans": [{"bbox": [71, 704, 103, 713], "score": 0.92, "content": "a=a^{\\prime}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [103, 702, 192, 717], "score": 1.0, "content": " also holds when ", "type": "text"}, {"bbox": [192, 704, 205, 713], "score": 0.88, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [205, 702, 459, 717], "score": 1.0, "content": " is even, and hence we may assume (hitting with ", "type": "text"}, {"bbox": [459, 703, 487, 716], "score": 0.84, "content": "\\pi[1]^{a}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [487, 702, 540, 717], "score": 1.0, "content": ") that also", "type": "text"}], "index": 32}, {"bbox": [71, 72, 542, 90], "spans": [{"bbox": [71, 75, 126, 84], "score": 0.93, "content": "a=a^{\\prime}=0", "type": "inline_equation", "height": 9, "width": 55, "cross_page": true}, {"bbox": [126, 72, 178, 90], "score": 1.0, "content": " holds for ", "type": "text", "cross_page": true}, {"bbox": [178, 75, 191, 84], "score": 0.39, "content": "k r", "type": "inline_equation", "height": 9, "width": 13, "cross_page": true}, {"bbox": [191, 72, 312, 90], "score": 1.0, "content": " even. From the fusion ", "type": "text", "cross_page": true}, {"bbox": [313, 75, 326, 86], "score": 0.64, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [327, 72, 344, 90], "score": 1.0, "content": " × ", "type": "text", "cross_page": true}, {"bbox": [344, 74, 358, 86], "score": 0.63, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [358, 72, 398, 90], "score": 1.0, "content": "we get ", "type": "text", "cross_page": true}, {"bbox": [398, 75, 529, 87], "score": 0.92, "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "type": "inline_equation", "height": 12, "width": 131, "cross_page": true}, {"bbox": [529, 72, 542, 90], "score": 1.0, "content": " if", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [71, 89, 408, 102], "spans": [{"bbox": [71, 90, 120, 100], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 10, "width": 49, "cross_page": true}, {"bbox": [121, 89, 145, 102], "score": 1.0, "content": "; for ", "type": "text", "cross_page": true}, {"bbox": [146, 90, 174, 99], "score": 0.92, "content": "r<k", "type": "inline_equation", "height": 9, "width": 28, "cross_page": true}, {"bbox": [174, 89, 408, 102], "score": 1.0, "content": " conclude the argument with the calculation", "type": "text", "cross_page": true}], "index": 1}], "index": 30.5, "bbox_fs": [70, 659, 542, 717]}]}
|
[{"type": "text", "bbox": [70, 93, 541, 122], "content": "is easy: and . is automatic. will be done later in this subsection. Assume now that .", "index": 0}, {"type": "text", "bbox": [70, 123, 541, 266], "content": "From Proposition 4.1(b) we can write and . We know , so (2.7b) says must take spinors to spinors, and nonspinors to nonspinors. Then we will have for any spinor . Now if , then will take the spinors which maximize , to those which minimize it. Both these maxima and minima can be easily found from (3.2). Thus we get that equals (when odd) or (when even). But the sets and have different cardi- nalities ( is a -fixed-point), and so can’t get mapped to each other. Also, the fusions × + and + + have different numbers of weights on their right sides, so also .", "index": 1}, {"type": "text", "bbox": [70, 267, 541, 295], "content": "Thus and for some . Similarly, . Hitting with , we may assume that fixes .", "index": 2}, {"type": "text", "bbox": [70, 296, 540, 326], "content": "Now assume fixes , for . Then the fusion × says that equals or . But from (3.2) we find", "index": 3}, {"type": "interline_equation", "bbox": [124, 340, 527, 401], "content": "", "index": 4}, {"type": "text", "bbox": [70, 409, 391, 425], "content": "Hence will fix if it fixes , concluding the argument.", "index": 5}, {"type": "text", "bbox": [70, 428, 541, 501], "content": "Now consider the more interesting case: . Then ; recall the weights in are the simple-currents , the -fixed-points (notation defined in §3.2), and the spinors . Because , we know both and must take -fixed-points to -fixed-points, i.e. and for some . It is easy to compute [25]", "index": 6}, {"type": "interline_equation", "bbox": [247, 515, 365, 547], "content": "", "index": 7}, {"type": "text", "bbox": [70, 560, 541, 589], "content": "From this we see (mod ), so is coprime to . Hitting it with the Galois fusion-symmetry , we see that we may assume .", "index": 8}, {"type": "text", "bbox": [71, 590, 541, 619], "content": "Now use (4.2) to get for all . Then equals the identity or , depending on what does to .", "index": 9}, {"type": "title", "bbox": [71, 634, 218, 649], "content": "4.4. The -series argument", "index": 10}, {"type": "text", "bbox": [70, 657, 541, 715], "content": "By rank-level duality, we may take . For now assume . Then we know and for some . Since , (2.7b) says if is odd. Since (using (3.3)), implies that also holds when is even, and hence we may assume (hitting with ) that also holds for even. From the fusion × we get if ; for conclude the argument with the calculation", "index": 11}]
|
[{"bbox": [95, 95, 541, 111], "content": "is easy: and . is automatic. will be done later in", "parent_index": 0, "line_index": 0}, {"bbox": [72, 111, 285, 124], "content": "this subsection. Assume now that .", "parent_index": 0, "line_index": 1}, {"bbox": [93, 124, 541, 140], "content": "From Proposition 4.1(b) we can write and . We know", "parent_index": 1, "line_index": 0}, {"bbox": [71, 140, 540, 155], "content": ", so (2.7b) says must take spinors to spinors, and nonspinors to nonspinors.", "parent_index": 1, "line_index": 1}, {"bbox": [70, 153, 542, 169], "content": "Then we will have for any spinor . Now if , then will", "parent_index": 1, "line_index": 2}, {"bbox": [70, 168, 542, 184], "content": "take the spinors which maximize , to those which minimize it. Both these maxima", "parent_index": 1, "line_index": 3}, {"bbox": [70, 182, 541, 198], "content": "and minima can be easily found from (3.2). Thus we get that equals (when", "parent_index": 1, "line_index": 4}, {"bbox": [70, 196, 541, 212], "content": "odd) or (when even). But the sets and have different cardi-", "parent_index": 1, "line_index": 5}, {"bbox": [71, 211, 541, 226], "content": "nalities ( is a -fixed-point), and so can’t get mapped to each other. Also, the fusions", "parent_index": 1, "line_index": 6}, {"bbox": [71, 224, 541, 241], "content": "× + and +", "parent_index": 1, "line_index": 7}, {"bbox": [70, 237, 541, 257], "content": "+ have different numbers of weights on their right sides,", "parent_index": 1, "line_index": 8}, {"bbox": [70, 254, 209, 269], "content": "so also .", "parent_index": 1, "line_index": 9}, {"bbox": [94, 268, 541, 285], "content": "Thus and for some . Similarly, . Hitting with , we", "parent_index": 2, "line_index": 0}, {"bbox": [71, 284, 220, 298], "content": "may assume that fixes .", "parent_index": 2, "line_index": 1}, {"bbox": [93, 297, 539, 315], "content": "Now assume fixes , for . Then the fusion × says that", "parent_index": 3, "line_index": 0}, {"bbox": [71, 313, 321, 327], "content": "equals or . But from (3.2) we find", "parent_index": 3, "line_index": 1}, {"bbox": [70, 411, 390, 426], "content": "Hence will fix if it fixes , concluding the argument.", "parent_index": 5, "line_index": 0}, {"bbox": [94, 430, 540, 446], "content": "Now consider the more interesting case: . Then ; recall the weights", "parent_index": 6, "line_index": 0}, {"bbox": [69, 443, 542, 460], "content": "in are the simple-currents , the -fixed-points (notation defined", "parent_index": 6, "line_index": 1}, {"bbox": [69, 458, 541, 474], "content": "in §3.2), and the spinors . Because , we know both and", "parent_index": 6, "line_index": 2}, {"bbox": [71, 472, 542, 489], "content": "must take -fixed-points to -fixed-points, i.e. and for some", "parent_index": 6, "line_index": 3}, {"bbox": [71, 487, 287, 504], "content": ". It is easy to compute [25]", "parent_index": 6, "line_index": 4}, {"bbox": [70, 561, 541, 578], "content": "From this we see (mod ), so is coprime to . Hitting it with the Galois", "parent_index": 8, "line_index": 0}, {"bbox": [70, 576, 445, 592], "content": "fusion-symmetry , we see that we may assume .", "parent_index": 8, "line_index": 1}, {"bbox": [94, 591, 540, 607], "content": "Now use (4.2) to get for all . Then equals the identity or , depending", "parent_index": 9, "line_index": 0}, {"bbox": [70, 605, 189, 621], "content": "on what does to .", "parent_index": 9, "line_index": 1}, {"bbox": [71, 637, 218, 649], "content": "4.4. The -series argument", "parent_index": 10, "line_index": 0}, {"bbox": [94, 659, 541, 674], "content": "By rank-level duality, we may take . For now assume . Then we", "parent_index": 11, "line_index": 0}, {"bbox": [70, 672, 540, 688], "content": "know and for some . Since , (2.7b) says", "parent_index": 11, "line_index": 1}, {"bbox": [71, 685, 542, 707], "content": "if is odd. Since (using (3.3)), implies that", "parent_index": 11, "line_index": 2}, {"bbox": [71, 702, 540, 717], "content": "also holds when is even, and hence we may assume (hitting with ) that also", "parent_index": 11, "line_index": 3}, {"bbox": [71, 72, 542, 90], "content": "holds for even. From the fusion × we get if", "parent_index": 11, "line_index": 4}, {"bbox": [71, 89, 408, 102], "content": "; for conclude the argument with the calculation", "parent_index": 11, "line_index": 5}]
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[]
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[{"bbox": [95, 98, 124, 107], "content": "k=1", "parent_index": 0, "subtype": "inline"}, {"bbox": [171, 97, 259, 110], "content": "P_{+}=\\{0,J0,\\Lambda_{r}\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [286, 98, 321, 107], "content": "\\pi=i d", "parent_index": 0, "subtype": "inline"}, {"bbox": [401, 98, 430, 107], "content": "k=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [252, 113, 281, 123], "content": "k\\geq3", "parent_index": 0, "subtype": "inline"}, {"bbox": [304, 127, 372, 138], "content": "\\pi\\Lambda_{1}\\,=\\,J^{a}\\Lambda_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [402, 124, 478, 138], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J^{a^{\\prime}}\\Lambda_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [71, 142, 125, 151], "content": "\\pi J0\\,=\\,J0", "parent_index": 1, "subtype": "inline"}, {"bbox": [209, 145, 217, 150], "content": "\\pi", "parent_index": 1, "subtype": "inline"}, {"bbox": [172, 153, 300, 168], "content": "\\chi_{\\Lambda_{1}}[\\psi]\\,=\\,(-1)^{a^{\\prime}}\\chi_{\\Lambda_{1}}[\\pi\\psi]", "parent_index": 1, "subtype": "inline"}, {"bbox": [382, 156, 390, 167], "content": "\\psi", "parent_index": 1, "subtype": "inline"}, {"bbox": [440, 155, 474, 165], "content": "a^{\\prime}=1", "parent_index": 1, "subtype": "inline"}, {"bbox": [509, 158, 516, 165], "content": "\\pi", "parent_index": 1, "subtype": "inline"}, {"bbox": [251, 173, 271, 182], "content": "\\chi_{\\Lambda_{1}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [403, 183, 442, 196], "content": "\\pi(S\\Lambda_{r})", "parent_index": 1, "subtype": "inline"}, {"bbox": [482, 182, 503, 195], "content": "k\\Lambda_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [70, 197, 79, 209], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [127, 196, 196, 211], "content": "S((k-1)\\Lambda_{r})", "parent_index": 1, "subtype": "inline"}, {"bbox": [236, 198, 244, 208], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [357, 199, 379, 210], "content": "S\\Lambda_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [408, 198, 429, 210], "content": "k\\Lambda_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [116, 211, 138, 224], "content": "k\\Lambda_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [163, 212, 171, 223], "content": "J", "parent_index": 1, "subtype": "inline"}, {"bbox": [71, 225, 85, 238], "content": "\\Lambda_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [102, 225, 147, 238], "content": "\\Lambda_{r}=\\Lambda_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [164, 225, 215, 239], "content": "(\\Lambda_{1}+\\Lambda_{r})", "parent_index": 1, "subtype": "inline"}, {"bbox": [241, 226, 457, 240], "content": "J^{a}\\Lambda_{1}\\boxtimes\\left(J^{i}(k-1)\\Lambda_{r}\\right)=\\left(J^{a+i}(k-1)\\Lambda_{r}\\right)", "parent_index": 1, "subtype": "inline"}, {"bbox": [477, 224, 541, 240], "content": "(J^{a+i+1}(k-", "parent_index": 1, "subtype": "inline"}, {"bbox": [70, 239, 99, 254], "content": "1)\\Lambda_{r})", "parent_index": 1, "subtype": "inline"}, {"bbox": [118, 240, 254, 254], "content": "J^{a+i+1}(\\Lambda_{r-1}+(k-3)\\Lambda_{r})", "parent_index": 1, "subtype": "inline"}, {"bbox": [110, 254, 204, 268], "content": "\\pi\\Lambda_{r}\\notin{\\cal S}(k-1)\\Lambda_{r}", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 269, 160, 281], "content": "a^{\\prime}=0", "parent_index": 2, "subtype": "inline"}, {"bbox": [186, 269, 251, 282], "content": "\\pi\\Lambda_{r}=J^{b}\\Lambda_{r}", "parent_index": 2, "subtype": "inline"}, {"bbox": [304, 271, 310, 280], "content": "b", "parent_index": 2, "subtype": "inline"}, {"bbox": [372, 270, 403, 281], "content": "a=0", "parent_index": 2, "subtype": "inline"}, {"bbox": [453, 272, 462, 280], "content": "\\pi", "parent_index": 2, "subtype": "inline"}, {"bbox": [493, 268, 518, 283], "content": "\\pi[1]^{b}", "parent_index": 2, "subtype": "inline"}, {"bbox": [164, 286, 172, 295], "content": "\\pi", "parent_index": 2, "subtype": "inline"}, {"bbox": [202, 285, 216, 296], "content": "\\Lambda_{r}", "parent_index": 2, "subtype": "inline"}, {"bbox": [163, 301, 172, 309], "content": "\\pi", "parent_index": 3, "subtype": "inline"}, {"bbox": [202, 300, 215, 311], "content": "\\Lambda_{\\ell}", "parent_index": 3, "subtype": "inline"}, {"bbox": [241, 300, 311, 311], "content": "1\\leq\\ell<r-1", "parent_index": 3, "subtype": "inline"}, {"bbox": [406, 298, 421, 311], "content": "\\Lambda_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [438, 298, 452, 311], "content": "\\Lambda_{\\ell}", "parent_index": 3, "subtype": "inline"}, {"bbox": [506, 299, 539, 312], "content": "\\pi\\Lambda_{\\ell+1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [107, 313, 133, 326], "content": "\\Lambda_{\\ell+1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [151, 313, 192, 325], "content": "\\Lambda_{1}+\\Lambda_{\\ell}", "parent_index": 3, "subtype": "inline"}, {"bbox": [124, 340, 527, 401], "content": "\\begin{array}{c}{{-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=2\\,\\{\\displaystyle\\mathrm{cos}(\\pi\\frac{2r-2\\ell+1}{\\kappa})-\\mathrm{cos}(\\pi\\frac{2r-2\\ell-1}{\\kappa})+\\mathrm{cos}(\\pi\\frac{2r+1}{\\kappa})\\}}}\\\\ {{-\\cos(\\pi\\frac{2r+3}{\\kappa})\\}=4\\cos(\\pi\\frac{2r-\\ell+1}{\\kappa})\\,\\{\\displaystyle\\mathrm{cos}(2\\pi\\frac{\\ell}{\\kappa})-\\mathrm{cos}(2\\pi\\frac{\\ell+1}{\\kappa})\\},}}\\end{array}", "parent_index": 4, "subtype": "interline"}, {"bbox": [107, 417, 114, 423], "content": "\\pi", "parent_index": 5, "subtype": "inline"}, {"bbox": [158, 413, 182, 425], "content": "\\Lambda_{\\ell+1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [236, 414, 249, 424], "content": "\\Lambda_{\\ell}", "parent_index": 5, "subtype": "inline"}, {"bbox": [311, 432, 341, 441], "content": "k=2", "parent_index": 6, "subtype": "inline"}, {"bbox": [382, 433, 440, 442], "content": "\\kappa=2r+1", "parent_index": 6, "subtype": "inline"}, {"bbox": [86, 446, 132, 459], "content": "P_{+}(B_{r,2})", "parent_index": 6, "subtype": "inline"}, {"bbox": [262, 447, 288, 458], "content": "0,J0", "parent_index": 6, "subtype": "inline"}, {"bbox": [316, 447, 325, 456], "content": "J", "parent_index": 6, "subtype": "inline"}, {"bbox": [393, 445, 444, 458], "content": "\\gamma^{1},\\ldots,\\gamma^{r}", "parent_index": 6, "subtype": "inline"}, {"bbox": [207, 461, 247, 472], "content": "\\Lambda_{r},J\\Lambda_{r}", "parent_index": 6, "subtype": "inline"}, {"bbox": [304, 460, 421, 473], "content": "\\pi(J0)\\,=\\,\\pi^{\\prime}(J0)\\,=\\,J0", "parent_index": 6, "subtype": "inline"}, {"bbox": [508, 464, 515, 470], "content": "\\pi", "parent_index": 6, "subtype": "inline"}, {"bbox": [71, 475, 82, 484], "content": "\\pi^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [145, 475, 153, 484], "content": "J", "parent_index": 6, "subtype": "inline"}, {"bbox": [238, 475, 246, 484], "content": "J", "parent_index": 6, "subtype": "inline"}, {"bbox": [340, 475, 397, 487], "content": "\\pi\\Lambda_{1}\\,=\\,\\gamma^{m}", "parent_index": 6, "subtype": "inline"}, {"bbox": [425, 473, 489, 487], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\gamma^{m^{\\prime}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [71, 489, 145, 501], "content": "1\\leq m,m^{\\prime}\\leq r", "parent_index": 6, "subtype": "inline"}, {"bbox": [247, 515, 365, 547], "content": "\\frac{S_{\\gamma^{a}\\gamma^{b}}}{S_{0\\gamma^{b}}}=2\\cos(2\\pi\\frac{a b}{\\kappa})\\ .", "parent_index": 7, "subtype": "interline"}, {"bbox": [165, 564, 225, 574], "content": "m\\,m^{\\prime}\\equiv\\pm1", "parent_index": 8, "subtype": "inline"}, {"bbox": [262, 567, 269, 573], "content": "\\kappa", "parent_index": 8, "subtype": "inline"}, {"bbox": [297, 568, 307, 573], "content": "^{\\prime\\prime}", "parent_index": 8, "subtype": "inline"}, {"bbox": [385, 566, 393, 573], "content": "\\kappa", "parent_index": 8, "subtype": "inline"}, {"bbox": [162, 578, 195, 591], "content": "\\pi\\{m^{\\prime}\\}", "parent_index": 8, "subtype": "inline"}, {"bbox": [350, 576, 441, 590], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [205, 591, 251, 605], "content": "\\pi\\gamma^{i}=\\gamma^{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [289, 594, 293, 603], "content": "i", "parent_index": 9, "subtype": "inline"}, {"bbox": [332, 594, 340, 603], "content": "\\pi", "parent_index": 9, "subtype": "inline"}, {"bbox": [459, 592, 479, 605], "content": "\\pi[1]", "parent_index": 9, "subtype": "inline"}, {"bbox": [118, 611, 125, 617], "content": "\\pi", "parent_index": 9, "subtype": "inline"}, {"bbox": [170, 608, 184, 619], "content": "\\Lambda_{r}", "parent_index": 9, "subtype": "inline"}, {"bbox": [119, 639, 129, 647], "content": "C", "parent_index": 10, "subtype": "inline"}, {"bbox": [284, 661, 314, 672], "content": "r\\le k", "parent_index": 11, "subtype": "inline"}, {"bbox": [413, 659, 484, 673], "content": "(r,k)\\neq(2,3)", "parent_index": 11, "subtype": "inline"}, {"bbox": [102, 676, 168, 686], "content": "\\pi\\Lambda_{1}=J^{a}\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [196, 672, 265, 686], "content": "\\pi\\Lambda_{1}=J^{a^{\\prime}}\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [318, 675, 340, 687], "content": "a,a^{\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [380, 674, 475, 686], "content": "\\pi J0=\\pi^{\\prime}J0=J0", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 689, 126, 699], "content": "a=a^{\\prime}=0", "parent_index": 11, "subtype": "inline"}, {"bbox": [140, 690, 153, 699], "content": "k r", "parent_index": 11, "subtype": "inline"}, {"bbox": [228, 689, 291, 702], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "parent_index": 11, "subtype": "inline"}, {"bbox": [367, 689, 473, 703], "content": "S_{\\Lambda_{1}\\Lambda_{1}}=S_{J^{a}\\Lambda_{1},J^{a^{\\prime}}\\Lambda_{1}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 704, 103, 713], "content": "a=a^{\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [192, 704, 205, 713], "content": "k r", "parent_index": 11, "subtype": "inline"}, {"bbox": [459, 703, 487, 716], "content": "\\pi[1]^{a}", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 75, 126, 84], "content": "a=a^{\\prime}=0", "parent_index": 11, "subtype": "inline"}, {"bbox": [178, 75, 191, 84], "content": "k r", "parent_index": 11, "subtype": "inline"}, {"bbox": [313, 75, 326, 86], "content": "\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [344, 74, 358, 86], "content": "\\Lambda_{\\ell}", "parent_index": 11, "subtype": "inline"}, {"bbox": [398, 75, 529, 87], "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 90, 120, 100], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "parent_index": 11, "subtype": "inline"}, {"bbox": [146, 90, 174, 99], "content": "r<k", "parent_index": 11, "subtype": "inline"}]
|
[]
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|
$$
\chi_{\Lambda_{1}}[\Lambda_{\ell+1}]-\chi_{\Lambda_{1}}[\Lambda_{1}+\Lambda_{\ell}]=4\cos(\pi\,\frac{2r+2-\ell}{2\kappa})\,\{\cos(\pi\,\frac{\ell}{2\kappa})-\cos(\pi\,\frac{\ell+2}{2\kappa})\}>
$$
as in $\S4.3$ . When $r=k$ , that inequality only holds for $\ell>1$ , but we can force $\pi\Lambda_{2}=\Lambda_{2}$ by hitting $\pi$ if necessary with $\pi_{\mathrm{rld}}$ .
The remaining case $C_{2,3}$ follows because $\pi^{\prime}J0\,=\,J0$ : by (2.7b) $\pi\Lambda_{1}\notin S\Lambda_{2}$ , and by (2.7a) $\pi\Lambda_{1}\neq3\Lambda_{1}$ ( $3\Lambda_{1}$ is a $J$ -fixed-point).
# 4.5. The $D$ -series argument
$k=1$ is trivial, and $k=2$ will be considered shortly. For $k>2$ , Proposition 4.1 tells us that $\pi\Lambda_{1}=J_{v}^{a}J_{s}^{b}\Lambda_{1}$ and $\pi^{\prime}\Lambda_{1}=J_{v}^{a^{\prime}}J_{s}^{b^{\prime}}\Lambda_{1}$ , for $a,a^{\prime},b,b^{\prime}\in\{0,1\}$ . Immediate from (3.4) is that $\chi_{\Lambda_{1}}[\Lambda_{1}]>0$ and that $\chi_{\Lambda_{1}}[\psi]$ , for a spinor $\psi$ , takes its maximum at $C^{i}J_{v}^{j}\Lambda_{r}$ . Our first step is to force $\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ . Unfortunately this requires a case analysis.
Consider first even $r\,\neq\,4$ , and even $k\ >\ 2$ . Now, $0\;\neq\;S_{\Lambda_{1}\Lambda_{1}}\;=\;S_{\pi\Lambda_{1},\pi^{\prime}\Lambda_{1}}$ forces $b=b^{\prime}$ ; hence hitting with the simple-current automorphism $\pi\left[{\begin{array}{l l}{0}&{a}\\ {a^{\prime}}&{b}\end{array}}\right]$ , we may assume $\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ .
Next consider even $r\neq4$ and odd $k\,>\,2$ . Either of $\pi\Lambda_{1}=J_{v}\Lambda_{1}$ or $\pi^{\prime}\Lambda_{1}\,=\,J_{v}\Lambda_{1}$ is impossible, by comparing $S_{\Lambda_{1},J_{s}0}$ and $S_{J_{v}\Lambda_{1},J0}$ for any simple-current $J$ . For any of the three remaining choices of $J_{v}^{a}J_{s}^{b}\Lambda_{1}$ , we can find a simple-current automorphism of the form $\pi\left[{\ast}\quad a\,\right];$ hitting its inverse onto $\pi$ allows us to take $a=b=0$ . Again $0\not=S_{\Lambda_{1}\Lambda_{1}}$ forces $b^{\prime}=0$ , and now $a^{\prime}=1$ is forbidden. Thus again $\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ .
As usual, $r=4$ is complicated by triality. We can force $\pi\Lambda_{1}=\Lambda_{1}$ . That we can also take $\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ , follows from the inequality $\chi_{\Lambda_{1}}[\Lambda_{1}]>\chi_{\Lambda_{1}}[\Lambda_{3}]=\chi_{\Lambda_{1}}[\Lambda_{4}]>0$ , valid for $k\geq3$ . Establishing that inequality from (3.4) is equivalent to showing
$1+\cos(x)+\cos(2x)+\cos(4x)>\cos(x/2)+\cos(3x/2)+\cos(5x/2)+\cos(7x/2)>0$ 2)
for $0<x\le2\pi/9$ , which can be shown e.g. using Taylor series.
For odd $r$ , the charge-conjugation $C$ equals $C_{1}$ . Since it must commute with $\pi$ , i.e. that $C_{1}\pi\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\Lambda_{1}$ must equal $\pi C_{1}\Lambda_{1}=J_{v}^{a}J_{s}^{b}\Lambda_{1}$ , we get that $b=0$ . Similarly $b^{\prime}=0$ . When $k$ is odd, eliminate $a=1$ and $a\prime=1$ by comparing $S_{\Lambda_{1},J_{s}0}$ and $S_{J_{v}\Lambda_{1},J0}$ as before. The hardest case is $k$ even. We can force $\pi\Lambda_{1}\,=\,\Lambda_{1}$ by hitting with $\pi[a]$ . Suppose for contradiction that $\pi^{\prime}\Lambda_{1}=J_{v}\Lambda_{1}$ . We know $\pi^{\prime}(J_{v}0)=J_{v}0$ (compare $S_{\Lambda_{1},J_{v}0}$ and $S_{\Lambda_{1},J0})$ , so by (2.7b) $\pi\Lambda_{r}$ must be a spinor. $\chi_{\Lambda_{1}}[\Lambda_{r}]\;=\;\chi_{J_{v}\Lambda_{1}}[\pi\Lambda_{r}]$ requires $\pi\Lambda_{r}\,=\,C_{1}^{i}J_{v}^{j}J_{s}\Lambda_{r}$ . From the $\Lambda_{1}\boxtimes\Lambda_{r}$ fusion we get $\pi\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\Lambda_{r-1}$ , but $C\pi=\pi C$ says that $\pi\Lambda_{r-1}=$ $C_{1}^{i}J_{v}^{j+1}J_{s}\Lambda_{r-1}$ — a contradiction.
Thus in all cases we have $\pi\Lambda_{1}\,=\,\pi^{\prime}\Lambda_{1}\,=\,\Lambda_{1}$ . We know $\pi^{\prime}(J_{v}0)\;=\;J_{v}0$ (compare $S_{\Lambda_{1},J_{v}0}$ and $S_{\Lambda_{1},J0})$ , so $\pi\Lambda_{r}$ is a spinor and in fact must equal $\pi\Lambda_{r}\,=\,C_{1}^{i}J_{v}^{j}\Lambda_{r}$ . Hitting with $(C_{1}^{i}\pi_{v}^{j})^{-1}$ , we can require $\pi\Lambda_{r}=\Lambda_{r}$ . That $\pi\Lambda_{r-1}$ must now equal $\Lambda_{r-1}$ follows from the $\Lambda_{1}\boxtimes\Lambda_{r}$ fusion.
|
<html><body>
<div class="equation" data-bbox="99 113 500 141">$$
\chi_{\Lambda_{1}}[\Lambda_{\ell+1}]-\chi_{\Lambda_{1}}[\Lambda_{1}+\Lambda_{\ell}]=4\cos(\pi\,\frac{2r+2-\ell}{2\kappa})\,\{\cos(\pi\,\frac{\ell}{2\kappa})-\cos(\pi\,\frac{\ell+2}{2\kappa})\}>
$$</div>
<p data-bbox="70 149 541 178">as in $\S4.3$ . When $r=k$ , that inequality only holds for $\ell>1$ , but we can force $\pi\Lambda_{2}=\Lambda_{2}$ by hitting $\pi$ if necessary with $\pi_{\mathrm{rld}}$ . </p>
<p data-bbox="70 179 541 208">The remaining case $C_{2,3}$ follows because $\pi^{\prime}J0\,=\,J0$ : by (2.7b) $\pi\Lambda_{1}\notin S\Lambda_{2}$ , and by (2.7a) $\pi\Lambda_{1}\neq3\Lambda_{1}$ ( $3\Lambda_{1}$ is a $J$ -fixed-point). </p>
<h1 data-bbox="71 221 219 236">4.5. The $D$ -series argument </h1>
<p data-bbox="70 243 541 300">$k=1$ is trivial, and $k=2$ will be considered shortly. For $k>2$ , Proposition 4.1 tells us that $\pi\Lambda_{1}=J_{v}^{a}J_{s}^{b}\Lambda_{1}$ and $\pi^{\prime}\Lambda_{1}=J_{v}^{a^{\prime}}J_{s}^{b^{\prime}}\Lambda_{1}$ , for $a,a^{\prime},b,b^{\prime}\in\{0,1\}$ . Immediate from (3.4) is that $\chi_{\Lambda_{1}}[\Lambda_{1}]>0$ and that $\chi_{\Lambda_{1}}[\psi]$ , for a spinor $\psi$ , takes its maximum at $C^{i}J_{v}^{j}\Lambda_{r}$ . Our first step is to force $\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ . Unfortunately this requires a case analysis. </p>
<p data-bbox="69 301 541 358">Consider first even $r\,\neq\,4$ , and even $k\ >\ 2$ . Now, $0\;\neq\;S_{\Lambda_{1}\Lambda_{1}}\;=\;S_{\pi\Lambda_{1},\pi^{\prime}\Lambda_{1}}$ forces $b=b^{\prime}$ ; hence hitting with the simple-current automorphism $\pi\left[{\begin{array}{l l}{0}&{a}\\ {a^{\prime}}&{b}\end{array}}\right]$ , we may assume $\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ . </p>
<p data-bbox="70 358 542 444">Next consider even $r\neq4$ and odd $k\,>\,2$ . Either of $\pi\Lambda_{1}=J_{v}\Lambda_{1}$ or $\pi^{\prime}\Lambda_{1}\,=\,J_{v}\Lambda_{1}$ is impossible, by comparing $S_{\Lambda_{1},J_{s}0}$ and $S_{J_{v}\Lambda_{1},J0}$ for any simple-current $J$ . For any of the three remaining choices of $J_{v}^{a}J_{s}^{b}\Lambda_{1}$ , we can find a simple-current automorphism of the form $\pi\left[{\ast}\quad a\,\right];$ hitting its inverse onto $\pi$ allows us to take $a=b=0$ . Again $0\not=S_{\Lambda_{1}\Lambda_{1}}$ forces $b^{\prime}=0$ , and now $a^{\prime}=1$ is forbidden. Thus again $\pi\Lambda_{1}=\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ . </p>
<p data-bbox="70 444 541 488">As usual, $r=4$ is complicated by triality. We can force $\pi\Lambda_{1}=\Lambda_{1}$ . That we can also take $\pi^{\prime}\Lambda_{1}=\Lambda_{1}$ , follows from the inequality $\chi_{\Lambda_{1}}[\Lambda_{1}]>\chi_{\Lambda_{1}}[\Lambda_{3}]=\chi_{\Lambda_{1}}[\Lambda_{4}]>0$ , valid for $k\geq3$ . Establishing that inequality from (3.4) is equivalent to showing </p>
<p data-bbox="98 500 514 516">$1+\cos(x)+\cos(2x)+\cos(4x)>\cos(x/2)+\cos(3x/2)+\cos(5x/2)+\cos(7x/2)>0$ 2) </p>
<p data-bbox="69 528 398 542">for $0<x\le2\pi/9$ , which can be shown e.g. using Taylor series. </p>
<p data-bbox="70 543 541 657">For odd $r$ , the charge-conjugation $C$ equals $C_{1}$ . Since it must commute with $\pi$ , i.e. that $C_{1}\pi\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\Lambda_{1}$ must equal $\pi C_{1}\Lambda_{1}=J_{v}^{a}J_{s}^{b}\Lambda_{1}$ , we get that $b=0$ . Similarly $b^{\prime}=0$ . When $k$ is odd, eliminate $a=1$ and $a\prime=1$ by comparing $S_{\Lambda_{1},J_{s}0}$ and $S_{J_{v}\Lambda_{1},J0}$ as before. The hardest case is $k$ even. We can force $\pi\Lambda_{1}\,=\,\Lambda_{1}$ by hitting with $\pi[a]$ . Suppose for contradiction that $\pi^{\prime}\Lambda_{1}=J_{v}\Lambda_{1}$ . We know $\pi^{\prime}(J_{v}0)=J_{v}0$ (compare $S_{\Lambda_{1},J_{v}0}$ and $S_{\Lambda_{1},J0})$ , so by (2.7b) $\pi\Lambda_{r}$ must be a spinor. $\chi_{\Lambda_{1}}[\Lambda_{r}]\;=\;\chi_{J_{v}\Lambda_{1}}[\pi\Lambda_{r}]$ requires $\pi\Lambda_{r}\,=\,C_{1}^{i}J_{v}^{j}J_{s}\Lambda_{r}$ . From the $\Lambda_{1}\boxtimes\Lambda_{r}$ fusion we get $\pi\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\Lambda_{r-1}$ , but $C\pi=\pi C$ says that $\pi\Lambda_{r-1}=$ $C_{1}^{i}J_{v}^{j+1}J_{s}\Lambda_{r-1}$ — a contradiction. </p>
<p data-bbox="70 658 541 715">Thus in all cases we have $\pi\Lambda_{1}\,=\,\pi^{\prime}\Lambda_{1}\,=\,\Lambda_{1}$ . We know $\pi^{\prime}(J_{v}0)\;=\;J_{v}0$ (compare $S_{\Lambda_{1},J_{v}0}$ and $S_{\Lambda_{1},J0})$ , so $\pi\Lambda_{r}$ is a spinor and in fact must equal $\pi\Lambda_{r}\,=\,C_{1}^{i}J_{v}^{j}\Lambda_{r}$ . Hitting with $(C_{1}^{i}\pi_{v}^{j})^{-1}$ , we can require $\pi\Lambda_{r}=\Lambda_{r}$ . That $\pi\Lambda_{r-1}$ must now equal $\Lambda_{r-1}$ follows from the $\Lambda_{1}\boxtimes\Lambda_{r}$ fusion. </p>
</body></html>
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0002044v1
| 18 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "", "page_idx": 18}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>\n$$", "text_format": "latex", "page_idx": 18}, {"type": "text", "text": "as in $\\S4.3$ . When $r=k$ , that inequality only holds for $\\ell>1$ , but we can force $\\pi\\Lambda_{2}=\\Lambda_{2}$ by hitting $\\pi$ if necessary with $\\pi_{\\mathrm{rld}}$ . ", "page_idx": 18}, {"type": "text", "text": "The remaining case $C_{2,3}$ follows because $\\pi^{\\prime}J0\\,=\\,J0$ : by (2.7b) $\\pi\\Lambda_{1}\\notin S\\Lambda_{2}$ , and by (2.7a) $\\pi\\Lambda_{1}\\neq3\\Lambda_{1}$ ( $3\\Lambda_{1}$ is a $J$ -fixed-point). ", "page_idx": 18}, {"type": "text", "text": "4.5. The $D$ -series argument ", "text_level": 1, "page_idx": 18}, {"type": "text", "text": "$k=1$ is trivial, and $k=2$ will be considered shortly. For $k>2$ , Proposition 4.1 tells us that $\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}$ and $\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}$ , for $a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}$ . Immediate from (3.4) is that $\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0$ and that $\\chi_{\\Lambda_{1}}[\\psi]$ , for a spinor $\\psi$ , takes its maximum at $C^{i}J_{v}^{j}\\Lambda_{r}$ . Our first step is to force $\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ . Unfortunately this requires a case analysis. ", "page_idx": 18}, {"type": "text", "text": "Consider first even $r\\,\\neq\\,4$ , and even $k\\ >\\ 2$ . Now, $0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}$ forces $b=b^{\\prime}$ ; hence hitting with the simple-current automorphism $\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]$ , we may assume $\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ . ", "page_idx": 18}, {"type": "text", "text": "Next consider even $r\\neq4$ and odd $k\\,>\\,2$ . Either of $\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}$ or $\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}$ is impossible, by comparing $S_{\\Lambda_{1},J_{s}0}$ and $S_{J_{v}\\Lambda_{1},J0}$ for any simple-current $J$ . For any of the three remaining choices of $J_{v}^{a}J_{s}^{b}\\Lambda_{1}$ , we can find a simple-current automorphism of the form $\\pi\\left[{\\ast}\\quad a\\,\\right];$ hitting its inverse onto $\\pi$ allows us to take $a=b=0$ . Again $0\\not=S_{\\Lambda_{1}\\Lambda_{1}}$ forces $b^{\\prime}=0$ , and now $a^{\\prime}=1$ is forbidden. Thus again $\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ . ", "page_idx": 18}, {"type": "text", "text": "As usual, $r=4$ is complicated by triality. We can force $\\pi\\Lambda_{1}=\\Lambda_{1}$ . That we can also take $\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}$ , follows from the inequality $\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0$ , valid for $k\\geq3$ . Establishing that inequality from (3.4) is equivalent to showing ", "page_idx": 18}, {"type": "text", "text": "$1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0$ 2) ", "page_idx": 18}, {"type": "text", "text": "for $0<x\\le2\\pi/9$ , which can be shown e.g. using Taylor series. ", "page_idx": 18}, {"type": "text", "text": "For odd $r$ , the charge-conjugation $C$ equals $C_{1}$ . Since it must commute with $\\pi$ , i.e. that $C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}$ must equal $\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}$ , we get that $b=0$ . Similarly $b^{\\prime}=0$ . When $k$ is odd, eliminate $a=1$ and $a\\prime=1$ by comparing $S_{\\Lambda_{1},J_{s}0}$ and $S_{J_{v}\\Lambda_{1},J0}$ as before. The hardest case is $k$ even. We can force $\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}$ by hitting with $\\pi[a]$ . Suppose for contradiction that $\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}$ . We know $\\pi^{\\prime}(J_{v}0)=J_{v}0$ (compare $S_{\\Lambda_{1},J_{v}0}$ and $S_{\\Lambda_{1},J0})$ , so by (2.7b) $\\pi\\Lambda_{r}$ must be a spinor. $\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]$ requires $\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}$ . From the $\\Lambda_{1}\\boxtimes\\Lambda_{r}$ fusion we get $\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}$ , but $C\\pi=\\pi C$ says that $\\pi\\Lambda_{r-1}=$ $C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}$ — a contradiction. ", "page_idx": 18}, {"type": "text", "text": "Thus in all cases we have $\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}$ . We know $\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0$ (compare $S_{\\Lambda_{1},J_{v}0}$ and $S_{\\Lambda_{1},J0})$ , so $\\pi\\Lambda_{r}$ is a spinor and in fact must equal $\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}$ . Hitting with $(C_{1}^{i}\\pi_{v}^{j})^{-1}$ , we can require $\\pi\\Lambda_{r}=\\Lambda_{r}$ . That $\\pi\\Lambda_{r-1}$ must now equal $\\Lambda_{r-1}$ follows from the $\\Lambda_{1}\\boxtimes\\Lambda_{r}$ fusion. ", "page_idx": 18}]
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From the fusion ", "type": "text"}, {"bbox": [313, 75, 326, 86], "score": 0.64, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [327, 72, 344, 90], "score": 1.0, "content": " × ", "type": "text"}, {"bbox": [344, 74, 358, 86], "score": 0.63, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [358, 72, 398, 90], "score": 1.0, "content": "we get ", "type": "text"}, {"bbox": [398, 75, 529, 87], "score": 0.92, "content": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}", "type": "inline_equation", "height": 12, "width": 131}, {"bbox": [529, 72, 542, 90], "score": 1.0, "content": " if", "type": "text"}], "index": 0}, {"bbox": [71, 89, 408, 102], "spans": [{"bbox": [71, 90, 120, 100], "score": 0.92, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [121, 89, 145, 102], "score": 1.0, "content": "; for ", "type": "text"}, {"bbox": [146, 90, 174, 99], "score": 0.92, "content": "r<k", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [174, 89, 408, 102], "score": 1.0, "content": " conclude the argument with the calculation", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [99, 113, 500, 141], "lines": [{"bbox": [99, 113, 500, 141], "spans": [{"bbox": [99, 113, 500, 141], "score": 0.9, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [70, 149, 541, 178], "lines": [{"bbox": [70, 151, 539, 168], "spans": [{"bbox": [70, 151, 100, 168], "score": 1.0, "content": "as in ", "type": "text"}, {"bbox": [100, 153, 121, 165], "score": 0.41, "content": "\\S4.3", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [122, 151, 165, 168], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [165, 154, 195, 163], "score": 0.92, "content": "r=k", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [195, 151, 360, 168], "score": 1.0, "content": ", that inequality only holds for ", "type": "text"}, {"bbox": [360, 153, 389, 164], "score": 0.88, "content": "\\ell>1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [390, 151, 487, 168], "score": 1.0, "content": ", but we can force ", "type": "text"}, {"bbox": [488, 154, 539, 165], "score": 0.91, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 51}], "index": 3}, {"bbox": [70, 166, 254, 183], "spans": [{"bbox": [70, 166, 127, 183], "score": 1.0, "content": "by hitting ", "type": "text"}, {"bbox": [127, 172, 135, 178], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [135, 166, 229, 183], "score": 1.0, "content": " if necessary with ", "type": "text"}, {"bbox": [229, 172, 249, 179], "score": 0.88, "content": "\\pi_{\\mathrm{rld}}", "type": "inline_equation", "height": 7, "width": 20}, {"bbox": [249, 166, 254, 183], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [70, 179, 541, 208], "lines": [{"bbox": [93, 180, 541, 197], "spans": [{"bbox": [93, 180, 203, 197], "score": 1.0, "content": "The remaining case ", "type": "text"}, {"bbox": [203, 182, 225, 195], "score": 0.91, "content": "C_{2,3}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [225, 180, 314, 197], "score": 1.0, "content": " follows because ", "type": "text"}, {"bbox": [314, 182, 371, 192], "score": 0.92, "content": "\\pi^{\\prime}J0\\,=\\,J0", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [372, 180, 434, 197], "score": 1.0, "content": ": by (2.7b) ", "type": "text"}, {"bbox": [434, 182, 495, 194], "score": 0.87, "content": "\\pi\\Lambda_{1}\\notin S\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [495, 180, 541, 197], "score": 1.0, "content": ", and by", "type": "text"}], "index": 5}, {"bbox": [72, 195, 292, 210], "spans": [{"bbox": [72, 196, 105, 210], "score": 1.0, "content": "(2.7a) ", "type": "text"}, {"bbox": [105, 195, 162, 208], "score": 0.89, "content": "\\pi\\Lambda_{1}\\neq3\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [163, 196, 170, 210], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [171, 195, 191, 208], "score": 0.88, "content": "3\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [191, 196, 216, 210], "score": 1.0, "content": " is a ", "type": "text"}, {"bbox": [216, 196, 225, 206], "score": 0.84, "content": "J", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [225, 196, 292, 210], "score": 1.0, "content": "-fixed-point).", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "title", "bbox": [71, 221, 219, 236], "lines": [{"bbox": [71, 225, 219, 236], "spans": [{"bbox": [71, 225, 119, 236], "score": 1.0, "content": "4.5. The ", "type": "text"}, {"bbox": [119, 225, 130, 235], "score": 0.8, "content": "D", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [131, 225, 219, 236], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 243, 541, 300], "lines": [{"bbox": [95, 244, 541, 259], "spans": [{"bbox": [95, 247, 124, 256], "score": 0.9, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [124, 244, 201, 259], "score": 1.0, "content": " is trivial, and ", "type": "text"}, {"bbox": [202, 245, 231, 256], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [232, 244, 398, 259], "score": 1.0, "content": " will be considered shortly. For ", "type": "text"}, {"bbox": [398, 247, 427, 256], "score": 0.9, "content": "k>2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [427, 244, 541, 259], "score": 1.0, "content": ", Proposition 4.1 tells", "type": "text"}], "index": 8}, {"bbox": [70, 258, 541, 274], "spans": [{"bbox": [70, 258, 111, 274], "score": 1.0, "content": "us that", "type": "text"}, {"bbox": [112, 259, 189, 273], "score": 0.93, "content": "\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 77}, {"bbox": [190, 258, 216, 274], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [216, 258, 303, 273], "score": 0.94, "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 15, "width": 87}, {"bbox": [303, 258, 329, 274], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [329, 260, 419, 273], "score": 0.93, "content": "a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [419, 258, 541, 274], "score": 1.0, "content": ". Immediate from (3.4)", "type": "text"}], "index": 9}, {"bbox": [69, 272, 542, 289], "spans": [{"bbox": [69, 272, 109, 289], "score": 1.0, "content": "is that ", "type": "text"}, {"bbox": [110, 274, 173, 288], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [173, 272, 226, 289], "score": 1.0, "content": " and that ", "type": "text"}, {"bbox": [226, 275, 261, 287], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\psi]", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [262, 272, 334, 289], "score": 1.0, "content": ", for a spinor ", "type": "text"}, {"bbox": [334, 275, 343, 287], "score": 0.89, "content": "\\psi", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [343, 272, 469, 289], "score": 1.0, "content": ", takes its maximum at ", "type": "text"}, {"bbox": [469, 274, 509, 287], "score": 0.93, "content": "C^{i}J_{v}^{j}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [509, 272, 542, 289], "score": 1.0, "content": ". Our", "type": "text"}], "index": 10}, {"bbox": [70, 288, 502, 303], "spans": [{"bbox": [70, 288, 176, 303], "score": 1.0, "content": "first step is to force ", "type": "text"}, {"bbox": [177, 289, 268, 301], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [268, 288, 502, 303], "score": 1.0, "content": ". Unfortunately this requires a case analysis.", "type": "text"}], "index": 11}], "index": 9.5}, {"type": "text", "bbox": [69, 301, 541, 358], "lines": [{"bbox": [94, 301, 542, 319], "spans": [{"bbox": [94, 301, 201, 319], "score": 1.0, "content": "Consider first even ", "type": "text"}, {"bbox": [201, 304, 235, 315], "score": 0.92, "content": "r\\,\\neq\\,4", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [235, 301, 297, 319], "score": 1.0, "content": ", and even ", "type": "text"}, {"bbox": [297, 304, 331, 313], "score": 0.88, "content": "k\\ >\\ 2", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [332, 301, 376, 319], "score": 1.0, "content": ". Now, ", "type": "text"}, {"bbox": [377, 304, 504, 316], "score": 0.92, "content": "0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}", "type": "inline_equation", "height": 12, "width": 127}, {"bbox": [504, 301, 542, 319], "score": 1.0, "content": " forces", "type": "text"}], "index": 12}, {"bbox": [71, 317, 541, 347], "spans": [{"bbox": [71, 325, 104, 335], "score": 0.9, "content": "b=b^{\\prime}", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [104, 324, 394, 339], "score": 1.0, "content": "; hence hitting with the simple-current automorphism ", "type": "text"}, {"bbox": [394, 317, 449, 347], "score": 0.94, "content": "\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]", "type": "inline_equation", "height": 30, "width": 55}, {"bbox": [450, 326, 541, 338], "score": 1.0, "content": ", we may assume", "type": "text"}], "index": 13}, {"bbox": [71, 347, 165, 361], "spans": [{"bbox": [71, 347, 162, 358], "score": 0.91, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [163, 347, 165, 361], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [70, 358, 542, 444], "lines": [{"bbox": [93, 359, 541, 375], "spans": [{"bbox": [93, 359, 200, 375], "score": 1.0, "content": "Next consider even ", "type": "text"}, {"bbox": [200, 362, 230, 373], "score": 0.93, "content": "r\\neq4", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [230, 359, 281, 375], "score": 1.0, "content": " and odd ", "type": "text"}, {"bbox": [282, 362, 312, 371], "score": 0.9, "content": "k\\,>\\,2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [313, 359, 374, 375], "score": 1.0, "content": ". Either of ", "type": "text"}, {"bbox": [374, 362, 439, 373], "score": 0.92, "content": "\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [439, 359, 458, 375], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [459, 360, 527, 373], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [527, 359, 541, 375], "score": 1.0, "content": " is", "type": "text"}], "index": 15}, {"bbox": [69, 373, 542, 392], "spans": [{"bbox": [69, 373, 210, 392], "score": 1.0, "content": "impossible, by comparing ", "type": "text"}, {"bbox": [210, 376, 246, 388], "score": 0.92, "content": "S_{\\Lambda_{1},J_{s}0}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [247, 373, 275, 392], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [275, 376, 318, 388], "score": 0.93, "content": "S_{J_{v}\\Lambda_{1},J0}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [319, 373, 444, 392], "score": 1.0, "content": " for any simple-current ", "type": "text"}, {"bbox": [444, 376, 453, 385], "score": 0.86, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [453, 373, 542, 392], "score": 1.0, "content": ". For any of the", "type": "text"}], "index": 16}, {"bbox": [70, 387, 541, 405], "spans": [{"bbox": [70, 387, 208, 405], "score": 1.0, "content": "three remaining choices of ", "type": "text"}, {"bbox": [209, 389, 249, 402], "score": 0.93, "content": "J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [249, 387, 541, 405], "score": 1.0, "content": ", we can find a simple-current automorphism of the form", "type": "text"}], "index": 17}, {"bbox": [71, 403, 541, 433], "spans": [{"bbox": [71, 403, 123, 433], "score": 0.95, "content": "\\pi\\left[{\\ast}\\quad a\\,\\right];", "type": "inline_equation", "height": 30, "width": 52}, {"bbox": [124, 409, 252, 425], "score": 1.0, "content": " hitting its inverse onto ", "type": "text"}, {"bbox": [253, 415, 260, 421], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [260, 409, 356, 425], "score": 1.0, "content": " allows us to take ", "type": "text"}, {"bbox": [357, 412, 408, 421], "score": 0.92, "content": "a=b=0", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [408, 409, 452, 425], "score": 1.0, "content": ". Again ", "type": "text"}, {"bbox": [452, 412, 505, 424], "score": 0.95, "content": "0\\not=S_{\\Lambda_{1}\\Lambda_{1}}", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [505, 409, 541, 425], "score": 1.0, "content": " forces", "type": "text"}], "index": 18}, {"bbox": [71, 431, 420, 447], "spans": [{"bbox": [71, 433, 102, 443], "score": 0.87, "content": "b^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [102, 431, 157, 447], "score": 1.0, "content": ", and now ", "type": "text"}, {"bbox": [157, 433, 189, 443], "score": 0.91, "content": "a^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [189, 431, 325, 447], "score": 1.0, "content": " is forbidden. Thus again ", "type": "text"}, {"bbox": [325, 433, 416, 444], "score": 0.94, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [416, 431, 420, 447], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 17}, {"type": "text", "bbox": [70, 444, 541, 488], "lines": [{"bbox": [95, 446, 541, 461], "spans": [{"bbox": [95, 446, 147, 461], "score": 1.0, "content": "As usual, ", "type": "text"}, {"bbox": [147, 448, 176, 457], "score": 0.9, "content": "r=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [176, 446, 389, 461], "score": 1.0, "content": " is complicated by triality. We can force ", "type": "text"}, {"bbox": [390, 448, 441, 459], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [441, 446, 541, 461], "score": 1.0, "content": ". That we can also", "type": "text"}], "index": 20}, {"bbox": [70, 460, 541, 477], "spans": [{"bbox": [70, 460, 97, 477], "score": 1.0, "content": "take ", "type": "text"}, {"bbox": [97, 461, 154, 473], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [154, 460, 307, 477], "score": 1.0, "content": ", follows from the inequality ", "type": "text"}, {"bbox": [307, 461, 488, 474], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0", "type": "inline_equation", "height": 13, "width": 181}, {"bbox": [488, 460, 541, 477], "score": 1.0, "content": ", valid for", "type": "text"}], "index": 21}, {"bbox": [71, 473, 442, 492], "spans": [{"bbox": [71, 476, 100, 487], "score": 0.89, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [100, 473, 442, 492], "score": 1.0, "content": ". Establishing that inequality from (3.4) is equivalent to showing", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [98, 500, 514, 516], "lines": [{"bbox": [98, 502, 512, 520], "spans": [{"bbox": [98, 503, 500, 516], "score": 0.77, "content": "1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0", "type": "inline_equation"}, {"bbox": [501, 502, 512, 520], "score": 1.0, "content": "2)", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [69, 528, 398, 542], "lines": [{"bbox": [70, 530, 397, 545], "spans": [{"bbox": [70, 530, 89, 545], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [89, 531, 160, 544], "score": 0.92, "content": "0<x\\le2\\pi/9", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [160, 530, 397, 545], "score": 1.0, "content": ", which can be shown e.g. using Taylor series.", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [70, 543, 541, 657], "lines": [{"bbox": [93, 543, 540, 560], "spans": [{"bbox": [93, 543, 141, 560], "score": 1.0, "content": "For odd ", "type": "text"}, {"bbox": [141, 550, 147, 556], "score": 0.8, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [147, 543, 279, 560], "score": 1.0, "content": ", the charge-conjugation ", "type": "text"}, {"bbox": [279, 547, 289, 556], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [289, 543, 329, 560], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [330, 547, 344, 558], "score": 0.91, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [344, 543, 508, 560], "score": 1.0, "content": ". Since it must commute with ", "type": "text"}, {"bbox": [509, 550, 516, 556], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [516, 543, 540, 560], "score": 1.0, "content": ", i.e.", "type": "text"}], "index": 25}, {"bbox": [69, 556, 541, 574], "spans": [{"bbox": [69, 556, 96, 574], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [96, 558, 198, 573], "score": 0.93, "content": "C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 15, "width": 102}, {"bbox": [199, 556, 261, 574], "score": 1.0, "content": " must equal ", "type": "text"}, {"bbox": [262, 559, 352, 573], "score": 0.95, "content": "\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [353, 556, 419, 574], "score": 1.0, "content": ", we get that", "type": "text"}, {"bbox": [420, 561, 447, 570], "score": 0.9, "content": "b=0", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [448, 556, 505, 574], "score": 1.0, "content": ". Similarly ", "type": "text"}, {"bbox": [506, 560, 536, 570], "score": 0.92, "content": "b^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [537, 556, 541, 574], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [70, 572, 541, 590], "spans": [{"bbox": [70, 572, 106, 590], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [106, 574, 113, 584], "score": 0.78, "content": "k", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [114, 572, 208, 590], "score": 1.0, "content": " is odd, eliminate ", "type": "text"}, {"bbox": [208, 574, 238, 584], "score": 0.86, "content": "a=1", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [238, 572, 265, 590], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [265, 574, 298, 584], "score": 0.92, "content": "a\\prime=1", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [298, 572, 377, 590], "score": 1.0, "content": " by comparing ", "type": "text"}, {"bbox": [377, 575, 414, 587], "score": 0.92, "content": "S_{\\Lambda_{1},J_{s}0}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [414, 572, 441, 590], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 575, 485, 587], "score": 0.93, "content": "S_{J_{v}\\Lambda_{1},J0}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [485, 572, 541, 590], "score": 1.0, "content": " as before.", "type": "text"}], "index": 27}, {"bbox": [70, 587, 541, 602], "spans": [{"bbox": [70, 587, 179, 602], "score": 1.0, "content": "The hardest case is ", "type": "text"}, {"bbox": [180, 588, 187, 599], "score": 0.85, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [187, 587, 300, 602], "score": 1.0, "content": " even. We can force ", "type": "text"}, {"bbox": [300, 590, 354, 600], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [354, 587, 445, 602], "score": 1.0, "content": " by hitting with ", "type": "text"}, {"bbox": [445, 589, 466, 601], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [466, 587, 541, 602], "score": 1.0, "content": ". Suppose for", "type": "text"}], "index": 28}, {"bbox": [70, 601, 541, 617], "spans": [{"bbox": [70, 601, 170, 617], "score": 1.0, "content": "contradiction that ", "type": "text"}, {"bbox": [171, 601, 239, 614], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [239, 601, 301, 617], "score": 1.0, "content": ". We know ", "type": "text"}, {"bbox": [302, 603, 376, 616], "score": 0.93, "content": "\\pi^{\\prime}(J_{v}0)=J_{v}0", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [376, 601, 433, 617], "score": 1.0, "content": " (compare ", "type": "text"}, {"bbox": [433, 604, 470, 616], "score": 0.93, "content": "S_{\\Lambda_{1},J_{v}0}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [470, 601, 498, 617], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [498, 603, 536, 616], "score": 0.89, "content": "S_{\\Lambda_{1},J0})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [537, 601, 541, 617], "score": 1.0, "content": ",", "type": "text"}], "index": 29}, {"bbox": [69, 615, 540, 633], "spans": [{"bbox": [69, 615, 141, 633], "score": 1.0, "content": "so by (2.7b) ", "type": "text"}, {"bbox": [141, 616, 163, 629], "score": 0.86, "content": "\\pi\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [163, 615, 272, 633], "score": 1.0, "content": " must be a spinor.", "type": "text"}, {"bbox": [273, 617, 390, 630], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [391, 615, 442, 633], "score": 1.0, "content": " requires ", "type": "text"}, {"bbox": [442, 617, 536, 630], "score": 0.93, "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [537, 615, 540, 633], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [69, 630, 541, 646], "spans": [{"bbox": [69, 630, 123, 646], "score": 1.0, "content": "From the ", "type": "text"}, {"bbox": [123, 631, 169, 644], "score": 0.52, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [169, 630, 245, 646], "score": 1.0, "content": " fusion we get ", "type": "text"}, {"bbox": [245, 631, 360, 644], "score": 0.92, "content": "\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [360, 630, 388, 646], "score": 1.0, "content": ", but ", "type": "text"}, {"bbox": [389, 633, 438, 641], "score": 0.92, "content": "C\\pi=\\pi C", "type": "inline_equation", "height": 8, "width": 49}, {"bbox": [438, 630, 493, 646], "score": 1.0, "content": " says that ", "type": "text"}, {"bbox": [493, 632, 541, 644], "score": 0.9, "content": "\\pi\\Lambda_{r-1}=", "type": "inline_equation", "height": 12, "width": 48}], "index": 31}, {"bbox": [71, 644, 251, 662], "spans": [{"bbox": [71, 644, 147, 659], "score": 0.91, "content": "C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}", "type": "inline_equation", "height": 15, "width": 76}, {"bbox": [148, 644, 251, 662], "score": 1.0, "content": " — a contradiction.", "type": "text"}], "index": 32}], "index": 28.5}, {"type": "text", "bbox": [70, 658, 541, 715], "lines": [{"bbox": [93, 658, 541, 675], "spans": [{"bbox": [93, 658, 239, 675], "score": 1.0, "content": "Thus in all cases we have ", "type": "text"}, {"bbox": [240, 660, 340, 672], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 100}, {"bbox": [341, 658, 407, 675], "score": 1.0, "content": ". We know ", "type": "text"}, {"bbox": [408, 660, 486, 673], "score": 0.93, "content": "\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [486, 658, 541, 675], "score": 1.0, "content": " (compare", "type": "text"}], "index": 33}, {"bbox": [71, 673, 541, 689], "spans": [{"bbox": [71, 673, 109, 687], "score": 0.91, "content": "S_{\\Lambda_{1},J_{v}0}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [109, 674, 137, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [137, 673, 175, 687], "score": 0.89, "content": "S_{\\Lambda_{1},J0})", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [175, 674, 198, 689], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [198, 675, 219, 686], "score": 0.9, "content": "\\pi\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [220, 674, 410, 689], "score": 1.0, "content": " is a spinor and in fact must equal ", "type": "text"}, {"bbox": [411, 674, 491, 687], "score": 0.93, "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [491, 674, 541, 689], "score": 1.0, "content": ". Hitting", "type": "text"}], "index": 34}, {"bbox": [70, 687, 542, 704], "spans": [{"bbox": [70, 687, 98, 704], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [99, 687, 147, 702], "score": 0.93, "content": "(C_{1}^{i}\\pi_{v}^{j})^{-1}", "type": "inline_equation", "height": 15, "width": 48}, {"bbox": [147, 687, 234, 704], "score": 1.0, "content": ", we can require ", "type": "text"}, {"bbox": [234, 690, 285, 700], "score": 0.91, "content": "\\pi\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [285, 687, 322, 704], "score": 1.0, "content": ". That ", "type": "text"}, {"bbox": [323, 690, 356, 701], "score": 0.92, "content": "\\pi\\Lambda_{r-1}", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [356, 687, 445, 704], "score": 1.0, "content": " must now equal ", "type": "text"}, {"bbox": [446, 690, 471, 701], "score": 0.92, "content": "\\Lambda_{r-1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [472, 687, 542, 704], "score": 1.0, "content": " follows from", "type": "text"}], "index": 35}, {"bbox": [71, 703, 177, 717], "spans": [{"bbox": [71, 703, 91, 717], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [92, 703, 137, 715], "score": 0.28, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [138, 703, 177, 717], "score": 1.0, "content": " fusion.", "type": "text"}], "index": 36}], "index": 34.5}], "layout_bboxes": [], "page_idx": 18, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [99, 113, 500, 141], "lines": [{"bbox": [99, 113, 500, 141], "spans": [{"bbox": [99, 113, 500, 141], "score": 0.9, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>", "type": "interline_equation"}], "index": 2}], "index": 2}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 741], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "19", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 70, 542, 100], "lines": [], "index": 0.5, "bbox_fs": [71, 72, 542, 102], "lines_deleted": true}, {"type": "interline_equation", "bbox": [99, 113, 500, 141], "lines": [{"bbox": [99, 113, 500, 141], "spans": [{"bbox": [99, 113, 500, 141], "score": 0.9, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [70, 149, 541, 178], "lines": [{"bbox": [70, 151, 539, 168], "spans": [{"bbox": [70, 151, 100, 168], "score": 1.0, "content": "as in ", "type": "text"}, {"bbox": [100, 153, 121, 165], "score": 0.41, "content": "\\S4.3", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [122, 151, 165, 168], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [165, 154, 195, 163], "score": 0.92, "content": "r=k", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [195, 151, 360, 168], "score": 1.0, "content": ", that inequality only holds for ", "type": "text"}, {"bbox": [360, 153, 389, 164], "score": 0.88, "content": "\\ell>1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [390, 151, 487, 168], "score": 1.0, "content": ", but we can force ", "type": "text"}, {"bbox": [488, 154, 539, 165], "score": 0.91, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 51}], "index": 3}, {"bbox": [70, 166, 254, 183], "spans": [{"bbox": [70, 166, 127, 183], "score": 1.0, "content": "by hitting ", "type": "text"}, {"bbox": [127, 172, 135, 178], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [135, 166, 229, 183], "score": 1.0, "content": " if necessary with ", "type": "text"}, {"bbox": [229, 172, 249, 179], "score": 0.88, "content": "\\pi_{\\mathrm{rld}}", "type": "inline_equation", "height": 7, "width": 20}, {"bbox": [249, 166, 254, 183], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5, "bbox_fs": [70, 151, 539, 183]}, {"type": "text", "bbox": [70, 179, 541, 208], "lines": [{"bbox": [93, 180, 541, 197], "spans": [{"bbox": [93, 180, 203, 197], "score": 1.0, "content": "The remaining case ", "type": "text"}, {"bbox": [203, 182, 225, 195], "score": 0.91, "content": "C_{2,3}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [225, 180, 314, 197], "score": 1.0, "content": " follows because ", "type": "text"}, {"bbox": [314, 182, 371, 192], "score": 0.92, "content": "\\pi^{\\prime}J0\\,=\\,J0", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [372, 180, 434, 197], "score": 1.0, "content": ": by (2.7b) ", "type": "text"}, {"bbox": [434, 182, 495, 194], "score": 0.87, "content": "\\pi\\Lambda_{1}\\notin S\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [495, 180, 541, 197], "score": 1.0, "content": ", and by", "type": "text"}], "index": 5}, {"bbox": [72, 195, 292, 210], "spans": [{"bbox": [72, 196, 105, 210], "score": 1.0, "content": "(2.7a) ", "type": "text"}, {"bbox": [105, 195, 162, 208], "score": 0.89, "content": "\\pi\\Lambda_{1}\\neq3\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [163, 196, 170, 210], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [171, 195, 191, 208], "score": 0.88, "content": "3\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [191, 196, 216, 210], "score": 1.0, "content": " is a ", "type": "text"}, {"bbox": [216, 196, 225, 206], "score": 0.84, "content": "J", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [225, 196, 292, 210], "score": 1.0, "content": "-fixed-point).", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [72, 180, 541, 210]}, {"type": "title", "bbox": [71, 221, 219, 236], "lines": [{"bbox": [71, 225, 219, 236], "spans": [{"bbox": [71, 225, 119, 236], "score": 1.0, "content": "4.5. The ", "type": "text"}, {"bbox": [119, 225, 130, 235], "score": 0.8, "content": "D", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [131, 225, 219, 236], "score": 1.0, "content": "-series argument", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 243, 541, 300], "lines": [{"bbox": [95, 244, 541, 259], "spans": [{"bbox": [95, 247, 124, 256], "score": 0.9, "content": "k=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [124, 244, 201, 259], "score": 1.0, "content": " is trivial, and ", "type": "text"}, {"bbox": [202, 245, 231, 256], "score": 0.88, "content": "k=2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [232, 244, 398, 259], "score": 1.0, "content": " will be considered shortly. For ", "type": "text"}, {"bbox": [398, 247, 427, 256], "score": 0.9, "content": "k>2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [427, 244, 541, 259], "score": 1.0, "content": ", Proposition 4.1 tells", "type": "text"}], "index": 8}, {"bbox": [70, 258, 541, 274], "spans": [{"bbox": [70, 258, 111, 274], "score": 1.0, "content": "us that", "type": "text"}, {"bbox": [112, 259, 189, 273], "score": 0.93, "content": "\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 77}, {"bbox": [190, 258, 216, 274], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [216, 258, 303, 273], "score": 0.94, "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}", "type": "inline_equation", "height": 15, "width": 87}, {"bbox": [303, 258, 329, 274], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [329, 260, 419, 273], "score": 0.93, "content": "a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [419, 258, 541, 274], "score": 1.0, "content": ". Immediate from (3.4)", "type": "text"}], "index": 9}, {"bbox": [69, 272, 542, 289], "spans": [{"bbox": [69, 272, 109, 289], "score": 1.0, "content": "is that ", "type": "text"}, {"bbox": [110, 274, 173, 288], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [173, 272, 226, 289], "score": 1.0, "content": " and that ", "type": "text"}, {"bbox": [226, 275, 261, 287], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\psi]", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [262, 272, 334, 289], "score": 1.0, "content": ", for a spinor ", "type": "text"}, {"bbox": [334, 275, 343, 287], "score": 0.89, "content": "\\psi", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [343, 272, 469, 289], "score": 1.0, "content": ", takes its maximum at ", "type": "text"}, {"bbox": [469, 274, 509, 287], "score": 0.93, "content": "C^{i}J_{v}^{j}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [509, 272, 542, 289], "score": 1.0, "content": ". Our", "type": "text"}], "index": 10}, {"bbox": [70, 288, 502, 303], "spans": [{"bbox": [70, 288, 176, 303], "score": 1.0, "content": "first step is to force ", "type": "text"}, {"bbox": [177, 289, 268, 301], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [268, 288, 502, 303], "score": 1.0, "content": ". Unfortunately this requires a case analysis.", "type": "text"}], "index": 11}], "index": 9.5, "bbox_fs": [69, 244, 542, 303]}, {"type": "text", "bbox": [69, 301, 541, 358], "lines": [{"bbox": [94, 301, 542, 319], "spans": [{"bbox": [94, 301, 201, 319], "score": 1.0, "content": "Consider first even ", "type": "text"}, {"bbox": [201, 304, 235, 315], "score": 0.92, "content": "r\\,\\neq\\,4", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [235, 301, 297, 319], "score": 1.0, "content": ", and even ", "type": "text"}, {"bbox": [297, 304, 331, 313], "score": 0.88, "content": "k\\ >\\ 2", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [332, 301, 376, 319], "score": 1.0, "content": ". Now, ", "type": "text"}, {"bbox": [377, 304, 504, 316], "score": 0.92, "content": "0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}", "type": "inline_equation", "height": 12, "width": 127}, {"bbox": [504, 301, 542, 319], "score": 1.0, "content": " forces", "type": "text"}], "index": 12}, {"bbox": [71, 317, 541, 347], "spans": [{"bbox": [71, 325, 104, 335], "score": 0.9, "content": "b=b^{\\prime}", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [104, 324, 394, 339], "score": 1.0, "content": "; hence hitting with the simple-current automorphism ", "type": "text"}, {"bbox": [394, 317, 449, 347], "score": 0.94, "content": "\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]", "type": "inline_equation", "height": 30, "width": 55}, {"bbox": [450, 326, 541, 338], "score": 1.0, "content": ", we may assume", "type": "text"}], "index": 13}, {"bbox": [71, 347, 165, 361], "spans": [{"bbox": [71, 347, 162, 358], "score": 0.91, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [163, 347, 165, 361], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13, "bbox_fs": [71, 301, 542, 361]}, {"type": "text", "bbox": [70, 358, 542, 444], "lines": [{"bbox": [93, 359, 541, 375], "spans": [{"bbox": [93, 359, 200, 375], "score": 1.0, "content": "Next consider even ", "type": "text"}, {"bbox": [200, 362, 230, 373], "score": 0.93, "content": "r\\neq4", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [230, 359, 281, 375], "score": 1.0, "content": " and odd ", "type": "text"}, {"bbox": [282, 362, 312, 371], "score": 0.9, "content": "k\\,>\\,2", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [313, 359, 374, 375], "score": 1.0, "content": ". Either of ", "type": "text"}, {"bbox": [374, 362, 439, 373], "score": 0.92, "content": "\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [439, 359, 458, 375], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [459, 360, 527, 373], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [527, 359, 541, 375], "score": 1.0, "content": " is", "type": "text"}], "index": 15}, {"bbox": [69, 373, 542, 392], "spans": [{"bbox": [69, 373, 210, 392], "score": 1.0, "content": "impossible, by comparing ", "type": "text"}, {"bbox": [210, 376, 246, 388], "score": 0.92, "content": "S_{\\Lambda_{1},J_{s}0}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [247, 373, 275, 392], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [275, 376, 318, 388], "score": 0.93, "content": "S_{J_{v}\\Lambda_{1},J0}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [319, 373, 444, 392], "score": 1.0, "content": " for any simple-current ", "type": "text"}, {"bbox": [444, 376, 453, 385], "score": 0.86, "content": "J", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [453, 373, 542, 392], "score": 1.0, "content": ". For any of the", "type": "text"}], "index": 16}, {"bbox": [70, 387, 541, 405], "spans": [{"bbox": [70, 387, 208, 405], "score": 1.0, "content": "three remaining choices of ", "type": "text"}, {"bbox": [209, 389, 249, 402], "score": 0.93, "content": "J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [249, 387, 541, 405], "score": 1.0, "content": ", we can find a simple-current automorphism of the form", "type": "text"}], "index": 17}, {"bbox": [71, 403, 541, 433], "spans": [{"bbox": [71, 403, 123, 433], "score": 0.95, "content": "\\pi\\left[{\\ast}\\quad a\\,\\right];", "type": "inline_equation", "height": 30, "width": 52}, {"bbox": [124, 409, 252, 425], "score": 1.0, "content": " hitting its inverse onto ", "type": "text"}, {"bbox": [253, 415, 260, 421], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [260, 409, 356, 425], "score": 1.0, "content": " allows us to take ", "type": "text"}, {"bbox": [357, 412, 408, 421], "score": 0.92, "content": "a=b=0", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [408, 409, 452, 425], "score": 1.0, "content": ". Again ", "type": "text"}, {"bbox": [452, 412, 505, 424], "score": 0.95, "content": "0\\not=S_{\\Lambda_{1}\\Lambda_{1}}", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [505, 409, 541, 425], "score": 1.0, "content": " forces", "type": "text"}], "index": 18}, {"bbox": [71, 431, 420, 447], "spans": [{"bbox": [71, 433, 102, 443], "score": 0.87, "content": "b^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [102, 431, 157, 447], "score": 1.0, "content": ", and now ", "type": "text"}, {"bbox": [157, 433, 189, 443], "score": 0.91, "content": "a^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [189, 431, 325, 447], "score": 1.0, "content": " is forbidden. Thus again ", "type": "text"}, {"bbox": [325, 433, 416, 444], "score": 0.94, "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [416, 431, 420, 447], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 17, "bbox_fs": [69, 359, 542, 447]}, {"type": "text", "bbox": [70, 444, 541, 488], "lines": [{"bbox": [95, 446, 541, 461], "spans": [{"bbox": [95, 446, 147, 461], "score": 1.0, "content": "As usual, ", "type": "text"}, {"bbox": [147, 448, 176, 457], "score": 0.9, "content": "r=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [176, 446, 389, 461], "score": 1.0, "content": " is complicated by triality. We can force ", "type": "text"}, {"bbox": [390, 448, 441, 459], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [441, 446, 541, 461], "score": 1.0, "content": ". That we can also", "type": "text"}], "index": 20}, {"bbox": [70, 460, 541, 477], "spans": [{"bbox": [70, 460, 97, 477], "score": 1.0, "content": "take ", "type": "text"}, {"bbox": [97, 461, 154, 473], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [154, 460, 307, 477], "score": 1.0, "content": ", follows from the inequality ", "type": "text"}, {"bbox": [307, 461, 488, 474], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0", "type": "inline_equation", "height": 13, "width": 181}, {"bbox": [488, 460, 541, 477], "score": 1.0, "content": ", valid for", "type": "text"}], "index": 21}, {"bbox": [71, 473, 442, 492], "spans": [{"bbox": [71, 476, 100, 487], "score": 0.89, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [100, 473, 442, 492], "score": 1.0, "content": ". Establishing that inequality from (3.4) is equivalent to showing", "type": "text"}], "index": 22}], "index": 21, "bbox_fs": [70, 446, 541, 492]}, {"type": "text", "bbox": [98, 500, 514, 516], "lines": [{"bbox": [98, 502, 512, 520], "spans": [{"bbox": [98, 503, 500, 516], "score": 0.77, "content": "1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0", "type": "inline_equation"}, {"bbox": [501, 502, 512, 520], "score": 1.0, "content": "2)", "type": "text"}], "index": 23}], "index": 23, "bbox_fs": [98, 502, 512, 520]}, {"type": "text", "bbox": [69, 528, 398, 542], "lines": [{"bbox": [70, 530, 397, 545], "spans": [{"bbox": [70, 530, 89, 545], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [89, 531, 160, 544], "score": 0.92, "content": "0<x\\le2\\pi/9", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [160, 530, 397, 545], "score": 1.0, "content": ", which can be shown e.g. using Taylor series.", "type": "text"}], "index": 24}], "index": 24, "bbox_fs": [70, 530, 397, 545]}, {"type": "text", "bbox": [70, 543, 541, 657], "lines": [{"bbox": [93, 543, 540, 560], "spans": [{"bbox": [93, 543, 141, 560], "score": 1.0, "content": "For odd ", "type": "text"}, {"bbox": [141, 550, 147, 556], "score": 0.8, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [147, 543, 279, 560], "score": 1.0, "content": ", the charge-conjugation ", "type": "text"}, {"bbox": [279, 547, 289, 556], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [289, 543, 329, 560], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [330, 547, 344, 558], "score": 0.91, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [344, 543, 508, 560], "score": 1.0, "content": ". Since it must commute with ", "type": "text"}, {"bbox": [509, 550, 516, 556], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [516, 543, 540, 560], "score": 1.0, "content": ", i.e.", "type": "text"}], "index": 25}, {"bbox": [69, 556, 541, 574], "spans": [{"bbox": [69, 556, 96, 574], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [96, 558, 198, 573], "score": 0.93, "content": "C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 15, "width": 102}, {"bbox": [199, 556, 261, 574], "score": 1.0, "content": " must equal ", "type": "text"}, {"bbox": [262, 559, 352, 573], "score": 0.95, "content": "\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [353, 556, 419, 574], "score": 1.0, "content": ", we get that", "type": "text"}, {"bbox": [420, 561, 447, 570], "score": 0.9, "content": "b=0", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [448, 556, 505, 574], "score": 1.0, "content": ". Similarly ", "type": "text"}, {"bbox": [506, 560, 536, 570], "score": 0.92, "content": "b^{\\prime}=0", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [537, 556, 541, 574], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [70, 572, 541, 590], "spans": [{"bbox": [70, 572, 106, 590], "score": 1.0, "content": "When ", "type": "text"}, {"bbox": [106, 574, 113, 584], "score": 0.78, "content": "k", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [114, 572, 208, 590], "score": 1.0, "content": " is odd, eliminate ", "type": "text"}, {"bbox": [208, 574, 238, 584], "score": 0.86, "content": "a=1", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [238, 572, 265, 590], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [265, 574, 298, 584], "score": 0.92, "content": "a\\prime=1", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [298, 572, 377, 590], "score": 1.0, "content": " by comparing ", "type": "text"}, {"bbox": [377, 575, 414, 587], "score": 0.92, "content": "S_{\\Lambda_{1},J_{s}0}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [414, 572, 441, 590], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 575, 485, 587], "score": 0.93, "content": "S_{J_{v}\\Lambda_{1},J0}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [485, 572, 541, 590], "score": 1.0, "content": " as before.", "type": "text"}], "index": 27}, {"bbox": [70, 587, 541, 602], "spans": [{"bbox": [70, 587, 179, 602], "score": 1.0, "content": "The hardest case is ", "type": "text"}, {"bbox": [180, 588, 187, 599], "score": 0.85, "content": "k", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [187, 587, 300, 602], "score": 1.0, "content": " even. We can force ", "type": "text"}, {"bbox": [300, 590, 354, 600], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [354, 587, 445, 602], "score": 1.0, "content": " by hitting with ", "type": "text"}, {"bbox": [445, 589, 466, 601], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [466, 587, 541, 602], "score": 1.0, "content": ". Suppose for", "type": "text"}], "index": 28}, {"bbox": [70, 601, 541, 617], "spans": [{"bbox": [70, 601, 170, 617], "score": 1.0, "content": "contradiction that ", "type": "text"}, {"bbox": [171, 601, 239, 614], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [239, 601, 301, 617], "score": 1.0, "content": ". We know ", "type": "text"}, {"bbox": [302, 603, 376, 616], "score": 0.93, "content": "\\pi^{\\prime}(J_{v}0)=J_{v}0", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [376, 601, 433, 617], "score": 1.0, "content": " (compare ", "type": "text"}, {"bbox": [433, 604, 470, 616], "score": 0.93, "content": "S_{\\Lambda_{1},J_{v}0}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [470, 601, 498, 617], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [498, 603, 536, 616], "score": 0.89, "content": "S_{\\Lambda_{1},J0})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [537, 601, 541, 617], "score": 1.0, "content": ",", "type": "text"}], "index": 29}, {"bbox": [69, 615, 540, 633], "spans": [{"bbox": [69, 615, 141, 633], "score": 1.0, "content": "so by (2.7b) ", "type": "text"}, {"bbox": [141, 616, 163, 629], "score": 0.86, "content": "\\pi\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [163, 615, 272, 633], "score": 1.0, "content": " must be a spinor.", "type": "text"}, {"bbox": [273, 617, 390, 630], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [391, 615, 442, 633], "score": 1.0, "content": " requires ", "type": "text"}, {"bbox": [442, 617, 536, 630], "score": 0.93, "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [537, 615, 540, 633], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [69, 630, 541, 646], "spans": [{"bbox": [69, 630, 123, 646], "score": 1.0, "content": "From the ", "type": "text"}, {"bbox": [123, 631, 169, 644], "score": 0.52, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [169, 630, 245, 646], "score": 1.0, "content": " fusion we get ", "type": "text"}, {"bbox": [245, 631, 360, 644], "score": 0.92, "content": "\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [360, 630, 388, 646], "score": 1.0, "content": ", but ", "type": "text"}, {"bbox": [389, 633, 438, 641], "score": 0.92, "content": "C\\pi=\\pi C", "type": "inline_equation", "height": 8, "width": 49}, {"bbox": [438, 630, 493, 646], "score": 1.0, "content": " says that ", "type": "text"}, {"bbox": [493, 632, 541, 644], "score": 0.9, "content": "\\pi\\Lambda_{r-1}=", "type": "inline_equation", "height": 12, "width": 48}], "index": 31}, {"bbox": [71, 644, 251, 662], "spans": [{"bbox": [71, 644, 147, 659], "score": 0.91, "content": "C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}", "type": "inline_equation", "height": 15, "width": 76}, {"bbox": [148, 644, 251, 662], "score": 1.0, "content": " — a contradiction.", "type": "text"}], "index": 32}], "index": 28.5, "bbox_fs": [69, 543, 541, 662]}, {"type": "text", "bbox": [70, 658, 541, 715], "lines": [{"bbox": [93, 658, 541, 675], "spans": [{"bbox": [93, 658, 239, 675], "score": 1.0, "content": "Thus in all cases we have ", "type": "text"}, {"bbox": [240, 660, 340, 672], "score": 0.92, "content": "\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 100}, {"bbox": [341, 658, 407, 675], "score": 1.0, "content": ". We know ", "type": "text"}, {"bbox": [408, 660, 486, 673], "score": 0.93, "content": "\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [486, 658, 541, 675], "score": 1.0, "content": " (compare", "type": "text"}], "index": 33}, {"bbox": [71, 673, 541, 689], "spans": [{"bbox": [71, 673, 109, 687], "score": 0.91, "content": "S_{\\Lambda_{1},J_{v}0}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [109, 674, 137, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [137, 673, 175, 687], "score": 0.89, "content": "S_{\\Lambda_{1},J0})", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [175, 674, 198, 689], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [198, 675, 219, 686], "score": 0.9, "content": "\\pi\\Lambda_{r}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [220, 674, 410, 689], "score": 1.0, "content": " is a spinor and in fact must equal ", "type": "text"}, {"bbox": [411, 674, 491, 687], "score": 0.93, "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [491, 674, 541, 689], "score": 1.0, "content": ". Hitting", "type": "text"}], "index": 34}, {"bbox": [70, 687, 542, 704], "spans": [{"bbox": [70, 687, 98, 704], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [99, 687, 147, 702], "score": 0.93, "content": "(C_{1}^{i}\\pi_{v}^{j})^{-1}", "type": "inline_equation", "height": 15, "width": 48}, {"bbox": [147, 687, 234, 704], "score": 1.0, "content": ", we can require ", "type": "text"}, {"bbox": [234, 690, 285, 700], "score": 0.91, "content": "\\pi\\Lambda_{r}=\\Lambda_{r}", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [285, 687, 322, 704], "score": 1.0, "content": ". That ", "type": "text"}, {"bbox": [323, 690, 356, 701], "score": 0.92, "content": "\\pi\\Lambda_{r-1}", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [356, 687, 445, 704], "score": 1.0, "content": " must now equal ", "type": "text"}, {"bbox": [446, 690, 471, 701], "score": 0.92, "content": "\\Lambda_{r-1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [472, 687, 542, 704], "score": 1.0, "content": " follows from", "type": "text"}], "index": 35}, {"bbox": [71, 703, 177, 717], "spans": [{"bbox": [71, 703, 91, 717], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [92, 703, 137, 715], "score": 0.28, "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [138, 703, 177, 717], "score": 1.0, "content": " fusion.", "type": "text"}], "index": 36}], "index": 34.5, "bbox_fs": [70, 658, 542, 717]}]}
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[{"type": "text", "bbox": [69, 70, 542, 100], "content": "", "index": 0}, {"type": "interline_equation", "bbox": [99, 113, 500, 141], "content": "", "index": 1}, {"type": "text", "bbox": [70, 149, 541, 178], "content": "as in . When , that inequality only holds for , but we can force by hitting if necessary with .", "index": 2}, {"type": "text", "bbox": [70, 179, 541, 208], "content": "The remaining case follows because : by (2.7b) , and by (2.7a) ( is a -fixed-point).", "index": 3}, {"type": "title", "bbox": [71, 221, 219, 236], "content": "4.5. The -series argument", "index": 4}, {"type": "text", "bbox": [70, 243, 541, 300], "content": "is trivial, and will be considered shortly. For , Proposition 4.1 tells us that and , for . Immediate from (3.4) is that and that , for a spinor , takes its maximum at . Our first step is to force . Unfortunately this requires a case analysis.", "index": 5}, {"type": "text", "bbox": [69, 301, 541, 358], "content": "Consider first even , and even . Now, forces ; hence hitting with the simple-current automorphism , we may assume .", "index": 6}, {"type": "text", "bbox": [70, 358, 542, 444], "content": "Next consider even and odd . Either of or is impossible, by comparing and for any simple-current . For any of the three remaining choices of , we can find a simple-current automorphism of the form hitting its inverse onto allows us to take . Again forces , and now is forbidden. Thus again .", "index": 7}, {"type": "text", "bbox": [70, 444, 541, 488], "content": "As usual, is complicated by triality. We can force . That we can also take , follows from the inequality , valid for . Establishing that inequality from (3.4) is equivalent to showing", "index": 8}, {"type": "text", "bbox": [98, 500, 514, 516], "content": "2)", "index": 9}, {"type": "text", "bbox": [69, 528, 398, 542], "content": "for , which can be shown e.g. using Taylor series.", "index": 10}, {"type": "text", "bbox": [70, 543, 541, 657], "content": "For odd , the charge-conjugation equals . Since it must commute with , i.e. that must equal , we get that . Similarly . When is odd, eliminate and by comparing and as before. The hardest case is even. We can force by hitting with . Suppose for contradiction that . We know (compare and , so by (2.7b) must be a spinor. requires . From the fusion we get , but says that — a contradiction.", "index": 11}, {"type": "text", "bbox": [70, 658, 541, 715], "content": "Thus in all cases we have . We know (compare and , so is a spinor and in fact must equal . Hitting with , we can require . That must now equal follows from the fusion.", "index": 12}]
|
[{"bbox": [70, 151, 539, 168], "content": "as in . When , that inequality only holds for , but we can force", "parent_index": 2, "line_index": 0}, {"bbox": [70, 166, 254, 183], "content": "by hitting if necessary with .", "parent_index": 2, "line_index": 1}, {"bbox": [93, 180, 541, 197], "content": "The remaining case follows because : by (2.7b) , and by", "parent_index": 3, "line_index": 0}, {"bbox": [72, 195, 292, 210], "content": "(2.7a) ( is a -fixed-point).", "parent_index": 3, "line_index": 1}, {"bbox": [71, 225, 219, 236], "content": "4.5. The -series argument", "parent_index": 4, "line_index": 0}, {"bbox": [95, 244, 541, 259], "content": "is trivial, and will be considered shortly. For , Proposition 4.1 tells", "parent_index": 5, "line_index": 0}, {"bbox": [70, 258, 541, 274], "content": "us that and , for . Immediate from (3.4)", "parent_index": 5, "line_index": 1}, {"bbox": [69, 272, 542, 289], "content": "is that and that , for a spinor , takes its maximum at . Our", "parent_index": 5, "line_index": 2}, {"bbox": [70, 288, 502, 303], "content": "first step is to force . Unfortunately this requires a case analysis.", "parent_index": 5, "line_index": 3}, {"bbox": [94, 301, 542, 319], "content": "Consider first even , and even . Now, forces", "parent_index": 6, "line_index": 0}, {"bbox": [71, 317, 541, 347], "content": "; hence hitting with the simple-current automorphism , we may assume", "parent_index": 6, "line_index": 1}, {"bbox": [71, 347, 165, 361], "content": ".", "parent_index": 6, "line_index": 2}, {"bbox": [93, 359, 541, 375], "content": "Next consider even and odd . Either of or is", "parent_index": 7, "line_index": 0}, {"bbox": [69, 373, 542, 392], "content": "impossible, by comparing and for any simple-current . For any of the", "parent_index": 7, "line_index": 1}, {"bbox": [70, 387, 541, 405], "content": "three remaining choices of , we can find a simple-current automorphism of the form", "parent_index": 7, "line_index": 2}, {"bbox": [71, 403, 541, 433], "content": "hitting its inverse onto allows us to take . Again forces", "parent_index": 7, "line_index": 3}, {"bbox": [71, 431, 420, 447], "content": ", and now is forbidden. Thus again .", "parent_index": 7, "line_index": 4}, {"bbox": [95, 446, 541, 461], "content": "As usual, is complicated by triality. We can force . That we can also", "parent_index": 8, "line_index": 0}, {"bbox": [70, 460, 541, 477], "content": "take , follows from the inequality , valid for", "parent_index": 8, "line_index": 1}, {"bbox": [71, 473, 442, 492], "content": ". Establishing that inequality from (3.4) is equivalent to showing", "parent_index": 8, "line_index": 2}, {"bbox": [98, 502, 512, 520], "content": "2)", "parent_index": 9, "line_index": 0}, {"bbox": [70, 530, 397, 545], "content": "for , which can be shown e.g. using Taylor series.", "parent_index": 10, "line_index": 0}, {"bbox": [93, 543, 540, 560], "content": "For odd , the charge-conjugation equals . Since it must commute with , i.e.", "parent_index": 11, "line_index": 0}, {"bbox": [69, 556, 541, 574], "content": "that must equal , we get that . Similarly .", "parent_index": 11, "line_index": 1}, {"bbox": [70, 572, 541, 590], "content": "When is odd, eliminate and by comparing and as before.", "parent_index": 11, "line_index": 2}, {"bbox": [70, 587, 541, 602], "content": "The hardest case is even. We can force by hitting with . Suppose for", "parent_index": 11, "line_index": 3}, {"bbox": [70, 601, 541, 617], "content": "contradiction that . We know (compare and ,", "parent_index": 11, "line_index": 4}, {"bbox": [69, 615, 540, 633], "content": "so by (2.7b) must be a spinor. requires .", "parent_index": 11, "line_index": 5}, {"bbox": [69, 630, 541, 646], "content": "From the fusion we get , but says that", "parent_index": 11, "line_index": 6}, {"bbox": [71, 644, 251, 662], "content": "— a contradiction.", "parent_index": 11, "line_index": 7}, {"bbox": [93, 658, 541, 675], "content": "Thus in all cases we have . We know (compare", "parent_index": 12, "line_index": 0}, {"bbox": [71, 673, 541, 689], "content": "and , so is a spinor and in fact must equal . Hitting", "parent_index": 12, "line_index": 1}, {"bbox": [70, 687, 542, 704], "content": "with , we can require . That must now equal follows from", "parent_index": 12, "line_index": 2}, {"bbox": [71, 703, 177, 717], "content": "the fusion.", "parent_index": 12, "line_index": 3}]
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[]
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[{"bbox": [99, 113, 500, 141], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r+2-\\ell}{2\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{2\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{2\\kappa})\\}>", "parent_index": 1, "subtype": "interline"}, {"bbox": [100, 153, 121, 165], "content": "\\S4.3", "parent_index": 2, "subtype": "inline"}, {"bbox": [165, 154, 195, 163], "content": "r=k", "parent_index": 2, "subtype": "inline"}, {"bbox": [360, 153, 389, 164], "content": "\\ell>1", "parent_index": 2, "subtype": "inline"}, {"bbox": [488, 154, 539, 165], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [127, 172, 135, 178], "content": "\\pi", "parent_index": 2, "subtype": "inline"}, {"bbox": [229, 172, 249, 179], "content": "\\pi_{\\mathrm{rld}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [203, 182, 225, 195], "content": "C_{2,3}", "parent_index": 3, "subtype": "inline"}, {"bbox": [314, 182, 371, 192], "content": "\\pi^{\\prime}J0\\,=\\,J0", "parent_index": 3, "subtype": "inline"}, {"bbox": [434, 182, 495, 194], "content": "\\pi\\Lambda_{1}\\notin S\\Lambda_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [105, 195, 162, 208], "content": "\\pi\\Lambda_{1}\\neq3\\Lambda_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [171, 195, 191, 208], "content": "3\\Lambda_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [216, 196, 225, 206], "content": "J", "parent_index": 3, "subtype": "inline"}, {"bbox": [119, 225, 130, 235], "content": "D", "parent_index": 4, "subtype": "inline"}, {"bbox": [95, 247, 124, 256], "content": "k=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [202, 245, 231, 256], "content": "k=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [398, 247, 427, 256], "content": "k>2", "parent_index": 5, "subtype": "inline"}, {"bbox": [112, 259, 189, 273], "content": "\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [216, 258, 303, 273], "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [329, 260, 419, 273], "content": "a,a^{\\prime},b,b^{\\prime}\\in\\{0,1\\}", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 274, 173, 288], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0", "parent_index": 5, "subtype": "inline"}, {"bbox": [226, 275, 261, 287], "content": "\\chi_{\\Lambda_{1}}[\\psi]", "parent_index": 5, "subtype": "inline"}, {"bbox": [334, 275, 343, 287], "content": "\\psi", "parent_index": 5, "subtype": "inline"}, {"bbox": [469, 274, 509, 287], "content": "C^{i}J_{v}^{j}\\Lambda_{r}", "parent_index": 5, "subtype": "inline"}, {"bbox": [177, 289, 268, 301], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [201, 304, 235, 315], "content": "r\\,\\neq\\,4", "parent_index": 6, "subtype": "inline"}, {"bbox": [297, 304, 331, 313], "content": "k\\ >\\ 2", "parent_index": 6, "subtype": "inline"}, {"bbox": [377, 304, 504, 316], "content": "0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [71, 325, 104, 335], "content": "b=b^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [394, 317, 449, 347], "content": "\\pi\\left[{\\begin{array}{l l}{0}&{a}\\\\ {a^{\\prime}}&{b}\\end{array}}\\right]", "parent_index": 6, "subtype": "inline"}, {"bbox": [71, 347, 162, 358], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [200, 362, 230, 373], "content": "r\\neq4", "parent_index": 7, "subtype": "inline"}, {"bbox": [282, 362, 312, 371], "content": "k\\,>\\,2", "parent_index": 7, "subtype": "inline"}, {"bbox": [374, 362, 439, 373], "content": "\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [459, 360, 527, 373], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [210, 376, 246, 388], "content": "S_{\\Lambda_{1},J_{s}0}", "parent_index": 7, "subtype": "inline"}, {"bbox": [275, 376, 318, 388], "content": "S_{J_{v}\\Lambda_{1},J0}", "parent_index": 7, "subtype": "inline"}, {"bbox": [444, 376, 453, 385], "content": "J", "parent_index": 7, "subtype": "inline"}, {"bbox": [209, 389, 249, 402], "content": "J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [71, 403, 123, 433], "content": "\\pi\\left[{\\ast}\\quad a\\,\\right];", "parent_index": 7, "subtype": "inline"}, {"bbox": [253, 415, 260, 421], "content": "\\pi", "parent_index": 7, "subtype": "inline"}, {"bbox": [357, 412, 408, 421], "content": "a=b=0", "parent_index": 7, "subtype": "inline"}, {"bbox": [452, 412, 505, 424], "content": "0\\not=S_{\\Lambda_{1}\\Lambda_{1}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [71, 433, 102, 443], "content": "b^{\\prime}=0", "parent_index": 7, "subtype": "inline"}, {"bbox": [157, 433, 189, 443], "content": "a^{\\prime}=1", "parent_index": 7, "subtype": "inline"}, {"bbox": [325, 433, 416, 444], "content": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [147, 448, 176, 457], "content": "r=4", "parent_index": 8, "subtype": "inline"}, {"bbox": [390, 448, 441, 459], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [97, 461, 154, 473], "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [307, 461, 488, 474], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>\\chi_{\\Lambda_{1}}[\\Lambda_{3}]=\\chi_{\\Lambda_{1}}[\\Lambda_{4}]>0", "parent_index": 8, "subtype": "inline"}, {"bbox": [71, 476, 100, 487], "content": "k\\geq3", "parent_index": 8, "subtype": "inline"}, {"bbox": [98, 503, 500, 516], "content": "1+\\cos(x)+\\cos(2x)+\\cos(4x)>\\cos(x/2)+\\cos(3x/2)+\\cos(5x/2)+\\cos(7x/2)>0", "parent_index": 9, "subtype": "inline"}, {"bbox": [89, 531, 160, 544], "content": "0<x\\le2\\pi/9", "parent_index": 10, "subtype": "inline"}, {"bbox": [141, 550, 147, 556], "content": "r", "parent_index": 11, "subtype": "inline"}, {"bbox": [279, 547, 289, 556], "content": "C", "parent_index": 11, "subtype": "inline"}, {"bbox": [330, 547, 344, 558], "content": "C_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [509, 550, 516, 556], "content": "\\pi", "parent_index": 11, "subtype": "inline"}, {"bbox": [96, 558, 198, 573], "content": "C_{1}\\pi\\Lambda_{1}=J_{v}^{a+b}J_{s}^{b}\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [262, 559, 352, 573], "content": "\\pi C_{1}\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [420, 561, 447, 570], "content": "b=0", "parent_index": 11, "subtype": "inline"}, {"bbox": [506, 560, 536, 570], "content": "b^{\\prime}=0", "parent_index": 11, "subtype": "inline"}, {"bbox": [106, 574, 113, 584], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [208, 574, 238, 584], "content": "a=1", "parent_index": 11, "subtype": "inline"}, {"bbox": [265, 574, 298, 584], "content": "a\\prime=1", "parent_index": 11, "subtype": "inline"}, {"bbox": [377, 575, 414, 587], "content": "S_{\\Lambda_{1},J_{s}0}", "parent_index": 11, "subtype": "inline"}, {"bbox": [442, 575, 485, 587], "content": "S_{J_{v}\\Lambda_{1},J0}", "parent_index": 11, "subtype": "inline"}, {"bbox": [180, 588, 187, 599], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [300, 590, 354, 600], "content": "\\pi\\Lambda_{1}\\,=\\,\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [445, 589, 466, 601], "content": "\\pi[a]", "parent_index": 11, "subtype": "inline"}, {"bbox": [171, 601, 239, 614], "content": "\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [302, 603, 376, 616], "content": "\\pi^{\\prime}(J_{v}0)=J_{v}0", "parent_index": 11, "subtype": "inline"}, {"bbox": [433, 604, 470, 616], "content": "S_{\\Lambda_{1},J_{v}0}", "parent_index": 11, "subtype": "inline"}, {"bbox": [498, 603, 536, 616], "content": "S_{\\Lambda_{1},J0})", "parent_index": 11, "subtype": "inline"}, {"bbox": [141, 616, 163, 629], "content": "\\pi\\Lambda_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [273, 617, 390, 630], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]", "parent_index": 11, "subtype": "inline"}, {"bbox": [442, 617, 536, 630], "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [123, 631, 169, 644], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "parent_index": 11, "subtype": "inline"}, {"bbox": [245, 631, 360, 644], "content": "\\pi\\Lambda_{r-1}=C_{1}^{i}J_{v}^{j}J_{s}\\Lambda_{r-1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [389, 633, 438, 641], "content": "C\\pi=\\pi C", "parent_index": 11, "subtype": "inline"}, {"bbox": [493, 632, 541, 644], "content": "\\pi\\Lambda_{r-1}=", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 644, 147, 659], "content": "C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [240, 660, 340, 672], "content": "\\pi\\Lambda_{1}\\,=\\,\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\Lambda_{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [408, 660, 486, 673], "content": "\\pi^{\\prime}(J_{v}0)\\;=\\;J_{v}0", "parent_index": 12, "subtype": "inline"}, {"bbox": [71, 673, 109, 687], "content": "S_{\\Lambda_{1},J_{v}0}", "parent_index": 12, "subtype": "inline"}, {"bbox": [137, 673, 175, 687], "content": "S_{\\Lambda_{1},J0})", "parent_index": 12, "subtype": "inline"}, {"bbox": [198, 675, 219, 686], "content": "\\pi\\Lambda_{r}", "parent_index": 12, "subtype": "inline"}, {"bbox": [411, 674, 491, 687], "content": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}", "parent_index": 12, "subtype": "inline"}, {"bbox": [99, 687, 147, 702], "content": "(C_{1}^{i}\\pi_{v}^{j})^{-1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [234, 690, 285, 700], "content": "\\pi\\Lambda_{r}=\\Lambda_{r}", "parent_index": 12, "subtype": "inline"}, {"bbox": [323, 690, 356, 701], "content": "\\pi\\Lambda_{r-1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [446, 690, 471, 701], "content": "\\Lambda_{r-1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [92, 703, 137, 715], "content": "\\Lambda_{1}\\boxtimes\\Lambda_{r}", "parent_index": 12, "subtype": "inline"}]
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Next, note that we know from $\Lambda_{1}$ × $\Lambda_{1}$ that $\pi\Lambda_{2}$ is $\Lambda_{2}$ or $2\Lambda_{1}$ . As in $\S4.2$ , the fusion $(2\Lambda_{1})$ × $\Lambda_{1}$ × · · · × $\Lambda_{1}$ ( $k{-}2$ times) contains the simple-current $J_{v}0$ , but $\Lambda_{2}$ × $\Lambda_{1}$ × · · · × $\Lambda_{1}$ ( $k-2$ times) doesn’t, and thus $\pi\Lambda_{2}=\Lambda_{2}$ .
Assume $\pi\Lambda_{\ell}=\Lambda_{\ell}$ . Using the fusions $\Lambda_{1}$ × $\Lambda_{\ell}$ (for $1<\ell<r-2$ ), and noting that
$$
\chi_{\Lambda_{1}}[\Lambda_{\ell+1}]-\chi_{\Lambda_{1}}[\Lambda_{1}+\Lambda_{\ell}]=4\cos(\pi\,\frac{2r-\ell}{\kappa})\,\{\cos(\pi\,\frac{\ell}{\kappa})-\cos(\pi\,\frac{\ell+2}{\kappa})\}
$$
equals 0 only when $\ell\,=\,r+1\,-\,k/2$ , we see that $\pi\Lambda_{\ell+1}\;=\;\Lambda_{\ell+1}$ except possibly for $\ell=r+1-k/2$ (hence $2r-2\geq k\geq4)$ ). For that $\ell$ , use q-dimensions:
$$
\frac{\mathcal{D}(\Lambda_{1}+\Lambda_{\ell})}{\mathcal{D}(\Lambda_{\ell+1})}=\frac{\sin(2\pi\left(k-2\right)/\kappa)}{\sin(2\pi/\kappa)}>1\ ,
$$
which is valid for these $k$ . So we also know $\pi\Lambda_{i}=\Lambda_{i}$ for all $i\le r-2$ , and we are done.
All that remains is $D_{r,2}$ . Recall the $\lambda^{i}$ defined in $\S3.4$ . Note that $\mathcal{D}(\Lambda_{r})\;=\;\sqrt{r}$ , $\mathcal{D}(\lambda^{a})=2$ , and $S_{\lambda^{a}\lambda^{b}}/S_{0\lambda^{b}}=2\cos(\pi a b/r)$ . For $r\neq4$ , the q-dimensions force $\pi\Lambda_{1}=\lambda^{m}$ and $\pi^{\prime}\Lambda_{1}\,=\,\lambda^{m^{\prime}}$ , and $S_{\Lambda_{1}\Lambda_{1}}\,=\,S_{\lambda^{m}\lambda^{m^{\prime}}}$ says $m m^{\prime}\,\equiv\,\pm1$ (mod $2r$ ). So without loss of generality we may take $m=m^{\prime}=1$ . The rest of the argument is easy.
For $D_{4,2}$ , we can force $\pi\Lambda_{1}=\Lambda_{1}$ , and then eliminate $\pi^{\prime}\Lambda_{1}=\Lambda_{r-1}$ or $\Lambda_{r}$ by $S_{\Lambda_{1}\Lambda_{1}}\ne$ $0=S_{\Lambda_{1}\Lambda_{r}}=S_{\Lambda_{1}\Lambda_{r-1}}$ . The rest of the argument is as before.
4.6. The arguments for the exceptional algebras
The exceptional algebras follow quickly from the fusions (and Dynkin diagram symmetries) given in §§3.5-3.9.
For example, consider $E_{6}^{(1)}$ for $k\geq2$ . Proposition 4.1 tells us $\pi\Lambda_{1}=C^{a}J^{b}\Lambda_{1}$ for some $a,b$ , and we know $\pi^{\prime}J0\,=\,J^{c}0$ for $c=\pm1$ . Hence from (2.7b) we get $k b\not\equiv-1$ (mod 3). Hitting $\pi$ with $\pi[-b]^{-1}C^{a}$ , we need consider only $\pi\Lambda_{1}=\Lambda_{1}$ . It is now immediate that $\pi\Lambda_{5}=\Lambda_{5}$ , by commuting $\pi$ with $C$ . From (3.6a) we get that $\pi$ must permute $\Lambda_{2}$ and $2\Lambda_{1}$ . Compare (3.6c) with (3.6d): since for any $k\geq2$ they have different numbers of summands, we find in fact that $\pi$ will fix both $\Lambda_{2}$ (hence $\Lambda_{4}$ ) and $2\Lambda_{1}$ . From (3.6b) we get that $\pi$ permutes $\Lambda_{6}$ and $\Lambda_{1}+\Lambda_{5}$ , and so (3.6d) now tells us $\pi\Lambda_{6}=\Lambda_{6}$ . Finally, (3.6c) implies (for $k\geq3$ ) $\pi\Lambda_{3}=\Lambda_{3}$ (since $C\pi=\pi C$ ), and we are done for $k\geq3$ . Since $\{\Lambda_{1},\Lambda_{2},\Lambda_{4},\Lambda_{5},\Lambda_{6}\}$ is a fusion-generator for $k=2$ (see $\S2.2)$ , we are also done for $k=2$ .
For ${E}_{8}^{(1)}$ when $k\geq7$ , (3.7a) tells us that $\Lambda_{2},\Lambda_{7},2\Lambda_{1}$ are permuted. For those $k$ , the highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so $\Lambda_{2},\Lambda_{7},2\Lambda_{1}$ must all be fixed. The fusion product (3.7c) also tells us that $\Lambda_{3},\Lambda_{6},\Lambda_{8},\Lambda_{1}+\Lambda_{7},2\Lambda_{7}$ are permuted; (3.7d) then says that the sets $\{\Lambda_{6},\Lambda_{8}\}$ , $\{\Lambda_{3},\Lambda_{1}+\Lambda_{7},2\Lambda_{7}\}$ , and $\{2\Lambda_{2},\Lambda_{2}+\Lambda_{7},3\Lambda_{1},2\Lambda_{1}+$ $\Lambda_{2},2\Lambda_{1}+\Lambda_{7},4\Lambda_{1}\right\}$ are stabilised. Now (3.7b) implies $\Lambda_{3},\Lambda_{6},\Lambda_{8},2\Lambda_{7}$ are all fixed, while the set $\{\Lambda_{4},\Lambda_{1}+\Lambda_{3}\}$ is stabilised. Comparing $(3.7\mathrm{e})$ and (3.7f), we get that $\Lambda_{4}$ is fixed and $\Lambda_{5},\Lambda_{7}+\Lambda_{8}$ are permuted. Finally, $\left(3.7\mathrm{g}\right)$ shows $\Lambda_{5}$ also is fixed. To do ${E}_{8}^{(1)}$ when $k\leq6$ , knowing q-dimensions really simplifies things.
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<html><body>
<p data-bbox="71 70 569 114">Next, note that we know from $\Lambda_{1}$ × $\Lambda_{1}$ that $\pi\Lambda_{2}$ is $\Lambda_{2}$ or $2\Lambda_{1}$ . As in $\S4.2$ , the fusion $(2\Lambda_{1})$ × $\Lambda_{1}$ × · · · × $\Lambda_{1}$ ( $k{-}2$ times) contains the simple-current $J_{v}0$ , but $\Lambda_{2}$ × $\Lambda_{1}$ × · · · × $\Lambda_{1}$ ( $k-2$ times) doesn’t, and thus $\pi\Lambda_{2}=\Lambda_{2}$ . </p>
<p data-bbox="93 114 527 129">Assume $\pi\Lambda_{\ell}=\Lambda_{\ell}$ . Using the fusions $\Lambda_{1}$ × $\Lambda_{\ell}$ (for $1<\ell<r-2$ ), and noting that </p>
<div class="equation" data-bbox="124 142 488 171">$$
\chi_{\Lambda_{1}}[\Lambda_{\ell+1}]-\chi_{\Lambda_{1}}[\Lambda_{1}+\Lambda_{\ell}]=4\cos(\pi\,\frac{2r-\ell}{\kappa})\,\{\cos(\pi\,\frac{\ell}{\kappa})-\cos(\pi\,\frac{\ell+2}{\kappa})\}
$$</div>
<p data-bbox="69 182 541 211">equals 0 only when $\ell\,=\,r+1\,-\,k/2$ , we see that $\pi\Lambda_{\ell+1}\;=\;\Lambda_{\ell+1}$ except possibly for $\ell=r+1-k/2$ (hence $2r-2\geq k\geq4)$ ). For that $\ell$ , use q-dimensions: </p>
<div class="equation" data-bbox="205 225 405 257">$$
\frac{\mathcal{D}(\Lambda_{1}+\Lambda_{\ell})}{\mathcal{D}(\Lambda_{\ell+1})}=\frac{\sin(2\pi\left(k-2\right)/\kappa)}{\sin(2\pi/\kappa)}>1\ ,
$$</div>
<p data-bbox="71 267 531 282">which is valid for these $k$ . So we also know $\pi\Lambda_{i}=\Lambda_{i}$ for all $i\le r-2$ , and we are done. </p>
<p data-bbox="70 284 542 342">All that remains is $D_{r,2}$ . Recall the $\lambda^{i}$ defined in $\S3.4$ . Note that $\mathcal{D}(\Lambda_{r})\;=\;\sqrt{r}$ , $\mathcal{D}(\lambda^{a})=2$ , and $S_{\lambda^{a}\lambda^{b}}/S_{0\lambda^{b}}=2\cos(\pi a b/r)$ . For $r\neq4$ , the q-dimensions force $\pi\Lambda_{1}=\lambda^{m}$ and $\pi^{\prime}\Lambda_{1}\,=\,\lambda^{m^{\prime}}$ , and $S_{\Lambda_{1}\Lambda_{1}}\,=\,S_{\lambda^{m}\lambda^{m^{\prime}}}$ says $m m^{\prime}\,\equiv\,\pm1$ (mod $2r$ ). So without loss of generality we may take $m=m^{\prime}=1$ . The rest of the argument is easy. </p>
<p data-bbox="70 343 541 372">For $D_{4,2}$ , we can force $\pi\Lambda_{1}=\Lambda_{1}$ , and then eliminate $\pi^{\prime}\Lambda_{1}=\Lambda_{r-1}$ or $\Lambda_{r}$ by $S_{\Lambda_{1}\Lambda_{1}}\ne$ $0=S_{\Lambda_{1}\Lambda_{r}}=S_{\Lambda_{1}\Lambda_{r-1}}$ . The rest of the argument is as before. </p>
<p data-bbox="72 385 321 399">4.6. The arguments for the exceptional algebras </p>
<p data-bbox="70 406 540 436">The exceptional algebras follow quickly from the fusions (and Dynkin diagram symmetries) given in §§3.5-3.9. </p>
<p data-bbox="70 437 541 567">For example, consider $E_{6}^{(1)}$ for $k\geq2$ . Proposition 4.1 tells us $\pi\Lambda_{1}=C^{a}J^{b}\Lambda_{1}$ for some $a,b$ , and we know $\pi^{\prime}J0\,=\,J^{c}0$ for $c=\pm1$ . Hence from (2.7b) we get $k b\not\equiv-1$ (mod 3). Hitting $\pi$ with $\pi[-b]^{-1}C^{a}$ , we need consider only $\pi\Lambda_{1}=\Lambda_{1}$ . It is now immediate that $\pi\Lambda_{5}=\Lambda_{5}$ , by commuting $\pi$ with $C$ . From (3.6a) we get that $\pi$ must permute $\Lambda_{2}$ and $2\Lambda_{1}$ . Compare (3.6c) with (3.6d): since for any $k\geq2$ they have different numbers of summands, we find in fact that $\pi$ will fix both $\Lambda_{2}$ (hence $\Lambda_{4}$ ) and $2\Lambda_{1}$ . From (3.6b) we get that $\pi$ permutes $\Lambda_{6}$ and $\Lambda_{1}+\Lambda_{5}$ , and so (3.6d) now tells us $\pi\Lambda_{6}=\Lambda_{6}$ . Finally, (3.6c) implies (for $k\geq3$ ) $\pi\Lambda_{3}=\Lambda_{3}$ (since $C\pi=\pi C$ ), and we are done for $k\geq3$ . Since $\{\Lambda_{1},\Lambda_{2},\Lambda_{4},\Lambda_{5},\Lambda_{6}\}$ is a fusion-generator for $k=2$ (see $\S2.2)$ , we are also done for $k=2$ . </p>
<p data-bbox="70 569 542 687">For ${E}_{8}^{(1)}$ when $k\geq7$ , (3.7a) tells us that $\Lambda_{2},\Lambda_{7},2\Lambda_{1}$ are permuted. For those $k$ , the highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so $\Lambda_{2},\Lambda_{7},2\Lambda_{1}$ must all be fixed. The fusion product (3.7c) also tells us that $\Lambda_{3},\Lambda_{6},\Lambda_{8},\Lambda_{1}+\Lambda_{7},2\Lambda_{7}$ are permuted; (3.7d) then says that the sets $\{\Lambda_{6},\Lambda_{8}\}$ , $\{\Lambda_{3},\Lambda_{1}+\Lambda_{7},2\Lambda_{7}\}$ , and $\{2\Lambda_{2},\Lambda_{2}+\Lambda_{7},3\Lambda_{1},2\Lambda_{1}+$ $\Lambda_{2},2\Lambda_{1}+\Lambda_{7},4\Lambda_{1}\right\}$ are stabilised. Now (3.7b) implies $\Lambda_{3},\Lambda_{6},\Lambda_{8},2\Lambda_{7}$ are all fixed, while the set $\{\Lambda_{4},\Lambda_{1}+\Lambda_{3}\}$ is stabilised. Comparing $(3.7\mathrm{e})$ and (3.7f), we get that $\Lambda_{4}$ is fixed and $\Lambda_{5},\Lambda_{7}+\Lambda_{8}$ are permuted. Finally, $\left(3.7\mathrm{g}\right)$ shows $\Lambda_{5}$ also is fixed. To do ${E}_{8}^{(1)}$ when $k\leq6$ , knowing q-dimensions really simplifies things. </p>
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0002044v1
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[{"type": "text", "text": "Next, note that we know from $\\Lambda_{1}$ × $\\Lambda_{1}$ that $\\pi\\Lambda_{2}$ is $\\Lambda_{2}$ or $2\\Lambda_{1}$ . As in $\\S4.2$ , the fusion $(2\\Lambda_{1})$ × $\\Lambda_{1}$ × · · · × $\\Lambda_{1}$ ( $k{-}2$ times) contains the simple-current $J_{v}0$ , but $\\Lambda_{2}$ × $\\Lambda_{1}$ × · · · × $\\Lambda_{1}$ ( $k-2$ times) doesn’t, and thus $\\pi\\Lambda_{2}=\\Lambda_{2}$ . ", "page_idx": 19}, {"type": "text", "text": "Assume $\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$ . Using the fusions $\\Lambda_{1}$ × $\\Lambda_{\\ell}$ (for $1<\\ell<r-2$ ), and noting that ", "page_idx": 19}, {"type": "equation", "text": "$$\n\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}\n$$", "text_format": "latex", "page_idx": 19}, {"type": "text", "text": "equals 0 only when $\\ell\\,=\\,r+1\\,-\\,k/2$ , we see that $\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}$ except possibly for $\\ell=r+1-k/2$ (hence $2r-2\\geq k\\geq4)$ ). For that $\\ell$ , use q-dimensions: ", "page_idx": 19}, {"type": "equation", "text": "$$\n\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,\n$$", "text_format": "latex", "page_idx": 19}, {"type": "text", "text": "which is valid for these $k$ . So we also know $\\pi\\Lambda_{i}=\\Lambda_{i}$ for all $i\\le r-2$ , and we are done. ", "page_idx": 19}, {"type": "text", "text": "All that remains is $D_{r,2}$ . Recall the $\\lambda^{i}$ defined in $\\S3.4$ . Note that $\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}$ , $\\mathcal{D}(\\lambda^{a})=2$ , and $S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)$ . For $r\\neq4$ , the q-dimensions force $\\pi\\Lambda_{1}=\\lambda^{m}$ and $\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}$ , and $S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}$ says $m m^{\\prime}\\,\\equiv\\,\\pm1$ (mod $2r$ ). So without loss of generality we may take $m=m^{\\prime}=1$ . The rest of the argument is easy. ", "page_idx": 19}, {"type": "text", "text": "For $D_{4,2}$ , we can force $\\pi\\Lambda_{1}=\\Lambda_{1}$ , and then eliminate $\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}$ or $\\Lambda_{r}$ by $S_{\\Lambda_{1}\\Lambda_{1}}\\ne$ $0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}$ . The rest of the argument is as before. ", "page_idx": 19}, {"type": "text", "text": "4.6. The arguments for the exceptional algebras ", "page_idx": 19}, {"type": "text", "text": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram symmetries) given in §§3.5-3.9. ", "page_idx": 19}, {"type": "text", "text": "For example, consider $E_{6}^{(1)}$ for $k\\geq2$ . Proposition 4.1 tells us $\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}$ for some $a,b$ , and we know $\\pi^{\\prime}J0\\,=\\,J^{c}0$ for $c=\\pm1$ . Hence from (2.7b) we get $k b\\not\\equiv-1$ (mod 3). Hitting $\\pi$ with $\\pi[-b]^{-1}C^{a}$ , we need consider only $\\pi\\Lambda_{1}=\\Lambda_{1}$ . It is now immediate that $\\pi\\Lambda_{5}=\\Lambda_{5}$ , by commuting $\\pi$ with $C$ . From (3.6a) we get that $\\pi$ must permute $\\Lambda_{2}$ and $2\\Lambda_{1}$ . Compare (3.6c) with (3.6d): since for any $k\\geq2$ they have different numbers of summands, we find in fact that $\\pi$ will fix both $\\Lambda_{2}$ (hence $\\Lambda_{4}$ ) and $2\\Lambda_{1}$ . From (3.6b) we get that $\\pi$ permutes $\\Lambda_{6}$ and $\\Lambda_{1}+\\Lambda_{5}$ , and so (3.6d) now tells us $\\pi\\Lambda_{6}=\\Lambda_{6}$ . Finally, (3.6c) implies (for $k\\geq3$ ) $\\pi\\Lambda_{3}=\\Lambda_{3}$ (since $C\\pi=\\pi C$ ), and we are done for $k\\geq3$ . Since $\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}$ is a fusion-generator for $k=2$ (see $\\S2.2)$ , we are also done for $k=2$ . ", "page_idx": 19}, {"type": "text", "text": "For ${E}_{8}^{(1)}$ when $k\\geq7$ , (3.7a) tells us that $\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}$ are permuted. For those $k$ , the highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so $\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}$ must all be fixed. The fusion product (3.7c) also tells us that $\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}$ are permuted; (3.7d) then says that the sets $\\{\\Lambda_{6},\\Lambda_{8}\\}$ , $\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}$ , and $\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+$ $\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}$ are stabilised. Now (3.7b) implies $\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}$ are all fixed, while the set $\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}$ is stabilised. Comparing $(3.7\\mathrm{e})$ and (3.7f), we get that $\\Lambda_{4}$ is fixed and $\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}$ are permuted. Finally, $\\left(3.7\\mathrm{g}\\right)$ shows $\\Lambda_{5}$ also is fixed. To do ${E}_{8}^{(1)}$ when $k\\leq6$ , knowing q-dimensions really simplifies things. ", "page_idx": 19}]
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{"preproc_blocks": [{"type": "text", "bbox": [71, 70, 569, 114], "lines": [{"bbox": [93, 72, 541, 89], "spans": [{"bbox": [93, 72, 255, 89], "score": 1.0, "content": "Next, note that we know from", "type": "text"}, {"bbox": [256, 73, 271, 86], "score": 0.85, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [271, 72, 287, 89], "score": 1.0, "content": " × ", "type": "text"}, {"bbox": [288, 73, 302, 86], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [303, 72, 331, 89], "score": 1.0, "content": " that ", "type": "text"}, {"bbox": [331, 73, 353, 86], "score": 0.9, "content": "\\pi\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [353, 72, 368, 89], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [368, 74, 382, 86], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [383, 72, 400, 89], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [400, 74, 420, 86], "score": 0.87, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [420, 72, 459, 89], "score": 1.0, "content": ". As in ", "type": "text"}, {"bbox": [460, 73, 482, 87], "score": 0.6, "content": "\\S4.2", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [482, 72, 541, 89], "score": 1.0, "content": ", the fusion", "type": "text"}], "index": 0}, {"bbox": [71, 86, 569, 104], "spans": [{"bbox": [71, 87, 100, 101], "score": 0.84, "content": "(2\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [101, 86, 117, 104], "score": 1.0, "content": " × ", "type": "text"}, {"bbox": [118, 87, 132, 100], "score": 0.87, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [133, 86, 185, 104], "score": 1.0, "content": " × · · · × ", "type": "text"}, {"bbox": [185, 87, 200, 101], "score": 0.85, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [201, 86, 207, 104], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [207, 88, 230, 100], "score": 0.53, "content": "k{-}2", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [230, 86, 409, 104], "score": 1.0, "content": " times) contains the simple-current ", "type": "text"}, {"bbox": [409, 90, 428, 100], "score": 0.91, "content": "J_{v}0", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [429, 86, 454, 104], "score": 1.0, "content": ", but", "type": "text"}, {"bbox": [455, 88, 469, 100], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [469, 86, 485, 104], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [486, 87, 501, 100], "score": 0.87, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [501, 86, 553, 104], "score": 1.0, "content": " × · · · ×", "type": "text"}, {"bbox": [554, 87, 569, 101], "score": 0.86, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 15}], "index": 1}, {"bbox": [71, 101, 294, 118], "spans": [{"bbox": [71, 101, 75, 118], "score": 1.0, "content": "(", "type": "text"}, {"bbox": [75, 102, 104, 114], "score": 0.68, "content": "k-2", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [104, 101, 237, 118], "score": 1.0, "content": " times) doesn’t, and thus ", "type": "text"}, {"bbox": [238, 102, 289, 115], "score": 0.92, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [289, 101, 294, 118], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [93, 114, 527, 129], "lines": [{"bbox": [94, 115, 528, 134], "spans": [{"bbox": [94, 115, 138, 134], "score": 1.0, "content": "Assume", "type": "text"}, {"bbox": [139, 117, 189, 129], "score": 0.9, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [189, 115, 290, 134], "score": 1.0, "content": ". Using the fusions ", "type": "text"}, {"bbox": [290, 116, 304, 129], "score": 0.83, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [305, 115, 321, 134], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [322, 116, 336, 129], "score": 0.77, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [336, 115, 362, 134], "score": 1.0, "content": "(for ", "type": "text"}, {"bbox": [362, 117, 433, 129], "score": 0.85, "content": "1<\\ell<r-2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [433, 115, 528, 134], "score": 1.0, "content": "), and noting that", "type": "text"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [124, 142, 488, 171], "lines": [{"bbox": [124, 142, 488, 171], "spans": [{"bbox": [124, 142, 488, 171], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [69, 182, 541, 211], "lines": [{"bbox": [70, 183, 542, 201], "spans": [{"bbox": [70, 183, 181, 201], "score": 1.0, "content": "equals 0 only when ", "type": "text"}, {"bbox": [182, 184, 275, 198], "score": 0.91, "content": "\\ell\\,=\\,r+1\\,-\\,k/2", "type": "inline_equation", "height": 14, "width": 93}, {"bbox": [275, 183, 352, 201], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [352, 184, 432, 198], "score": 0.91, "content": "\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 14, "width": 80}, {"bbox": [432, 183, 542, 201], "score": 1.0, "content": " except possibly for", "type": "text"}], "index": 5}, {"bbox": [71, 199, 436, 213], "spans": [{"bbox": [71, 200, 152, 212], "score": 0.92, "content": "\\ell=r+1-k/2", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [152, 199, 193, 213], "score": 1.0, "content": " (hence ", "type": "text"}, {"bbox": [193, 199, 273, 211], "score": 0.85, "content": "2r-2\\geq k\\geq4)", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [273, 199, 330, 213], "score": 1.0, "content": "). For that ", "type": "text"}, {"bbox": [330, 199, 337, 209], "score": 0.81, "content": "\\ell", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [337, 199, 436, 213], "score": 1.0, "content": ", use q-dimensions:", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "interline_equation", "bbox": [205, 225, 405, 257], "lines": [{"bbox": [205, 225, 405, 257], "spans": [{"bbox": [205, 225, 405, 257], "score": 0.92, "content": "\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [71, 267, 531, 282], "lines": [{"bbox": [70, 270, 529, 285], "spans": [{"bbox": [70, 270, 195, 285], "score": 1.0, "content": "which is valid for these ", "type": "text"}, {"bbox": [195, 271, 202, 280], "score": 0.85, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [203, 270, 299, 285], "score": 1.0, "content": ". So we also know ", "type": "text"}, {"bbox": [300, 271, 348, 282], "score": 0.92, "content": "\\pi\\Lambda_{i}=\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [348, 270, 386, 285], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [386, 270, 434, 282], "score": 0.88, "content": "i\\le r-2", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [434, 270, 529, 285], "score": 1.0, "content": ", and we are done.", "type": "text"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [70, 284, 542, 342], "lines": [{"bbox": [94, 286, 540, 302], "spans": [{"bbox": [94, 286, 203, 302], "score": 1.0, "content": "All that remains is ", "type": "text"}, {"bbox": [204, 289, 226, 301], "score": 0.91, "content": "D_{r,2}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [226, 286, 299, 302], "score": 1.0, "content": ". Recall the ", "type": "text"}, {"bbox": [299, 288, 310, 298], "score": 0.89, "content": "\\lambda^{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [311, 286, 374, 302], "score": 1.0, "content": " defined in ", "type": "text"}, {"bbox": [374, 287, 396, 300], "score": 0.3, "content": "\\S3.4", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [396, 286, 466, 302], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [466, 288, 537, 301], "score": 0.92, "content": "\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [537, 286, 540, 302], "score": 1.0, "content": ",", "type": "text"}], "index": 9}, {"bbox": [71, 299, 539, 318], "spans": [{"bbox": [71, 303, 126, 315], "score": 0.93, "content": "\\mathcal{D}(\\lambda^{a})=2", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [127, 299, 157, 318], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [157, 302, 296, 315], "score": 0.91, "content": "S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)", "type": "inline_equation", "height": 13, "width": 139}, {"bbox": [296, 299, 326, 318], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [327, 303, 356, 315], "score": 0.92, "content": "r\\neq4", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [356, 299, 484, 318], "score": 1.0, "content": ", the q-dimensions force ", "type": "text"}, {"bbox": [485, 303, 539, 314], "score": 0.89, "content": "\\pi\\Lambda_{1}=\\lambda^{m}", "type": "inline_equation", "height": 11, "width": 54}], "index": 10}, {"bbox": [68, 311, 545, 335], "spans": [{"bbox": [68, 311, 95, 335], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [96, 315, 160, 329], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [160, 311, 193, 335], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [193, 318, 281, 330], "score": 0.92, "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [281, 311, 314, 335], "score": 1.0, "content": " says ", "type": "text"}, {"bbox": [315, 317, 375, 327], "score": 0.89, "content": "m m^{\\prime}\\,\\equiv\\,\\pm1", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [375, 311, 412, 335], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [412, 316, 425, 327], "score": 0.56, "content": "2r", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [425, 311, 545, 335], "score": 1.0, "content": "). So without loss of", "type": "text"}], "index": 11}, {"bbox": [70, 329, 442, 346], "spans": [{"bbox": [70, 329, 195, 346], "score": 1.0, "content": "generality we may take ", "type": "text"}, {"bbox": [195, 331, 258, 341], "score": 0.91, "content": "m=m^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [259, 329, 442, 346], "score": 1.0, "content": ". The rest of the argument is easy.", "type": "text"}], "index": 12}], "index": 10.5}, {"type": "text", "bbox": [70, 343, 541, 372], "lines": [{"bbox": [92, 343, 542, 361], "spans": [{"bbox": [92, 343, 117, 361], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 347, 140, 359], "score": 0.92, "content": "D_{4,2}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [140, 343, 215, 361], "score": 1.0, "content": ", we can force ", "type": "text"}, {"bbox": [216, 347, 267, 357], "score": 0.94, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [267, 343, 376, 361], "score": 1.0, "content": ", and then eliminate ", "type": "text"}, {"bbox": [376, 345, 443, 358], "score": 0.91, "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [443, 343, 461, 361], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [461, 345, 475, 357], "score": 0.89, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [476, 343, 496, 361], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [496, 345, 542, 358], "score": 0.89, "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\ne", "type": "inline_equation", "height": 13, "width": 46}], "index": 13}, {"bbox": [71, 358, 389, 375], "spans": [{"bbox": [71, 361, 181, 373], "score": 0.92, "content": "0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}", "type": "inline_equation", "height": 12, "width": 110}, {"bbox": [181, 358, 389, 375], "score": 1.0, "content": ". The rest of the argument is as before.", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "text", "bbox": [72, 385, 321, 399], "lines": [{"bbox": [71, 387, 322, 402], "spans": [{"bbox": [71, 387, 322, 402], "score": 1.0, "content": "4.6. The arguments for the exceptional algebras", "type": "text"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [70, 406, 540, 436], "lines": [{"bbox": [94, 408, 540, 425], "spans": [{"bbox": [94, 408, 540, 425], "score": 1.0, "content": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym-", "type": "text"}], "index": 16}, {"bbox": [71, 424, 213, 439], "spans": [{"bbox": [71, 424, 213, 439], "score": 1.0, "content": "metries) given in §§3.5-3.9.", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "text", "bbox": [70, 437, 541, 567], "lines": [{"bbox": [91, 435, 544, 459], "spans": [{"bbox": [91, 435, 211, 459], "score": 1.0, "content": "For example, consider ", "type": "text"}, {"bbox": [212, 438, 234, 454], "score": 0.94, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [234, 435, 254, 459], "score": 1.0, "content": "for", "type": "text"}, {"bbox": [255, 442, 284, 452], "score": 0.91, "content": "k\\geq2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [284, 435, 414, 459], "score": 1.0, "content": ". Proposition 4.1 tells us ", "type": "text"}, {"bbox": [414, 440, 492, 452], "score": 0.94, "content": "\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [492, 435, 544, 459], "score": 1.0, "content": " for some", "type": "text"}], "index": 18}, {"bbox": [71, 454, 541, 469], "spans": [{"bbox": [71, 456, 89, 467], "score": 0.88, "content": "a,b", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [89, 454, 170, 469], "score": 1.0, "content": ", and we know ", "type": "text"}, {"bbox": [170, 455, 232, 466], "score": 0.93, "content": "\\pi^{\\prime}J0\\,=\\,J^{c}0", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [232, 454, 254, 469], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [255, 457, 293, 466], "score": 0.91, "content": "c=\\pm1", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [294, 454, 443, 469], "score": 1.0, "content": ". Hence from (2.7b) we get ", "type": "text"}, {"bbox": [443, 456, 489, 468], "score": 0.9, "content": "k b\\not\\equiv-1", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [489, 454, 541, 469], "score": 1.0, "content": " (mod 3).", "type": "text"}], "index": 19}, {"bbox": [71, 469, 542, 483], "spans": [{"bbox": [71, 469, 113, 483], "score": 1.0, "content": "Hitting ", "type": "text"}, {"bbox": [114, 474, 121, 480], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [121, 469, 153, 483], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [154, 469, 210, 482], "score": 0.94, "content": "\\pi[-b]^{-1}C^{a}", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [211, 469, 340, 483], "score": 1.0, "content": ", we need consider only ", "type": "text"}, {"bbox": [340, 470, 394, 481], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [394, 469, 542, 483], "score": 1.0, "content": ". It is now immediate that", "type": "text"}], "index": 20}, {"bbox": [71, 484, 540, 498], "spans": [{"bbox": [71, 485, 122, 496], "score": 0.92, "content": "\\pi\\Lambda_{5}=\\Lambda_{5}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [122, 484, 206, 498], "score": 1.0, "content": ", by commuting ", "type": "text"}, {"bbox": [207, 488, 214, 494], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 484, 244, 498], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [244, 485, 254, 494], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [254, 484, 389, 498], "score": 1.0, "content": ". From (3.6a) we get that ", "type": "text"}, {"bbox": [389, 488, 397, 494], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [397, 484, 476, 498], "score": 1.0, "content": " must permute ", "type": "text"}, {"bbox": [476, 485, 490, 496], "score": 0.91, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [490, 484, 516, 498], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [516, 485, 536, 496], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [536, 484, 540, 498], "score": 1.0, "content": ".", "type": "text"}], "index": 21}, {"bbox": [71, 498, 541, 512], "spans": [{"bbox": [71, 498, 288, 512], "score": 1.0, "content": "Compare (3.6c) with (3.6d): since for any", "type": "text"}, {"bbox": [289, 499, 318, 510], "score": 0.92, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [318, 498, 541, 512], "score": 1.0, "content": " they have different numbers of summands,", "type": "text"}], "index": 22}, {"bbox": [71, 513, 539, 526], "spans": [{"bbox": [71, 513, 180, 526], "score": 1.0, "content": "we find in fact that ", "type": "text"}, {"bbox": [180, 517, 187, 523], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [188, 513, 262, 526], "score": 1.0, "content": " will fix both ", "type": "text"}, {"bbox": [262, 514, 276, 525], "score": 0.89, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [277, 513, 319, 526], "score": 1.0, "content": " (hence ", "type": "text"}, {"bbox": [320, 514, 334, 525], "score": 0.86, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [334, 513, 366, 526], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [367, 514, 387, 524], "score": 0.89, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [387, 513, 532, 526], "score": 1.0, "content": ". From (3.6b) we get that ", "type": "text"}, {"bbox": [532, 517, 539, 523], "score": 0.82, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}], "index": 23}, {"bbox": [70, 526, 541, 541], "spans": [{"bbox": [70, 526, 122, 541], "score": 1.0, "content": "permutes ", "type": "text"}, {"bbox": [122, 528, 136, 538], "score": 0.92, "content": "\\Lambda_{6}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [137, 526, 162, 541], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 528, 203, 539], "score": 0.93, "content": "\\Lambda_{1}+\\Lambda_{5}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [203, 526, 345, 541], "score": 1.0, "content": ", and so (3.6d) now tells us ", "type": "text"}, {"bbox": [345, 528, 396, 539], "score": 0.92, "content": "\\pi\\Lambda_{6}=\\Lambda_{6}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [396, 526, 541, 541], "score": 1.0, "content": ". Finally, (3.6c) implies (for", "type": "text"}], "index": 24}, {"bbox": [71, 540, 540, 555], "spans": [{"bbox": [71, 542, 101, 553], "score": 0.88, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [101, 540, 108, 555], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [109, 542, 160, 553], "score": 0.92, "content": "\\pi\\Lambda_{3}=\\Lambda_{3}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [160, 540, 198, 555], "score": 1.0, "content": " (since ", "type": "text"}, {"bbox": [198, 542, 248, 551], "score": 0.88, "content": "C\\pi=\\pi C", "type": "inline_equation", "height": 9, "width": 50}, {"bbox": [249, 540, 367, 555], "score": 1.0, "content": "), and we are done for", "type": "text"}, {"bbox": [368, 542, 398, 553], "score": 0.9, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [398, 540, 437, 555], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [438, 542, 540, 554], "score": 0.93, "content": "\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}", "type": "inline_equation", "height": 12, "width": 102}], "index": 25}, {"bbox": [69, 555, 431, 569], "spans": [{"bbox": [69, 555, 199, 569], "score": 1.0, "content": "is a fusion-generator for ", "type": "text"}, {"bbox": [200, 557, 228, 565], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [229, 555, 255, 569], "score": 1.0, "content": " (see ", "type": "text"}, {"bbox": [256, 555, 280, 568], "score": 0.39, "content": "\\S2.2)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [280, 555, 397, 569], "score": 1.0, "content": ", we are also done for ", "type": "text"}, {"bbox": [397, 556, 426, 566], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [426, 555, 431, 569], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 22}, {"type": "text", "bbox": [70, 569, 542, 687], "lines": [{"bbox": [89, 564, 545, 589], "spans": [{"bbox": [89, 564, 117, 589], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 569, 139, 585], "score": 0.94, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [140, 564, 174, 589], "score": 1.0, "content": "when ", "type": "text"}, {"bbox": [175, 573, 205, 584], "score": 0.91, "content": "k\\geq7", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [205, 564, 313, 589], "score": 1.0, "content": ", (3.7a) tells us that ", "type": "text"}, {"bbox": [313, 571, 372, 585], "score": 0.91, "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [372, 564, 508, 589], "score": 1.0, "content": " are permuted. For those ", "type": "text"}, {"bbox": [508, 573, 515, 582], "score": 0.82, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [516, 564, 545, 589], "score": 1.0, "content": ", the", "type": "text"}], "index": 27}, {"bbox": [71, 586, 541, 601], "spans": [{"bbox": [71, 586, 417, 601], "score": 1.0, "content": "highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so ", "type": "text"}, {"bbox": [417, 586, 475, 599], "score": 0.91, "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [476, 586, 541, 601], "score": 1.0, "content": " must all be", "type": "text"}], "index": 28}, {"bbox": [70, 600, 541, 615], "spans": [{"bbox": [70, 600, 336, 615], "score": 1.0, "content": "fixed. The fusion product (3.7c) also tells us that ", "type": "text"}, {"bbox": [337, 600, 462, 613], "score": 0.92, "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [462, 600, 541, 615], "score": 1.0, "content": " are permuted;", "type": "text"}], "index": 29}, {"bbox": [71, 614, 541, 630], "spans": [{"bbox": [71, 615, 227, 630], "score": 1.0, "content": "(3.7d) then says that the sets ", "type": "text"}, {"bbox": [227, 615, 272, 628], "score": 0.91, "content": "\\{\\Lambda_{6},\\Lambda_{8}\\}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [273, 615, 279, 630], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [279, 615, 376, 628], "score": 0.91, "content": "\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [377, 615, 406, 630], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [406, 614, 541, 628], "score": 0.9, "content": "\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+", "type": "inline_equation", "height": 14, "width": 135}], "index": 30}, {"bbox": [71, 629, 541, 643], "spans": [{"bbox": [71, 630, 171, 642], "score": 0.9, "content": "\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}", "type": "inline_equation", "height": 12, "width": 100}, {"bbox": [171, 630, 359, 643], "score": 1.0, "content": " are stabilised. Now (3.7b) implies ", "type": "text"}, {"bbox": [360, 629, 437, 642], "score": 0.91, "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [437, 630, 541, 643], "score": 1.0, "content": " are all fixed, while", "type": "text"}], "index": 31}, {"bbox": [70, 643, 541, 659], "spans": [{"bbox": [70, 644, 111, 659], "score": 1.0, "content": "the set ", "type": "text"}, {"bbox": [111, 644, 186, 657], "score": 0.92, "content": "\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [186, 644, 323, 659], "score": 1.0, "content": " is stabilised. Comparing ", "type": "text"}, {"bbox": [324, 643, 353, 657], "score": 0.25, "content": "(3.7\\mathrm{e})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [354, 644, 483, 659], "score": 1.0, "content": " and (3.7f), we get that ", "type": "text"}, {"bbox": [483, 643, 497, 656], "score": 0.87, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [498, 644, 541, 659], "score": 1.0, "content": " is fixed", "type": "text"}], "index": 32}, {"bbox": [66, 656, 540, 680], "spans": [{"bbox": [66, 656, 95, 680], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 662, 157, 673], "score": 0.92, "content": "\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [158, 656, 287, 680], "score": 1.0, "content": " are permuted. Finally, ", "type": "text"}, {"bbox": [288, 660, 318, 673], "score": 0.49, "content": "\\left(3.7\\mathrm{g}\\right)", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [319, 656, 357, 680], "score": 1.0, "content": " shows ", "type": "text"}, {"bbox": [357, 660, 371, 672], "score": 0.88, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [372, 656, 484, 680], "score": 1.0, "content": " also is fixed. To do ", "type": "text"}, {"bbox": [484, 658, 507, 674], "score": 0.93, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [508, 656, 540, 680], "score": 1.0, "content": "when", "type": "text"}], "index": 33}, {"bbox": [71, 674, 348, 691], "spans": [{"bbox": [71, 676, 100, 687], "score": 0.91, "content": "k\\leq6", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [100, 674, 348, 691], "score": 1.0, "content": ", knowing q-dimensions really simplifies things.", "type": "text"}], "index": 34}], "index": 30.5}], "layout_bboxes": [], "page_idx": 19, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [124, 142, 488, 171], "lines": [{"bbox": [124, 142, 488, 171], "spans": [{"bbox": [124, 142, 488, 171], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [205, 225, 405, 257], "lines": [{"bbox": [205, 225, 405, 257], "spans": [{"bbox": [205, 225, 405, 257], "score": 0.92, "content": "\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,", "type": "interline_equation"}], "index": 7}], "index": 7}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "20", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [71, 70, 569, 114], "lines": [{"bbox": [93, 72, 541, 89], "spans": [{"bbox": [93, 72, 255, 89], "score": 1.0, "content": "Next, note that we know from", "type": "text"}, {"bbox": [256, 73, 271, 86], "score": 0.85, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [271, 72, 287, 89], "score": 1.0, "content": " × ", "type": "text"}, {"bbox": [288, 73, 302, 86], "score": 0.84, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [303, 72, 331, 89], "score": 1.0, "content": " that ", "type": "text"}, {"bbox": [331, 73, 353, 86], "score": 0.9, "content": "\\pi\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [353, 72, 368, 89], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [368, 74, 382, 86], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [383, 72, 400, 89], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [400, 74, 420, 86], "score": 0.87, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [420, 72, 459, 89], "score": 1.0, "content": ". As in ", "type": "text"}, {"bbox": [460, 73, 482, 87], "score": 0.6, "content": "\\S4.2", "type": "inline_equation", "height": 14, "width": 22}, {"bbox": [482, 72, 541, 89], "score": 1.0, "content": ", the fusion", "type": "text"}], "index": 0}, {"bbox": [71, 86, 569, 104], "spans": [{"bbox": [71, 87, 100, 101], "score": 0.84, "content": "(2\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [101, 86, 117, 104], "score": 1.0, "content": " × ", "type": "text"}, {"bbox": [118, 87, 132, 100], "score": 0.87, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [133, 86, 185, 104], "score": 1.0, "content": " × · · · × ", "type": "text"}, {"bbox": [185, 87, 200, 101], "score": 0.85, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 15}, {"bbox": [201, 86, 207, 104], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [207, 88, 230, 100], "score": 0.53, "content": "k{-}2", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [230, 86, 409, 104], "score": 1.0, "content": " times) contains the simple-current ", "type": "text"}, {"bbox": [409, 90, 428, 100], "score": 0.91, "content": "J_{v}0", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [429, 86, 454, 104], "score": 1.0, "content": ", but", "type": "text"}, {"bbox": [455, 88, 469, 100], "score": 0.88, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [469, 86, 485, 104], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [486, 87, 501, 100], "score": 0.87, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [501, 86, 553, 104], "score": 1.0, "content": " × · · · ×", "type": "text"}, {"bbox": [554, 87, 569, 101], "score": 0.86, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 15}], "index": 1}, {"bbox": [71, 101, 294, 118], "spans": [{"bbox": [71, 101, 75, 118], "score": 1.0, "content": "(", "type": "text"}, {"bbox": [75, 102, 104, 114], "score": 0.68, "content": "k-2", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [104, 101, 237, 118], "score": 1.0, "content": " times) doesn’t, and thus ", "type": "text"}, {"bbox": [238, 102, 289, 115], "score": 0.92, "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [289, 101, 294, 118], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [71, 72, 569, 118]}, {"type": "text", "bbox": [93, 114, 527, 129], "lines": [{"bbox": [94, 115, 528, 134], "spans": [{"bbox": [94, 115, 138, 134], "score": 1.0, "content": "Assume", "type": "text"}, {"bbox": [139, 117, 189, 129], "score": 0.9, "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [189, 115, 290, 134], "score": 1.0, "content": ". Using the fusions ", "type": "text"}, {"bbox": [290, 116, 304, 129], "score": 0.83, "content": "\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [305, 115, 321, 134], "score": 1.0, "content": " ×", "type": "text"}, {"bbox": [322, 116, 336, 129], "score": 0.77, "content": "\\Lambda_{\\ell}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [336, 115, 362, 134], "score": 1.0, "content": "(for ", "type": "text"}, {"bbox": [362, 117, 433, 129], "score": 0.85, "content": "1<\\ell<r-2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [433, 115, 528, 134], "score": 1.0, "content": "), and noting that", "type": "text"}], "index": 3}], "index": 3, "bbox_fs": [94, 115, 528, 134]}, {"type": "interline_equation", "bbox": [124, 142, 488, 171], "lines": [{"bbox": [124, 142, 488, 171], "spans": [{"bbox": [124, 142, 488, 171], "score": 0.92, "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [69, 182, 541, 211], "lines": [{"bbox": [70, 183, 542, 201], "spans": [{"bbox": [70, 183, 181, 201], "score": 1.0, "content": "equals 0 only when ", "type": "text"}, {"bbox": [182, 184, 275, 198], "score": 0.91, "content": "\\ell\\,=\\,r+1\\,-\\,k/2", "type": "inline_equation", "height": 14, "width": 93}, {"bbox": [275, 183, 352, 201], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [352, 184, 432, 198], "score": 0.91, "content": "\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}", "type": "inline_equation", "height": 14, "width": 80}, {"bbox": [432, 183, 542, 201], "score": 1.0, "content": " except possibly for", "type": "text"}], "index": 5}, {"bbox": [71, 199, 436, 213], "spans": [{"bbox": [71, 200, 152, 212], "score": 0.92, "content": "\\ell=r+1-k/2", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [152, 199, 193, 213], "score": 1.0, "content": " (hence ", "type": "text"}, {"bbox": [193, 199, 273, 211], "score": 0.85, "content": "2r-2\\geq k\\geq4)", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [273, 199, 330, 213], "score": 1.0, "content": "). For that ", "type": "text"}, {"bbox": [330, 199, 337, 209], "score": 0.81, "content": "\\ell", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [337, 199, 436, 213], "score": 1.0, "content": ", use q-dimensions:", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [70, 183, 542, 213]}, {"type": "interline_equation", "bbox": [205, 225, 405, 257], "lines": [{"bbox": [205, 225, 405, 257], "spans": [{"bbox": [205, 225, 405, 257], "score": 0.92, "content": "\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [71, 267, 531, 282], "lines": [{"bbox": [70, 270, 529, 285], "spans": [{"bbox": [70, 270, 195, 285], "score": 1.0, "content": "which is valid for these ", "type": "text"}, {"bbox": [195, 271, 202, 280], "score": 0.85, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [203, 270, 299, 285], "score": 1.0, "content": ". So we also know ", "type": "text"}, {"bbox": [300, 271, 348, 282], "score": 0.92, "content": "\\pi\\Lambda_{i}=\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [348, 270, 386, 285], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [386, 270, 434, 282], "score": 0.88, "content": "i\\le r-2", "type": "inline_equation", "height": 12, "width": 48}, {"bbox": [434, 270, 529, 285], "score": 1.0, "content": ", and we are done.", "type": "text"}], "index": 8}], "index": 8, "bbox_fs": [70, 270, 529, 285]}, {"type": "text", "bbox": [70, 284, 542, 342], "lines": [{"bbox": [94, 286, 540, 302], "spans": [{"bbox": [94, 286, 203, 302], "score": 1.0, "content": "All that remains is ", "type": "text"}, {"bbox": [204, 289, 226, 301], "score": 0.91, "content": "D_{r,2}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [226, 286, 299, 302], "score": 1.0, "content": ". Recall the ", "type": "text"}, {"bbox": [299, 288, 310, 298], "score": 0.89, "content": "\\lambda^{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [311, 286, 374, 302], "score": 1.0, "content": " defined in ", "type": "text"}, {"bbox": [374, 287, 396, 300], "score": 0.3, "content": "\\S3.4", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [396, 286, 466, 302], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [466, 288, 537, 301], "score": 0.92, "content": "\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [537, 286, 540, 302], "score": 1.0, "content": ",", "type": "text"}], "index": 9}, {"bbox": [71, 299, 539, 318], "spans": [{"bbox": [71, 303, 126, 315], "score": 0.93, "content": "\\mathcal{D}(\\lambda^{a})=2", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [127, 299, 157, 318], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [157, 302, 296, 315], "score": 0.91, "content": "S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)", "type": "inline_equation", "height": 13, "width": 139}, {"bbox": [296, 299, 326, 318], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [327, 303, 356, 315], "score": 0.92, "content": "r\\neq4", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [356, 299, 484, 318], "score": 1.0, "content": ", the q-dimensions force ", "type": "text"}, {"bbox": [485, 303, 539, 314], "score": 0.89, "content": "\\pi\\Lambda_{1}=\\lambda^{m}", "type": "inline_equation", "height": 11, "width": 54}], "index": 10}, {"bbox": [68, 311, 545, 335], "spans": [{"bbox": [68, 311, 95, 335], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [96, 315, 160, 329], "score": 0.92, "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [160, 311, 193, 335], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [193, 318, 281, 330], "score": 0.92, "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [281, 311, 314, 335], "score": 1.0, "content": " says ", "type": "text"}, {"bbox": [315, 317, 375, 327], "score": 0.89, "content": "m m^{\\prime}\\,\\equiv\\,\\pm1", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [375, 311, 412, 335], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [412, 316, 425, 327], "score": 0.56, "content": "2r", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [425, 311, 545, 335], "score": 1.0, "content": "). So without loss of", "type": "text"}], "index": 11}, {"bbox": [70, 329, 442, 346], "spans": [{"bbox": [70, 329, 195, 346], "score": 1.0, "content": "generality we may take ", "type": "text"}, {"bbox": [195, 331, 258, 341], "score": 0.91, "content": "m=m^{\\prime}=1", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [259, 329, 442, 346], "score": 1.0, "content": ". The rest of the argument is easy.", "type": "text"}], "index": 12}], "index": 10.5, "bbox_fs": [68, 286, 545, 346]}, {"type": "text", "bbox": [70, 343, 541, 372], "lines": [{"bbox": [92, 343, 542, 361], "spans": [{"bbox": [92, 343, 117, 361], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 347, 140, 359], "score": 0.92, "content": "D_{4,2}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [140, 343, 215, 361], "score": 1.0, "content": ", we can force ", "type": "text"}, {"bbox": [216, 347, 267, 357], "score": 0.94, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [267, 343, 376, 361], "score": 1.0, "content": ", and then eliminate ", "type": "text"}, {"bbox": [376, 345, 443, 358], "score": 0.91, "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [443, 343, 461, 361], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [461, 345, 475, 357], "score": 0.89, "content": "\\Lambda_{r}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [476, 343, 496, 361], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [496, 345, 542, 358], "score": 0.89, "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\ne", "type": "inline_equation", "height": 13, "width": 46}], "index": 13}, {"bbox": [71, 358, 389, 375], "spans": [{"bbox": [71, 361, 181, 373], "score": 0.92, "content": "0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}", "type": "inline_equation", "height": 12, "width": 110}, {"bbox": [181, 358, 389, 375], "score": 1.0, "content": ". The rest of the argument is as before.", "type": "text"}], "index": 14}], "index": 13.5, "bbox_fs": [71, 343, 542, 375]}, {"type": "text", "bbox": [72, 385, 321, 399], "lines": [{"bbox": [71, 387, 322, 402], "spans": [{"bbox": [71, 387, 322, 402], "score": 1.0, "content": "4.6. The arguments for the exceptional algebras", "type": "text"}], "index": 15}], "index": 15, "bbox_fs": [71, 387, 322, 402]}, {"type": "text", "bbox": [70, 406, 540, 436], "lines": [{"bbox": [94, 408, 540, 425], "spans": [{"bbox": [94, 408, 540, 425], "score": 1.0, "content": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym-", "type": "text"}], "index": 16}, {"bbox": [71, 424, 213, 439], "spans": [{"bbox": [71, 424, 213, 439], "score": 1.0, "content": "metries) given in §§3.5-3.9.", "type": "text"}], "index": 17}], "index": 16.5, "bbox_fs": [71, 408, 540, 439]}, {"type": "text", "bbox": [70, 437, 541, 567], "lines": [{"bbox": [91, 435, 544, 459], "spans": [{"bbox": [91, 435, 211, 459], "score": 1.0, "content": "For example, consider ", "type": "text"}, {"bbox": [212, 438, 234, 454], "score": 0.94, "content": "E_{6}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [234, 435, 254, 459], "score": 1.0, "content": "for", "type": "text"}, {"bbox": [255, 442, 284, 452], "score": 0.91, "content": "k\\geq2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [284, 435, 414, 459], "score": 1.0, "content": ". Proposition 4.1 tells us ", "type": "text"}, {"bbox": [414, 440, 492, 452], "score": 0.94, "content": "\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [492, 435, 544, 459], "score": 1.0, "content": " for some", "type": "text"}], "index": 18}, {"bbox": [71, 454, 541, 469], "spans": [{"bbox": [71, 456, 89, 467], "score": 0.88, "content": "a,b", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [89, 454, 170, 469], "score": 1.0, "content": ", and we know ", "type": "text"}, {"bbox": [170, 455, 232, 466], "score": 0.93, "content": "\\pi^{\\prime}J0\\,=\\,J^{c}0", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [232, 454, 254, 469], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [255, 457, 293, 466], "score": 0.91, "content": "c=\\pm1", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [294, 454, 443, 469], "score": 1.0, "content": ". Hence from (2.7b) we get ", "type": "text"}, {"bbox": [443, 456, 489, 468], "score": 0.9, "content": "k b\\not\\equiv-1", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [489, 454, 541, 469], "score": 1.0, "content": " (mod 3).", "type": "text"}], "index": 19}, {"bbox": [71, 469, 542, 483], "spans": [{"bbox": [71, 469, 113, 483], "score": 1.0, "content": "Hitting ", "type": "text"}, {"bbox": [114, 474, 121, 480], "score": 0.77, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [121, 469, 153, 483], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [154, 469, 210, 482], "score": 0.94, "content": "\\pi[-b]^{-1}C^{a}", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [211, 469, 340, 483], "score": 1.0, "content": ", we need consider only ", "type": "text"}, {"bbox": [340, 470, 394, 481], "score": 0.93, "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [394, 469, 542, 483], "score": 1.0, "content": ". It is now immediate that", "type": "text"}], "index": 20}, {"bbox": [71, 484, 540, 498], "spans": [{"bbox": [71, 485, 122, 496], "score": 0.92, "content": "\\pi\\Lambda_{5}=\\Lambda_{5}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [122, 484, 206, 498], "score": 1.0, "content": ", by commuting ", "type": "text"}, {"bbox": [207, 488, 214, 494], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 484, 244, 498], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [244, 485, 254, 494], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [254, 484, 389, 498], "score": 1.0, "content": ". From (3.6a) we get that ", "type": "text"}, {"bbox": [389, 488, 397, 494], "score": 0.87, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [397, 484, 476, 498], "score": 1.0, "content": " must permute ", "type": "text"}, {"bbox": [476, 485, 490, 496], "score": 0.91, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [490, 484, 516, 498], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [516, 485, 536, 496], "score": 0.9, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [536, 484, 540, 498], "score": 1.0, "content": ".", "type": "text"}], "index": 21}, {"bbox": [71, 498, 541, 512], "spans": [{"bbox": [71, 498, 288, 512], "score": 1.0, "content": "Compare (3.6c) with (3.6d): since for any", "type": "text"}, {"bbox": [289, 499, 318, 510], "score": 0.92, "content": "k\\geq2", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [318, 498, 541, 512], "score": 1.0, "content": " they have different numbers of summands,", "type": "text"}], "index": 22}, {"bbox": [71, 513, 539, 526], "spans": [{"bbox": [71, 513, 180, 526], "score": 1.0, "content": "we find in fact that ", "type": "text"}, {"bbox": [180, 517, 187, 523], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [188, 513, 262, 526], "score": 1.0, "content": " will fix both ", "type": "text"}, {"bbox": [262, 514, 276, 525], "score": 0.89, "content": "\\Lambda_{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [277, 513, 319, 526], "score": 1.0, "content": " (hence ", "type": "text"}, {"bbox": [320, 514, 334, 525], "score": 0.86, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [334, 513, 366, 526], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [367, 514, 387, 524], "score": 0.89, "content": "2\\Lambda_{1}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [387, 513, 532, 526], "score": 1.0, "content": ". From (3.6b) we get that ", "type": "text"}, {"bbox": [532, 517, 539, 523], "score": 0.82, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}], "index": 23}, {"bbox": [70, 526, 541, 541], "spans": [{"bbox": [70, 526, 122, 541], "score": 1.0, "content": "permutes ", "type": "text"}, {"bbox": [122, 528, 136, 538], "score": 0.92, "content": "\\Lambda_{6}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [137, 526, 162, 541], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 528, 203, 539], "score": 0.93, "content": "\\Lambda_{1}+\\Lambda_{5}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [203, 526, 345, 541], "score": 1.0, "content": ", and so (3.6d) now tells us ", "type": "text"}, {"bbox": [345, 528, 396, 539], "score": 0.92, "content": "\\pi\\Lambda_{6}=\\Lambda_{6}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [396, 526, 541, 541], "score": 1.0, "content": ". Finally, (3.6c) implies (for", "type": "text"}], "index": 24}, {"bbox": [71, 540, 540, 555], "spans": [{"bbox": [71, 542, 101, 553], "score": 0.88, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [101, 540, 108, 555], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [109, 542, 160, 553], "score": 0.92, "content": "\\pi\\Lambda_{3}=\\Lambda_{3}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [160, 540, 198, 555], "score": 1.0, "content": " (since ", "type": "text"}, {"bbox": [198, 542, 248, 551], "score": 0.88, "content": "C\\pi=\\pi C", "type": "inline_equation", "height": 9, "width": 50}, {"bbox": [249, 540, 367, 555], "score": 1.0, "content": "), and we are done for", "type": "text"}, {"bbox": [368, 542, 398, 553], "score": 0.9, "content": "k\\geq3", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [398, 540, 437, 555], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [438, 542, 540, 554], "score": 0.93, "content": "\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}", "type": "inline_equation", "height": 12, "width": 102}], "index": 25}, {"bbox": [69, 555, 431, 569], "spans": [{"bbox": [69, 555, 199, 569], "score": 1.0, "content": "is a fusion-generator for ", "type": "text"}, {"bbox": [200, 557, 228, 565], "score": 0.91, "content": "k=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [229, 555, 255, 569], "score": 1.0, "content": " (see ", "type": "text"}, {"bbox": [256, 555, 280, 568], "score": 0.39, "content": "\\S2.2)", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [280, 555, 397, 569], "score": 1.0, "content": ", we are also done for ", "type": "text"}, {"bbox": [397, 556, 426, 566], "score": 0.9, "content": "k=2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [426, 555, 431, 569], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 22, "bbox_fs": [69, 435, 544, 569]}, {"type": "text", "bbox": [70, 569, 542, 687], "lines": [{"bbox": [89, 564, 545, 589], "spans": [{"bbox": [89, 564, 117, 589], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [117, 569, 139, 585], "score": 0.94, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 16, "width": 22}, {"bbox": [140, 564, 174, 589], "score": 1.0, "content": "when ", "type": "text"}, {"bbox": [175, 573, 205, 584], "score": 0.91, "content": "k\\geq7", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [205, 564, 313, 589], "score": 1.0, "content": ", (3.7a) tells us that ", "type": "text"}, {"bbox": [313, 571, 372, 585], "score": 0.91, "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [372, 564, 508, 589], "score": 1.0, "content": " are permuted. For those ", "type": "text"}, {"bbox": [508, 573, 515, 582], "score": 0.82, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [516, 564, 545, 589], "score": 1.0, "content": ", the", "type": "text"}], "index": 27}, {"bbox": [71, 586, 541, 601], "spans": [{"bbox": [71, 586, 417, 601], "score": 1.0, "content": "highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so ", "type": "text"}, {"bbox": [417, 586, 475, 599], "score": 0.91, "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [476, 586, 541, 601], "score": 1.0, "content": " must all be", "type": "text"}], "index": 28}, {"bbox": [70, 600, 541, 615], "spans": [{"bbox": [70, 600, 336, 615], "score": 1.0, "content": "fixed. The fusion product (3.7c) also tells us that ", "type": "text"}, {"bbox": [337, 600, 462, 613], "score": 0.92, "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [462, 600, 541, 615], "score": 1.0, "content": " are permuted;", "type": "text"}], "index": 29}, {"bbox": [71, 614, 541, 630], "spans": [{"bbox": [71, 615, 227, 630], "score": 1.0, "content": "(3.7d) then says that the sets ", "type": "text"}, {"bbox": [227, 615, 272, 628], "score": 0.91, "content": "\\{\\Lambda_{6},\\Lambda_{8}\\}", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [273, 615, 279, 630], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [279, 615, 376, 628], "score": 0.91, "content": "\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [377, 615, 406, 630], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [406, 614, 541, 628], "score": 0.9, "content": "\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+", "type": "inline_equation", "height": 14, "width": 135}], "index": 30}, {"bbox": [71, 629, 541, 643], "spans": [{"bbox": [71, 630, 171, 642], "score": 0.9, "content": "\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}", "type": "inline_equation", "height": 12, "width": 100}, {"bbox": [171, 630, 359, 643], "score": 1.0, "content": " are stabilised. Now (3.7b) implies ", "type": "text"}, {"bbox": [360, 629, 437, 642], "score": 0.91, "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [437, 630, 541, 643], "score": 1.0, "content": " are all fixed, while", "type": "text"}], "index": 31}, {"bbox": [70, 643, 541, 659], "spans": [{"bbox": [70, 644, 111, 659], "score": 1.0, "content": "the set ", "type": "text"}, {"bbox": [111, 644, 186, 657], "score": 0.92, "content": "\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [186, 644, 323, 659], "score": 1.0, "content": " is stabilised. Comparing ", "type": "text"}, {"bbox": [324, 643, 353, 657], "score": 0.25, "content": "(3.7\\mathrm{e})", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [354, 644, 483, 659], "score": 1.0, "content": " and (3.7f), we get that ", "type": "text"}, {"bbox": [483, 643, 497, 656], "score": 0.87, "content": "\\Lambda_{4}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [498, 644, 541, 659], "score": 1.0, "content": " is fixed", "type": "text"}], "index": 32}, {"bbox": [66, 656, 540, 680], "spans": [{"bbox": [66, 656, 95, 680], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 662, 157, 673], "score": 0.92, "content": "\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [158, 656, 287, 680], "score": 1.0, "content": " are permuted. Finally, ", "type": "text"}, {"bbox": [288, 660, 318, 673], "score": 0.49, "content": "\\left(3.7\\mathrm{g}\\right)", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [319, 656, 357, 680], "score": 1.0, "content": " shows ", "type": "text"}, {"bbox": [357, 660, 371, 672], "score": 0.88, "content": "\\Lambda_{5}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [372, 656, 484, 680], "score": 1.0, "content": " also is fixed. To do ", "type": "text"}, {"bbox": [484, 658, 507, 674], "score": 0.93, "content": "{E}_{8}^{(1)}", "type": "inline_equation", "height": 16, "width": 23}, {"bbox": [508, 656, 540, 680], "score": 1.0, "content": "when", "type": "text"}], "index": 33}, {"bbox": [71, 674, 348, 691], "spans": [{"bbox": [71, 676, 100, 687], "score": 0.91, "content": "k\\leq6", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [100, 674, 348, 691], "score": 1.0, "content": ", knowing q-dimensions really simplifies things.", "type": "text"}], "index": 34}], "index": 30.5, "bbox_fs": [66, 564, 545, 691]}]}
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[{"type": "text", "bbox": [71, 70, 569, 114], "content": "Next, note that we know from × that is or . As in , the fusion × × · · · × ( times) contains the simple-current , but × × · · · × ( times) doesn’t, and thus .", "index": 0}, {"type": "text", "bbox": [93, 114, 527, 129], "content": "Assume . Using the fusions × (for ), and noting that", "index": 1}, {"type": "interline_equation", "bbox": [124, 142, 488, 171], "content": "", "index": 2}, {"type": "text", "bbox": [69, 182, 541, 211], "content": "equals 0 only when , we see that except possibly for (hence ). For that , use q-dimensions:", "index": 3}, {"type": "interline_equation", "bbox": [205, 225, 405, 257], "content": "", "index": 4}, {"type": "text", "bbox": [71, 267, 531, 282], "content": "which is valid for these . So we also know for all , and we are done.", "index": 5}, {"type": "text", "bbox": [70, 284, 542, 342], "content": "All that remains is . Recall the defined in . Note that , , and . For , the q-dimensions force and , and says (mod ). So without loss of generality we may take . The rest of the argument is easy.", "index": 6}, {"type": "text", "bbox": [70, 343, 541, 372], "content": "For , we can force , and then eliminate or by . The rest of the argument is as before.", "index": 7}, {"type": "text", "bbox": [72, 385, 321, 399], "content": "4.6. The arguments for the exceptional algebras", "index": 8}, {"type": "text", "bbox": [70, 406, 540, 436], "content": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym- metries) given in §§3.5-3.9.", "index": 9}, {"type": "text", "bbox": [70, 437, 541, 567], "content": "For example, consider for . Proposition 4.1 tells us for some , and we know for . Hence from (2.7b) we get (mod 3). Hitting with , we need consider only . It is now immediate that , by commuting with . From (3.6a) we get that must permute and . Compare (3.6c) with (3.6d): since for any they have different numbers of summands, we find in fact that will fix both (hence ) and . From (3.6b) we get that permutes and , and so (3.6d) now tells us . Finally, (3.6c) implies (for ) (since ), and we are done for . Since is a fusion-generator for (see , we are also done for .", "index": 10}, {"type": "text", "bbox": [70, 569, 542, 687], "content": "For when , (3.7a) tells us that are permuted. For those , the highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so must all be fixed. The fusion product (3.7c) also tells us that are permuted; (3.7d) then says that the sets , , and are stabilised. Now (3.7b) implies are all fixed, while the set is stabilised. Comparing and (3.7f), we get that is fixed and are permuted. Finally, shows also is fixed. To do when , knowing q-dimensions really simplifies things.", "index": 11}]
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[{"bbox": [93, 72, 541, 89], "content": "Next, note that we know from × that is or . As in , the fusion", "parent_index": 0, "line_index": 0}, {"bbox": [71, 86, 569, 104], "content": "× × · · · × ( times) contains the simple-current , but × × · · · ×", "parent_index": 0, "line_index": 1}, {"bbox": [71, 101, 294, 118], "content": "( times) doesn’t, and thus .", "parent_index": 0, "line_index": 2}, {"bbox": [94, 115, 528, 134], "content": "Assume . Using the fusions × (for ), and noting that", "parent_index": 1, "line_index": 0}, {"bbox": [70, 183, 542, 201], "content": "equals 0 only when , we see that except possibly for", "parent_index": 3, "line_index": 0}, {"bbox": [71, 199, 436, 213], "content": "(hence ). For that , use q-dimensions:", "parent_index": 3, "line_index": 1}, {"bbox": [70, 270, 529, 285], "content": "which is valid for these . So we also know for all , and we are done.", "parent_index": 5, "line_index": 0}, {"bbox": [94, 286, 540, 302], "content": "All that remains is . Recall the defined in . Note that ,", "parent_index": 6, "line_index": 0}, {"bbox": [71, 299, 539, 318], "content": ", and . For , the q-dimensions force", "parent_index": 6, "line_index": 1}, {"bbox": [68, 311, 545, 335], "content": "and , and says (mod ). So without loss of", "parent_index": 6, "line_index": 2}, {"bbox": [70, 329, 442, 346], "content": "generality we may take . The rest of the argument is easy.", "parent_index": 6, "line_index": 3}, {"bbox": [92, 343, 542, 361], "content": "For , we can force , and then eliminate or by", "parent_index": 7, "line_index": 0}, {"bbox": [71, 358, 389, 375], "content": ". The rest of the argument is as before.", "parent_index": 7, "line_index": 1}, {"bbox": [71, 387, 322, 402], "content": "4.6. The arguments for the exceptional algebras", "parent_index": 8, "line_index": 0}, {"bbox": [94, 408, 540, 425], "content": "The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym-", "parent_index": 9, "line_index": 0}, {"bbox": [71, 424, 213, 439], "content": "metries) given in §§3.5-3.9.", "parent_index": 9, "line_index": 1}, {"bbox": [91, 435, 544, 459], "content": "For example, consider for . Proposition 4.1 tells us for some", "parent_index": 10, "line_index": 0}, {"bbox": [71, 454, 541, 469], "content": ", and we know for . Hence from (2.7b) we get (mod 3).", "parent_index": 10, "line_index": 1}, {"bbox": [71, 469, 542, 483], "content": "Hitting with , we need consider only . It is now immediate that", "parent_index": 10, "line_index": 2}, {"bbox": [71, 484, 540, 498], "content": ", by commuting with . From (3.6a) we get that must permute and .", "parent_index": 10, "line_index": 3}, {"bbox": [71, 498, 541, 512], "content": "Compare (3.6c) with (3.6d): since for any they have different numbers of summands,", "parent_index": 10, "line_index": 4}, {"bbox": [71, 513, 539, 526], "content": "we find in fact that will fix both (hence ) and . From (3.6b) we get that", "parent_index": 10, "line_index": 5}, {"bbox": [70, 526, 541, 541], "content": "permutes and , and so (3.6d) now tells us . Finally, (3.6c) implies (for", "parent_index": 10, "line_index": 6}, {"bbox": [71, 540, 540, 555], "content": ") (since ), and we are done for . Since", "parent_index": 10, "line_index": 7}, {"bbox": [69, 555, 431, 569], "content": "is a fusion-generator for (see , we are also done for .", "parent_index": 10, "line_index": 8}, {"bbox": [89, 564, 545, 589], "content": "For when , (3.7a) tells us that are permuted. For those , the", "parent_index": 11, "line_index": 0}, {"bbox": [71, 586, 541, 601], "content": "highest multiplicities in (3.7b)–(3.7d) are 3, 1, 2, respectively, so must all be", "parent_index": 11, "line_index": 1}, {"bbox": [70, 600, 541, 615], "content": "fixed. The fusion product (3.7c) also tells us that are permuted;", "parent_index": 11, "line_index": 2}, {"bbox": [71, 614, 541, 630], "content": "(3.7d) then says that the sets , , and", "parent_index": 11, "line_index": 3}, {"bbox": [71, 629, 541, 643], "content": "are stabilised. Now (3.7b) implies are all fixed, while", "parent_index": 11, "line_index": 4}, {"bbox": [70, 643, 541, 659], "content": "the set is stabilised. Comparing and (3.7f), we get that is fixed", "parent_index": 11, "line_index": 5}, {"bbox": [66, 656, 540, 680], "content": "and are permuted. Finally, shows also is fixed. To do when", "parent_index": 11, "line_index": 6}, {"bbox": [71, 674, 348, 691], "content": ", knowing q-dimensions really simplifies things.", "parent_index": 11, "line_index": 7}]
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[]
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[{"bbox": [256, 73, 271, 86], "content": "\\Lambda_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [288, 73, 302, 86], "content": "\\Lambda_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [331, 73, 353, 86], "content": "\\pi\\Lambda_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [368, 74, 382, 86], "content": "\\Lambda_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [400, 74, 420, 86], "content": "2\\Lambda_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [460, 73, 482, 87], "content": "\\S4.2", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 87, 100, 101], "content": "(2\\Lambda_{1})", "parent_index": 0, "subtype": "inline"}, {"bbox": [118, 87, 132, 100], "content": "\\Lambda_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [185, 87, 200, 101], "content": "\\Lambda_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [207, 88, 230, 100], "content": "k{-}2", "parent_index": 0, "subtype": "inline"}, {"bbox": [409, 90, 428, 100], "content": "J_{v}0", "parent_index": 0, "subtype": "inline"}, {"bbox": [455, 88, 469, 100], "content": "\\Lambda_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [486, 87, 501, 100], "content": "\\Lambda_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [554, 87, 569, 101], "content": "\\Lambda_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [75, 102, 104, 114], "content": "k-2", "parent_index": 0, "subtype": "inline"}, {"bbox": [238, 102, 289, 115], "content": "\\pi\\Lambda_{2}=\\Lambda_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [139, 117, 189, 129], "content": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}", "parent_index": 1, "subtype": "inline"}, {"bbox": [290, 116, 304, 129], "content": "\\Lambda_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [322, 116, 336, 129], "content": "\\Lambda_{\\ell}", "parent_index": 1, "subtype": "inline"}, {"bbox": [362, 117, 433, 129], "content": "1<\\ell<r-2", "parent_index": 1, "subtype": "inline"}, {"bbox": [124, 142, 488, 171], "content": "\\chi_{\\Lambda_{1}}[\\Lambda_{\\ell+1}]-\\chi_{\\Lambda_{1}}[\\Lambda_{1}+\\Lambda_{\\ell}]=4\\cos(\\pi\\,\\frac{2r-\\ell}{\\kappa})\\,\\{\\cos(\\pi\\,\\frac{\\ell}{\\kappa})-\\cos(\\pi\\,\\frac{\\ell+2}{\\kappa})\\}", "parent_index": 2, "subtype": "interline"}, {"bbox": [182, 184, 275, 198], "content": "\\ell\\,=\\,r+1\\,-\\,k/2", "parent_index": 3, "subtype": "inline"}, {"bbox": [352, 184, 432, 198], "content": "\\pi\\Lambda_{\\ell+1}\\;=\\;\\Lambda_{\\ell+1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [71, 200, 152, 212], "content": "\\ell=r+1-k/2", "parent_index": 3, "subtype": "inline"}, {"bbox": [193, 199, 273, 211], "content": "2r-2\\geq k\\geq4)", "parent_index": 3, "subtype": "inline"}, {"bbox": [330, 199, 337, 209], "content": "\\ell", "parent_index": 3, "subtype": "inline"}, {"bbox": [205, 225, 405, 257], "content": "\\frac{\\mathcal{D}(\\Lambda_{1}+\\Lambda_{\\ell})}{\\mathcal{D}(\\Lambda_{\\ell+1})}=\\frac{\\sin(2\\pi\\left(k-2\\right)/\\kappa)}{\\sin(2\\pi/\\kappa)}>1\\ ,", "parent_index": 4, "subtype": "interline"}, {"bbox": [195, 271, 202, 280], "content": "k", "parent_index": 5, "subtype": "inline"}, {"bbox": [300, 271, 348, 282], "content": "\\pi\\Lambda_{i}=\\Lambda_{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [386, 270, 434, 282], "content": "i\\le r-2", "parent_index": 5, "subtype": "inline"}, {"bbox": [204, 289, 226, 301], "content": "D_{r,2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [299, 288, 310, 298], "content": "\\lambda^{i}", "parent_index": 6, "subtype": "inline"}, {"bbox": [374, 287, 396, 300], "content": "\\S3.4", "parent_index": 6, "subtype": "inline"}, {"bbox": [466, 288, 537, 301], "content": "\\mathcal{D}(\\Lambda_{r})\\;=\\;\\sqrt{r}", "parent_index": 6, "subtype": "inline"}, {"bbox": [71, 303, 126, 315], "content": "\\mathcal{D}(\\lambda^{a})=2", "parent_index": 6, "subtype": "inline"}, {"bbox": [157, 302, 296, 315], "content": "S_{\\lambda^{a}\\lambda^{b}}/S_{0\\lambda^{b}}=2\\cos(\\pi a b/r)", "parent_index": 6, "subtype": "inline"}, {"bbox": [327, 303, 356, 315], "content": "r\\neq4", "parent_index": 6, "subtype": "inline"}, {"bbox": [485, 303, 539, 314], "content": "\\pi\\Lambda_{1}=\\lambda^{m}", "parent_index": 6, "subtype": "inline"}, {"bbox": [96, 315, 160, 329], "content": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,\\lambda^{m^{\\prime}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [193, 318, 281, 330], "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\,=\\,S_{\\lambda^{m}\\lambda^{m^{\\prime}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [315, 317, 375, 327], "content": "m m^{\\prime}\\,\\equiv\\,\\pm1", "parent_index": 6, "subtype": "inline"}, {"bbox": [412, 316, 425, 327], "content": "2r", "parent_index": 6, "subtype": "inline"}, {"bbox": [195, 331, 258, 341], "content": "m=m^{\\prime}=1", "parent_index": 6, "subtype": "inline"}, {"bbox": [117, 347, 140, 359], "content": "D_{4,2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [216, 347, 267, 357], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [376, 345, 443, 358], "content": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{r-1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [461, 345, 475, 357], "content": "\\Lambda_{r}", "parent_index": 7, "subtype": "inline"}, {"bbox": [496, 345, 542, 358], "content": "S_{\\Lambda_{1}\\Lambda_{1}}\\ne", "parent_index": 7, "subtype": "inline"}, {"bbox": [71, 361, 181, 373], "content": "0=S_{\\Lambda_{1}\\Lambda_{r}}=S_{\\Lambda_{1}\\Lambda_{r-1}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [212, 438, 234, 454], "content": "E_{6}^{(1)}", "parent_index": 10, "subtype": "inline"}, {"bbox": [255, 442, 284, 452], "content": "k\\geq2", "parent_index": 10, "subtype": "inline"}, {"bbox": [414, 440, 492, 452], "content": "\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [71, 456, 89, 467], "content": "a,b", "parent_index": 10, "subtype": "inline"}, {"bbox": [170, 455, 232, 466], "content": "\\pi^{\\prime}J0\\,=\\,J^{c}0", "parent_index": 10, "subtype": "inline"}, {"bbox": [255, 457, 293, 466], "content": "c=\\pm1", "parent_index": 10, "subtype": "inline"}, {"bbox": [443, 456, 489, 468], "content": "k b\\not\\equiv-1", "parent_index": 10, "subtype": "inline"}, {"bbox": [114, 474, 121, 480], "content": "\\pi", "parent_index": 10, "subtype": "inline"}, {"bbox": [154, 469, 210, 482], "content": "\\pi[-b]^{-1}C^{a}", "parent_index": 10, "subtype": "inline"}, {"bbox": [340, 470, 394, 481], "content": "\\pi\\Lambda_{1}=\\Lambda_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [71, 485, 122, 496], "content": "\\pi\\Lambda_{5}=\\Lambda_{5}", "parent_index": 10, "subtype": "inline"}, {"bbox": [207, 488, 214, 494], "content": "\\pi", "parent_index": 10, "subtype": "inline"}, {"bbox": [244, 485, 254, 494], "content": "C", "parent_index": 10, "subtype": "inline"}, {"bbox": [389, 488, 397, 494], "content": "\\pi", "parent_index": 10, "subtype": "inline"}, {"bbox": [476, 485, 490, 496], "content": "\\Lambda_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [516, 485, 536, 496], "content": "2\\Lambda_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [289, 499, 318, 510], "content": "k\\geq2", "parent_index": 10, "subtype": "inline"}, {"bbox": [180, 517, 187, 523], "content": "\\pi", "parent_index": 10, "subtype": "inline"}, {"bbox": [262, 514, 276, 525], "content": "\\Lambda_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [320, 514, 334, 525], "content": "\\Lambda_{4}", "parent_index": 10, "subtype": "inline"}, {"bbox": [367, 514, 387, 524], "content": "2\\Lambda_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [532, 517, 539, 523], "content": "\\pi", "parent_index": 10, "subtype": "inline"}, {"bbox": [122, 528, 136, 538], "content": "\\Lambda_{6}", "parent_index": 10, "subtype": "inline"}, {"bbox": [162, 528, 203, 539], "content": "\\Lambda_{1}+\\Lambda_{5}", "parent_index": 10, "subtype": "inline"}, {"bbox": [345, 528, 396, 539], "content": "\\pi\\Lambda_{6}=\\Lambda_{6}", "parent_index": 10, "subtype": "inline"}, {"bbox": [71, 542, 101, 553], "content": "k\\geq3", "parent_index": 10, "subtype": "inline"}, {"bbox": [109, 542, 160, 553], "content": "\\pi\\Lambda_{3}=\\Lambda_{3}", "parent_index": 10, "subtype": "inline"}, {"bbox": [198, 542, 248, 551], "content": "C\\pi=\\pi C", "parent_index": 10, "subtype": "inline"}, {"bbox": [368, 542, 398, 553], "content": "k\\geq3", "parent_index": 10, "subtype": "inline"}, {"bbox": [438, 542, 540, 554], "content": "\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}", "parent_index": 10, "subtype": "inline"}, {"bbox": [200, 557, 228, 565], "content": "k=2", "parent_index": 10, "subtype": "inline"}, {"bbox": [256, 555, 280, 568], "content": "\\S2.2)", "parent_index": 10, "subtype": "inline"}, {"bbox": [397, 556, 426, 566], "content": "k=2", "parent_index": 10, "subtype": "inline"}, {"bbox": [117, 569, 139, 585], "content": "{E}_{8}^{(1)}", "parent_index": 11, "subtype": "inline"}, {"bbox": [175, 573, 205, 584], "content": "k\\geq7", "parent_index": 11, "subtype": "inline"}, {"bbox": [313, 571, 372, 585], "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [508, 573, 515, 582], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [417, 586, 475, 599], "content": "\\Lambda_{2},\\Lambda_{7},2\\Lambda_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [337, 600, 462, 613], "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}", "parent_index": 11, "subtype": "inline"}, {"bbox": [227, 615, 272, 628], "content": "\\{\\Lambda_{6},\\Lambda_{8}\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [279, 615, 376, 628], "content": "\\{\\Lambda_{3},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [406, 614, 541, 628], "content": "\\{2\\Lambda_{2},\\Lambda_{2}+\\Lambda_{7},3\\Lambda_{1},2\\Lambda_{1}+", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 630, 171, 642], "content": "\\Lambda_{2},2\\Lambda_{1}+\\Lambda_{7},4\\Lambda_{1}\\right\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [360, 629, 437, 642], "content": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},2\\Lambda_{7}", "parent_index": 11, "subtype": "inline"}, {"bbox": [111, 644, 186, 657], "content": "\\{\\Lambda_{4},\\Lambda_{1}+\\Lambda_{3}\\}", "parent_index": 11, "subtype": "inline"}, {"bbox": [324, 643, 353, 657], "content": "(3.7\\mathrm{e})", "parent_index": 11, "subtype": "inline"}, {"bbox": [483, 643, 497, 656], "content": "\\Lambda_{4}", "parent_index": 11, "subtype": "inline"}, {"bbox": [95, 662, 157, 673], "content": "\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}", "parent_index": 11, "subtype": "inline"}, {"bbox": [288, 660, 318, 673], "content": "\\left(3.7\\mathrm{g}\\right)", "parent_index": 11, "subtype": "inline"}, {"bbox": [357, 660, 371, 672], "content": "\\Lambda_{5}", "parent_index": 11, "subtype": "inline"}, {"bbox": [484, 658, 507, 674], "content": "{E}_{8}^{(1)}", "parent_index": 11, "subtype": "inline"}, {"bbox": [71, 676, 100, 687], "content": "k\\leq6", "parent_index": 11, "subtype": "inline"}]
|
[]
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# 5. Affine fusion ring isomorphisms
We conclude the paper with the determination of all isomorphisms among the affine fusion rings $\mathcal{R}(X_{r,k})$ . Recall Definition 2.1 and the discussion in $\S2.2$ .
Theorem 5.1. The complete list of fusion ring isomorphisms $\mathscr{R}(X_{r,k})\,\cong\,\mathscr{R}(Y_{s,m})$ when $X_{r,k}\neq Y_{s,m}$ (where $X_{r},Y_{s}$ are simple) is: rank-level duality $\mathcal{R}(C_{r,k})\cong\mathcal{R}(C_{k,r})$ for all $r,k$ , as well as $\mathcal{R}(A_{1,k})\cong\mathcal{R}(C_{k,1})$ ; $\mathcal{R}(B_{r,1})\cong\mathcal{R}(A_{1,2})\cong\mathcal{R}(C_{2,1})\cong\mathcal{R}(E_{8,2})$ for all $r\geq3$ ; $\mathcal{R}(A_{3,1})\cong\mathcal{R}(D_{o d d,1})$ ; $\mathcal{R}(D_{r,1})\cong\mathcal{R}(D_{s,1})$ whenever $r\equiv s$ (mod 2); $\mathscr{R}(A_{2,1})\cong\mathscr{R}(E_{6,1})$ and $\mathcal{R}(A_{1,1})\cong\mathcal{R}(E_{7,1})$ ; $\mathcal{R}(F_{4,1})\cong\mathcal{R}(G_{2,1})$ , $\mathcal{R}(F_{4,2})\cong\mathcal{R}(E_{8,3})$ , and $\mathcal{R}(F_{4,3})\cong\mathcal{R}(G_{2,4})$ .
The isomorphism $\mathcal{R}(A_{1,k})\cong\mathcal{R}(C_{k,1})$ takes $a\Lambda_{1}$ to $\widetilde{\Lambda}_{a}$ . The isomorphism $\mathcal{R}(F_{4,2})\cong$ $\mathcal{R}(E_{8,3})$ was found in [14]; it relates $\Lambda_{1}\leftrightarrow\tilde{\Lambda}_{8}$ , $2\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{2}$ , $\Lambda_{3}\,\leftrightarrow\,\widetilde{\Lambda}_{1}$ , $\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{7}$ . The isomorphism $\mathcal{R}(F_{4,3})\,\cong\,\mathcal{R}(G_{2,4})$ was found i n [34,14]; a corresponde nce which works is $\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{1}$ , $\Lambda_{1}\leftrightarrow2\widetilde{\Lambda}_{1}$ , $\Lambda_{3}\leftrightarrow3\widetilde{\Lambda}_{2}$ , $2\Lambda_{4}\leftrightarrow2\widetilde{\Lambda}_{2}$ , $\Lambda_{1}+\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{1}+2\widetilde{\Lambda}_{2}$ , $\Lambda_{2}\leftrightarrow4\widetilde{\Lambda}_{2}$ , $3\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{2}$ , and $\Lambda_{3}+\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{1}+\tilde{\Lambda}_{2}$ .
We will sket ch th e proof here. The idea is to compare invariants for the various fusion rings, case by case. For example, suppose $\mathcal{R}(A_{r,k})$ and $\mathcal{R}(A_{s,m})$ are isomorphic. Then their simple-current groups $\mathbb{Z}_{r+1}$ and $\mathbb{Z}_{s+1}$ must be isomorphic (since simple-currents must get mapped to simple-currents), so $r=s$ . Now compare the numbers $||P_{+}||$ of highest-weights: $\big(\begin{array}{c}{{r+k}}\\ {{r}}\end{array}\big)=\big(\begin{array}{c}{{r+m}}\\ {{r}}\end{array}\big)$ , which forces $m=k$ .
It is also quite useful here to know those weights with second smallest q-dimension. This is a by-product of the proof of Proposition 4.1, and the complete answer is given in [18, Table 3]. Here we will simply state that those weights in $P_{+}^{k}(X_{r}^{(1)})$ with second smallest q-dimension are precisely the orbit $S\Lambda_{\star}$ , except for: $A_{r,1}$ ; $B_{r,k}$ for $k\leq3$ ; $C_{2,2},C_{2,3},C_{3,2}$ ; $D_{r,k}$ for $k\leq2$ ; $E_{6,k}$ for $k\leq2$ ; and $E_{7,k},E_{8,k},F_{4,k},G_{2,k}$ for $k\leq4$ .
$C_{r,k}$ and $B_{s,m}$ both have two simple-currents, but their fusion rings can’t be isomorphic (generically) because the orbit $J^{i}\Lambda_{1}$ has the second smallest q-dimension for both algebras at generic rank/level, but the numbers $Q_{j}(J^{i}\Lambda_{1})$ for the two algebras are different.
Another useful invariant involves the set of integers $\ell$ coprime to $\kappa N$ for which $0^{(\ell)}$ is a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign, the q-dimension of $0^{(\ell)}$ ( $\ell$ coprime to $2\kappa$ ) for the algebras $B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}$ is, respectively,
$$
\begin{array}{r l}&{\displaystyle\prod_{a=0}^{r-1}\frac{\sin(\pi\ell\,(2a+1)/2\kappa)}{\sin(\pi\,(2a+1)/2\kappa)}\,\prod_{b=1}^{2r-2}\frac{\sin(\pi\ell b/\kappa)^{\left[\frac{2r-b}{2}\right]}}{\sin(\pi b/\kappa)^{\left[\frac{2r-b}{2}\right]}}\,\,,}\\ &{\displaystyle\prod_{a=1}^{r-1}\frac{\sin(\pi\ell a/\kappa)^{r-a}\,\sin(\pi\ell\,(2a-1)/2\kappa)^{r-a}}{\sin(\pi a/\kappa)^{r-a}\,\sin(\pi\,(2a-1)/2\kappa)^{r-a}}\,\prod_{b=r}^{2r-1}\frac{\sin(\pi\ell b/2\kappa)}{\sin(\pi b/2\kappa)}\,\,,}\\ &{\displaystyle\prod_{a=1}^{r-1}\frac{\sin(\pi\ell a/\kappa)^{\left[\frac{2r-a+1}{2}\right]}}{\sin(\pi a/\kappa)^{\left[\frac{2r-a+1}{2}\right]}}\,\prod_{b=r}^{2r-3}\frac{\sin(\pi\ell b/\kappa)^{\left[\frac{2r-b-1}{2}\right]}}{\sin(\pi b/\kappa)^{\left[\frac{2r-b-1}{2}\right]}}\,\,,}\end{array}
$$
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<html><body>
<h1 data-bbox="200 71 410 86">5. Affine fusion ring isomorphisms </h1>
<p data-bbox="70 102 542 132">We conclude the paper with the determination of all isomorphisms among the affine fusion rings $\mathcal{R}(X_{r,k})$ . Recall Definition 2.1 and the discussion in $\S2.2$ . </p>
<p data-bbox="70 138 541 259">Theorem 5.1. The complete list of fusion ring isomorphisms $\mathscr{R}(X_{r,k})\,\cong\,\mathscr{R}(Y_{s,m})$ when $X_{r,k}\neq Y_{s,m}$ (where $X_{r},Y_{s}$ are simple) is: rank-level duality $\mathcal{R}(C_{r,k})\cong\mathcal{R}(C_{k,r})$ for all $r,k$ , as well as $\mathcal{R}(A_{1,k})\cong\mathcal{R}(C_{k,1})$ ; $\mathcal{R}(B_{r,1})\cong\mathcal{R}(A_{1,2})\cong\mathcal{R}(C_{2,1})\cong\mathcal{R}(E_{8,2})$ for all $r\geq3$ ; $\mathcal{R}(A_{3,1})\cong\mathcal{R}(D_{o d d,1})$ ; $\mathcal{R}(D_{r,1})\cong\mathcal{R}(D_{s,1})$ whenever $r\equiv s$ (mod 2); $\mathscr{R}(A_{2,1})\cong\mathscr{R}(E_{6,1})$ and $\mathcal{R}(A_{1,1})\cong\mathcal{R}(E_{7,1})$ ; $\mathcal{R}(F_{4,1})\cong\mathcal{R}(G_{2,1})$ , $\mathcal{R}(F_{4,2})\cong\mathcal{R}(E_{8,3})$ , and $\mathcal{R}(F_{4,3})\cong\mathcal{R}(G_{2,4})$ . </p>
<p data-bbox="70 266 541 343">The isomorphism $\mathcal{R}(A_{1,k})\cong\mathcal{R}(C_{k,1})$ takes $a\Lambda_{1}$ to $\widetilde{\Lambda}_{a}$ . The isomorphism $\mathcal{R}(F_{4,2})\cong$ $\mathcal{R}(E_{8,3})$ was found in [14]; it relates $\Lambda_{1}\leftrightarrow\tilde{\Lambda}_{8}$ , $2\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{2}$ , $\Lambda_{3}\,\leftrightarrow\,\widetilde{\Lambda}_{1}$ , $\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{7}$ . The isomorphism $\mathcal{R}(F_{4,3})\,\cong\,\mathcal{R}(G_{2,4})$ was found i n [34,14]; a corresponde nce which works is $\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{1}$ , $\Lambda_{1}\leftrightarrow2\widetilde{\Lambda}_{1}$ , $\Lambda_{3}\leftrightarrow3\widetilde{\Lambda}_{2}$ , $2\Lambda_{4}\leftrightarrow2\widetilde{\Lambda}_{2}$ , $\Lambda_{1}+\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{1}+2\widetilde{\Lambda}_{2}$ , $\Lambda_{2}\leftrightarrow4\widetilde{\Lambda}_{2}$ , $3\Lambda_{4}\leftrightarrow\widetilde{\Lambda}_{2}$ , and $\Lambda_{3}+\Lambda_{4}\leftrightarrow\tilde{\Lambda}_{1}+\tilde{\Lambda}_{2}$ . </p>
<p data-bbox="70 344 541 416">We will sket ch th e proof here. The idea is to compare invariants for the various fusion rings, case by case. For example, suppose $\mathcal{R}(A_{r,k})$ and $\mathcal{R}(A_{s,m})$ are isomorphic. Then their simple-current groups $\mathbb{Z}_{r+1}$ and $\mathbb{Z}_{s+1}$ must be isomorphic (since simple-currents must get mapped to simple-currents), so $r=s$ . Now compare the numbers $||P_{+}||$ of highest-weights: $\big(\begin{array}{c}{{r+k}}\\ {{r}}\end{array}\big)=\big(\begin{array}{c}{{r+m}}\\ {{r}}\end{array}\big)$ , which forces $m=k$ . </p>
<p data-bbox="70 416 541 490">It is also quite useful here to know those weights with second smallest q-dimension. This is a by-product of the proof of Proposition 4.1, and the complete answer is given in [18, Table 3]. Here we will simply state that those weights in $P_{+}^{k}(X_{r}^{(1)})$ with second smallest q-dimension are precisely the orbit $S\Lambda_{\star}$ , except for: $A_{r,1}$ ; $B_{r,k}$ for $k\leq3$ ; $C_{2,2},C_{2,3},C_{3,2}$ ; $D_{r,k}$ for $k\leq2$ ; $E_{6,k}$ for $k\leq2$ ; and $E_{7,k},E_{8,k},F_{4,k},G_{2,k}$ for $k\leq4$ . </p>
<p data-bbox="71 490 541 533">$C_{r,k}$ and $B_{s,m}$ both have two simple-currents, but their fusion rings can’t be isomorphic (generically) because the orbit $J^{i}\Lambda_{1}$ has the second smallest q-dimension for both algebras at generic rank/level, but the numbers $Q_{j}(J^{i}\Lambda_{1})$ for the two algebras are different. </p>
<p data-bbox="70 534 541 579">Another useful invariant involves the set of integers $\ell$ coprime to $\kappa N$ for which $0^{(\ell)}$ is a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign, the q-dimension of $0^{(\ell)}$ ( $\ell$ coprime to $2\kappa$ ) for the algebras $B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}$ is, respectively, </p>
<div class="equation" data-bbox="145 595 464 720">$$
\begin{array}{r l}&{\displaystyle\prod_{a=0}^{r-1}\frac{\sin(\pi\ell\,(2a+1)/2\kappa)}{\sin(\pi\,(2a+1)/2\kappa)}\,\prod_{b=1}^{2r-2}\frac{\sin(\pi\ell b/\kappa)^{\left[\frac{2r-b}{2}\right]}}{\sin(\pi b/\kappa)^{\left[\frac{2r-b}{2}\right]}}\,\,,}\\ &{\displaystyle\prod_{a=1}^{r-1}\frac{\sin(\pi\ell a/\kappa)^{r-a}\,\sin(\pi\ell\,(2a-1)/2\kappa)^{r-a}}{\sin(\pi a/\kappa)^{r-a}\,\sin(\pi\,(2a-1)/2\kappa)^{r-a}}\,\prod_{b=r}^{2r-1}\frac{\sin(\pi\ell b/2\kappa)}{\sin(\pi b/2\kappa)}\,\,,}\\ &{\displaystyle\prod_{a=1}^{r-1}\frac{\sin(\pi\ell a/\kappa)^{\left[\frac{2r-a+1}{2}\right]}}{\sin(\pi a/\kappa)^{\left[\frac{2r-a+1}{2}\right]}}\,\prod_{b=r}^{2r-3}\frac{\sin(\pi\ell b/\kappa)^{\left[\frac{2r-b-1}{2}\right]}}{\sin(\pi b/\kappa)^{\left[\frac{2r-b-1}{2}\right]}}\,\,,}\end{array}
$$</div>
</body></html>
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0002044v1
| 20 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "5. Affine fusion ring isomorphisms ", "text_level": 1, "page_idx": 20}, {"type": "text", "text": "We conclude the paper with the determination of all isomorphisms among the affine fusion rings $\\mathcal{R}(X_{r,k})$ . Recall Definition 2.1 and the discussion in $\\S2.2$ . ", "page_idx": 20}, {"type": "text", "text": "Theorem 5.1. The complete list of fusion ring isomorphisms $\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})$ when $X_{r,k}\\neq Y_{s,m}$ (where $X_{r},Y_{s}$ are simple) is: rank-level duality $\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})$ for all $r,k$ , as well as $\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})$ ; $\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})$ for all $r\\geq3$ ; $\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})$ ; $\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})$ whenever $r\\equiv s$ (mod 2); $\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})$ and $\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})$ ; $\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})$ , $\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})$ , and $\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})$ . ", "page_idx": 20}, {"type": "text", "text": "The isomorphism $\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})$ takes $a\\Lambda_{1}$ to $\\widetilde{\\Lambda}_{a}$ . The isomorphism $\\mathcal{R}(F_{4,2})\\cong$ $\\mathcal{R}(E_{8,3})$ was found in [14]; it relates $\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}$ , $2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}$ , $\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}$ , $\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}$ . The isomorphism $\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})$ was found i n [34,14]; a corresponde nce which works is $\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}$ , $\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}$ , $\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}$ , $2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}$ , $\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}$ , $\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}$ , $3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}$ , and $\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}$ . ", "page_idx": 20}, {"type": "text", "text": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion rings, case by case. For example, suppose $\\mathcal{R}(A_{r,k})$ and $\\mathcal{R}(A_{s,m})$ are isomorphic. Then their simple-current groups $\\mathbb{Z}_{r+1}$ and $\\mathbb{Z}_{s+1}$ must be isomorphic (since simple-currents must get mapped to simple-currents), so $r=s$ . Now compare the numbers $||P_{+}||$ of highest-weights: $\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)$ , which forces $m=k$ . ", "page_idx": 20}, {"type": "text", "text": "It is also quite useful here to know those weights with second smallest q-dimension. This is a by-product of the proof of Proposition 4.1, and the complete answer is given in [18, Table 3]. Here we will simply state that those weights in $P_{+}^{k}(X_{r}^{(1)})$ with second smallest q-dimension are precisely the orbit $S\\Lambda_{\\star}$ , except for: $A_{r,1}$ ; $B_{r,k}$ for $k\\leq3$ ; $C_{2,2},C_{2,3},C_{3,2}$ ; $D_{r,k}$ for $k\\leq2$ ; $E_{6,k}$ for $k\\leq2$ ; and $E_{7,k},E_{8,k},F_{4,k},G_{2,k}$ for $k\\leq4$ . ", "page_idx": 20}, {"type": "text", "text": "$C_{r,k}$ and $B_{s,m}$ both have two simple-currents, but their fusion rings can’t be isomorphic (generically) because the orbit $J^{i}\\Lambda_{1}$ has the second smallest q-dimension for both algebras at generic rank/level, but the numbers $Q_{j}(J^{i}\\Lambda_{1})$ for the two algebras are different. ", "page_idx": 20}, {"type": "text", "text": "Another useful invariant involves the set of integers $\\ell$ coprime to $\\kappa N$ for which $0^{(\\ell)}$ is a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign, the q-dimension of $0^{(\\ell)}$ ( $\\ell$ coprime to $2\\kappa$ ) for the algebras $B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}$ is, respectively, ", "page_idx": 20}, {"type": "equation", "text": "$$\n\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}\n$$", "text_format": "latex", "page_idx": 20}]
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Recall Definition 2.1 and the discussion in ", "type": "text"}, {"bbox": [411, 119, 432, 132], "score": 0.3, "content": "\\S2.2", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [433, 119, 435, 133], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1.5}, {"type": "text", "bbox": [70, 138, 541, 259], "lines": [{"bbox": [94, 140, 540, 159], "spans": [{"bbox": [94, 140, 434, 159], "score": 1.0, "content": "Theorem 5.1. 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0.91, "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 99}, {"bbox": [263, 169, 302, 187], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [303, 171, 321, 184], "score": 0.88, "content": "r,k", "type": "inline_equation", "height": 13, "width": 18}, {"bbox": [321, 169, 382, 187], "score": 1.0, "content": ", as well as ", "type": "text"}, {"bbox": [383, 171, 483, 185], "score": 0.92, "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [484, 169, 489, 187], "score": 1.0, "content": ";", "type": "text"}], "index": 5}, {"bbox": [71, 185, 358, 203], "spans": [{"bbox": [71, 185, 284, 200], "score": 0.86, "content": "\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})", "type": "inline_equation", "height": 15, "width": 213}, {"bbox": [284, 185, 323, 203], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [323, 186, 352, 199], "score": 0.89, "content": "r\\geq3", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [353, 185, 358, 203], "score": 1.0, "content": ";", "type": "text"}], "index": 6}, {"bbox": [71, 200, 188, 217], "spans": [{"bbox": [71, 200, 182, 214], "score": 0.89, "content": "\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})", "type": "inline_equation", "height": 14, "width": 111}, {"bbox": [182, 200, 188, 217], "score": 1.0, "content": ";", "type": "text"}], "index": 7}, {"bbox": [71, 214, 307, 231], "spans": [{"bbox": [71, 215, 171, 229], "score": 0.9, "content": "\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [171, 214, 226, 231], "score": 1.0, "content": " whenever ", "type": "text"}, {"bbox": [226, 216, 255, 227], "score": 0.48, "content": "r\\equiv s", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [256, 214, 307, 231], "score": 1.0, "content": " (mod 2);", "type": "text"}], "index": 8}, {"bbox": [71, 229, 302, 245], "spans": [{"bbox": [71, 230, 171, 244], "score": 0.89, "content": "\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [171, 230, 197, 245], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [197, 229, 296, 244], "score": 0.87, "content": "\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})", "type": "inline_equation", "height": 15, "width": 99}, {"bbox": [296, 230, 302, 245], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [71, 244, 408, 261], "spans": [{"bbox": [71, 245, 169, 259], "score": 0.89, "content": "\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [170, 244, 176, 261], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [177, 244, 275, 259], "score": 0.85, "content": "\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [275, 244, 304, 261], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [305, 244, 405, 259], "score": 0.9, "content": "\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})", "type": "inline_equation", "height": 15, "width": 100}, {"bbox": [405, 244, 408, 261], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 6.5}, {"type": "text", "bbox": [70, 266, 541, 343], "lines": [{"bbox": [93, 267, 541, 285], "spans": [{"bbox": [93, 267, 190, 285], "score": 1.0, "content": "The isomorphism ", "type": "text"}, {"bbox": [190, 269, 291, 284], "score": 0.91, "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "type": "inline_equation", "height": 15, "width": 101}, {"bbox": [292, 267, 326, 285], "score": 1.0, "content": " takes ", "type": "text"}, {"bbox": [326, 269, 347, 282], "score": 0.86, "content": "a\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [348, 267, 365, 285], "score": 1.0, "content": " to", "type": "text"}, {"bbox": [366, 267, 381, 282], "score": 0.9, "content": "\\widetilde{\\Lambda}_{a}", "type": "inline_equation", "height": 15, "width": 15}, {"bbox": [381, 267, 485, 285], "score": 1.0, "content": ". The isomorphism ", "type": "text"}, {"bbox": [486, 270, 541, 284], "score": 0.89, "content": "\\mathcal{R}(F_{4,2})\\cong", "type": "inline_equation", "height": 14, "width": 55}], "index": 11}, {"bbox": [71, 283, 540, 300], "spans": [{"bbox": [71, 288, 113, 300], "score": 0.93, "content": "\\mathcal{R}(E_{8,3})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [113, 285, 272, 300], "score": 1.0, "content": " was found in [14]; it relates ", "type": "text"}, {"bbox": [272, 283, 323, 299], "score": 0.89, "content": "\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [323, 285, 331, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [331, 284, 388, 299], "score": 0.88, "content": "2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 57}, {"bbox": [389, 285, 396, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [397, 285, 447, 299], "score": 0.93, "content": "\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [448, 285, 455, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [456, 285, 506, 299], "score": 0.93, "content": "\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [507, 285, 540, 300], "score": 1.0, "content": ". The", "type": "text"}], "index": 12}, {"bbox": [70, 299, 542, 315], "spans": [{"bbox": [70, 299, 142, 315], "score": 1.0, "content": "isomorphism ", "type": "text"}, {"bbox": [142, 300, 244, 314], "score": 0.91, "content": "\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})", "type": "inline_equation", "height": 14, "width": 102}, {"bbox": [244, 299, 542, 315], "score": 1.0, "content": " was found i n [34,14]; a corresponde nce which works is", "type": "text"}], "index": 13}, {"bbox": [71, 314, 540, 331], "spans": [{"bbox": [71, 315, 118, 330], "score": 0.92, "content": "\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 15, "width": 47}, {"bbox": [118, 315, 124, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [124, 315, 177, 329], "score": 0.88, "content": "\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [177, 315, 183, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [184, 315, 236, 330], "score": 0.87, "content": "\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 52}, {"bbox": [236, 315, 242, 331], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [243, 314, 302, 329], "score": 0.9, "content": "2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [302, 315, 308, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [308, 315, 417, 330], "score": 0.89, "content": "\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 109}, {"bbox": [418, 315, 424, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [424, 315, 477, 330], "score": 0.9, "content": "\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [477, 315, 483, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [484, 315, 536, 329], "score": 0.93, "content": "3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [536, 315, 540, 331], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [70, 329, 201, 345], "spans": [{"bbox": [70, 329, 94, 345], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 330, 197, 344], "score": 0.91, "content": "\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}", "type": "inline_equation", "height": 14, "width": 102}, {"bbox": [198, 329, 201, 345], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13}, {"type": "text", "bbox": [70, 344, 541, 416], "lines": [{"bbox": [95, 345, 540, 361], "spans": [{"bbox": [95, 345, 540, 361], "score": 1.0, "content": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion", "type": "text"}], "index": 16}, {"bbox": [69, 359, 541, 376], "spans": [{"bbox": [69, 359, 286, 376], "score": 1.0, "content": "rings, case by case. For example, suppose ", "type": "text"}, {"bbox": [286, 361, 328, 374], "score": 0.94, "content": "\\mathcal{R}(A_{r,k})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [328, 359, 353, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [353, 361, 398, 374], "score": 0.94, "content": "\\mathcal{R}(A_{s,m})", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [398, 359, 541, 376], "score": 1.0, "content": " are isomorphic. Then their", "type": "text"}], "index": 17}, {"bbox": [69, 374, 542, 390], "spans": [{"bbox": [69, 374, 188, 390], "score": 1.0, "content": "simple-current groups ", "type": "text"}, {"bbox": [188, 376, 214, 388], "score": 0.92, "content": "\\mathbb{Z}_{r+1}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [214, 374, 241, 390], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [241, 376, 266, 388], "score": 0.93, "content": "\\mathbb{Z}_{s+1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [266, 374, 542, 390], "score": 1.0, "content": " must be isomorphic (since simple-currents must get", "type": "text"}], "index": 18}, {"bbox": [70, 389, 541, 404], "spans": [{"bbox": [70, 389, 235, 404], "score": 1.0, "content": "mapped to simple-currents), so ", "type": "text"}, {"bbox": [235, 394, 263, 399], "score": 0.88, "content": "r=s", "type": "inline_equation", "height": 5, "width": 28}, {"bbox": [263, 389, 412, 404], "score": 1.0, "content": ". Now compare the numbers ", "type": "text"}, {"bbox": [412, 390, 440, 403], "score": 0.94, "content": "||P_{+}||", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [440, 389, 541, 404], "score": 1.0, "content": "of highest-weights:", "type": "text"}], "index": 19}, {"bbox": [72, 399, 264, 420], "spans": [{"bbox": [72, 403, 150, 418], "score": 0.93, "content": "\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)", "type": "inline_equation", "height": 15, "width": 78}, {"bbox": [150, 399, 224, 420], "score": 1.0, "content": ", which forces ", "type": "text"}, {"bbox": [225, 405, 258, 414], "score": 0.93, "content": "m=k", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [258, 399, 264, 420], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 18}, {"type": "text", "bbox": [70, 416, 541, 490], "lines": [{"bbox": [93, 418, 540, 433], "spans": [{"bbox": [93, 418, 540, 433], "score": 1.0, "content": "It is also quite useful here to know those weights with second smallest q-dimension.", "type": "text"}], "index": 21}, {"bbox": [70, 432, 541, 448], "spans": [{"bbox": [70, 432, 541, 448], "score": 1.0, "content": "This is a by-product of the proof of Proposition 4.1, and the complete answer is given in", "type": "text"}], "index": 22}, {"bbox": [69, 442, 543, 466], "spans": [{"bbox": [69, 442, 382, 466], "score": 1.0, "content": "[18, Table 3]. Here we will simply state that those weights in ", "type": "text"}, {"bbox": [382, 446, 431, 463], "score": 0.94, "content": "P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 49}, {"bbox": [431, 442, 543, 466], "score": 1.0, "content": " with second smallest", "type": "text"}], "index": 23}, {"bbox": [69, 462, 541, 479], "spans": [{"bbox": [69, 462, 258, 479], "score": 1.0, "content": "q-dimension are precisely the orbit ", "type": "text"}, {"bbox": [258, 465, 280, 475], "score": 0.92, "content": "S\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [281, 462, 349, 479], "score": 1.0, "content": ", except for: ", "type": "text"}, {"bbox": [349, 465, 371, 477], "score": 0.84, "content": "A_{r,1}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [371, 462, 378, 479], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [378, 465, 401, 477], "score": 0.91, "content": "B_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [401, 462, 423, 479], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [424, 465, 453, 475], "score": 0.91, "content": "k\\leq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [453, 462, 460, 479], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [461, 465, 536, 477], "score": 0.93, "content": "C_{2,2},C_{2,3},C_{3,2}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [537, 462, 541, 479], "score": 1.0, "content": ";", "type": "text"}], "index": 24}, {"bbox": [71, 478, 416, 492], "spans": [{"bbox": [71, 479, 94, 491], "score": 0.92, "content": "D_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [94, 478, 116, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [116, 479, 145, 489], "score": 0.9, "content": "k\\leq2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [145, 478, 151, 492], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [152, 479, 174, 491], "score": 0.92, "content": "E_{6,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [175, 478, 196, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [196, 479, 226, 489], "score": 0.92, "content": "k\\leq2", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [226, 478, 255, 492], "score": 1.0, "content": "; and ", "type": "text"}, {"bbox": [255, 479, 360, 491], "score": 0.93, "content": "E_{7,k},E_{8,k},F_{4,k},G_{2,k}", "type": "inline_equation", "height": 12, "width": 105}, {"bbox": [361, 478, 382, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [383, 479, 412, 489], "score": 0.92, "content": "k\\leq4", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [412, 478, 416, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 23}, {"type": "text", "bbox": [71, 490, 541, 533], "lines": [{"bbox": [95, 492, 540, 506], "spans": [{"bbox": [95, 494, 116, 506], "score": 0.93, "content": "C_{r,k}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [117, 492, 141, 506], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [141, 494, 167, 506], "score": 0.94, "content": "B_{s,m}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [167, 492, 540, 506], "score": 1.0, "content": " both have two simple-currents, but their fusion rings can’t be isomorphic", "type": "text"}], "index": 26}, {"bbox": [71, 506, 540, 521], "spans": [{"bbox": [71, 506, 232, 521], "score": 1.0, "content": "(generically) because the orbit ", "type": "text"}, {"bbox": [232, 507, 257, 519], "score": 0.93, "content": "J^{i}\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [258, 506, 540, 521], "score": 1.0, "content": " has the second smallest q-dimension for both algebras", "type": "text"}], "index": 27}, {"bbox": [70, 521, 506, 535], "spans": [{"bbox": [70, 522, 277, 535], "score": 1.0, "content": "at generic rank/level, but the numbers ", "type": "text"}, {"bbox": [277, 521, 327, 535], "score": 0.95, "content": "Q_{j}(J^{i}\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [327, 522, 506, 535], "score": 1.0, "content": " for the two algebras are different.", "type": "text"}], "index": 28}], "index": 27}, {"type": "text", "bbox": [70, 534, 541, 579], "lines": [{"bbox": [93, 534, 542, 551], "spans": [{"bbox": [93, 534, 367, 551], "score": 1.0, "content": "Another useful invariant involves the set of integers ", "type": "text"}, {"bbox": [367, 537, 373, 546], "score": 0.85, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [373, 534, 435, 551], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [435, 537, 453, 546], "score": 0.89, "content": "\\kappa N", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [454, 534, 509, 551], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [509, 535, 527, 546], "score": 0.88, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [528, 534, 542, 551], "score": 1.0, "content": " is", "type": "text"}], "index": 29}, {"bbox": [70, 551, 539, 565], "spans": [{"bbox": [70, 551, 539, 565], "score": 1.0, "content": "a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign,", "type": "text"}], "index": 30}, {"bbox": [68, 561, 539, 583], "spans": [{"bbox": [68, 561, 172, 583], "score": 1.0, "content": "the q-dimension of ", "type": "text"}, {"bbox": [172, 566, 190, 577], "score": 0.91, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [191, 561, 199, 583], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [199, 568, 204, 577], "score": 0.76, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [205, 561, 267, 583], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [268, 569, 281, 577], "score": 0.78, "content": "2\\kappa", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [281, 561, 374, 583], "score": 1.0, "content": ") for the algebras ", "type": "text"}, {"bbox": [375, 564, 453, 579], "score": 0.94, "content": "B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 78}, {"bbox": [453, 561, 539, 583], "score": 1.0, "content": "is, respectively,", "type": "text"}], "index": 31}], "index": 30}, {"type": "interline_equation", "bbox": [145, 595, 464, 720], "lines": [{"bbox": [145, 595, 464, 720], "spans": [{"bbox": [145, 595, 464, 720], "score": 0.94, "content": "\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}", "type": "interline_equation"}], "index": 32}], "index": 32}], "layout_bboxes": [], "page_idx": 20, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [145, 595, 464, 720], "lines": [{"bbox": [145, 595, 464, 720], "spans": [{"bbox": [145, 595, 464, 720], "score": 0.94, "content": "\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}", "type": "interline_equation"}], "index": 32}], "index": 32}], "discarded_blocks": [{"type": "discarded", "bbox": [298, 731, 311, 742], "lines": [{"bbox": [298, 731, 313, 745], "spans": [{"bbox": [298, 731, 313, 745], "score": 1.0, "content": "21", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [200, 71, 410, 86], "lines": [{"bbox": [201, 74, 409, 88], "spans": [{"bbox": [201, 74, 409, 88], "score": 1.0, "content": "5. Affine fusion ring isomorphisms", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [70, 102, 542, 132], "lines": [{"bbox": [94, 103, 541, 120], "spans": [{"bbox": [94, 103, 541, 120], "score": 1.0, "content": "We conclude the paper with the determination of all isomorphisms among the affine", "type": "text"}], "index": 1}, {"bbox": [72, 119, 435, 133], "spans": [{"bbox": [72, 119, 135, 133], "score": 1.0, "content": "fusion rings ", "type": "text"}, {"bbox": [135, 120, 178, 133], "score": 0.93, "content": "\\mathcal{R}(X_{r,k})", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [178, 119, 410, 133], "score": 1.0, "content": ". Recall Definition 2.1 and the discussion in ", "type": "text"}, {"bbox": [411, 119, 432, 132], "score": 0.3, "content": "\\S2.2", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [433, 119, 435, 133], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1.5, "bbox_fs": [72, 103, 541, 133]}, {"type": "text", "bbox": [70, 138, 541, 259], "lines": [{"bbox": [94, 140, 540, 159], "spans": [{"bbox": [94, 140, 434, 159], "score": 1.0, "content": "Theorem 5.1. The complete list of fusion ring isomorphisms ", "type": "text"}, {"bbox": [434, 143, 540, 156], "score": 0.92, "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})", "type": "inline_equation", "height": 13, "width": 106}], "index": 3}, {"bbox": [72, 156, 321, 172], "spans": [{"bbox": [72, 156, 101, 172], "score": 1.0, "content": "when ", "type": "text"}, {"bbox": [101, 156, 164, 170], "score": 0.92, "content": "X_{r,k}\\neq Y_{s,m}", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [164, 156, 206, 172], "score": 1.0, "content": " (where ", "type": "text"}, {"bbox": [206, 156, 240, 169], "score": 0.9, "content": "X_{r},Y_{s}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [240, 156, 321, 172], "score": 1.0, "content": " are simple) is:", "type": "text"}], "index": 4}, {"bbox": [70, 169, 489, 187], "spans": [{"bbox": [70, 169, 164, 187], "score": 1.0, "content": "rank-level duality ", "type": "text"}, {"bbox": [164, 171, 263, 185], "score": 0.91, "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "type": "inline_equation", "height": 14, "width": 99}, {"bbox": [263, 169, 302, 187], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [303, 171, 321, 184], "score": 0.88, "content": "r,k", "type": "inline_equation", "height": 13, "width": 18}, {"bbox": [321, 169, 382, 187], "score": 1.0, "content": ", as well as ", "type": "text"}, {"bbox": [383, 171, 483, 185], "score": 0.92, "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [484, 169, 489, 187], "score": 1.0, "content": ";", "type": "text"}], "index": 5}, {"bbox": [71, 185, 358, 203], "spans": [{"bbox": [71, 185, 284, 200], "score": 0.86, "content": "\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})", "type": "inline_equation", "height": 15, "width": 213}, {"bbox": [284, 185, 323, 203], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [323, 186, 352, 199], "score": 0.89, "content": "r\\geq3", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [353, 185, 358, 203], "score": 1.0, "content": ";", "type": "text"}], "index": 6}, {"bbox": [71, 200, 188, 217], "spans": [{"bbox": [71, 200, 182, 214], "score": 0.89, "content": "\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})", "type": "inline_equation", "height": 14, "width": 111}, {"bbox": [182, 200, 188, 217], "score": 1.0, "content": ";", "type": "text"}], "index": 7}, {"bbox": [71, 214, 307, 231], "spans": [{"bbox": [71, 215, 171, 229], "score": 0.9, "content": "\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [171, 214, 226, 231], "score": 1.0, "content": " whenever ", "type": "text"}, {"bbox": [226, 216, 255, 227], "score": 0.48, "content": "r\\equiv s", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [256, 214, 307, 231], "score": 1.0, "content": " (mod 2);", "type": "text"}], "index": 8}, {"bbox": [71, 229, 302, 245], "spans": [{"bbox": [71, 230, 171, 244], "score": 0.89, "content": "\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})", "type": "inline_equation", "height": 14, "width": 100}, {"bbox": [171, 230, 197, 245], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [197, 229, 296, 244], "score": 0.87, "content": "\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})", "type": "inline_equation", "height": 15, "width": 99}, {"bbox": [296, 230, 302, 245], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [71, 244, 408, 261], "spans": [{"bbox": [71, 245, 169, 259], "score": 0.89, "content": "\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [170, 244, 176, 261], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [177, 244, 275, 259], "score": 0.85, "content": "\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [275, 244, 304, 261], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [305, 244, 405, 259], "score": 0.9, "content": "\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})", "type": "inline_equation", "height": 15, "width": 100}, {"bbox": [405, 244, 408, 261], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 6.5, "bbox_fs": [70, 140, 540, 261]}, {"type": "text", "bbox": [70, 266, 541, 343], "lines": [{"bbox": [93, 267, 541, 285], "spans": [{"bbox": [93, 267, 190, 285], "score": 1.0, "content": "The isomorphism ", "type": "text"}, {"bbox": [190, 269, 291, 284], "score": 0.91, "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "type": "inline_equation", "height": 15, "width": 101}, {"bbox": [292, 267, 326, 285], "score": 1.0, "content": " takes ", "type": "text"}, {"bbox": [326, 269, 347, 282], "score": 0.86, "content": "a\\Lambda_{1}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [348, 267, 365, 285], "score": 1.0, "content": " to", "type": "text"}, {"bbox": [366, 267, 381, 282], "score": 0.9, "content": "\\widetilde{\\Lambda}_{a}", "type": "inline_equation", "height": 15, "width": 15}, {"bbox": [381, 267, 485, 285], "score": 1.0, "content": ". The isomorphism ", "type": "text"}, {"bbox": [486, 270, 541, 284], "score": 0.89, "content": "\\mathcal{R}(F_{4,2})\\cong", "type": "inline_equation", "height": 14, "width": 55}], "index": 11}, {"bbox": [71, 283, 540, 300], "spans": [{"bbox": [71, 288, 113, 300], "score": 0.93, "content": "\\mathcal{R}(E_{8,3})", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [113, 285, 272, 300], "score": 1.0, "content": " was found in [14]; it relates ", "type": "text"}, {"bbox": [272, 283, 323, 299], "score": 0.89, "content": "\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}", "type": "inline_equation", "height": 16, "width": 51}, {"bbox": [323, 285, 331, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [331, 284, 388, 299], "score": 0.88, "content": "2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 57}, {"bbox": [389, 285, 396, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [397, 285, 447, 299], "score": 0.93, "content": "\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [448, 285, 455, 300], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [456, 285, 506, 299], "score": 0.93, "content": "\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [507, 285, 540, 300], "score": 1.0, "content": ". The", "type": "text"}], "index": 12}, {"bbox": [70, 299, 542, 315], "spans": [{"bbox": [70, 299, 142, 315], "score": 1.0, "content": "isomorphism ", "type": "text"}, {"bbox": [142, 300, 244, 314], "score": 0.91, "content": "\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})", "type": "inline_equation", "height": 14, "width": 102}, {"bbox": [244, 299, 542, 315], "score": 1.0, "content": " was found i n [34,14]; a corresponde nce which works is", "type": "text"}], "index": 13}, {"bbox": [71, 314, 540, 331], "spans": [{"bbox": [71, 315, 118, 330], "score": 0.92, "content": "\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 15, "width": 47}, {"bbox": [118, 315, 124, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [124, 315, 177, 329], "score": 0.88, "content": "\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}", "type": "inline_equation", "height": 14, "width": 53}, {"bbox": [177, 315, 183, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [184, 315, 236, 330], "score": 0.87, "content": "\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 52}, {"bbox": [236, 315, 242, 331], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [243, 314, 302, 329], "score": 0.9, "content": "2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [302, 315, 308, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [308, 315, 417, 330], "score": 0.89, "content": "\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 109}, {"bbox": [418, 315, 424, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [424, 315, 477, 330], "score": 0.9, "content": "\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [477, 315, 483, 331], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [484, 315, 536, 329], "score": 0.93, "content": "3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [536, 315, 540, 331], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [70, 329, 201, 345], "spans": [{"bbox": [70, 329, 94, 345], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [95, 330, 197, 344], "score": 0.91, "content": "\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}", "type": "inline_equation", "height": 14, "width": 102}, {"bbox": [198, 329, 201, 345], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13, "bbox_fs": [70, 267, 542, 345]}, {"type": "text", "bbox": [70, 344, 541, 416], "lines": [{"bbox": [95, 345, 540, 361], "spans": [{"bbox": [95, 345, 540, 361], "score": 1.0, "content": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion", "type": "text"}], "index": 16}, {"bbox": [69, 359, 541, 376], "spans": [{"bbox": [69, 359, 286, 376], "score": 1.0, "content": "rings, case by case. For example, suppose ", "type": "text"}, {"bbox": [286, 361, 328, 374], "score": 0.94, "content": "\\mathcal{R}(A_{r,k})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [328, 359, 353, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [353, 361, 398, 374], "score": 0.94, "content": "\\mathcal{R}(A_{s,m})", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [398, 359, 541, 376], "score": 1.0, "content": " are isomorphic. Then their", "type": "text"}], "index": 17}, {"bbox": [69, 374, 542, 390], "spans": [{"bbox": [69, 374, 188, 390], "score": 1.0, "content": "simple-current groups ", "type": "text"}, {"bbox": [188, 376, 214, 388], "score": 0.92, "content": "\\mathbb{Z}_{r+1}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [214, 374, 241, 390], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [241, 376, 266, 388], "score": 0.93, "content": "\\mathbb{Z}_{s+1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [266, 374, 542, 390], "score": 1.0, "content": " must be isomorphic (since simple-currents must get", "type": "text"}], "index": 18}, {"bbox": [70, 389, 541, 404], "spans": [{"bbox": [70, 389, 235, 404], "score": 1.0, "content": "mapped to simple-currents), so ", "type": "text"}, {"bbox": [235, 394, 263, 399], "score": 0.88, "content": "r=s", "type": "inline_equation", "height": 5, "width": 28}, {"bbox": [263, 389, 412, 404], "score": 1.0, "content": ". Now compare the numbers ", "type": "text"}, {"bbox": [412, 390, 440, 403], "score": 0.94, "content": "||P_{+}||", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [440, 389, 541, 404], "score": 1.0, "content": "of highest-weights:", "type": "text"}], "index": 19}, {"bbox": [72, 399, 264, 420], "spans": [{"bbox": [72, 403, 150, 418], "score": 0.93, "content": "\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)", "type": "inline_equation", "height": 15, "width": 78}, {"bbox": [150, 399, 224, 420], "score": 1.0, "content": ", which forces ", "type": "text"}, {"bbox": [225, 405, 258, 414], "score": 0.93, "content": "m=k", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [258, 399, 264, 420], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 18, "bbox_fs": [69, 345, 542, 420]}, {"type": "text", "bbox": [70, 416, 541, 490], "lines": [{"bbox": [93, 418, 540, 433], "spans": [{"bbox": [93, 418, 540, 433], "score": 1.0, "content": "It is also quite useful here to know those weights with second smallest q-dimension.", "type": "text"}], "index": 21}, {"bbox": [70, 432, 541, 448], "spans": [{"bbox": [70, 432, 541, 448], "score": 1.0, "content": "This is a by-product of the proof of Proposition 4.1, and the complete answer is given in", "type": "text"}], "index": 22}, {"bbox": [69, 442, 543, 466], "spans": [{"bbox": [69, 442, 382, 466], "score": 1.0, "content": "[18, Table 3]. Here we will simply state that those weights in ", "type": "text"}, {"bbox": [382, 446, 431, 463], "score": 0.94, "content": "P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 49}, {"bbox": [431, 442, 543, 466], "score": 1.0, "content": " with second smallest", "type": "text"}], "index": 23}, {"bbox": [69, 462, 541, 479], "spans": [{"bbox": [69, 462, 258, 479], "score": 1.0, "content": "q-dimension are precisely the orbit ", "type": "text"}, {"bbox": [258, 465, 280, 475], "score": 0.92, "content": "S\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [281, 462, 349, 479], "score": 1.0, "content": ", except for: ", "type": "text"}, {"bbox": [349, 465, 371, 477], "score": 0.84, "content": "A_{r,1}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [371, 462, 378, 479], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [378, 465, 401, 477], "score": 0.91, "content": "B_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [401, 462, 423, 479], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [424, 465, 453, 475], "score": 0.91, "content": "k\\leq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [453, 462, 460, 479], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [461, 465, 536, 477], "score": 0.93, "content": "C_{2,2},C_{2,3},C_{3,2}", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [537, 462, 541, 479], "score": 1.0, "content": ";", "type": "text"}], "index": 24}, {"bbox": [71, 478, 416, 492], "spans": [{"bbox": [71, 479, 94, 491], "score": 0.92, "content": "D_{r,k}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [94, 478, 116, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [116, 479, 145, 489], "score": 0.9, "content": "k\\leq2", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [145, 478, 151, 492], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [152, 479, 174, 491], "score": 0.92, "content": "E_{6,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [175, 478, 196, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [196, 479, 226, 489], "score": 0.92, "content": "k\\leq2", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [226, 478, 255, 492], "score": 1.0, "content": "; and ", "type": "text"}, {"bbox": [255, 479, 360, 491], "score": 0.93, "content": "E_{7,k},E_{8,k},F_{4,k},G_{2,k}", "type": "inline_equation", "height": 12, "width": 105}, {"bbox": [361, 478, 382, 492], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [383, 479, 412, 489], "score": 0.92, "content": "k\\leq4", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [412, 478, 416, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 23, "bbox_fs": [69, 418, 543, 492]}, {"type": "text", "bbox": [71, 490, 541, 533], "lines": [{"bbox": [95, 492, 540, 506], "spans": [{"bbox": [95, 494, 116, 506], "score": 0.93, "content": "C_{r,k}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [117, 492, 141, 506], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [141, 494, 167, 506], "score": 0.94, "content": "B_{s,m}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [167, 492, 540, 506], "score": 1.0, "content": " both have two simple-currents, but their fusion rings can’t be isomorphic", "type": "text"}], "index": 26}, {"bbox": [71, 506, 540, 521], "spans": [{"bbox": [71, 506, 232, 521], "score": 1.0, "content": "(generically) because the orbit ", "type": "text"}, {"bbox": [232, 507, 257, 519], "score": 0.93, "content": "J^{i}\\Lambda_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [258, 506, 540, 521], "score": 1.0, "content": " has the second smallest q-dimension for both algebras", "type": "text"}], "index": 27}, {"bbox": [70, 521, 506, 535], "spans": [{"bbox": [70, 522, 277, 535], "score": 1.0, "content": "at generic rank/level, but the numbers ", "type": "text"}, {"bbox": [277, 521, 327, 535], "score": 0.95, "content": "Q_{j}(J^{i}\\Lambda_{1})", "type": "inline_equation", "height": 14, "width": 50}, {"bbox": [327, 522, 506, 535], "score": 1.0, "content": " for the two algebras are different.", "type": "text"}], "index": 28}], "index": 27, "bbox_fs": [70, 492, 540, 535]}, {"type": "text", "bbox": [70, 534, 541, 579], "lines": [{"bbox": [93, 534, 542, 551], "spans": [{"bbox": [93, 534, 367, 551], "score": 1.0, "content": "Another useful invariant involves the set of integers ", "type": "text"}, {"bbox": [367, 537, 373, 546], "score": 0.85, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [373, 534, 435, 551], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [435, 537, 453, 546], "score": 0.89, "content": "\\kappa N", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [454, 534, 509, 551], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [509, 535, 527, 546], "score": 0.88, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [528, 534, 542, 551], "score": 1.0, "content": " is", "type": "text"}], "index": 29}, {"bbox": [70, 551, 539, 565], "spans": [{"bbox": [70, 551, 539, 565], "score": 1.0, "content": "a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign,", "type": "text"}], "index": 30}, {"bbox": [68, 561, 539, 583], "spans": [{"bbox": [68, 561, 172, 583], "score": 1.0, "content": "the q-dimension of ", "type": "text"}, {"bbox": [172, 566, 190, 577], "score": 0.91, "content": "0^{(\\ell)}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [191, 561, 199, 583], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [199, 568, 204, 577], "score": 0.76, "content": "\\ell", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [205, 561, 267, 583], "score": 1.0, "content": "coprime to ", "type": "text"}, {"bbox": [268, 569, 281, 577], "score": 0.78, "content": "2\\kappa", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [281, 561, 374, 583], "score": 1.0, "content": ") for the algebras ", "type": "text"}, {"bbox": [375, 564, 453, 579], "score": 0.94, "content": "B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 78}, {"bbox": [453, 561, 539, 583], "score": 1.0, "content": "is, respectively,", "type": "text"}], "index": 31}], "index": 30, "bbox_fs": [68, 534, 542, 583]}, {"type": "interline_equation", "bbox": [145, 595, 464, 720], "lines": [{"bbox": [145, 595, 464, 720], "spans": [{"bbox": [145, 595, 464, 720], "score": 0.94, "content": "\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}", "type": "interline_equation"}], "index": 32}], "index": 32}]}
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[{"type": "title", "bbox": [200, 71, 410, 86], "content": "5. Affine fusion ring isomorphisms", "index": 0}, {"type": "text", "bbox": [70, 102, 542, 132], "content": "We conclude the paper with the determination of all isomorphisms among the affine fusion rings . Recall Definition 2.1 and the discussion in .", "index": 1}, {"type": "text", "bbox": [70, 138, 541, 259], "content": "Theorem 5.1. The complete list of fusion ring isomorphisms when (where are simple) is: rank-level duality for all , as well as ; for all ; ; whenever (mod 2); and ; , , and .", "index": 2}, {"type": "text", "bbox": [70, 266, 541, 343], "content": "The isomorphism takes to . The isomorphism was found in [14]; it relates , , , . The isomorphism was found i n [34,14]; a corresponde nce which works is , , , , , , , and .", "index": 3}, {"type": "text", "bbox": [70, 344, 541, 416], "content": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion rings, case by case. For example, suppose and are isomorphic. Then their simple-current groups and must be isomorphic (since simple-currents must get mapped to simple-currents), so . Now compare the numbers of highest-weights: , which forces .", "index": 4}, {"type": "text", "bbox": [70, 416, 541, 490], "content": "It is also quite useful here to know those weights with second smallest q-dimension. This is a by-product of the proof of Proposition 4.1, and the complete answer is given in [18, Table 3]. Here we will simply state that those weights in with second smallest q-dimension are precisely the orbit , except for: ; for ; ; for ; for ; and for .", "index": 5}, {"type": "text", "bbox": [71, 490, 541, 533], "content": "and both have two simple-currents, but their fusion rings can’t be isomorphic (generically) because the orbit has the second smallest q-dimension for both algebras at generic rank/level, but the numbers for the two algebras are different.", "index": 6}, {"type": "text", "bbox": [70, 534, 541, 579], "content": "Another useful invariant involves the set of integers coprime to for which is a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign, the q-dimension of ( coprime to ) for the algebras is, respectively,", "index": 7}, {"type": "interline_equation", "bbox": [145, 595, 464, 720], "content": "", "index": 8}]
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[{"bbox": [201, 74, 409, 88], "content": "5. Affine fusion ring isomorphisms", "parent_index": 0, "line_index": 0}, {"bbox": [94, 103, 541, 120], "content": "We conclude the paper with the determination of all isomorphisms among the affine", "parent_index": 1, "line_index": 0}, {"bbox": [72, 119, 435, 133], "content": "fusion rings . Recall Definition 2.1 and the discussion in .", "parent_index": 1, "line_index": 1}, {"bbox": [94, 140, 540, 159], "content": "Theorem 5.1. The complete list of fusion ring isomorphisms", "parent_index": 2, "line_index": 0}, {"bbox": [72, 156, 321, 172], "content": "when (where are simple) is:", "parent_index": 2, "line_index": 1}, {"bbox": [70, 169, 489, 187], "content": "rank-level duality for all , as well as ;", "parent_index": 2, "line_index": 2}, {"bbox": [71, 185, 358, 203], "content": "for all ;", "parent_index": 2, "line_index": 3}, {"bbox": [71, 200, 188, 217], "content": ";", "parent_index": 2, "line_index": 4}, {"bbox": [71, 214, 307, 231], "content": "whenever (mod 2);", "parent_index": 2, "line_index": 5}, {"bbox": [71, 229, 302, 245], "content": "and ;", "parent_index": 2, "line_index": 6}, {"bbox": [71, 244, 408, 261], "content": ", , and .", "parent_index": 2, "line_index": 7}, {"bbox": [93, 267, 541, 285], "content": "The isomorphism takes to . The isomorphism", "parent_index": 3, "line_index": 0}, {"bbox": [71, 283, 540, 300], "content": "was found in [14]; it relates , , , . The", "parent_index": 3, "line_index": 1}, {"bbox": [70, 299, 542, 315], "content": "isomorphism was found i n [34,14]; a corresponde nce which works is", "parent_index": 3, "line_index": 2}, {"bbox": [71, 314, 540, 331], "content": ", , , , , , ,", "parent_index": 3, "line_index": 3}, {"bbox": [70, 329, 201, 345], "content": "and .", "parent_index": 3, "line_index": 4}, {"bbox": [95, 345, 540, 361], "content": "We will sket ch th e proof here. The idea is to compare invariants for the various fusion", "parent_index": 4, "line_index": 0}, {"bbox": [69, 359, 541, 376], "content": "rings, case by case. For example, suppose and are isomorphic. Then their", "parent_index": 4, "line_index": 1}, {"bbox": [69, 374, 542, 390], "content": "simple-current groups and must be isomorphic (since simple-currents must get", "parent_index": 4, "line_index": 2}, {"bbox": [70, 389, 541, 404], "content": "mapped to simple-currents), so . Now compare the numbers of highest-weights:", "parent_index": 4, "line_index": 3}, {"bbox": [72, 399, 264, 420], "content": ", which forces .", "parent_index": 4, "line_index": 4}, {"bbox": [93, 418, 540, 433], "content": "It is also quite useful here to know those weights with second smallest q-dimension.", "parent_index": 5, "line_index": 0}, {"bbox": [70, 432, 541, 448], "content": "This is a by-product of the proof of Proposition 4.1, and the complete answer is given in", "parent_index": 5, "line_index": 1}, {"bbox": [69, 442, 543, 466], "content": "[18, Table 3]. Here we will simply state that those weights in with second smallest", "parent_index": 5, "line_index": 2}, {"bbox": [69, 462, 541, 479], "content": "q-dimension are precisely the orbit , except for: ; for ; ;", "parent_index": 5, "line_index": 3}, {"bbox": [71, 478, 416, 492], "content": "for ; for ; and for .", "parent_index": 5, "line_index": 4}, {"bbox": [95, 492, 540, 506], "content": "and both have two simple-currents, but their fusion rings can’t be isomorphic", "parent_index": 6, "line_index": 0}, {"bbox": [71, 506, 540, 521], "content": "(generically) because the orbit has the second smallest q-dimension for both algebras", "parent_index": 6, "line_index": 1}, {"bbox": [70, 521, 506, 535], "content": "at generic rank/level, but the numbers for the two algebras are different.", "parent_index": 6, "line_index": 2}, {"bbox": [93, 534, 542, 551], "content": "Another useful invariant involves the set of integers coprime to for which is", "parent_index": 7, "line_index": 0}, {"bbox": [70, 551, 539, 565], "content": "a simple-current. For the classical algebras this is easy to find, using (2.1c): Up to a sign,", "parent_index": 7, "line_index": 1}, {"bbox": [68, 561, 539, 583], "content": "the q-dimension of ( coprime to ) for the algebras is, respectively,", "parent_index": 7, "line_index": 2}]
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[]
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[{"bbox": [135, 120, 178, 133], "content": "\\mathcal{R}(X_{r,k})", "parent_index": 1, "subtype": "inline"}, {"bbox": [411, 119, 432, 132], "content": "\\S2.2", "parent_index": 1, "subtype": "inline"}, {"bbox": [434, 143, 540, 156], "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(Y_{s,m})", "parent_index": 2, "subtype": "inline"}, {"bbox": [101, 156, 164, 170], "content": "X_{r,k}\\neq Y_{s,m}", "parent_index": 2, "subtype": "inline"}, {"bbox": [206, 156, 240, 169], "content": "X_{r},Y_{s}", "parent_index": 2, "subtype": "inline"}, {"bbox": [164, 171, 263, 185], "content": "\\mathcal{R}(C_{r,k})\\cong\\mathcal{R}(C_{k,r})", "parent_index": 2, "subtype": "inline"}, {"bbox": [303, 171, 321, 184], "content": "r,k", "parent_index": 2, "subtype": "inline"}, {"bbox": [383, 171, 483, 185], "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 185, 284, 200], "content": "\\mathcal{R}(B_{r,1})\\cong\\mathcal{R}(A_{1,2})\\cong\\mathcal{R}(C_{2,1})\\cong\\mathcal{R}(E_{8,2})", "parent_index": 2, "subtype": "inline"}, {"bbox": [323, 186, 352, 199], "content": "r\\geq3", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 200, 182, 214], "content": "\\mathcal{R}(A_{3,1})\\cong\\mathcal{R}(D_{o d d,1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 215, 171, 229], "content": "\\mathcal{R}(D_{r,1})\\cong\\mathcal{R}(D_{s,1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [226, 216, 255, 227], "content": "r\\equiv s", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 230, 171, 244], "content": "\\mathscr{R}(A_{2,1})\\cong\\mathscr{R}(E_{6,1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [197, 229, 296, 244], "content": "\\mathcal{R}(A_{1,1})\\cong\\mathcal{R}(E_{7,1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 245, 169, 259], "content": "\\mathcal{R}(F_{4,1})\\cong\\mathcal{R}(G_{2,1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [177, 244, 275, 259], "content": "\\mathcal{R}(F_{4,2})\\cong\\mathcal{R}(E_{8,3})", "parent_index": 2, "subtype": "inline"}, {"bbox": [305, 244, 405, 259], "content": "\\mathcal{R}(F_{4,3})\\cong\\mathcal{R}(G_{2,4})", "parent_index": 2, "subtype": "inline"}, {"bbox": [190, 269, 291, 284], "content": "\\mathcal{R}(A_{1,k})\\cong\\mathcal{R}(C_{k,1})", "parent_index": 3, "subtype": "inline"}, {"bbox": [326, 269, 347, 282], "content": "a\\Lambda_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [366, 267, 381, 282], "content": "\\widetilde{\\Lambda}_{a}", "parent_index": 3, "subtype": "inline"}, {"bbox": [486, 270, 541, 284], "content": "\\mathcal{R}(F_{4,2})\\cong", "parent_index": 3, "subtype": "inline"}, {"bbox": [71, 288, 113, 300], "content": "\\mathcal{R}(E_{8,3})", "parent_index": 3, "subtype": "inline"}, {"bbox": [272, 283, 323, 299], "content": "\\Lambda_{1}\\leftrightarrow\\tilde{\\Lambda}_{8}", "parent_index": 3, "subtype": "inline"}, {"bbox": [331, 284, 388, 299], "content": "2\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [397, 285, 447, 299], "content": "\\Lambda_{3}\\,\\leftrightarrow\\,\\widetilde{\\Lambda}_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [456, 285, 506, 299], "content": "\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{7}", "parent_index": 3, "subtype": "inline"}, {"bbox": [142, 300, 244, 314], "content": "\\mathcal{R}(F_{4,3})\\,\\cong\\,\\mathcal{R}(G_{2,4})", "parent_index": 3, "subtype": "inline"}, {"bbox": [71, 315, 118, 330], "content": "\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [124, 315, 177, 329], "content": "\\Lambda_{1}\\leftrightarrow2\\widetilde{\\Lambda}_{1}", "parent_index": 3, "subtype": "inline"}, {"bbox": [184, 315, 236, 330], "content": "\\Lambda_{3}\\leftrightarrow3\\widetilde{\\Lambda}_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [243, 314, 302, 329], "content": "2\\Lambda_{4}\\leftrightarrow2\\widetilde{\\Lambda}_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [308, 315, 417, 330], "content": "\\Lambda_{1}+\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{1}+2\\widetilde{\\Lambda}_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [424, 315, 477, 330], "content": "\\Lambda_{2}\\leftrightarrow4\\widetilde{\\Lambda}_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [484, 315, 536, 329], "content": "3\\Lambda_{4}\\leftrightarrow\\widetilde{\\Lambda}_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [95, 330, 197, 344], "content": "\\Lambda_{3}+\\Lambda_{4}\\leftrightarrow\\tilde{\\Lambda}_{1}+\\tilde{\\Lambda}_{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [286, 361, 328, 374], "content": "\\mathcal{R}(A_{r,k})", "parent_index": 4, "subtype": "inline"}, {"bbox": [353, 361, 398, 374], "content": "\\mathcal{R}(A_{s,m})", "parent_index": 4, "subtype": "inline"}, {"bbox": [188, 376, 214, 388], "content": "\\mathbb{Z}_{r+1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [241, 376, 266, 388], "content": "\\mathbb{Z}_{s+1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [235, 394, 263, 399], "content": "r=s", "parent_index": 4, "subtype": "inline"}, {"bbox": [412, 390, 440, 403], "content": "||P_{+}||", "parent_index": 4, "subtype": "inline"}, {"bbox": [72, 403, 150, 418], "content": "\\big(\\begin{array}{c}{{r+k}}\\\\ {{r}}\\end{array}\\big)=\\big(\\begin{array}{c}{{r+m}}\\\\ {{r}}\\end{array}\\big)", "parent_index": 4, "subtype": "inline"}, {"bbox": [225, 405, 258, 414], "content": "m=k", "parent_index": 4, "subtype": "inline"}, {"bbox": [382, 446, 431, 463], "content": "P_{+}^{k}(X_{r}^{(1)})", "parent_index": 5, "subtype": "inline"}, {"bbox": [258, 465, 280, 475], "content": "S\\Lambda_{\\star}", "parent_index": 5, "subtype": "inline"}, {"bbox": [349, 465, 371, 477], "content": "A_{r,1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [378, 465, 401, 477], "content": "B_{r,k}", "parent_index": 5, "subtype": "inline"}, {"bbox": [424, 465, 453, 475], "content": "k\\leq3", "parent_index": 5, "subtype": "inline"}, {"bbox": [461, 465, 536, 477], "content": "C_{2,2},C_{2,3},C_{3,2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [71, 479, 94, 491], "content": "D_{r,k}", "parent_index": 5, "subtype": "inline"}, {"bbox": [116, 479, 145, 489], "content": "k\\leq2", "parent_index": 5, "subtype": "inline"}, {"bbox": [152, 479, 174, 491], "content": "E_{6,k}", "parent_index": 5, "subtype": "inline"}, {"bbox": [196, 479, 226, 489], "content": "k\\leq2", "parent_index": 5, "subtype": "inline"}, {"bbox": [255, 479, 360, 491], "content": "E_{7,k},E_{8,k},F_{4,k},G_{2,k}", "parent_index": 5, "subtype": "inline"}, {"bbox": [383, 479, 412, 489], "content": "k\\leq4", "parent_index": 5, "subtype": "inline"}, {"bbox": [95, 494, 116, 506], "content": "C_{r,k}", "parent_index": 6, "subtype": "inline"}, {"bbox": [141, 494, 167, 506], "content": "B_{s,m}", "parent_index": 6, "subtype": "inline"}, {"bbox": [232, 507, 257, 519], "content": "J^{i}\\Lambda_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [277, 521, 327, 535], "content": "Q_{j}(J^{i}\\Lambda_{1})", "parent_index": 6, "subtype": "inline"}, {"bbox": [367, 537, 373, 546], "content": "\\ell", "parent_index": 7, "subtype": "inline"}, {"bbox": [435, 537, 453, 546], "content": "\\kappa N", "parent_index": 7, "subtype": "inline"}, {"bbox": [509, 535, 527, 546], "content": "0^{(\\ell)}", "parent_index": 7, "subtype": "inline"}, {"bbox": [172, 566, 190, 577], "content": "0^{(\\ell)}", "parent_index": 7, "subtype": "inline"}, {"bbox": [199, 568, 204, 577], "content": "\\ell", "parent_index": 7, "subtype": "inline"}, {"bbox": [268, 569, 281, 577], "content": "2\\kappa", "parent_index": 7, "subtype": "inline"}, {"bbox": [375, 564, 453, 579], "content": "B_{r}^{(1)},C_{r}^{(1)},D_{r}^{(1)}", "parent_index": 7, "subtype": "inline"}, {"bbox": [145, 595, 464, 720], "content": "\\begin{array}{r l}&{\\displaystyle\\prod_{a=0}^{r-1}\\frac{\\sin(\\pi\\ell\\,(2a+1)/2\\kappa)}{\\sin(\\pi\\,(2a+1)/2\\kappa)}\\,\\prod_{b=1}^{2r-2}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b}{2}\\right]}}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{r-a}\\,\\sin(\\pi\\ell\\,(2a-1)/2\\kappa)^{r-a}}{\\sin(\\pi a/\\kappa)^{r-a}\\,\\sin(\\pi\\,(2a-1)/2\\kappa)^{r-a}}\\,\\prod_{b=r}^{2r-1}\\frac{\\sin(\\pi\\ell b/2\\kappa)}{\\sin(\\pi b/2\\kappa)}\\,\\,,}\\\\ &{\\displaystyle\\prod_{a=1}^{r-1}\\frac{\\sin(\\pi\\ell a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}{\\sin(\\pi a/\\kappa)^{\\left[\\frac{2r-a+1}{2}\\right]}}\\,\\prod_{b=r}^{2r-3}\\frac{\\sin(\\pi\\ell b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}{\\sin(\\pi b/\\kappa)^{\\left[\\frac{2r-b-1}{2}\\right]}}\\,\\,,}\\end{array}", "parent_index": 8, "subtype": "interline"}]
|
[]
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|
where $[x]$ here denotes the greatest integer not more than $x$ . The absolute value of each of these is quickly seen to be greater than 1 unless $\ell\,\equiv\,\pm1$ (mod $2\kappa$ ), except for the orthogonal algebras when $k\ \leq\ 2$ . An isomorphism $\mathscr{R}(X_{r,k})\,\cong\,\mathscr{R}(X_{r^{\prime},k^{\prime}})$ would require then that whenever $\ell\equiv\pm1$ (mod $2\kappa$ ) is coprime to $\kappa^{\prime}$ , it must also satisfy $\ell\equiv\pm1$ (mod $2\kappa^{\prime}$ ), and conversely. This forces $\kappa=\kappa^{\prime}$ , for $X=B$ or $D$ and $k>2$ , or $X=C$ and any $k$ .
If $\mathscr{R}(C_{r,k})\cong\mathscr{R}(C_{s,m})$ , then that Galois argument implies $r+k+1=s+m+1$ , so compare numbers of highest-weights: $\big(\mathbf{\Lambda}_{r}^{r+k}\big)=\big(\mathbf{\Lambda}_{s}^{r+k}\big)$ .
A similar argument works for the orthogonal algebras. For instance suppose $\mathcal{R}(B_{r,k})\cong$ $\mathcal{R}(B_{s,m})$ but $B_{r,k}\ne B_{s,m}$ , and that $k,m\,>\,2$ . Then Galois implies $2r+k\,=\,2s\,+\,m$ . Comparing the value of $\mathcal{D}(\Lambda_{1})$ (the second smallest q-dimension when $k>3$ ), using (3.2) with $\lambda=0$ , tells us that $2s+1=k,2r+1=m$ . Now count the number of fixed-points of $J$ in both cases: $\binom{\kappa/2-1}{r-1}=\binom{\kappa/2-1}{s-1}$ , i.e. $s-1=(k-1)/2$ , a contradiction.
For comparing classical algebras with exceptional algebras, a useful device is to count the number of weights appearing in the fusion $\Lambda_{\star}$ × $\Lambda_{\star}$ (when $\Lambda_{\star}$ has second smallest q-dimension). For example, for $A_{1,k}$ $\left(k>1\right)$ , $C_{r,k}$ ( $k>1$ , except for $C_{2,2},C_{2,3},C_{3,2})$ , and $E_{7,k}$ ( $k>4)$ ), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can be isomorphic.
For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively $r$ and $r-1$ for $B_{r,2}$ and $D_{r,2}$ , except for $D_{4,2}$ ).
For the exceptional algebras, comparing $\mathcal{D}(\boldsymbol{\Lambda}_{\star})$ and the number of highest-weights is effective. Recall that both $||P_{+}||$ and $\mathcal{D}(\boldsymbol{\Lambda}_{\star})$ for a fixed algebra monotonically increase with $k$ to (respectively) $\infty$ and the Weyl dimension of $\Lambda_{\star}$ , which is 7, 26, and 248 for $G_{2},F_{4},E_{8}$ respectively. For $E_{8,k}$ , $\mathcal{D}(\Lambda_{1})$ exceeds 7 for $k\geq5$ , and exceeds 26 for $k\geq11$ , while $F_{4,k}$ exceeds 7 for $k\geq4$ . The number of highest-weights of $E_{8,4},E_{8,10}$ , and $F_{4,3}$ are 10, 135, and 9, so only a small number of possibilities need be considered.
# References
1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the classification of bulk and boundary conformal field theories, Phys. Lett. B444 (1998),
163–166; J. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds, Nucl. Phys. B530 (1998), 99–136.
2. D. Bernard, String characters from Kac–Moody automorphisms, Nucl. Phys. B288 (1987), 628–648.
3. J. B¨ockenhauer and D. E. Evans, Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, preprint math.OA/9911239, 1999.
4. A. Coste and T. Gannon, Remarks on Galois in rational conformal field theories, Phys. Lett. B323 (1994), 316–321.
5. A. Coste, T. Gannon and P. Ruelle, Finite group modular data, preprint hepth/0001158, 2000.
6. Ph. Di Francesco, P. Mathieu and D. S´en´echal, “Conformal Field Theory”, Springer-Verlag, New York, 1997.
7. G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347–374. 8. M. Finkelberg, An equivalence of fusion categories, Geom. Func. Anal. 6 (1996), 249–267. 9. I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteg-de Vries type equations, in: “Lie algebras and related topics”, Lecture Notes in Math, Vol. 933, Springer-Verlag, New York, 1982.
10. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, “On axiomatic approaches to vertex operator algebras and modules”, Memoirs Amer. Math. Soc. 104 (1993).
11. J. Fr¨ohlich and T. Kerler, “Quantum groups, quantum categories and quantum field theory”, Lecture Notes in Mathematics, Vol. 1542, Springer-Verlag, Berlin, 1993.
12. J. Fuchs, Simple WZW currents, Commun. Math. Phys. 136 (1991), 345–356.
13. J. Fuchs, B. Gato-Rivera, A. N. Schellekens and C. Schweigert, Modular invariants and fusion automorphisms from Galois theory, Phys. Lett. B334 (1994), 113–120.
14. J. Fuchs and P. van Driel, WZW fusion rules, quantum groups, and the modular matrix $S$ , Nucl. Phys. B346 (1990), 632–648.
15. J. Fuchs and P. van Driel, Fusion rule engineering, Lett. Math. Phys. 23 (1991), 11–18.
16. T. Gannon, WZW commutants, lattices, and level-one partition functions, Nucl. Phys. B396 (1993), 708–736; P. Ruelle, E. Thiran and J. Weyers, Implications of an arithmetical symmetry of the commutant for modular invariants, Nucl. Phys. B402 (1993), 693–708.
17. T. Gannon, Symmetries of the Kac-Peterson modular matrices of affine algebras, Invent. math. 122 (1995), 341–357.
18. T. Gannon, Ph. Ruelle and M. A. Walton, Automorphism modular invariants of current algebras, Commun. Math. Phys. 179 (1996), 121–156.
19. G. Georgiev and O. Mathieu, Cat´egorie de fusion pour les groupes de Chevalley, C. R. Acad. Sci. Paris 315 (1992), 659–662.
20. F. M. Goodman and H. Wenzl, Littlewood-Richardson coefficients for Hecke algebras at roots of unity, Adv. Math. 82 (1990), 244–265.
21. Y.-Z. Huang and J. Lepowsky, Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke Math. J. 99 (1999), 113–134.
22. V. G. Kac, Simple Lie groups and the Legendre symbol, in: “Lie algebras, group theory, and partially ordered algebraic structures”, Lecture Notes in Math, Vol. 848, Springer-Verlag, Berlin, 1981.
23. V. G. Kac, “Infinite Dimensional Lie algebras”, 3rd edition, Cambridge University Press, Cambridge, 1990.
24. V. G. Kac and D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984), 125–264.
25. V. G. Kac and M. Wakimoto, Modular and conformal constraints in representation theory of affine algebras, Adv. Math. 70 (1988), 156–236.
26. G. Lusztig, Exotic Fourier transform, Duke Math. J. 73 (1994), 227–241.
27. I. G. Macdonald, “Symmetric functions and Hall polynomials”, 2nd edition, Oxford University Press, New York, 1995.
28. W. G. McKay, R. V. Moody and J. Patera, Decomposition of tensor products of $E_{8}$ representations, Alg., Groups Geom. 3 (1986), 286–328.
29. W. G. McKay, J. Patera and D. W. Rand, “Tables of representations of simple Lie algebras”, Vol. 1, Centre de Recherches Math´ematiques, Univ´ersit´e de Montr´eal, 1990.
30. W. Nahm, Conformal field theories, dilogarithms, and 3-dimensional manifolds, in: “Interface between physics and mathematics”, World-Scientific, 1994, (W. Nahm and J.-M. Shen, Eds.), World-Scientific, Singapore, 1994.
31. A. N. Schellekens, Fusion rule automorphisms from integer spin simple currents, Phys. Lett. B244 (1990), 255–260.
32. V. G. Turaev, “Quantum invariants of knots and 3-manifolds”, Walter de Gruyter, Berlin, 1994.
33. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. 300 (1988), 360–376.
34. D. Verstegen, New exceptional modular invariant partition functions for simple Kac–Moody algebras, Nucl. Phys. B346 (1990), 349–386.
35. M. A. Walton, Algorithm for WZW fusion rules: a proof, Phys. Lett. B241 (1990), 365–368.
36. A. J. Wassermann, Operator algebras and conformal field theory, in: “Proc. ICM, Zurich", Birkhauser, Basel, 1995.
37. E. Witten, The Verlinde formula and the cohomology of the Grassmannian, in: “Geometry, Topology and Physics”, International Press, Cambridge, MA, 1995.
|
<html><body>
<p data-bbox="71 70 541 143">where $[x]$ here denotes the greatest integer not more than $x$ . The absolute value of each of these is quickly seen to be greater than 1 unless $\ell\,\equiv\,\pm1$ (mod $2\kappa$ ), except for the orthogonal algebras when $k\ \leq\ 2$ . An isomorphism $\mathscr{R}(X_{r,k})\,\cong\,\mathscr{R}(X_{r^{\prime},k^{\prime}})$ would require then that whenever $\ell\equiv\pm1$ (mod $2\kappa$ ) is coprime to $\kappa^{\prime}$ , it must also satisfy $\ell\equiv\pm1$ (mod $2\kappa^{\prime}$ ), and conversely. This forces $\kappa=\kappa^{\prime}$ , for $X=B$ or $D$ and $k>2$ , or $X=C$ and any $k$ . </p>
<p data-bbox="70 144 541 175">If $\mathscr{R}(C_{r,k})\cong\mathscr{R}(C_{s,m})$ , then that Galois argument implies $r+k+1=s+m+1$ , so compare numbers of highest-weights: $\big(\mathbf{\Lambda}_{r}^{r+k}\big)=\big(\mathbf{\Lambda}_{s}^{r+k}\big)$ . </p>
<p data-bbox="70 176 541 248">A similar argument works for the orthogonal algebras. For instance suppose $\mathcal{R}(B_{r,k})\cong$ $\mathcal{R}(B_{s,m})$ but $B_{r,k}\ne B_{s,m}$ , and that $k,m\,>\,2$ . Then Galois implies $2r+k\,=\,2s\,+\,m$ . Comparing the value of $\mathcal{D}(\Lambda_{1})$ (the second smallest q-dimension when $k>3$ ), using (3.2) with $\lambda=0$ , tells us that $2s+1=k,2r+1=m$ . Now count the number of fixed-points of $J$ in both cases: $\binom{\kappa/2-1}{r-1}=\binom{\kappa/2-1}{s-1}$ , i.e. $s-1=(k-1)/2$ , a contradiction. </p>
<p data-bbox="71 249 541 319">For comparing classical algebras with exceptional algebras, a useful device is to count the number of weights appearing in the fusion $\Lambda_{\star}$ × $\Lambda_{\star}$ (when $\Lambda_{\star}$ has second smallest q-dimension). For example, for $A_{1,k}$ $\left(k>1\right)$ , $C_{r,k}$ ( $k>1$ , except for $C_{2,2},C_{2,3},C_{3,2})$ , and $E_{7,k}$ ( $k>4)$ ), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can be isomorphic. </p>
<p data-bbox="70 321 541 349">For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively $r$ and $r-1$ for $B_{r,2}$ and $D_{r,2}$ , except for $D_{4,2}$ ). </p>
<p data-bbox="70 351 541 436">For the exceptional algebras, comparing $\mathcal{D}(\boldsymbol{\Lambda}_{\star})$ and the number of highest-weights is effective. Recall that both $||P_{+}||$ and $\mathcal{D}(\boldsymbol{\Lambda}_{\star})$ for a fixed algebra monotonically increase with $k$ to (respectively) $\infty$ and the Weyl dimension of $\Lambda_{\star}$ , which is 7, 26, and 248 for $G_{2},F_{4},E_{8}$ respectively. For $E_{8,k}$ , $\mathcal{D}(\Lambda_{1})$ exceeds 7 for $k\geq5$ , and exceeds 26 for $k\geq11$ , while $F_{4,k}$ exceeds 7 for $k\geq4$ . The number of highest-weights of $E_{8,4},E_{8,10}$ , and $F_{4,3}$ are 10, 135, and 9, so only a small number of possibilities need be considered. </p>
<h1 data-bbox="270 455 342 468">References </h1>
<p data-bbox="78 478 541 716">1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the classification of bulk and boundary conformal field theories, Phys. Lett. B444 (1998),
163–166; J. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds, Nucl. Phys. B530 (1998), 99–136.
2. D. Bernard, String characters from Kac–Moody automorphisms, Nucl. Phys. B288 (1987), 628–648.
3. J. B¨ockenhauer and D. E. Evans, Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, preprint math.OA/9911239, 1999.
4. A. Coste and T. Gannon, Remarks on Galois in rational conformal field theories, Phys. Lett. B323 (1994), 316–321.
5. A. Coste, T. Gannon and P. Ruelle, Finite group modular data, preprint hepth/0001158, 2000.
6. Ph. Di Francesco, P. Mathieu and D. S´en´echal, “Conformal Field Theory”, Springer-Verlag, New York, 1997.
7. G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347–374. 8. M. Finkelberg, An equivalence of fusion categories, Geom. Func. Anal. 6 (1996), 249–267. 9. I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteg-de Vries type equations, in: “Lie algebras and related topics”, Lecture Notes in Math, Vol. 933, Springer-Verlag, New York, 1982.
10. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, “On axiomatic approaches to vertex operator algebras and modules”, Memoirs Amer. Math. Soc. 104 (1993).
11. J. Fr¨ohlich and T. Kerler, “Quantum groups, quantum categories and quantum field theory”, Lecture Notes in Mathematics, Vol. 1542, Springer-Verlag, Berlin, 1993.
12. J. Fuchs, Simple WZW currents, Commun. Math. Phys. 136 (1991), 345–356.
13. J. Fuchs, B. Gato-Rivera, A. N. Schellekens and C. Schweigert, Modular invariants and fusion automorphisms from Galois theory, Phys. Lett. B334 (1994), 113–120.
14. J. Fuchs and P. van Driel, WZW fusion rules, quantum groups, and the modular matrix $S$ , Nucl. Phys. B346 (1990), 632–648.
15. J. Fuchs and P. van Driel, Fusion rule engineering, Lett. Math. Phys. 23 (1991), 11–18.
16. T. Gannon, WZW commutants, lattices, and level-one partition functions, Nucl. Phys. B396 (1993), 708–736; P. Ruelle, E. Thiran and J. Weyers, Implications of an arithmetical symmetry of the commutant for modular invariants, Nucl. Phys. B402 (1993), 693–708.
17. T. Gannon, Symmetries of the Kac-Peterson modular matrices of affine algebras, Invent. math. 122 (1995), 341–357.
18. T. Gannon, Ph. Ruelle and M. A. Walton, Automorphism modular invariants of current algebras, Commun. Math. Phys. 179 (1996), 121–156.
19. G. Georgiev and O. Mathieu, Cat´egorie de fusion pour les groupes de Chevalley, C. R. Acad. Sci. Paris 315 (1992), 659–662.
20. F. M. Goodman and H. Wenzl, Littlewood-Richardson coefficients for Hecke algebras at roots of unity, Adv. Math. 82 (1990), 244–265.
21. Y.-Z. Huang and J. Lepowsky, Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke Math. J. 99 (1999), 113–134.
22. V. G. Kac, Simple Lie groups and the Legendre symbol, in: “Lie algebras, group theory, and partially ordered algebraic structures”, Lecture Notes in Math, Vol. 848, Springer-Verlag, Berlin, 1981.
23. V. G. Kac, “Infinite Dimensional Lie algebras”, 3rd edition, Cambridge University Press, Cambridge, 1990.
24. V. G. Kac and D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984), 125–264.
25. V. G. Kac and M. Wakimoto, Modular and conformal constraints in representation theory of affine algebras, Adv. Math. 70 (1988), 156–236.
26. G. Lusztig, Exotic Fourier transform, Duke Math. J. 73 (1994), 227–241.
27. I. G. Macdonald, “Symmetric functions and Hall polynomials”, 2nd edition, Oxford University Press, New York, 1995.
28. W. G. McKay, R. V. Moody and J. Patera, Decomposition of tensor products of $E_{8}$ representations, Alg., Groups Geom. 3 (1986), 286–328.
29. W. G. McKay, J. Patera and D. W. Rand, “Tables of representations of simple Lie algebras”, Vol. 1, Centre de Recherches Math´ematiques, Univ´ersit´e de Montr´eal, 1990.
30. W. Nahm, Conformal field theories, dilogarithms, and 3-dimensional manifolds, in: “Interface between physics and mathematics”, World-Scientific, 1994, (W. Nahm and J.-M. Shen, Eds.), World-Scientific, Singapore, 1994.
31. A. N. Schellekens, Fusion rule automorphisms from integer spin simple currents, Phys. Lett. B244 (1990), 255–260.
32. V. G. Turaev, “Quantum invariants of knots and 3-manifolds”, Walter de Gruyter, Berlin, 1994.
33. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. 300 (1988), 360–376.
34. D. Verstegen, New exceptional modular invariant partition functions for simple Kac–Moody algebras, Nucl. Phys. B346 (1990), 349–386.
35. M. A. Walton, Algorithm for WZW fusion rules: a proof, Phys. Lett. B241 (1990), 365–368.
36. A. J. Wassermann, Operator algebras and conformal field theory, in: “Proc. ICM, Zurich", Birkhauser, Basel, 1995.
37. E. Witten, The Verlinde formula and the cohomology of the Grassmannian, in: “Geometry, Topology and Physics”, International Press, Cambridge, MA, 1995. </p>
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0002044v1
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[{"type": "text", "text": "where $[x]$ here denotes the greatest integer not more than $x$ . The absolute value of each of these is quickly seen to be greater than 1 unless $\\ell\\,\\equiv\\,\\pm1$ (mod $2\\kappa$ ), except for the orthogonal algebras when $k\\ \\leq\\ 2$ . An isomorphism $\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})$ would require then that whenever $\\ell\\equiv\\pm1$ (mod $2\\kappa$ ) is coprime to $\\kappa^{\\prime}$ , it must also satisfy $\\ell\\equiv\\pm1$ (mod $2\\kappa^{\\prime}$ ), and conversely. This forces $\\kappa=\\kappa^{\\prime}$ , for $X=B$ or $D$ and $k>2$ , or $X=C$ and any $k$ . ", "page_idx": 21}, {"type": "text", "text": "If $\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})$ , then that Galois argument implies $r+k+1=s+m+1$ , so compare numbers of highest-weights: $\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)$ . ", "page_idx": 21}, {"type": "text", "text": "A similar argument works for the orthogonal algebras. For instance suppose $\\mathcal{R}(B_{r,k})\\cong$ $\\mathcal{R}(B_{s,m})$ but $B_{r,k}\\ne B_{s,m}$ , and that $k,m\\,>\\,2$ . Then Galois implies $2r+k\\,=\\,2s\\,+\\,m$ . Comparing the value of $\\mathcal{D}(\\Lambda_{1})$ (the second smallest q-dimension when $k>3$ ), using (3.2) with $\\lambda=0$ , tells us that $2s+1=k,2r+1=m$ . Now count the number of fixed-points of $J$ in both cases: $\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}$ , i.e. $s-1=(k-1)/2$ , a contradiction. ", "page_idx": 21}, {"type": "text", "text": "For comparing classical algebras with exceptional algebras, a useful device is to count the number of weights appearing in the fusion $\\Lambda_{\\star}$ × $\\Lambda_{\\star}$ (when $\\Lambda_{\\star}$ has second smallest q-dimension). For example, for $A_{1,k}$ $\\left(k>1\\right)$ , $C_{r,k}$ ( $k>1$ , except for $C_{2,2},C_{2,3},C_{3,2})$ , and $E_{7,k}$ ( $k>4)$ ), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can be isomorphic. ", "page_idx": 21}, {"type": "text", "text": "For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively $r$ and $r-1$ for $B_{r,2}$ and $D_{r,2}$ , except for $D_{4,2}$ ). ", "page_idx": 21}, {"type": "text", "text": "For the exceptional algebras, comparing $\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})$ and the number of highest-weights is effective. Recall that both $||P_{+}||$ and $\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})$ for a fixed algebra monotonically increase with $k$ to (respectively) $\\infty$ and the Weyl dimension of $\\Lambda_{\\star}$ , which is 7, 26, and 248 for $G_{2},F_{4},E_{8}$ respectively. For $E_{8,k}$ , $\\mathcal{D}(\\Lambda_{1})$ exceeds 7 for $k\\geq5$ , and exceeds 26 for $k\\geq11$ , while $F_{4,k}$ exceeds 7 for $k\\geq4$ . The number of highest-weights of $E_{8,4},E_{8,10}$ , and $F_{4,3}$ are 10, 135, and 9, so only a small number of possibilities need be considered. ", "page_idx": 21}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 21}, {"type": "text", "text": "1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the classification of bulk and boundary conformal field theories, Phys. Lett. B444 (1998), \n163–166; J. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds, Nucl. Phys. B530 (1998), 99–136. \n2. D. Bernard, String characters from Kac–Moody automorphisms, Nucl. Phys. B288 (1987), 628–648. \n3. J. B¨ockenhauer and D. E. Evans, Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors, preprint math.OA/9911239, 1999. \n4. A. Coste and T. Gannon, Remarks on Galois in rational conformal field theories, Phys. Lett. B323 (1994), 316–321. \n5. A. Coste, T. Gannon and P. Ruelle, Finite group modular data, preprint hepth/0001158, 2000. \n6. Ph. Di Francesco, P. Mathieu and D. S´en´echal, “Conformal Field Theory”, Springer-Verlag, New York, 1997. \n7. G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347–374. 8. M. Finkelberg, An equivalence of fusion categories, Geom. Func. Anal. 6 (1996), 249–267. 9. I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteg-de Vries type equations, in: “Lie algebras and related topics”, Lecture Notes in Math, Vol. 933, Springer-Verlag, New York, 1982. \n10. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, “On axiomatic approaches to vertex operator algebras and modules”, Memoirs Amer. Math. Soc. 104 (1993). \n11. J. Fr¨ohlich and T. 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{"preproc_blocks": [{"type": "text", "bbox": [71, 70, 541, 143], "lines": [{"bbox": [71, 73, 541, 89], "spans": [{"bbox": [71, 73, 106, 89], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 75, 120, 87], "score": 0.9, "content": "[x]", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [120, 73, 381, 89], "score": 1.0, "content": " here denotes the greatest integer not more than ", "type": "text"}, {"bbox": [381, 78, 388, 84], "score": 0.89, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [389, 73, 541, 89], "score": 1.0, "content": ". The absolute value of each", "type": "text"}], "index": 0}, {"bbox": [70, 88, 541, 104], "spans": [{"bbox": [70, 88, 356, 104], "score": 1.0, "content": "of these is quickly seen to be greater than 1 unless ", "type": "text"}, {"bbox": [357, 90, 398, 100], "score": 0.91, "content": "\\ell\\,\\equiv\\,\\pm1", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [399, 88, 437, 104], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [437, 90, 450, 99], "score": 0.72, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [451, 88, 541, 104], "score": 1.0, "content": "), except for the", "type": "text"}], "index": 1}, {"bbox": [71, 102, 541, 118], "spans": [{"bbox": [71, 102, 212, 118], "score": 1.0, "content": "orthogonal algebras when ", "type": "text"}, {"bbox": [213, 104, 246, 115], "score": 0.92, "content": "k\\ \\leq\\ 2", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [246, 102, 349, 118], "score": 1.0, "content": ". An isomorphism ", "type": "text"}, {"bbox": [350, 104, 462, 116], "score": 0.94, "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})", "type": "inline_equation", "height": 12, "width": 112}, {"bbox": [462, 102, 541, 118], "score": 1.0, "content": " would require", "type": "text"}], "index": 2}, {"bbox": [71, 117, 540, 131], "spans": [{"bbox": [71, 117, 177, 131], "score": 1.0, "content": "then that whenever ", "type": "text"}, {"bbox": [178, 118, 216, 128], "score": 0.9, "content": "\\ell\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [216, 117, 252, 131], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [252, 119, 266, 127], "score": 0.66, "content": "2\\kappa", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [266, 117, 347, 131], "score": 1.0, "content": ") is coprime to ", "type": "text"}, {"bbox": [347, 118, 357, 127], "score": 0.89, "content": "\\kappa^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 117, 469, 131], "score": 1.0, "content": ", it must also satisfy ", "type": "text"}, {"bbox": [470, 118, 507, 128], "score": 0.91, "content": "\\ell\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [508, 117, 540, 131], "score": 1.0, "content": " (mod", "type": "text"}], "index": 3}, {"bbox": [71, 131, 540, 145], "spans": [{"bbox": [71, 132, 88, 142], "score": 0.77, "content": "2\\kappa^{\\prime}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [88, 131, 243, 145], "score": 1.0, "content": "), and conversely. This forces ", "type": "text"}, {"bbox": [244, 132, 277, 142], "score": 0.91, "content": "\\kappa=\\kappa^{\\prime}", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [277, 131, 302, 145], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [302, 133, 339, 142], "score": 0.92, "content": "X=B", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [339, 131, 356, 145], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [357, 133, 367, 142], "score": 0.91, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 131, 393, 145], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 133, 423, 142], "score": 0.91, "content": "k>2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [423, 131, 443, 145], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [444, 133, 480, 142], "score": 0.92, "content": "X=C", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [480, 131, 529, 145], "score": 1.0, "content": " and any ", "type": "text"}, {"bbox": [529, 133, 536, 142], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [537, 131, 540, 145], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [70, 144, 541, 175], "lines": [{"bbox": [94, 146, 542, 163], "spans": [{"bbox": [94, 146, 107, 163], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [107, 147, 210, 160], "score": 0.93, "content": "\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [210, 146, 403, 163], "score": 1.0, "content": ", then that Galois argument implies ", "type": "text"}, {"bbox": [403, 148, 521, 158], "score": 0.92, "content": "r+k+1=s+m+1", "type": "inline_equation", "height": 10, "width": 118}, {"bbox": [522, 146, 542, 163], "score": 1.0, "content": ", so", "type": "text"}], "index": 5}, {"bbox": [70, 160, 349, 179], "spans": [{"bbox": [70, 160, 269, 179], "score": 1.0, "content": "compare numbers of highest-weights: ", "type": "text"}, {"bbox": [270, 161, 345, 177], "score": 0.96, "content": "\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [345, 160, 349, 179], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [70, 176, 541, 248], "lines": [{"bbox": [94, 176, 542, 193], "spans": [{"bbox": [94, 176, 485, 193], "score": 1.0, "content": "A similar argument works for the orthogonal algebras. For instance suppose ", "type": "text"}, {"bbox": [485, 178, 542, 191], "score": 0.93, "content": "\\mathcal{R}(B_{r,k})\\cong", "type": "inline_equation", "height": 13, "width": 57}], "index": 7}, {"bbox": [71, 191, 541, 207], "spans": [{"bbox": [71, 193, 117, 205], "score": 0.93, "content": "\\mathcal{R}(B_{s,m})", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [117, 191, 144, 207], "score": 1.0, "content": " but ", "type": "text"}, {"bbox": [144, 194, 212, 205], "score": 0.93, "content": "B_{r,k}\\ne B_{s,m}", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [212, 191, 271, 207], "score": 1.0, "content": ", and that ", "type": "text"}, {"bbox": [271, 193, 320, 205], "score": 0.89, "content": "k,m\\,>\\,2", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [321, 191, 443, 207], "score": 1.0, "content": ". Then Galois implies ", "type": "text"}, {"bbox": [444, 194, 536, 204], "score": 0.91, "content": "2r+k\\,=\\,2s\\,+\\,m", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [536, 191, 541, 207], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [72, 207, 541, 221], "spans": [{"bbox": [72, 207, 198, 221], "score": 1.0, "content": "Comparing the value of ", "type": "text"}, {"bbox": [198, 207, 231, 220], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [231, 207, 442, 221], "score": 1.0, "content": " (the second smallest q-dimension when ", "type": "text"}, {"bbox": [443, 208, 472, 217], "score": 0.9, "content": "k>3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [472, 207, 541, 221], "score": 1.0, "content": "), using (3.2)", "type": "text"}], "index": 9}, {"bbox": [71, 221, 543, 235], "spans": [{"bbox": [71, 221, 98, 235], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [98, 222, 127, 231], "score": 0.9, "content": "\\lambda=0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [128, 221, 200, 235], "score": 1.0, "content": ", tells us that ", "type": "text"}, {"bbox": [201, 222, 320, 233], "score": 0.47, "content": "2s+1=k,2r+1=m", "type": "inline_equation", "height": 11, "width": 119}, {"bbox": [320, 221, 543, 235], "score": 1.0, "content": ". Now count the number of fixed-points of", "type": "text"}], "index": 10}, {"bbox": [71, 227, 467, 258], "spans": [{"bbox": [71, 237, 79, 246], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [80, 227, 159, 258], "score": 1.0, "content": " in both cases: ", "type": "text"}, {"bbox": [160, 234, 255, 250], "score": 0.93, "content": "\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}", "type": "inline_equation", "height": 16, "width": 95}, {"bbox": [255, 227, 281, 258], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [281, 235, 372, 249], "score": 0.92, "content": "s-1=(k-1)/2", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [373, 227, 467, 258], "score": 1.0, "content": ", a contradiction.", "type": "text"}], "index": 11}], "index": 9}, {"type": "text", "bbox": [71, 249, 541, 319], "lines": [{"bbox": [93, 248, 542, 266], "spans": [{"bbox": [93, 248, 542, 266], "score": 1.0, "content": "For comparing classical algebras with exceptional algebras, a useful device is to count", "type": "text"}], "index": 12}, {"bbox": [70, 264, 541, 280], "spans": [{"bbox": [70, 264, 326, 280], "score": 1.0, "content": "the number of weights appearing in the fusion ", "type": "text"}, {"bbox": [326, 264, 341, 277], "score": 0.77, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [342, 264, 358, 280], "score": 1.0, "content": "× ", "type": "text"}, {"bbox": [358, 264, 373, 277], "score": 0.73, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [374, 264, 415, 280], "score": 1.0, "content": "(when ", "type": "text"}, {"bbox": [415, 264, 429, 277], "score": 0.86, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [430, 264, 541, 280], "score": 1.0, "content": "has second smallest", "type": "text"}], "index": 13}, {"bbox": [70, 278, 542, 294], "spans": [{"bbox": [70, 278, 237, 294], "score": 1.0, "content": "q-dimension). For example, for ", "type": "text"}, {"bbox": [238, 280, 261, 293], "score": 0.84, "content": "A_{1,k}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [261, 278, 267, 294], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [267, 278, 302, 292], "score": 0.43, "content": "\\left(k>1\\right)", "type": "inline_equation", "height": 14, "width": 35}, {"bbox": [302, 278, 309, 294], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [309, 278, 332, 293], "score": 0.69, "content": "C_{r,k}", "type": "inline_equation", "height": 15, "width": 23}, {"bbox": [333, 278, 339, 294], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [340, 279, 370, 291], "score": 0.82, "content": "k>1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [370, 278, 433, 294], "score": 1.0, "content": ", except for ", "type": "text"}, {"bbox": [433, 280, 513, 293], "score": 0.92, "content": "C_{2,2},C_{2,3},C_{3,2})", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [513, 278, 542, 294], "score": 1.0, "content": ", and", "type": "text"}], "index": 14}, {"bbox": [71, 294, 541, 308], "spans": [{"bbox": [71, 295, 93, 307], "score": 0.86, "content": "E_{7,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [94, 294, 101, 308], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [101, 294, 133, 306], "score": 0.51, "content": "k>4)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [133, 294, 541, 308], "score": 1.0, "content": "), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can", "type": "text"}], "index": 15}, {"bbox": [70, 308, 148, 322], "spans": [{"bbox": [70, 308, 148, 322], "score": 1.0, "content": "be isomorphic.", "type": "text"}], "index": 16}], "index": 14}, {"type": "text", "bbox": [70, 321, 541, 349], "lines": [{"bbox": [95, 323, 541, 337], "spans": [{"bbox": [95, 323, 541, 337], "score": 1.0, "content": "For the orthogonal algebras at level 2, useful is the number of weights with second", "type": "text"}], "index": 17}, {"bbox": [70, 337, 502, 353], "spans": [{"bbox": [70, 338, 254, 353], "score": 1.0, "content": "smallest q-dimension (respectively ", "type": "text"}, {"bbox": [254, 342, 260, 348], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [260, 338, 286, 353], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [286, 337, 313, 349], "score": 0.86, "content": "r-1", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [314, 338, 335, 353], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [335, 338, 357, 351], "score": 0.91, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [357, 338, 383, 353], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [384, 339, 407, 351], "score": 0.92, "content": "D_{r,2}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [407, 338, 469, 353], "score": 1.0, "content": ", except for ", "type": "text"}, {"bbox": [470, 339, 492, 351], "score": 0.92, "content": "D_{4,2}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [493, 338, 502, 353], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "text", "bbox": [70, 351, 541, 436], "lines": [{"bbox": [93, 352, 541, 367], "spans": [{"bbox": [93, 352, 308, 367], "score": 1.0, "content": "For the exceptional algebras, comparing ", "type": "text"}, {"bbox": [309, 352, 342, 366], "score": 0.92, "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [342, 352, 541, 367], "score": 1.0, "content": " and the number of highest-weights is", "type": "text"}], "index": 19}, {"bbox": [70, 365, 541, 383], "spans": [{"bbox": [70, 365, 210, 383], "score": 1.0, "content": "effective. Recall that both ", "type": "text"}, {"bbox": [210, 368, 237, 380], "score": 0.94, "content": "||P_{+}||", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [237, 365, 263, 383], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [263, 368, 296, 380], "score": 0.94, "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [296, 365, 541, 383], "score": 1.0, "content": " for a fixed algebra monotonically increase with", "type": "text"}], "index": 20}, {"bbox": [71, 381, 540, 396], "spans": [{"bbox": [71, 383, 78, 392], "score": 0.86, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 381, 169, 396], "score": 1.0, "content": " to (respectively) ", "type": "text"}, {"bbox": [170, 386, 182, 392], "score": 0.86, "content": "\\infty", "type": "inline_equation", "height": 6, "width": 12}, {"bbox": [182, 381, 326, 396], "score": 1.0, "content": "and the Weyl dimension of ", "type": "text"}, {"bbox": [327, 383, 341, 393], "score": 0.91, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [341, 381, 486, 396], "score": 1.0, "content": ", which is 7, 26, and 248 for ", "type": "text"}, {"bbox": [487, 383, 540, 394], "score": 0.93, "content": "G_{2},F_{4},E_{8}", "type": "inline_equation", "height": 11, "width": 53}], "index": 21}, {"bbox": [70, 396, 539, 411], "spans": [{"bbox": [70, 396, 163, 411], "score": 1.0, "content": "respectively. For ", "type": "text"}, {"bbox": [163, 397, 186, 410], "score": 0.92, "content": "E_{8,k}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [186, 396, 193, 411], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [194, 397, 226, 409], "score": 0.93, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [226, 396, 302, 411], "score": 1.0, "content": " exceeds 7 for ", "type": "text"}, {"bbox": [303, 397, 333, 408], "score": 0.91, "content": "k\\geq5", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [333, 396, 442, 411], "score": 1.0, "content": ", and exceeds 26 for ", "type": "text"}, {"bbox": [442, 397, 479, 408], "score": 0.88, "content": "k\\geq11", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [479, 396, 518, 411], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [518, 397, 539, 409], "score": 0.92, "content": "F_{4,k}", "type": "inline_equation", "height": 12, "width": 21}], "index": 22}, {"bbox": [70, 409, 541, 425], "spans": [{"bbox": [70, 409, 144, 425], "score": 1.0, "content": "exceeds 7 for ", "type": "text"}, {"bbox": [144, 412, 175, 422], "score": 0.92, "content": "k\\geq4", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [175, 409, 366, 425], "score": 1.0, "content": ". The number of highest-weights of ", "type": "text"}, {"bbox": [367, 411, 421, 424], "score": 0.92, "content": "E_{8,4},E_{8,10}", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [421, 409, 452, 425], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [452, 412, 473, 424], "score": 0.92, "content": "F_{4,3}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [473, 409, 541, 425], "score": 1.0, "content": " are 10, 135,", "type": "text"}], "index": 23}, {"bbox": [70, 424, 415, 438], "spans": [{"bbox": [70, 424, 415, 438], "score": 1.0, "content": "and 9, so only a small number of possibilities need be considered.", "type": "text"}], "index": 24}], "index": 21.5}, {"type": "title", "bbox": [270, 455, 342, 468], "lines": [{"bbox": [270, 457, 342, 469], "spans": [{"bbox": [270, 457, 342, 469], "score": 1.0, "content": "References", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [78, 478, 541, 716], "lines": [{"bbox": [79, 481, 539, 495], "spans": [{"bbox": [79, 481, 539, 495], "score": 1.0, "content": "1. 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Geom. 3 (1994), 347–374.", "type": "text"}], "index": 41}], "index": 33.5}], "layout_bboxes": [], "page_idx": 21, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 731, 312, 742], "lines": [{"bbox": [298, 731, 314, 744], "spans": [{"bbox": [298, 731, 314, 744], "score": 1.0, "content": "22", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [71, 70, 541, 143], "lines": [{"bbox": [71, 73, 541, 89], "spans": [{"bbox": [71, 73, 106, 89], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 75, 120, 87], "score": 0.9, "content": "[x]", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [120, 73, 381, 89], "score": 1.0, "content": " here denotes the greatest integer not more than ", "type": "text"}, {"bbox": [381, 78, 388, 84], "score": 0.89, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [389, 73, 541, 89], "score": 1.0, "content": ". The absolute value of each", "type": "text"}], "index": 0}, {"bbox": [70, 88, 541, 104], "spans": [{"bbox": [70, 88, 356, 104], "score": 1.0, "content": "of these is quickly seen to be greater than 1 unless ", "type": "text"}, {"bbox": [357, 90, 398, 100], "score": 0.91, "content": "\\ell\\,\\equiv\\,\\pm1", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [399, 88, 437, 104], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [437, 90, 450, 99], "score": 0.72, "content": "2\\kappa", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [451, 88, 541, 104], "score": 1.0, "content": "), except for the", "type": "text"}], "index": 1}, {"bbox": [71, 102, 541, 118], "spans": [{"bbox": [71, 102, 212, 118], "score": 1.0, "content": "orthogonal algebras when ", "type": "text"}, {"bbox": [213, 104, 246, 115], "score": 0.92, "content": "k\\ \\leq\\ 2", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [246, 102, 349, 118], "score": 1.0, "content": ". An isomorphism ", "type": "text"}, {"bbox": [350, 104, 462, 116], "score": 0.94, "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})", "type": "inline_equation", "height": 12, "width": 112}, {"bbox": [462, 102, 541, 118], "score": 1.0, "content": " would require", "type": "text"}], "index": 2}, {"bbox": [71, 117, 540, 131], "spans": [{"bbox": [71, 117, 177, 131], "score": 1.0, "content": "then that whenever ", "type": "text"}, {"bbox": [178, 118, 216, 128], "score": 0.9, "content": "\\ell\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [216, 117, 252, 131], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [252, 119, 266, 127], "score": 0.66, "content": "2\\kappa", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [266, 117, 347, 131], "score": 1.0, "content": ") is coprime to ", "type": "text"}, {"bbox": [347, 118, 357, 127], "score": 0.89, "content": "\\kappa^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 117, 469, 131], "score": 1.0, "content": ", it must also satisfy ", "type": "text"}, {"bbox": [470, 118, 507, 128], "score": 0.91, "content": "\\ell\\equiv\\pm1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [508, 117, 540, 131], "score": 1.0, "content": " (mod", "type": "text"}], "index": 3}, {"bbox": [71, 131, 540, 145], "spans": [{"bbox": [71, 132, 88, 142], "score": 0.77, "content": "2\\kappa^{\\prime}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [88, 131, 243, 145], "score": 1.0, "content": "), and conversely. This forces ", "type": "text"}, {"bbox": [244, 132, 277, 142], "score": 0.91, "content": "\\kappa=\\kappa^{\\prime}", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [277, 131, 302, 145], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [302, 133, 339, 142], "score": 0.92, "content": "X=B", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [339, 131, 356, 145], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [357, 133, 367, 142], "score": 0.91, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 131, 393, 145], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 133, 423, 142], "score": 0.91, "content": "k>2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [423, 131, 443, 145], "score": 1.0, "content": ", or ", "type": "text"}, {"bbox": [444, 133, 480, 142], "score": 0.92, "content": "X=C", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [480, 131, 529, 145], "score": 1.0, "content": " and any ", "type": "text"}, {"bbox": [529, 133, 536, 142], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [537, 131, 540, 145], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 2, "bbox_fs": [70, 73, 541, 145]}, {"type": "text", "bbox": [70, 144, 541, 175], "lines": [{"bbox": [94, 146, 542, 163], "spans": [{"bbox": [94, 146, 107, 163], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [107, 147, 210, 160], "score": 0.93, "content": "\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [210, 146, 403, 163], "score": 1.0, "content": ", then that Galois argument implies ", "type": "text"}, {"bbox": [403, 148, 521, 158], "score": 0.92, "content": "r+k+1=s+m+1", "type": "inline_equation", "height": 10, "width": 118}, {"bbox": [522, 146, 542, 163], "score": 1.0, "content": ", so", "type": "text"}], "index": 5}, {"bbox": [70, 160, 349, 179], "spans": [{"bbox": [70, 160, 269, 179], "score": 1.0, "content": "compare numbers of highest-weights: ", "type": "text"}, {"bbox": [270, 161, 345, 177], "score": 0.96, "content": "\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [345, 160, 349, 179], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [70, 146, 542, 179]}, {"type": "text", "bbox": [70, 176, 541, 248], "lines": [{"bbox": [94, 176, 542, 193], "spans": [{"bbox": [94, 176, 485, 193], "score": 1.0, "content": "A similar argument works for the orthogonal algebras. For instance suppose ", "type": "text"}, {"bbox": [485, 178, 542, 191], "score": 0.93, "content": "\\mathcal{R}(B_{r,k})\\cong", "type": "inline_equation", "height": 13, "width": 57}], "index": 7}, {"bbox": [71, 191, 541, 207], "spans": [{"bbox": [71, 193, 117, 205], "score": 0.93, "content": "\\mathcal{R}(B_{s,m})", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [117, 191, 144, 207], "score": 1.0, "content": " but ", "type": "text"}, {"bbox": [144, 194, 212, 205], "score": 0.93, "content": "B_{r,k}\\ne B_{s,m}", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [212, 191, 271, 207], "score": 1.0, "content": ", and that ", "type": "text"}, {"bbox": [271, 193, 320, 205], "score": 0.89, "content": "k,m\\,>\\,2", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [321, 191, 443, 207], "score": 1.0, "content": ". Then Galois implies ", "type": "text"}, {"bbox": [444, 194, 536, 204], "score": 0.91, "content": "2r+k\\,=\\,2s\\,+\\,m", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [536, 191, 541, 207], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [72, 207, 541, 221], "spans": [{"bbox": [72, 207, 198, 221], "score": 1.0, "content": "Comparing the value of ", "type": "text"}, {"bbox": [198, 207, 231, 220], "score": 0.94, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [231, 207, 442, 221], "score": 1.0, "content": " (the second smallest q-dimension when ", "type": "text"}, {"bbox": [443, 208, 472, 217], "score": 0.9, "content": "k>3", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [472, 207, 541, 221], "score": 1.0, "content": "), using (3.2)", "type": "text"}], "index": 9}, {"bbox": [71, 221, 543, 235], "spans": [{"bbox": [71, 221, 98, 235], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [98, 222, 127, 231], "score": 0.9, "content": "\\lambda=0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [128, 221, 200, 235], "score": 1.0, "content": ", tells us that ", "type": "text"}, {"bbox": [201, 222, 320, 233], "score": 0.47, "content": "2s+1=k,2r+1=m", "type": "inline_equation", "height": 11, "width": 119}, {"bbox": [320, 221, 543, 235], "score": 1.0, "content": ". Now count the number of fixed-points of", "type": "text"}], "index": 10}, {"bbox": [71, 227, 467, 258], "spans": [{"bbox": [71, 237, 79, 246], "score": 0.89, "content": "J", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [80, 227, 159, 258], "score": 1.0, "content": " in both cases: ", "type": "text"}, {"bbox": [160, 234, 255, 250], "score": 0.93, "content": "\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}", "type": "inline_equation", "height": 16, "width": 95}, {"bbox": [255, 227, 281, 258], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [281, 235, 372, 249], "score": 0.92, "content": "s-1=(k-1)/2", "type": "inline_equation", "height": 14, "width": 91}, {"bbox": [373, 227, 467, 258], "score": 1.0, "content": ", a contradiction.", "type": "text"}], "index": 11}], "index": 9, "bbox_fs": [71, 176, 543, 258]}, {"type": "text", "bbox": [71, 249, 541, 319], "lines": [{"bbox": [93, 248, 542, 266], "spans": [{"bbox": [93, 248, 542, 266], "score": 1.0, "content": "For comparing classical algebras with exceptional algebras, a useful device is to count", "type": "text"}], "index": 12}, {"bbox": [70, 264, 541, 280], "spans": [{"bbox": [70, 264, 326, 280], "score": 1.0, "content": "the number of weights appearing in the fusion ", "type": "text"}, {"bbox": [326, 264, 341, 277], "score": 0.77, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [342, 264, 358, 280], "score": 1.0, "content": "× ", "type": "text"}, {"bbox": [358, 264, 373, 277], "score": 0.73, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [374, 264, 415, 280], "score": 1.0, "content": "(when ", "type": "text"}, {"bbox": [415, 264, 429, 277], "score": 0.86, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [430, 264, 541, 280], "score": 1.0, "content": "has second smallest", "type": "text"}], "index": 13}, {"bbox": [70, 278, 542, 294], "spans": [{"bbox": [70, 278, 237, 294], "score": 1.0, "content": "q-dimension). For example, for ", "type": "text"}, {"bbox": [238, 280, 261, 293], "score": 0.84, "content": "A_{1,k}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [261, 278, 267, 294], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [267, 278, 302, 292], "score": 0.43, "content": "\\left(k>1\\right)", "type": "inline_equation", "height": 14, "width": 35}, {"bbox": [302, 278, 309, 294], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [309, 278, 332, 293], "score": 0.69, "content": "C_{r,k}", "type": "inline_equation", "height": 15, "width": 23}, {"bbox": [333, 278, 339, 294], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [340, 279, 370, 291], "score": 0.82, "content": "k>1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [370, 278, 433, 294], "score": 1.0, "content": ", except for ", "type": "text"}, {"bbox": [433, 280, 513, 293], "score": 0.92, "content": "C_{2,2},C_{2,3},C_{3,2})", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [513, 278, 542, 294], "score": 1.0, "content": ", and", "type": "text"}], "index": 14}, {"bbox": [71, 294, 541, 308], "spans": [{"bbox": [71, 295, 93, 307], "score": 0.86, "content": "E_{7,k}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [94, 294, 101, 308], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [101, 294, 133, 306], "score": 0.51, "content": "k>4)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [133, 294, 541, 308], "score": 1.0, "content": "), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can", "type": "text"}], "index": 15}, {"bbox": [70, 308, 148, 322], "spans": [{"bbox": [70, 308, 148, 322], "score": 1.0, "content": "be isomorphic.", "type": "text"}], "index": 16}], "index": 14, "bbox_fs": [70, 248, 542, 322]}, {"type": "text", "bbox": [70, 321, 541, 349], "lines": [{"bbox": [95, 323, 541, 337], "spans": [{"bbox": [95, 323, 541, 337], "score": 1.0, "content": "For the orthogonal algebras at level 2, useful is the number of weights with second", "type": "text"}], "index": 17}, {"bbox": [70, 337, 502, 353], "spans": [{"bbox": [70, 338, 254, 353], "score": 1.0, "content": "smallest q-dimension (respectively ", "type": "text"}, {"bbox": [254, 342, 260, 348], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [260, 338, 286, 353], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [286, 337, 313, 349], "score": 0.86, "content": "r-1", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [314, 338, 335, 353], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [335, 338, 357, 351], "score": 0.91, "content": "B_{r,2}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [357, 338, 383, 353], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [384, 339, 407, 351], "score": 0.92, "content": "D_{r,2}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [407, 338, 469, 353], "score": 1.0, "content": ", except for ", "type": "text"}, {"bbox": [470, 339, 492, 351], "score": 0.92, "content": "D_{4,2}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [493, 338, 502, 353], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 17.5, "bbox_fs": [70, 323, 541, 353]}, {"type": "text", "bbox": [70, 351, 541, 436], "lines": [{"bbox": [93, 352, 541, 367], "spans": [{"bbox": [93, 352, 308, 367], "score": 1.0, "content": "For the exceptional algebras, comparing ", "type": "text"}, {"bbox": [309, 352, 342, 366], "score": 0.92, "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [342, 352, 541, 367], "score": 1.0, "content": " and the number of highest-weights is", "type": "text"}], "index": 19}, {"bbox": [70, 365, 541, 383], "spans": [{"bbox": [70, 365, 210, 383], "score": 1.0, "content": "effective. Recall that both ", "type": "text"}, {"bbox": [210, 368, 237, 380], "score": 0.94, "content": "||P_{+}||", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [237, 365, 263, 383], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [263, 368, 296, 380], "score": 0.94, "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [296, 365, 541, 383], "score": 1.0, "content": " for a fixed algebra monotonically increase with", "type": "text"}], "index": 20}, {"bbox": [71, 381, 540, 396], "spans": [{"bbox": [71, 383, 78, 392], "score": 0.86, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [78, 381, 169, 396], "score": 1.0, "content": " to (respectively) ", "type": "text"}, {"bbox": [170, 386, 182, 392], "score": 0.86, "content": "\\infty", "type": "inline_equation", "height": 6, "width": 12}, {"bbox": [182, 381, 326, 396], "score": 1.0, "content": "and the Weyl dimension of ", "type": "text"}, {"bbox": [327, 383, 341, 393], "score": 0.91, "content": "\\Lambda_{\\star}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [341, 381, 486, 396], "score": 1.0, "content": ", which is 7, 26, and 248 for ", "type": "text"}, {"bbox": [487, 383, 540, 394], "score": 0.93, "content": "G_{2},F_{4},E_{8}", "type": "inline_equation", "height": 11, "width": 53}], "index": 21}, {"bbox": [70, 396, 539, 411], "spans": [{"bbox": [70, 396, 163, 411], "score": 1.0, "content": "respectively. For ", "type": "text"}, {"bbox": [163, 397, 186, 410], "score": 0.92, "content": "E_{8,k}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [186, 396, 193, 411], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [194, 397, 226, 409], "score": 0.93, "content": "\\mathcal{D}(\\Lambda_{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [226, 396, 302, 411], "score": 1.0, "content": " exceeds 7 for ", "type": "text"}, {"bbox": [303, 397, 333, 408], "score": 0.91, "content": "k\\geq5", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [333, 396, 442, 411], "score": 1.0, "content": ", and exceeds 26 for ", "type": "text"}, {"bbox": [442, 397, 479, 408], "score": 0.88, "content": "k\\geq11", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [479, 396, 518, 411], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [518, 397, 539, 409], "score": 0.92, "content": "F_{4,k}", "type": "inline_equation", "height": 12, "width": 21}], "index": 22}, {"bbox": [70, 409, 541, 425], "spans": [{"bbox": [70, 409, 144, 425], "score": 1.0, "content": "exceeds 7 for ", "type": "text"}, {"bbox": [144, 412, 175, 422], "score": 0.92, "content": "k\\geq4", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [175, 409, 366, 425], "score": 1.0, "content": ". The number of highest-weights of ", "type": "text"}, {"bbox": [367, 411, 421, 424], "score": 0.92, "content": "E_{8,4},E_{8,10}", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [421, 409, 452, 425], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [452, 412, 473, 424], "score": 0.92, "content": "F_{4,3}", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [473, 409, 541, 425], "score": 1.0, "content": " are 10, 135,", "type": "text"}], "index": 23}, {"bbox": [70, 424, 415, 438], "spans": [{"bbox": [70, 424, 415, 438], "score": 1.0, "content": "and 9, so only a small number of possibilities need be considered.", "type": "text"}], "index": 24}], "index": 21.5, "bbox_fs": [70, 352, 541, 438]}, {"type": "title", "bbox": [270, 455, 342, 468], "lines": [{"bbox": [270, 457, 342, 469], "spans": [{"bbox": [270, 457, 342, 469], "score": 1.0, "content": "References", "type": "text"}], "index": 25}], "index": 25}, {"type": "list", "bbox": [78, 478, 541, 716], "lines": [{"bbox": [79, 481, 539, 495], "spans": [{"bbox": [79, 481, 539, 495], "score": 1.0, "content": "1. 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[{"type": "text", "bbox": [71, 70, 541, 143], "content": "where here denotes the greatest integer not more than . The absolute value of each of these is quickly seen to be greater than 1 unless (mod ), except for the orthogonal algebras when . An isomorphism would require then that whenever (mod ) is coprime to , it must also satisfy (mod ), and conversely. This forces , for or and , or and any .", "index": 0}, {"type": "text", "bbox": [70, 144, 541, 175], "content": "If , then that Galois argument implies , so compare numbers of highest-weights: .", "index": 1}, {"type": "text", "bbox": [70, 176, 541, 248], "content": "A similar argument works for the orthogonal algebras. For instance suppose but , and that . Then Galois implies . Comparing the value of (the second smallest q-dimension when ), using (3.2) with , tells us that . Now count the number of fixed-points of in both cases: , i.e. , a contradiction.", "index": 2}, {"type": "text", "bbox": [71, 249, 541, 319], "content": "For comparing classical algebras with exceptional algebras, a useful device is to count the number of weights appearing in the fusion × (when has second smallest q-dimension). For example, for , ( , except for , and ( ), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can be isomorphic.", "index": 3}, {"type": "text", "bbox": [70, 321, 541, 349], "content": "For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively and for and , except for ).", "index": 4}, {"type": "text", "bbox": [70, 351, 541, 436], "content": "For the exceptional algebras, comparing and the number of highest-weights is effective. Recall that both and for a fixed algebra monotonically increase with to (respectively) and the Weyl dimension of , which is 7, 26, and 248 for respectively. For , exceeds 7 for , and exceeds 26 for , while exceeds 7 for . The number of highest-weights of , and are 10, 135, and 9, so only a small number of possibilities need be considered.", "index": 5}, {"type": "title", "bbox": [270, 455, 342, 468], "content": "References", "index": 6}, {"type": "list", "bbox": [78, 478, 541, 716], "content": "", "index": 7}]
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[{"bbox": [71, 73, 541, 89], "content": "where here denotes the greatest integer not more than . The absolute value of each", "parent_index": 0, "line_index": 0}, {"bbox": [70, 88, 541, 104], "content": "of these is quickly seen to be greater than 1 unless (mod ), except for the", "parent_index": 0, "line_index": 1}, {"bbox": [71, 102, 541, 118], "content": "orthogonal algebras when . An isomorphism would require", "parent_index": 0, "line_index": 2}, {"bbox": [71, 117, 540, 131], "content": "then that whenever (mod ) is coprime to , it must also satisfy (mod", "parent_index": 0, "line_index": 3}, {"bbox": [71, 131, 540, 145], "content": "), and conversely. This forces , for or and , or and any .", "parent_index": 0, "line_index": 4}, {"bbox": [94, 146, 542, 163], "content": "If , then that Galois argument implies , so", "parent_index": 1, "line_index": 0}, {"bbox": [70, 160, 349, 179], "content": "compare numbers of highest-weights: .", "parent_index": 1, "line_index": 1}, {"bbox": [94, 176, 542, 193], "content": "A similar argument works for the orthogonal algebras. For instance suppose", "parent_index": 2, "line_index": 0}, {"bbox": [71, 191, 541, 207], "content": "but , and that . Then Galois implies .", "parent_index": 2, "line_index": 1}, {"bbox": [72, 207, 541, 221], "content": "Comparing the value of (the second smallest q-dimension when ), using (3.2)", "parent_index": 2, "line_index": 2}, {"bbox": [71, 221, 543, 235], "content": "with , tells us that . Now count the number of fixed-points of", "parent_index": 2, "line_index": 3}, {"bbox": [71, 227, 467, 258], "content": "in both cases: , i.e. , a contradiction.", "parent_index": 2, "line_index": 4}, {"bbox": [93, 248, 542, 266], "content": "For comparing classical algebras with exceptional algebras, a useful device is to count", "parent_index": 3, "line_index": 0}, {"bbox": [70, 264, 541, 280], "content": "the number of weights appearing in the fusion × (when has second smallest", "parent_index": 3, "line_index": 1}, {"bbox": [70, 278, 542, 294], "content": "q-dimension). For example, for , ( , except for , and", "parent_index": 3, "line_index": 2}, {"bbox": [71, 294, 541, 308], "content": "( ), we learned in §3 that this number is 2, 3, 4 respectively, so none of these can", "parent_index": 3, "line_index": 3}, {"bbox": [70, 308, 148, 322], "content": "be isomorphic.", "parent_index": 3, "line_index": 4}, {"bbox": [95, 323, 541, 337], "content": "For the orthogonal algebras at level 2, useful is the number of weights with second", "parent_index": 4, "line_index": 0}, {"bbox": [70, 337, 502, 353], "content": "smallest q-dimension (respectively and for and , except for ).", "parent_index": 4, "line_index": 1}, {"bbox": [93, 352, 541, 367], "content": "For the exceptional algebras, comparing and the number of highest-weights is", "parent_index": 5, "line_index": 0}, {"bbox": [70, 365, 541, 383], "content": "effective. Recall that both and for a fixed algebra monotonically increase with", "parent_index": 5, "line_index": 1}, {"bbox": [71, 381, 540, 396], "content": "to (respectively) and the Weyl dimension of , which is 7, 26, and 248 for", "parent_index": 5, "line_index": 2}, {"bbox": [70, 396, 539, 411], "content": "respectively. For , exceeds 7 for , and exceeds 26 for , while", "parent_index": 5, "line_index": 3}, {"bbox": [70, 409, 541, 425], "content": "exceeds 7 for . The number of highest-weights of , and are 10, 135,", "parent_index": 5, "line_index": 4}, {"bbox": [70, 424, 415, 438], "content": "and 9, so only a small number of possibilities need be considered.", "parent_index": 5, "line_index": 5}, {"bbox": [270, 457, 342, 469], "content": "References", "parent_index": 6, "line_index": 0}, {"bbox": [79, 481, 539, 495], "content": "1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. 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[{"bbox": [106, 75, 120, 87], "content": "[x]", "parent_index": 0, "subtype": "inline"}, {"bbox": [381, 78, 388, 84], "content": "x", "parent_index": 0, "subtype": "inline"}, {"bbox": [357, 90, 398, 100], "content": "\\ell\\,\\equiv\\,\\pm1", "parent_index": 0, "subtype": "inline"}, {"bbox": [437, 90, 450, 99], "content": "2\\kappa", "parent_index": 0, "subtype": "inline"}, {"bbox": [213, 104, 246, 115], "content": "k\\ \\leq\\ 2", "parent_index": 0, "subtype": "inline"}, {"bbox": [350, 104, 462, 116], "content": "\\mathscr{R}(X_{r,k})\\,\\cong\\,\\mathscr{R}(X_{r^{\\prime},k^{\\prime}})", "parent_index": 0, "subtype": "inline"}, {"bbox": [178, 118, 216, 128], "content": "\\ell\\equiv\\pm1", "parent_index": 0, "subtype": "inline"}, {"bbox": [252, 119, 266, 127], "content": "2\\kappa", "parent_index": 0, "subtype": "inline"}, {"bbox": [347, 118, 357, 127], "content": "\\kappa^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [470, 118, 507, 128], "content": "\\ell\\equiv\\pm1", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 132, 88, 142], "content": "2\\kappa^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [244, 132, 277, 142], "content": "\\kappa=\\kappa^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [302, 133, 339, 142], "content": "X=B", "parent_index": 0, "subtype": "inline"}, {"bbox": [357, 133, 367, 142], "content": "D", "parent_index": 0, "subtype": "inline"}, {"bbox": [394, 133, 423, 142], "content": "k>2", "parent_index": 0, "subtype": "inline"}, {"bbox": [444, 133, 480, 142], "content": "X=C", "parent_index": 0, "subtype": "inline"}, {"bbox": [529, 133, 536, 142], "content": "k", "parent_index": 0, "subtype": "inline"}, {"bbox": [107, 147, 210, 160], "content": "\\mathscr{R}(C_{r,k})\\cong\\mathscr{R}(C_{s,m})", "parent_index": 1, "subtype": "inline"}, {"bbox": [403, 148, 521, 158], "content": "r+k+1=s+m+1", "parent_index": 1, "subtype": "inline"}, {"bbox": [270, 161, 345, 177], "content": "\\big(\\mathbf{\\Lambda}_{r}^{r+k}\\big)=\\big(\\mathbf{\\Lambda}_{s}^{r+k}\\big)", "parent_index": 1, "subtype": "inline"}, {"bbox": [485, 178, 542, 191], "content": "\\mathcal{R}(B_{r,k})\\cong", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 193, 117, 205], "content": "\\mathcal{R}(B_{s,m})", "parent_index": 2, "subtype": "inline"}, {"bbox": [144, 194, 212, 205], "content": "B_{r,k}\\ne B_{s,m}", "parent_index": 2, "subtype": "inline"}, {"bbox": [271, 193, 320, 205], "content": "k,m\\,>\\,2", "parent_index": 2, "subtype": "inline"}, {"bbox": [444, 194, 536, 204], "content": "2r+k\\,=\\,2s\\,+\\,m", "parent_index": 2, "subtype": "inline"}, {"bbox": [198, 207, 231, 220], "content": "\\mathcal{D}(\\Lambda_{1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [443, 208, 472, 217], "content": "k>3", "parent_index": 2, "subtype": "inline"}, {"bbox": [98, 222, 127, 231], "content": "\\lambda=0", "parent_index": 2, "subtype": "inline"}, {"bbox": [201, 222, 320, 233], "content": "2s+1=k,2r+1=m", "parent_index": 2, "subtype": "inline"}, {"bbox": [71, 237, 79, 246], "content": "J", "parent_index": 2, "subtype": "inline"}, {"bbox": [160, 234, 255, 250], "content": "\\binom{\\kappa/2-1}{r-1}=\\binom{\\kappa/2-1}{s-1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [281, 235, 372, 249], "content": "s-1=(k-1)/2", "parent_index": 2, "subtype": "inline"}, {"bbox": [326, 264, 341, 277], "content": "\\Lambda_{\\star}", "parent_index": 3, "subtype": "inline"}, {"bbox": [358, 264, 373, 277], "content": "\\Lambda_{\\star}", "parent_index": 3, "subtype": "inline"}, {"bbox": [415, 264, 429, 277], "content": "\\Lambda_{\\star}", "parent_index": 3, "subtype": "inline"}, {"bbox": [238, 280, 261, 293], "content": "A_{1,k}", "parent_index": 3, "subtype": "inline"}, {"bbox": [267, 278, 302, 292], "content": "\\left(k>1\\right)", "parent_index": 3, "subtype": "inline"}, {"bbox": [309, 278, 332, 293], "content": "C_{r,k}", "parent_index": 3, "subtype": "inline"}, {"bbox": [340, 279, 370, 291], "content": "k>1", "parent_index": 3, "subtype": "inline"}, {"bbox": [433, 280, 513, 293], "content": "C_{2,2},C_{2,3},C_{3,2})", "parent_index": 3, "subtype": "inline"}, {"bbox": [71, 295, 93, 307], "content": "E_{7,k}", "parent_index": 3, "subtype": "inline"}, {"bbox": [101, 294, 133, 306], "content": "k>4)", "parent_index": 3, "subtype": "inline"}, {"bbox": [254, 342, 260, 348], "content": "r", "parent_index": 4, "subtype": "inline"}, {"bbox": [286, 337, 313, 349], "content": "r-1", "parent_index": 4, "subtype": "inline"}, {"bbox": [335, 338, 357, 351], "content": "B_{r,2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [384, 339, 407, 351], "content": "D_{r,2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [470, 339, 492, 351], "content": "D_{4,2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [309, 352, 342, 366], "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "parent_index": 5, "subtype": "inline"}, {"bbox": [210, 368, 237, 380], "content": "||P_{+}||", "parent_index": 5, "subtype": "inline"}, {"bbox": [263, 368, 296, 380], "content": "\\mathcal{D}(\\boldsymbol{\\Lambda}_{\\star})", "parent_index": 5, "subtype": "inline"}, {"bbox": [71, 383, 78, 392], "content": "k", "parent_index": 5, "subtype": "inline"}, {"bbox": [170, 386, 182, 392], "content": "\\infty", "parent_index": 5, "subtype": "inline"}, {"bbox": [327, 383, 341, 393], "content": "\\Lambda_{\\star}", "parent_index": 5, "subtype": "inline"}, {"bbox": [487, 383, 540, 394], "content": "G_{2},F_{4},E_{8}", "parent_index": 5, "subtype": "inline"}, {"bbox": [163, 397, 186, 410], "content": "E_{8,k}", "parent_index": 5, "subtype": "inline"}, {"bbox": [194, 397, 226, 409], "content": "\\mathcal{D}(\\Lambda_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [303, 397, 333, 408], "content": "k\\geq5", "parent_index": 5, "subtype": "inline"}, {"bbox": [442, 397, 479, 408], "content": "k\\geq11", "parent_index": 5, "subtype": "inline"}, {"bbox": [518, 397, 539, 409], "content": "F_{4,k}", "parent_index": 5, "subtype": "inline"}, {"bbox": [144, 412, 175, 422], "content": "k\\geq4", "parent_index": 5, "subtype": "inline"}, {"bbox": [367, 411, 421, 424], "content": "E_{8,4},E_{8,10}", "parent_index": 5, "subtype": "inline"}, {"bbox": [452, 412, 473, 424], "content": "F_{4,3}", "parent_index": 5, "subtype": "inline"}, {"bbox": [134, 284, 142, 293], "content": "S", "parent_index": 7, "subtype": "inline"}, {"bbox": [109, 90, 123, 101], "content": "E_{8}", "parent_index": 7, "subtype": "inline"}]
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# Three generation neutrino mixing is compatible with all experiments
B. Hoeneisen and C. Marı´n
Universidad San Francisco de Quito 2 February 2000
# Abstract
We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff
Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of $\nu_{e}$ in a beam of $\nu_{\mu}$ by the LSND Collaboration. The invisible width of the $Z$ implies that the number of massless, or light Dirac, or light Majorana neutrino species is $N_{\nu}\,=\,2.993\pm0.011.$ .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates $\nu_{l}$ are superpositions of the neutrino mass eigenstates $\nu_{m}$ :
$$
|\nu_{l}\rangle=\sum_{m}U_{l m}|\nu_{m}\rangle
$$
We consider the “standard” parametrization of the unitary matrix $U_{l m}$ [1]:
$$
\left(\begin{array}{c}{{\nu_{e}}}\\ {{\nu_{\mu}}}\\ {{\nu_{\tau}}}\end{array}\right)=\left(\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\delta}}}\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}}}&{{s_{23}c_{13}}}\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}}}&{{c_{23}c_{13}}}\end{array}\right)\left(\begin{array}{c}{{\nu_{1}}}\\ {{\nu_{2}}}\\ {{\nu_{3}}}\end{array}\right)
$$
where $c_{i j}~\equiv~c o s{\theta}_{i j}$ , $s_{i j}~\equiv~s i n\theta_{i j}$ , $\begin{array}{r}{0\ \leq\ \theta_{i j}\ \leq\ \frac{\pi}{2}}\end{array}$ and $-\pi\ \leq\ \delta\ <\ \pi$ . The probability that an ultrarelativistic neutrino produced as $\nu_{l}$ decays as $\nu_{l^{\prime}}$
|
<html><body>
<h1 data-bbox="149 169 447 217">Three generation neutrino mixing is compatible with all experiments </h1>
<p data-bbox="212 235 382 251">B. Hoeneisen and C. Marı´n </p>
<p data-bbox="210 262 384 290">Universidad San Francisco de Quito 2 February 2000 </p>
<h1 data-bbox="273 320 321 333">Abstract </h1>
<p data-bbox="131 341 463 394">We consider the minimal extension of the Standard Model with three generations of massive neutrinos that mix. We then determine the parameters of the model that satisfy all experimental constraints. PACS 14.60.Pq, 12.15.Ff </p>
<p data-bbox="101 407 493 537">Three observables in disagreement with the Standard Model of Quarks and Leptons are: i) A deficit of electron-type solar neutrinos; ii) A deficit of muon-type atmospheric neutrinos; and, possibly, iii) The observation of the apearance of $\nu_{e}$ in a beam of $\nu_{\mu}$ by the LSND Collaboration. The invisible width of the $Z$ implies that the number of massless, or light Dirac, or light Majorana neutrino species is $N_{\nu}\,=\,2.993\pm0.011.$ .[1] To account for these observations we consider the minimal extension of the Standard Model with three massive neutrinos that mix. The neutrino interaction eigenstates $\nu_{l}$ are superpositions of the neutrino mass eigenstates $\nu_{m}$ : </p>
<div class="equation" data-bbox="250 548 344 578">$$
|\nu_{l}\rangle=\sum_{m}U_{l m}|\nu_{m}\rangle
$$</div>
<p data-bbox="101 587 483 602">We consider the “standard” parametrization of the unitary matrix $U_{l m}$ [1]: </p>
<div class="equation" data-bbox="101 613 500 661">$$
\left(\begin{array}{c}{{\nu_{e}}}\\ {{\nu_{\mu}}}\\ {{\nu_{\tau}}}\end{array}\right)=\left(\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\delta}}}\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}}}&{{s_{23}c_{13}}}\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}}}&{{c_{23}c_{13}}}\end{array}\right)\left(\begin{array}{c}{{\nu_{1}}}\\ {{\nu_{2}}}\\ {{\nu_{3}}}\end{array}\right)
$$</div>
<p data-bbox="101 683 492 713">where $c_{i j}~\equiv~c o s{\theta}_{i j}$ , $s_{i j}~\equiv~s i n\theta_{i j}$ , $\begin{array}{r}{0\ \leq\ \theta_{i j}\ \leq\ \frac{\pi}{2}}\end{array}$ and $-\pi\ \leq\ \delta\ <\ \pi$ . The probability that an ultrarelativistic neutrino produced as $\nu_{l}$ decays as $\nu_{l^{\prime}}$ </p>
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0002004v1
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[]
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[{"bbox": [172, 456, 182, 465], "content": "\\nu_{e}", "parent_index": 5, "subtype": "inline"}, {"bbox": [256, 456, 267, 466], "content": "\\nu_{\\mu}", "parent_index": 5, "subtype": "inline"}, {"bbox": [170, 469, 179, 478], "content": "Z", "parent_index": 5, "subtype": "inline"}, {"bbox": [258, 483, 363, 494], "content": "N_{\\nu}\\,=\\,2.993\\pm0.011.", "parent_index": 5, "subtype": "inline"}, {"bbox": [463, 515, 472, 523], "content": "\\nu_{l}", "parent_index": 5, "subtype": "inline"}, {"bbox": [348, 530, 362, 537], "content": "\\nu_{m}", "parent_index": 5, "subtype": "inline"}, {"bbox": [250, 548, 344, 578], "content": "|\\nu_{l}\\rangle=\\sum_{m}U_{l m}|\\nu_{m}\\rangle", "parent_index": 6, "subtype": "interline"}, {"bbox": [446, 591, 465, 603], "content": "U_{l m}", "parent_index": 7, "subtype": "inline"}, {"bbox": [101, 613, 500, 661], "content": "\\left(\\begin{array}{c}{{\\nu_{e}}}\\\\ {{\\nu_{\\mu}}}\\\\ {{\\nu_{\\tau}}}\\end{array}\\right)=\\left(\\begin{array}{c c c}{{c_{12}c_{13}}}&{{s_{12}c_{13}}}&{{s_{13}e^{-i\\delta}}}\\\\ {{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\\delta}}}&{{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\\delta}}}&{{s_{23}c_{13}}}\\\\ {{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta}}}&{{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta}}}&{{c_{23}c_{13}}}\\end{array}\\right)\\left(\\begin{array}{c}{{\\nu_{1}}}\\\\ {{\\nu_{2}}}\\\\ {{\\nu_{3}}}\\end{array}\\right)", "parent_index": 8, "subtype": "interline"}, {"bbox": [137, 687, 201, 700], "content": "c_{i j}~\\equiv~c o s{\\theta}_{i j}", "parent_index": 9, "subtype": "inline"}, {"bbox": [210, 687, 275, 700], "content": "s_{i j}~\\equiv~s i n\\theta_{i j}", "parent_index": 9, "subtype": "inline"}, {"bbox": [283, 687, 354, 700], "content": "\\begin{array}{r}{0\\ \\leq\\ \\theta_{i j}\\ \\leq\\ \\frac{\\pi}{2}}\\end{array}", "parent_index": 9, "subtype": "inline"}, {"bbox": [384, 687, 457, 698], "content": "-\\pi\\ \\leq\\ \\delta\\ <\\ \\pi", "parent_index": 9, "subtype": "inline"}, {"bbox": [410, 705, 419, 712], "content": "\\nu_{l}", "parent_index": 9, "subtype": "inline"}, {"bbox": [479, 705, 490, 712], "content": "\\nu_{l^{\\prime}}", "parent_index": 9, "subtype": "inline"}]
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[]
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<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 ± 0.23</td><td>21-2 2</td><td>0.33±0.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67±8</td><td></td><td>0.52 ± 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78±8</td><td></td><td>0.60 ± 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 ± 0.38</td><td></td><td>0.53 ± 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 ± 0.08</td></tr></table></body></html>
Table 1: Observed solar electron-type neutrino flux, compared to the Standard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is $10^{-36}$ captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of $10^{6}\mathrm{cm}^{-2}\mathrm{s}^{-1}$ at Earth above 7MeV (6.5MeV).
is[1]:
$$
P(\nu_{l}\to\nu_{l^{\prime}})=|\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\prime}m}^{*}|^{2}=P(\bar{\nu}_{l^{\prime}}\to\bar{\nu}_{l})
$$
where $E$ and $L$ are the energy and traveling distance of $\nu_{l}$ , and $M_{m}$ is the mass of $\nu_{m}$ . We choose $M_{1}\,\leq\,M_{2}\,\leq\,M_{3}$ . This extension of the Standard Model introduces six parameters: $S_{12}$ , $s_{23}$ , $s_{13}$ , $\delta$ , and two mass-squared differences, e.g. $\Delta M_{21}^{2}\equiv M_{2}^{2}-M_{1}^{2}$ and $\Delta M_{32}^{2}\equiv M_{3}^{2}-M_{2}^{2}$ . We vary these parameters to minimize a $\chi^{2}$ . This $\chi^{2}$ has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: $P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})\,=\,\textstyle{\frac{1}{2}}\sin^{2}(2\theta)\,=\,0.0031\pm0.0013$ for $L[\mathrm{km}]/E[\mathrm{GeV}]\!=[P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})]^{1/2}/1.27\cdot\Delta M^{2}[\mathrm{eV^{2}}]\!\approx\,0.73$ (here $\sin^{2}(2\theta)$ corresponds to “large” $\Delta M^{2}$ , and $\Delta M^{2}$ corresponds to $\sin^{2}(2\theta)\,=\,1$ , see discussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been confirmed by an independent experiment, we multiply the error by 1.5 and take $P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})=0.0031\pm0.0020$ . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3.
The $\chi^{2}$ has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of $\chi^{2}$ , a few of which are listed in Table 4. With $90\%$ confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations.
If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,
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<html><body>
<div class="table" data-bbox="112 125 481 231"><table data-bbox="112 125 481 231"><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 ± 0.23</td><td>21-2 2</td><td>0.33±0.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67±8</td><td></td><td>0.52 ± 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78±8</td><td></td><td>0.60 ± 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 ± 0.38</td><td></td><td>0.53 ± 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 ± 0.08</td></tr></table></div>
<p data-bbox="100 250 492 323">Table 1: Observed solar electron-type neutrino flux, compared to the Standard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is $10^{-36}$ captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of $10^{6}\mathrm{cm}^{-2}\mathrm{s}^{-1}$ at Earth above 7MeV (6.5MeV). </p>
<p data-bbox="101 341 127 357">is[1]: </p>
<div class="equation" data-bbox="142 366 451 397">$$
P(\nu_{l}\to\nu_{l^{\prime}})=|\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\prime}m}^{*}|^{2}=P(\bar{\nu}_{l^{\prime}}\to\bar{\nu}_{l})
$$</div>
<p data-bbox="101 402 492 619">where $E$ and $L$ are the energy and traveling distance of $\nu_{l}$ , and $M_{m}$ is the mass of $\nu_{m}$ . We choose $M_{1}\,\leq\,M_{2}\,\leq\,M_{3}$ . This extension of the Standard Model introduces six parameters: $S_{12}$ , $s_{23}$ , $s_{13}$ , $\delta$ , and two mass-squared differences, e.g. $\Delta M_{21}^{2}\equiv M_{2}^{2}-M_{1}^{2}$ and $\Delta M_{32}^{2}\equiv M_{3}^{2}-M_{2}^{2}$ . We vary these parameters to minimize a $\chi^{2}$ . This $\chi^{2}$ has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: $P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})\,=\,\textstyle{\frac{1}{2}}\sin^{2}(2\theta)\,=\,0.0031\pm0.0013$ for $L[\mathrm{km}]/E[\mathrm{GeV}]\!=[P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})]^{1/2}/1.27\cdot\Delta M^{2}[\mathrm{eV^{2}}]\!\approx\,0.73$ (here $\sin^{2}(2\theta)$ corresponds to “large” $\Delta M^{2}$ , and $\Delta M^{2}$ corresponds to $\sin^{2}(2\theta)\,=\,1$ , see discussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been confirmed by an independent experiment, we multiply the error by 1.5 and take $P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})=0.0031\pm0.0020$ . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3. </p>
<p data-bbox="101 620 492 705">The $\chi^{2}$ has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of $\chi^{2}$ , a few of which are listed in Table 4. With $90\%$ confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations. </p>
<p data-bbox="117 707 491 721">If neutrinos have a hierarchy of masses (as the charged leptons, up quarks, </p>
</body></html>
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0002004v1
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Note that one of the mass-squared differences", "type": "text"}], "index": 28}, {"bbox": [101, 680, 492, 694], "spans": [{"bbox": [101, 680, 492, 694], "score": 1.0, "content": "is determined by the solar neutrino experiments and the other one by the", "type": "text"}], "index": 29}, {"bbox": [102, 695, 281, 708], "spans": [{"bbox": [102, 695, 281, 708], "score": 1.0, "content": "atmospheric neutrino observations.", "type": "text"}], "index": 30}], "index": 27.5}, {"type": "text", "bbox": [117, 707, 491, 721], "lines": [{"bbox": [118, 708, 490, 722], "spans": [{"bbox": [118, 708, 490, 722], "score": 1.0, "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "type": "text"}], "index": 31}], "index": 31}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [112, 125, 481, 231], "blocks": [{"type": "table_body", "bbox": [112, 125, 481, 231], "group_id": 0, "lines": [{"bbox": [112, 125, 481, 231], "spans": [{"bbox": [112, 125, 481, 231], "score": 0.976, "html": "<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 ± 0.23</td><td>21-2 2</td><td>0.33±0.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67±8</td><td></td><td>0.52 ± 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78±8</td><td></td><td>0.60 ± 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 ± 0.38</td><td></td><td>0.53 ± 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 ± 0.08</td></tr></table></body></html>", "type": "table", "image_path": "58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg"}]}], "index": 1, "virtual_lines": [{"bbox": [112, 125, 481, 160.33333333333334], "spans": [], "index": 0}, {"bbox": [112, 160.33333333333334, 481, 195.66666666666669], "spans": [], "index": 1}, {"bbox": [112, 195.66666666666669, 481, 231.00000000000003], "spans": [], "index": 2}]}], "index": 1}], "interline_equations": [{"type": "interline_equation", "bbox": [142, 366, 451, 397], "lines": [{"bbox": [142, 366, 451, 397], "spans": [{"bbox": [142, 366, 451, 397], "score": 0.92, "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "type": "interline_equation"}], "index": 9}], "index": 9}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 738, 301, 748], "lines": [{"bbox": [293, 739, 301, 751], "spans": [{"bbox": [293, 739, 301, 751], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "table", "bbox": [112, 125, 481, 231], "blocks": [{"type": "table_body", "bbox": [112, 125, 481, 231], "group_id": 0, "lines": [{"bbox": [112, 125, 481, 231], "spans": [{"bbox": [112, 125, 481, 231], "score": 0.976, "html": "<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 ± 0.23</td><td>21-2 2</td><td>0.33±0.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67±8</td><td></td><td>0.52 ± 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78±8</td><td></td><td>0.60 ± 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 ± 0.38</td><td></td><td>0.53 ± 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 ± 0.08</td></tr></table></body></html>", "type": "table", "image_path": "58f4e515357520a89a5c257eeb2486ac2b1e29825cfbe284d732c4c616013e28.jpg"}]}], "index": 1, "virtual_lines": [{"bbox": [112, 125, 481, 160.33333333333334], "spans": [], "index": 0}, {"bbox": [112, 160.33333333333334, 481, 195.66666666666669], "spans": [], "index": 1}, {"bbox": [112, 195.66666666666669, 481, 231.00000000000003], "spans": [], "index": 2}]}], "index": 1}, {"type": "text", "bbox": [100, 250, 492, 323], "lines": [{"bbox": [101, 252, 492, 268], "spans": [{"bbox": [101, 252, 492, 268], "score": 1.0, "content": "Table 1: Observed solar electron-type neutrino flux, compared to the Stan-", "type": "text"}], "index": 3}, {"bbox": [102, 268, 492, 281], "spans": [{"bbox": [102, 268, 492, 281], "score": 1.0, "content": "dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their", "type": "text"}], "index": 4}, {"bbox": [101, 281, 492, 297], "spans": [{"bbox": [101, 281, 308, 297], "score": 1.0, "content": "ratio. The Solar Neutrino Unit (SNU) is ", "type": "text"}, {"bbox": [309, 283, 336, 293], "score": 0.92, "content": "10^{-36}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [336, 281, 492, 297], "score": 1.0, "content": " captures per atom per second.", "type": "text"}], "index": 5}, {"bbox": [101, 296, 493, 310], "spans": [{"bbox": [101, 296, 417, 310], "score": 1.0, "content": "For Kamiokande (Super Kamiokande) the flux is in units of ", "type": "text"}, {"bbox": [417, 297, 476, 307], "score": 0.92, "content": "10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [477, 296, 493, 310], "score": 1.0, "content": " at", "type": "text"}], "index": 6}, {"bbox": [101, 311, 256, 325], "spans": [{"bbox": [101, 311, 256, 325], "score": 1.0, "content": "Earth above 7MeV (6.5MeV).", "type": "text"}], "index": 7}], "index": 5, "bbox_fs": [101, 252, 493, 325]}, {"type": "text", "bbox": [101, 341, 127, 357], "lines": [{"bbox": [100, 342, 128, 361], "spans": [{"bbox": [100, 342, 128, 361], "score": 1.0, "content": "is[1]:", "type": "text"}], "index": 8}], "index": 8, "bbox_fs": [100, 342, 128, 361]}, {"type": "interline_equation", "bbox": [142, 366, 451, 397], "lines": [{"bbox": [142, 366, 451, 397], "spans": [{"bbox": [142, 366, 451, 397], "score": 0.92, "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [101, 402, 492, 619], "lines": [{"bbox": [102, 406, 492, 418], "spans": [{"bbox": [102, 406, 134, 418], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [135, 407, 144, 415], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [145, 406, 168, 418], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [168, 407, 177, 415], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [177, 406, 378, 418], "score": 1.0, "content": " are the energy and traveling distance of ", "type": "text"}, {"bbox": [378, 410, 387, 417], "score": 0.89, "content": "\\nu_{l}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [387, 406, 414, 418], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [415, 407, 434, 417], "score": 0.93, "content": "M_{m}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [434, 406, 492, 418], "score": 1.0, "content": " is the mass", "type": "text"}], "index": 10}, {"bbox": [101, 418, 492, 433], "spans": [{"bbox": [101, 418, 116, 433], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [116, 424, 130, 432], "score": 0.9, "content": "\\nu_{m}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [131, 418, 198, 433], "score": 1.0, "content": ". 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We vary these parameters", "type": "text"}], "index": 13}, {"bbox": [100, 462, 493, 477], "spans": [{"bbox": [100, 462, 179, 477], "score": 1.0, "content": "to minimize a ", "type": "text"}, {"bbox": [179, 464, 191, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [192, 462, 230, 477], "score": 1.0, "content": ". This ", "type": "text"}, {"bbox": [230, 463, 243, 476], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [243, 462, 493, 477], "score": 1.0, "content": " has 14 terms obtained from the solar neutrino", "type": "text"}], "index": 14}, {"bbox": [101, 477, 492, 491], "spans": [{"bbox": [101, 477, 492, 491], "score": 1.0, "content": "data summarized in Table 1, the atmospheric neutrino data shown in Table", "type": "text"}], "index": 15}, {"bbox": [100, 490, 493, 506], "spans": [{"bbox": [100, 490, 242, 506], "score": 1.0, "content": "2, and the LSND data[9]: ", "type": "text"}, {"bbox": [243, 492, 472, 505], "score": 0.86, "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013", "type": "inline_equation", "height": 13, "width": 229}, {"bbox": [473, 490, 493, 506], "score": 1.0, "content": " for", "type": "text"}], "index": 16}, {"bbox": [102, 506, 492, 520], "spans": [{"bbox": [102, 506, 392, 520], "score": 0.86, "content": "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73", "type": "inline_equation", "height": 14, "width": 290}, {"bbox": [392, 506, 426, 520], "score": 1.0, "content": " (here ", "type": "text"}, {"bbox": [426, 506, 466, 520], "score": 0.85, "content": "\\sin^{2}(2\\theta)", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [467, 506, 492, 520], "score": 1.0, "content": " cor-", "type": "text"}], "index": 17}, {"bbox": [102, 521, 491, 534], "spans": [{"bbox": [102, 521, 208, 534], "score": 1.0, "content": "responds to “large” ", "type": "text"}, {"bbox": [208, 521, 235, 531], "score": 0.89, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [236, 521, 267, 534], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [267, 521, 294, 531], "score": 0.9, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [295, 521, 379, 534], "score": 1.0, "content": " corresponds to ", "type": "text"}, {"bbox": [380, 521, 444, 534], "score": 0.93, "content": "\\sin^{2}(2\\theta)\\,=\\,1", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [444, 521, 491, 534], "score": 1.0, "content": ", see dis-", "type": "text"}], "index": 18}, {"bbox": [101, 536, 492, 548], "spans": [{"bbox": [101, 536, 492, 548], "score": 1.0, "content": "cussion in [1]). 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The most stringent of these limits are", "type": "text"}], "index": 23}, {"bbox": [101, 608, 189, 620], "spans": [{"bbox": [101, 608, 189, 620], "score": 1.0, "content": "listed in Table 3.", "type": "text"}], "index": 24}], "index": 17, "bbox_fs": [99, 406, 495, 620]}, {"type": "text", "bbox": [101, 620, 492, 705], "lines": [{"bbox": [119, 620, 492, 637], "spans": [{"bbox": [119, 620, 143, 637], "score": 1.0, "content": "The ", "type": "text"}, {"bbox": [144, 622, 155, 635], "score": 0.92, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 11}, {"bbox": [156, 620, 492, 637], "score": 1.0, "content": " has 8 degrees of freedom (14 terms minus 6 parameters). Varying", "type": "text"}], "index": 25}, {"bbox": [102, 636, 492, 650], "spans": [{"bbox": [102, 636, 304, 650], "score": 1.0, "content": "the parameters we obtain minimums of ", "type": "text"}, {"bbox": [304, 636, 316, 649], "score": 0.92, "content": "\\chi^{2}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [317, 636, 492, 650], "score": 1.0, "content": ", a few of which are listed in Table", "type": "text"}], "index": 26}, {"bbox": [101, 650, 492, 665], "spans": [{"bbox": [101, 650, 153, 665], "score": 1.0, "content": "4. With ", "type": "text"}, {"bbox": [153, 651, 176, 662], "score": 0.28, "content": "90\\%", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [176, 650, 492, 665], "score": 1.0, "content": " confidence the neutrino mass-squared differences lie within", "type": "text"}], "index": 27}, {"bbox": [101, 666, 492, 679], "spans": [{"bbox": [101, 666, 492, 679], "score": 1.0, "content": "the dots shown in Figure 1. Note that one of the mass-squared differences", "type": "text"}], "index": 28}, {"bbox": [101, 680, 492, 694], "spans": [{"bbox": [101, 680, 492, 694], "score": 1.0, "content": "is determined by the solar neutrino experiments and the other one by the", "type": "text"}], "index": 29}, {"bbox": [102, 695, 281, 708], "spans": [{"bbox": [102, 695, 281, 708], "score": 1.0, "content": "atmospheric neutrino observations.", "type": "text"}], "index": 30}], "index": 27.5, "bbox_fs": [101, 620, 492, 708]}, {"type": "text", "bbox": [117, 707, 491, 721], "lines": [{"bbox": [118, 708, 490, 722], "spans": [{"bbox": [118, 708, 490, 722], "score": 1.0, "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "type": "text"}], "index": 31}], "index": 31, "bbox_fs": [118, 708, 490, 722]}]}
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[{"type": "table", "bbox": [112, 125, 481, 231], "content": "", "index": 0}, {"type": "text", "bbox": [100, 250, 492, 323], "content": "Table 1: Observed solar electron-type neutrino flux, compared to the Stan- dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their ratio. The Solar Neutrino Unit (SNU) is captures per atom per second. For Kamiokande (Super Kamiokande) the flux is in units of at Earth above 7MeV (6.5MeV).", "index": 1}, {"type": "text", "bbox": [101, 341, 127, 357], "content": "is[1]:", "index": 2}, {"type": "interline_equation", "bbox": [142, 366, 451, 397], "content": "", "index": 3}, {"type": "text", "bbox": [101, 402, 492, 619], "content": "where and are the energy and traveling distance of , and is the mass of . We choose . This extension of the Standard Model introduces six parameters: , , , , and two mass-squared differences, e.g. and . We vary these parameters to minimize a . This has 14 terms obtained from the solar neutrino data summarized in Table 1, the atmospheric neutrino data shown in Table 2, and the LSND data[9]: for (here cor- responds to “large” , and corresponds to , see dis- cussion in [1]). Because one author[10] of the LSND Collaboration is in disagreement with the conclusion, and because the result has not been con- firmed by an independent experiment, we multiply the error by 1.5 and take . We require that the astrophysical, reactor and accelerator limits be satisfied. The most stringent of these limits are listed in Table 3.", "index": 4}, {"type": "text", "bbox": [101, 620, 492, 705], "content": "The has 8 degrees of freedom (14 terms minus 6 parameters). Varying the parameters we obtain minimums of , a few of which are listed in Table 4. With confidence the neutrino mass-squared differences lie within the dots shown in Figure 1. Note that one of the mass-squared differences is determined by the solar neutrino experiments and the other one by the atmospheric neutrino observations.", "index": 5}, {"type": "text", "bbox": [117, 707, 491, 721], "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "index": 6}]
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[{"bbox": [101, 252, 492, 268], "content": "Table 1: Observed solar electron-type neutrino flux, compared to the Stan-", "parent_index": 1, "line_index": 0}, {"bbox": [102, 268, 492, 281], "content": "dard Solar Model (SSM) predictions asuming no neutrino mixing[7], and their", "parent_index": 1, "line_index": 1}, {"bbox": [101, 281, 492, 297], "content": "ratio. The Solar Neutrino Unit (SNU) is captures per atom per second.", "parent_index": 1, "line_index": 2}, {"bbox": [101, 296, 493, 310], "content": "For Kamiokande (Super Kamiokande) the flux is in units of at", "parent_index": 1, "line_index": 3}, {"bbox": [101, 311, 256, 325], "content": "Earth above 7MeV (6.5MeV).", "parent_index": 1, "line_index": 4}, {"bbox": [100, 342, 128, 361], "content": "is[1]:", "parent_index": 2, "line_index": 0}, {"bbox": [102, 406, 492, 418], "content": "where and are the energy and traveling distance of , and is the mass", "parent_index": 4, "line_index": 0}, {"bbox": [101, 418, 492, 433], "content": "of . We choose . This extension of the Standard Model", "parent_index": 4, "line_index": 1}, {"bbox": [101, 434, 493, 449], "content": "introduces six parameters: , , , , and two mass-squared differences,", "parent_index": 4, "line_index": 2}, {"bbox": [99, 443, 495, 469], "content": "e.g. and . We vary these parameters", "parent_index": 4, "line_index": 3}, {"bbox": [100, 462, 493, 477], "content": "to minimize a . This has 14 terms obtained from the solar neutrino", "parent_index": 4, "line_index": 4}, {"bbox": [101, 477, 492, 491], "content": "data summarized in Table 1, the atmospheric neutrino data shown in Table", "parent_index": 4, "line_index": 5}, {"bbox": [100, 490, 493, 506], "content": "2, and the LSND data[9]: for", "parent_index": 4, "line_index": 6}, {"bbox": [102, 506, 492, 520], "content": "(here cor-", "parent_index": 4, "line_index": 7}, {"bbox": [102, 521, 491, 534], "content": "responds to “large” , and corresponds to , see dis-", "parent_index": 4, "line_index": 8}, {"bbox": [101, 536, 492, 548], "content": "cussion in [1]). Because one author[10] of the LSND Collaboration is in", "parent_index": 4, "line_index": 9}, {"bbox": [102, 551, 492, 563], "content": "disagreement with the conclusion, and because the result has not been con-", "parent_index": 4, "line_index": 10}, {"bbox": [101, 564, 492, 578], "content": "firmed by an independent experiment, we multiply the error by 1.5 and take", "parent_index": 4, "line_index": 11}, {"bbox": [102, 578, 492, 593], "content": ". We require that the astrophysical, reactor", "parent_index": 4, "line_index": 12}, {"bbox": [101, 593, 492, 606], "content": "and accelerator limits be satisfied. The most stringent of these limits are", "parent_index": 4, "line_index": 13}, {"bbox": [101, 608, 189, 620], "content": "listed in Table 3.", "parent_index": 4, "line_index": 14}, {"bbox": [119, 620, 492, 637], "content": "The has 8 degrees of freedom (14 terms minus 6 parameters). Varying", "parent_index": 5, "line_index": 0}, {"bbox": [102, 636, 492, 650], "content": "the parameters we obtain minimums of , a few of which are listed in Table", "parent_index": 5, "line_index": 1}, {"bbox": [101, 650, 492, 665], "content": "4. With confidence the neutrino mass-squared differences lie within", "parent_index": 5, "line_index": 2}, {"bbox": [101, 666, 492, 679], "content": "the dots shown in Figure 1. Note that one of the mass-squared differences", "parent_index": 5, "line_index": 3}, {"bbox": [101, 680, 492, 694], "content": "is determined by the solar neutrino experiments and the other one by the", "parent_index": 5, "line_index": 4}, {"bbox": [102, 695, 281, 708], "content": "atmospheric neutrino observations.", "parent_index": 5, "line_index": 5}, {"bbox": [118, 708, 490, 722], "content": "If neutrinos have a hierarchy of masses (as the charged leptons, up quarks,", "parent_index": 6, "line_index": 0}]
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[]
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[{"bbox": [309, 283, 336, 293], "content": "10^{-36}", "parent_index": 1, "subtype": "inline"}, {"bbox": [417, 297, 476, 307], "content": "10^{6}\\mathrm{cm}^{-2}\\mathrm{s}^{-1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [142, 366, 451, 397], "content": "P(\\nu_{l}\\to\\nu_{l^{\\prime}})=|\\sum_{m}U_{l m}e x p(-i L M_{m}^{2}/2E)U_{l^{\\prime}m}^{*}|^{2}=P(\\bar{\\nu}_{l^{\\prime}}\\to\\bar{\\nu}_{l})", "parent_index": 3, "subtype": "interline"}, {"bbox": [135, 407, 144, 415], "content": "E", "parent_index": 4, "subtype": "inline"}, {"bbox": [168, 407, 177, 415], "content": "L", "parent_index": 4, "subtype": "inline"}, {"bbox": [378, 410, 387, 417], "content": "\\nu_{l}", "parent_index": 4, "subtype": "inline"}, {"bbox": [415, 407, 434, 417], "content": "M_{m}", "parent_index": 4, "subtype": "inline"}, {"bbox": [116, 424, 130, 432], "content": "\\nu_{m}", "parent_index": 4, "subtype": "inline"}, {"bbox": [198, 421, 283, 432], "content": "M_{1}\\,\\leq\\,M_{2}\\,\\leq\\,M_{3}", "parent_index": 4, "subtype": "inline"}, {"bbox": [241, 439, 256, 446], "content": "S_{12}", "parent_index": 4, "subtype": "inline"}, {"bbox": [263, 439, 277, 446], "content": "s_{23}", "parent_index": 4, "subtype": "inline"}, {"bbox": [284, 439, 299, 446], "content": "s_{13}", "parent_index": 4, "subtype": "inline"}, {"bbox": [306, 436, 312, 445], "content": "\\delta", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 449, 223, 462], "content": "\\Delta M_{21}^{2}\\equiv M_{2}^{2}-M_{1}^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [250, 449, 347, 462], "content": "\\Delta M_{32}^{2}\\equiv M_{3}^{2}-M_{2}^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [179, 464, 191, 476], "content": "\\chi^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [230, 463, 243, 476], "content": "\\chi^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [243, 492, 472, 505], "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})\\,=\\,\\textstyle{\\frac{1}{2}}\\sin^{2}(2\\theta)\\,=\\,0.0031\\pm0.0013", "parent_index": 4, "subtype": "inline"}, {"bbox": [102, 506, 392, 520], "content": "L[\\mathrm{km}]/E[\\mathrm{GeV}]\\!=[P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})]^{1/2}/1.27\\cdot\\Delta M^{2}[\\mathrm{eV^{2}}]\\!\\approx\\,0.73", "parent_index": 4, "subtype": "inline"}, {"bbox": [426, 506, 466, 520], "content": "\\sin^{2}(2\\theta)", "parent_index": 4, "subtype": "inline"}, {"bbox": [208, 521, 235, 531], "content": "\\Delta M^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [267, 521, 294, 531], "content": "\\Delta M^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [380, 521, 444, 534], "content": "\\sin^{2}(2\\theta)\\,=\\,1", "parent_index": 4, "subtype": "inline"}, {"bbox": [102, 579, 262, 592], "content": "P(\\bar{\\nu}_{\\mu}\\to\\bar{\\nu}_{e})=0.0031\\pm0.0020", "parent_index": 4, "subtype": "inline"}, {"bbox": [144, 622, 155, 635], "content": "\\chi^{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [304, 636, 316, 649], "content": "\\chi^{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [153, 651, 176, 662], "content": "90\\%", "parent_index": 5, "subtype": "inline"}]
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[{"bbox": [112, 125, 481, 231], "content": "<html><body><table><tr><td>Experiment</td><td>Energy [MeV]</td><td>Observed flux (SNU)</td><td>SSM1 predic- tion (SNU)</td><td>Ratio</td></tr><tr><td>Homestake[2]</td><td>0.87</td><td>2.56 ± 0.23</td><td>21-2 2</td><td>0.33±0.05</td></tr><tr><td>Sage[3]</td><td>0.233 - 0.4</td><td>67±8</td><td></td><td>0.52 ± 0.07</td></tr><tr><td>Gallex[4]</td><td>0.233 - 0.4</td><td>78±8</td><td></td><td>0.60 ± 0.07</td></tr><tr><td>Kamiokande[5]</td><td>7-13</td><td>2.80 ± 0.38</td><td></td><td>0.53 ± 0.11</td></tr><tr><td>Super-Kam.[6]</td><td>6 - 13</td><td>2.42 +0.12 -0.09</td><td></td><td>0.46 ± 0.08</td></tr></table></body></html>", "parent_index": 0, "subtype": "body"}]
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<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>Rμ</td></tr><tr><td>10</td><td>1.20 ± 0.15</td><td>1.00 ± 0.15</td></tr><tr><td>100</td><td>1.20 ± 0.15</td><td>0.85 ± 0.12</td></tr><tr><td>1000</td><td>1.20 ± 0.15</td><td>0.70± 0.10</td></tr><tr><td>10000</td><td>1.20 ± 0.15</td><td>0.60± 0.08</td></tr></table></body></html>
Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as measured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because $R_{e}$ is observed to be independent of $L/E$ , we divide the numbers in this table by 1.15 so that $R_{e}\approx1$ .
or down quarks), then the “upper island” in Figure 1 applies, and $M_{3}\,\approx$ 0.07eV, $M_{2}\approx10^{-5}\mathrm{eV}$ and $M_{1}<M_{2}$ , with large uncertainties.
Note in Table 1 that the ratio of the observed-to-predicted solar neutrino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a $^7B e$ line. For the Homestake experiment the spread in $L/E$ is due to the spread in $L$ , which in turn is due to the excentricity of the orbit of the Earth. Therefore the interference is coherent for up to $L\Delta M^{2}/(2E\!\cdot\!2\pi)\approx30$ oscillations from the Sun to the Earth (here $\Delta M^{2}$ is either $\Delta M_{21}^{2}$ or $\Delta M_{32}^{2}$ ). Due to this coherence at “small” $\Delta M^{2}$ it is possible to find acceptable solutions with $\chi^{2}<13.4$ as shown in Figure 1. For larger values of $\Delta M^{2}$ coherence is lost and we find solutions with $\chi^{2}>18$ which are unacceptable if the Homestake experimental and theoretical errors are correct.
An important test of the model would be to observe seasonal variations of the neutrino flux of the $^7B e$ line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the $^7B e$ line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the $^7B e$ neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is $0.09\pm0.10$ . This observation implies that there are $\leq8.5$ periods of oscillation from Sun to Earth at 90% confidence level. With a $\chi^{2}$
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<html><body>
<div class="table" data-bbox="181 125 412 202"><table data-bbox="181 125 412 202"><tr><td>L/E [km/GeV]</td><td>Re</td><td>Rμ</td></tr><tr><td>10</td><td>1.20 ± 0.15</td><td>1.00 ± 0.15</td></tr><tr><td>100</td><td>1.20 ± 0.15</td><td>0.85 ± 0.12</td></tr><tr><td>1000</td><td>1.20 ± 0.15</td><td>0.70± 0.10</td></tr><tr><td>10000</td><td>1.20 ± 0.15</td><td>0.60± 0.08</td></tr></table></div>
<p data-bbox="101 221 492 294">Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as measured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because $R_{e}$ is observed to be independent of $L/E$ , we divide the numbers in this table by 1.15 so that $R_{e}\approx1$ . </p>
<p data-bbox="101 315 492 343">or down quarks), then the “upper island” in Figure 1 applies, and $M_{3}\,\approx$ 0.07eV, $M_{2}\approx10^{-5}\mathrm{eV}$ and $M_{1}<M_{2}$ , with large uncertainties. </p>
<p data-bbox="100 344 492 545">Note in Table 1 that the ratio of the observed-to-predicted solar neutrino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a $^7B e$ line. For the Homestake experiment the spread in $L/E$ is due to the spread in $L$ , which in turn is due to the excentricity of the orbit of the Earth. Therefore the interference is coherent for up to $L\Delta M^{2}/(2E\!\cdot\!2\pi)\approx30$ oscillations from the Sun to the Earth (here $\Delta M^{2}$ is either $\Delta M_{21}^{2}$ or $\Delta M_{32}^{2}$ ). Due to this coherence at “small” $\Delta M^{2}$ it is possible to find acceptable solutions with $\chi^{2}<13.4$ as shown in Figure 1. For larger values of $\Delta M^{2}$ coherence is lost and we find solutions with $\chi^{2}>18$ which are unacceptable if the Homestake experimental and theoretical errors are correct. </p>
<p data-bbox="100 546 493 719">An important test of the model would be to observe seasonal variations of the neutrino flux of the $^7B e$ line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the $^7B e$ line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the $^7B e$ neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is $0.09\pm0.10$ . This observation implies that there are $\leq8.5$ periods of oscillation from Sun to Earth at 90% confidence level. With a $\chi^{2}$ </p>
</body></html>
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0002004v1
| 2 | 612 | 792 | 1,275 | 1,650 |
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Therefore the interference is coherent for up", "type": "text"}], "index": 20}, {"bbox": [101, 461, 492, 475], "spans": [{"bbox": [101, 461, 115, 475], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [115, 462, 227, 475], "score": 0.93, "content": "L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [227, 461, 452, 475], "score": 1.0, "content": " oscillations from the Sun to the Earth (here ", "type": "text"}, {"bbox": [453, 462, 479, 472], "score": 0.93, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 26}, {"bbox": [480, 461, 492, 475], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [101, 475, 492, 490], "spans": [{"bbox": [101, 475, 135, 490], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [135, 477, 165, 489], "score": 0.94, "content": "\\Delta M_{21}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [165, 475, 183, 490], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [183, 477, 213, 489], "score": 0.92, "content": "\\Delta M_{32}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [214, 475, 396, 490], "score": 1.0, "content": "). Due to this coherence at “small” ", "type": "text"}, {"bbox": [397, 477, 424, 486], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [424, 475, 492, 490], "score": 1.0, "content": " it is possible", "type": "text"}], "index": 22}, {"bbox": [101, 490, 492, 504], "spans": [{"bbox": [101, 490, 272, 504], "score": 1.0, "content": "to find acceptable solutions with ", "type": "text"}, {"bbox": [273, 491, 322, 503], "score": 0.93, "content": "\\chi^{2}<13.4", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [322, 490, 492, 504], "score": 1.0, "content": " as shown in Figure 1. For larger", "type": "text"}], "index": 23}, {"bbox": [102, 505, 491, 518], "spans": [{"bbox": [102, 505, 151, 518], "score": 1.0, "content": "values of ", "type": "text"}, {"bbox": [152, 505, 179, 515], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [179, 505, 414, 518], "score": 1.0, "content": " coherence is lost and we find solutions with ", "type": "text"}, {"bbox": [415, 505, 457, 518], "score": 0.95, "content": "\\chi^{2}>18", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [457, 505, 491, 518], "score": 1.0, "content": " which", "type": "text"}], "index": 24}, {"bbox": [102, 519, 492, 532], "spans": [{"bbox": [102, 519, 492, 532], "score": 1.0, "content": "are unacceptable if the Homestake experimental and theoretical errors are", "type": "text"}], "index": 25}, {"bbox": [101, 535, 141, 546], "spans": [{"bbox": [101, 535, 141, 546], "score": 1.0, "content": "correct.", "type": "text"}], "index": 26}], "index": 19.5}, {"type": "text", "bbox": [100, 546, 493, 719], "lines": [{"bbox": [120, 549, 492, 561], "spans": [{"bbox": [120, 549, 492, 561], "score": 1.0, "content": "An important test of the model would be to observe seasonal variations", "type": "text"}], "index": 27}, {"bbox": [101, 562, 492, 576], "spans": [{"bbox": [101, 562, 248, 576], "score": 1.0, "content": "of the neutrino flux of the ", "type": "text"}, {"bbox": [249, 563, 269, 573], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [269, 562, 492, 576], "score": 1.0, "content": " line. If the lower ratio measured by the", "type": "text"}], "index": 28}, {"bbox": [101, 577, 492, 591], "spans": [{"bbox": [101, 577, 492, 591], "score": 1.0, "content": "Homestake experiment is real, we expect that the electron-type neutrino flux", "type": "text"}], "index": 29}, {"bbox": [101, 591, 492, 605], "spans": [{"bbox": [101, 591, 134, 605], "score": 1.0, "content": "of the ", "type": "text"}, {"bbox": [134, 592, 154, 602], "score": 0.9, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [155, 591, 492, 605], "score": 1.0, "content": " line is near a minimum of the oscillation at the average Sun-Earth", "type": "text"}], "index": 30}, {"bbox": [101, 605, 492, 619], "spans": [{"bbox": [101, 605, 492, 619], "score": 1.0, "content": "distance. In other words, there are an odd number of half-wavelengths from", "type": "text"}], "index": 31}, {"bbox": [101, 620, 493, 633], "spans": [{"bbox": [101, 620, 365, 633], "score": 1.0, "content": "Sun to Earth. Then we expect a modulation of the ", "type": "text"}, {"bbox": [366, 621, 385, 631], "score": 0.88, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [386, 620, 493, 633], "score": 1.0, "content": " neutrino flux with a", "type": "text"}], "index": 32}, {"bbox": [100, 635, 493, 649], "spans": [{"bbox": [100, 635, 493, 649], "score": 1.0, "content": "period of half a year, with maximums occurring at the perihelion and aphelion", "type": "text"}], "index": 33}, {"bbox": [101, 649, 492, 663], "spans": [{"bbox": [101, 649, 492, 663], "score": 1.0, "content": "of the Earth orbit. We see no statistically significant Fourier component", "type": "text"}], "index": 34}, {"bbox": [101, 664, 492, 677], "spans": [{"bbox": [101, 664, 492, 677], "score": 1.0, "content": "of the time dependent Homestake data from 1970.281 to 1994.388.[12] In", "type": "text"}], "index": 35}, {"bbox": [101, 679, 494, 691], "spans": [{"bbox": [101, 679, 494, 691], "score": 1.0, "content": "particular the amplitude relative to the mean of a Fourier component of", "type": "text"}], "index": 36}, {"bbox": [101, 693, 491, 707], "spans": [{"bbox": [101, 693, 198, 707], "score": 1.0, "content": "period 0.5 years is ", "type": "text"}, {"bbox": [198, 695, 253, 704], "score": 0.87, "content": "0.09\\pm0.10", "type": "inline_equation", "height": 9, "width": 55}, {"bbox": [254, 693, 463, 707], "score": 1.0, "content": ". This observation implies that there are ", "type": "text"}, {"bbox": [463, 695, 491, 705], "score": 0.44, "content": "\\leq8.5", "type": "inline_equation", "height": 10, "width": 28}], "index": 37}, {"bbox": [101, 706, 491, 721], "spans": [{"bbox": [101, 706, 478, 721], "score": 1.0, "content": "periods of oscillation from Sun to Earth at 90% confidence level. With a ", "type": "text"}, {"bbox": [479, 708, 491, 720], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}], "index": 38}], "index": 32.5}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [181, 125, 412, 202], "blocks": [{"type": "table_body", "bbox": [181, 125, 412, 202], "group_id": 0, "lines": [{"bbox": [181, 125, 412, 202], "spans": [{"bbox": [181, 125, 412, 202], "score": 0.976, "html": "<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>Rμ</td></tr><tr><td>10</td><td>1.20 ± 0.15</td><td>1.00 ± 0.15</td></tr><tr><td>100</td><td>1.20 ± 0.15</td><td>0.85 ± 0.12</td></tr><tr><td>1000</td><td>1.20 ± 0.15</td><td>0.70± 0.10</td></tr><tr><td>10000</td><td>1.20 ± 0.15</td><td>0.60± 0.08</td></tr></table></body></html>", "type": "table", "image_path": "77d9acb6b86f9c9a84ea1fbf27649993b710e71540165032c1de6955bb3136fb.jpg"}]}], "index": 2.5, 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0.15</td><td>1.00 ± 0.15</td></tr><tr><td>100</td><td>1.20 ± 0.15</td><td>0.85 ± 0.12</td></tr><tr><td>1000</td><td>1.20 ± 0.15</td><td>0.70± 0.10</td></tr><tr><td>10000</td><td>1.20 ± 0.15</td><td>0.60± 0.08</td></tr></table></body></html>", "type": "table", "image_path": "77d9acb6b86f9c9a84ea1fbf27649993b710e71540165032c1de6955bb3136fb.jpg"}]}], "index": 2.5, "virtual_lines": [{"bbox": [181, 125, 412, 139], "spans": [], "index": 0}, {"bbox": [181, 139, 412, 153], "spans": [], "index": 1}, {"bbox": [181, 153, 412, 167], "spans": [], "index": 2}, {"bbox": [181, 167, 412, 181], "spans": [], "index": 3}, {"bbox": [181, 181, 412, 195], "spans": [], "index": 4}, {"bbox": [181, 195, 412, 209], "spans": [], "index": 5}]}], "index": 2.5}, {"type": "text", "bbox": [101, 221, 492, 294], "lines": [{"bbox": [101, 224, 492, 238], "spans": [{"bbox": [101, 224, 492, 238], "score": 1.0, "content": "Table 2: Ratio of the numbers of observed and predicted electron-type and", "type": "text"}], "index": 6}, {"bbox": [101, 238, 492, 254], "spans": [{"bbox": [101, 238, 492, 254], "score": 1.0, "content": "muon-type neutrinos as a function of the flight length-to-energy ratio as mea-", "type": "text"}], "index": 7}, {"bbox": [102, 254, 491, 267], "spans": [{"bbox": [102, 254, 491, 267], "score": 1.0, "content": "sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty", "type": "text"}], "index": 8}, {"bbox": [102, 268, 491, 281], "spans": [{"bbox": [102, 268, 321, 281], "score": 1.0, "content": "on the absolute neutrino flux and because ", "type": "text"}, {"bbox": [321, 270, 334, 280], "score": 0.93, "content": "R_{e}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [335, 268, 491, 281], "score": 1.0, "content": " is observed to be independent", "type": "text"}], "index": 9}, {"bbox": [102, 282, 452, 296], "spans": [{"bbox": [102, 282, 115, 296], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [115, 283, 139, 296], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [139, 282, 411, 296], "score": 1.0, "content": ", we divide the numbers in this table by 1.15 so that ", "type": "text"}, {"bbox": [412, 284, 447, 294], "score": 0.92, "content": "R_{e}\\approx1", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 282, 452, 296], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 8, "bbox_fs": [101, 224, 492, 296]}, {"type": "text", "bbox": [101, 315, 492, 343], "lines": [{"bbox": [102, 317, 492, 331], "spans": [{"bbox": [102, 317, 460, 331], "score": 1.0, "content": "or down quarks), then the “upper island” in Figure 1 applies, and ", "type": "text"}, {"bbox": [460, 318, 492, 329], "score": 0.88, "content": "M_{3}\\,\\approx", "type": "inline_equation", "height": 11, "width": 32}], "index": 11}, {"bbox": [101, 331, 418, 345], "spans": [{"bbox": [101, 331, 144, 345], "score": 1.0, "content": "0.07eV, ", "type": "text"}, {"bbox": [144, 332, 213, 344], "score": 0.92, "content": "M_{2}\\approx10^{-5}\\mathrm{eV}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [214, 331, 239, 345], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 333, 288, 344], "score": 0.94, "content": "M_{1}<M_{2}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [288, 331, 418, 345], "score": 1.0, "content": ", with large uncertainties.", "type": "text"}], "index": 12}], "index": 11.5, "bbox_fs": [101, 317, 492, 345]}, {"type": "text", "bbox": [100, 344, 492, 545], "lines": [{"bbox": [119, 345, 491, 359], "spans": [{"bbox": [119, 345, 491, 359], "score": 1.0, "content": "Note in Table 1 that the ratio of the observed-to-predicted solar neu-", "type": "text"}], "index": 13}, {"bbox": [101, 359, 492, 375], "spans": [{"bbox": [101, 359, 492, 375], "score": 1.0, "content": "trino flux is significantly lower for the Homestake experiment than for Sage,", "type": "text"}], "index": 14}, {"bbox": [102, 376, 491, 388], "spans": [{"bbox": [102, 376, 491, 388], "score": 1.0, "content": "Gallex, Kamiokande and Super-Kamiokande which are all compatible with", "type": "text"}], "index": 15}, {"bbox": [101, 390, 491, 403], "spans": [{"bbox": [101, 390, 491, 403], "score": 1.0, "content": "0.5. These latter experiments observe neutrinos within wide energy bands,", "type": "text"}], "index": 16}, {"bbox": [101, 403, 492, 417], "spans": [{"bbox": [101, 403, 492, 417], "score": 1.0, "content": "while the chlorine detector in the Homestake mine observes monochromatic", "type": "text"}], "index": 17}, {"bbox": [101, 418, 492, 432], "spans": [{"bbox": [101, 418, 262, 432], "score": 1.0, "content": "electron-type neutrinos from a ", "type": "text"}, {"bbox": [262, 419, 282, 428], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [282, 418, 492, 432], "score": 1.0, "content": " line. For the Homestake experiment the", "type": "text"}], "index": 18}, {"bbox": [101, 433, 492, 447], "spans": [{"bbox": [101, 433, 153, 447], "score": 1.0, "content": "spread in ", "type": "text"}, {"bbox": [153, 434, 177, 446], "score": 0.94, "content": "L/E", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [177, 433, 300, 447], "score": 1.0, "content": " is due to the spread in ", "type": "text"}, {"bbox": [300, 434, 309, 443], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [309, 433, 492, 447], "score": 1.0, "content": ", which in turn is due to the excen-", "type": "text"}], "index": 19}, {"bbox": [101, 446, 492, 462], "spans": [{"bbox": [101, 446, 492, 462], "score": 1.0, "content": "tricity of the orbit of the Earth. Therefore the interference is coherent for up", "type": "text"}], "index": 20}, {"bbox": [101, 461, 492, 475], "spans": [{"bbox": [101, 461, 115, 475], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [115, 462, 227, 475], "score": 0.93, "content": "L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [227, 461, 452, 475], "score": 1.0, "content": " oscillations from the Sun to the Earth (here ", "type": "text"}, {"bbox": [453, 462, 479, 472], "score": 0.93, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 26}, {"bbox": [480, 461, 492, 475], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [101, 475, 492, 490], "spans": [{"bbox": [101, 475, 135, 490], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [135, 477, 165, 489], "score": 0.94, "content": "\\Delta M_{21}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [165, 475, 183, 490], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [183, 477, 213, 489], "score": 0.92, "content": "\\Delta M_{32}^{2}", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [214, 475, 396, 490], "score": 1.0, "content": "). Due to this coherence at “small” ", "type": "text"}, {"bbox": [397, 477, 424, 486], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [424, 475, 492, 490], "score": 1.0, "content": " it is possible", "type": "text"}], "index": 22}, {"bbox": [101, 490, 492, 504], "spans": [{"bbox": [101, 490, 272, 504], "score": 1.0, "content": "to find acceptable solutions with ", "type": "text"}, {"bbox": [273, 491, 322, 503], "score": 0.93, "content": "\\chi^{2}<13.4", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [322, 490, 492, 504], "score": 1.0, "content": " as shown in Figure 1. For larger", "type": "text"}], "index": 23}, {"bbox": [102, 505, 491, 518], "spans": [{"bbox": [102, 505, 151, 518], "score": 1.0, "content": "values of ", "type": "text"}, {"bbox": [152, 505, 179, 515], "score": 0.92, "content": "\\Delta M^{2}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [179, 505, 414, 518], "score": 1.0, "content": " coherence is lost and we find solutions with ", "type": "text"}, {"bbox": [415, 505, 457, 518], "score": 0.95, "content": "\\chi^{2}>18", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [457, 505, 491, 518], "score": 1.0, "content": " which", "type": "text"}], "index": 24}, {"bbox": [102, 519, 492, 532], "spans": [{"bbox": [102, 519, 492, 532], "score": 1.0, "content": "are unacceptable if the Homestake experimental and theoretical errors are", "type": "text"}], "index": 25}, {"bbox": [101, 535, 141, 546], "spans": [{"bbox": [101, 535, 141, 546], "score": 1.0, "content": "correct.", "type": "text"}], "index": 26}], "index": 19.5, "bbox_fs": [101, 345, 492, 546]}, {"type": "text", "bbox": [100, 546, 493, 719], "lines": [{"bbox": [120, 549, 492, 561], "spans": [{"bbox": [120, 549, 492, 561], "score": 1.0, "content": "An important test of the model would be to observe seasonal variations", "type": "text"}], "index": 27}, {"bbox": [101, 562, 492, 576], "spans": [{"bbox": [101, 562, 248, 576], "score": 1.0, "content": "of the neutrino flux of the ", "type": "text"}, {"bbox": [249, 563, 269, 573], "score": 0.92, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [269, 562, 492, 576], "score": 1.0, "content": " line. If the lower ratio measured by the", "type": "text"}], "index": 28}, {"bbox": [101, 577, 492, 591], "spans": [{"bbox": [101, 577, 492, 591], "score": 1.0, "content": "Homestake experiment is real, we expect that the electron-type neutrino flux", "type": "text"}], "index": 29}, {"bbox": [101, 591, 492, 605], "spans": [{"bbox": [101, 591, 134, 605], "score": 1.0, "content": "of the ", "type": "text"}, {"bbox": [134, 592, 154, 602], "score": 0.9, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [155, 591, 492, 605], "score": 1.0, "content": " line is near a minimum of the oscillation at the average Sun-Earth", "type": "text"}], "index": 30}, {"bbox": [101, 605, 492, 619], "spans": [{"bbox": [101, 605, 492, 619], "score": 1.0, "content": "distance. In other words, there are an odd number of half-wavelengths from", "type": "text"}], "index": 31}, {"bbox": [101, 620, 493, 633], "spans": [{"bbox": [101, 620, 365, 633], "score": 1.0, "content": "Sun to Earth. Then we expect a modulation of the ", "type": "text"}, {"bbox": [366, 621, 385, 631], "score": 0.88, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [386, 620, 493, 633], "score": 1.0, "content": " neutrino flux with a", "type": "text"}], "index": 32}, {"bbox": [100, 635, 493, 649], "spans": [{"bbox": [100, 635, 493, 649], "score": 1.0, "content": "period of half a year, with maximums occurring at the perihelion and aphelion", "type": "text"}], "index": 33}, {"bbox": [101, 649, 492, 663], "spans": [{"bbox": [101, 649, 492, 663], "score": 1.0, "content": "of the Earth orbit. We see no statistically significant Fourier component", "type": "text"}], "index": 34}, {"bbox": [101, 664, 492, 677], "spans": [{"bbox": [101, 664, 492, 677], "score": 1.0, "content": "of the time dependent Homestake data from 1970.281 to 1994.388.[12] In", "type": "text"}], "index": 35}, {"bbox": [101, 679, 494, 691], "spans": [{"bbox": [101, 679, 494, 691], "score": 1.0, "content": "particular the amplitude relative to the mean of a Fourier component of", "type": "text"}], "index": 36}, {"bbox": [101, 693, 491, 707], "spans": [{"bbox": [101, 693, 198, 707], "score": 1.0, "content": "period 0.5 years is ", "type": "text"}, {"bbox": [198, 695, 253, 704], "score": 0.87, "content": "0.09\\pm0.10", "type": "inline_equation", "height": 9, "width": 55}, {"bbox": [254, 693, 463, 707], "score": 1.0, "content": ". This observation implies that there are ", "type": "text"}, {"bbox": [463, 695, 491, 705], "score": 0.44, "content": "\\leq8.5", "type": "inline_equation", "height": 10, "width": 28}], "index": 37}, {"bbox": [101, 706, 491, 721], "spans": [{"bbox": [101, 706, 478, 721], "score": 1.0, "content": "periods of oscillation from Sun to Earth at 90% confidence level. With a ", "type": "text"}, {"bbox": [479, 708, 491, 720], "score": 0.91, "content": "\\chi^{2}", "type": "inline_equation", "height": 12, "width": 12}], "index": 38}], "index": 32.5, "bbox_fs": [100, 549, 494, 721]}]}
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[{"type": "table", "bbox": [181, 125, 412, 202], "content": "", "index": 0}, {"type": "text", "bbox": [101, 221, 492, 294], "content": "Table 2: Ratio of the numbers of observed and predicted electron-type and muon-type neutrinos as a function of the flight length-to-energy ratio as mea- sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty on the absolute neutrino flux and because is observed to be independent of , we divide the numbers in this table by 1.15 so that .", "index": 1}, {"type": "text", "bbox": [101, 315, 492, 343], "content": "or down quarks), then the “upper island” in Figure 1 applies, and 0.07eV, and , with large uncertainties.", "index": 2}, {"type": "text", "bbox": [100, 344, 492, 545], "content": "Note in Table 1 that the ratio of the observed-to-predicted solar neu- trino flux is significantly lower for the Homestake experiment than for Sage, Gallex, Kamiokande and Super-Kamiokande which are all compatible with 0.5. These latter experiments observe neutrinos within wide energy bands, while the chlorine detector in the Homestake mine observes monochromatic electron-type neutrinos from a line. For the Homestake experiment the spread in is due to the spread in , which in turn is due to the excen- tricity of the orbit of the Earth. Therefore the interference is coherent for up to oscillations from the Sun to the Earth (here is either or ). Due to this coherence at “small” it is possible to find acceptable solutions with as shown in Figure 1. For larger values of coherence is lost and we find solutions with which are unacceptable if the Homestake experimental and theoretical errors are correct.", "index": 3}, {"type": "text", "bbox": [100, 546, 493, 719], "content": "An important test of the model would be to observe seasonal variations of the neutrino flux of the line. If the lower ratio measured by the Homestake experiment is real, we expect that the electron-type neutrino flux of the line is near a minimum of the oscillation at the average Sun-Earth distance. In other words, there are an odd number of half-wavelengths from Sun to Earth. Then we expect a modulation of the neutrino flux with a period of half a year, with maximums occurring at the perihelion and aphelion of the Earth orbit. We see no statistically significant Fourier component of the time dependent Homestake data from 1970.281 to 1994.388.[12] In particular the amplitude relative to the mean of a Fourier component of period 0.5 years is . This observation implies that there are periods of oscillation from Sun to Earth at 90% confidence level. With a", "index": 4}]
|
[{"bbox": [101, 224, 492, 238], "content": "Table 2: Ratio of the numbers of observed and predicted electron-type and", "parent_index": 1, "line_index": 0}, {"bbox": [101, 238, 492, 254], "content": "muon-type neutrinos as a function of the flight length-to-energy ratio as mea-", "parent_index": 1, "line_index": 1}, {"bbox": [102, 254, 491, 267], "content": "sured by the Super-Kamiokande Collaboration.[8] Because of the uncertainty", "parent_index": 1, "line_index": 2}, {"bbox": [102, 268, 491, 281], "content": "on the absolute neutrino flux and because is observed to be independent", "parent_index": 1, "line_index": 3}, {"bbox": [102, 282, 452, 296], "content": "of , we divide the numbers in this table by 1.15 so that .", "parent_index": 1, "line_index": 4}, {"bbox": [102, 317, 492, 331], "content": "or down quarks), then the “upper island” in Figure 1 applies, and", "parent_index": 2, "line_index": 0}, {"bbox": [101, 331, 418, 345], "content": "0.07eV, and , with large uncertainties.", "parent_index": 2, "line_index": 1}, {"bbox": [119, 345, 491, 359], "content": "Note in Table 1 that the ratio of the observed-to-predicted solar neu-", "parent_index": 3, "line_index": 0}, {"bbox": [101, 359, 492, 375], "content": "trino flux is significantly lower for the Homestake experiment than for Sage,", "parent_index": 3, "line_index": 1}, {"bbox": [102, 376, 491, 388], "content": "Gallex, Kamiokande and Super-Kamiokande which are all compatible with", "parent_index": 3, "line_index": 2}, {"bbox": [101, 390, 491, 403], "content": "0.5. These latter experiments observe neutrinos within wide energy bands,", "parent_index": 3, "line_index": 3}, {"bbox": [101, 403, 492, 417], "content": "while the chlorine detector in the Homestake mine observes monochromatic", "parent_index": 3, "line_index": 4}, {"bbox": [101, 418, 492, 432], "content": "electron-type neutrinos from a line. For the Homestake experiment the", "parent_index": 3, "line_index": 5}, {"bbox": [101, 433, 492, 447], "content": "spread in is due to the spread in , which in turn is due to the excen-", "parent_index": 3, "line_index": 6}, {"bbox": [101, 446, 492, 462], "content": "tricity of the orbit of the Earth. Therefore the interference is coherent for up", "parent_index": 3, "line_index": 7}, {"bbox": [101, 461, 492, 475], "content": "to oscillations from the Sun to the Earth (here is", "parent_index": 3, "line_index": 8}, {"bbox": [101, 475, 492, 490], "content": "either or ). Due to this coherence at “small” it is possible", "parent_index": 3, "line_index": 9}, {"bbox": [101, 490, 492, 504], "content": "to find acceptable solutions with as shown in Figure 1. For larger", "parent_index": 3, "line_index": 10}, {"bbox": [102, 505, 491, 518], "content": "values of coherence is lost and we find solutions with which", "parent_index": 3, "line_index": 11}, {"bbox": [102, 519, 492, 532], "content": "are unacceptable if the Homestake experimental and theoretical errors are", "parent_index": 3, "line_index": 12}, {"bbox": [101, 535, 141, 546], "content": "correct.", "parent_index": 3, "line_index": 13}, {"bbox": [120, 549, 492, 561], "content": "An important test of the model would be to observe seasonal variations", "parent_index": 4, "line_index": 0}, {"bbox": [101, 562, 492, 576], "content": "of the neutrino flux of the line. If the lower ratio measured by the", "parent_index": 4, "line_index": 1}, {"bbox": [101, 577, 492, 591], "content": "Homestake experiment is real, we expect that the electron-type neutrino flux", "parent_index": 4, "line_index": 2}, {"bbox": [101, 591, 492, 605], "content": "of the line is near a minimum of the oscillation at the average Sun-Earth", "parent_index": 4, "line_index": 3}, {"bbox": [101, 605, 492, 619], "content": "distance. In other words, there are an odd number of half-wavelengths from", "parent_index": 4, "line_index": 4}, {"bbox": [101, 620, 493, 633], "content": "Sun to Earth. Then we expect a modulation of the neutrino flux with a", "parent_index": 4, "line_index": 5}, {"bbox": [100, 635, 493, 649], "content": "period of half a year, with maximums occurring at the perihelion and aphelion", "parent_index": 4, "line_index": 6}, {"bbox": [101, 649, 492, 663], "content": "of the Earth orbit. We see no statistically significant Fourier component", "parent_index": 4, "line_index": 7}, {"bbox": [101, 664, 492, 677], "content": "of the time dependent Homestake data from 1970.281 to 1994.388.[12] In", "parent_index": 4, "line_index": 8}, {"bbox": [101, 679, 494, 691], "content": "particular the amplitude relative to the mean of a Fourier component of", "parent_index": 4, "line_index": 9}, {"bbox": [101, 693, 491, 707], "content": "period 0.5 years is . This observation implies that there are", "parent_index": 4, "line_index": 10}, {"bbox": [101, 706, 491, 721], "content": "periods of oscillation from Sun to Earth at 90% confidence level. With a", "parent_index": 4, "line_index": 11}]
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[]
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[{"bbox": [321, 270, 334, 280], "content": "R_{e}", "parent_index": 1, "subtype": "inline"}, {"bbox": [115, 283, 139, 296], "content": "L/E", "parent_index": 1, "subtype": "inline"}, {"bbox": [412, 284, 447, 294], "content": "R_{e}\\approx1", "parent_index": 1, "subtype": "inline"}, {"bbox": [460, 318, 492, 329], "content": "M_{3}\\,\\approx", "parent_index": 2, "subtype": "inline"}, {"bbox": [144, 332, 213, 344], "content": "M_{2}\\approx10^{-5}\\mathrm{eV}", "parent_index": 2, "subtype": "inline"}, {"bbox": [240, 333, 288, 344], "content": "M_{1}<M_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [262, 419, 282, 428], "content": "^7B e", "parent_index": 3, "subtype": "inline"}, {"bbox": [153, 434, 177, 446], "content": "L/E", "parent_index": 3, "subtype": "inline"}, {"bbox": [300, 434, 309, 443], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [115, 462, 227, 475], "content": "L\\Delta M^{2}/(2E\\!\\cdot\\!2\\pi)\\approx30", "parent_index": 3, "subtype": "inline"}, {"bbox": [453, 462, 479, 472], "content": "\\Delta M^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [135, 477, 165, 489], "content": "\\Delta M_{21}^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [183, 477, 213, 489], "content": "\\Delta M_{32}^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [397, 477, 424, 486], "content": "\\Delta M^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [273, 491, 322, 503], "content": "\\chi^{2}<13.4", "parent_index": 3, "subtype": "inline"}, {"bbox": [152, 505, 179, 515], "content": "\\Delta M^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [415, 505, 457, 518], "content": "\\chi^{2}>18", "parent_index": 3, "subtype": "inline"}, {"bbox": [249, 563, 269, 573], "content": "^7B e", "parent_index": 4, "subtype": "inline"}, {"bbox": [134, 592, 154, 602], "content": "^7B e", "parent_index": 4, "subtype": "inline"}, {"bbox": [366, 621, 385, 631], "content": "^7B e", "parent_index": 4, "subtype": "inline"}, {"bbox": [198, 695, 253, 704], "content": "0.09\\pm0.10", "parent_index": 4, "subtype": "inline"}, {"bbox": [463, 695, 491, 705], "content": "\\leq8.5", "parent_index": 4, "subtype": "inline"}, {"bbox": [479, 708, 491, 720], "content": "\\chi^{2}", "parent_index": 4, "subtype": "inline"}]
|
[{"bbox": [181, 125, 412, 202], "content": "<html><body><table><tr><td>L/E [km/GeV]</td><td>Re</td><td>Rμ</td></tr><tr><td>10</td><td>1.20 ± 0.15</td><td>1.00 ± 0.15</td></tr><tr><td>100</td><td>1.20 ± 0.15</td><td>0.85 ± 0.12</td></tr><tr><td>1000</td><td>1.20 ± 0.15</td><td>0.70± 0.10</td></tr><tr><td>10000</td><td>1.20 ± 0.15</td><td>0.60± 0.08</td></tr></table></body></html>", "parent_index": 0, "subtype": "body"}]
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|
<html><body><table><tr><td>Probability</td><td>L/E [km/GeV]</td></tr><tr><td>P(ve → ve) > 0.99 P(vμ→vμ) >0.99 P(vμ(vμ)→ ve(ve)) < 0.90 · 10-3 P(vμ→v)< 0.002 P(vμ ←→v)<0.35 P(vμ →v)< 0.022 P(ve →v)< 0.125 P(ve ←> vμ)< 0.25 P(ve →ve) > 0.75[11]</td><td>88 0.34 0.31 0.039 31 0.053 0.031 40 232</td></tr></table></body></html>
Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1]
Table 4: Parameters at local minima of $\chi^{2}$ for 8 degrees of freedom.
<html><body><table><tr><td></td><td>M2 - M² [eV2]</td><td>M3 - M2 [eV2]</td><td>S23</td><td>S13</td><td>S12</td><td></td></tr><tr><td>7.0</td><td>4.9· 10-11</td><td>5.0·10-3</td><td>0.83</td><td>0.08</td><td>0.50</td><td>-3.14</td></tr><tr><td>7.2</td><td>1.6 · 10-10</td><td>5.0·10-3</td><td>0.57</td><td>0.00</td><td>0.74</td><td>-3.14</td></tr><tr><td>7.2</td><td>4.3· 10-10</td><td>5.0·10-3</td><td>0.57</td><td>0.00</td><td>0.74</td><td>3.14</td></tr></table></body></html>
Figure 1: The mass-squared differences $\left(M_{2}^{2}-M_{1}^{2},M_{3}^{2}-M_{2}^{2}\right)$ lie within the dots with $90\%$ confidence.
|
<html><body>
<div class="table" data-bbox="162 141 432 290"><table data-bbox="162 141 432 290"><tr><td>Probability</td><td>L/E [km/GeV]</td></tr><tr><td>P(ve → ve) > 0.99 P(vμ→vμ) >0.99 P(vμ(vμ)→ ve(ve)) < 0.90 · 10-3 P(vμ→v)< 0.002 P(vμ ←→v)<0.35 P(vμ →v)< 0.022 P(ve →v)< 0.125 P(ve ←> vμ)< 0.25 P(ve →ve) > 0.75[11]</td><td>88 0.34 0.31 0.039 31 0.053 0.031 40 232</td></tr></table></div>
<p data-bbox="101 309 493 339">Table 3: Limits on the mixing probabilities from astrophysical, accelerator and reactor experiments.[1] </p>
<div class="table" data-bbox="123 376 471 438"><p class="caption" data-bbox="122 459 471 474">Table 4: Parameters at local minima of $\chi^{2}$ for 8 degrees of freedom. </p><table data-bbox="123 376 471 438"><tr><td></td><td>M2 - M² [eV2]</td><td>M3 - M2 [eV2]</td><td>S23</td><td>S13</td><td>S12</td><td></td></tr><tr><td>7.0</td><td>4.9· 10-11</td><td>5.0·10-3</td><td>0.83</td><td>0.08</td><td>0.50</td><td>-3.14</td></tr><tr><td>7.2</td><td>1.6 · 10-10</td><td>5.0·10-3</td><td>0.57</td><td>0.00</td><td>0.74</td><td>-3.14</td></tr><tr><td>7.2</td><td>4.3· 10-10</td><td>5.0·10-3</td><td>0.57</td><td>0.00</td><td>0.74</td><td>3.14</td></tr></table></div>
<div class="image" data-bbox="178 503 406 659"><img data-bbox="178 503 406 659"/><p class="caption" data-bbox="100 662 494 692">Figure 1: The mass-squared differences $\left(M_{2}^{2}-M_{1}^{2},M_{3}^{2}-M_{2}^{2}\right)$ lie within the dots with $90\%$ confidence. </p></div>
</body></html>
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0002004v1
| 3 | 612 | 792 | 1,275 | 1,650 |
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[]
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[{"bbox": [162, 141, 432, 290], "content": "<html><body><table><tr><td>Probability</td><td>L/E [km/GeV]</td></tr><tr><td>P(ve → ve) > 0.99 P(vμ→vμ) >0.99 P(vμ(vμ)→ ve(ve)) < 0.90 · 10-3 P(vμ→v)< 0.002 P(vμ ←→v)<0.35 P(vμ →v)< 0.022 P(ve →v)< 0.125 P(ve ←> vμ)< 0.25 P(ve →ve) > 0.75[11]</td><td>88 0.34 0.31 0.039 31 0.053 0.031 40 232</td></tr></table></body></html>", "parent_index": 0, "subtype": "body"}, {"bbox": [123, 376, 471, 438], "content": "<html><body><table><tr><td></td><td>M2 - M² [eV2]</td><td>M3 - M2 [eV2]</td><td>S23</td><td>S13</td><td>S12</td><td></td></tr><tr><td>7.0</td><td>4.9· 10-11</td><td>5.0·10-3</td><td>0.83</td><td>0.08</td><td>0.50</td><td>-3.14</td></tr><tr><td>7.2</td><td>1.6 · 10-10</td><td>5.0·10-3</td><td>0.57</td><td>0.00</td><td>0.74</td><td>-3.14</td></tr><tr><td>7.2</td><td>4.3· 10-10</td><td>5.0·10-3</td><td>0.57</td><td>0.00</td><td>0.74</td><td>3.14</td></tr></table></body></html>", "parent_index": 2, "subtype": "body"}, {"bbox": [122, 459, 471, 474], "content": "", "parent_index": 2, "subtype": "caption"}]
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Figure 2: Detail of the “upper island” of Figure 1 for the fit with 116 degrees of freedom (see text) at $90\%$ confidence level. The “lower island” is symmetrical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the $^7B e$ line.
with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2.
The reliability of $M_{2}^{2}\mathrm{~-~}M_{1}^{2}$ depends on the correctness of the error assigned to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3.
In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from antineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge.
Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements $D$ , $^{3}H e$ , $^4H e$ and $^7L i$ . These abundances are determined by the temperatures of freezout $T_{f}\approx1M e V$ when the reaction rates $\propto T_{f}^{5}$ become comparable to the expansion rate $\propto T_{f}^{2}\times(5.5+\textstyle{\frac{7}{4}}{N_{\nu}})^{1/2}$ . Here $N_{\nu}$ is the equivalent number of massless neutrino flavors that are ultrarelativistic at $T_{f}$ and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if $1.6\,\leq\,N_{\nu}\,\leq\,4.0$ at $95\%$ confidence level.[1] For three generations of Majorana neutrinos, $N_{\nu}=3$ . For three generations of Dirac neutrinos, $N_{\nu}\,=\,6$ while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to $Z$ , $W^{+}$ and $W^{-}$ . Right-handed neutrinos are not in thermal equilibrium at
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<html><body>
<div class="image" data-bbox="179 119 407 273"><img data-bbox="179 119 407 273"/><p class="caption" data-bbox="100 277 493 335">Figure 2: Detail of the “upper island” of Figure 1 for the fit with 116 degrees of freedom (see text) at $90\%$ confidence level. The “lower island” is symmetrical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the $^7B e$ line. </p></div>
<p data-bbox="102 366 491 409">with 116 degrees of freedom, including the 8 discussed earlier plus the 108 measurements by the Homestake Collaboration from 1970.281 to 1994.388[12] we obtain the allowed region shown in Figure 2. </p>
<p data-bbox="101 410 492 467">The reliability of $M_{2}^{2}\mathrm{~-~}M_{1}^{2}$ depends on the correctness of the error assigned to the Homestake observed-to-predicted flux ratio. For example, if the Homestake error listed in Table 1 is doubled we obtain the solutions shown in Figure 3. </p>
<p data-bbox="101 468 492 539">In view of the preceeding results let us assume that neutrinos indeed have mass. The question then arizes wether neutrinos are distinct from antineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles (Majorana neutrinos). This latter possibility arizes because neutrinos have no electric charge. </p>
<p data-bbox="101 540 493 715">Let us consider Big-Bang nucleosynthesis that determines the abundances of the light elements $D$ , $^{3}H e$ , $^4H e$ and $^7L i$ . These abundances are determined by the temperatures of freezout $T_{f}\approx1M e V$ when the reaction rates $\propto T_{f}^{5}$ become comparable to the expansion rate $\propto T_{f}^{2}\times(5.5+\textstyle{\frac{7}{4}}{N_{\nu}})^{1/2}$ . Here $N_{\nu}$ is the equivalent number of massless neutrino flavors that are ultrarelativistic at $T_{f}$ and are still in thermal equilibrium with photons and electrons at that temperature. The calculated abundances of the light elements are in agreement with observations if $1.6\,\leq\,N_{\nu}\,\leq\,4.0$ at $95\%$ confidence level.[1] For three generations of Majorana neutrinos, $N_{\nu}=3$ . For three generations of Dirac neutrinos, $N_{\nu}\,=\,6$ while in thermal equilibrium. However, in the Standard Model only the left-handed component of neutrinos couple to $Z$ , $W^{+}$ and $W^{-}$ . Right-handed neutrinos are not in thermal equilibrium at </p>
</body></html>
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0002004v1
| 4 | 612 | 792 | 1,275 | 1,650 |
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The “lower island” is symmet-", "type": "text"}], "index": 13}, {"bbox": [101, 309, 492, 324], "spans": [{"bbox": [101, 309, 492, 324], "score": 1.0, "content": "rical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5", "type": "text"}], "index": 14}, {"bbox": [101, 323, 381, 337], "spans": [{"bbox": [101, 323, 335, 337], "score": 1.0, "content": "and 7.5 oscillations from Sun to Earth of the ", "type": "text"}, {"bbox": [336, 324, 356, 334], "score": 0.89, "content": "^7B e", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [356, 323, 381, 337], "score": 1.0, "content": " line.", "type": "text"}], "index": 15}], "index": 13.5}], "index": 9.5}, {"type": "text", "bbox": [102, 366, 491, 409], "lines": [{"bbox": [103, 369, 492, 383], "spans": [{"bbox": [103, 369, 492, 383], "score": 1.0, "content": "with 116 degrees of freedom, including the 8 discussed earlier plus the 108", "type": "text"}], "index": 16}, {"bbox": [101, 383, 490, 396], "spans": [{"bbox": [101, 383, 490, 396], "score": 1.0, "content": "measurements by the Homestake Collaboration from 1970.281 to 1994.388[12]", "type": "text"}], "index": 17}, {"bbox": [102, 397, 348, 411], "spans": [{"bbox": [102, 397, 348, 411], "score": 1.0, "content": "we obtain the allowed region shown in Figure 2.", "type": "text"}], "index": 18}], "index": 17}, {"type": "text", "bbox": [101, 410, 492, 467], "lines": [{"bbox": [119, 411, 492, 426], "spans": [{"bbox": [119, 411, 212, 426], "score": 1.0, "content": "The reliability of ", "type": "text"}, {"bbox": [212, 412, 262, 425], "score": 0.95, "content": "M_{2}^{2}\\mathrm{~-~}M_{1}^{2}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [263, 411, 492, 426], "score": 1.0, "content": " depends on the correctness of the error as-", "type": "text"}], "index": 19}, {"bbox": [102, 427, 491, 439], "spans": [{"bbox": [102, 427, 491, 439], "score": 1.0, "content": "signed to the Homestake observed-to-predicted flux ratio. For example, if the", "type": "text"}], "index": 20}, {"bbox": [102, 441, 492, 453], "spans": [{"bbox": [102, 441, 492, 453], "score": 1.0, "content": "Homestake error listed in Table 1 is doubled we obtain the solutions shown", "type": "text"}], "index": 21}, {"bbox": [101, 455, 163, 468], "spans": [{"bbox": [101, 455, 163, 468], "score": 1.0, "content": "in Figure 3.", "type": "text"}], "index": 22}], "index": 20.5}, {"type": "text", "bbox": [101, 468, 492, 539], "lines": [{"bbox": [118, 470, 492, 483], "spans": [{"bbox": [118, 470, 492, 483], "score": 1.0, "content": "In view of the preceeding results let us assume that neutrinos indeed", "type": "text"}], "index": 23}, {"bbox": [100, 483, 492, 497], "spans": [{"bbox": [100, 483, 492, 497], "score": 1.0, "content": "have mass. The question then arizes wether neutrinos are distinct from an-", "type": "text"}], "index": 24}, {"bbox": [102, 499, 492, 512], "spans": [{"bbox": [102, 499, 492, 512], "score": 1.0, "content": "tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles", "type": "text"}], "index": 25}, {"bbox": [103, 513, 492, 527], "spans": [{"bbox": [103, 513, 492, 527], "score": 1.0, "content": "(Majorana neutrinos). This latter possibility arizes because neutrinos have", "type": "text"}], "index": 26}, {"bbox": [102, 529, 195, 541], "spans": [{"bbox": [102, 529, 195, 541], "score": 1.0, "content": "no electric charge.", "type": "text"}], "index": 27}], "index": 25}, {"type": "text", "bbox": [101, 540, 493, 715], "lines": [{"bbox": [118, 540, 493, 556], "spans": [{"bbox": [118, 540, 493, 556], "score": 1.0, "content": "Let us consider Big-Bang nucleosynthesis that determines the abundances", "type": "text"}], "index": 28}, {"bbox": [101, 556, 492, 569], "spans": [{"bbox": [101, 556, 206, 569], "score": 1.0, "content": "of the light elements ", "type": "text"}, {"bbox": [207, 558, 217, 567], "score": 0.83, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [217, 556, 222, 569], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [222, 557, 244, 567], "score": 0.78, "content": "^{3}H e", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [244, 556, 249, 569], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [249, 557, 271, 567], "score": 0.63, "content": "^4H e", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [271, 556, 294, 569], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [295, 557, 312, 567], "score": 0.71, "content": "^7L i", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [312, 556, 492, 569], "score": 1.0, "content": ". These abundances are determined", "type": "text"}], "index": 29}, {"bbox": [101, 569, 491, 586], "spans": [{"bbox": [101, 569, 270, 586], "score": 1.0, "content": "by the temperatures of freezout ", "type": "text"}, {"bbox": [270, 572, 334, 584], "score": 0.93, "content": "T_{f}\\approx1M e V", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [334, 569, 464, 586], "score": 1.0, "content": " when the reaction rates ", "type": "text"}, {"bbox": [464, 571, 491, 586], "score": 0.93, "content": "\\propto T_{f}^{5}", "type": "inline_equation", "height": 15, "width": 27}], "index": 30}, {"bbox": [99, 584, 495, 605], "spans": [{"bbox": [99, 584, 317, 605], "score": 1.0, "content": "become comparable to the expansion rate ", "type": "text"}, {"bbox": [318, 586, 429, 601], "score": 0.94, "content": "\\propto T_{f}^{2}\\times(5.5+\\textstyle{\\frac{7}{4}}{N_{\\nu}})^{1/2}", "type": "inline_equation", "height": 15, "width": 111}, {"bbox": [429, 584, 464, 605], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [465, 588, 479, 599], "score": 0.93, "content": "N_{\\nu}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [480, 584, 495, 605], "score": 1.0, "content": " is", "type": "text"}], "index": 31}, {"bbox": [101, 601, 492, 615], "spans": [{"bbox": [101, 601, 492, 615], "score": 1.0, "content": "the equivalent number of massless neutrino flavors that are ultrarelativistic", "type": "text"}], "index": 32}, {"bbox": [101, 616, 493, 630], "spans": [{"bbox": [101, 616, 118, 629], "score": 1.0, "content": "at ", "type": "text"}, {"bbox": [118, 618, 131, 630], "score": 0.92, "content": "T_{f}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [131, 616, 493, 629], "score": 1.0, "content": " and are still in thermal equilibrium with photons and electrons at", "type": "text"}], "index": 33}, {"bbox": [101, 631, 492, 644], "spans": [{"bbox": [101, 631, 492, 644], "score": 1.0, "content": "that temperature. The calculated abundances of the light elements are in", "type": "text"}], "index": 34}, {"bbox": [101, 645, 491, 658], "spans": [{"bbox": [101, 645, 266, 658], "score": 1.0, "content": "agreement with observations if ", "type": "text"}, {"bbox": [266, 646, 349, 657], "score": 0.92, "content": "1.6\\,\\leq\\,N_{\\nu}\\,\\leq\\,4.0", "type": "inline_equation", "height": 11, "width": 83}, {"bbox": [349, 645, 368, 658], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [369, 645, 390, 656], "score": 0.43, "content": "95\\%", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [391, 645, 491, 658], "score": 1.0, "content": " confidence level.[1]", "type": "text"}], "index": 35}, {"bbox": [101, 658, 492, 673], "spans": [{"bbox": [101, 658, 336, 673], "score": 1.0, "content": "For three generations of Majorana neutrinos, ", "type": "text"}, {"bbox": [337, 660, 373, 671], "score": 0.93, "content": "N_{\\nu}=3", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [374, 658, 492, 673], "score": 1.0, "content": ". For three generations", "type": "text"}], "index": 36}, {"bbox": [101, 673, 492, 687], "spans": [{"bbox": [101, 673, 204, 687], "score": 1.0, "content": "of Dirac neutrinos, ", "type": "text"}, {"bbox": [205, 675, 244, 685], "score": 0.93, "content": "N_{\\nu}\\,=\\,6", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [244, 673, 492, 687], "score": 1.0, "content": " while in thermal equilibrium. However, in the", "type": "text"}], "index": 37}, {"bbox": [101, 687, 492, 702], "spans": [{"bbox": [101, 687, 478, 702], "score": 1.0, "content": "Standard Model only the left-handed component of neutrinos couple to ", "type": "text"}, {"bbox": [479, 689, 488, 698], "score": 0.89, "content": "Z", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 687, 492, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 38}, {"bbox": [102, 702, 492, 716], "spans": [{"bbox": [102, 703, 122, 712], "score": 0.92, "content": "W^{+}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [123, 702, 151, 716], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [151, 703, 172, 713], "score": 0.92, "content": "W^{-}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [172, 702, 492, 716], "score": 1.0, "content": ". 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The question then arizes wether neutrinos are distinct from an-", "type": "text"}], "index": 24}, {"bbox": [102, 499, 492, 512], "spans": [{"bbox": [102, 499, 492, 512], "score": 1.0, "content": "tineutrinos (Dirac neutrinos) or wether neutrinos are their own antiparticles", "type": "text"}], "index": 25}, {"bbox": [103, 513, 492, 527], "spans": [{"bbox": [103, 513, 492, 527], "score": 1.0, "content": "(Majorana neutrinos). 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These abundances are determined", "type": "text"}], "index": 29}, {"bbox": [101, 569, 491, 586], "spans": [{"bbox": [101, 569, 270, 586], "score": 1.0, "content": "by the temperatures of freezout ", "type": "text"}, {"bbox": [270, 572, 334, 584], "score": 0.93, "content": "T_{f}\\approx1M e V", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [334, 569, 464, 586], "score": 1.0, "content": " when the reaction rates ", "type": "text"}, {"bbox": [464, 571, 491, 586], "score": 0.93, "content": "\\propto T_{f}^{5}", "type": "inline_equation", "height": 15, "width": 27}], "index": 30}, {"bbox": [99, 584, 495, 605], "spans": [{"bbox": [99, 584, 317, 605], "score": 1.0, "content": "become comparable to the expansion rate ", "type": "text"}, {"bbox": [318, 586, 429, 601], "score": 0.94, "content": "\\propto T_{f}^{2}\\times(5.5+\\textstyle{\\frac{7}{4}}{N_{\\nu}})^{1/2}", "type": "inline_equation", "height": 15, "width": 111}, {"bbox": [429, 584, 464, 605], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [465, 588, 479, 599], "score": 0.93, "content": "N_{\\nu}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [480, 584, 495, 605], "score": 1.0, "content": " is", "type": "text"}], "index": 31}, {"bbox": [101, 601, 492, 615], "spans": [{"bbox": [101, 601, 492, 615], "score": 1.0, "content": "the equivalent number of massless neutrino flavors that are ultrarelativistic", "type": "text"}], "index": 32}, {"bbox": [101, 616, 493, 630], "spans": [{"bbox": [101, 616, 118, 629], "score": 1.0, "content": "at ", "type": "text"}, {"bbox": [118, 618, 131, 630], "score": 0.92, "content": "T_{f}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [131, 616, 493, 629], "score": 1.0, "content": " and are still in thermal equilibrium with photons and electrons at", "type": "text"}], "index": 33}, {"bbox": [101, 631, 492, 644], "spans": [{"bbox": [101, 631, 492, 644], "score": 1.0, "content": "that temperature. The calculated abundances of the light elements are in", "type": "text"}], "index": 34}, {"bbox": [101, 645, 491, 658], "spans": [{"bbox": [101, 645, 266, 658], "score": 1.0, "content": "agreement with observations if ", "type": "text"}, {"bbox": [266, 646, 349, 657], "score": 0.92, "content": "1.6\\,\\leq\\,N_{\\nu}\\,\\leq\\,4.0", "type": "inline_equation", "height": 11, "width": 83}, {"bbox": [349, 645, 368, 658], "score": 1.0, "content": " at ", "type": "text"}, {"bbox": [369, 645, 390, 656], "score": 0.43, "content": "95\\%", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [391, 645, 491, 658], "score": 1.0, "content": " confidence level.[1]", "type": "text"}], "index": 35}, {"bbox": [101, 658, 492, 673], "spans": [{"bbox": [101, 658, 336, 673], "score": 1.0, "content": "For three generations of Majorana neutrinos, ", "type": "text"}, {"bbox": [337, 660, 373, 671], "score": 0.93, "content": "N_{\\nu}=3", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [374, 658, 492, 673], "score": 1.0, "content": ". For three generations", "type": "text"}], "index": 36}, {"bbox": [101, 673, 492, 687], "spans": [{"bbox": [101, 673, 204, 687], "score": 1.0, "content": "of Dirac neutrinos, ", "type": "text"}, {"bbox": [205, 675, 244, 685], "score": 0.93, "content": "N_{\\nu}\\,=\\,6", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [244, 673, 492, 687], "score": 1.0, "content": " while in thermal equilibrium. However, in the", "type": "text"}], "index": 37}, {"bbox": [101, 687, 492, 702], "spans": [{"bbox": [101, 687, 478, 702], "score": 1.0, "content": "Standard Model only the left-handed component of neutrinos couple to ", "type": "text"}, {"bbox": [479, 689, 488, 698], "score": 0.89, "content": "Z", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 687, 492, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 38}, {"bbox": [102, 702, 492, 716], "spans": [{"bbox": [102, 703, 122, 712], "score": 0.92, "content": "W^{+}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [123, 702, 151, 716], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [151, 703, 172, 713], "score": 0.92, "content": "W^{-}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [172, 702, 492, 716], "score": 1.0, "content": ". Right-handed neutrinos are not in thermal equilibrium at", "type": "text"}], "index": 39}], "index": 33.5, "bbox_fs": [99, 540, 495, 716]}]}
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[{"bbox": [179, 119, 407, 273], "content": "", "parent_index": 0, "subtype": "body"}, {"bbox": [100, 277, 493, 335], "content": "Figure 2: Detail of the “upper island” of Figure 1 for the fit with 116 degrees of freedom (see text) at confidence level. The “lower island” is symmet- rical. The vertical bands correspond, from left to right, to 2.5, 3.5, 4.5, 6.5 and 7.5 oscillations from Sun to Earth of the line.", "parent_index": 0, "subtype": "caption"}]
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[{"bbox": [212, 412, 262, 425], "content": "M_{2}^{2}\\mathrm{~-~}M_{1}^{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [207, 558, 217, 567], "content": "D", "parent_index": 4, "subtype": "inline"}, {"bbox": [222, 557, 244, 567], "content": "^{3}H e", "parent_index": 4, "subtype": "inline"}, {"bbox": [249, 557, 271, 567], "content": "^4H e", "parent_index": 4, "subtype": "inline"}, {"bbox": [295, 557, 312, 567], "content": "^7L i", "parent_index": 4, "subtype": "inline"}, {"bbox": [270, 572, 334, 584], "content": "T_{f}\\approx1M e V", "parent_index": 4, "subtype": "inline"}, {"bbox": [464, 571, 491, 586], "content": "\\propto T_{f}^{5}", "parent_index": 4, "subtype": "inline"}, {"bbox": [318, 586, 429, 601], "content": "\\propto T_{f}^{2}\\times(5.5+\\textstyle{\\frac{7}{4}}{N_{\\nu}})^{1/2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [465, 588, 479, 599], "content": "N_{\\nu}", "parent_index": 4, "subtype": "inline"}, {"bbox": [118, 618, 131, 630], "content": "T_{f}", "parent_index": 4, "subtype": "inline"}, {"bbox": [266, 646, 349, 657], "content": "1.6\\,\\leq\\,N_{\\nu}\\,\\leq\\,4.0", "parent_index": 4, "subtype": "inline"}, {"bbox": [369, 645, 390, 656], "content": "95\\%", "parent_index": 4, "subtype": "inline"}, {"bbox": [337, 660, 373, 671], "content": "N_{\\nu}=3", "parent_index": 4, "subtype": "inline"}, {"bbox": [205, 675, 244, 685], "content": "N_{\\nu}\\,=\\,6", "parent_index": 4, "subtype": "inline"}, {"bbox": [479, 689, 488, 698], "content": "Z", "parent_index": 4, "subtype": "inline"}, {"bbox": [102, 703, 122, 712], "content": "W^{+}", "parent_index": 4, "subtype": "inline"}, {"bbox": [151, 703, 172, 713], "content": "W^{-}", "parent_index": 4, "subtype": "inline"}]
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[]
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Figure 3: Same as Figure 1 but we have doubled the error of the Homestake experiment, i.e. $R=0.33\pm0.10$ .
$T_{f}$ : their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the righthanded neutrinos. Therefore for Dirac neutrinos at $T_{f}$ we have $N_{\nu}\approx3$ . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis.
In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the $^7B e$ spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar $p p$ reaction.
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[3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca.
[4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364
[5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683
[6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158
[7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1
[8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562
[9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650
[10] Hill, Phys. Rev. Lett. 75 (1995) 2654
[11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN ’99 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disapearence experiment, unpublished.
[12] Cleveland et.al., Astrophysical Journal 496 (1998), 505
|
<html><body>
<div class="image" data-bbox="179 119 407 273"><img data-bbox="179 119 407 273"/><p class="caption" data-bbox="101 278 492 306">Figure 3: Same as Figure 1 but we have doubled the error of the Homestake experiment, i.e. $R=0.33\pm0.10$ . </p></div>
<p data-bbox="101 339 492 411">$T_{f}$ : their temperature has lagged below the temperature of photons due to the anihilation of particle-antiparticle pairs after the decoupling of the righthanded neutrinos. Therefore for Dirac neutrinos at $T_{f}$ we have $N_{\nu}\approx3$ . So we can not distinguish Dirac from Majorana neutrinos using available data on nucleosynthesis. </p>
<p data-bbox="101 412 492 498">In conclusion, the minimal extension of the Standard Model with three massive Majorana or Dirac neutrinos that mix is in good agreement with all experimental constraints. However, confirmation of the model is needed, e.g. by the observation of seasonal variations of the $^7B e$ spectral lines with a period of 0.5 years, or spectral distortions and seasonal variations of the low energy neutrinos from the solar $p p$ reaction. </p>
<h1 data-bbox="101 520 194 538">References </h1>
<p data-bbox="100 550 493 711">] Review of Particle Physics, The European Physical Journal C3 (1998) 1
[2] B.T. Cleveland et al., Astrophys. J. 496 (1998) 505; B.T. Cleveland et al., Nucl. Phys. B (Proc. Suppl.) 38 (1995) 47; R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13
[3] SAGE Collab., V. Gavrin et al., in Neutrino 98, Proceedings of the XVIII International Conference on Neutrino Physics and Astrophysics, Takayama, Japan, 1998, edited by Y. Suzuki and Y. Totsuca.
[4] GALLEX Collab., P. Anselmann et al., Phys. Lett. B342, (1995) 440; GALLEX Collaboration, W. Hampel et al., Phys. Lett. B388 (1996) 364
[5] KAMIOKANDE Collab., Y. Fukuda et al., Phys. Rev. Lett. 77 (1996) 1683
[6] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1158
[7] J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B433 (1998) 1
[8] Super-Kamiokande Collab., Phys. Rev. Lett. 81 (1998) 1562
[9] LSND Collab., Phys. Rev. Lett. 75 (1995) 2650
[10] Hill, Phys. Rev. Lett. 75 (1995) 2654
[11] “Neutrino Oscillation Experiments at Nuclear Reactors”, G. Gratta. 17th International Workshop on Weak Interactions and Neutrinos : WIN ’99 Cape Town, South Africa ; 24-30 Jan 1999 . Palo Verde reactor disapearence experiment, unpublished.
[12] Cleveland et.al., Astrophysical Journal 496 (1998), 505 </p>
</body></html>
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0002004v1
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So", "type": "text"}], "index": 16}, {"bbox": [102, 385, 492, 400], "spans": [{"bbox": [102, 385, 492, 400], "score": 1.0, "content": "we can not distinguish Dirac from Majorana neutrinos using available data", "type": "text"}], "index": 17}, {"bbox": [102, 399, 199, 414], "spans": [{"bbox": [102, 399, 199, 414], "score": 1.0, "content": "on nucleosynthesis.", "type": "text"}], "index": 18}], "index": 16}, {"type": "text", "bbox": [101, 412, 492, 498], "lines": [{"bbox": [118, 415, 492, 426], "spans": [{"bbox": [118, 415, 492, 426], "score": 1.0, "content": "In conclusion, the minimal extension of the Standard Model with three", "type": "text"}], "index": 19}, {"bbox": [102, 429, 492, 442], "spans": [{"bbox": [102, 429, 492, 442], "score": 1.0, "content": "massive Majorana or Dirac neutrinos that mix is in good agreement with", "type": "text"}], "index": 20}, {"bbox": [102, 443, 491, 455], "spans": [{"bbox": [102, 443, 491, 455], "score": 1.0, "content": "all experimental constraints. 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[]
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# Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper
# Percolation properties of the 2D Heisenberg model
B. Allés $\mathrm{a}$ , J. J. Alonso $\mathrm{b}$ , C. Criado $\mathrm{b}$ , M. Pepe $\mathrm{c}$ $\mathrm{a}$ Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy
$^\mathrm{b}$ Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain
The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims.
Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless $[K T]$ phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for $T\,>\,0.5$ ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate.
As for the points that appear in the Comment:
• (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This scenario is based mainly on three hypotheses, the third one (which states the non–percolation of the $\boldsymbol{S}$ –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for $\epsilon=1/3$ ,... There is no indication of percolation...”. On the contrary, such interesting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temperature $T$ and consequently it requires a huge lattice size. If “Everybody agrees that at $\beta\:=\:2.0$ the standard action model has a finite correlation length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature.
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<html><body>
<h1 data-bbox="134 142 464 157">Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper </h1>
<h1 data-bbox="110 170 487 206">Percolation properties of the 2D Heisenberg model </h1>
<p data-bbox="163 245 434 261">B. Allés $\mathrm{a}$ , J. J. Alonso $\mathrm{b}$ , C. Criado $\mathrm{b}$ , M. Pepe $\mathrm{c}$ $\mathrm{a}$ Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy </p>
<p data-bbox="87 310 509 324">$^\mathrm{b}$ Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain </p>
<p data-bbox="99 385 497 443">The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims. </p>
<p data-bbox="99 448 498 564">Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless $[K T]$ phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for $T\,>\,0.5$ ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate. </p>
<p data-bbox="117 583 358 597">As for the points that appear in the Comment: </p>
<p data-bbox="116 615 497 717">• (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This scenario is based mainly on three hypotheses, the third one (which states the non–percolation of the $\boldsymbol{S}$ –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for $\epsilon=1/3$ ,... There is no indication of percolation...”. On the contrary, such interesting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temperature $T$ and consequently it requires a huge lattice size. If “Everybody agrees that at $\beta\:=\:2.0$ the standard action model has a finite correlation length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature. </p>
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0002014v1
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[2] are based on claims which have not been written in [2], for instance", "type": "text"}], "index": 8}, {"bbox": [100, 417, 497, 431], "spans": [{"bbox": [100, 417, 497, 431], "score": 1.0, "content": "almost all the introduction and the point (1) in [1] are based on such non–", "type": "text"}], "index": 9}, {"bbox": [100, 432, 179, 444], "spans": [{"bbox": [100, 432, 179, 444], "score": 1.0, "content": "existent claims.", "type": "text"}], "index": 10}], "index": 8.5, "bbox_fs": [99, 387, 498, 444]}, {"type": "text", "bbox": [99, 448, 498, 564], "lines": [{"bbox": [117, 451, 497, 464], "spans": [{"bbox": [117, 451, 497, 464], "score": 1.0, "content": "Instead in Ref. [2] we avoid to make claims not based on well–founded", "type": "text"}], "index": 11}, {"bbox": [99, 464, 497, 479], "spans": [{"bbox": [99, 464, 497, 479], "score": 1.0, "content": "results. For instance in the abstract we write “... 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To back up that hypothesis a numerical", "type": "text"}], "index": 26}, {"bbox": [128, 94, 499, 107], "spans": [{"bbox": [128, 94, 499, 107], "score": 1.0, "content": "test was cited in [3] but the details of the numerics (temperature, size of", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [128, 108, 497, 122], "spans": [{"bbox": [128, 108, 497, 122], "score": 1.0, "content": "the lattice, etc.) and several data concerning the percolation properties", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [128, 123, 498, 137], "spans": [{"bbox": [128, 123, 498, 137], "score": 1.0, "content": "of the system, were completely skipped. The only quoted result was (see", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [128, 136, 497, 152], "spans": [{"bbox": [128, 136, 445, 152], "score": 1.0, "content": "beginning of section 4 in [3]) “We also tested numerically for ", "type": "text", "cross_page": true}, {"bbox": [445, 138, 484, 151], "score": 0.81, "content": "\\epsilon=1/3", "type": "inline_equation", "height": 13, "width": 39, "cross_page": true}, {"bbox": [484, 136, 497, 152], "score": 1.0, "content": ",...", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [128, 151, 496, 165], "spans": [{"bbox": [128, 151, 496, 165], "score": 1.0, "content": "There is no indication of percolation...”. On the contrary, such inter-", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [129, 167, 499, 180], "spans": [{"bbox": [129, 167, 499, 180], "score": 1.0, "content": "esting results about the critical properties should be put forward with a", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [128, 181, 497, 194], "spans": [{"bbox": [128, 181, 497, 194], "score": 1.0, "content": "thorough description of the hypotheses involved. Moreover, one would", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [128, 194, 497, 209], "spans": [{"bbox": [128, 194, 497, 209], "score": 1.0, "content": "like to understand how was possible to use the small value of epsilon", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [128, 209, 497, 223], "spans": [{"bbox": [128, 209, 497, 223], "score": 1.0, "content": "mentioned in Ref. [3], because that value implies a really tiny temper-", "type": "text", "cross_page": true}], "index": 8}, {"bbox": [129, 224, 497, 237], "spans": [{"bbox": [129, 224, 159, 237], "score": 1.0, "content": "ature ", "type": "text", "cross_page": true}, {"bbox": [159, 226, 168, 234], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9, "cross_page": true}, {"bbox": [169, 224, 497, 237], "score": 1.0, "content": " and consequently it requires a huge lattice size. If “Everybody", "type": "text", "cross_page": true}], "index": 9}, {"bbox": [128, 239, 497, 251], "spans": [{"bbox": [128, 239, 206, 251], "score": 1.0, "content": "agrees that at ", "type": "text", "cross_page": true}, {"bbox": [206, 240, 248, 251], "score": 0.92, "content": "\\beta\\:=\\:2.0", "type": "inline_equation", "height": 11, "width": 42, "cross_page": true}, {"bbox": [248, 239, 497, 251], "score": 1.0, "content": " the standard action model has a finite correla-", "type": "text", "cross_page": true}], "index": 10}, {"bbox": [128, 253, 499, 268], "spans": [{"bbox": [128, 253, 499, 268], "score": 1.0, "content": "tion length”, see [1], also everybody would like to know details about the", "type": "text", "cross_page": true}], "index": 11}, {"bbox": [128, 268, 498, 280], "spans": [{"bbox": [128, 268, 498, 280], "score": 1.0, "content": "numerics and the computer used to simulate the model at such a small", "type": "text", "cross_page": true}], "index": 12}, {"bbox": [128, 282, 196, 296], "spans": [{"bbox": [128, 282, 196, 296], "score": 1.0, "content": "temperature.", "type": "text", "cross_page": true}], "index": 13}], "index": 23, "bbox_fs": [118, 618, 498, 718]}]}
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[{"type": "title", "bbox": [134, 142, 464, 157], "content": "Reply to A. Patrascioiu’s and E. Seiler’s comment on our paper", "index": 0}, {"type": "title", "bbox": [110, 170, 487, 206], "content": "Percolation properties of the 2D Heisenberg model", "index": 1}, {"type": "text", "bbox": [163, 245, 434, 261], "content": "B. Allés , J. J. Alonso , C. Criado , M. Pepe Dipartimento di Fisica, Universita di Milano-Bicocca and INFN Sezione di Milano, Milano, Italy", "index": 2}, {"type": "text", "bbox": [160, 275, 437, 304], "content": "", "index": 3}, {"type": "text", "bbox": [87, 310, 509, 324], "content": "Departamento de Fisica Aplicada I, Facultad de Ciencias, 29071 Malaga, Spain", "index": 4}, {"type": "text", "bbox": [99, 385, 497, 443], "content": "The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims.", "index": 5}, {"type": "text", "bbox": [99, 448, 498, 564], "content": "Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write “... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless phase transition...” and in the conclusive part we notice that “Our results exclude this massless phase for ”. Therefore it seems to us that the opening sentence in the Comment [1] “In a recent letter All´es et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase.” is strongly inadequate.", "index": 6}, {"type": "text", "bbox": [117, 583, 358, 597], "content": "As for the points that appear in the Comment:", "index": 7}, {"type": "text", "bbox": [116, 615, 497, 717], "content": "• (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This sce- nario is based mainly on three hypotheses, the third one (which states the non–percolation of the –type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical test was cited in [3] but the details of the numerics (temperature, size of the lattice, etc.) and several data concerning the percolation properties of the system, were completely skipped. The only quoted result was (see beginning of section 4 in [3]) “We also tested numerically for ,... There is no indication of percolation...”. On the contrary, such inter- esting results about the critical properties should be put forward with a thorough description of the hypotheses involved. Moreover, one would like to understand how was possible to use the small value of epsilon mentioned in Ref. [3], because that value implies a really tiny temper- ature and consequently it requires a huge lattice size. If “Everybody agrees that at the standard action model has a finite correla- tion length”, see [1], also everybody would like to know details about the numerics and the computer used to simulate the model at such a small temperature.", "index": 8}]
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[2] are based on claims which have not been written in [2], for instance", "parent_index": 5, "line_index": 1}, {"bbox": [100, 417, 497, 431], "content": "almost all the introduction and the point (1) in [1] are based on such non–", "parent_index": 5, "line_index": 2}, {"bbox": [100, 432, 179, 444], "content": "existent claims.", "parent_index": 5, "line_index": 3}, {"bbox": [117, 451, 497, 464], "content": "Instead in Ref. [2] we avoid to make claims not based on well–founded", "parent_index": 6, "line_index": 0}, {"bbox": [99, 464, 497, 479], "content": "results. For instance in the abstract we write “... This result indicates how", "parent_index": 6, "line_index": 1}, {"bbox": [99, 479, 498, 494], "content": "the model can avoid a previously conjectured Kosterlitz–Thouless phase", "parent_index": 6, "line_index": 2}, {"bbox": [99, 493, 498, 509], "content": "transition...” and in the conclusive part we notice that “Our results exclude", "parent_index": 6, "line_index": 3}, {"bbox": [99, 507, 498, 524], "content": "this massless phase for ”. 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• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor $\vec{n}$ of the internal symmetry space $O(3)$ . In particular, a percolating equatorial cluster is found for every $\vec{n}$ . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the $O(3)$ symmetry.
On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying $\epsilon$ around the value $\epsilon=1$ (for $T=$ 0.5), one can make the data for $\langle M_{S}\rangle/L^{2}$ in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like $T=0.5$ ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated.
• (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the $X Y$ model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length $L]$ the $O(2)$ model has a KT phase transition for any finite $L^{\gamma}$ is surprising. In Ref. [5] it is shown that for any finite $L$ the KT transition is still present but it approaches $T=0$ as $L$ becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten.
(4) We agree with one of the sentences of this point: “It would be interesting to verify this [the existence of a KT transition for $X Y$ models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim.
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<p data-bbox="117 306 498 379">• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor $\vec{n}$ of the internal symmetry space $O(3)$ . In particular, a percolating equatorial cluster is found for every $\vec{n}$ . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the $O(3)$ symmetry. </p>
<p data-bbox="127 384 499 542">On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying $\epsilon$ around the value $\epsilon=1$ (for $T=$ 0.5), one can make the data for $\langle M_{S}\rangle/L^{2}$ in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like $T=0.5$ ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated. </p>
<p data-bbox="117 555 500 691">• (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the $X Y$ model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length $L]$ the $O(2)$ model has a KT phase transition for any finite $L^{\gamma}$ is surprising. In Ref. [5] it is shown that for any finite $L$ the KT transition is still present but it approaches $T=0$ as $L$ becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten. </p>
<p data-bbox="118 703 497 717">(4) We agree with one of the sentences of this point: “It would be interesting to verify this [the existence of a KT transition for $X Y$ models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim. </p>
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0002014v1
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[{"type": "text", "text": "", "page_idx": 1}, {"type": "text", "text": "• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor $\\vec{n}$ of the internal symmetry space $O(3)$ . In particular, a percolating equatorial cluster is found for every $\\vec{n}$ . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the $O(3)$ symmetry. ", "page_idx": 1}, {"type": "text", "text": "On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying $\\epsilon$ around the value $\\epsilon=1$ (for $T=$ 0.5), one can make the data for $\\langle M_{S}\\rangle/L^{2}$ in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like $T=0.5$ ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated. ", "page_idx": 1}, {"type": "text", "text": "• (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the $X Y$ model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length $L]$ the $O(2)$ model has a KT phase transition for any finite $L^{\\gamma}$ is surprising. In Ref. [5] it is shown that for any finite $L$ the KT transition is still present but it approaches $T=0$ as $L$ becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten. ", "page_idx": 1}, {"type": "text", "text": "(4) We agree with one of the sentences of this point: “It would be interesting to verify this [the existence of a KT transition for $X Y$ models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim. ", "page_idx": 1}]
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{"preproc_blocks": [{"type": "text", "bbox": [128, 91, 498, 293], "lines": [{"bbox": [128, 94, 499, 107], "spans": [{"bbox": [128, 94, 499, 107], "score": 1.0, "content": "test was cited in [3] but the details of the numerics (temperature, size of", "type": "text"}], "index": 0}, {"bbox": [128, 108, 497, 122], "spans": [{"bbox": [128, 108, 497, 122], "score": 1.0, "content": "the lattice, etc.) and several data concerning the percolation properties", "type": "text"}], "index": 1}, {"bbox": [128, 123, 498, 137], "spans": [{"bbox": [128, 123, 498, 137], "score": 1.0, "content": "of the system, were completely skipped. The only quoted result was (see", "type": "text"}], "index": 2}, {"bbox": [128, 136, 497, 152], "spans": [{"bbox": [128, 136, 445, 152], "score": 1.0, "content": "beginning of section 4 in [3]) “We also tested numerically for ", "type": "text"}, {"bbox": [445, 138, 484, 151], "score": 0.81, "content": "\\epsilon=1/3", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [484, 136, 497, 152], "score": 1.0, "content": ",...", "type": "text"}], "index": 3}, {"bbox": [128, 151, 496, 165], "spans": [{"bbox": [128, 151, 496, 165], "score": 1.0, "content": "There is no indication of percolation...”. On the contrary, such inter-", "type": "text"}], "index": 4}, {"bbox": [129, 167, 499, 180], "spans": [{"bbox": [129, 167, 499, 180], "score": 1.0, "content": "esting results about the critical properties should be put forward with a", "type": "text"}], "index": 5}, {"bbox": [128, 181, 497, 194], "spans": [{"bbox": [128, 181, 497, 194], "score": 1.0, "content": "thorough description of the hypotheses involved. Moreover, one would", "type": "text"}], "index": 6}, {"bbox": [128, 194, 497, 209], "spans": [{"bbox": [128, 194, 497, 209], "score": 1.0, "content": "like to understand how was possible to use the small value of epsilon", "type": "text"}], "index": 7}, {"bbox": [128, 209, 497, 223], "spans": [{"bbox": [128, 209, 497, 223], "score": 1.0, "content": "mentioned in Ref. [3], because that value implies a really tiny temper-", "type": "text"}], "index": 8}, {"bbox": [129, 224, 497, 237], "spans": [{"bbox": [129, 224, 159, 237], "score": 1.0, "content": "ature ", "type": "text"}, {"bbox": [159, 226, 168, 234], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [169, 224, 497, 237], "score": 1.0, "content": " and consequently it requires a huge lattice size. If “Everybody", "type": "text"}], "index": 9}, {"bbox": [128, 239, 497, 251], "spans": [{"bbox": [128, 239, 206, 251], "score": 1.0, "content": "agrees that at ", "type": "text"}, {"bbox": [206, 240, 248, 251], "score": 0.92, "content": "\\beta\\:=\\:2.0", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [248, 239, 497, 251], "score": 1.0, "content": " the standard action model has a finite correla-", "type": "text"}], "index": 10}, {"bbox": [128, 253, 499, 268], "spans": [{"bbox": [128, 253, 499, 268], "score": 1.0, "content": "tion length”, see [1], also everybody would like to know details about the", "type": "text"}], "index": 11}, {"bbox": [128, 268, 498, 280], "spans": [{"bbox": [128, 268, 498, 280], "score": 1.0, "content": "numerics and the computer used to simulate the model at such a small", "type": "text"}], "index": 12}, {"bbox": [128, 282, 196, 296], "spans": [{"bbox": [128, 282, 196, 296], "score": 1.0, "content": "temperature.", "type": "text"}], "index": 13}], "index": 6.5}, {"type": "text", "bbox": [117, 306, 498, 379], "lines": [{"bbox": [118, 309, 498, 322], "spans": [{"bbox": [118, 309, 498, 322], "score": 1.0, "content": "• (2) There is a statement in [2] which is repeated several times: all results", "type": "text"}], "index": 14}, {"bbox": [128, 324, 497, 338], "spans": [{"bbox": [128, 324, 252, 338], "score": 1.0, "content": "are valid for any versor ", "type": "text"}, {"bbox": [252, 325, 259, 334], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [260, 324, 427, 338], "score": 1.0, "content": " of the internal symmetry space ", "type": "text"}, {"bbox": [427, 325, 452, 337], "score": 0.95, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [452, 324, 497, 338], "score": 1.0, "content": ". In par-", "type": "text"}], "index": 15}, {"bbox": [128, 338, 497, 352], "spans": [{"bbox": [128, 338, 420, 352], "score": 1.0, "content": "ticular, a percolating equatorial cluster is found for every ", "type": "text"}, {"bbox": [420, 339, 428, 348], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 338, 497, 352], "score": 1.0, "content": ". Under these", "type": "text"}], "index": 16}, {"bbox": [129, 353, 497, 366], "spans": [{"bbox": [129, 353, 497, 366], "score": 1.0, "content": "conditions, we do not see how the percolation of the equatorial cluster", "type": "text"}], "index": 17}, {"bbox": [129, 367, 366, 381], "spans": [{"bbox": [129, 367, 284, 381], "score": 1.0, "content": "may lead to a breaking of the ", "type": "text"}, {"bbox": [284, 368, 309, 380], "score": 0.94, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [309, 367, 366, 381], "score": 1.0, "content": " symmetry.", "type": "text"}], "index": 18}], "index": 16}, {"type": "text", "bbox": [127, 384, 499, 542], "lines": [{"bbox": [129, 386, 498, 399], "spans": [{"bbox": [129, 386, 498, 399], "score": 1.0, "content": "On the other hand, the fractal properties of a cluster are very sensitive to", "type": "text"}], "index": 19}, {"bbox": [128, 399, 498, 416], "spans": [{"bbox": [128, 399, 323, 416], "score": 1.0, "content": "the choice of parameters. By varying ", "type": "text"}, {"bbox": [323, 405, 328, 411], "score": 0.85, "content": "\\epsilon", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [329, 399, 422, 416], "score": 1.0, "content": " around the value ", "type": "text"}, {"bbox": [422, 402, 449, 411], "score": 0.91, "content": "\\epsilon=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [450, 399, 475, 416], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [475, 402, 498, 412], "score": 0.83, "content": "T=", "type": "inline_equation", "height": 10, "width": 23}], "index": 20}, {"bbox": [128, 415, 497, 429], "spans": [{"bbox": [128, 415, 286, 429], "score": 1.0, "content": "0.5), one can make the data for ", "type": "text"}, {"bbox": [286, 415, 332, 428], "score": 0.94, "content": "\\langle M_{S}\\rangle/L^{2}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [332, 415, 497, 429], "score": 1.0, "content": " in Table 1 of [2] to change rather", "type": "text"}], "index": 21}, {"bbox": [127, 428, 498, 444], "spans": [{"bbox": [127, 428, 498, 444], "score": 1.0, "content": "dramatically. It is important (even in the case of a high temperature", "type": "text"}], "index": 22}, {"bbox": [129, 444, 498, 458], "spans": [{"bbox": [129, 444, 192, 458], "score": 1.0, "content": "regime, like ", "type": "text"}, {"bbox": [192, 445, 232, 454], "score": 0.86, "content": "T=0.5", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [233, 444, 498, 458], "score": 1.0, "content": ") to study this dependence. It is sensible to expect", "type": "text"}], "index": 23}, {"bbox": [127, 457, 500, 472], "spans": [{"bbox": [127, 457, 500, 472], "score": 1.0, "content": "that the fractal properties of the cluster show up at the threshold of", "type": "text"}], "index": 24}, {"bbox": [127, 473, 498, 486], "spans": [{"bbox": [127, 473, 498, 486], "score": 1.0, "content": "percolation. Again in [2] we do not claim that the cluster is a fractal, but", "type": "text"}], "index": 25}, {"bbox": [127, 487, 498, 502], "spans": [{"bbox": [127, 487, 498, 502], "score": 1.0, "content": "just write “... [the equatorial clusters] present a high degree of roughness", "type": "text"}], "index": 26}, {"bbox": [128, 501, 498, 516], "spans": [{"bbox": [128, 501, 498, 516], "score": 1.0, "content": "recalling a fractal structure”. To state any firmer claim, a deep analysis", "type": "text"}], "index": 27}, {"bbox": [128, 516, 498, 529], "spans": [{"bbox": [128, 516, 498, 529], "score": 1.0, "content": "of the errors and better statistics in Table 1 should be performed. All", "type": "text"}], "index": 28}, {"bbox": [128, 530, 342, 544], "spans": [{"bbox": [128, 530, 342, 544], "score": 1.0, "content": "these problems are currently investigated.", "type": "text"}], "index": 29}], "index": 24}, {"type": "text", "bbox": [117, 555, 500, 691], "lines": [{"bbox": [120, 558, 497, 571], "spans": [{"bbox": [120, 558, 497, 571], "score": 1.0, "content": "• (3) It is true that not all flimsy clusters can avoid a KT transition.", "type": "text"}], "index": 30}, {"bbox": [128, 572, 498, 586], "spans": [{"bbox": [128, 572, 498, 586], "score": 1.0, "content": "However this trivial truth proves nothing. Other kinds of lattices can", "type": "text"}], "index": 31}, {"bbox": [128, 586, 494, 600], "spans": [{"bbox": [128, 586, 232, 600], "score": 1.0, "content": "hold versions of the ", "type": "text"}, {"bbox": [232, 588, 253, 597], "score": 0.9, "content": "X Y", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [253, 586, 494, 600], "score": 1.0, "content": " model with no transition (see for instance [4]).", "type": "text"}], "index": 32}, {"bbox": [129, 606, 498, 619], "spans": [{"bbox": [129, 606, 498, 619], "score": 1.0, "content": "On the other hand, the statement “... there should be no doubt that on", "type": "text"}], "index": 33}, {"bbox": [128, 620, 497, 634], "spans": [{"bbox": [128, 620, 347, 634], "score": 1.0, "content": "such a lattice [square holes of side length ", "type": "text"}, {"bbox": [347, 621, 358, 633], "score": 0.5, "content": "L]", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 620, 383, 634], "score": 1.0, "content": " the ", "type": "text"}, {"bbox": [383, 621, 408, 634], "score": 0.94, "content": "O(2)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [408, 620, 497, 634], "score": 1.0, "content": " model has a KT", "type": "text"}], "index": 34}, {"bbox": [128, 635, 498, 649], "spans": [{"bbox": [128, 635, 289, 649], "score": 1.0, "content": "phase transition for any finite ", "type": "text"}, {"bbox": [289, 636, 302, 645], "score": 0.88, "content": "L^{\\gamma}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 635, 498, 649], "score": 1.0, "content": " is surprising. In Ref. [5] it is shown", "type": "text"}], "index": 35}, {"bbox": [128, 649, 497, 662], "spans": [{"bbox": [128, 649, 224, 662], "score": 1.0, "content": "that for any finite ", "type": "text"}, {"bbox": [225, 651, 232, 659], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [233, 649, 497, 662], "score": 1.0, "content": " the KT transition is still present but it approaches", "type": "text"}], "index": 36}, {"bbox": [129, 663, 498, 677], "spans": [{"bbox": [129, 665, 162, 674], "score": 0.93, "content": "T=0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [162, 663, 181, 677], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [181, 665, 189, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [190, 663, 498, 677], "score": 1.0, "content": " becomes larger. The idea of a fractal as the limit of some", "type": "text"}], "index": 37}, {"bbox": [127, 677, 330, 692], "spans": [{"bbox": [127, 677, 330, 692], "score": 1.0, "content": "kind of cluster should not be forgotten.", "type": "text"}], "index": 38}], "index": 34}, {"type": "text", "bbox": [118, 703, 497, 717], "lines": [{"bbox": [121, 704, 497, 719], "spans": [{"bbox": [121, 704, 497, 719], "score": 1.0, "content": " (4) We agree with one of the sentences of this point: “It would be in-", "type": "text"}], "index": 39}], "index": 39}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [294, 735, 302, 745], "lines": [{"bbox": [294, 735, 303, 748], "spans": [{"bbox": [294, 735, 303, 748], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [128, 91, 498, 293], "lines": [], "index": 6.5, "bbox_fs": [128, 94, 499, 296], "lines_deleted": true}, {"type": "text", "bbox": [117, 306, 498, 379], "lines": [{"bbox": [118, 309, 498, 322], "spans": [{"bbox": [118, 309, 498, 322], "score": 1.0, "content": "• (2) There is a statement in [2] which is repeated several times: all results", "type": "text"}], "index": 14}, {"bbox": [128, 324, 497, 338], "spans": [{"bbox": [128, 324, 252, 338], "score": 1.0, "content": "are valid for any versor ", "type": "text"}, {"bbox": [252, 325, 259, 334], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [260, 324, 427, 338], "score": 1.0, "content": " of the internal symmetry space ", "type": "text"}, {"bbox": [427, 325, 452, 337], "score": 0.95, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [452, 324, 497, 338], "score": 1.0, "content": ". In par-", "type": "text"}], "index": 15}, {"bbox": [128, 338, 497, 352], "spans": [{"bbox": [128, 338, 420, 352], "score": 1.0, "content": "ticular, a percolating equatorial cluster is found for every ", "type": "text"}, {"bbox": [420, 339, 428, 348], "score": 0.9, "content": "\\vec{n}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 338, 497, 352], "score": 1.0, "content": ". Under these", "type": "text"}], "index": 16}, {"bbox": [129, 353, 497, 366], "spans": [{"bbox": [129, 353, 497, 366], "score": 1.0, "content": "conditions, we do not see how the percolation of the equatorial cluster", "type": "text"}], "index": 17}, {"bbox": [129, 367, 366, 381], "spans": [{"bbox": [129, 367, 284, 381], "score": 1.0, "content": "may lead to a breaking of the ", "type": "text"}, {"bbox": [284, 368, 309, 380], "score": 0.94, "content": "O(3)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [309, 367, 366, 381], "score": 1.0, "content": " symmetry.", "type": "text"}], "index": 18}], "index": 16, "bbox_fs": [118, 309, 498, 381]}, {"type": "text", "bbox": [127, 384, 499, 542], "lines": [{"bbox": [129, 386, 498, 399], "spans": [{"bbox": [129, 386, 498, 399], "score": 1.0, "content": "On the other hand, the fractal properties of a cluster are very sensitive to", "type": "text"}], "index": 19}, {"bbox": [128, 399, 498, 416], "spans": [{"bbox": [128, 399, 323, 416], "score": 1.0, "content": "the choice of parameters. By varying ", "type": "text"}, {"bbox": [323, 405, 328, 411], "score": 0.85, "content": "\\epsilon", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [329, 399, 422, 416], "score": 1.0, "content": " around the value ", "type": "text"}, {"bbox": [422, 402, 449, 411], "score": 0.91, "content": "\\epsilon=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [450, 399, 475, 416], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [475, 402, 498, 412], "score": 0.83, "content": "T=", "type": "inline_equation", "height": 10, "width": 23}], "index": 20}, {"bbox": [128, 415, 497, 429], "spans": [{"bbox": [128, 415, 286, 429], "score": 1.0, "content": "0.5), one can make the data for ", "type": "text"}, {"bbox": [286, 415, 332, 428], "score": 0.94, "content": "\\langle M_{S}\\rangle/L^{2}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [332, 415, 497, 429], "score": 1.0, "content": " in Table 1 of [2] to change rather", "type": "text"}], "index": 21}, {"bbox": [127, 428, 498, 444], "spans": [{"bbox": [127, 428, 498, 444], "score": 1.0, "content": "dramatically. It is important (even in the case of a high temperature", "type": "text"}], "index": 22}, {"bbox": [129, 444, 498, 458], "spans": [{"bbox": [129, 444, 192, 458], "score": 1.0, "content": "regime, like ", "type": "text"}, {"bbox": [192, 445, 232, 454], "score": 0.86, "content": "T=0.5", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [233, 444, 498, 458], "score": 1.0, "content": ") to study this dependence. It is sensible to expect", "type": "text"}], "index": 23}, {"bbox": [127, 457, 500, 472], "spans": [{"bbox": [127, 457, 500, 472], "score": 1.0, "content": "that the fractal properties of the cluster show up at the threshold of", "type": "text"}], "index": 24}, {"bbox": [127, 473, 498, 486], "spans": [{"bbox": [127, 473, 498, 486], "score": 1.0, "content": "percolation. Again in [2] we do not claim that the cluster is a fractal, but", "type": "text"}], "index": 25}, {"bbox": [127, 487, 498, 502], "spans": [{"bbox": [127, 487, 498, 502], "score": 1.0, "content": "just write “... [the equatorial clusters] present a high degree of roughness", "type": "text"}], "index": 26}, {"bbox": [128, 501, 498, 516], "spans": [{"bbox": [128, 501, 498, 516], "score": 1.0, "content": "recalling a fractal structure”. To state any firmer claim, a deep analysis", "type": "text"}], "index": 27}, {"bbox": [128, 516, 498, 529], "spans": [{"bbox": [128, 516, 498, 529], "score": 1.0, "content": "of the errors and better statistics in Table 1 should be performed. All", "type": "text"}], "index": 28}, {"bbox": [128, 530, 342, 544], "spans": [{"bbox": [128, 530, 342, 544], "score": 1.0, "content": "these problems are currently investigated.", "type": "text"}], "index": 29}], "index": 24, "bbox_fs": [127, 386, 500, 544]}, {"type": "text", "bbox": [117, 555, 500, 691], "lines": [{"bbox": [120, 558, 497, 571], "spans": [{"bbox": [120, 558, 497, 571], "score": 1.0, "content": "• (3) It is true that not all flimsy clusters can avoid a KT transition.", "type": "text"}], "index": 30}, {"bbox": [128, 572, 498, 586], "spans": [{"bbox": [128, 572, 498, 586], "score": 1.0, "content": "However this trivial truth proves nothing. Other kinds of lattices can", "type": "text"}], "index": 31}, {"bbox": [128, 586, 494, 600], "spans": [{"bbox": [128, 586, 232, 600], "score": 1.0, "content": "hold versions of the ", "type": "text"}, {"bbox": [232, 588, 253, 597], "score": 0.9, "content": "X Y", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [253, 586, 494, 600], "score": 1.0, "content": " model with no transition (see for instance [4]).", "type": "text"}], "index": 32}, {"bbox": [129, 606, 498, 619], "spans": [{"bbox": [129, 606, 498, 619], "score": 1.0, "content": "On the other hand, the statement “... there should be no doubt that on", "type": "text"}], "index": 33}, {"bbox": [128, 620, 497, 634], "spans": [{"bbox": [128, 620, 347, 634], "score": 1.0, "content": "such a lattice [square holes of side length ", "type": "text"}, {"bbox": [347, 621, 358, 633], "score": 0.5, "content": "L]", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 620, 383, 634], "score": 1.0, "content": " the ", "type": "text"}, {"bbox": [383, 621, 408, 634], "score": 0.94, "content": "O(2)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [408, 620, 497, 634], "score": 1.0, "content": " model has a KT", "type": "text"}], "index": 34}, {"bbox": [128, 635, 498, 649], "spans": [{"bbox": [128, 635, 289, 649], "score": 1.0, "content": "phase transition for any finite ", "type": "text"}, {"bbox": [289, 636, 302, 645], "score": 0.88, "content": "L^{\\gamma}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 635, 498, 649], "score": 1.0, "content": " is surprising. In Ref. [5] it is shown", "type": "text"}], "index": 35}, {"bbox": [128, 649, 497, 662], "spans": [{"bbox": [128, 649, 224, 662], "score": 1.0, "content": "that for any finite ", "type": "text"}, {"bbox": [225, 651, 232, 659], "score": 0.9, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [233, 649, 497, 662], "score": 1.0, "content": " the KT transition is still present but it approaches", "type": "text"}], "index": 36}, {"bbox": [129, 663, 498, 677], "spans": [{"bbox": [129, 665, 162, 674], "score": 0.93, "content": "T=0", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [162, 663, 181, 677], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [181, 665, 189, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [190, 663, 498, 677], "score": 1.0, "content": " becomes larger. The idea of a fractal as the limit of some", "type": "text"}], "index": 37}, {"bbox": [127, 677, 330, 692], "spans": [{"bbox": [127, 677, 330, 692], "score": 1.0, "content": "kind of cluster should not be forgotten.", "type": "text"}], "index": 38}], "index": 34, "bbox_fs": [120, 558, 498, 692]}, {"type": "text", "bbox": [118, 703, 497, 717], "lines": [{"bbox": [121, 704, 497, 719], "spans": [{"bbox": [121, 704, 497, 719], "score": 1.0, "content": " (4) We agree with one of the sentences of this point: “It would be in-", "type": "text"}], "index": 39}, {"bbox": [128, 94, 498, 106], "spans": [{"bbox": [128, 94, 436, 106], "score": 1.0, "content": "teresting to verify this [the existence of a KT transition for ", "type": "text", "cross_page": true}, {"bbox": [437, 95, 457, 104], "score": 0.47, "content": "X Y", "type": "inline_equation", "height": 9, "width": 20, "cross_page": true}, {"bbox": [457, 94, 498, 106], "score": 1.0, "content": " models", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [129, 108, 498, 121], "spans": [{"bbox": [129, 108, 498, 121], "score": 1.0, "content": "on a fractal lattice]”. Yet we do not see the relevance of such an obvious", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [127, 123, 160, 136], "spans": [{"bbox": [127, 123, 160, 136], "score": 1.0, "content": "claim.", "type": "text", "cross_page": true}], "index": 2}], "index": 39, "bbox_fs": [121, 704, 497, 719]}]}
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[{"type": "text", "bbox": [128, 91, 498, 293], "content": "", "index": 0}, {"type": "text", "bbox": [117, 306, 498, 379], "content": "• (2) There is a statement in [2] which is repeated several times: all results are valid for any versor of the internal symmetry space . In par- ticular, a percolating equatorial cluster is found for every . Under these conditions, we do not see how the percolation of the equatorial cluster may lead to a breaking of the symmetry.", "index": 1}, {"type": "text", "bbox": [127, 384, 499, 542], "content": "On the other hand, the fractal properties of a cluster are very sensitive to the choice of parameters. By varying around the value (for 0.5), one can make the data for in Table 1 of [2] to change rather dramatically. It is important (even in the case of a high temperature regime, like ) to study this dependence. It is sensible to expect that the fractal properties of the cluster show up at the threshold of percolation. Again in [2] we do not claim that the cluster is a fractal, but just write “... [the equatorial clusters] present a high degree of roughness recalling a fractal structure”. To state any firmer claim, a deep analysis of the errors and better statistics in Table 1 should be performed. All these problems are currently investigated.", "index": 2}, {"type": "text", "bbox": [117, 555, 500, 691], "content": "• (3) It is true that not all flimsy clusters can avoid a KT transition. However this trivial truth proves nothing. Other kinds of lattices can hold versions of the model with no transition (see for instance [4]). On the other hand, the statement “... there should be no doubt that on such a lattice [square holes of side length the model has a KT phase transition for any finite is surprising. In Ref. [5] it is shown that for any finite the KT transition is still present but it approaches as becomes larger. The idea of a fractal as the limit of some kind of cluster should not be forgotten.", "index": 3}, {"type": "text", "bbox": [118, 703, 497, 717], "content": "(4) We agree with one of the sentences of this point: “It would be in- teresting to verify this [the existence of a KT transition for models on a fractal lattice]”. Yet we do not see the relevance of such an obvious claim.", "index": 4}]
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We disagree however with the authors of [1] when they say “our argument does not depend on the existence of such a transition on that particular percolating cluster”. Instead, after the conclusions of Ref. [2], we think that the non–rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized.
# References
[1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1).
[2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669.
[3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184.
[4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145.
[5] P. Minnhagen and H. Weber, Physica B152 (1988) 50.
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<p data-bbox="128 140 498 226">We disagree however with the authors of [1] when they say “our argument does not depend on the existence of such a transition on that particular percolating cluster”. Instead, after the conclusions of Ref. [2], we think that the non–rigorous proof proposed in [3] for the case when the equatorial cluster does percolate, heavily lies on whether or not such a transition is realized. </p>
<h1 data-bbox="99 294 191 312">References </h1>
<p data-bbox="104 327 498 458">[1] A. Patrascioiu and E. Seiler, unpublished report hep–lat/9912014 (v1).
[2] B. All´es, J.J. Alonso, C. Criado and M. Pepe, Phys. Rev. Lett. 83 (1999) 3669.
[3] A. Patrascioiu and E. Seiler, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 184.
[4] Yu.E. Lozovik and L.M. Pomirchy, Solid State Comm. 89 (1994) 145.
[5] P. Minnhagen and H. Weber, Physica B152 (1988) 50. </p>
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# IMAGINARY QUADRATIC FIELDS $k$ WITH $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ AND RANK $\mathrm{Cl}_{2}(k^{1})=2$
E. BENJAMIN, F. LEMMERMEYER, C. SNYDER
Abstract. Let $k$ be an imaginary quadratic number field and $k^{1}$ the Hilbert 2-class field of $k$ . We give a characterization of those $k$ with $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ such that $\mathrm{Cl}_{2}(k^{1})$ has 2 generators.
# 1. Introduction
Let $k$ be an algebraic number field with $\mathrm{Cl_{2}}(k)$ , the Sylow 2-subgroup of its ideal class group, $\operatorname{Cl}(k)$ . Denote by $k^{1}$ the Hilbert 2-class field of $k$ (in the wide sense). Also let $k^{n}$ (for $n$ a nonnegative integer) be defined inductively as: $k^{0}\,=\,k$ and kn+1 = (kn)1. Then
$$
k^{0}\subseteq k^{1}\subseteq k^{2}\subseteq\cdots\subseteq k^{n}\subseteq\cdot\cdot.
$$
is called the 2-class field tower of $k$ . If $n$ is the minimal integer such that $k^{n}=k^{n+1}$ , then $n$ is called the length of the tower. If no such $n$ exists, then the tower is said to be of infinite length.
At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field $k$ is infinite. However, it is known by group theoretic results (see [2]) that if r $\mathrm{ank\,Cl_{2}}(k^{1})\,\leq\,2$ , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank $\mathrm{Cl}_{2}(k^{1})\,\geq\,3$ (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank $\mathrm{Cl}_{2}(k^{1})\leq2$ . We have already started this program for imaginary quadratic number fields $k$ . In [1] we classified all imaginary quadratic fields whose 2-class field $k^{1}$ has cyclic 2-class group. In this paper we determine when $\mathrm{Cl_{2}}(k^{1})$ has rank 2 for imaginary quadratic fields $k$ with $\mathrm{Cl_{2}}(k)$ of type $(2,2^{m})$ . (The notation $(2,2^{m})$ means the direct sum of a group of order 2 and a cyclic group of order $2^{m}$ .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2.
From a classification of imaginary quadratic number fields $k$ with $\mathrm{Cl_{2}}(k)\simeq$ $(2,2^{m})$ and our results from [1] we see that it suffices to consider discriminants $d=d_{1}d_{2}d_{3}$ with prime discriminants $d_{1},d_{2}>0$ , $d_{3}<0$ such that exactly one of the $\left(d_{i}/p_{j}\right)$ equals $^{-1}$ (we let $p_{j}$ denote the prime dividing $d_{j}$ ); thus there are only two cases:
1991 Mathematics Subject Classification. Primary 11R37.
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<html><body>
<h1 data-bbox="129 142 482 171">IMAGINARY QUADRATIC FIELDS $k$ WITH $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ AND RANK $\mathrm{Cl}_{2}(k^{1})=2$ </h1>
<p data-bbox="205 189 406 199">E. BENJAMIN, F. LEMMERMEYER, C. SNYDER </p>
<p data-bbox="160 218 450 248">Abstract. Let $k$ be an imaginary quadratic number field and $k^{1}$ the Hilbert 2-class field of $k$ . We give a characterization of those $k$ with $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ such that $\mathrm{Cl}_{2}(k^{1})$ has 2 generators. </p>
<h1 data-bbox="265 282 346 294">1. Introduction </h1>
<p data-bbox="125 300 485 348">Let $k$ be an algebraic number field with $\mathrm{Cl_{2}}(k)$ , the Sylow 2-subgroup of its ideal class group, $\operatorname{Cl}(k)$ . Denote by $k^{1}$ the Hilbert 2-class field of $k$ (in the wide sense). Also let $k^{n}$ (for $n$ a nonnegative integer) be defined inductively as: $k^{0}\,=\,k$ and kn+1 = (kn)1. Then </p>
<div class="equation" data-bbox="240 354 371 365">$$
k^{0}\subseteq k^{1}\subseteq k^{2}\subseteq\cdots\subseteq k^{n}\subseteq\cdot\cdot.
$$</div>
<p data-bbox="125 368 486 404">is called the 2-class field tower of $k$ . If $n$ is the minimal integer such that $k^{n}=k^{n+1}$ , then $n$ is called the length of the tower. If no such $n$ exists, then the tower is said to be of infinite length. </p>
<p data-bbox="126 405 486 607">At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field $k$ is infinite. However, it is known by group theoretic results (see [2]) that if r $\mathrm{ank\,Cl_{2}}(k^{1})\,\leq\,2$ , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank $\mathrm{Cl}_{2}(k^{1})\,\geq\,3$ (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank $\mathrm{Cl}_{2}(k^{1})\leq2$ . We have already started this program for imaginary quadratic number fields $k$ . In [1] we classified all imaginary quadratic fields whose 2-class field $k^{1}$ has cyclic 2-class group. In this paper we determine when $\mathrm{Cl_{2}}(k^{1})$ has rank 2 for imaginary quadratic fields $k$ with $\mathrm{Cl_{2}}(k)$ of type $(2,2^{m})$ . (The notation $(2,2^{m})$ means the direct sum of a group of order 2 and a cyclic group of order $2^{m}$ .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2. </p>
<p data-bbox="126 608 486 667">From a classification of imaginary quadratic number fields $k$ with $\mathrm{Cl_{2}}(k)\simeq$ $(2,2^{m})$ and our results from [1] we see that it suffices to consider discriminants $d=d_{1}d_{2}d_{3}$ with prime discriminants $d_{1},d_{2}>0$ , $d_{3}<0$ such that exactly one of the $\left(d_{i}/p_{j}\right)$ equals $^{-1}$ (we let $p_{j}$ denote the prime dividing $d_{j}$ ); thus there are only two cases: </p>
<p data-bbox="137 689 353 700">1991 Mathematics Subject Classification. Primary 11R37. </p>
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0003244v1
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Denote by ", "type": "text"}, {"bbox": [260, 315, 270, 323], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [270, 314, 390, 327], "score": 1.0, "content": " the Hilbert 2-class field of ", "type": "text"}, {"bbox": [390, 316, 396, 323], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [396, 314, 486, 327], "score": 1.0, "content": " (in the wide sense).", "type": "text"}], "index": 8}, {"bbox": [126, 325, 487, 339], "spans": [{"bbox": [126, 325, 164, 339], "score": 1.0, "content": "Also let ", "type": "text"}, {"bbox": [164, 328, 176, 335], "score": 0.9, "content": "k^{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [176, 325, 199, 339], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [200, 330, 206, 335], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [206, 325, 432, 339], "score": 1.0, "content": " a nonnegative integer) be defined inductively as: ", "type": "text"}, {"bbox": [432, 327, 465, 335], "score": 0.93, "content": "k^{0}\\,=\\,k", "type": "inline_equation", "height": 8, "width": 33}, {"bbox": [465, 325, 487, 339], "score": 1.0, "content": " and", "type": "text"}], "index": 9}, {"bbox": [125, 336, 215, 351], "spans": [{"bbox": [125, 336, 215, 351], "score": 1.0, "content": "kn+1 = (kn)1. Then", "type": "text"}], "index": 10}], "index": 8.5}, {"type": "interline_equation", "bbox": [240, 354, 371, 365], "lines": [{"bbox": [240, 354, 371, 365], "spans": [{"bbox": [240, 354, 371, 365], "score": 0.91, "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 368, 486, 404], "lines": [{"bbox": [124, 369, 484, 383], "spans": [{"bbox": [124, 369, 268, 383], "score": 1.0, "content": "is called the 2-class field tower of ", "type": "text"}, {"bbox": [268, 372, 274, 379], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [274, 369, 289, 383], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [289, 375, 296, 379], "score": 0.89, "content": "n", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [296, 369, 436, 383], "score": 1.0, "content": " is the minimal integer such that", "type": "text"}, {"bbox": [437, 371, 482, 379], "score": 0.92, "content": "k^{n}=k^{n+1}", "type": "inline_equation", "height": 8, "width": 45}, {"bbox": [482, 369, 484, 383], "score": 1.0, "content": ",", "type": "text"}], "index": 12}, {"bbox": [126, 383, 486, 394], "spans": [{"bbox": [126, 383, 148, 394], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [149, 387, 155, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [155, 383, 350, 394], "score": 1.0, "content": " is called the length of the tower. If no such ", "type": "text"}, {"bbox": [350, 387, 357, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [357, 383, 486, 394], "score": 1.0, "content": " exists, then the tower is said", "type": "text"}], "index": 13}, {"bbox": [126, 395, 226, 406], "spans": [{"bbox": [126, 395, 226, 406], "score": 1.0, "content": "to be of infinite length.", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [126, 405, 486, 607], "lines": [{"bbox": [138, 407, 485, 417], "spans": [{"bbox": [138, 407, 485, 417], "score": 1.0, "content": "At present there is no known decision procedure to determine whether or not", "type": "text"}], "index": 15}, {"bbox": [126, 419, 485, 429], "spans": [{"bbox": [126, 419, 309, 429], "score": 1.0, "content": "the (2-)class field tower of a given field ", "type": "text"}, {"bbox": [310, 420, 316, 427], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [316, 419, 485, 429], "score": 1.0, "content": " is infinite. However, it is known by", "type": "text"}], "index": 16}, {"bbox": [125, 430, 486, 442], "spans": [{"bbox": [125, 430, 303, 442], "score": 1.0, "content": "group theoretic results (see [2]) that if r", "type": "text"}, {"bbox": [303, 430, 374, 442], "score": 0.69, "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [374, 430, 486, 442], "score": 1.0, "content": ", then the tower is finite,", "type": "text"}], "index": 17}, {"bbox": [125, 442, 486, 454], "spans": [{"bbox": [125, 442, 486, 454], "score": 1.0, "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "type": "text"}], "index": 18}, {"bbox": [125, 454, 487, 467], "spans": [{"bbox": [125, 454, 487, 467], "score": 1.0, "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "type": "text"}], "index": 19}, {"bbox": [124, 465, 487, 479], "spans": [{"bbox": [124, 465, 487, 479], "score": 1.0, "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "type": "text"}], "index": 20}, {"bbox": [125, 478, 486, 490], "spans": [{"bbox": [125, 478, 198, 490], "score": 1.0, "content": "fields with rank ", "type": "text"}, {"bbox": [198, 479, 252, 489], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [253, 478, 486, 490], "score": 1.0, "content": " (let us mention in particular Schmithals [13]) have", "type": "text"}], "index": 21}, {"bbox": [125, 490, 485, 502], "spans": [{"bbox": [125, 490, 485, 502], "score": 1.0, "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "type": "text"}], "index": 22}, {"bbox": [124, 502, 487, 515], "spans": [{"bbox": [124, 502, 487, 515], "score": 1.0, "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "type": "text"}], "index": 23}, {"bbox": [125, 514, 487, 526], "spans": [{"bbox": [125, 514, 487, 526], "score": 1.0, "content": "good starting point would be to find a procedure for sieving out those fields with", "type": "text"}], "index": 24}, {"bbox": [124, 525, 487, 539], "spans": [{"bbox": [124, 525, 147, 539], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 526, 200, 537], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [200, 525, 487, 539], "score": 1.0, "content": ". We have already started this program for imaginary quadratic", "type": "text"}], "index": 25}, {"bbox": [126, 538, 486, 550], "spans": [{"bbox": [126, 538, 186, 550], "score": 1.0, "content": "number fields ", "type": "text"}, {"bbox": [186, 540, 192, 547], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [192, 538, 486, 550], "score": 1.0, "content": ". In [1] we classified all imaginary quadratic fields whose 2-class field", "type": "text"}], "index": 26}, {"bbox": [126, 548, 487, 563], "spans": [{"bbox": [126, 550, 136, 559], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 548, 402, 563], "score": 1.0, "content": " has cyclic 2-class group. In this paper we determine when ", "type": "text"}, {"bbox": [402, 550, 434, 561], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [434, 548, 487, 563], "score": 1.0, "content": " has rank 2", "type": "text"}], "index": 27}, {"bbox": [125, 562, 485, 574], "spans": [{"bbox": [125, 562, 258, 574], "score": 1.0, "content": "for imaginary quadratic fields ", "type": "text"}, {"bbox": [258, 564, 263, 571], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [264, 562, 289, 574], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [289, 563, 317, 573], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [317, 562, 353, 574], "score": 1.0, "content": " of type ", "type": "text"}, {"bbox": [354, 563, 384, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [384, 562, 455, 574], "score": 1.0, "content": ". (The notation ", "type": "text"}, {"bbox": [455, 563, 485, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}], "index": 28}, {"bbox": [125, 574, 487, 586], "spans": [{"bbox": [125, 574, 442, 586], "score": 1.0, "content": "means the direct sum of a group of order 2 and a cyclic group of order ", "type": "text"}, {"bbox": [443, 576, 456, 583], "score": 0.9, "content": "2^{m}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [456, 574, 487, 586], "score": 1.0, "content": ".) The", "type": "text"}], "index": 29}, {"bbox": [125, 586, 486, 597], "spans": [{"bbox": [125, 586, 486, 597], "score": 1.0, "content": "group theoretic results mentioned above also show that such fields have 2-class field", "type": "text"}], "index": 30}, {"bbox": [125, 598, 204, 610], "spans": [{"bbox": [125, 598, 204, 610], "score": 1.0, "content": "tower of length 2.", "type": "text"}], "index": 31}], "index": 23}, {"type": "text", "bbox": [126, 608, 486, 667], "lines": [{"bbox": [136, 608, 486, 623], "spans": [{"bbox": [136, 608, 408, 623], "score": 1.0, "content": "From a classification of imaginary quadratic number fields ", "type": "text"}, {"bbox": [409, 611, 415, 618], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [415, 608, 443, 623], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 610, 486, 621], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)\\simeq", "type": "inline_equation", "height": 11, "width": 42}], "index": 32}, {"bbox": [126, 622, 487, 633], "spans": [{"bbox": [126, 623, 156, 633], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 622, 487, 633], "score": 1.0, "content": " and our results from [1] we see that it suffices to consider discriminants", "type": "text"}], "index": 33}, {"bbox": [126, 633, 487, 645], "spans": [{"bbox": [126, 635, 175, 644], "score": 0.93, "content": "d=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [176, 633, 293, 645], "score": 1.0, "content": " with prime discriminants ", "type": "text"}, {"bbox": [293, 635, 337, 644], "score": 0.93, "content": "d_{1},d_{2}>0", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [338, 633, 344, 645], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [344, 635, 374, 644], "score": 0.93, "content": "d_{3}<0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [374, 633, 487, 645], "score": 1.0, "content": " such that exactly one of", "type": "text"}], "index": 34}, {"bbox": [125, 645, 485, 658], "spans": [{"bbox": [125, 645, 143, 658], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [143, 646, 173, 657], "score": 0.92, "content": "\\left(d_{i}/p_{j}\\right)", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [174, 645, 207, 658], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [207, 648, 220, 655], "score": 0.9, "content": "^{-1}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [220, 645, 255, 658], "score": 1.0, "content": " (we let ", "type": "text"}, {"bbox": [255, 650, 264, 657], "score": 0.89, "content": "p_{j}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [265, 645, 383, 658], "score": 1.0, "content": " denote the prime dividing ", "type": "text"}, {"bbox": [383, 647, 393, 657], "score": 0.89, "content": "d_{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [393, 645, 485, 658], "score": 1.0, "content": "); thus there are only", "type": "text"}], "index": 35}, {"bbox": [125, 658, 171, 670], "spans": [{"bbox": [125, 658, 171, 670], "score": 1.0, "content": "two cases:", "type": "text"}], "index": 36}], "index": 34}, {"type": "text", "bbox": [137, 689, 353, 700], "lines": [{"bbox": [138, 691, 353, 701], "spans": [{"bbox": [138, 691, 353, 701], "score": 1.0, "content": "1991 Mathematics Subject Classification. 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Introduction", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [125, 300, 485, 348], "lines": [{"bbox": [137, 302, 486, 315], "spans": [{"bbox": [137, 302, 155, 315], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [155, 304, 161, 311], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 302, 310, 315], "score": 1.0, "content": " be an algebraic number field with ", "type": "text"}, {"bbox": [310, 303, 338, 314], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [339, 302, 486, 315], "score": 1.0, "content": ", the Sylow 2-subgroup of its ideal", "type": "text"}], "index": 7}, {"bbox": [124, 314, 486, 327], "spans": [{"bbox": [124, 314, 180, 327], "score": 1.0, "content": "class group, ", "type": "text"}, {"bbox": [181, 315, 204, 326], "score": 0.85, "content": "\\operatorname{Cl}(k)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [204, 314, 259, 327], "score": 1.0, "content": ". Denote by ", "type": "text"}, {"bbox": [260, 315, 270, 323], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [270, 314, 390, 327], "score": 1.0, "content": " the Hilbert 2-class field of ", "type": "text"}, {"bbox": [390, 316, 396, 323], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [396, 314, 486, 327], "score": 1.0, "content": " (in the wide sense).", "type": "text"}], "index": 8}, {"bbox": [126, 325, 487, 339], "spans": [{"bbox": [126, 325, 164, 339], "score": 1.0, "content": "Also let ", "type": "text"}, {"bbox": [164, 328, 176, 335], "score": 0.9, "content": "k^{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [176, 325, 199, 339], "score": 1.0, "content": " (for ", "type": "text"}, {"bbox": [200, 330, 206, 335], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [206, 325, 432, 339], "score": 1.0, "content": " a nonnegative integer) be defined inductively as: ", "type": "text"}, {"bbox": [432, 327, 465, 335], "score": 0.93, "content": "k^{0}\\,=\\,k", "type": "inline_equation", "height": 8, "width": 33}, {"bbox": [465, 325, 487, 339], "score": 1.0, "content": " and", "type": "text"}], "index": 9}, {"bbox": [125, 336, 215, 351], "spans": [{"bbox": [125, 336, 215, 351], "score": 1.0, "content": "kn+1 = (kn)1. Then", "type": "text"}], "index": 10}], "index": 8.5, "bbox_fs": [124, 302, 487, 351]}, {"type": "interline_equation", "bbox": [240, 354, 371, 365], "lines": [{"bbox": [240, 354, 371, 365], "spans": [{"bbox": [240, 354, 371, 365], "score": 0.91, "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 368, 486, 404], "lines": [{"bbox": [124, 369, 484, 383], "spans": [{"bbox": [124, 369, 268, 383], "score": 1.0, "content": "is called the 2-class field tower of ", "type": "text"}, {"bbox": [268, 372, 274, 379], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [274, 369, 289, 383], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [289, 375, 296, 379], "score": 0.89, "content": "n", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [296, 369, 436, 383], "score": 1.0, "content": " is the minimal integer such that", "type": "text"}, {"bbox": [437, 371, 482, 379], "score": 0.92, "content": "k^{n}=k^{n+1}", "type": "inline_equation", "height": 8, "width": 45}, {"bbox": [482, 369, 484, 383], "score": 1.0, "content": ",", "type": "text"}], "index": 12}, {"bbox": [126, 383, 486, 394], "spans": [{"bbox": [126, 383, 148, 394], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [149, 387, 155, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [155, 383, 350, 394], "score": 1.0, "content": " is called the length of the tower. If no such ", "type": "text"}, {"bbox": [350, 387, 357, 392], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [357, 383, 486, 394], "score": 1.0, "content": " exists, then the tower is said", "type": "text"}], "index": 13}, {"bbox": [126, 395, 226, 406], "spans": [{"bbox": [126, 395, 226, 406], "score": 1.0, "content": "to be of infinite length.", "type": "text"}], "index": 14}], "index": 13, "bbox_fs": [124, 369, 486, 406]}, {"type": "text", "bbox": [126, 405, 486, 607], "lines": [{"bbox": [138, 407, 485, 417], "spans": [{"bbox": [138, 407, 485, 417], "score": 1.0, "content": "At present there is no known decision procedure to determine whether or not", "type": "text"}], "index": 15}, {"bbox": [126, 419, 485, 429], "spans": [{"bbox": [126, 419, 309, 429], "score": 1.0, "content": "the (2-)class field tower of a given field ", "type": "text"}, {"bbox": [310, 420, 316, 427], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [316, 419, 485, 429], "score": 1.0, "content": " is infinite. However, it is known by", "type": "text"}], "index": 16}, {"bbox": [125, 430, 486, 442], "spans": [{"bbox": [125, 430, 303, 442], "score": 1.0, "content": "group theoretic results (see [2]) that if r", "type": "text"}, {"bbox": [303, 430, 374, 442], "score": 0.69, "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [374, 430, 486, 442], "score": 1.0, "content": ", then the tower is finite,", "type": "text"}], "index": 17}, {"bbox": [125, 442, 486, 454], "spans": [{"bbox": [125, 442, 486, 454], "score": 1.0, "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "type": "text"}], "index": 18}, {"bbox": [125, 454, 487, 467], "spans": [{"bbox": [125, 454, 487, 467], "score": 1.0, "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "type": "text"}], "index": 19}, {"bbox": [124, 465, 487, 479], "spans": [{"bbox": [124, 465, 487, 479], "score": 1.0, "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "type": "text"}], "index": 20}, {"bbox": [125, 478, 486, 490], "spans": [{"bbox": [125, 478, 198, 490], "score": 1.0, "content": "fields with rank ", "type": "text"}, {"bbox": [198, 479, 252, 489], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [253, 478, 486, 490], "score": 1.0, "content": " (let us mention in particular Schmithals [13]) have", "type": "text"}], "index": 21}, {"bbox": [125, 490, 485, 502], "spans": [{"bbox": [125, 490, 485, 502], "score": 1.0, "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "type": "text"}], "index": 22}, {"bbox": [124, 502, 487, 515], "spans": [{"bbox": [124, 502, 487, 515], "score": 1.0, "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "type": "text"}], "index": 23}, {"bbox": [125, 514, 487, 526], "spans": [{"bbox": [125, 514, 487, 526], "score": 1.0, "content": "good starting point would be to find a procedure for sieving out those fields with", "type": "text"}], "index": 24}, {"bbox": [124, 525, 487, 539], "spans": [{"bbox": [124, 525, 147, 539], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 526, 200, 537], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [200, 525, 487, 539], "score": 1.0, "content": ". We have already started this program for imaginary quadratic", "type": "text"}], "index": 25}, {"bbox": [126, 538, 486, 550], "spans": [{"bbox": [126, 538, 186, 550], "score": 1.0, "content": "number fields ", "type": "text"}, {"bbox": [186, 540, 192, 547], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [192, 538, 486, 550], "score": 1.0, "content": ". In [1] we classified all imaginary quadratic fields whose 2-class field", "type": "text"}], "index": 26}, {"bbox": [126, 548, 487, 563], "spans": [{"bbox": [126, 550, 136, 559], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 548, 402, 563], "score": 1.0, "content": " has cyclic 2-class group. In this paper we determine when ", "type": "text"}, {"bbox": [402, 550, 434, 561], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [434, 548, 487, 563], "score": 1.0, "content": " has rank 2", "type": "text"}], "index": 27}, {"bbox": [125, 562, 485, 574], "spans": [{"bbox": [125, 562, 258, 574], "score": 1.0, "content": "for imaginary quadratic fields ", "type": "text"}, {"bbox": [258, 564, 263, 571], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [264, 562, 289, 574], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [289, 563, 317, 573], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [317, 562, 353, 574], "score": 1.0, "content": " of type ", "type": "text"}, {"bbox": [354, 563, 384, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [384, 562, 455, 574], "score": 1.0, "content": ". (The notation ", "type": "text"}, {"bbox": [455, 563, 485, 573], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}], "index": 28}, {"bbox": [125, 574, 487, 586], "spans": [{"bbox": [125, 574, 442, 586], "score": 1.0, "content": "means the direct sum of a group of order 2 and a cyclic group of order ", "type": "text"}, {"bbox": [443, 576, 456, 583], "score": 0.9, "content": "2^{m}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [456, 574, 487, 586], "score": 1.0, "content": ".) The", "type": "text"}], "index": 29}, {"bbox": [125, 586, 486, 597], "spans": [{"bbox": [125, 586, 486, 597], "score": 1.0, "content": "group theoretic results mentioned above also show that such fields have 2-class field", "type": "text"}], "index": 30}, {"bbox": [125, 598, 204, 610], "spans": [{"bbox": [125, 598, 204, 610], "score": 1.0, "content": "tower of length 2.", "type": "text"}], "index": 31}], "index": 23, "bbox_fs": [124, 407, 487, 610]}, {"type": "text", "bbox": [126, 608, 486, 667], "lines": [{"bbox": [136, 608, 486, 623], "spans": [{"bbox": [136, 608, 408, 623], "score": 1.0, "content": "From a classification of imaginary quadratic number fields ", "type": "text"}, {"bbox": [409, 611, 415, 618], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [415, 608, 443, 623], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 610, 486, 621], "score": 0.91, "content": "\\mathrm{Cl_{2}}(k)\\simeq", "type": "inline_equation", "height": 11, "width": 42}], "index": 32}, {"bbox": [126, 622, 487, 633], "spans": [{"bbox": [126, 623, 156, 633], "score": 0.93, "content": "(2,2^{m})", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 622, 487, 633], "score": 1.0, "content": " and our results from [1] we see that it suffices to consider discriminants", "type": "text"}], "index": 33}, {"bbox": [126, 633, 487, 645], "spans": [{"bbox": [126, 635, 175, 644], "score": 0.93, "content": "d=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [176, 633, 293, 645], "score": 1.0, "content": " with prime discriminants ", "type": "text"}, {"bbox": [293, 635, 337, 644], "score": 0.93, "content": "d_{1},d_{2}>0", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [338, 633, 344, 645], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [344, 635, 374, 644], "score": 0.93, "content": "d_{3}<0", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [374, 633, 487, 645], "score": 1.0, "content": " such that exactly one of", "type": "text"}], "index": 34}, {"bbox": [125, 645, 485, 658], "spans": [{"bbox": [125, 645, 143, 658], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [143, 646, 173, 657], "score": 0.92, "content": "\\left(d_{i}/p_{j}\\right)", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [174, 645, 207, 658], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [207, 648, 220, 655], "score": 0.9, "content": "^{-1}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [220, 645, 255, 658], "score": 1.0, "content": " (we let ", "type": "text"}, {"bbox": [255, 650, 264, 657], "score": 0.89, "content": "p_{j}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [265, 645, 383, 658], "score": 1.0, "content": " denote the prime dividing ", "type": "text"}, {"bbox": [383, 647, 393, 657], "score": 0.89, "content": "d_{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [393, 645, 485, 658], "score": 1.0, "content": "); thus there are only", "type": "text"}], "index": 35}, {"bbox": [125, 658, 171, 670], "spans": [{"bbox": [125, 658, 171, 670], "score": 1.0, "content": "two cases:", "type": "text"}], "index": 36}], "index": 34, "bbox_fs": [125, 608, 487, 670]}, {"type": "text", "bbox": [137, 689, 353, 700], "lines": [{"bbox": [138, 691, 353, 701], "spans": [{"bbox": [138, 691, 353, 701], "score": 1.0, "content": "1991 Mathematics Subject Classification. Primary 11R37.", "type": "text"}], "index": 37}], "index": 37, "bbox_fs": [138, 691, 353, 701]}]}
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[{"type": "title", "bbox": [129, 142, 482, 171], "content": "IMAGINARY QUADRATIC FIELDS WITH AND RANK", "index": 0}, {"type": "text", "bbox": [205, 189, 406, 199], "content": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER", "index": 1}, {"type": "text", "bbox": [160, 218, 450, 248], "content": "Abstract. Let be an imaginary quadratic number field and the Hilbert 2-class field of . We give a characterization of those with such that has 2 generators.", "index": 2}, {"type": "title", "bbox": [265, 282, 346, 294], "content": "1. Introduction", "index": 3}, {"type": "text", "bbox": [125, 300, 485, 348], "content": "Let be an algebraic number field with , the Sylow 2-subgroup of its ideal class group, . Denote by the Hilbert 2-class field of (in the wide sense). Also let (for a nonnegative integer) be defined inductively as: and kn+1 = (kn)1. Then", "index": 4}, {"type": "interline_equation", "bbox": [240, 354, 371, 365], "content": "", "index": 5}, {"type": "text", "bbox": [125, 368, 486, 404], "content": "is called the 2-class field tower of . If is the minimal integer such that , then is called the length of the tower. If no such exists, then the tower is said to be of infinite length.", "index": 6}, {"type": "text", "bbox": [126, 405, 486, 607], "content": "At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field is infinite. However, it is known by group theoretic results (see [2]) that if r , then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank . We have already started this program for imaginary quadratic number fields . In [1] we classified all imaginary quadratic fields whose 2-class field has cyclic 2-class group. In this paper we determine when has rank 2 for imaginary quadratic fields with of type . (The notation means the direct sum of a group of order 2 and a cyclic group of order .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2.", "index": 7}, {"type": "text", "bbox": [126, 608, 486, 667], "content": "From a classification of imaginary quadratic number fields with and our results from [1] we see that it suffices to consider discriminants with prime discriminants , such that exactly one of the equals (we let denote the prime dividing ); thus there are only two cases:", "index": 8}, {"type": "text", "bbox": [137, 689, 353, 700], "content": "1991 Mathematics Subject Classification. Primary 11R37.", "index": 9}]
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[{"bbox": [130, 145, 480, 158], "content": "IMAGINARY QUADRATIC FIELDS WITH AND", "parent_index": 0, "line_index": 0}, {"bbox": [259, 159, 351, 172], "content": "RANK", "parent_index": 0, "line_index": 1}, {"bbox": [205, 192, 406, 201], "content": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER", "parent_index": 1, "line_index": 0}, {"bbox": [162, 220, 450, 230], "content": "Abstract. Let be an imaginary quadratic number field and the Hilbert", "parent_index": 2, "line_index": 0}, {"bbox": [161, 230, 449, 240], "content": "2-class field of . We give a characterization of those with", "parent_index": 2, "line_index": 1}, {"bbox": [161, 240, 294, 250], "content": "such that has 2 generators.", "parent_index": 2, "line_index": 2}, {"bbox": [264, 284, 348, 296], "content": "1. Introduction", "parent_index": 3, "line_index": 0}, {"bbox": [137, 302, 486, 315], "content": "Let be an algebraic number field with , the Sylow 2-subgroup of its ideal", "parent_index": 4, "line_index": 0}, {"bbox": [124, 314, 486, 327], "content": "class group, . Denote by the Hilbert 2-class field of (in the wide sense).", "parent_index": 4, "line_index": 1}, {"bbox": [126, 325, 487, 339], "content": "Also let (for a nonnegative integer) be defined inductively as: and", "parent_index": 4, "line_index": 2}, {"bbox": [125, 336, 215, 351], "content": "kn+1 = (kn)1. Then", "parent_index": 4, "line_index": 3}, {"bbox": [124, 369, 484, 383], "content": "is called the 2-class field tower of . If is the minimal integer such that ,", "parent_index": 6, "line_index": 0}, {"bbox": [126, 383, 486, 394], "content": "then is called the length of the tower. If no such exists, then the tower is said", "parent_index": 6, "line_index": 1}, {"bbox": [126, 395, 226, 406], "content": "to be of infinite length.", "parent_index": 6, "line_index": 2}, {"bbox": [138, 407, 485, 417], "content": "At present there is no known decision procedure to determine whether or not", "parent_index": 7, "line_index": 0}, {"bbox": [126, 419, 485, 429], "content": "the (2-)class field tower of a given field is infinite. However, it is known by", "parent_index": 7, "line_index": 1}, {"bbox": [125, 430, 486, 442], "content": "group theoretic results (see [2]) that if r , then the tower is finite,", "parent_index": 7, "line_index": 2}, {"bbox": [125, 442, 486, 454], "content": "in fact of length at most 3. (Here the rank means minimal number of generators.)", "parent_index": 7, "line_index": 3}, {"bbox": [125, 454, 487, 467], "content": "On the other hand, until now (see Table 1 and the penultimate paragraph of this", "parent_index": 7, "line_index": 4}, {"bbox": [124, 465, 487, 479], "content": "introduction) all examples in the mathematical literature of imaginary quadratic", "parent_index": 7, "line_index": 5}, {"bbox": [125, 478, 486, 490], "content": "fields with rank (let us mention in particular Schmithals [13]) have", "parent_index": 7, "line_index": 6}, {"bbox": [125, 490, 485, 502], "content": "infinite 2-class field tower. Nevertheless, if we are interested in developing a decision", "parent_index": 7, "line_index": 7}, {"bbox": [124, 502, 487, 515], "content": "procedure for determining if the 2-class field tower of a field is infinite, then a", "parent_index": 7, "line_index": 8}, {"bbox": [125, 514, 487, 526], "content": "good starting point would be to find a procedure for sieving out those fields with", "parent_index": 7, "line_index": 9}, {"bbox": [124, 525, 487, 539], "content": "rank . We have already started this program for imaginary quadratic", "parent_index": 7, "line_index": 10}, {"bbox": [126, 538, 486, 550], "content": "number fields . In [1] we classified all imaginary quadratic fields whose 2-class field", "parent_index": 7, "line_index": 11}, {"bbox": [126, 548, 487, 563], "content": "has cyclic 2-class group. In this paper we determine when has rank 2", "parent_index": 7, "line_index": 12}, {"bbox": [125, 562, 485, 574], "content": "for imaginary quadratic fields with of type . (The notation", "parent_index": 7, "line_index": 13}, {"bbox": [125, 574, 487, 586], "content": "means the direct sum of a group of order 2 and a cyclic group of order .) The", "parent_index": 7, "line_index": 14}, {"bbox": [125, 586, 486, 597], "content": "group theoretic results mentioned above also show that such fields have 2-class field", "parent_index": 7, "line_index": 15}, {"bbox": [125, 598, 204, 610], "content": "tower of length 2.", "parent_index": 7, "line_index": 16}, {"bbox": [136, 608, 486, 623], "content": "From a classification of imaginary quadratic number fields with", "parent_index": 8, "line_index": 0}, {"bbox": [126, 622, 487, 633], "content": "and our results from [1] we see that it suffices to consider discriminants", "parent_index": 8, "line_index": 1}, {"bbox": [126, 633, 487, 645], "content": "with prime discriminants , such that exactly one of", "parent_index": 8, "line_index": 2}, {"bbox": [125, 645, 485, 658], "content": "the equals (we let denote the prime dividing ); thus there are only", "parent_index": 8, "line_index": 3}, {"bbox": [125, 658, 171, 670], "content": "two cases:", "parent_index": 8, "line_index": 4}, {"bbox": [138, 691, 353, 701], "content": "1991 Mathematics Subject Classification. Primary 11R37.", "parent_index": 9, "line_index": 0}]
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[{"bbox": [329, 148, 335, 155], "content": "k", "parent_index": 0, "subtype": "inline"}, {"bbox": [378, 147, 448, 158], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "parent_index": 0, "subtype": "inline"}, {"bbox": [300, 160, 351, 172], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [223, 222, 227, 227], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [396, 220, 405, 227], "content": "k^{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [217, 231, 222, 237], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [361, 231, 365, 237], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [388, 231, 449, 239], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "parent_index": 2, "subtype": "inline"}, {"bbox": [199, 240, 227, 249], "content": "\\mathrm{Cl}_{2}(k^{1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [155, 304, 161, 311], "content": "k", "parent_index": 4, "subtype": "inline"}, {"bbox": [310, 303, 338, 314], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 4, "subtype": "inline"}, {"bbox": [181, 315, 204, 326], "content": "\\operatorname{Cl}(k)", "parent_index": 4, "subtype": "inline"}, {"bbox": [260, 315, 270, 323], "content": "k^{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [390, 316, 396, 323], "content": "k", "parent_index": 4, "subtype": "inline"}, {"bbox": [164, 328, 176, 335], "content": "k^{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [200, 330, 206, 335], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [432, 327, 465, 335], "content": "k^{0}\\,=\\,k", "parent_index": 4, "subtype": "inline"}, {"bbox": [240, 354, 371, 365], "content": "k^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.", "parent_index": 5, "subtype": "interline"}, {"bbox": [268, 372, 274, 379], "content": "k", "parent_index": 6, "subtype": "inline"}, {"bbox": [289, 375, 296, 379], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [437, 371, 482, 379], "content": "k^{n}=k^{n+1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [149, 387, 155, 392], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [350, 387, 357, 392], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [310, 420, 316, 427], "content": "k", "parent_index": 7, "subtype": "inline"}, {"bbox": [303, 430, 374, 442], "content": "\\mathrm{ank\\,Cl_{2}}(k^{1})\\,\\leq\\,2", "parent_index": 7, "subtype": "inline"}, {"bbox": [198, 479, 252, 489], "content": "\\mathrm{Cl}_{2}(k^{1})\\,\\geq\\,3", "parent_index": 7, "subtype": "inline"}, {"bbox": [147, 526, 200, 537], "content": "\\mathrm{Cl}_{2}(k^{1})\\leq2", "parent_index": 7, "subtype": "inline"}, {"bbox": [186, 540, 192, 547], "content": "k", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 550, 136, 559], "content": "k^{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [402, 550, 434, 561], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 7, "subtype": "inline"}, {"bbox": [258, 564, 263, 571], "content": "k", "parent_index": 7, "subtype": "inline"}, {"bbox": [289, 563, 317, 573], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 7, "subtype": "inline"}, {"bbox": [354, 563, 384, 573], "content": "(2,2^{m})", "parent_index": 7, "subtype": "inline"}, {"bbox": [455, 563, 485, 573], "content": "(2,2^{m})", "parent_index": 7, "subtype": "inline"}, {"bbox": [443, 576, 456, 583], "content": "2^{m}", "parent_index": 7, "subtype": "inline"}, {"bbox": [409, 611, 415, 618], "content": "k", "parent_index": 8, "subtype": "inline"}, {"bbox": [444, 610, 486, 621], "content": "\\mathrm{Cl_{2}}(k)\\simeq", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 623, 156, 633], "content": "(2,2^{m})", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 635, 175, 644], "content": "d=d_{1}d_{2}d_{3}", "parent_index": 8, "subtype": "inline"}, {"bbox": [293, 635, 337, 644], "content": "d_{1},d_{2}>0", "parent_index": 8, "subtype": "inline"}, {"bbox": [344, 635, 374, 644], "content": "d_{3}<0", "parent_index": 8, "subtype": "inline"}, {"bbox": [143, 646, 173, 657], "content": "\\left(d_{i}/p_{j}\\right)", "parent_index": 8, "subtype": "inline"}, {"bbox": [207, 648, 220, 655], "content": "^{-1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [255, 650, 264, 657], "content": "p_{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [383, 647, 393, 657], "content": "d_{j}", "parent_index": 8, "subtype": "inline"}]
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The $C_{4}$ -factorization corresponding to the nontrivial 4-part of $\mathrm{Cl_{2}}(k)$ is $d=\boldsymbol{d}_{1}\cdot\boldsymbol{d}_{2}\boldsymbol{d}_{3}$ in case A) and $d=d_{1}d_{2}\cdot d_{3}$ in case B). Note that, by our results from [1], some of these fields have cyclic $\mathrm{Cl_{2}}(k^{1})$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work.
The main result of the paper is that rank $\mathrm{Cl}_{2}(k^{1})=2$ only occurs for fields of type B); more precisely, we prove the following
Theorem 1. Let $k$ be a complex quadratic number field with $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ , and let $k^{1}$ be its 2-class field. Then rank $\mathrm{Cl}_{2}(k^{1})=2$ if and only if disc $k=d_{1}d_{2}d_{3}$ is the product of three prime discriminants $d_{1},d_{2}\,>\,0$ and $-4\,\ne\,d_{3}\,<\,0$ such that $(d_{1}/p_{3})=(d_{2}/p_{3})=+1$ , $(d_{1}/p_{2})=-1$ , and $h_{2}(K)=2$ , where $K$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $k$ such that $\mathbb{Q}({\sqrt{d_{1}d_{2}}}\,)\subset K$ .
This result is the first step in the classification of imaginary quadratic number fields $k$ with rank $\mathrm{Cl}_{2}(k^{1})\,=\,2$ ; it remains to solve these problems for fields with rank $\mathrm{Cl}_{2}(k)=3$ and those with $\mathrm{Cl}_{2}(k)\supseteq(4,4)$ since we know that rank $\mathrm{Cl}_{2}(k^{1})\geq5$ whenever rank $\mathrm{Cl}_{2}(k)\geq4$ (using Schur multipliers as in [1]).
As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $k$ , arranged by decreasing value of their discriminants, with ran $\mathinner{\mathrm{\Omega}\mathopen{\left(\lambda\right)}}\mathinner{\mathrm{\Omega}\mathopen{\left(k\right)}}=2$ and noncyclic $\mathrm{Cl_{2}}(k^{1})$ .
Table 1
<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>
Here $f$ denotes a generating polynomial for a field $K$ as in Theorem 1, $r$ denotes the rank of $\mathrm{Cl_{2}}(k^{1})$ . The cases where $r=3$ follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups $G$ with $G/G^{\prime}\simeq(2,4)$ satisfy $\mathrm{rank}\,G^{\prime}\leq3$ ), see [3].
In order to verify that $\mathrm{Cl_{2}}(k^{1})$ has rank at least 3 for $k\,=\,\mathbb{Q}({\sqrt{-2379}}\,)$ it is sufficient to show that its genus class field $k_{\mathrm{gen}}$ has class group $(4,4,8)$ : in fact, $\mathrm{Cl_{2}}(k^{1})$ then contains a quotient of $(4,4,8)$ by $(2,2)\simeq\mathrm{Gal}(k^{1}/k_{\mathrm{gen}})$ , and the claim follows.
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<p data-bbox="125 127 486 186">The $C_{4}$ -factorization corresponding to the nontrivial 4-part of $\mathrm{Cl_{2}}(k)$ is $d=\boldsymbol{d}_{1}\cdot\boldsymbol{d}_{2}\boldsymbol{d}_{3}$ in case A) and $d=d_{1}d_{2}\cdot d_{3}$ in case B). Note that, by our results from [1], some of these fields have cyclic $\mathrm{Cl_{2}}(k^{1})$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work. </p>
<p data-bbox="125 187 487 211">The main result of the paper is that rank $\mathrm{Cl}_{2}(k^{1})=2$ only occurs for fields of type B); more precisely, we prove the following </p>
<p data-bbox="125 217 487 290">Theorem 1. Let $k$ be a complex quadratic number field with $\mathrm{Cl}_{2}(k)\simeq(2,2^{m})$ , and let $k^{1}$ be its 2-class field. Then rank $\mathrm{Cl}_{2}(k^{1})=2$ if and only if disc $k=d_{1}d_{2}d_{3}$ is the product of three prime discriminants $d_{1},d_{2}\,>\,0$ and $-4\,\ne\,d_{3}\,<\,0$ such that $(d_{1}/p_{3})=(d_{2}/p_{3})=+1$ , $(d_{1}/p_{2})=-1$ , and $h_{2}(K)=2$ , where $K$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $k$ such that $\mathbb{Q}({\sqrt{d_{1}d_{2}}}\,)\subset K$ . </p>
<p data-bbox="125 296 486 344">This result is the first step in the classification of imaginary quadratic number fields $k$ with rank $\mathrm{Cl}_{2}(k^{1})\,=\,2$ ; it remains to solve these problems for fields with rank $\mathrm{Cl}_{2}(k)=3$ and those with $\mathrm{Cl}_{2}(k)\supseteq(4,4)$ since we know that rank $\mathrm{Cl}_{2}(k^{1})\geq5$ whenever rank $\mathrm{Cl}_{2}(k)\geq4$ (using Schur multipliers as in [1]). </p>
<p data-bbox="124 344 486 380">As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $k$ , arranged by decreasing value of their discriminants, with ran $\mathinner{\mathrm{\Omega}\mathopen{\left(\lambda\right)}}\mathinner{\mathrm{\Omega}\mathopen{\left(k\right)}}=2$ and noncyclic $\mathrm{Cl_{2}}(k^{1})$ . </p>
<div class="table" data-bbox="126 429 489 588"><p class="caption" data-bbox="285 392 325 403">Table 1 </p><table data-bbox="126 429 489 588"><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></div>
<p data-bbox="125 603 486 651">Here $f$ denotes a generating polynomial for a field $K$ as in Theorem 1, $r$ denotes the rank of $\mathrm{Cl_{2}}(k^{1})$ . The cases where $r=3$ follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups $G$ with $G/G^{\prime}\simeq(2,4)$ satisfy $\mathrm{rank}\,G^{\prime}\leq3$ ), see [3]. </p>
<p data-bbox="124 651 486 699">In order to verify that $\mathrm{Cl_{2}}(k^{1})$ has rank at least 3 for $k\,=\,\mathbb{Q}({\sqrt{-2379}}\,)$ it is sufficient to show that its genus class field $k_{\mathrm{gen}}$ has class group $(4,4,8)$ : in fact, $\mathrm{Cl_{2}}(k^{1})$ then contains a quotient of $(4,4,8)$ by $(2,2)\simeq\mathrm{Gal}(k^{1}/k_{\mathrm{gen}})$ , and the claim follows. </p>
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0003244v1
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[{"type": "text", "text": "The $C_{4}$ -factorization corresponding to the nontrivial 4-part of $\\mathrm{Cl_{2}}(k)$ is $d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}$ in case A) and $d=d_{1}d_{2}\\cdot d_{3}$ in case B). Note that, by our results from [1], some of these fields have cyclic $\\mathrm{Cl_{2}}(k^{1})$ ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work. ", "page_idx": 1}, {"type": "text", "text": "The main result of the paper is that rank $\\mathrm{Cl}_{2}(k^{1})=2$ only occurs for fields of type B); more precisely, we prove the following ", "page_idx": 1}, {"type": "text", "text": "Theorem 1. Let $k$ be a complex quadratic number field with $\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$ , and let $k^{1}$ be its 2-class field. Then rank $\\mathrm{Cl}_{2}(k^{1})=2$ if and only if disc $k=d_{1}d_{2}d_{3}$ is the product of three prime discriminants $d_{1},d_{2}\\,>\\,0$ and $-4\\,\\ne\\,d_{3}\\,<\\,0$ such that $(d_{1}/p_{3})=(d_{2}/p_{3})=+1$ , $(d_{1}/p_{2})=-1$ , and $h_{2}(K)=2$ , where $K$ is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of $k$ such that $\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K$ . ", "page_idx": 1}, {"type": "text", "text": "This result is the first step in the classification of imaginary quadratic number fields $k$ with rank $\\mathrm{Cl}_{2}(k^{1})\\,=\\,2$ ; it remains to solve these problems for fields with rank $\\mathrm{Cl}_{2}(k)=3$ and those with $\\mathrm{Cl}_{2}(k)\\supseteq(4,4)$ since we know that rank $\\mathrm{Cl}_{2}(k^{1})\\geq5$ whenever rank $\\mathrm{Cl}_{2}(k)\\geq4$ (using Schur multipliers as in [1]). ", "page_idx": 1}, {"type": "text", "text": "As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields $k$ , arranged by decreasing value of their discriminants, with ran $\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2$ and noncyclic $\\mathrm{Cl_{2}}(k^{1})$ . ", "page_idx": 1}, {"type": "table", "img_path": "images/582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg", "table_caption": ["Table 1 "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>\n\n", "page_idx": 1}, {"type": "text", "text": "Here $f$ denotes a generating polynomial for a field $K$ as in Theorem 1, $r$ denotes the rank of $\\mathrm{Cl_{2}}(k^{1})$ . The cases where $r=3$ follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups $G$ with $G/G^{\\prime}\\simeq(2,4)$ satisfy $\\mathrm{rank}\\,G^{\\prime}\\leq3$ ), see [3]. ", "page_idx": 1}, {"type": "text", "text": "In order to verify that $\\mathrm{Cl_{2}}(k^{1})$ has rank at least 3 for $k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)$ it is sufficient to show that its genus class field $k_{\\mathrm{gen}}$ has class group $(4,4,8)$ : in fact, $\\mathrm{Cl_{2}}(k^{1})$ then contains a quotient of $(4,4,8)$ by $(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})$ , and the claim follows. ", "page_idx": 1}]
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B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>"}, {"category_id": 1, "poly": [347, 1811, 1351, 1811, 1351, 1943, 347, 1943], "score": 0.956}, {"category_id": 1, "poly": [348, 521, 1353, 521, 1353, 588, 348, 588], "score": 0.893}, {"category_id": 6, "poly": [793, 1089, 904, 1089, 904, 1122, 793, 1122], "score": 0.862}, {"category_id": 2, "poly": [348, 253, 366, 253, 366, 275, 348, 275], "score": 0.783}, {"category_id": 8, "poly": [367, 311, 895, 311, 895, 349, 367, 349], "score": 0.653}, {"category_id": 1, "poly": [349, 521, 1353, 521, 1353, 588, 349, 588], "score": 0.383}, {"category_id": 13, "poly": [351, 778, 539, 778, 539, 810, 351, 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{"preproc_blocks": [{"type": "text", "bbox": [125, 127, 486, 186], "lines": [{"bbox": [126, 128, 484, 141], "spans": [{"bbox": [126, 128, 145, 141], "score": 1.0, "content": "The", "type": "text"}, {"bbox": [146, 131, 157, 140], "score": 0.93, "content": "C_{4}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [158, 128, 393, 141], "score": 1.0, "content": "-factorization corresponding to the nontrivial 4-part of ", "type": "text"}, {"bbox": [393, 130, 421, 141], "score": 0.89, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [421, 128, 433, 141], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [433, 131, 484, 140], "score": 0.93, "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "type": "inline_equation", "height": 9, "width": 51}], "index": 0}, {"bbox": [126, 141, 486, 153], "spans": [{"bbox": [126, 141, 195, 153], "score": 1.0, "content": "in case A) and ", "type": "text"}, {"bbox": [195, 143, 252, 151], "score": 0.94, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 57}, {"bbox": [252, 141, 486, 153], "score": 1.0, "content": " in case B). Note that, by our results from [1], some", "type": "text"}], "index": 1}, {"bbox": [126, 153, 486, 165], "spans": [{"bbox": [126, 153, 237, 165], "score": 1.0, "content": "of these fields have cyclic ", "type": "text"}, {"bbox": [238, 154, 270, 164], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [271, 153, 486, 165], "score": 1.0, "content": "; however, we do not exclude them right from the", "type": "text"}], "index": 2}, {"bbox": [126, 166, 486, 177], "spans": [{"bbox": [126, 166, 486, 177], "score": 1.0, "content": "start since there is no extra work involved and since it provides a welcome check", "type": "text"}], "index": 3}, {"bbox": [126, 178, 213, 188], "spans": [{"bbox": [126, 178, 213, 188], "score": 1.0, "content": "on our earlier work.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [125, 187, 487, 211], "lines": [{"bbox": [137, 188, 488, 201], "spans": [{"bbox": [137, 188, 325, 201], "score": 1.0, "content": "The main result of the paper is that rank ", "type": "text"}, {"bbox": [325, 189, 377, 200], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [378, 188, 488, 201], "score": 1.0, "content": " only occurs for fields of", "type": "text"}], "index": 5}, {"bbox": [125, 200, 333, 214], "spans": [{"bbox": [125, 200, 333, 214], "score": 1.0, "content": "type B); more precisely, we prove the following", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [125, 217, 487, 290], "lines": [{"bbox": [126, 219, 487, 232], "spans": [{"bbox": [126, 219, 205, 232], "score": 1.0, "content": "Theorem 1. Let ", "type": "text"}, {"bbox": [205, 221, 211, 229], "score": 0.58, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [212, 219, 392, 232], "score": 1.0, "content": " be a complex quadratic number field with ", "type": "text"}, {"bbox": [392, 221, 463, 231], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [463, 219, 487, 232], "score": 1.0, "content": ", and", "type": "text"}], "index": 7}, {"bbox": [125, 231, 487, 244], "spans": [{"bbox": [125, 231, 140, 244], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 232, 150, 241], "score": 0.88, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [151, 231, 288, 244], "score": 1.0, "content": " be its 2-class field. Then rank", "type": "text"}, {"bbox": [289, 232, 340, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [341, 231, 423, 244], "score": 1.0, "content": " if and only if disc", "type": "text"}, {"bbox": [423, 233, 474, 242], "score": 0.84, "content": "k=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [474, 231, 487, 244], "score": 1.0, "content": " is", "type": "text"}], "index": 8}, {"bbox": [126, 244, 487, 255], "spans": [{"bbox": [126, 244, 309, 255], "score": 1.0, "content": "the product of three prime discriminants ", "type": "text"}, {"bbox": [310, 245, 356, 254], "score": 0.92, "content": "d_{1},d_{2}\\,>\\,0", "type": "inline_equation", "height": 9, "width": 46}, {"bbox": [356, 244, 379, 255], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [380, 245, 440, 254], "score": 0.91, "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "type": "inline_equation", "height": 9, "width": 60}, {"bbox": [441, 244, 487, 255], "score": 1.0, "content": " such that", "type": "text"}], "index": 9}, {"bbox": [126, 256, 487, 268], "spans": [{"bbox": [126, 257, 231, 267], "score": 0.89, "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "type": "inline_equation", "height": 10, "width": 105}, {"bbox": [232, 256, 237, 268], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [238, 257, 297, 267], "score": 0.92, "content": "(d_{1}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [297, 256, 322, 268], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [323, 257, 370, 267], "score": 0.93, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [370, 256, 404, 268], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [405, 257, 414, 264], "score": 0.89, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [415, 256, 487, 268], "score": 1.0, "content": " is a nonnormal", "type": "text"}], "index": 10}, {"bbox": [126, 268, 487, 281], "spans": [{"bbox": [126, 268, 437, 281], "score": 1.0, "content": "quartic subfield of one of the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [437, 269, 443, 277], "score": 0.61, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [443, 268, 487, 281], "score": 1.0, "content": " such that", "type": "text"}], "index": 11}, {"bbox": [126, 280, 198, 291], "spans": [{"bbox": [126, 280, 194, 291], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [194, 280, 198, 291], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5}, {"type": "text", "bbox": [125, 296, 486, 344], "lines": [{"bbox": [137, 298, 486, 311], "spans": [{"bbox": [137, 298, 486, 311], "score": 1.0, "content": "This result is the first step in the classification of imaginary quadratic number", "type": "text"}], "index": 13}, {"bbox": [126, 310, 486, 322], "spans": [{"bbox": [126, 310, 151, 322], "score": 1.0, "content": "fields ", "type": "text"}, {"bbox": [152, 312, 158, 319], "score": 0.86, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [158, 310, 206, 322], "score": 1.0, "content": " with rank", "type": "text"}, {"bbox": [207, 311, 259, 322], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [260, 310, 486, 322], "score": 1.0, "content": "; it remains to solve these problems for fields with", "type": "text"}], "index": 14}, {"bbox": [126, 322, 485, 335], "spans": [{"bbox": [126, 322, 147, 335], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 324, 193, 334], "score": 0.88, "content": "\\mathrm{Cl}_{2}(k)=3", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [194, 322, 262, 335], "score": 1.0, "content": " and those with ", "type": "text"}, {"bbox": [262, 324, 326, 334], "score": 0.94, "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [326, 322, 434, 335], "score": 1.0, "content": " since we know that rank", "type": "text"}, {"bbox": [434, 323, 485, 334], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "type": "inline_equation", "height": 11, "width": 51}], "index": 15}, {"bbox": [127, 335, 388, 346], "spans": [{"bbox": [127, 335, 191, 346], "score": 1.0, "content": "whenever rank ", "type": "text"}, {"bbox": [191, 335, 237, 346], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\geq4", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [237, 335, 388, 346], "score": 1.0, "content": " (using Schur multipliers as in [1]).", "type": "text"}], "index": 16}], "index": 14.5}, {"type": "text", "bbox": [124, 344, 486, 380], "lines": [{"bbox": [137, 346, 486, 359], "spans": [{"bbox": [137, 346, 486, 359], "score": 1.0, "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "type": "text"}], "index": 17}, {"bbox": [125, 357, 487, 372], "spans": [{"bbox": [125, 357, 308, 372], "score": 1.0, "content": "of the first 12 imaginary quadratic fields ", "type": "text"}, {"bbox": [309, 360, 314, 367], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [315, 357, 487, 372], "score": 1.0, "content": ", arranged by decreasing value of their", "type": "text"}], "index": 18}, {"bbox": [126, 370, 382, 382], "spans": [{"bbox": [126, 370, 229, 382], "score": 1.0, "content": "discriminants, with ran", "type": "text"}, {"bbox": [229, 371, 280, 381], "score": 0.78, "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [280, 370, 346, 382], "score": 1.0, "content": " and noncyclic ", "type": "text"}, {"bbox": [346, 371, 378, 381], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [379, 370, 382, 382], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0}, {"type": "text", "bbox": [125, 603, 486, 651], "lines": [{"bbox": [137, 605, 486, 618], "spans": [{"bbox": [137, 605, 161, 618], "score": 1.0, "content": "Here ", "type": "text"}, {"bbox": [161, 607, 167, 617], "score": 0.89, "content": "f", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [167, 605, 356, 618], "score": 1.0, "content": " denotes a generating polynomial for a field ", "type": "text"}, {"bbox": [357, 607, 366, 614], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [366, 605, 444, 618], "score": 1.0, "content": " as in Theorem 1, ", "type": "text"}, {"bbox": [444, 610, 449, 614], "score": 0.85, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [450, 605, 486, 618], "score": 1.0, "content": " denotes", "type": "text"}], "index": 24}, {"bbox": [126, 618, 484, 629], "spans": [{"bbox": [126, 618, 176, 629], "score": 1.0, "content": "the rank of ", "type": "text"}, {"bbox": [176, 618, 208, 629], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [209, 618, 288, 629], "score": 1.0, "content": ". The cases where ", "type": "text"}, {"bbox": [288, 619, 311, 626], "score": 0.91, "content": "r=3", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [312, 618, 484, 629], "score": 1.0, "content": " follow from our theorem combined with", "type": "text"}], "index": 25}, {"bbox": [126, 629, 485, 642], "spans": [{"bbox": [126, 629, 485, 642], "score": 1.0, "content": "Blackburn’s upper bound for the number of generators of derived groups (it implies", "type": "text"}], "index": 26}, {"bbox": [126, 641, 431, 653], "spans": [{"bbox": [126, 641, 213, 653], "score": 1.0, "content": "that finite 2-groups ", "type": "text"}, {"bbox": [213, 643, 221, 650], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [222, 641, 247, 653], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [247, 642, 306, 653], "score": 0.94, "content": "G/G^{\\prime}\\simeq(2,4)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [307, 641, 340, 653], "score": 1.0, "content": " satisfy ", "type": "text"}, {"bbox": [340, 642, 391, 651], "score": 0.84, "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [392, 641, 431, 653], "score": 1.0, "content": "), see [3].", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "text", "bbox": [124, 651, 486, 699], "lines": [{"bbox": [136, 652, 486, 665], "spans": [{"bbox": [136, 652, 243, 665], "score": 1.0, "content": "In order to verify that ", "type": "text"}, {"bbox": [243, 654, 276, 665], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [276, 652, 386, 665], "score": 1.0, "content": " has rank at least 3 for ", "type": "text"}, {"bbox": [387, 653, 462, 665], "score": 0.91, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [463, 652, 486, 665], "score": 1.0, "content": " it is", "type": "text"}], "index": 28}, {"bbox": [126, 664, 486, 679], "spans": [{"bbox": [126, 664, 319, 679], "score": 1.0, "content": "sufficient to show that its genus class field ", "type": "text"}, {"bbox": [319, 667, 337, 677], "score": 0.92, "content": "k_{\\mathrm{gen}}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [337, 664, 412, 679], "score": 1.0, "content": " has class group ", "type": "text"}, {"bbox": [413, 666, 444, 677], "score": 0.91, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [445, 664, 486, 679], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 29}, {"bbox": [126, 676, 487, 690], "spans": [{"bbox": [126, 677, 158, 689], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [159, 676, 280, 690], "score": 1.0, "content": " then contains a quotient of ", "type": "text"}, {"bbox": [280, 678, 312, 689], "score": 0.92, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [312, 676, 328, 690], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [329, 677, 420, 689], "score": 0.93, "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [420, 676, 487, 690], "score": 1.0, "content": ", and the claim", "type": "text"}], "index": 30}, {"bbox": [125, 689, 159, 702], "spans": [{"bbox": [125, 689, 159, 702], "score": 1.0, "content": "follows.", "type": "text"}], "index": 31}], "index": 29.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [126, 93, 131, 102], "spans": [{"bbox": [126, 93, 131, 102], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 127, 486, 186], "lines": [{"bbox": [126, 128, 484, 141], "spans": [{"bbox": [126, 128, 145, 141], "score": 1.0, "content": "The", "type": "text"}, {"bbox": [146, 131, 157, 140], "score": 0.93, "content": "C_{4}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [158, 128, 393, 141], "score": 1.0, "content": "-factorization corresponding to the nontrivial 4-part of ", "type": "text"}, {"bbox": [393, 130, 421, 141], "score": 0.89, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [421, 128, 433, 141], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [433, 131, 484, 140], "score": 0.93, "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "type": "inline_equation", "height": 9, "width": 51}], "index": 0}, {"bbox": [126, 141, 486, 153], "spans": [{"bbox": [126, 141, 195, 153], "score": 1.0, "content": "in case A) and ", "type": "text"}, {"bbox": [195, 143, 252, 151], "score": 0.94, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 57}, {"bbox": [252, 141, 486, 153], "score": 1.0, "content": " in case B). Note that, by our results from [1], some", "type": "text"}], "index": 1}, {"bbox": [126, 153, 486, 165], "spans": [{"bbox": [126, 153, 237, 165], "score": 1.0, "content": "of these fields have cyclic ", "type": "text"}, {"bbox": [238, 154, 270, 164], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [271, 153, 486, 165], "score": 1.0, "content": "; however, we do not exclude them right from the", "type": "text"}], "index": 2}, {"bbox": [126, 166, 486, 177], "spans": [{"bbox": [126, 166, 486, 177], "score": 1.0, "content": "start since there is no extra work involved and since it provides a welcome check", "type": "text"}], "index": 3}, {"bbox": [126, 178, 213, 188], "spans": [{"bbox": [126, 178, 213, 188], "score": 1.0, "content": "on our earlier work.", "type": "text"}], "index": 4}], "index": 2, "bbox_fs": [126, 128, 486, 188]}, {"type": "text", "bbox": [125, 187, 487, 211], "lines": [{"bbox": [137, 188, 488, 201], "spans": [{"bbox": [137, 188, 325, 201], "score": 1.0, "content": "The main result of the paper is that rank ", "type": "text"}, {"bbox": [325, 189, 377, 200], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [378, 188, 488, 201], "score": 1.0, "content": " only occurs for fields of", "type": "text"}], "index": 5}, {"bbox": [125, 200, 333, 214], "spans": [{"bbox": [125, 200, 333, 214], "score": 1.0, "content": "type B); more precisely, we prove the following", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [125, 188, 488, 214]}, {"type": "text", "bbox": [125, 217, 487, 290], "lines": [{"bbox": [126, 219, 487, 232], "spans": [{"bbox": [126, 219, 205, 232], "score": 1.0, "content": "Theorem 1. Let ", "type": "text"}, {"bbox": [205, 221, 211, 229], "score": 0.58, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [212, 219, 392, 232], "score": 1.0, "content": " be a complex quadratic number field with ", "type": "text"}, {"bbox": [392, 221, 463, 231], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [463, 219, 487, 232], "score": 1.0, "content": ", and", "type": "text"}], "index": 7}, {"bbox": [125, 231, 487, 244], "spans": [{"bbox": [125, 231, 140, 244], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 232, 150, 241], "score": 0.88, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [151, 231, 288, 244], "score": 1.0, "content": " be its 2-class field. Then rank", "type": "text"}, {"bbox": [289, 232, 340, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [341, 231, 423, 244], "score": 1.0, "content": " if and only if disc", "type": "text"}, {"bbox": [423, 233, 474, 242], "score": 0.84, "content": "k=d_{1}d_{2}d_{3}", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [474, 231, 487, 244], "score": 1.0, "content": " is", "type": "text"}], "index": 8}, {"bbox": [126, 244, 487, 255], "spans": [{"bbox": [126, 244, 309, 255], "score": 1.0, "content": "the product of three prime discriminants ", "type": "text"}, {"bbox": [310, 245, 356, 254], "score": 0.92, "content": "d_{1},d_{2}\\,>\\,0", "type": "inline_equation", "height": 9, "width": 46}, {"bbox": [356, 244, 379, 255], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [380, 245, 440, 254], "score": 0.91, "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "type": "inline_equation", "height": 9, "width": 60}, {"bbox": [441, 244, 487, 255], "score": 1.0, "content": " such that", "type": "text"}], "index": 9}, {"bbox": [126, 256, 487, 268], "spans": [{"bbox": [126, 257, 231, 267], "score": 0.89, "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "type": "inline_equation", "height": 10, "width": 105}, {"bbox": [232, 256, 237, 268], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [238, 257, 297, 267], "score": 0.92, "content": "(d_{1}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [297, 256, 322, 268], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [323, 257, 370, 267], "score": 0.93, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [370, 256, 404, 268], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [405, 257, 414, 264], "score": 0.89, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [415, 256, 487, 268], "score": 1.0, "content": " is a nonnormal", "type": "text"}], "index": 10}, {"bbox": [126, 268, 487, 281], "spans": [{"bbox": [126, 268, 437, 281], "score": 1.0, "content": "quartic subfield of one of the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [437, 269, 443, 277], "score": 0.61, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [443, 268, 487, 281], "score": 1.0, "content": " such that", "type": "text"}], "index": 11}, {"bbox": [126, 280, 198, 291], "spans": [{"bbox": [126, 280, 194, 291], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [194, 280, 198, 291], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 9.5, "bbox_fs": [125, 219, 487, 291]}, {"type": "text", "bbox": [125, 296, 486, 344], "lines": [{"bbox": [137, 298, 486, 311], "spans": [{"bbox": [137, 298, 486, 311], "score": 1.0, "content": "This result is the first step in the classification of imaginary quadratic number", "type": "text"}], "index": 13}, {"bbox": [126, 310, 486, 322], "spans": [{"bbox": [126, 310, 151, 322], "score": 1.0, "content": "fields ", "type": "text"}, {"bbox": [152, 312, 158, 319], "score": 0.86, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [158, 310, 206, 322], "score": 1.0, "content": " with rank", "type": "text"}, {"bbox": [207, 311, 259, 322], "score": 0.9, "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [260, 310, 486, 322], "score": 1.0, "content": "; it remains to solve these problems for fields with", "type": "text"}], "index": 14}, {"bbox": [126, 322, 485, 335], "spans": [{"bbox": [126, 322, 147, 335], "score": 1.0, "content": "rank", "type": "text"}, {"bbox": [147, 324, 193, 334], "score": 0.88, "content": "\\mathrm{Cl}_{2}(k)=3", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [194, 322, 262, 335], "score": 1.0, "content": " and those with ", "type": "text"}, {"bbox": [262, 324, 326, 334], "score": 0.94, "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [326, 322, 434, 335], "score": 1.0, "content": " since we know that rank", "type": "text"}, {"bbox": [434, 323, 485, 334], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "type": "inline_equation", "height": 11, "width": 51}], "index": 15}, {"bbox": [127, 335, 388, 346], "spans": [{"bbox": [127, 335, 191, 346], "score": 1.0, "content": "whenever rank ", "type": "text"}, {"bbox": [191, 335, 237, 346], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)\\geq4", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [237, 335, 388, 346], "score": 1.0, "content": " (using Schur multipliers as in [1]).", "type": "text"}], "index": 16}], "index": 14.5, "bbox_fs": [126, 298, 486, 346]}, {"type": "text", "bbox": [124, 344, 486, 380], "lines": [{"bbox": [137, 346, 486, 359], "spans": [{"bbox": [137, 346, 486, 359], "score": 1.0, "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "type": "text"}], "index": 17}, {"bbox": [125, 357, 487, 372], "spans": [{"bbox": [125, 357, 308, 372], "score": 1.0, "content": "of the first 12 imaginary quadratic fields ", "type": "text"}, {"bbox": [309, 360, 314, 367], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [315, 357, 487, 372], "score": 1.0, "content": ", arranged by decreasing value of their", "type": "text"}], "index": 18}, {"bbox": [126, 370, 382, 382], "spans": [{"bbox": [126, 370, 229, 382], "score": 1.0, "content": "discriminants, with ran", "type": "text"}, {"bbox": [229, 371, 280, 381], "score": 0.78, "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "type": "inline_equation", "height": 10, "width": 51}, {"bbox": [280, 370, 346, 382], "score": 1.0, "content": " and noncyclic ", "type": "text"}, {"bbox": [346, 371, 378, 381], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [379, 370, 382, 382], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [125, 346, 487, 382]}, {"type": "table", "bbox": [126, 429, 489, 588], "blocks": [{"type": "table_caption", "bbox": [285, 392, 325, 403], "group_id": 0, "lines": [{"bbox": [285, 393, 326, 405], "spans": [{"bbox": [285, 393, 326, 405], "score": 1.0, "content": "Table 1", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [126, 429, 489, 588], "group_id": 0, "lines": [{"bbox": [126, 429, 489, 588], "spans": [{"bbox": [126, 429, 489, 588], "score": 0.964, "html": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "type": "table", "image_path": "582709e690d2b641037bcb2cc5f471d2d0153fd447f9e0f673a5dd38693c6836.jpg"}]}], "index": 22, "virtual_lines": [{"bbox": [126, 429, 489, 482.0], "spans": [], "index": 21}, {"bbox": [126, 482.0, 489, 535.0], "spans": [], "index": 22}, {"bbox": [126, 535.0, 489, 588.0], "spans": [], "index": 23}]}], "index": 21.0}, {"type": "text", "bbox": [125, 603, 486, 651], "lines": [{"bbox": [137, 605, 486, 618], "spans": [{"bbox": [137, 605, 161, 618], "score": 1.0, "content": "Here ", "type": "text"}, {"bbox": [161, 607, 167, 617], "score": 0.89, "content": "f", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [167, 605, 356, 618], "score": 1.0, "content": " denotes a generating polynomial for a field ", "type": "text"}, {"bbox": [357, 607, 366, 614], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [366, 605, 444, 618], "score": 1.0, "content": " as in Theorem 1, ", "type": "text"}, {"bbox": [444, 610, 449, 614], "score": 0.85, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [450, 605, 486, 618], "score": 1.0, "content": " denotes", "type": "text"}], "index": 24}, {"bbox": [126, 618, 484, 629], "spans": [{"bbox": [126, 618, 176, 629], "score": 1.0, "content": "the rank of ", "type": "text"}, {"bbox": [176, 618, 208, 629], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [209, 618, 288, 629], "score": 1.0, "content": ". The cases where ", "type": "text"}, {"bbox": [288, 619, 311, 626], "score": 0.91, "content": "r=3", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [312, 618, 484, 629], "score": 1.0, "content": " follow from our theorem combined with", "type": "text"}], "index": 25}, {"bbox": [126, 629, 485, 642], "spans": [{"bbox": [126, 629, 485, 642], "score": 1.0, "content": "Blackburn’s upper bound for the number of generators of derived groups (it implies", "type": "text"}], "index": 26}, {"bbox": [126, 641, 431, 653], "spans": [{"bbox": [126, 641, 213, 653], "score": 1.0, "content": "that finite 2-groups ", "type": "text"}, {"bbox": [213, 643, 221, 650], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [222, 641, 247, 653], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [247, 642, 306, 653], "score": 0.94, "content": "G/G^{\\prime}\\simeq(2,4)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [307, 641, 340, 653], "score": 1.0, "content": " satisfy ", "type": "text"}, {"bbox": [340, 642, 391, 651], "score": 0.84, "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "type": "inline_equation", "height": 9, "width": 51}, {"bbox": [392, 641, 431, 653], "score": 1.0, "content": "), see [3].", "type": "text"}], "index": 27}], "index": 25.5, "bbox_fs": [126, 605, 486, 653]}, {"type": "text", "bbox": [124, 651, 486, 699], "lines": [{"bbox": [136, 652, 486, 665], "spans": [{"bbox": [136, 652, 243, 665], "score": 1.0, "content": "In order to verify that ", "type": "text"}, {"bbox": [243, 654, 276, 665], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [276, 652, 386, 665], "score": 1.0, "content": " has rank at least 3 for ", "type": "text"}, {"bbox": [387, 653, 462, 665], "score": 0.91, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "type": "inline_equation", "height": 12, "width": 75}, {"bbox": [463, 652, 486, 665], "score": 1.0, "content": " it is", "type": "text"}], "index": 28}, {"bbox": [126, 664, 486, 679], "spans": [{"bbox": [126, 664, 319, 679], "score": 1.0, "content": "sufficient to show that its genus class field ", "type": "text"}, {"bbox": [319, 667, 337, 677], "score": 0.92, "content": "k_{\\mathrm{gen}}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [337, 664, 412, 679], "score": 1.0, "content": " has class group ", "type": "text"}, {"bbox": [413, 666, 444, 677], "score": 0.91, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [445, 664, 486, 679], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 29}, {"bbox": [126, 676, 487, 690], "spans": [{"bbox": [126, 677, 158, 689], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [159, 676, 280, 690], "score": 1.0, "content": " then contains a quotient of ", "type": "text"}, {"bbox": [280, 678, 312, 689], "score": 0.92, "content": "(4,4,8)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [312, 676, 328, 690], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [329, 677, 420, 689], "score": 0.93, "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "type": "inline_equation", "height": 12, "width": 91}, {"bbox": [420, 676, 487, 690], "score": 1.0, "content": ", and the claim", "type": "text"}], "index": 30}, {"bbox": [125, 689, 159, 702], "spans": [{"bbox": [125, 689, 159, 702], "score": 1.0, "content": "follows.", "type": "text"}], "index": 31}], "index": 29.5, "bbox_fs": [125, 652, 487, 702]}]}
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[{"type": "text", "bbox": [125, 127, 486, 186], "content": "The -factorization corresponding to the nontrivial 4-part of is in case A) and in case B). Note that, by our results from [1], some of these fields have cyclic ; however, we do not exclude them right from the start since there is no extra work involved and since it provides a welcome check on our earlier work.", "index": 0}, {"type": "text", "bbox": [125, 187, 487, 211], "content": "The main result of the paper is that rank only occurs for fields of type B); more precisely, we prove the following", "index": 1}, {"type": "text", "bbox": [125, 217, 487, 290], "content": "Theorem 1. Let be a complex quadratic number field with , and let be its 2-class field. Then rank if and only if disc is the product of three prime discriminants and such that , , and , where is a nonnormal quartic subfield of one of the two unramified cyclic quartic extensions of such that .", "index": 2}, {"type": "text", "bbox": [125, 296, 486, 344], "content": "This result is the first step in the classification of imaginary quadratic number fields with rank ; it remains to solve these problems for fields with rank and those with since we know that rank whenever rank (using Schur multipliers as in [1]).", "index": 3}, {"type": "text", "bbox": [124, 344, 486, 380], "content": "As a demonstration of the utility of our results, we give in Table 1 below a list of the first 12 imaginary quadratic fields , arranged by decreasing value of their discriminants, with ran and noncyclic .", "index": 4}, {"type": "table", "bbox": [126, 429, 489, 588], "content": "", "index": 5}, {"type": "text", "bbox": [125, 603, 486, 651], "content": "Here denotes a generating polynomial for a field as in Theorem 1, denotes the rank of . The cases where follow from our theorem combined with Blackburn’s upper bound for the number of generators of derived groups (it implies that finite 2-groups with satisfy ), see [3].", "index": 6}, {"type": "text", "bbox": [124, 651, 486, 699], "content": "In order to verify that has rank at least 3 for it is sufficient to show that its genus class field has class group : in fact, then contains a quotient of by , and the claim follows.", "index": 7}]
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[{"bbox": [126, 128, 484, 141], "content": "The -factorization corresponding to the nontrivial 4-part of is", "parent_index": 0, "line_index": 0}, {"bbox": [126, 141, 486, 153], "content": "in case A) and in case B). Note that, by our results from [1], some", "parent_index": 0, "line_index": 1}, {"bbox": [126, 153, 486, 165], "content": "of these fields have cyclic ; however, we do not exclude them right from the", "parent_index": 0, "line_index": 2}, {"bbox": [126, 166, 486, 177], "content": "start since there is no extra work involved and since it provides a welcome check", "parent_index": 0, "line_index": 3}, {"bbox": [126, 178, 213, 188], "content": "on our earlier work.", "parent_index": 0, "line_index": 4}, {"bbox": [137, 188, 488, 201], "content": "The main result of the paper is that rank only occurs for fields of", "parent_index": 1, "line_index": 0}, {"bbox": [125, 200, 333, 214], "content": "type B); more precisely, we prove the following", "parent_index": 1, "line_index": 1}, {"bbox": [126, 219, 487, 232], "content": "Theorem 1. Let be a complex quadratic number field with , and", "parent_index": 2, "line_index": 0}, {"bbox": [125, 231, 487, 244], "content": "let be its 2-class field. Then rank if and only if disc is", "parent_index": 2, "line_index": 1}, {"bbox": [126, 244, 487, 255], "content": "the product of three prime discriminants and such that", "parent_index": 2, "line_index": 2}, {"bbox": [126, 256, 487, 268], "content": ", , and , where is a nonnormal", "parent_index": 2, "line_index": 3}, {"bbox": [126, 268, 487, 281], "content": "quartic subfield of one of the two unramified cyclic quartic extensions of such that", "parent_index": 2, "line_index": 4}, {"bbox": [126, 280, 198, 291], "content": ".", "parent_index": 2, "line_index": 5}, {"bbox": [137, 298, 486, 311], "content": "This result is the first step in the classification of imaginary quadratic number", "parent_index": 3, "line_index": 0}, {"bbox": [126, 310, 486, 322], "content": "fields with rank ; it remains to solve these problems for fields with", "parent_index": 3, "line_index": 1}, {"bbox": [126, 322, 485, 335], "content": "rank and those with since we know that rank", "parent_index": 3, "line_index": 2}, {"bbox": [127, 335, 388, 346], "content": "whenever rank (using Schur multipliers as in [1]).", "parent_index": 3, "line_index": 3}, {"bbox": [137, 346, 486, 359], "content": "As a demonstration of the utility of our results, we give in Table 1 below a list", "parent_index": 4, "line_index": 0}, {"bbox": [125, 357, 487, 372], "content": "of the first 12 imaginary quadratic fields , arranged by decreasing value of their", "parent_index": 4, "line_index": 1}, {"bbox": [126, 370, 382, 382], "content": "discriminants, with ran and noncyclic .", "parent_index": 4, "line_index": 2}, {"bbox": [137, 605, 486, 618], "content": "Here denotes a generating polynomial for a field as in Theorem 1, denotes", "parent_index": 6, "line_index": 0}, {"bbox": [126, 618, 484, 629], "content": "the rank of . The cases where follow from our theorem combined with", "parent_index": 6, "line_index": 1}, {"bbox": [126, 629, 485, 642], "content": "Blackburn’s upper bound for the number of generators of derived groups (it implies", "parent_index": 6, "line_index": 2}, {"bbox": [126, 641, 431, 653], "content": "that finite 2-groups with satisfy ), see [3].", "parent_index": 6, "line_index": 3}, {"bbox": [136, 652, 486, 665], "content": "In order to verify that has rank at least 3 for it is", "parent_index": 7, "line_index": 0}, {"bbox": [126, 664, 486, 679], "content": "sufficient to show that its genus class field has class group : in fact,", "parent_index": 7, "line_index": 1}, {"bbox": [126, 676, 487, 690], "content": "then contains a quotient of by , and the claim", "parent_index": 7, "line_index": 2}, {"bbox": [125, 689, 159, 702], "content": "follows.", "parent_index": 7, "line_index": 3}]
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[]
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[{"bbox": [146, 131, 157, 140], "content": "C_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [393, 130, 421, 141], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 0, "subtype": "inline"}, {"bbox": [433, 131, 484, 140], "content": "d=\\boldsymbol{d}_{1}\\cdot\\boldsymbol{d}_{2}\\boldsymbol{d}_{3}", "parent_index": 0, "subtype": "inline"}, {"bbox": [195, 143, 252, 151], "content": "d=d_{1}d_{2}\\cdot d_{3}", "parent_index": 0, "subtype": "inline"}, {"bbox": [238, 154, 270, 164], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 0, "subtype": "inline"}, {"bbox": [325, 189, 377, 200], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "parent_index": 1, "subtype": "inline"}, {"bbox": [205, 221, 211, 229], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [392, 221, 463, 231], "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "parent_index": 2, "subtype": "inline"}, {"bbox": [140, 232, 150, 241], "content": "k^{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [289, 232, 340, 243], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "parent_index": 2, "subtype": "inline"}, {"bbox": [423, 233, 474, 242], "content": "k=d_{1}d_{2}d_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [310, 245, 356, 254], "content": "d_{1},d_{2}\\,>\\,0", "parent_index": 2, "subtype": "inline"}, {"bbox": [380, 245, 440, 254], "content": "-4\\,\\ne\\,d_{3}\\,<\\,0", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 257, 231, 267], "content": "(d_{1}/p_{3})=(d_{2}/p_{3})=+1", "parent_index": 2, "subtype": "inline"}, {"bbox": [238, 257, 297, 267], "content": "(d_{1}/p_{2})=-1", "parent_index": 2, "subtype": "inline"}, {"bbox": [323, 257, 370, 267], "content": "h_{2}(K)=2", "parent_index": 2, "subtype": "inline"}, {"bbox": [405, 257, 414, 264], "content": "K", "parent_index": 2, "subtype": "inline"}, {"bbox": [437, 269, 443, 277], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 280, 194, 291], "content": "\\mathbb{Q}({\\sqrt{d_{1}d_{2}}}\\,)\\subset K", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 312, 158, 319], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [207, 311, 259, 322], "content": "\\mathrm{Cl}_{2}(k^{1})\\,=\\,2", "parent_index": 3, "subtype": "inline"}, {"bbox": [147, 324, 193, 334], "content": "\\mathrm{Cl}_{2}(k)=3", "parent_index": 3, "subtype": "inline"}, {"bbox": [262, 324, 326, 334], "content": "\\mathrm{Cl}_{2}(k)\\supseteq(4,4)", "parent_index": 3, "subtype": "inline"}, {"bbox": [434, 323, 485, 334], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq5", "parent_index": 3, "subtype": "inline"}, {"bbox": [191, 335, 237, 346], "content": "\\mathrm{Cl}_{2}(k)\\geq4", "parent_index": 3, "subtype": "inline"}, {"bbox": [309, 360, 314, 367], "content": "k", "parent_index": 4, "subtype": "inline"}, {"bbox": [229, 371, 280, 381], "content": "\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(\\lambda\\right)}}\\mathinner{\\mathrm{\\Omega}\\mathopen{\\left(k\\right)}}=2", "parent_index": 4, "subtype": "inline"}, {"bbox": [346, 371, 378, 381], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [161, 607, 167, 617], "content": "f", "parent_index": 6, "subtype": "inline"}, {"bbox": [357, 607, 366, 614], "content": "K", "parent_index": 6, "subtype": "inline"}, {"bbox": [444, 610, 449, 614], "content": "r", "parent_index": 6, "subtype": "inline"}, {"bbox": [176, 618, 208, 629], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 6, "subtype": "inline"}, {"bbox": [288, 619, 311, 626], "content": "r=3", "parent_index": 6, "subtype": "inline"}, {"bbox": [213, 643, 221, 650], "content": "G", "parent_index": 6, "subtype": "inline"}, {"bbox": [247, 642, 306, 653], "content": "G/G^{\\prime}\\simeq(2,4)", "parent_index": 6, "subtype": "inline"}, {"bbox": [340, 642, 391, 651], "content": "\\mathrm{rank}\\,G^{\\prime}\\leq3", "parent_index": 6, "subtype": "inline"}, {"bbox": [243, 654, 276, 665], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 7, "subtype": "inline"}, {"bbox": [387, 653, 462, 665], "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{-2379}}\\,)", "parent_index": 7, "subtype": "inline"}, {"bbox": [319, 667, 337, 677], "content": "k_{\\mathrm{gen}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [413, 666, 444, 677], "content": "(4,4,8)", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 677, 158, 689], "content": "\\mathrm{Cl_{2}}(k^{1})", "parent_index": 7, "subtype": "inline"}, {"bbox": [280, 678, 312, 689], "content": "(4,4,8)", "parent_index": 7, "subtype": "inline"}, {"bbox": [329, 677, 420, 689], "content": "(2,2)\\simeq\\mathrm{Gal}(k^{1}/k_{\\mathrm{gen}})", "parent_index": 7, "subtype": "inline"}]
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[{"bbox": [285, 392, 325, 403], "content": "", "parent_index": 5, "subtype": "caption"}, {"bbox": [126, 429, 489, 588], "content": "<html><body><table><tr><td>disc k</td><td>factors</td><td>Cl2(k)</td><td>type</td><td>f</td><td>Cl2(K)</td><td>r</td><td>Cl2 (kgen)</td></tr><tr><td>-1015 -1240 -1443 -1595 -1615 -1624 -1780 -2035 -2067 -2072 -2379 -2392</td><td>-7.5·29 -31.8.5 -3·13·37 -11·5·29 -19·5·17 -7.8.29 -4·5.89 -11·5·37 -3.13.53 -7.8·37 -3·13.61 -23·8·13</td><td>(2,8) (2,4) (2,4) (2,8) (2,4) (2,8) (2,4) (2,4) (2,4) (2,8) (4, 4) (2,4)</td><td>A B B A B B A B A B B</td><td>x4 -22x²+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x² - 171 x4 -30x²- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x²-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>≥3 2 2 N3 2 2 3 3 3 N3 >3 2</td><td>(2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,8) (2,2,4) (2,2,16) (2,2,4) (2,2,8) (4, 4, 8) (2,2,8)</td></tr></table></body></html>", "parent_index": 5, "subtype": "body"}]
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We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those quadratic fields with rank $\mathrm{Cl}_{2}(k^{1})\geq3$ and discriminant $0>d>-2000$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $k$ with discriminants $-1015$ , $-1595$ and $-1780$ have finite (2-)class field tower even though rank $\mathrm{Cl}_{2}(k^{1})\geq3$ . Of course, it would be interesting to determine the length of their towers.
The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $\mathrm{Cl}_{2}(k^{1})=2$ from the field $k^{1}$ with degree $2^{m+2}$ to a subfield $L$ of $k^{1}$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $K$ of degree 4 occurring in Theorem 1.
# 2. Group Theoretic Preliminaries
Let $G$ be a group. If $x,y\ \in\ G$ , then we let $[x,y]~=~x^{-1}y^{-1}x y$ denote the commutator of $x$ and $_y$ . If $A$ and $B$ are nonempty subsets of $G$ , then $[A,B]$ denotes the subgroup of $G$ generated by the set $\{[a,b]:a\in A,b\in B\}$ . The lower central series $\{G_{j}\}$ of $G$ is defined inductively by: $G_{1}\,=\,G$ and $G_{j+1}\,=\,[G,G_{j}]$ for $j~\geq~1$ . The derived series $\{G^{(n)}\}$ is defined inductively by: $G^{(0)}\ =\ G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for $n\geq0$ . Notice that $G^{(1)}=G_{2}=[G,G]$ the commutator subgroup, $G^{\prime}$ , of $G$ .
Throughout this section, we assume that $G$ is a finite, nonmetacyclic, 2-group such that its abelianization $G^{\mathrm{ab}}=G/G^{\prime}$ is of type $(2,2^{m})$ for some positive integer ${\boldsymbol{r}}n$ (necessarily $\geq2$ ). Let $G=\langle a,b\rangle$ , where $a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$ mod $G_{2}$ (actually $\mathrm{mod}G_{3}$ since $G$ is nonmetacyclic, cf. [1]); $c_{2}=[a,b]$ and $c_{j+1}=[b,c_{j}]$ for $j\geq2$ .
Lemma 1. Let $G$ be as above (but not necessarily metabelian). Suppose that $d(G^{\prime})\,=\,n$ where $d(G^{\prime})$ denotes the minimal number of generators of the derived group $G^{\prime}=G_{2}$ of $G$ . Then
$$
G^{\prime}=\langle c_{2},c_{3},\cdot\cdot\cdot,c_{n+1}\rangle;
$$
moreover,
$$
G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle.
$$
Proof. By the Burnside Basis Theorem, $d(G_{2})=d(G_{2}/\Phi(G))$ , where $\Phi(G)$ is the Frattini subgroup of $G$ , i.e. the intersection of all maximal subgroups of $G$ , see [5]. But in the case of a 2-group, $\Phi(G)=G^{2}$ , see [8]. By Blackburn, [3], since $G/G_{2}^{2}$ has elementary derived group, we know that $G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle$ . Again, by the Burnside Basis Theorem, $G_{2}=\langle c_{2},\cdots,c_{n+1}\rangle$ . 口
Lemma 2. Let $G$ be as above. Moreover, assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\prime}$ is cyclic, and denote the index $\left(G^{\prime}:H^{\prime}\right)$ by $2^{\kappa}$ . Then $G^{\prime}$ contains an element of order $2^{\kappa}$ .
Proof. Without loss of generality, let $H\,=\,\left\langle{b,G^{\prime}}\right\rangle$ . Notice that $G^{\prime}\,=\,\langle c_{2},c_{3},\cdot\cdot\cdot\rangle$ and by our presentation of $H$ , $H^{\prime}=\langle c_{3},c_{4},\cdot\cdot\cdot\rangle$ . Thus, $G^{\prime}/H^{\prime}=\langle c_{2}H^{\prime}\rangle$ . But since $(G^{\prime}:H^{\prime})=2^{\kappa}\,$ , the order of $c_{2}$ is $\geq2^{\kappa}$ . This establishes the lemma. 口
Lemma 3. Let $G$ be as above and again assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\prime}$ is cyclic, and assume that $(G^{\prime}\,:\,H^{\prime})\;\equiv\;$ 0 mod 4. If $d(G^{\prime})=2$ , then $G_{2}=\langle c_{2},c_{3}\rangle$ and $G_{j}=\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\rangle$ for $j>2$ .
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<html><body>
<p data-bbox="124 111 486 195">We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those quadratic fields with rank $\mathrm{Cl}_{2}(k^{1})\geq3$ and discriminant $0>d>-2000$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $k$ with discriminants $-1015$ , $-1595$ and $-1780$ have finite (2-)class field tower even though rank $\mathrm{Cl}_{2}(k^{1})\geq3$ . Of course, it would be interesting to determine the length of their towers. </p>
<p data-bbox="124 196 486 255">The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $\mathrm{Cl}_{2}(k^{1})=2$ from the field $k^{1}$ with degree $2^{m+2}$ to a subfield $L$ of $k^{1}$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $K$ of degree 4 occurring in Theorem 1. </p>
<h1 data-bbox="218 263 393 276">2. Group Theoretic Preliminaries </h1>
<p data-bbox="124 281 486 367">Let $G$ be a group. If $x,y\ \in\ G$ , then we let $[x,y]~=~x^{-1}y^{-1}x y$ denote the commutator of $x$ and $_y$ . If $A$ and $B$ are nonempty subsets of $G$ , then $[A,B]$ denotes the subgroup of $G$ generated by the set $\{[a,b]:a\in A,b\in B\}$ . The lower central series $\{G_{j}\}$ of $G$ is defined inductively by: $G_{1}\,=\,G$ and $G_{j+1}\,=\,[G,G_{j}]$ for $j~\geq~1$ . The derived series $\{G^{(n)}\}$ is defined inductively by: $G^{(0)}\ =\ G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for $n\geq0$ . Notice that $G^{(1)}=G_{2}=[G,G]$ the commutator subgroup, $G^{\prime}$ , of $G$ . </p>
<p data-bbox="125 368 486 417">Throughout this section, we assume that $G$ is a finite, nonmetacyclic, 2-group such that its abelianization $G^{\mathrm{ab}}=G/G^{\prime}$ is of type $(2,2^{m})$ for some positive integer ${\boldsymbol{r}}n$ (necessarily $\geq2$ ). Let $G=\langle a,b\rangle$ , where $a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$ mod $G_{2}$ (actually $\mathrm{mod}G_{3}$ since $G$ is nonmetacyclic, cf. [1]); $c_{2}=[a,b]$ and $c_{j+1}=[b,c_{j}]$ for $j\geq2$ . </p>
<p data-bbox="124 421 486 457">Lemma 1. Let $G$ be as above (but not necessarily metabelian). Suppose that $d(G^{\prime})\,=\,n$ where $d(G^{\prime})$ denotes the minimal number of generators of the derived group $G^{\prime}=G_{2}$ of $G$ . Then </p>
<div class="equation" data-bbox="255 464 355 475">$$
G^{\prime}=\langle c_{2},c_{3},\cdot\cdot\cdot,c_{n+1}\rangle;
$$</div>
<p data-bbox="126 479 169 489">moreover, </p>
<div class="equation" data-bbox="230 495 381 507">$$
G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle.
$$</div>
<p data-bbox="125 511 486 573">Proof. By the Burnside Basis Theorem, $d(G_{2})=d(G_{2}/\Phi(G))$ , where $\Phi(G)$ is the Frattini subgroup of $G$ , i.e. the intersection of all maximal subgroups of $G$ , see [5]. But in the case of a 2-group, $\Phi(G)=G^{2}$ , see [8]. By Blackburn, [3], since $G/G_{2}^{2}$ has elementary derived group, we know that $G_{2}/G_{2}^{2}\simeq\langle c_{2}G_{2}^{2}\rangle\oplus\cdots\oplus\langle c_{n+1}G_{2}^{2}\rangle$ . Again, by the Burnside Basis Theorem, $G_{2}=\langle c_{2},\cdots,c_{n+1}\rangle$ . 口 </p>
<p data-bbox="125 577 487 614">Lemma 2. Let $G$ be as above. Moreover, assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\prime}$ is cyclic, and denote the index $\left(G^{\prime}:H^{\prime}\right)$ by $2^{\kappa}$ . Then $G^{\prime}$ contains an element of order $2^{\kappa}$ . </p>
<p data-bbox="125 619 487 657">Proof. Without loss of generality, let $H\,=\,\left\langle{b,G^{\prime}}\right\rangle$ . Notice that $G^{\prime}\,=\,\langle c_{2},c_{3},\cdot\cdot\cdot\rangle$ and by our presentation of $H$ , $H^{\prime}=\langle c_{3},c_{4},\cdot\cdot\cdot\rangle$ . Thus, $G^{\prime}/H^{\prime}=\langle c_{2}H^{\prime}\rangle$ . But since $(G^{\prime}:H^{\prime})=2^{\kappa}\,$ , the order of $c_{2}$ is $\geq2^{\kappa}$ . This establishes the lemma. 口 </p>
<p data-bbox="125 661 486 701">Lemma 3. Let $G$ be as above and again assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\prime}$ is cyclic, and assume that $(G^{\prime}\,:\,H^{\prime})\;\equiv\;$ 0 mod 4. If $d(G^{\prime})=2$ , then $G_{2}=\langle c_{2},c_{3}\rangle$ and $G_{j}=\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\rangle$ for $j>2$ . </p>
</body></html>
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0003244v1
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[{"type": "text", "text": "We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those quadratic fields with rank $\\mathrm{Cl}_{2}(k^{1})\\geq3$ and discriminant $0>d>-2000$ have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those $k$ with discriminants $-1015$ , $-1595$ and $-1780$ have finite (2-)class field tower even though rank $\\mathrm{Cl}_{2}(k^{1})\\geq3$ . Of course, it would be interesting to determine the length of their towers. ", "page_idx": 2}, {"type": "text", "text": "The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank $\\mathrm{Cl}_{2}(k^{1})=2$ from the field $k^{1}$ with degree $2^{m+2}$ to a subfield $L$ of $k^{1}$ with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field $K$ of degree 4 occurring in Theorem 1. ", "page_idx": 2}, {"type": "text", "text": "2. Group Theoretic Preliminaries ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "Let $G$ be a group. If $x,y\\ \\in\\ G$ , then we let $[x,y]~=~x^{-1}y^{-1}x y$ denote the commutator of $x$ and $_y$ . If $A$ and $B$ are nonempty subsets of $G$ , then $[A,B]$ denotes the subgroup of $G$ generated by the set $\\{[a,b]:a\\in A,b\\in B\\}$ . The lower central series $\\{G_{j}\\}$ of $G$ is defined inductively by: $G_{1}\\,=\\,G$ and $G_{j+1}\\,=\\,[G,G_{j}]$ for $j~\\geq~1$ . The derived series $\\{G^{(n)}\\}$ is defined inductively by: $G^{(0)}\\ =\\ G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for $n\\geq0$ . Notice that $G^{(1)}=G_{2}=[G,G]$ the commutator subgroup, $G^{\\prime}$ , of $G$ . ", "page_idx": 2}, {"type": "text", "text": "Throughout this section, we assume that $G$ is a finite, nonmetacyclic, 2-group such that its abelianization $G^{\\mathrm{ab}}=G/G^{\\prime}$ is of type $(2,2^{m})$ for some positive integer ${\\boldsymbol{r}}n$ (necessarily $\\geq2$ ). Let $G=\\langle a,b\\rangle$ , where $a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$ mod $G_{2}$ (actually $\\mathrm{mod}G_{3}$ since $G$ is nonmetacyclic, cf. [1]); $c_{2}=[a,b]$ and $c_{j+1}=[b,c_{j}]$ for $j\\geq2$ . ", "page_idx": 2}, {"type": "text", "text": "Lemma 1. Let $G$ be as above (but not necessarily metabelian). Suppose that $d(G^{\\prime})\\,=\\,n$ where $d(G^{\\prime})$ denotes the minimal number of generators of the derived group $G^{\\prime}=G_{2}$ of $G$ . Then ", "page_idx": 2}, {"type": "equation", "text": "$$\nG^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "moreover, ", "page_idx": 2}, {"type": "equation", "text": "$$\nG_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Proof. By the Burnside Basis Theorem, $d(G_{2})=d(G_{2}/\\Phi(G))$ , where $\\Phi(G)$ is the Frattini subgroup of $G$ , i.e. the intersection of all maximal subgroups of $G$ , see [5]. But in the case of a 2-group, $\\Phi(G)=G^{2}$ , see [8]. By Blackburn, [3], since $G/G_{2}^{2}$ has elementary derived group, we know that $G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle$ . Again, by the Burnside Basis Theorem, $G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle$ . 口 ", "page_idx": 2}, {"type": "text", "text": "Lemma 2. Let $G$ be as above. Moreover, assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\\prime}$ is cyclic, and denote the index $\\left(G^{\\prime}:H^{\\prime}\\right)$ by $2^{\\kappa}$ . Then $G^{\\prime}$ contains an element of order $2^{\\kappa}$ . ", "page_idx": 2}, {"type": "text", "text": "Proof. Without loss of generality, let $H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle$ . Notice that $G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle$ and by our presentation of $H$ , $H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle$ . Thus, $G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle$ . But since $(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$ , the order of $c_{2}$ is $\\geq2^{\\kappa}$ . This establishes the lemma. 口 ", "page_idx": 2}, {"type": "text", "text": "Lemma 3. Let $G$ be as above and again assume $G$ is metabelian. Let $H$ be a maximal subgroup of $G$ such that $H/G^{\\prime}$ is cyclic, and assume that $(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;$ 0 mod 4. If $d(G^{\\prime})=2$ , then $G_{2}=\\langle c_{2},c_{3}\\rangle$ and $G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle$ for $j>2$ . ", "page_idx": 2}]
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It follows from conditional", "type": "text"}], "index": 0}, {"bbox": [125, 126, 486, 139], "spans": [{"bbox": [125, 126, 486, 139], "score": 1.0, "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 222, 150], "score": 1.0, "content": "dratic fields with rank", "type": "text"}, {"bbox": [222, 139, 273, 150], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [273, 138, 350, 150], "score": 1.0, "content": " and discriminant ", "type": "text"}, {"bbox": [351, 140, 415, 148], "score": 0.9, "content": "0>d>-2000", "type": "inline_equation", "height": 8, "width": 64}, {"bbox": [415, 138, 486, 150], "score": 1.0, "content": " have finite class", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 162], "spans": [{"bbox": [125, 150, 486, 162], "score": 1.0, "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "type": "text"}], "index": 3}, {"bbox": [126, 163, 485, 174], "spans": [{"bbox": [126, 163, 173, 174], "score": 1.0, "content": "that those ", "type": "text"}, {"bbox": [173, 164, 179, 171], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [179, 163, 266, 174], "score": 1.0, "content": " with discriminants ", "type": "text"}, {"bbox": [266, 164, 294, 172], "score": 0.47, "content": "-1015", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [294, 163, 298, 174], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [298, 164, 326, 172], "score": 0.51, "content": "-1595", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [326, 163, 348, 174], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 164, 376, 172], "score": 0.81, "content": "-1780", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [376, 163, 485, 174], "score": 1.0, "content": " have finite (2-)class field", "type": "text"}], "index": 4}, {"bbox": [125, 174, 485, 186], "spans": [{"bbox": [125, 174, 228, 186], "score": 1.0, "content": "tower even though rank", "type": "text"}, {"bbox": [228, 175, 279, 186], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [279, 174, 485, 186], "score": 1.0, "content": ". Of course, it would be interesting to determine", "type": "text"}], "index": 5}, {"bbox": [126, 186, 240, 198], "spans": [{"bbox": [126, 186, 240, 198], "score": 1.0, "content": "the length of their towers.", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [124, 196, 486, 255], "lines": [{"bbox": [137, 197, 485, 210], "spans": [{"bbox": [137, 197, 485, 210], "score": 1.0, "content": "The structure of this paper is as follows: we use results from group theory", "type": "text"}], "index": 7}, {"bbox": [125, 209, 487, 222], "spans": [{"bbox": [125, 209, 370, 222], "score": 1.0, "content": "developed in Section 2 to pull down the condition rank", "type": "text"}, {"bbox": [370, 210, 422, 221], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [422, 209, 487, 222], "score": 1.0, "content": " from the field", "type": "text"}], "index": 8}, {"bbox": [126, 220, 488, 235], "spans": [{"bbox": [126, 222, 136, 231], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 220, 194, 235], "score": 1.0, "content": " with degree ", "type": "text"}, {"bbox": [194, 223, 217, 231], "score": 0.92, "content": "2^{m+2}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [217, 220, 278, 235], "score": 1.0, "content": " to a subfield ", "type": "text"}, {"bbox": [279, 224, 286, 231], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 220, 300, 235], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [301, 222, 311, 231], "score": 0.92, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [311, 220, 488, 235], "score": 1.0, "content": " with degree 8. Using the arithmetic of", "type": "text"}], "index": 9}, {"bbox": [125, 233, 486, 247], "spans": [{"bbox": [125, 233, 381, 247], "score": 1.0, "content": "dihedral fields from Section 4 we then go down to the field ", "type": "text"}, {"bbox": [381, 236, 391, 243], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [392, 233, 486, 247], "score": 1.0, "content": " of degree 4 occurring", "type": "text"}], "index": 10}, {"bbox": [126, 246, 188, 257], "spans": [{"bbox": [126, 246, 188, 257], "score": 1.0, "content": "in Theorem 1.", "type": "text"}], "index": 11}], "index": 9}, {"type": "title", "bbox": [218, 263, 393, 276], "lines": [{"bbox": [217, 266, 394, 277], "spans": [{"bbox": [217, 266, 394, 277], "score": 1.0, "content": "2. Group Theoretic Preliminaries", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 281, 486, 367], "lines": [{"bbox": [136, 284, 486, 296], "spans": [{"bbox": [136, 284, 157, 296], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [157, 286, 165, 293], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [166, 284, 243, 296], "score": 1.0, "content": " be a group. If ", "type": "text"}, {"bbox": [244, 286, 285, 295], "score": 0.93, "content": "x,y\\ \\in\\ G", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [285, 284, 349, 296], "score": 1.0, "content": ", then we let ", "type": "text"}, {"bbox": [349, 285, 433, 296], "score": 0.92, "content": "[x,y]~=~x^{-1}y^{-1}x y", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [433, 284, 486, 296], "score": 1.0, "content": " denote the", "type": "text"}], "index": 13}, {"bbox": [125, 296, 484, 309], "spans": [{"bbox": [125, 296, 196, 309], "score": 1.0, "content": "commutator of ", "type": "text"}, {"bbox": [196, 300, 203, 305], "score": 0.89, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [203, 296, 228, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [228, 300, 234, 307], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [234, 296, 257, 309], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [257, 298, 265, 305], "score": 0.89, "content": "A", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [265, 296, 290, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [290, 298, 299, 305], "score": 0.9, "content": "B", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [299, 296, 419, 309], "score": 1.0, "content": " are nonempty subsets of ", "type": "text"}, {"bbox": [419, 298, 427, 305], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [427, 296, 459, 309], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [459, 297, 484, 308], "score": 0.93, "content": "[A,B]", "type": "inline_equation", "height": 11, "width": 25}], "index": 14}, {"bbox": [126, 308, 486, 320], "spans": [{"bbox": [126, 308, 234, 320], "score": 1.0, "content": "denotes the subgroup of ", "type": "text"}, {"bbox": [235, 310, 243, 317], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 308, 339, 320], "score": 1.0, "content": " generated by the set", "type": "text"}, {"bbox": [339, 309, 433, 320], "score": 0.92, "content": "\\{[a,b]:a\\in A,b\\in B\\}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [434, 308, 486, 320], "score": 1.0, "content": ". The lower", "type": "text"}], "index": 15}, {"bbox": [124, 320, 485, 334], "spans": [{"bbox": [124, 320, 187, 334], "score": 1.0, "content": "central series ", "type": "text"}, {"bbox": [187, 321, 209, 332], "score": 0.94, "content": "\\{G_{j}\\}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [210, 320, 226, 334], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [226, 322, 234, 329], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [234, 320, 355, 334], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [356, 322, 392, 331], "score": 0.92, "content": "G_{1}\\,=\\,G", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [393, 320, 416, 334], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 321, 485, 332], "score": 0.92, "content": "G_{j+1}\\,=\\,[G,G_{j}]", "type": "inline_equation", "height": 11, "width": 69}], "index": 16}, {"bbox": [124, 332, 487, 345], "spans": [{"bbox": [124, 332, 142, 345], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [142, 335, 170, 344], "score": 0.92, "content": "j~\\geq~1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [170, 332, 266, 345], "score": 1.0, "content": ". The derived series ", "type": "text"}, {"bbox": [267, 333, 297, 344], "score": 0.94, "content": "\\{G^{(n)}\\}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [297, 332, 420, 345], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [420, 333, 464, 342], "score": 0.9, "content": "G^{(0)}\\ =\\ G", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [465, 332, 487, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [126, 342, 487, 359], "spans": [{"bbox": [126, 345, 219, 357], "score": 0.92, "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [219, 342, 238, 359], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [238, 347, 264, 356], "score": 0.92, "content": "n\\geq0", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [264, 342, 325, 359], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [325, 345, 410, 357], "score": 0.92, "content": "G^{(1)}=G_{2}=[G,G]", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [411, 342, 487, 359], "score": 1.0, "content": " the commutator", "type": "text"}], "index": 18}, {"bbox": [125, 357, 212, 370], "spans": [{"bbox": [125, 357, 172, 370], "score": 1.0, "content": "subgroup, ", "type": "text"}, {"bbox": [172, 358, 183, 366], "score": 0.89, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [183, 357, 200, 370], "score": 1.0, "content": ", of ", "type": "text"}, {"bbox": [200, 359, 208, 366], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 357, 212, 370], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 16}, {"type": "text", "bbox": [125, 368, 486, 417], "lines": [{"bbox": [137, 368, 486, 383], "spans": [{"bbox": [137, 368, 322, 383], "score": 1.0, "content": "Throughout this section, we assume that ", "type": "text"}, {"bbox": [323, 371, 331, 378], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [331, 368, 486, 383], "score": 1.0, "content": " is a finite, nonmetacyclic, 2-group", "type": "text"}], "index": 20}, {"bbox": [125, 379, 486, 394], "spans": [{"bbox": [125, 379, 246, 394], "score": 1.0, "content": "such that its abelianization ", "type": "text"}, {"bbox": [246, 382, 300, 393], "score": 0.93, "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [300, 379, 346, 394], "score": 1.0, "content": " is of type ", "type": "text"}, {"bbox": [346, 382, 376, 393], "score": 0.91, "content": "(2,2^{m})", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [376, 379, 486, 394], "score": 1.0, "content": " for some positive integer", "type": "text"}], "index": 21}, {"bbox": [126, 392, 484, 406], "spans": [{"bbox": [126, 398, 135, 402], "score": 0.86, "content": "{\\boldsymbol{r}}n", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [135, 392, 191, 406], "score": 1.0, "content": " (necessarily ", "type": "text"}, {"bbox": [191, 396, 207, 403], "score": 0.88, "content": "\\geq2", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [207, 392, 235, 406], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [236, 394, 279, 405], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [279, 392, 313, 406], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [313, 393, 370, 402], "score": 0.9, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 57}, {"bbox": [370, 392, 394, 406], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [394, 395, 407, 404], "score": 0.9, "content": "G_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [407, 392, 452, 406], "score": 1.0, "content": " (actually ", "type": "text"}, {"bbox": [452, 395, 484, 404], "score": 0.63, "content": "\\mathrm{mod}G_{3}", "type": "inline_equation", "height": 9, "width": 32}], "index": 22}, {"bbox": [124, 404, 440, 419], "spans": [{"bbox": [124, 404, 150, 419], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [150, 407, 158, 414], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [159, 404, 274, 419], "score": 1.0, "content": " is nonmetacyclic, cf. [1]); ", "type": "text"}, {"bbox": [275, 406, 316, 417], "score": 0.94, "content": "c_{2}=[a,b]", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [317, 404, 338, 419], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 406, 394, 417], "score": 0.95, "content": "c_{j+1}=[b,c_{j}]", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [394, 404, 411, 419], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [412, 407, 435, 416], "score": 0.92, "content": "j\\geq2", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [435, 404, 440, 419], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 21.5}, {"type": "text", "bbox": [124, 421, 486, 457], "lines": [{"bbox": [125, 423, 487, 435], "spans": [{"bbox": [125, 423, 199, 435], "score": 1.0, "content": "Lemma 1. Let ", "type": "text"}, {"bbox": [200, 425, 208, 432], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 423, 487, 435], "score": 1.0, "content": " be as above (but not necessarily metabelian). Suppose that", "type": "text"}], "index": 24}, {"bbox": [126, 434, 487, 448], "spans": [{"bbox": [126, 436, 172, 447], "score": 0.93, "content": "d(G^{\\prime})\\,=\\,n", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [172, 434, 204, 448], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [205, 436, 228, 447], "score": 0.93, "content": "d(G^{\\prime})", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [228, 434, 487, 448], "score": 1.0, "content": " denotes the minimal number of generators of the derived", "type": "text"}], "index": 25}, {"bbox": [126, 447, 245, 460], "spans": [{"bbox": [126, 447, 153, 460], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [153, 449, 190, 458], "score": 0.93, "content": "G^{\\prime}=G_{2}", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [190, 447, 204, 460], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [205, 449, 213, 456], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [213, 447, 245, 460], "score": 1.0, "content": ". Then", "type": "text"}], "index": 26}], "index": 25}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [126, 479, 169, 489], "lines": [{"bbox": [126, 481, 169, 491], "spans": [{"bbox": [126, 481, 169, 491], "score": 1.0, "content": "moreover,", "type": "text"}], "index": 28}], "index": 28}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 511, 486, 573], "lines": [{"bbox": [126, 514, 486, 526], "spans": [{"bbox": [126, 514, 305, 526], "score": 1.0, "content": "Proof. By the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [305, 515, 398, 525], "score": 0.93, "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "type": "inline_equation", "height": 10, "width": 93}, {"bbox": [398, 514, 433, 526], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [434, 515, 457, 525], "score": 0.95, "content": "\\Phi(G)", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [457, 514, 486, 526], "score": 1.0, "content": " is the", "type": "text"}], "index": 30}, {"bbox": [126, 526, 485, 538], "spans": [{"bbox": [126, 526, 216, 538], "score": 1.0, "content": "Frattini subgroup of ", "type": "text"}, {"bbox": [216, 528, 224, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [225, 526, 442, 538], "score": 1.0, "content": " , i.e. the intersection of all maximal subgroups of ", "type": "text"}, {"bbox": [442, 528, 450, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [451, 526, 485, 538], "score": 1.0, "content": ", see [5].", "type": "text"}], "index": 31}, {"bbox": [124, 537, 485, 551], "spans": [{"bbox": [124, 537, 258, 551], "score": 1.0, "content": "But in the case of a 2-group, ", "type": "text"}, {"bbox": [258, 538, 308, 549], "score": 0.94, "content": "\\Phi(G)=G^{2}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [308, 537, 459, 551], "score": 1.0, "content": ", see [8]. By Blackburn, [3], since ", "type": "text"}, {"bbox": [460, 538, 485, 549], "score": 0.94, "content": "G/G_{2}^{2}", "type": "inline_equation", "height": 11, "width": 25}], "index": 32}, {"bbox": [124, 549, 485, 564], "spans": [{"bbox": [124, 549, 329, 564], "score": 1.0, "content": "has elementary derived group, we know that ", "type": "text"}, {"bbox": [329, 550, 482, 561], "score": 0.91, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 153}, {"bbox": [483, 549, 485, 564], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [126, 560, 486, 575], "spans": [{"bbox": [126, 560, 301, 575], "score": 1.0, "content": "Again, by the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [302, 563, 388, 573], "score": 0.9, "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "type": "inline_equation", "height": 10, "width": 86}, {"bbox": [388, 560, 392, 575], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 562, 486, 572], "score": 0.9934230446815491, "content": "口", "type": "text"}], "index": 34}], "index": 32}, {"type": "text", "bbox": [125, 577, 487, 614], "lines": [{"bbox": [126, 580, 486, 592], "spans": [{"bbox": [126, 580, 198, 592], "score": 1.0, "content": "Lemma 2. Let ", "type": "text"}, {"bbox": [199, 582, 207, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 580, 354, 592], "score": 1.0, "content": " be as above. Moreover, assume ", "type": "text"}, {"bbox": [355, 582, 363, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [363, 580, 452, 592], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [453, 582, 462, 589], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [462, 580, 486, 592], "score": 1.0, "content": " be a", "type": "text"}], "index": 35}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 218, 604], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [218, 594, 226, 601], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 592, 271, 604], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [272, 593, 295, 603], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [296, 592, 434, 604], "score": 1.0, "content": " is cyclic, and denote the index ", "type": "text"}, {"bbox": [434, 592, 472, 603], "score": 0.8, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [472, 592, 486, 604], "score": 1.0, "content": " by", "type": "text"}], "index": 36}, {"bbox": [126, 603, 327, 615], "spans": [{"bbox": [126, 605, 136, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [137, 603, 169, 615], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [170, 605, 180, 613], "score": 0.9, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [181, 603, 312, 615], "score": 1.0, "content": " contains an element of order ", "type": "text"}, {"bbox": [312, 605, 323, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [323, 603, 327, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36}, {"type": "text", "bbox": [125, 619, 487, 657], "lines": [{"bbox": [126, 621, 485, 634], "spans": [{"bbox": [126, 621, 294, 634], "score": 1.0, "content": "Proof. Without loss of generality, let ", "type": "text"}, {"bbox": [294, 622, 347, 633], "score": 0.93, "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [348, 621, 411, 634], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [412, 621, 485, 633], "score": 0.92, "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 12, "width": 73}], "index": 38}, {"bbox": [126, 633, 487, 646], "spans": [{"bbox": [126, 633, 245, 646], "score": 1.0, "content": "and by our presentation of ", "type": "text"}, {"bbox": [245, 636, 254, 643], "score": 0.87, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 633, 260, 646], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [260, 635, 332, 645], "score": 0.92, "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [332, 633, 366, 646], "score": 1.0, "content": ". Thus,", "type": "text"}, {"bbox": [367, 634, 437, 645], "score": 0.93, "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [437, 633, 487, 646], "score": 1.0, "content": ". But since", "type": "text"}], "index": 39}, {"bbox": [126, 646, 486, 658], "spans": [{"bbox": [126, 647, 188, 657], "score": 0.92, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [188, 646, 248, 658], "score": 1.0, "content": ", the order of ", "type": "text"}, {"bbox": [249, 650, 258, 656], "score": 0.88, "content": "c_{2}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [258, 646, 270, 658], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [271, 648, 291, 656], "score": 0.9, "content": "\\geq2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [292, 646, 421, 658], "score": 1.0, "content": ". This establishes the lemma.", "type": "text"}, {"bbox": [475, 646, 486, 656], "score": 0.991905927658081, "content": "口", "type": "text"}], "index": 40}], "index": 39}, {"type": "text", "bbox": [125, 661, 486, 701], "lines": [{"bbox": [125, 664, 487, 676], "spans": [{"bbox": [125, 664, 199, 676], "score": 1.0, "content": "Lemma 3. Let ", "type": "text"}, {"bbox": [199, 666, 207, 673], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 664, 351, 676], "score": 1.0, "content": " be as above and again assume ", "type": "text"}, {"bbox": [352, 664, 361, 673], "score": 0.76, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [361, 664, 451, 676], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [452, 664, 462, 673], "score": 0.76, "content": "H", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [462, 664, 487, 676], "score": 1.0, "content": " be a", "type": "text"}], "index": 41}, {"bbox": [126, 675, 486, 687], "spans": [{"bbox": [126, 676, 221, 687], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [222, 677, 230, 685], "score": 0.85, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [230, 676, 279, 687], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [279, 676, 304, 687], "score": 0.89, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [304, 676, 430, 687], "score": 1.0, "content": " is cyclic, and assume that ", "type": "text"}, {"bbox": [430, 675, 486, 687], "score": 0.87, "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "type": "inline_equation", "height": 12, "width": 56}], "index": 42}, {"bbox": [124, 687, 454, 702], "spans": [{"bbox": [124, 689, 178, 702], "score": 1.0, "content": "0 mod 4. If ", "type": "text"}, {"bbox": [179, 690, 221, 701], "score": 0.93, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [221, 689, 249, 702], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [249, 690, 305, 701], "score": 0.93, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [305, 689, 327, 702], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [327, 687, 408, 701], "score": 0.93, "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [408, 689, 426, 702], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [426, 689, 451, 700], "score": 0.88, "content": "j>2", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [451, 689, 454, 702], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 42}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 98], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 486, 195], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 486, 126], "score": 1.0, "content": "We mention one last feature gleaned from the table. It follows from conditional", "type": "text"}], "index": 0}, {"bbox": [125, 126, 486, 139], "spans": [{"bbox": [125, 126, 486, 139], "score": 1.0, "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 222, 150], "score": 1.0, "content": "dratic fields with rank", "type": "text"}, {"bbox": [222, 139, 273, 150], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [273, 138, 350, 150], "score": 1.0, "content": " and discriminant ", "type": "text"}, {"bbox": [351, 140, 415, 148], "score": 0.9, "content": "0>d>-2000", "type": "inline_equation", "height": 8, "width": 64}, {"bbox": [415, 138, 486, 150], "score": 1.0, "content": " have finite class", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 162], "spans": [{"bbox": [125, 150, 486, 162], "score": 1.0, "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "type": "text"}], "index": 3}, {"bbox": [126, 163, 485, 174], "spans": [{"bbox": [126, 163, 173, 174], "score": 1.0, "content": "that those ", "type": "text"}, {"bbox": [173, 164, 179, 171], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [179, 163, 266, 174], "score": 1.0, "content": " with discriminants ", "type": "text"}, {"bbox": [266, 164, 294, 172], "score": 0.47, "content": "-1015", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [294, 163, 298, 174], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [298, 164, 326, 172], "score": 0.51, "content": "-1595", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [326, 163, 348, 174], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 164, 376, 172], "score": 0.81, "content": "-1780", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [376, 163, 485, 174], "score": 1.0, "content": " have finite (2-)class field", "type": "text"}], "index": 4}, {"bbox": [125, 174, 485, 186], "spans": [{"bbox": [125, 174, 228, 186], "score": 1.0, "content": "tower even though rank", "type": "text"}, {"bbox": [228, 175, 279, 186], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [279, 174, 485, 186], "score": 1.0, "content": ". Of course, it would be interesting to determine", "type": "text"}], "index": 5}, {"bbox": [126, 186, 240, 198], "spans": [{"bbox": [126, 186, 240, 198], "score": 1.0, "content": "the length of their towers.", "type": "text"}], "index": 6}], "index": 3, "bbox_fs": [125, 114, 486, 198]}, {"type": "text", "bbox": [124, 196, 486, 255], "lines": [{"bbox": [137, 197, 485, 210], "spans": [{"bbox": [137, 197, 485, 210], "score": 1.0, "content": "The structure of this paper is as follows: we use results from group theory", "type": "text"}], "index": 7}, {"bbox": [125, 209, 487, 222], "spans": [{"bbox": [125, 209, 370, 222], "score": 1.0, "content": "developed in Section 2 to pull down the condition rank", "type": "text"}, {"bbox": [370, 210, 422, 221], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k^{1})=2", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [422, 209, 487, 222], "score": 1.0, "content": " from the field", "type": "text"}], "index": 8}, {"bbox": [126, 220, 488, 235], "spans": [{"bbox": [126, 222, 136, 231], "score": 0.91, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 220, 194, 235], "score": 1.0, "content": " with degree ", "type": "text"}, {"bbox": [194, 223, 217, 231], "score": 0.92, "content": "2^{m+2}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [217, 220, 278, 235], "score": 1.0, "content": " to a subfield ", "type": "text"}, {"bbox": [279, 224, 286, 231], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 220, 300, 235], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [301, 222, 311, 231], "score": 0.92, "content": "k^{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [311, 220, 488, 235], "score": 1.0, "content": " with degree 8. Using the arithmetic of", "type": "text"}], "index": 9}, {"bbox": [125, 233, 486, 247], "spans": [{"bbox": [125, 233, 381, 247], "score": 1.0, "content": "dihedral fields from Section 4 we then go down to the field ", "type": "text"}, {"bbox": [381, 236, 391, 243], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [392, 233, 486, 247], "score": 1.0, "content": " of degree 4 occurring", "type": "text"}], "index": 10}, {"bbox": [126, 246, 188, 257], "spans": [{"bbox": [126, 246, 188, 257], "score": 1.0, "content": "in Theorem 1.", "type": "text"}], "index": 11}], "index": 9, "bbox_fs": [125, 197, 488, 257]}, {"type": "title", "bbox": [218, 263, 393, 276], "lines": [{"bbox": [217, 266, 394, 277], "spans": [{"bbox": [217, 266, 394, 277], "score": 1.0, "content": "2. Group Theoretic Preliminaries", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 281, 486, 367], "lines": [{"bbox": [136, 284, 486, 296], "spans": [{"bbox": [136, 284, 157, 296], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [157, 286, 165, 293], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [166, 284, 243, 296], "score": 1.0, "content": " be a group. If ", "type": "text"}, {"bbox": [244, 286, 285, 295], "score": 0.93, "content": "x,y\\ \\in\\ G", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [285, 284, 349, 296], "score": 1.0, "content": ", then we let ", "type": "text"}, {"bbox": [349, 285, 433, 296], "score": 0.92, "content": "[x,y]~=~x^{-1}y^{-1}x y", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [433, 284, 486, 296], "score": 1.0, "content": " denote the", "type": "text"}], "index": 13}, {"bbox": [125, 296, 484, 309], "spans": [{"bbox": [125, 296, 196, 309], "score": 1.0, "content": "commutator of ", "type": "text"}, {"bbox": [196, 300, 203, 305], "score": 0.89, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [203, 296, 228, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [228, 300, 234, 307], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [234, 296, 257, 309], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [257, 298, 265, 305], "score": 0.89, "content": "A", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [265, 296, 290, 309], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [290, 298, 299, 305], "score": 0.9, "content": "B", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [299, 296, 419, 309], "score": 1.0, "content": " are nonempty subsets of ", "type": "text"}, {"bbox": [419, 298, 427, 305], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [427, 296, 459, 309], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [459, 297, 484, 308], "score": 0.93, "content": "[A,B]", "type": "inline_equation", "height": 11, "width": 25}], "index": 14}, {"bbox": [126, 308, 486, 320], "spans": [{"bbox": [126, 308, 234, 320], "score": 1.0, "content": "denotes the subgroup of ", "type": "text"}, {"bbox": [235, 310, 243, 317], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 308, 339, 320], "score": 1.0, "content": " generated by the set", "type": "text"}, {"bbox": [339, 309, 433, 320], "score": 0.92, "content": "\\{[a,b]:a\\in A,b\\in B\\}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [434, 308, 486, 320], "score": 1.0, "content": ". The lower", "type": "text"}], "index": 15}, {"bbox": [124, 320, 485, 334], "spans": [{"bbox": [124, 320, 187, 334], "score": 1.0, "content": "central series ", "type": "text"}, {"bbox": [187, 321, 209, 332], "score": 0.94, "content": "\\{G_{j}\\}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [210, 320, 226, 334], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [226, 322, 234, 329], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [234, 320, 355, 334], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [356, 322, 392, 331], "score": 0.92, "content": "G_{1}\\,=\\,G", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [393, 320, 416, 334], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 321, 485, 332], "score": 0.92, "content": "G_{j+1}\\,=\\,[G,G_{j}]", "type": "inline_equation", "height": 11, "width": 69}], "index": 16}, {"bbox": [124, 332, 487, 345], "spans": [{"bbox": [124, 332, 142, 345], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [142, 335, 170, 344], "score": 0.92, "content": "j~\\geq~1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [170, 332, 266, 345], "score": 1.0, "content": ". The derived series ", "type": "text"}, {"bbox": [267, 333, 297, 344], "score": 0.94, "content": "\\{G^{(n)}\\}", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [297, 332, 420, 345], "score": 1.0, "content": " is defined inductively by: ", "type": "text"}, {"bbox": [420, 333, 464, 342], "score": 0.9, "content": "G^{(0)}\\ =\\ G", "type": "inline_equation", "height": 9, "width": 44}, {"bbox": [465, 332, 487, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [126, 342, 487, 359], "spans": [{"bbox": [126, 345, 219, 357], "score": 0.92, "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [219, 342, 238, 359], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [238, 347, 264, 356], "score": 0.92, "content": "n\\geq0", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [264, 342, 325, 359], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [325, 345, 410, 357], "score": 0.92, "content": "G^{(1)}=G_{2}=[G,G]", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [411, 342, 487, 359], "score": 1.0, "content": " the commutator", "type": "text"}], "index": 18}, {"bbox": [125, 357, 212, 370], "spans": [{"bbox": [125, 357, 172, 370], "score": 1.0, "content": "subgroup, ", "type": "text"}, {"bbox": [172, 358, 183, 366], "score": 0.89, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [183, 357, 200, 370], "score": 1.0, "content": ", of ", "type": "text"}, {"bbox": [200, 359, 208, 366], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 357, 212, 370], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 16, "bbox_fs": [124, 284, 487, 370]}, {"type": "text", "bbox": [125, 368, 486, 417], "lines": [{"bbox": [137, 368, 486, 383], "spans": [{"bbox": [137, 368, 322, 383], "score": 1.0, "content": "Throughout this section, we assume that ", "type": "text"}, {"bbox": [323, 371, 331, 378], "score": 0.9, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [331, 368, 486, 383], "score": 1.0, "content": " is a finite, nonmetacyclic, 2-group", "type": "text"}], "index": 20}, {"bbox": [125, 379, 486, 394], "spans": [{"bbox": [125, 379, 246, 394], "score": 1.0, "content": "such that its abelianization ", "type": "text"}, {"bbox": [246, 382, 300, 393], "score": 0.93, "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [300, 379, 346, 394], "score": 1.0, "content": " is of type ", "type": "text"}, {"bbox": [346, 382, 376, 393], "score": 0.91, "content": "(2,2^{m})", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [376, 379, 486, 394], "score": 1.0, "content": " for some positive integer", "type": "text"}], "index": 21}, {"bbox": [126, 392, 484, 406], "spans": [{"bbox": [126, 398, 135, 402], "score": 0.86, "content": "{\\boldsymbol{r}}n", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [135, 392, 191, 406], "score": 1.0, "content": " (necessarily ", "type": "text"}, {"bbox": [191, 396, 207, 403], "score": 0.88, "content": "\\geq2", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [207, 392, 235, 406], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [236, 394, 279, 405], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [279, 392, 313, 406], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [313, 393, 370, 402], "score": 0.9, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 57}, {"bbox": [370, 392, 394, 406], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [394, 395, 407, 404], "score": 0.9, "content": "G_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [407, 392, 452, 406], "score": 1.0, "content": " (actually ", "type": "text"}, {"bbox": [452, 395, 484, 404], "score": 0.63, "content": "\\mathrm{mod}G_{3}", "type": "inline_equation", "height": 9, "width": 32}], "index": 22}, {"bbox": [124, 404, 440, 419], "spans": [{"bbox": [124, 404, 150, 419], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [150, 407, 158, 414], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [159, 404, 274, 419], "score": 1.0, "content": " is nonmetacyclic, cf. [1]); ", "type": "text"}, {"bbox": [275, 406, 316, 417], "score": 0.94, "content": "c_{2}=[a,b]", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [317, 404, 338, 419], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [339, 406, 394, 417], "score": 0.95, "content": "c_{j+1}=[b,c_{j}]", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [394, 404, 411, 419], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [412, 407, 435, 416], "score": 0.92, "content": "j\\geq2", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [435, 404, 440, 419], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 21.5, "bbox_fs": [124, 368, 486, 419]}, {"type": "text", "bbox": [124, 421, 486, 457], "lines": [{"bbox": [125, 423, 487, 435], "spans": [{"bbox": [125, 423, 199, 435], "score": 1.0, "content": "Lemma 1. Let ", "type": "text"}, {"bbox": [200, 425, 208, 432], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [208, 423, 487, 435], "score": 1.0, "content": " be as above (but not necessarily metabelian). Suppose that", "type": "text"}], "index": 24}, {"bbox": [126, 434, 487, 448], "spans": [{"bbox": [126, 436, 172, 447], "score": 0.93, "content": "d(G^{\\prime})\\,=\\,n", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [172, 434, 204, 448], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [205, 436, 228, 447], "score": 0.93, "content": "d(G^{\\prime})", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [228, 434, 487, 448], "score": 1.0, "content": " denotes the minimal number of generators of the derived", "type": "text"}], "index": 25}, {"bbox": [126, 447, 245, 460], "spans": [{"bbox": [126, 447, 153, 460], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [153, 449, 190, 458], "score": 0.93, "content": "G^{\\prime}=G_{2}", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [190, 447, 204, 460], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [205, 449, 213, 456], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [213, 447, 245, 460], "score": 1.0, "content": ". Then", "type": "text"}], "index": 26}], "index": 25, "bbox_fs": [125, 423, 487, 460]}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "lines": [{"bbox": [255, 464, 355, 475], "spans": [{"bbox": [255, 464, 355, 475], "score": 0.89, "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [126, 479, 169, 489], "lines": [{"bbox": [126, 481, 169, 491], "spans": [{"bbox": [126, 481, 169, 491], "score": 1.0, "content": "moreover,", "type": "text"}], "index": 28}], "index": 28, "bbox_fs": [126, 481, 169, 491]}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "lines": [{"bbox": [230, 495, 381, 507], "spans": [{"bbox": [230, 495, 381, 507], "score": 0.9, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "type": "interline_equation"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 511, 486, 573], "lines": [{"bbox": [126, 514, 486, 526], "spans": [{"bbox": [126, 514, 305, 526], "score": 1.0, "content": "Proof. By the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [305, 515, 398, 525], "score": 0.93, "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "type": "inline_equation", "height": 10, "width": 93}, {"bbox": [398, 514, 433, 526], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [434, 515, 457, 525], "score": 0.95, "content": "\\Phi(G)", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [457, 514, 486, 526], "score": 1.0, "content": " is the", "type": "text"}], "index": 30}, {"bbox": [126, 526, 485, 538], "spans": [{"bbox": [126, 526, 216, 538], "score": 1.0, "content": "Frattini subgroup of ", "type": "text"}, {"bbox": [216, 528, 224, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [225, 526, 442, 538], "score": 1.0, "content": " , i.e. the intersection of all maximal subgroups of ", "type": "text"}, {"bbox": [442, 528, 450, 535], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [451, 526, 485, 538], "score": 1.0, "content": ", see [5].", "type": "text"}], "index": 31}, {"bbox": [124, 537, 485, 551], "spans": [{"bbox": [124, 537, 258, 551], "score": 1.0, "content": "But in the case of a 2-group, ", "type": "text"}, {"bbox": [258, 538, 308, 549], "score": 0.94, "content": "\\Phi(G)=G^{2}", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [308, 537, 459, 551], "score": 1.0, "content": ", see [8]. By Blackburn, [3], since ", "type": "text"}, {"bbox": [460, 538, 485, 549], "score": 0.94, "content": "G/G_{2}^{2}", "type": "inline_equation", "height": 11, "width": 25}], "index": 32}, {"bbox": [124, 549, 485, 564], "spans": [{"bbox": [124, 549, 329, 564], "score": 1.0, "content": "has elementary derived group, we know that ", "type": "text"}, {"bbox": [329, 550, 482, 561], "score": 0.91, "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 153}, {"bbox": [483, 549, 485, 564], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [126, 560, 486, 575], "spans": [{"bbox": [126, 560, 301, 575], "score": 1.0, "content": "Again, by the Burnside Basis Theorem, ", "type": "text"}, {"bbox": [302, 563, 388, 573], "score": 0.9, "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "type": "inline_equation", "height": 10, "width": 86}, {"bbox": [388, 560, 392, 575], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 562, 486, 572], "score": 0.9934230446815491, "content": "口", "type": "text"}], "index": 34}], "index": 32, "bbox_fs": [124, 514, 486, 575]}, {"type": "text", "bbox": [125, 577, 487, 614], "lines": [{"bbox": [126, 580, 486, 592], "spans": [{"bbox": [126, 580, 198, 592], "score": 1.0, "content": "Lemma 2. Let ", "type": "text"}, {"bbox": [199, 582, 207, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 580, 354, 592], "score": 1.0, "content": " be as above. Moreover, assume ", "type": "text"}, {"bbox": [355, 582, 363, 589], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [363, 580, 452, 592], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [453, 582, 462, 589], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [462, 580, 486, 592], "score": 1.0, "content": " be a", "type": "text"}], "index": 35}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 218, 604], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [218, 594, 226, 601], "score": 0.87, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 592, 271, 604], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [272, 593, 295, 603], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [296, 592, 434, 604], "score": 1.0, "content": " is cyclic, and denote the index ", "type": "text"}, {"bbox": [434, 592, 472, 603], "score": 0.8, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [472, 592, 486, 604], "score": 1.0, "content": " by", "type": "text"}], "index": 36}, {"bbox": [126, 603, 327, 615], "spans": [{"bbox": [126, 605, 136, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [137, 603, 169, 615], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [170, 605, 180, 613], "score": 0.9, "content": "G^{\\prime}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [181, 603, 312, 615], "score": 1.0, "content": " contains an element of order ", "type": "text"}, {"bbox": [312, 605, 323, 613], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [323, 603, 327, 615], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36, "bbox_fs": [126, 580, 486, 615]}, {"type": "text", "bbox": [125, 619, 487, 657], "lines": [{"bbox": [126, 621, 485, 634], "spans": [{"bbox": [126, 621, 294, 634], "score": 1.0, "content": "Proof. Without loss of generality, let ", "type": "text"}, {"bbox": [294, 622, 347, 633], "score": 0.93, "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 53}, {"bbox": [348, 621, 411, 634], "score": 1.0, "content": ". Notice that ", "type": "text"}, {"bbox": [412, 621, 485, 633], "score": 0.92, "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 12, "width": 73}], "index": 38}, {"bbox": [126, 633, 487, 646], "spans": [{"bbox": [126, 633, 245, 646], "score": 1.0, "content": "and by our presentation of ", "type": "text"}, {"bbox": [245, 636, 254, 643], "score": 0.87, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 633, 260, 646], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [260, 635, 332, 645], "score": 0.92, "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [332, 633, 366, 646], "score": 1.0, "content": ". Thus,", "type": "text"}, {"bbox": [367, 634, 437, 645], "score": 0.93, "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [437, 633, 487, 646], "score": 1.0, "content": ". But since", "type": "text"}], "index": 39}, {"bbox": [126, 646, 486, 658], "spans": [{"bbox": [126, 647, 188, 657], "score": 0.92, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [188, 646, 248, 658], "score": 1.0, "content": ", the order of ", "type": "text"}, {"bbox": [249, 650, 258, 656], "score": 0.88, "content": "c_{2}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [258, 646, 270, 658], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [271, 648, 291, 656], "score": 0.9, "content": "\\geq2^{\\kappa}", "type": "inline_equation", "height": 8, "width": 20}, {"bbox": [292, 646, 421, 658], "score": 1.0, "content": ". This establishes the lemma.", "type": "text"}, {"bbox": [475, 646, 486, 656], "score": 0.991905927658081, "content": "口", "type": "text"}], "index": 40}], "index": 39, "bbox_fs": [126, 621, 487, 658]}, {"type": "text", "bbox": [125, 661, 486, 701], "lines": [{"bbox": [125, 664, 487, 676], "spans": [{"bbox": [125, 664, 199, 676], "score": 1.0, "content": "Lemma 3. Let ", "type": "text"}, {"bbox": [199, 666, 207, 673], "score": 0.86, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [207, 664, 351, 676], "score": 1.0, "content": " be as above and again assume ", "type": "text"}, {"bbox": [352, 664, 361, 673], "score": 0.76, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [361, 664, 451, 676], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [452, 664, 462, 673], "score": 0.76, "content": "H", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [462, 664, 487, 676], "score": 1.0, "content": " be a", "type": "text"}], "index": 41}, {"bbox": [126, 675, 486, 687], "spans": [{"bbox": [126, 676, 221, 687], "score": 1.0, "content": "maximal subgroup of ", "type": "text"}, {"bbox": [222, 677, 230, 685], "score": 0.85, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [230, 676, 279, 687], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [279, 676, 304, 687], "score": 0.89, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [304, 676, 430, 687], "score": 1.0, "content": " is cyclic, and assume that ", "type": "text"}, {"bbox": [430, 675, 486, 687], "score": 0.87, "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "type": "inline_equation", "height": 12, "width": 56}], "index": 42}, {"bbox": [124, 687, 454, 702], "spans": [{"bbox": [124, 689, 178, 702], "score": 1.0, "content": "0 mod 4. If ", "type": "text"}, {"bbox": [179, 690, 221, 701], "score": 0.93, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [221, 689, 249, 702], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [249, 690, 305, 701], "score": 0.93, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [305, 689, 327, 702], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [327, 687, 408, 701], "score": 0.93, "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [408, 689, 426, 702], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [426, 689, 451, 700], "score": 0.88, "content": "j>2", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [451, 689, 454, 702], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 42, "bbox_fs": [124, 664, 487, 702]}]}
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[{"type": "text", "bbox": [124, 111, 486, 195], "content": "We mention one last feature gleaned from the table. It follows from conditional Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua- dratic fields with rank and discriminant have finite class field tower; unconditional proofs are not known. Hence, conditionally, we conclude that those with discriminants , and have finite (2-)class field tower even though rank . Of course, it would be interesting to determine the length of their towers.", "index": 0}, {"type": "text", "bbox": [124, 196, 486, 255], "content": "The structure of this paper is as follows: we use results from group theory developed in Section 2 to pull down the condition rank from the field with degree to a subfield of with degree 8. Using the arithmetic of dihedral fields from Section 4 we then go down to the field of degree 4 occurring in Theorem 1.", "index": 1}, {"type": "title", "bbox": [218, 263, 393, 276], "content": "2. Group Theoretic Preliminaries", "index": 2}, {"type": "text", "bbox": [124, 281, 486, 367], "content": "Let be a group. If , then we let denote the commutator of and . If and are nonempty subsets of , then denotes the subgroup of generated by the set . The lower central series of is defined inductively by: and for . The derived series is defined inductively by: and for . Notice that the commutator subgroup, , of .", "index": 3}, {"type": "text", "bbox": [125, 368, 486, 417], "content": "Throughout this section, we assume that is a finite, nonmetacyclic, 2-group such that its abelianization is of type for some positive integer (necessarily ). Let , where mod (actually since is nonmetacyclic, cf. [1]); and for .", "index": 4}, {"type": "text", "bbox": [124, 421, 486, 457], "content": "Lemma 1. Let be as above (but not necessarily metabelian). Suppose that where denotes the minimal number of generators of the derived group of . Then", "index": 5}, {"type": "interline_equation", "bbox": [255, 464, 355, 475], "content": "", "index": 6}, {"type": "text", "bbox": [126, 479, 169, 489], "content": "moreover,", "index": 7}, {"type": "interline_equation", "bbox": [230, 495, 381, 507], "content": "", "index": 8}, {"type": "text", "bbox": [125, 511, 486, 573], "content": "Proof. By the Burnside Basis Theorem, , where is the Frattini subgroup of , i.e. the intersection of all maximal subgroups of , see [5]. But in the case of a 2-group, , see [8]. By Blackburn, [3], since has elementary derived group, we know that . Again, by the Burnside Basis Theorem, . 口", "index": 9}, {"type": "text", "bbox": [125, 577, 487, 614], "content": "Lemma 2. Let be as above. Moreover, assume is metabelian. Let be a maximal subgroup of such that is cyclic, and denote the index by . Then contains an element of order .", "index": 10}, {"type": "text", "bbox": [125, 619, 487, 657], "content": "Proof. Without loss of generality, let . Notice that and by our presentation of , . Thus, . But since , the order of is . This establishes the lemma. 口", "index": 11}, {"type": "text", "bbox": [125, 661, 486, 701], "content": "Lemma 3. Let be as above and again assume is metabelian. Let be a maximal subgroup of such that is cyclic, and assume that 0 mod 4. If , then and for .", "index": 12}]
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[{"bbox": [137, 114, 486, 126], "content": "We mention one last feature gleaned from the table. It follows from conditional", "parent_index": 0, "line_index": 0}, {"bbox": [125, 126, 486, 139], "content": "Odlyzko bounds (assuming the Generalized Riemann Hypothesis) that those qua-", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 486, 150], "content": "dratic fields with rank and discriminant have finite class", "parent_index": 0, "line_index": 2}, {"bbox": [125, 150, 486, 162], "content": "field tower; unconditional proofs are not known. Hence, conditionally, we conclude", "parent_index": 0, "line_index": 3}, {"bbox": [126, 163, 485, 174], "content": "that those with discriminants , and have finite (2-)class field", "parent_index": 0, "line_index": 4}, {"bbox": [125, 174, 485, 186], "content": "tower even though rank . Of course, it would be interesting to determine", "parent_index": 0, "line_index": 5}, {"bbox": [126, 186, 240, 198], "content": "the length of their towers.", "parent_index": 0, "line_index": 6}, {"bbox": [137, 197, 485, 210], "content": "The structure of this paper is as follows: we use results from group theory", "parent_index": 1, "line_index": 0}, {"bbox": [125, 209, 487, 222], "content": "developed in Section 2 to pull down the condition rank from the field", "parent_index": 1, "line_index": 1}, {"bbox": [126, 220, 488, 235], "content": "with degree to a subfield of with degree 8. Using the arithmetic of", "parent_index": 1, "line_index": 2}, {"bbox": [125, 233, 486, 247], "content": "dihedral fields from Section 4 we then go down to the field of degree 4 occurring", "parent_index": 1, "line_index": 3}, {"bbox": [126, 246, 188, 257], "content": "in Theorem 1.", "parent_index": 1, "line_index": 4}, {"bbox": [217, 266, 394, 277], "content": "2. Group Theoretic Preliminaries", "parent_index": 2, "line_index": 0}, {"bbox": [136, 284, 486, 296], "content": "Let be a group. If , then we let denote the", "parent_index": 3, "line_index": 0}, {"bbox": [125, 296, 484, 309], "content": "commutator of and . If and are nonempty subsets of , then", "parent_index": 3, "line_index": 1}, {"bbox": [126, 308, 486, 320], "content": "denotes the subgroup of generated by the set . The lower", "parent_index": 3, "line_index": 2}, {"bbox": [124, 320, 485, 334], "content": "central series of is defined inductively by: and", "parent_index": 3, "line_index": 3}, {"bbox": [124, 332, 487, 345], "content": "for . The derived series is defined inductively by: and", "parent_index": 3, "line_index": 4}, {"bbox": [126, 342, 487, 359], "content": "for . Notice that the commutator", "parent_index": 3, "line_index": 5}, {"bbox": [125, 357, 212, 370], "content": "subgroup, , of .", "parent_index": 3, "line_index": 6}, {"bbox": [137, 368, 486, 383], "content": "Throughout this section, we assume that is a finite, nonmetacyclic, 2-group", "parent_index": 4, "line_index": 0}, {"bbox": [125, 379, 486, 394], "content": "such that its abelianization is of type for some positive integer", "parent_index": 4, "line_index": 1}, {"bbox": [126, 392, 484, 406], "content": "(necessarily ). Let , where mod (actually", "parent_index": 4, "line_index": 2}, {"bbox": [124, 404, 440, 419], "content": "since is nonmetacyclic, cf. [1]); and for .", "parent_index": 4, "line_index": 3}, {"bbox": [125, 423, 487, 435], "content": "Lemma 1. Let be as above (but not necessarily metabelian). Suppose that", "parent_index": 5, "line_index": 0}, {"bbox": [126, 434, 487, 448], "content": "where denotes the minimal number of generators of the derived", "parent_index": 5, "line_index": 1}, {"bbox": [126, 447, 245, 460], "content": "group of . Then", "parent_index": 5, "line_index": 2}, {"bbox": [126, 481, 169, 491], "content": "moreover,", "parent_index": 7, "line_index": 0}, {"bbox": [126, 514, 486, 526], "content": "Proof. By the Burnside Basis Theorem, , where is the", "parent_index": 9, "line_index": 0}, {"bbox": [126, 526, 485, 538], "content": "Frattini subgroup of , i.e. the intersection of all maximal subgroups of , see [5].", "parent_index": 9, "line_index": 1}, {"bbox": [124, 537, 485, 551], "content": "But in the case of a 2-group, , see [8]. By Blackburn, [3], since", "parent_index": 9, "line_index": 2}, {"bbox": [124, 549, 485, 564], "content": "has elementary derived group, we know that .", "parent_index": 9, "line_index": 3}, {"bbox": [126, 560, 486, 575], "content": "Again, by the Burnside Basis Theorem, . 口", "parent_index": 9, "line_index": 4}, {"bbox": [126, 580, 486, 592], "content": "Lemma 2. Let be as above. Moreover, assume is metabelian. Let be a", "parent_index": 10, "line_index": 0}, {"bbox": [126, 592, 486, 604], "content": "maximal subgroup of such that is cyclic, and denote the index by", "parent_index": 10, "line_index": 1}, {"bbox": [126, 603, 327, 615], "content": ". Then contains an element of order .", "parent_index": 10, "line_index": 2}, {"bbox": [126, 621, 485, 634], "content": "Proof. Without loss of generality, let . Notice that", "parent_index": 11, "line_index": 0}, {"bbox": [126, 633, 487, 646], "content": "and by our presentation of , . Thus, . But since", "parent_index": 11, "line_index": 1}, {"bbox": [126, 646, 486, 658], "content": ", the order of is . This establishes the lemma. 口", "parent_index": 11, "line_index": 2}, {"bbox": [125, 664, 487, 676], "content": "Lemma 3. Let be as above and again assume is metabelian. Let be a", "parent_index": 12, "line_index": 0}, {"bbox": [126, 675, 486, 687], "content": "maximal subgroup of such that is cyclic, and assume that", "parent_index": 12, "line_index": 1}, {"bbox": [124, 687, 454, 702], "content": "0 mod 4. If , then and for .", "parent_index": 12, "line_index": 2}]
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[{"bbox": [222, 139, 273, 150], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "parent_index": 0, "subtype": "inline"}, {"bbox": [351, 140, 415, 148], "content": "0>d>-2000", "parent_index": 0, "subtype": "inline"}, {"bbox": [173, 164, 179, 171], "content": "k", "parent_index": 0, "subtype": "inline"}, {"bbox": [266, 164, 294, 172], "content": "-1015", "parent_index": 0, "subtype": "inline"}, {"bbox": [298, 164, 326, 172], "content": "-1595", "parent_index": 0, "subtype": "inline"}, {"bbox": [348, 164, 376, 172], "content": "-1780", "parent_index": 0, "subtype": "inline"}, {"bbox": [228, 175, 279, 186], "content": "\\mathrm{Cl}_{2}(k^{1})\\geq3", "parent_index": 0, "subtype": "inline"}, {"bbox": [370, 210, 422, 221], "content": "\\mathrm{Cl}_{2}(k^{1})=2", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 222, 136, 231], "content": "k^{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [194, 223, 217, 231], "content": "2^{m+2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [279, 224, 286, 231], "content": "L", "parent_index": 1, "subtype": "inline"}, {"bbox": [301, 222, 311, 231], "content": "k^{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [381, 236, 391, 243], "content": "K", "parent_index": 1, "subtype": "inline"}, {"bbox": [157, 286, 165, 293], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [244, 286, 285, 295], "content": "x,y\\ \\in\\ G", "parent_index": 3, "subtype": "inline"}, {"bbox": [349, 285, 433, 296], "content": "[x,y]~=~x^{-1}y^{-1}x y", "parent_index": 3, "subtype": "inline"}, {"bbox": [196, 300, 203, 305], "content": "x", "parent_index": 3, "subtype": "inline"}, {"bbox": [228, 300, 234, 307], "content": "_y", "parent_index": 3, "subtype": "inline"}, {"bbox": [257, 298, 265, 305], "content": "A", "parent_index": 3, "subtype": "inline"}, {"bbox": [290, 298, 299, 305], "content": "B", "parent_index": 3, "subtype": "inline"}, {"bbox": [419, 298, 427, 305], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [459, 297, 484, 308], "content": "[A,B]", "parent_index": 3, "subtype": "inline"}, {"bbox": [235, 310, 243, 317], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [339, 309, 433, 320], "content": "\\{[a,b]:a\\in A,b\\in B\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [187, 321, 209, 332], "content": "\\{G_{j}\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [226, 322, 234, 329], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [356, 322, 392, 331], "content": "G_{1}\\,=\\,G", "parent_index": 3, "subtype": "inline"}, {"bbox": [416, 321, 485, 332], "content": "G_{j+1}\\,=\\,[G,G_{j}]", "parent_index": 3, "subtype": "inline"}, {"bbox": [142, 335, 170, 344], "content": "j~\\geq~1", "parent_index": 3, "subtype": "inline"}, {"bbox": [267, 333, 297, 344], "content": "\\{G^{(n)}\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [420, 333, 464, 342], "content": "G^{(0)}\\ =\\ G", "parent_index": 3, "subtype": "inline"}, {"bbox": [126, 345, 219, 357], "content": "G^{(n+1)}=[G^{(n)},G^{(n)}]", "parent_index": 3, "subtype": "inline"}, {"bbox": [238, 347, 264, 356], "content": "n\\geq0", "parent_index": 3, "subtype": "inline"}, {"bbox": [325, 345, 410, 357], "content": "G^{(1)}=G_{2}=[G,G]", "parent_index": 3, "subtype": "inline"}, {"bbox": [172, 358, 183, 366], "content": "G^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [200, 359, 208, 366], "content": "G", "parent_index": 3, "subtype": "inline"}, {"bbox": [323, 371, 331, 378], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [246, 382, 300, 393], "content": "G^{\\mathrm{ab}}=G/G^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [346, 382, 376, 393], "content": "(2,2^{m})", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 398, 135, 402], "content": "{\\boldsymbol{r}}n", "parent_index": 4, "subtype": "inline"}, {"bbox": [191, 396, 207, 403], "content": "\\geq2", "parent_index": 4, "subtype": "inline"}, {"bbox": [236, 394, 279, 405], "content": "G=\\langle a,b\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [313, 393, 370, 402], "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "parent_index": 4, "subtype": "inline"}, {"bbox": [394, 395, 407, 404], "content": "G_{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [452, 395, 484, 404], "content": "\\mathrm{mod}G_{3}", "parent_index": 4, "subtype": "inline"}, {"bbox": [150, 407, 158, 414], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [275, 406, 316, 417], "content": "c_{2}=[a,b]", "parent_index": 4, "subtype": "inline"}, {"bbox": [339, 406, 394, 417], "content": "c_{j+1}=[b,c_{j}]", "parent_index": 4, "subtype": "inline"}, {"bbox": [412, 407, 435, 416], "content": "j\\geq2", "parent_index": 4, "subtype": "inline"}, {"bbox": [200, 425, 208, 432], "content": "G", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 436, 172, 447], "content": "d(G^{\\prime})\\,=\\,n", "parent_index": 5, "subtype": "inline"}, {"bbox": [205, 436, 228, 447], "content": "d(G^{\\prime})", "parent_index": 5, "subtype": "inline"}, {"bbox": [153, 449, 190, 458], "content": "G^{\\prime}=G_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [205, 449, 213, 456], "content": "G", "parent_index": 5, "subtype": "inline"}, {"bbox": [255, 464, 355, 475], "content": "G^{\\prime}=\\langle c_{2},c_{3},\\cdot\\cdot\\cdot,c_{n+1}\\rangle;", "parent_index": 6, "subtype": "interline"}, {"bbox": [230, 495, 381, 507], "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle.", "parent_index": 8, "subtype": "interline"}, {"bbox": [305, 515, 398, 525], "content": "d(G_{2})=d(G_{2}/\\Phi(G))", "parent_index": 9, "subtype": "inline"}, {"bbox": [434, 515, 457, 525], "content": "\\Phi(G)", "parent_index": 9, "subtype": "inline"}, {"bbox": [216, 528, 224, 535], "content": "G", "parent_index": 9, "subtype": "inline"}, {"bbox": [442, 528, 450, 535], "content": "G", "parent_index": 9, "subtype": "inline"}, {"bbox": [258, 538, 308, 549], "content": "\\Phi(G)=G^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [460, 538, 485, 549], "content": "G/G_{2}^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [329, 550, 482, 561], "content": "G_{2}/G_{2}^{2}\\simeq\\langle c_{2}G_{2}^{2}\\rangle\\oplus\\cdots\\oplus\\langle c_{n+1}G_{2}^{2}\\rangle", "parent_index": 9, "subtype": "inline"}, {"bbox": [302, 563, 388, 573], "content": "G_{2}=\\langle c_{2},\\cdots,c_{n+1}\\rangle", "parent_index": 9, "subtype": "inline"}, {"bbox": [199, 582, 207, 589], "content": "G", "parent_index": 10, "subtype": "inline"}, {"bbox": [355, 582, 363, 589], "content": "G", "parent_index": 10, "subtype": "inline"}, {"bbox": [453, 582, 462, 589], "content": "H", "parent_index": 10, "subtype": "inline"}, {"bbox": [218, 594, 226, 601], "content": "G", "parent_index": 10, "subtype": "inline"}, {"bbox": [272, 593, 295, 603], "content": "H/G^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [434, 592, 472, 603], "content": "\\left(G^{\\prime}:H^{\\prime}\\right)", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 605, 136, 613], "content": "2^{\\kappa}", "parent_index": 10, "subtype": "inline"}, {"bbox": [170, 605, 180, 613], "content": "G^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [312, 605, 323, 613], "content": "2^{\\kappa}", "parent_index": 10, "subtype": "inline"}, {"bbox": [294, 622, 347, 633], "content": "H\\,=\\,\\left\\langle{b,G^{\\prime}}\\right\\rangle", "parent_index": 11, "subtype": "inline"}, {"bbox": [412, 621, 485, 633], "content": "G^{\\prime}\\,=\\,\\langle c_{2},c_{3},\\cdot\\cdot\\cdot\\rangle", "parent_index": 11, "subtype": "inline"}, {"bbox": [245, 636, 254, 643], "content": "H", "parent_index": 11, "subtype": "inline"}, {"bbox": [260, 635, 332, 645], "content": "H^{\\prime}=\\langle c_{3},c_{4},\\cdot\\cdot\\cdot\\rangle", "parent_index": 11, "subtype": "inline"}, {"bbox": [367, 634, 437, 645], "content": "G^{\\prime}/H^{\\prime}=\\langle c_{2}H^{\\prime}\\rangle", "parent_index": 11, "subtype": "inline"}, {"bbox": [126, 647, 188, 657], "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "parent_index": 11, "subtype": "inline"}, {"bbox": [249, 650, 258, 656], "content": "c_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [271, 648, 291, 656], "content": "\\geq2^{\\kappa}", "parent_index": 11, "subtype": "inline"}, {"bbox": [199, 666, 207, 673], "content": "G", "parent_index": 12, "subtype": "inline"}, {"bbox": [352, 664, 361, 673], "content": "G", "parent_index": 12, "subtype": "inline"}, {"bbox": [452, 664, 462, 673], "content": "H", "parent_index": 12, "subtype": "inline"}, {"bbox": [222, 677, 230, 685], "content": "G", "parent_index": 12, "subtype": "inline"}, {"bbox": [279, 676, 304, 687], "content": "H/G^{\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [430, 675, 486, 687], "content": "(G^{\\prime}\\,:\\,H^{\\prime})\\;\\equiv\\;", "parent_index": 12, "subtype": "inline"}, {"bbox": [179, 690, 221, 701], "content": "d(G^{\\prime})=2", "parent_index": 12, "subtype": "inline"}, {"bbox": [249, 690, 305, 701], "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "parent_index": 12, "subtype": "inline"}, {"bbox": [327, 687, 408, 701], "content": "G_{j}=\\langle{c_{2}^{2^{j-2}},c_{3}^{2^{j-3}}}\\rangle", "parent_index": 12, "subtype": "inline"}, {"bbox": [426, 689, 451, 700], "content": "j>2", "parent_index": 12, "subtype": "inline"}]
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Proof. Assume that $d(G^{\prime})=2$ . By Lemma 1, $G_{2}=\langle c_{2},c_{3}\rangle$ and hence $c_{4}\in\langle c_{2},c_{3}\rangle$ . Write $c_{4}\:=\:c_{2}^{x}c_{3}^{y}$ where $x,y$ are positive integers. Without loss of generality, let $H=\langle b,c_{2},c_{3}\rangle$ and write $(G^{\prime}:H^{\prime})=2^{\kappa}\,$ for some $\kappa\geq2$ . Since $c_{3},c_{4}\in H^{\prime}$ we have, $c_{2}^{x}\equiv1$ mod $H^{\prime}$ . By the proof of Lemma 2, this implies that $x\equiv0$ mod $2^{\kappa}$ . Write $x\,=\,2^{\kappa}x_{1}$ for some positive integer $x_{1}$ . On the other hand, since $c_{4},c_{2}^{2^{\kappa}x_{1}}\,\in\,G_{4}$ we see that $c_{3}^{y}\equiv1$ mod $G_{4}$ . If $_y$ were odd, then $c_{3}\in G_{4}$ . This, however, implies that $G_{2}=\langle c_{2}\rangle$ , contrary to our assumptions. Thus $_y$ is even, say $y=2y_{1}$ . From all of this we see that c4 = $c_{4}\,=\,c_{2}^{2^{\kappa}x_{1}}c_{3}^{2y_{1}}$ c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− $G_{j}=\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\cdot\cdot\cdot,c_{j-1}^{2},c_{j},c_{j+1},\cdot\cdot\cdot\rangle$ , cf. [1], we obtain the lemma. 口
Let us translate the above into the field-theoretic language. Let $k$ be an imaginary quadratic number field of type A) or B) (see the Introduction), and let $M/k$ be one of the two quadratic subextensions of $k^{1}/k$ over which $k^{1}$ is cyclic. If $h_{2}(M)=2^{m+\kappa}$ and $\mathrm{Cl}_{2}(k)=(2,2^{m})$ , then Lemma 2 implies that $\mathrm{Cl_{2}}(k^{1})$ contains an element of order $2^{\kappa}$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $M/\mathbb{Q}$ (see e.g. Proposition 3 below) shows immediately that $h_{2}(M)=2^{m+\kappa}$ , where $2^{\kappa}$ is the class number of the quadratic subfield $\mathbb{Q}({\sqrt{d_{i}d_{j}}}\,)$ of $M$ , where $(d_{i}/p_{j})=+1$ ; in particular, we always have $\kappa\geq2$ , and the assumption $\left(G^{\prime}:H^{\prime}\right)\geq4$ is always satisfied for the fields that we consider.
Table 2
<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>
We now use the above results to prove the following useful proposition.
Proposition 1. Let $G$ be a nonmetacyclic 2-group such that $G/G^{\prime}\;\simeq\;(2,2^{m})$ ; (hence $m>1$ ). Let $H$ and $K$ be the two maximal subgroups of $G$ such that $H/G^{\prime}$ and $K/G^{\prime}$ are cyclic. Moreover, assume that $(G^{\prime}:H^{\prime})\equiv0$ mod 4. Finally, assume that $N$ is a subgroup of index 4 in $G$ not contained in $H$ or $K$ Then
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<html><body>
<p data-bbox="125 111 487 237">Proof. Assume that $d(G^{\prime})=2$ . By Lemma 1, $G_{2}=\langle c_{2},c_{3}\rangle$ and hence $c_{4}\in\langle c_{2},c_{3}\rangle$ . Write $c_{4}\:=\:c_{2}^{x}c_{3}^{y}$ where $x,y$ are positive integers. Without loss of generality, let $H=\langle b,c_{2},c_{3}\rangle$ and write $(G^{\prime}:H^{\prime})=2^{\kappa}\,$ for some $\kappa\geq2$ . Since $c_{3},c_{4}\in H^{\prime}$ we have, $c_{2}^{x}\equiv1$ mod $H^{\prime}$ . By the proof of Lemma 2, this implies that $x\equiv0$ mod $2^{\kappa}$ . Write $x\,=\,2^{\kappa}x_{1}$ for some positive integer $x_{1}$ . On the other hand, since $c_{4},c_{2}^{2^{\kappa}x_{1}}\,\in\,G_{4}$ we see that $c_{3}^{y}\equiv1$ mod $G_{4}$ . If $_y$ were odd, then $c_{3}\in G_{4}$ . This, however, implies that $G_{2}=\langle c_{2}\rangle$ , contrary to our assumptions. Thus $_y$ is even, say $y=2y_{1}$ . From all of this we see that c4 = $c_{4}\,=\,c_{2}^{2^{\kappa}x_{1}}c_{3}^{2y_{1}}$ c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− $G_{j}=\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\cdot\cdot\cdot,c_{j-1}^{2},c_{j},c_{j+1},\cdot\cdot\cdot\rangle$ , cf. [1], we obtain the lemma. 口 </p>
<p data-bbox="125 246 487 366">Let us translate the above into the field-theoretic language. Let $k$ be an imaginary quadratic number field of type A) or B) (see the Introduction), and let $M/k$ be one of the two quadratic subextensions of $k^{1}/k$ over which $k^{1}$ is cyclic. If $h_{2}(M)=2^{m+\kappa}$ and $\mathrm{Cl}_{2}(k)=(2,2^{m})$ , then Lemma 2 implies that $\mathrm{Cl_{2}}(k^{1})$ contains an element of order $2^{\kappa}$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $M/\mathbb{Q}$ (see e.g. Proposition 3 below) shows immediately that $h_{2}(M)=2^{m+\kappa}$ , where $2^{\kappa}$ is the class number of the quadratic subfield $\mathbb{Q}({\sqrt{d_{i}d_{j}}}\,)$ of $M$ , where $(d_{i}/p_{j})=+1$ ; in particular, we always have $\kappa\geq2$ , and the assumption $\left(G^{\prime}:H^{\prime}\right)\geq4$ is always satisfied for the fields that we consider. </p>
<div class="table" data-bbox="167 416 444 622"><p class="caption" data-bbox="285 377 327 389">Table 2 </p><table data-bbox="167 416 444 622"><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></div>
<p data-bbox="137 643 450 656">We now use the above results to prove the following useful proposition. </p>
<p data-bbox="125 662 486 700">Proposition 1. Let $G$ be a nonmetacyclic 2-group such that $G/G^{\prime}\;\simeq\;(2,2^{m})$ ; (hence $m>1$ ). Let $H$ and $K$ be the two maximal subgroups of $G$ such that $H/G^{\prime}$ and $K/G^{\prime}$ are cyclic. Moreover, assume that $(G^{\prime}:H^{\prime})\equiv0$ mod 4. Finally, assume that $N$ is a subgroup of index 4 in $G$ not contained in $H$ or $K$ Then </p>
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0003244v1
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[{"type": "text", "text": "Proof. Assume that $d(G^{\\prime})=2$ . By Lemma 1, $G_{2}=\\langle c_{2},c_{3}\\rangle$ and hence $c_{4}\\in\\langle c_{2},c_{3}\\rangle$ . Write $c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}$ where $x,y$ are positive integers. Without loss of generality, let $H=\\langle b,c_{2},c_{3}\\rangle$ and write $(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,$ for some $\\kappa\\geq2$ . Since $c_{3},c_{4}\\in H^{\\prime}$ we have, $c_{2}^{x}\\equiv1$ mod $H^{\\prime}$ . By the proof of Lemma 2, this implies that $x\\equiv0$ mod $2^{\\kappa}$ . Write $x\\,=\\,2^{\\kappa}x_{1}$ for some positive integer $x_{1}$ . On the other hand, since $c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}$ we see that $c_{3}^{y}\\equiv1$ mod $G_{4}$ . If $_y$ were odd, then $c_{3}\\in G_{4}$ . This, however, implies that $G_{2}=\\langle c_{2}\\rangle$ , contrary to our assumptions. Thus $_y$ is even, say $y=2y_{1}$ . From all of this we see that c4 = $c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}$ c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− $G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle$ , cf. [1], we obtain the lemma. 口 ", "page_idx": 3}, {"type": "text", "text": "Let us translate the above into the field-theoretic language. Let $k$ be an imaginary quadratic number field of type A) or B) (see the Introduction), and let $M/k$ be one of the two quadratic subextensions of $k^{1}/k$ over which $k^{1}$ is cyclic. If $h_{2}(M)=2^{m+\\kappa}$ and $\\mathrm{Cl}_{2}(k)=(2,2^{m})$ , then Lemma 2 implies that $\\mathrm{Cl_{2}}(k^{1})$ contains an element of order $2^{\\kappa}$ . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to $M/\\mathbb{Q}$ (see e.g. Proposition 3 below) shows immediately that $h_{2}(M)=2^{m+\\kappa}$ , where $2^{\\kappa}$ is the class number of the quadratic subfield $\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)$ of $M$ , where $(d_{i}/p_{j})=+1$ ; in particular, we always have $\\kappa\\geq2$ , and the assumption $\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4$ is always satisfied for the fields that we consider. ", "page_idx": 3}, {"type": "table", "img_path": "images/20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg", "table_caption": ["Table 2 "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>\n\n", "page_idx": 3}, {"type": "text", "text": "We now use the above results to prove the following useful proposition. ", "page_idx": 3}, {"type": "text", "text": "Proposition 1. Let $G$ be a nonmetacyclic 2-group such that $G/G^{\\prime}\\;\\simeq\\;(2,2^{m})$ ; (hence $m>1$ ). Let $H$ and $K$ be the two maximal subgroups of $G$ such that $H/G^{\\prime}$ and $K/G^{\\prime}$ are cyclic. Moreover, assume that $(G^{\\prime}:H^{\\prime})\\equiv0$ mod 4. Finally, assume that $N$ is a subgroup of index 4 in $G$ not contained in $H$ or $K$ Then ", "page_idx": 3}]
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29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>"}, {"category_id": 1, "poly": [382, 1788, 1252, 1788, 1252, 1823, 382, 1823], "score": 0.9}, {"category_id": 6, "poly": [793, 1049, 909, 1049, 909, 1083, 793, 1083], "score": 0.808}, {"category_id": 2, "poly": [348, 254, 366, 254, 366, 275, 348, 275], "score": 0.753}, {"category_id": 13, "poly": [431, 355, 552, 355, 552, 384, 431, 384], "score": 0.95, "latex": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}"}, {"category_id": 13, "poly": [907, 322, 1061, 322, 1061, 351, 907, 351], "score": 0.95, "latex": "G_{2}=\\langle 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1248.0, 1793.0, 1248.0, 1827.0, 385.0, 1827.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [793.0, 1052.0, 909.0, 1052.0, 909.0, 1088.0, 793.0, 1088.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [349.0, 263.5, 367.0, 263.5, 367.0, 276.5, 349.0, 276.5], "score": 1.0, "text": ""}], "page_info": {"page_no": 3, "height": 2200, "width": 1700}}
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 487, 237], "lines": [{"bbox": [126, 114, 485, 126], "spans": [{"bbox": [126, 114, 215, 126], "score": 1.0, "content": "Proof. Assume that ", "type": "text"}, {"bbox": [215, 115, 258, 126], "score": 0.94, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [258, 114, 326, 126], "score": 1.0, "content": ". By Lemma 1, ", "type": "text"}, {"bbox": [326, 115, 381, 126], "score": 0.95, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [382, 114, 431, 126], "score": 1.0, "content": "and hence ", "type": "text"}, {"bbox": [431, 115, 482, 126], "score": 0.94, "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [483, 114, 485, 126], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [123, 122, 489, 143], "spans": [{"bbox": [123, 122, 154, 143], "score": 1.0, "content": "Write ", "type": "text"}, {"bbox": [155, 127, 198, 138], "score": 0.95, "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [199, 122, 232, 143], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [232, 131, 248, 137], "score": 0.89, "content": "x,y", "type": "inline_equation", "height": 6, "width": 16}, {"bbox": [248, 122, 489, 143], "score": 1.0, "content": " are positive integers. Without loss of generality, let", "type": "text"}], "index": 1}, {"bbox": [126, 136, 487, 152], "spans": [{"bbox": [126, 139, 187, 150], "score": 0.93, "content": "H=\\langle b,c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [187, 136, 235, 152], "score": 1.0, "content": "and write ", "type": "text"}, {"bbox": [235, 139, 297, 150], "score": 0.93, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [297, 136, 340, 152], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [340, 140, 365, 149], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [365, 136, 398, 152], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [398, 139, 444, 149], "score": 0.93, "content": "c_{3},c_{4}\\in H^{\\prime}", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [444, 136, 487, 152], "score": 1.0, "content": " we have,", "type": "text"}], "index": 2}, {"bbox": [126, 150, 487, 162], "spans": [{"bbox": [126, 152, 155, 162], "score": 0.86, "content": "c_{2}^{x}\\equiv1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [155, 150, 179, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [179, 151, 191, 159], "score": 0.84, "content": "H^{\\prime}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [191, 150, 392, 162], "score": 1.0, "content": ". By the proof of Lemma 2, this implies that ", "type": "text"}, {"bbox": [393, 152, 417, 159], "score": 0.73, "content": "x\\equiv0", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [418, 150, 442, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [442, 152, 452, 159], "score": 0.89, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [453, 150, 487, 162], "score": 1.0, "content": ". Write", "type": "text"}], "index": 3}, {"bbox": [126, 160, 482, 178], "spans": [{"bbox": [126, 165, 168, 174], "score": 0.92, "content": "x\\,=\\,2^{\\kappa}x_{1}", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [168, 160, 284, 178], "score": 1.0, "content": " for some positive integer ", "type": "text"}, {"bbox": [285, 168, 295, 174], "score": 0.89, "content": "x_{1}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [295, 160, 420, 178], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [420, 162, 482, 175], "score": 0.93, "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "type": "inline_equation", "height": 13, "width": 62}], "index": 4}, {"bbox": [125, 175, 486, 187], "spans": [{"bbox": [125, 175, 179, 187], "score": 1.0, "content": "we see that ", "type": "text"}, {"bbox": [180, 176, 209, 187], "score": 0.93, "content": "c_{3}^{y}\\equiv1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [209, 175, 233, 187], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [233, 177, 245, 186], "score": 0.91, "content": "G_{4}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [246, 175, 264, 187], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [264, 180, 270, 186], "score": 0.89, "content": "_y", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 175, 343, 187], "score": 1.0, "content": " were odd, then ", "type": "text"}, {"bbox": [343, 177, 378, 186], "score": 0.93, "content": "c_{3}\\in G_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 175, 486, 187], "score": 1.0, "content": ". This, however, implies", "type": "text"}], "index": 5}, {"bbox": [124, 186, 486, 200], "spans": [{"bbox": [124, 186, 147, 200], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 188, 191, 199], "score": 0.94, "content": "G_{2}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [192, 186, 356, 200], "score": 1.0, "content": ", contrary to our assumptions. Thus ", "type": "text"}, {"bbox": [356, 191, 362, 198], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [362, 186, 419, 200], "score": 1.0, "content": " is even, say ", "type": "text"}, {"bbox": [419, 189, 453, 198], "score": 0.93, "content": "y=2y_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [454, 186, 486, 200], "score": 1.0, "content": ". From", "type": "text"}], "index": 6}, {"bbox": [123, 196, 487, 215], "spans": [{"bbox": [123, 197, 265, 214], "score": 1.0, "content": "all of this we see that c4 = ", "type": "text"}, {"bbox": [228, 199, 291, 212], "score": 0.94, "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [253, 196, 487, 215], "score": 1.0, "content": "c22κx1c23y1. Consequently, by induction we have cj ∈", "type": "text"}], "index": 7}, {"bbox": [123, 208, 485, 232], "spans": [{"bbox": [123, 208, 345, 232], "score": 1.0, "content": "⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j−", "type": "text"}, {"bbox": [272, 212, 449, 227], "score": 0.92, "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 15, "width": 177}, {"bbox": [450, 213, 485, 227], "score": 1.0, "content": ", cf. [1],", "type": "text"}], "index": 8}, {"bbox": [125, 226, 486, 238], "spans": [{"bbox": [125, 226, 221, 238], "score": 1.0, "content": "we obtain the lemma.", "type": "text"}, {"bbox": [475, 226, 486, 236], "score": 0.9939806461334229, "content": "口", "type": "text"}], "index": 9}], "index": 4.5}, {"type": "text", "bbox": [125, 246, 487, 366], "lines": [{"bbox": [137, 248, 485, 261], "spans": [{"bbox": [137, 248, 420, 261], "score": 1.0, "content": "Let us translate the above into the field-theoretic language. Let ", "type": "text"}, {"bbox": [421, 250, 426, 257], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [427, 248, 485, 261], "score": 1.0, "content": " be an imagi-", "type": "text"}], "index": 10}, {"bbox": [124, 261, 485, 272], "spans": [{"bbox": [124, 261, 464, 272], "score": 1.0, "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "type": "text"}, {"bbox": [464, 262, 485, 272], "score": 0.92, "content": "M/k", "type": "inline_equation", "height": 10, "width": 21}], "index": 11}, {"bbox": [125, 273, 487, 284], "spans": [{"bbox": [125, 273, 337, 284], "score": 1.0, "content": "be one of the two quadratic subextensions of ", "type": "text"}, {"bbox": [337, 273, 358, 284], "score": 0.94, "content": "k^{1}/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [358, 273, 415, 284], "score": 1.0, "content": " over which ", "type": "text"}, {"bbox": [415, 273, 425, 281], "score": 0.9, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [426, 273, 487, 284], "score": 1.0, "content": " is cyclic. If", "type": "text"}], "index": 12}, {"bbox": [126, 282, 487, 298], "spans": [{"bbox": [126, 285, 191, 296], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [192, 282, 214, 298], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 285, 285, 296], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [286, 282, 413, 298], "score": 1.0, "content": ", then Lemma 2 implies that ", "type": "text"}, {"bbox": [413, 285, 446, 296], "score": 0.94, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [446, 282, 487, 298], "score": 1.0, "content": " contains", "type": "text"}], "index": 13}, {"bbox": [126, 296, 486, 308], "spans": [{"bbox": [126, 296, 218, 308], "score": 1.0, "content": "an element of order ", "type": "text"}, {"bbox": [218, 298, 228, 305], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [228, 296, 486, 308], "score": 1.0, "content": ". Table 2 contains the relevant information for the fields", "type": "text"}], "index": 14}, {"bbox": [125, 307, 486, 321], "spans": [{"bbox": [125, 307, 424, 321], "score": 1.0, "content": "occurring in Table 1. An application of the class number formula to ", "type": "text"}, {"bbox": [424, 309, 447, 320], "score": 0.94, "content": "M/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [447, 307, 486, 321], "score": 1.0, "content": " (see e.g.", "type": "text"}], "index": 15}, {"bbox": [124, 320, 487, 333], "spans": [{"bbox": [124, 320, 325, 333], "score": 1.0, "content": "Proposition 3 below) shows immediately that ", "type": "text"}, {"bbox": [325, 321, 390, 332], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [391, 320, 425, 333], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [425, 322, 435, 329], "score": 0.91, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [435, 320, 487, 333], "score": 1.0, "content": " is the class", "type": "text"}], "index": 16}, {"bbox": [124, 332, 486, 346], "spans": [{"bbox": [124, 332, 271, 346], "score": 1.0, "content": "number of the quadratic subfield ", "type": "text"}, {"bbox": [271, 332, 316, 345], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [317, 332, 330, 346], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [331, 334, 342, 342], "score": 0.89, "content": "M", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [342, 332, 376, 346], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [376, 334, 433, 344], "score": 0.94, "content": "(d_{i}/p_{j})=+1", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [433, 332, 486, 346], "score": 1.0, "content": "; in particu-", "type": "text"}], "index": 17}, {"bbox": [125, 344, 486, 357], "spans": [{"bbox": [125, 344, 213, 357], "score": 1.0, "content": "lar, we always have ", "type": "text"}, {"bbox": [213, 347, 237, 354], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [238, 344, 332, 357], "score": 1.0, "content": ", and the assumption ", "type": "text"}, {"bbox": [332, 345, 389, 356], "score": 0.93, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [390, 344, 486, 357], "score": 1.0, "content": " is always satisfied for", "type": "text"}], "index": 18}, {"bbox": [125, 356, 244, 368], "spans": [{"bbox": [125, 356, 244, 368], "score": 1.0, "content": "the fields that we consider.", "type": "text"}], "index": 19}], "index": 14.5}, {"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}, {"type": "text", "bbox": [137, 643, 450, 656], "lines": [{"bbox": [138, 645, 449, 657], "spans": [{"bbox": [138, 645, 449, 657], "score": 1.0, "content": "We now use the above results to prove the following useful proposition.", "type": "text"}], "index": 37}], "index": 37}, {"type": "text", "bbox": [125, 662, 486, 700], "lines": [{"bbox": [125, 665, 486, 677], "spans": [{"bbox": [125, 665, 220, 677], "score": 1.0, "content": "Proposition 1. Let ", "type": "text"}, {"bbox": [221, 667, 229, 674], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [229, 665, 409, 677], "score": 1.0, "content": " be a nonmetacyclic 2-group such that ", "type": "text"}, {"bbox": [409, 666, 482, 677], "score": 0.92, "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [483, 665, 486, 677], "score": 1.0, "content": ";", "type": "text"}], "index": 38}, {"bbox": [127, 677, 484, 689], "spans": [{"bbox": [127, 677, 157, 689], "score": 1.0, "content": "(hence ", "type": "text"}, {"bbox": [158, 679, 186, 687], "score": 0.83, "content": "m>1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [186, 677, 214, 689], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [215, 679, 224, 686], "score": 0.84, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [225, 677, 247, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [247, 679, 257, 686], "score": 0.85, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [257, 677, 405, 689], "score": 1.0, "content": " be the two maximal subgroups of ", "type": "text"}, {"bbox": [406, 679, 414, 686], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [414, 677, 460, 689], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [460, 678, 484, 689], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}], "index": 39}, {"bbox": [127, 689, 487, 701], "spans": [{"bbox": [127, 689, 144, 701], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 690, 169, 701], "score": 0.91, "content": "K/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [169, 689, 322, 701], "score": 1.0, "content": " are cyclic. Moreover, assume that ", "type": "text"}, {"bbox": [322, 690, 379, 701], "score": 0.91, "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [380, 689, 487, 701], "score": 1.0, "content": " mod 4. Finally, assume", "type": "text"}], "index": 40}], "index": 39}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [125, 94, 132, 99], "spans": [{"bbox": [125, 94, 132, 99], "score": 1.0, "content": "4", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 487, 237], "lines": [{"bbox": [126, 114, 485, 126], "spans": [{"bbox": [126, 114, 215, 126], "score": 1.0, "content": "Proof. Assume that ", "type": "text"}, {"bbox": [215, 115, 258, 126], "score": 0.94, "content": "d(G^{\\prime})=2", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [258, 114, 326, 126], "score": 1.0, "content": ". By Lemma 1, ", "type": "text"}, {"bbox": [326, 115, 381, 126], "score": 0.95, "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [382, 114, 431, 126], "score": 1.0, "content": "and hence ", "type": "text"}, {"bbox": [431, 115, 482, 126], "score": 0.94, "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [483, 114, 485, 126], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [123, 122, 489, 143], "spans": [{"bbox": [123, 122, 154, 143], "score": 1.0, "content": "Write ", "type": "text"}, {"bbox": [155, 127, 198, 138], "score": 0.95, "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [199, 122, 232, 143], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [232, 131, 248, 137], "score": 0.89, "content": "x,y", "type": "inline_equation", "height": 6, "width": 16}, {"bbox": [248, 122, 489, 143], "score": 1.0, "content": " are positive integers. Without loss of generality, let", "type": "text"}], "index": 1}, {"bbox": [126, 136, 487, 152], "spans": [{"bbox": [126, 139, 187, 150], "score": 0.93, "content": "H=\\langle b,c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [187, 136, 235, 152], "score": 1.0, "content": "and write ", "type": "text"}, {"bbox": [235, 139, 297, 150], "score": 0.93, "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [297, 136, 340, 152], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [340, 140, 365, 149], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [365, 136, 398, 152], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [398, 139, 444, 149], "score": 0.93, "content": "c_{3},c_{4}\\in H^{\\prime}", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [444, 136, 487, 152], "score": 1.0, "content": " we have,", "type": "text"}], "index": 2}, {"bbox": [126, 150, 487, 162], "spans": [{"bbox": [126, 152, 155, 162], "score": 0.86, "content": "c_{2}^{x}\\equiv1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [155, 150, 179, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [179, 151, 191, 159], "score": 0.84, "content": "H^{\\prime}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [191, 150, 392, 162], "score": 1.0, "content": ". By the proof of Lemma 2, this implies that ", "type": "text"}, {"bbox": [393, 152, 417, 159], "score": 0.73, "content": "x\\equiv0", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [418, 150, 442, 162], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [442, 152, 452, 159], "score": 0.89, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [453, 150, 487, 162], "score": 1.0, "content": ". Write", "type": "text"}], "index": 3}, {"bbox": [126, 160, 482, 178], "spans": [{"bbox": [126, 165, 168, 174], "score": 0.92, "content": "x\\,=\\,2^{\\kappa}x_{1}", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [168, 160, 284, 178], "score": 1.0, "content": " for some positive integer ", "type": "text"}, {"bbox": [285, 168, 295, 174], "score": 0.89, "content": "x_{1}", "type": "inline_equation", "height": 6, "width": 10}, {"bbox": [295, 160, 420, 178], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [420, 162, 482, 175], "score": 0.93, "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "type": "inline_equation", "height": 13, "width": 62}], "index": 4}, {"bbox": [125, 175, 486, 187], "spans": [{"bbox": [125, 175, 179, 187], "score": 1.0, "content": "we see that ", "type": "text"}, {"bbox": [180, 176, 209, 187], "score": 0.93, "content": "c_{3}^{y}\\equiv1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [209, 175, 233, 187], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [233, 177, 245, 186], "score": 0.91, "content": "G_{4}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [246, 175, 264, 187], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [264, 180, 270, 186], "score": 0.89, "content": "_y", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 175, 343, 187], "score": 1.0, "content": " were odd, then ", "type": "text"}, {"bbox": [343, 177, 378, 186], "score": 0.93, "content": "c_{3}\\in G_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 175, 486, 187], "score": 1.0, "content": ". This, however, implies", "type": "text"}], "index": 5}, {"bbox": [124, 186, 486, 200], "spans": [{"bbox": [124, 186, 147, 200], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 188, 191, 199], "score": 0.94, "content": "G_{2}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [192, 186, 356, 200], "score": 1.0, "content": ", contrary to our assumptions. Thus ", "type": "text"}, {"bbox": [356, 191, 362, 198], "score": 0.88, "content": "_y", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [362, 186, 419, 200], "score": 1.0, "content": " is even, say ", "type": "text"}, {"bbox": [419, 189, 453, 198], "score": 0.93, "content": "y=2y_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [454, 186, 486, 200], "score": 1.0, "content": ". From", "type": "text"}], "index": 6}, {"bbox": [123, 196, 487, 215], "spans": [{"bbox": [123, 197, 265, 214], "score": 1.0, "content": "all of this we see that c4 = ", "type": "text"}, {"bbox": [228, 199, 291, 212], "score": 0.94, "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [253, 196, 487, 215], "score": 1.0, "content": "c22κx1c23y1. Consequently, by induction we have cj ∈", "type": "text"}], "index": 7}, {"bbox": [123, 208, 485, 232], "spans": [{"bbox": [123, 208, 345, 232], "score": 1.0, "content": "⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j−", "type": "text"}, {"bbox": [272, 212, 449, 227], "score": 0.92, "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "type": "inline_equation", "height": 15, "width": 177}, {"bbox": [450, 213, 485, 227], "score": 1.0, "content": ", cf. [1],", "type": "text"}], "index": 8}, {"bbox": [125, 226, 486, 238], "spans": [{"bbox": [125, 226, 221, 238], "score": 1.0, "content": "we obtain the lemma.", "type": "text"}, {"bbox": [475, 226, 486, 236], "score": 0.9939806461334229, "content": "口", "type": "text"}], "index": 9}], "index": 4.5, "bbox_fs": [123, 114, 489, 238]}, {"type": "text", "bbox": [125, 246, 487, 366], "lines": [{"bbox": [137, 248, 485, 261], "spans": [{"bbox": [137, 248, 420, 261], "score": 1.0, "content": "Let us translate the above into the field-theoretic language. Let ", "type": "text"}, {"bbox": [421, 250, 426, 257], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [427, 248, 485, 261], "score": 1.0, "content": " be an imagi-", "type": "text"}], "index": 10}, {"bbox": [124, 261, 485, 272], "spans": [{"bbox": [124, 261, 464, 272], "score": 1.0, "content": "nary quadratic number field of type A) or B) (see the Introduction), and let ", "type": "text"}, {"bbox": [464, 262, 485, 272], "score": 0.92, "content": "M/k", "type": "inline_equation", "height": 10, "width": 21}], "index": 11}, {"bbox": [125, 273, 487, 284], "spans": [{"bbox": [125, 273, 337, 284], "score": 1.0, "content": "be one of the two quadratic subextensions of ", "type": "text"}, {"bbox": [337, 273, 358, 284], "score": 0.94, "content": "k^{1}/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [358, 273, 415, 284], "score": 1.0, "content": " over which ", "type": "text"}, {"bbox": [415, 273, 425, 281], "score": 0.9, "content": "k^{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [426, 273, 487, 284], "score": 1.0, "content": " is cyclic. If", "type": "text"}], "index": 12}, {"bbox": [126, 282, 487, 298], "spans": [{"bbox": [126, 285, 191, 296], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [192, 282, 214, 298], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 285, 285, 296], "score": 0.93, "content": "\\mathrm{Cl}_{2}(k)=(2,2^{m})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [286, 282, 413, 298], "score": 1.0, "content": ", then Lemma 2 implies that ", "type": "text"}, {"bbox": [413, 285, 446, 296], "score": 0.94, "content": "\\mathrm{Cl_{2}}(k^{1})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [446, 282, 487, 298], "score": 1.0, "content": " contains", "type": "text"}], "index": 13}, {"bbox": [126, 296, 486, 308], "spans": [{"bbox": [126, 296, 218, 308], "score": 1.0, "content": "an element of order ", "type": "text"}, {"bbox": [218, 298, 228, 305], "score": 0.88, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [228, 296, 486, 308], "score": 1.0, "content": ". Table 2 contains the relevant information for the fields", "type": "text"}], "index": 14}, {"bbox": [125, 307, 486, 321], "spans": [{"bbox": [125, 307, 424, 321], "score": 1.0, "content": "occurring in Table 1. An application of the class number formula to ", "type": "text"}, {"bbox": [424, 309, 447, 320], "score": 0.94, "content": "M/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [447, 307, 486, 321], "score": 1.0, "content": " (see e.g.", "type": "text"}], "index": 15}, {"bbox": [124, 320, 487, 333], "spans": [{"bbox": [124, 320, 325, 333], "score": 1.0, "content": "Proposition 3 below) shows immediately that ", "type": "text"}, {"bbox": [325, 321, 390, 332], "score": 0.93, "content": "h_{2}(M)=2^{m+\\kappa}", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [391, 320, 425, 333], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [425, 322, 435, 329], "score": 0.91, "content": "2^{\\kappa}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [435, 320, 487, 333], "score": 1.0, "content": " is the class", "type": "text"}], "index": 16}, {"bbox": [124, 332, 486, 346], "spans": [{"bbox": [124, 332, 271, 346], "score": 1.0, "content": "number of the quadratic subfield ", "type": "text"}, {"bbox": [271, 332, 316, 345], "score": 0.94, "content": "\\mathbb{Q}({\\sqrt{d_{i}d_{j}}}\\,)", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [317, 332, 330, 346], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [331, 334, 342, 342], "score": 0.89, "content": "M", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [342, 332, 376, 346], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [376, 334, 433, 344], "score": 0.94, "content": "(d_{i}/p_{j})=+1", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [433, 332, 486, 346], "score": 1.0, "content": "; in particu-", "type": "text"}], "index": 17}, {"bbox": [125, 344, 486, 357], "spans": [{"bbox": [125, 344, 213, 357], "score": 1.0, "content": "lar, we always have ", "type": "text"}, {"bbox": [213, 347, 237, 354], "score": 0.92, "content": "\\kappa\\geq2", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [238, 344, 332, 357], "score": 1.0, "content": ", and the assumption ", "type": "text"}, {"bbox": [332, 345, 389, 356], "score": 0.93, "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [390, 344, 486, 357], "score": 1.0, "content": " is always satisfied for", "type": "text"}], "index": 18}, {"bbox": [125, 356, 244, 368], "spans": [{"bbox": [125, 356, 244, 368], "score": 1.0, "content": "the fields that we consider.", "type": "text"}], "index": 19}], "index": 14.5, "bbox_fs": [124, 248, 487, 368]}, {"type": "table", "bbox": [167, 416, 444, 622], "blocks": [{"type": "table_caption", "bbox": [285, 377, 327, 389], "group_id": 0, "lines": [{"bbox": [285, 378, 327, 391], "spans": [{"bbox": [285, 378, 327, 391], "score": 1.0, "content": "Table 2", "type": "text"}], "index": 20}], "index": 20}, {"type": "table_body", "bbox": [167, 416, 444, 622], "group_id": 0, "lines": [{"bbox": [167, 416, 444, 622], "spans": [{"bbox": [167, 416, 444, 622], "score": 0.912, "html": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>", "type": "table", "image_path": "20908e76a81bf659100a97a9d08d0e64405e7d94b7aaa0b27349fad02b929aae.jpg"}]}], "index": 28.5, "virtual_lines": [{"bbox": [167, 416, 444, 429], "spans": [], "index": 21}, {"bbox": [167, 429, 444, 442], "spans": [], "index": 22}, {"bbox": [167, 442, 444, 455], "spans": [], "index": 23}, {"bbox": [167, 455, 444, 468], "spans": [], "index": 24}, {"bbox": [167, 468, 444, 481], "spans": [], "index": 25}, {"bbox": [167, 481, 444, 494], "spans": [], "index": 26}, {"bbox": [167, 494, 444, 507], "spans": [], "index": 27}, {"bbox": [167, 507, 444, 520], "spans": [], "index": 28}, {"bbox": [167, 520, 444, 533], "spans": [], "index": 29}, {"bbox": [167, 533, 444, 546], "spans": [], "index": 30}, {"bbox": [167, 546, 444, 559], "spans": [], "index": 31}, {"bbox": [167, 559, 444, 572], "spans": [], "index": 32}, {"bbox": [167, 572, 444, 585], "spans": [], "index": 33}, {"bbox": [167, 585, 444, 598], "spans": [], "index": 34}, {"bbox": [167, 598, 444, 611], "spans": [], "index": 35}, {"bbox": [167, 611, 444, 624], "spans": [], "index": 36}]}], "index": 24.25}, {"type": "text", "bbox": [137, 643, 450, 656], "lines": [{"bbox": [138, 645, 449, 657], "spans": [{"bbox": [138, 645, 449, 657], "score": 1.0, "content": "We now use the above results to prove the following useful proposition.", "type": "text"}], "index": 37}], "index": 37, "bbox_fs": [138, 645, 449, 657]}, {"type": "text", "bbox": [125, 662, 486, 700], "lines": [{"bbox": [125, 665, 486, 677], "spans": [{"bbox": [125, 665, 220, 677], "score": 1.0, "content": "Proposition 1. Let ", "type": "text"}, {"bbox": [221, 667, 229, 674], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [229, 665, 409, 677], "score": 1.0, "content": " be a nonmetacyclic 2-group such that ", "type": "text"}, {"bbox": [409, 666, 482, 677], "score": 0.92, "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [483, 665, 486, 677], "score": 1.0, "content": ";", "type": "text"}], "index": 38}, {"bbox": [127, 677, 484, 689], "spans": [{"bbox": [127, 677, 157, 689], "score": 1.0, "content": "(hence ", "type": "text"}, {"bbox": [158, 679, 186, 687], "score": 0.83, "content": "m>1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [186, 677, 214, 689], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [215, 679, 224, 686], "score": 0.84, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [225, 677, 247, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [247, 679, 257, 686], "score": 0.85, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [257, 677, 405, 689], "score": 1.0, "content": " be the two maximal subgroups of ", "type": "text"}, {"bbox": [406, 679, 414, 686], "score": 0.88, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [414, 677, 460, 689], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [460, 678, 484, 689], "score": 0.92, "content": "H/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}], "index": 39}, {"bbox": [127, 689, 487, 701], "spans": [{"bbox": [127, 689, 144, 701], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 690, 169, 701], "score": 0.91, "content": "K/G^{\\prime}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [169, 689, 322, 701], "score": 1.0, "content": " are cyclic. Moreover, assume that ", "type": "text"}, {"bbox": [322, 690, 379, 701], "score": 0.91, "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [380, 689, 487, 701], "score": 1.0, "content": " mod 4. Finally, assume", "type": "text"}], "index": 40}, {"bbox": [127, 115, 426, 126], "spans": [{"bbox": [127, 115, 146, 126], "score": 1.0, "content": "that ", "type": "text", "cross_page": true}, {"bbox": [146, 116, 156, 123], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10, "cross_page": true}, {"bbox": [156, 115, 277, 126], "score": 1.0, "content": " is a subgroup of index 4 in", "type": "text", "cross_page": true}, {"bbox": [278, 115, 286, 124], "score": 0.8, "content": "G", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [287, 115, 363, 126], "score": 1.0, "content": " not contained in ", "type": "text", "cross_page": true}, {"bbox": [364, 115, 374, 123], "score": 0.76, "content": "H", "type": "inline_equation", "height": 8, "width": 10, "cross_page": true}, {"bbox": [374, 115, 389, 126], "score": 1.0, "content": " or ", "type": "text", "cross_page": true}, {"bbox": [389, 116, 399, 123], "score": 0.79, "content": "K", "type": "inline_equation", "height": 7, "width": 10, "cross_page": true}, {"bbox": [399, 115, 426, 126], "score": 1.0, "content": " Then", "type": "text", "cross_page": true}], "index": 0}], "index": 39, "bbox_fs": [125, 665, 487, 701]}]}
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[{"type": "text", "bbox": [125, 111, 487, 237], "content": "Proof. Assume that . By Lemma 1, and hence . Write where are positive integers. Without loss of generality, let and write for some . Since we have, mod . By the proof of Lemma 2, this implies that mod . Write for some positive integer . On the other hand, since we see that mod . If were odd, then . This, however, implies that , contrary to our assumptions. Thus is even, say . From all of this we see that c4 = c22κx1c23y1. Consequently, by induction we have cj ∈ ⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− , cf. [1], we obtain the lemma. 口", "index": 0}, {"type": "text", "bbox": [125, 246, 487, 366], "content": "Let us translate the above into the field-theoretic language. Let be an imagi- nary quadratic number field of type A) or B) (see the Introduction), and let be one of the two quadratic subextensions of over which is cyclic. If and , then Lemma 2 implies that contains an element of order . Table 2 contains the relevant information for the fields occurring in Table 1. An application of the class number formula to (see e.g. Proposition 3 below) shows immediately that , where is the class number of the quadratic subfield of , where ; in particu- lar, we always have , and the assumption is always satisfied for the fields that we consider.", "index": 1}, {"type": "table", "bbox": [167, 416, 444, 622], "content": "", "index": 2}, {"type": "text", "bbox": [137, 643, 450, 656], "content": "We now use the above results to prove the following useful proposition.", "index": 3}, {"type": "text", "bbox": [125, 662, 486, 700], "content": "Proposition 1. Let be a nonmetacyclic 2-group such that ; (hence ). Let and be the two maximal subgroups of such that and are cyclic. Moreover, assume that mod 4. Finally, assume that is a subgroup of index 4 in not contained in or Then", "index": 4}]
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[{"bbox": [126, 114, 485, 126], "content": "Proof. Assume that . By Lemma 1, and hence .", "parent_index": 0, "line_index": 0}, {"bbox": [123, 122, 489, 143], "content": "Write where are positive integers. Without loss of generality, let", "parent_index": 0, "line_index": 1}, {"bbox": [126, 136, 487, 152], "content": "and write for some . Since we have,", "parent_index": 0, "line_index": 2}, {"bbox": [126, 150, 487, 162], "content": "mod . By the proof of Lemma 2, this implies that mod . Write", "parent_index": 0, "line_index": 3}, {"bbox": [126, 160, 482, 178], "content": "for some positive integer . On the other hand, since", "parent_index": 0, "line_index": 4}, {"bbox": [125, 175, 486, 187], "content": "we see that mod . If were odd, then . This, however, implies", "parent_index": 0, "line_index": 5}, {"bbox": [124, 186, 486, 200], "content": "that , contrary to our assumptions. Thus is even, say . From", "parent_index": 0, "line_index": 6}, {"bbox": [123, 196, 487, 215], "content": "all of this we see that c4 = c22κx1c23y1. Consequently, by induction we have cj ∈", "parent_index": 0, "line_index": 7}, {"bbox": [123, 208, 485, 232], "content": "⟨c22j−2 , c32j−3 ⟩ for all j ≥ 4. Since Gj = ⟨c22j−2 , c23j− , cf. [1],", "parent_index": 0, "line_index": 8}, {"bbox": [125, 226, 486, 238], "content": "we obtain the lemma. 口", "parent_index": 0, "line_index": 9}, {"bbox": [137, 248, 485, 261], "content": "Let us translate the above into the field-theoretic language. Let be an imagi-", "parent_index": 1, "line_index": 0}, {"bbox": [124, 261, 485, 272], "content": "nary quadratic number field of type A) or B) (see the Introduction), and let", "parent_index": 1, "line_index": 1}, {"bbox": [125, 273, 487, 284], "content": "be one of the two quadratic subextensions of over which is cyclic. If", "parent_index": 1, "line_index": 2}, {"bbox": [126, 282, 487, 298], "content": "and , then Lemma 2 implies that contains", "parent_index": 1, "line_index": 3}, {"bbox": [126, 296, 486, 308], "content": "an element of order . Table 2 contains the relevant information for the fields", "parent_index": 1, "line_index": 4}, {"bbox": [125, 307, 486, 321], "content": "occurring in Table 1. An application of the class number formula to (see e.g.", "parent_index": 1, "line_index": 5}, {"bbox": [124, 320, 487, 333], "content": "Proposition 3 below) shows immediately that , where is the class", "parent_index": 1, "line_index": 6}, {"bbox": [124, 332, 486, 346], "content": "number of the quadratic subfield of , where ; in particu-", "parent_index": 1, "line_index": 7}, {"bbox": [125, 344, 486, 357], "content": "lar, we always have , and the assumption is always satisfied for", "parent_index": 1, "line_index": 8}, {"bbox": [125, 356, 244, 368], "content": "the fields that we consider.", "parent_index": 1, "line_index": 9}, {"bbox": [138, 645, 449, 657], "content": "We now use the above results to prove the following useful proposition.", "parent_index": 3, "line_index": 0}, {"bbox": [125, 665, 486, 677], "content": "Proposition 1. Let be a nonmetacyclic 2-group such that ;", "parent_index": 4, "line_index": 0}, {"bbox": [127, 677, 484, 689], "content": "(hence ). Let and be the two maximal subgroups of such that", "parent_index": 4, "line_index": 1}, {"bbox": [127, 689, 487, 701], "content": "and are cyclic. Moreover, assume that mod 4. Finally, assume", "parent_index": 4, "line_index": 2}, {"bbox": [127, 115, 426, 126], "content": "that is a subgroup of index 4 in not contained in or Then", "parent_index": 4, "line_index": 3}]
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[]
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[{"bbox": [215, 115, 258, 126], "content": "d(G^{\\prime})=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [326, 115, 381, 126], "content": "G_{2}=\\langle c_{2},c_{3}\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [431, 115, 482, 126], "content": "c_{4}\\in\\langle c_{2},c_{3}\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [155, 127, 198, 138], "content": "c_{4}\\:=\\:c_{2}^{x}c_{3}^{y}", "parent_index": 0, "subtype": "inline"}, {"bbox": [232, 131, 248, 137], "content": "x,y", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 139, 187, 150], "content": "H=\\langle b,c_{2},c_{3}\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [235, 139, 297, 150], "content": "(G^{\\prime}:H^{\\prime})=2^{\\kappa}\\,", "parent_index": 0, "subtype": "inline"}, {"bbox": [340, 140, 365, 149], "content": "\\kappa\\geq2", "parent_index": 0, "subtype": "inline"}, {"bbox": [398, 139, 444, 149], "content": "c_{3},c_{4}\\in H^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 152, 155, 162], "content": "c_{2}^{x}\\equiv1", "parent_index": 0, "subtype": "inline"}, {"bbox": [179, 151, 191, 159], "content": "H^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [393, 152, 417, 159], "content": "x\\equiv0", "parent_index": 0, "subtype": "inline"}, {"bbox": [442, 152, 452, 159], "content": "2^{\\kappa}", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 165, 168, 174], "content": "x\\,=\\,2^{\\kappa}x_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [285, 168, 295, 174], "content": "x_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [420, 162, 482, 175], "content": "c_{4},c_{2}^{2^{\\kappa}x_{1}}\\,\\in\\,G_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [180, 176, 209, 187], "content": "c_{3}^{y}\\equiv1", "parent_index": 0, "subtype": "inline"}, {"bbox": [233, 177, 245, 186], "content": "G_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [264, 180, 270, 186], "content": "_y", "parent_index": 0, "subtype": "inline"}, {"bbox": [343, 177, 378, 186], "content": "c_{3}\\in G_{4}", "parent_index": 0, "subtype": "inline"}, {"bbox": [148, 188, 191, 199], "content": "G_{2}=\\langle c_{2}\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [356, 191, 362, 198], "content": "_y", "parent_index": 0, "subtype": "inline"}, {"bbox": [419, 189, 453, 198], "content": "y=2y_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [228, 199, 291, 212], "content": "c_{4}\\,=\\,c_{2}^{2^{\\kappa}x_{1}}c_{3}^{2y_{1}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [272, 212, 449, 227], "content": "G_{j}=\\langle c_{2}^{2^{j-2}},c_{3}^{2^{j-3}},\\cdot\\cdot\\cdot,c_{j-1}^{2},c_{j},c_{j+1},\\cdot\\cdot\\cdot\\rangle", "parent_index": 0, "subtype": "inline"}, {"bbox": [421, 250, 426, 257], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [464, 262, 485, 272], "content": "M/k", "parent_index": 1, "subtype": "inline"}, {"bbox": [337, 273, 358, 284], 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"subtype": "inline"}, {"bbox": [331, 334, 342, 342], "content": "M", "parent_index": 1, "subtype": "inline"}, {"bbox": [376, 334, 433, 344], "content": "(d_{i}/p_{j})=+1", "parent_index": 1, "subtype": "inline"}, {"bbox": [213, 347, 237, 354], "content": "\\kappa\\geq2", "parent_index": 1, "subtype": "inline"}, {"bbox": [332, 345, 389, 356], "content": "\\left(G^{\\prime}:H^{\\prime}\\right)\\geq4", "parent_index": 1, "subtype": "inline"}, {"bbox": [221, 667, 229, 674], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [409, 666, 482, 677], "content": "G/G^{\\prime}\\;\\simeq\\;(2,2^{m})", "parent_index": 4, "subtype": "inline"}, {"bbox": [158, 679, 186, 687], "content": "m>1", "parent_index": 4, "subtype": "inline"}, {"bbox": [215, 679, 224, 686], "content": "H", "parent_index": 4, "subtype": "inline"}, {"bbox": [247, 679, 257, 686], "content": "K", "parent_index": 4, "subtype": "inline"}, {"bbox": [406, 679, 414, 686], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [460, 678, 484, 689], "content": "H/G^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [145, 690, 169, 701], "content": "K/G^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [322, 690, 379, 701], "content": "(G^{\\prime}:H^{\\prime})\\equiv0", "parent_index": 4, "subtype": "inline"}, {"bbox": [146, 116, 156, 123], "content": "N", "parent_index": 4, "subtype": "inline"}, {"bbox": [278, 115, 286, 124], "content": "G", "parent_index": 4, "subtype": "inline"}, {"bbox": [364, 115, 374, 123], "content": "H", "parent_index": 4, "subtype": "inline"}, {"bbox": [389, 116, 399, 123], "content": "K", "parent_index": 4, "subtype": "inline"}]
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[{"bbox": [285, 377, 327, 389], "content": "", "parent_index": 2, "subtype": "caption"}, {"bbox": [167, 416, 444, 622], "content": "<html><body><table><tr><td>M1</td><td>Cl2(M1)</td><td>M2</td><td>Cl2(M2)</td></tr><tr><td>Q(v5, V-7 . 29 ) Q(V2,√-5 · 31)</td><td>(2, 16)</td><td>Q(V5 · 29, √-7)</td><td>(2, 16)</td></tr><tr><td>Q(V13,√-3 ·37)</td><td>(4, 4) (2,16)</td><td>Q(v5,√-2 : 31) Q(V37,√-3. 13</td><td>(2, 16) (2,16)</td></tr><tr><td></td><td></td><td></td><td>(2,16)</td></tr><tr><td>Q(V-11,v5 · 29)</td><td>(2,16)</td><td>Q(V29, √-5 · 11)</td><td>(2, 16)</td></tr><tr><td>Q(V5, V-17 · 19)</td><td>(4, 4)</td><td>Q(v17, V-5 · 19 )</td><td></td></tr><tr><td>Q(V29, √-2 . 7)</td><td>(2,16)</td><td>Q(V2, √-7 . 29)</td><td>(2,16)</td></tr><tr><td>Q(v5 . 89, √-1)</td><td>(4, 4)</td><td>Q(V5, √-89 )</td><td>(2,8)</td></tr><tr><td>Q(V37, √-5 · 11)</td><td>(4, 4)</td><td>Q(v5, √-37 · 11)</td><td>(2,32)</td></tr><tr><td>Q(V53,√-3·13)</td><td>(4, 4)</td><td>Q(V13 · 53, √-3)</td><td>(2,2,4)</td></tr><tr><td>Q(V37, √-2 . 7)</td><td>(2, 16)</td><td>Q(v2,√-7 . 37)</td><td>(2, 16)</td></tr><tr><td>Q(√13,√-2 · 23）</td><td>(4, 4)</td><td>Q(v2,√-13 · 23</td><td>(2, 16)</td></tr></table></body></html>", "parent_index": 2, "subtype": "body"}]
|
|
$$
(N:N^{\prime})\;\left\{\begin{array}{l l}{{=}}&{{2^{m}\;\;\;\;\;i f\,d(G^{\prime})=1}}\\ {{=}}&{{2^{m+1}\;\;\;\;i f\,d(G^{\prime})=2}}\\ {{\geq}}&{{2^{m+2}\;\;\;\;i f\,d(G^{\prime})\geq3}}\end{array}\right..
$$
Proof. Without loss of generality we assume that $G$ is metabelian. Let $G=\langle a,b\rangle$ , where $a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$ mod $G_{3}$ . Also let $H=\langle b,G^{\prime}\rangle$ and $K=\langle a b,G^{\prime}\rangle$ (without loss of generality). Then $N=\langle a b^{2},G^{\prime}\rangle$ or $N=\langle a,b^{4},G^{\prime}\rangle$ .
Suppose that $N=\left\langle a b^{2},G^{\prime}\right\rangle$ .
First assume $d(G^{\prime})=1$ . Then $G^{\prime}=\langle c_{2}\rangle$ and thus $N^{\prime}=\left\langle[a b^{2},c_{2}]\right\rangle$ . But $[a b^{2},c_{2}]=$ $c_{2}^{2}\eta_{4}$ for some $\eta_{4}\,\in\,G_{4}\,=\,\langle c_{2}^{4}\rangle$ (cf. Lemma $^{1}$ of [1]). Hence, $N^{\prime}\,=\,\left\langle c_{2}^{2}\right\rangle$ , and so $\left(G^{\prime}:N^{\prime}\right)=2$ . Since $(N:G^{\prime})=2^{m-1}$ , we get $(N:N^{\prime})=2^{m}$ as desired.
Next, assume that $d(G^{\prime})\;=\;2$ . Then $N\,=\,\langle a b^{2},c_{2},c_{3}\rangle$ by Lemma 1. Notice that $[a b^{2},c_{2}]~=~c_{2}^{2}\eta_{4}$ and $[a b^{2},c_{3}]\;=\;c_{3}^{2}\eta_{5}$ where $\eta_{j}~\in~G_{j}$ for $j~=~4,5$ . Hence $N^{\prime}=\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},N_{3}\rangle$ and so $\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5}\rangle\subseteq N^{\prime}$ . But then $N^{\prime}G_{5}\supseteq\langle c_{2}^{4},c_{3}^{2}\rangle=G_{4}$ by Lemma 3. Therefore, by [5], $N^{\prime}\supseteq G_{4}$ . But notice that $N_{3}\subseteq G_{4}$ . Thus $N^{\prime}=\left\langle c_{2}^{2},c_{3}^{2}\right\rangle$ and so $(G^{\prime}:N^{\prime})=4$ which in turn implies that $(N:N^{\prime})=2^{m+1}$ , as desired.
Finally, assume $d(G^{\prime})\geq3$ . Then $d(G^{\prime}/G_{5})=3$ . Moreover there exists an exact sequence
$$
N/N^{\prime}\longrightarrow(N/G_{5})/(N/G_{5})^{\prime}\longrightarrow1,
$$
and thus $\#N^{\mathrm{ab}}\,\geq\,\#(N/G_{5})^{\mathrm{ab}}$ . Hence it suffices to prove the result for $G_{5}\,=\,1$ which we now assume. $N=\langle a b^{2},c_{2},c_{3},c_{4}\rangle$ and so, arguing as above, we have $N^{\prime}=$ $\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},c_{4}^{2}\eta_{6},N_{3}\rangle\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2},N_{3}\rangle$ , where $\eta_{j}\;\in\;G_{j}$ . But $N_{3}\,=\,\langle[a b^{2},c_{2}^{2}\eta_{4}]\rangle\,=$ $\langle c_{2}^{4}\rangle$ . Therefore, $N^{\prime}\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2}\rangle$ . From this we see that $(G^{\prime}\,:\,N^{\prime})\,=\,8$ and thus $(N:N^{\prime})=2^{m+2}$ as desired.
Now suppose that $N\,=\,\langle a,b^{4},G^{\prime}\rangle$ . Then the proof is essentially the same as above once we notice that $[a,b^{4}]\equiv c_{3}{}^{2}c_{2}{}^{-4}$ mod $G_{5}$ .
This establishes the proposition.
# 3. Number Theoretic Preliminaries
Proposition 2. Let $K/k$ be a quadratic extension, and assume that the class number of $k$ , $h(k)$ , is odd. If $K$ has an unramified cyclic extension M of order 4, then $M/k$ is normal and $\operatorname{Gal}(M/k)\simeq D_{4}$ .
Proof. R´edei and Reichardt [12] proved this for $k=\mathbb{Q}$ ; the general case is analogous.
We shall make extensive use of the class number formula for extensions of type $(2,2)$ :
Proposition 3. Let $K/k$ be a normal quartic extension with Galois group of type $(2,2)$ , and let $k_{j}$ $(j=1,2,3)$ ) denote the quadratic subextensions. Then
$$
h(K)=2^{d-\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},
$$
where $q(K)=(E_{K}:E_{1}E_{2}E_{3})$ denotes the unit index of $K/k$ $(E_{j}$ is the unit group of $k_{j}$ ), $d$ is the number of infinite primes in $k$ that ramify in $K/k$ , $\kappa$ is the $\mathbb{Z}$ -rank of the unit group $E_{k}$ of $k$ , and $\upsilon=0$ except when $K\subseteq k(\sqrt{E_{k}}\,)$ , where $\upsilon=1$ .
Proof. See [10].
|
<html><body>
<div class="equation" data-bbox="219 129 391 169">$$
(N:N^{\prime})\;\left\{\begin{array}{l l}{{=}}&{{2^{m}\;\;\;\;\;i f\,d(G^{\prime})=1}}\\ {{=}}&{{2^{m+1}\;\;\;\;i f\,d(G^{\prime})=2}}\\ {{\geq}}&{{2^{m+2}\;\;\;\;i f\,d(G^{\prime})\geq3}}\end{array}\right..
$$</div>
<p data-bbox="124 171 486 208">Proof. Without loss of generality we assume that $G$ is metabelian. Let $G=\langle a,b\rangle$ , where $a^{2}\equiv b^{2^{\prime\prime\prime}}\equiv1$ mod $G_{3}$ . Also let $H=\langle b,G^{\prime}\rangle$ and $K=\langle a b,G^{\prime}\rangle$ (without loss of generality). Then $N=\langle a b^{2},G^{\prime}\rangle$ or $N=\langle a,b^{4},G^{\prime}\rangle$ . </p>
<p data-bbox="135 209 261 220">Suppose that $N=\left\langle a b^{2},G^{\prime}\right\rangle$ . </p>
<p data-bbox="125 220 487 256">First assume $d(G^{\prime})=1$ . Then $G^{\prime}=\langle c_{2}\rangle$ and thus $N^{\prime}=\left\langle[a b^{2},c_{2}]\right\rangle$ . But $[a b^{2},c_{2}]=$ $c_{2}^{2}\eta_{4}$ for some $\eta_{4}\,\in\,G_{4}\,=\,\langle c_{2}^{4}\rangle$ (cf. Lemma $^{1}$ of [1]). Hence, $N^{\prime}\,=\,\left\langle c_{2}^{2}\right\rangle$ , and so $\left(G^{\prime}:N^{\prime}\right)=2$ . Since $(N:G^{\prime})=2^{m-1}$ , we get $(N:N^{\prime})=2^{m}$ as desired. </p>
<p data-bbox="124 256 486 316">Next, assume that $d(G^{\prime})\;=\;2$ . Then $N\,=\,\langle a b^{2},c_{2},c_{3}\rangle$ by Lemma 1. Notice that $[a b^{2},c_{2}]~=~c_{2}^{2}\eta_{4}$ and $[a b^{2},c_{3}]\;=\;c_{3}^{2}\eta_{5}$ where $\eta_{j}~\in~G_{j}$ for $j~=~4,5$ . Hence $N^{\prime}=\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},N_{3}\rangle$ and so $\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5}\rangle\subseteq N^{\prime}$ . But then $N^{\prime}G_{5}\supseteq\langle c_{2}^{4},c_{3}^{2}\rangle=G_{4}$ by Lemma 3. Therefore, by [5], $N^{\prime}\supseteq G_{4}$ . But notice that $N_{3}\subseteq G_{4}$ . Thus $N^{\prime}=\left\langle c_{2}^{2},c_{3}^{2}\right\rangle$ and so $(G^{\prime}:N^{\prime})=4$ which in turn implies that $(N:N^{\prime})=2^{m+1}$ , as desired. </p>
<p data-bbox="125 316 486 340">Finally, assume $d(G^{\prime})\geq3$ . Then $d(G^{\prime}/G_{5})=3$ . Moreover there exists an exact sequence </p>
<div class="equation" data-bbox="230 348 380 359">$$
N/N^{\prime}\longrightarrow(N/G_{5})/(N/G_{5})^{\prime}\longrightarrow1,
$$</div>
<p data-bbox="124 362 487 422">and thus $\#N^{\mathrm{ab}}\,\geq\,\#(N/G_{5})^{\mathrm{ab}}$ . Hence it suffices to prove the result for $G_{5}\,=\,1$ which we now assume. $N=\langle a b^{2},c_{2},c_{3},c_{4}\rangle$ and so, arguing as above, we have $N^{\prime}=$ $\langle c_{2}^{2}\eta_{4},c_{3}^{2}\eta_{5},c_{4}^{2}\eta_{6},N_{3}\rangle\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2},N_{3}\rangle$ , where $\eta_{j}\;\in\;G_{j}$ . But $N_{3}\,=\,\langle[a b^{2},c_{2}^{2}\eta_{4}]\rangle\,=$ $\langle c_{2}^{4}\rangle$ . Therefore, $N^{\prime}\,=\,\langle c_{2}^{2}\eta_{4},c_{3}^{2}\rangle$ . From this we see that $(G^{\prime}\,:\,N^{\prime})\,=\,8$ and thus $(N:N^{\prime})=2^{m+2}$ as desired. </p>
<p data-bbox="126 423 486 447">Now suppose that $N\,=\,\langle a,b^{4},G^{\prime}\rangle$ . Then the proof is essentially the same as above once we notice that $[a,b^{4}]\equiv c_{3}{}^{2}c_{2}{}^{-4}$ mod $G_{5}$ . </p>
<p data-bbox="137 447 281 459">This establishes the proposition. </p>
<h1 data-bbox="213 471 397 484">3. Number Theoretic Preliminaries </h1>
<p data-bbox="125 489 486 526">Proposition 2. Let $K/k$ be a quadratic extension, and assume that the class number of $k$ , $h(k)$ , is odd. If $K$ has an unramified cyclic extension M of order 4, then $M/k$ is normal and $\operatorname{Gal}(M/k)\simeq D_{4}$ . </p>
<p data-bbox="125 532 486 545">Proof. R´edei and Reichardt [12] proved this for $k=\mathbb{Q}$ ; the general case is analogous. </p>
<p data-bbox="125 564 487 589">We shall make extensive use of the class number formula for extensions of type $(2,2)$ : </p>
<p data-bbox="125 595 486 619">Proposition 3. Let $K/k$ be a normal quartic extension with Galois group of type $(2,2)$ , and let $k_{j}$ $(j=1,2,3)$ ) denote the quadratic subextensions. Then </p>
<div class="equation" data-bbox="203 626 404 639">$$
h(K)=2^{d-\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},
$$</div>
<p data-bbox="125 644 486 681">where $q(K)=(E_{K}:E_{1}E_{2}E_{3})$ denotes the unit index of $K/k$ $(E_{j}$ is the unit group of $k_{j}$ ), $d$ is the number of infinite primes in $k$ that ramify in $K/k$ , $\kappa$ is the $\mathbb{Z}$ -rank of the unit group $E_{k}$ of $k$ , and $\upsilon=0$ except when $K\subseteq k(\sqrt{E_{k}}\,)$ , where $\upsilon=1$ . </p>
<p data-bbox="126 687 192 700">Proof. See [10]. </p>
</body></html>
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0003244v1
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[{"type": "text", "text": "", "page_idx": 4}, {"type": "equation", "text": "$$\n(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Proof. Without loss of generality we assume that $G$ is metabelian. Let $G=\\langle a,b\\rangle$ , where $a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1$ mod $G_{3}$ . Also let $H=\\langle b,G^{\\prime}\\rangle$ and $K=\\langle a b,G^{\\prime}\\rangle$ (without loss of generality). Then $N=\\langle a b^{2},G^{\\prime}\\rangle$ or $N=\\langle a,b^{4},G^{\\prime}\\rangle$ . ", "page_idx": 4}, {"type": "text", "text": "Suppose that $N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle$ . ", "page_idx": 4}, {"type": "text", "text": "First assume $d(G^{\\prime})=1$ . Then $G^{\\prime}=\\langle c_{2}\\rangle$ and thus $N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle$ . But $[a b^{2},c_{2}]=$ $c_{2}^{2}\\eta_{4}$ for some $\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle$ (cf. Lemma $^{1}$ of [1]). Hence, $N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle$ , and so $\\left(G^{\\prime}:N^{\\prime}\\right)=2$ . Since $(N:G^{\\prime})=2^{m-1}$ , we get $(N:N^{\\prime})=2^{m}$ as desired. ", "page_idx": 4}, {"type": "text", "text": "Next, assume that $d(G^{\\prime})\\;=\\;2$ . Then $N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle$ by Lemma 1. Notice that $[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}$ and $[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}$ where $\\eta_{j}~\\in~G_{j}$ for $j~=~4,5$ . Hence $N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle$ and so $\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}$ . But then $N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}$ by Lemma 3. Therefore, by [5], $N^{\\prime}\\supseteq G_{4}$ . But notice that $N_{3}\\subseteq G_{4}$ . Thus $N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle$ and so $(G^{\\prime}:N^{\\prime})=4$ which in turn implies that $(N:N^{\\prime})=2^{m+1}$ , as desired. ", "page_idx": 4}, {"type": "text", "text": "Finally, assume $d(G^{\\prime})\\geq3$ . Then $d(G^{\\prime}/G_{5})=3$ . Moreover there exists an exact sequence ", "page_idx": 4}, {"type": "equation", "text": "$$\nN/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "and thus $\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}$ . Hence it suffices to prove the result for $G_{5}\\,=\\,1$ which we now assume. $N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle$ and so, arguing as above, we have $N^{\\prime}=$ $\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle$ , where $\\eta_{j}\\;\\in\\;G_{j}$ . But $N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=$ $\\langle c_{2}^{4}\\rangle$ . Therefore, $N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle$ . From this we see that $(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8$ and thus $(N:N^{\\prime})=2^{m+2}$ as desired. ", "page_idx": 4}, {"type": "text", "text": "Now suppose that $N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle$ . Then the proof is essentially the same as above once we notice that $[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}$ mod $G_{5}$ . ", "page_idx": 4}, {"type": "text", "text": "This establishes the proposition. ", "page_idx": 4}, {"type": "text", "text": "3. Number Theoretic Preliminaries ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "Proposition 2. Let $K/k$ be a quadratic extension, and assume that the class number of $k$ , $h(k)$ , is odd. If $K$ has an unramified cyclic extension M of order 4, then $M/k$ is normal and $\\operatorname{Gal}(M/k)\\simeq D_{4}$ . ", "page_idx": 4}, {"type": "text", "text": "Proof. R´edei and Reichardt [12] proved this for $k=\\mathbb{Q}$ ; the general case is analogous. ", "page_idx": 4}, {"type": "text", "text": "We shall make extensive use of the class number formula for extensions of type $(2,2)$ : ", "page_idx": 4}, {"type": "text", "text": "Proposition 3. Let $K/k$ be a normal quartic extension with Galois group of type $(2,2)$ , and let $k_{j}$ $(j=1,2,3)$ ) denote the quadratic subextensions. Then ", "page_idx": 4}, {"type": "equation", "text": "$$\nh(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "where $q(K)=(E_{K}:E_{1}E_{2}E_{3})$ denotes the unit index of $K/k$ $(E_{j}$ is the unit group of $k_{j}$ ), $d$ is the number of infinite primes in $k$ that ramify in $K/k$ , $\\kappa$ is the $\\mathbb{Z}$ -rank of the unit group $E_{k}$ of $k$ , and $\\upsilon=0$ except when $K\\subseteq k(\\sqrt{E_{k}}\\,)$ , where $\\upsilon=1$ . ", "page_idx": 4}, {"type": "text", "text": "Proof. See [10]. ", "page_idx": 4}]
|
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 427, 124], "lines": [{"bbox": [127, 115, 426, 126], "spans": [{"bbox": [127, 115, 146, 126], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 116, 156, 123], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [156, 115, 277, 126], "score": 1.0, "content": " is a subgroup of index 4 in", "type": "text"}, {"bbox": [278, 115, 286, 124], "score": 0.8, "content": "G", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [287, 115, 363, 126], "score": 1.0, "content": " not contained in ", "type": "text"}, {"bbox": [364, 115, 374, 123], "score": 0.76, "content": "H", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [374, 115, 389, 126], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [389, 116, 399, 123], "score": 0.79, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [399, 115, 426, 126], "score": 1.0, "content": " Then", "type": "text"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [124, 171, 486, 208], "lines": [{"bbox": [126, 174, 485, 187], "spans": [{"bbox": [126, 174, 344, 187], "score": 1.0, "content": "Proof. Without loss of generality we assume that ", "type": "text"}, {"bbox": [344, 176, 352, 183], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [353, 174, 438, 187], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [439, 175, 482, 186], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [482, 174, 485, 187], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [126, 186, 487, 199], "spans": [{"bbox": [126, 186, 154, 199], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [155, 186, 213, 195], "score": 0.92, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [214, 186, 237, 199], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [238, 188, 250, 197], "score": 0.9, "content": "G_{3}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [250, 186, 295, 199], "score": 1.0, "content": ". Also let ", "type": "text"}, {"bbox": [295, 187, 345, 198], "score": 0.92, "content": "H=\\langle b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [346, 186, 368, 199], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [369, 187, 424, 198], "score": 0.93, "content": "K=\\langle a b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [425, 186, 487, 199], "score": 1.0, "content": "(without loss", "type": "text"}], "index": 3}, {"bbox": [126, 198, 359, 210], "spans": [{"bbox": [126, 198, 217, 210], "score": 1.0, "content": "of generality). Then ", "type": "text"}, {"bbox": [217, 199, 276, 210], "score": 0.95, "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [277, 198, 291, 210], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [292, 199, 356, 210], "score": 0.94, "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [356, 198, 359, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [135, 209, 261, 220], "lines": [{"bbox": [137, 209, 261, 222], "spans": [{"bbox": [137, 209, 198, 221], "score": 1.0, "content": "Suppose that ", "type": "text"}, {"bbox": [198, 211, 258, 222], "score": 0.93, "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [258, 209, 261, 221], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 220, 487, 256], "lines": [{"bbox": [137, 220, 487, 235], "spans": [{"bbox": [137, 220, 194, 235], "score": 1.0, "content": "First assume", "type": "text"}, {"bbox": [195, 223, 236, 234], "score": 0.92, "content": "d(G^{\\prime})=1", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [237, 220, 268, 235], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [268, 223, 308, 234], "score": 0.93, "content": "G^{\\prime}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [309, 220, 349, 235], "score": 1.0, "content": "and thus", "type": "text"}, {"bbox": [350, 222, 416, 234], "score": 0.92, "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [416, 220, 441, 235], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [442, 223, 487, 234], "score": 0.9, "content": "[a b^{2},c_{2}]=", "type": "inline_equation", "height": 11, "width": 45}], "index": 6}, {"bbox": [126, 234, 486, 246], "spans": [{"bbox": [126, 235, 144, 246], "score": 0.92, "content": "c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [145, 234, 190, 246], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [191, 235, 261, 246], "score": 0.92, "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [262, 234, 323, 246], "score": 1.0, "content": "(cf. Lemma ", "type": "text"}, {"bbox": [323, 236, 329, 243], "score": 0.26, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [329, 234, 402, 246], "score": 1.0, "content": " of [1]). Hence, ", "type": "text"}, {"bbox": [403, 235, 448, 246], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [448, 234, 486, 246], "score": 1.0, "content": ", and so", "type": "text"}], "index": 7}, {"bbox": [126, 244, 438, 259], "spans": [{"bbox": [126, 247, 183, 258], "score": 0.92, "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [183, 244, 216, 259], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [216, 247, 288, 258], "score": 0.91, "content": "(N:G^{\\prime})=2^{m-1}", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [288, 244, 325, 259], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [325, 247, 388, 258], "score": 0.91, "content": "(N:N^{\\prime})=2^{m}", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [388, 244, 438, 259], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [124, 256, 486, 316], "lines": [{"bbox": [136, 256, 487, 270], "spans": [{"bbox": [136, 256, 225, 270], "score": 1.0, "content": "Next, assume that ", "type": "text"}, {"bbox": [225, 259, 271, 270], "score": 0.93, "content": "d(G^{\\prime})\\;=\\;2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [272, 256, 309, 270], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [309, 258, 385, 270], "score": 0.93, "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [385, 256, 487, 270], "score": 1.0, "content": "by Lemma 1. Notice", "type": "text"}], "index": 9}, {"bbox": [124, 268, 487, 284], "spans": [{"bbox": [124, 268, 149, 284], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [149, 270, 218, 282], "score": 0.92, "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [219, 268, 244, 284], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 270, 314, 282], "score": 0.92, "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [314, 268, 348, 284], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [349, 272, 387, 282], "score": 0.93, "content": "\\eta_{j}~\\in~G_{j}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [388, 268, 408, 284], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [409, 272, 447, 281], "score": 0.91, "content": "j~=~4,5", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [447, 268, 487, 284], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 10}, {"bbox": [126, 280, 487, 295], "spans": [{"bbox": [126, 283, 218, 294], "score": 0.92, "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [218, 280, 252, 295], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [253, 282, 326, 294], "score": 0.92, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [327, 280, 376, 295], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [376, 282, 471, 294], "score": 0.91, "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [471, 280, 487, 295], "score": 1.0, "content": " by", "type": "text"}], "index": 11}, {"bbox": [124, 293, 485, 307], "spans": [{"bbox": [124, 293, 248, 307], "score": 1.0, "content": "Lemma 3. Therefore, by [5], ", "type": "text"}, {"bbox": [248, 295, 286, 304], "score": 0.92, "content": "N^{\\prime}\\supseteq G_{4}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 293, 360, 307], "score": 1.0, "content": ". But notice that", "type": "text"}, {"bbox": [361, 296, 399, 304], "score": 0.94, "content": "N_{3}\\subseteq G_{4}", "type": "inline_equation", "height": 8, "width": 38}, {"bbox": [399, 293, 429, 307], "score": 1.0, "content": ". Thus", "type": "text"}, {"bbox": [430, 295, 485, 306], "score": 0.93, "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 55}], "index": 12}, {"bbox": [124, 304, 462, 318], "spans": [{"bbox": [124, 304, 157, 318], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [157, 307, 214, 317], "score": 0.93, "content": "(G^{\\prime}:N^{\\prime})=4", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [215, 304, 335, 318], "score": 1.0, "content": " which in turn implies that ", "type": "text"}, {"bbox": [335, 307, 408, 317], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+1}", "type": "inline_equation", "height": 10, "width": 73}, {"bbox": [409, 304, 462, 318], "score": 1.0, "content": ", as desired.", "type": "text"}], "index": 13}], "index": 11}, {"type": "text", "bbox": [125, 316, 486, 340], "lines": [{"bbox": [137, 318, 487, 330], "spans": [{"bbox": [137, 318, 208, 330], "score": 1.0, "content": "Finally, assume ", "type": "text"}, {"bbox": [208, 319, 250, 329], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [251, 318, 284, 330], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [284, 319, 343, 329], "score": 0.94, "content": "d(G^{\\prime}/G_{5})=3", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [344, 318, 487, 330], "score": 1.0, "content": ". Moreover there exists an exact", "type": "text"}], "index": 14}, {"bbox": [125, 331, 167, 343], "spans": [{"bbox": [125, 331, 167, 343], "score": 1.0, "content": "sequence", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 362, 487, 422], "lines": [{"bbox": [125, 363, 485, 378], "spans": [{"bbox": [125, 363, 168, 378], "score": 1.0, "content": "and thus ", "type": "text"}, {"bbox": [169, 366, 263, 377], "score": 0.91, "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [263, 363, 451, 378], "score": 1.0, "content": ". Hence it suffices to prove the result for ", "type": "text"}, {"bbox": [451, 367, 485, 376], "score": 0.92, "content": "G_{5}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 34}], "index": 17}, {"bbox": [126, 376, 487, 390], "spans": [{"bbox": [126, 376, 227, 390], "score": 1.0, "content": "which we now assume. ", "type": "text"}, {"bbox": [227, 378, 311, 389], "score": 0.91, "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [311, 376, 462, 390], "score": 1.0, "content": "and so, arguing as above, we have ", "type": "text"}, {"bbox": [462, 378, 487, 388], "score": 0.87, "content": "N^{\\prime}=", "type": "inline_equation", "height": 10, "width": 25}], "index": 18}, {"bbox": [126, 388, 485, 402], "spans": [{"bbox": [126, 389, 287, 401], "score": 0.9, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [287, 388, 324, 402], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [325, 391, 361, 401], "score": 0.94, "content": "\\eta_{j}\\;\\in\\;G_{j}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [362, 388, 393, 402], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [393, 389, 485, 401], "score": 0.91, "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "type": "inline_equation", "height": 12, "width": 92}], "index": 19}, {"bbox": [126, 399, 487, 415], "spans": [{"bbox": [126, 402, 142, 412], "score": 0.92, "content": "\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [143, 399, 201, 415], "score": 1.0, "content": ". Therefore, ", "type": "text"}, {"bbox": [201, 401, 268, 412], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [268, 399, 379, 415], "score": 1.0, "content": ". From this we see that ", "type": "text"}, {"bbox": [379, 402, 442, 412], "score": 0.9, "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [442, 399, 487, 415], "score": 1.0, "content": " and thus", "type": "text"}], "index": 20}, {"bbox": [126, 410, 250, 426], "spans": [{"bbox": [126, 414, 199, 425], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+2}", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [199, 410, 250, 426], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [126, 423, 486, 447], "lines": [{"bbox": [136, 424, 487, 437], "spans": [{"bbox": [136, 424, 222, 437], "score": 1.0, "content": "Now suppose that ", "type": "text"}, {"bbox": [222, 425, 290, 437], "score": 0.94, "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [290, 424, 487, 437], "score": 1.0, "content": ". Then the proof is essentially the same as", "type": "text"}], "index": 22}, {"bbox": [124, 434, 355, 450], "spans": [{"bbox": [124, 434, 242, 450], "score": 1.0, "content": "above once we notice that ", "type": "text"}, {"bbox": [243, 437, 312, 448], "score": 0.91, "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [313, 434, 337, 450], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [337, 439, 349, 448], "score": 0.91, "content": "G_{5}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 434, 355, 450], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [137, 447, 281, 459], "lines": [{"bbox": [137, 448, 279, 460], "spans": [{"bbox": [137, 448, 279, 460], "score": 1.0, "content": "This establishes the proposition.", "type": "text"}], "index": 24}], "index": 24}, {"type": "title", "bbox": [213, 471, 397, 484], "lines": [{"bbox": [214, 474, 397, 484], "spans": [{"bbox": [214, 474, 397, 484], "score": 1.0, "content": "3. Number Theoretic Preliminaries", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [125, 489, 486, 526], "lines": [{"bbox": [126, 492, 486, 505], "spans": [{"bbox": [126, 492, 218, 505], "score": 1.0, "content": "Proposition 2. Let ", "type": "text"}, {"bbox": [218, 493, 238, 503], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [238, 492, 486, 505], "score": 1.0, "content": " be a quadratic extension, and assume that the class num-", "type": "text"}], "index": 26}, {"bbox": [126, 502, 487, 517], "spans": [{"bbox": [126, 502, 154, 517], "score": 1.0, "content": "ber of ", "type": "text"}, {"bbox": [154, 505, 160, 513], "score": 0.76, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [160, 502, 166, 517], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [166, 505, 185, 515], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [186, 502, 235, 517], "score": 1.0, "content": ", is odd. If ", "type": "text"}, {"bbox": [236, 505, 245, 513], "score": 0.82, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [245, 502, 487, 517], "score": 1.0, "content": " has an unramified cyclic extension M of order 4, then", "type": "text"}], "index": 27}, {"bbox": [126, 515, 288, 528], "spans": [{"bbox": [126, 516, 147, 527], "score": 0.87, "content": "M/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [147, 515, 214, 528], "score": 1.0, "content": " is normal and ", "type": "text"}, {"bbox": [215, 516, 285, 527], "score": 0.93, "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [285, 515, 288, 528], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27}, {"type": "text", "bbox": [125, 532, 486, 545], "lines": [{"bbox": [126, 534, 486, 547], "spans": [{"bbox": [126, 534, 329, 547], "score": 1.0, "content": "Proof. R´edei and Reichardt [12] proved this for ", "type": "text"}, {"bbox": [329, 536, 356, 545], "score": 0.92, "content": "k=\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [356, 534, 486, 547], "score": 1.0, "content": "; the general case is analogous.", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 564, 487, 589], "lines": [{"bbox": [137, 565, 486, 579], "spans": [{"bbox": [137, 565, 486, 579], "score": 1.0, "content": "We shall make extensive use of the class number formula for extensions of type", "type": "text"}], "index": 30}, {"bbox": [126, 578, 153, 591], "spans": [{"bbox": [126, 579, 148, 590], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 578, 153, 591], "score": 1.0, "content": ":", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [125, 595, 486, 619], "lines": [{"bbox": [125, 597, 486, 609], "spans": [{"bbox": [125, 597, 218, 609], "score": 1.0, "content": "Proposition 3. Let", "type": "text"}, {"bbox": [219, 597, 239, 609], "score": 0.87, "content": "K/k", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [239, 597, 486, 609], "score": 1.0, "content": " be a normal quartic extension with Galois group of type", "type": "text"}], "index": 32}, {"bbox": [126, 610, 437, 621], "spans": [{"bbox": [126, 610, 148, 621], "score": 0.91, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 610, 187, 621], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [188, 610, 198, 621], "score": 0.86, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [198, 610, 202, 621], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [202, 610, 248, 621], "score": 0.77, "content": "(j=1,2,3)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [248, 610, 437, 621], "score": 1.0, "content": ") denote the quadratic subextensions. Then", "type": "text"}], "index": 33}], "index": 32.5}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}, {"type": "text", "bbox": [125, 644, 486, 681], "lines": [{"bbox": [127, 646, 487, 658], "spans": [{"bbox": [127, 646, 154, 658], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 647, 255, 658], "score": 0.91, "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "type": "inline_equation", "height": 11, "width": 101}, {"bbox": [256, 646, 370, 658], "score": 1.0, "content": " denotes the unit index of ", "type": "text"}, {"bbox": [371, 647, 390, 658], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [391, 646, 396, 658], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [396, 647, 409, 658], "score": 0.69, "content": "(E_{j}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [410, 646, 487, 658], "score": 1.0, "content": " is the unit group", "type": "text"}], "index": 35}, {"bbox": [126, 658, 487, 670], "spans": [{"bbox": [126, 658, 137, 669], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 659, 147, 670], "score": 0.8, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [147, 658, 157, 669], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [158, 659, 163, 667], "score": 0.8, "content": "d", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [164, 658, 319, 669], "score": 1.0, "content": " is the number of infinite primes in", "type": "text"}, {"bbox": [320, 658, 326, 667], "score": 0.74, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [327, 658, 393, 669], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [393, 659, 413, 669], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [413, 658, 418, 669], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [418, 660, 425, 667], "score": 0.44, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [425, 658, 455, 669], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [455, 659, 462, 667], "score": 0.8, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [463, 658, 487, 669], "score": 1.0, "content": "-rank", "type": "text"}], "index": 36}, {"bbox": [126, 670, 466, 682], "spans": [{"bbox": [126, 670, 202, 682], "score": 1.0, "content": "of the unit group ", "type": "text"}, {"bbox": [202, 671, 215, 681], "score": 0.87, "content": "E_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [215, 670, 230, 682], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [230, 671, 236, 680], "score": 0.77, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [236, 670, 261, 682], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [261, 670, 286, 680], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [286, 670, 344, 682], "score": 1.0, "content": " except when ", "type": "text"}, {"bbox": [344, 670, 403, 682], "score": 0.93, "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [403, 670, 436, 682], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [437, 672, 461, 680], "score": 0.88, "content": "\\upsilon=1", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [462, 670, 466, 682], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36}, {"type": "text", "bbox": [126, 687, 192, 700], "lines": [{"bbox": [126, 687, 194, 702], "spans": [{"bbox": [126, 687, 194, 702], "score": 1.0, "content": "Proof. See [10].", "type": "text"}], "index": 38}], "index": 38}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}], "discarded_blocks": [{"type": "discarded", "bbox": [237, 90, 374, 100], "lines": [{"bbox": [239, 92, 372, 101], "spans": [{"bbox": [239, 92, 372, 101], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 687, 486, 698], "lines": [{"bbox": [475, 689, 487, 700], "spans": [{"bbox": [475, 689, 487, 700], "score": 0.9910128712654114, "content": "口", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 90, 486, 99], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "5", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 447, 486, 458], "lines": [{"bbox": [475, 450, 487, 460], "spans": [{"bbox": [475, 450, 487, 460], "score": 0.9908492565155029, "content": "口", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 427, 124], "lines": [], "index": 0, "bbox_fs": [127, 115, 426, 126], "lines_deleted": true}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "lines": [{"bbox": [219, 129, 391, 169], "spans": [{"bbox": [219, 129, 391, 169], "score": 0.92, "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [124, 171, 486, 208], "lines": [{"bbox": [126, 174, 485, 187], "spans": [{"bbox": [126, 174, 344, 187], "score": 1.0, "content": "Proof. Without loss of generality we assume that ", "type": "text"}, {"bbox": [344, 176, 352, 183], "score": 0.89, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [353, 174, 438, 187], "score": 1.0, "content": " is metabelian. Let ", "type": "text"}, {"bbox": [439, 175, 482, 186], "score": 0.94, "content": "G=\\langle a,b\\rangle", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [482, 174, 485, 187], "score": 1.0, "content": ",", "type": "text"}], "index": 2}, {"bbox": [126, 186, 487, 199], "spans": [{"bbox": [126, 186, 154, 199], "score": 1.0, "content": "where", "type": "text"}, {"bbox": [155, 186, 213, 195], "score": 0.92, "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "type": "inline_equation", "height": 9, "width": 58}, {"bbox": [214, 186, 237, 199], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [238, 188, 250, 197], "score": 0.9, "content": "G_{3}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [250, 186, 295, 199], "score": 1.0, "content": ". Also let ", "type": "text"}, {"bbox": [295, 187, 345, 198], "score": 0.92, "content": "H=\\langle b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [346, 186, 368, 199], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [369, 187, 424, 198], "score": 0.93, "content": "K=\\langle a b,G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [425, 186, 487, 199], "score": 1.0, "content": "(without loss", "type": "text"}], "index": 3}, {"bbox": [126, 198, 359, 210], "spans": [{"bbox": [126, 198, 217, 210], "score": 1.0, "content": "of generality). Then ", "type": "text"}, {"bbox": [217, 199, 276, 210], "score": 0.95, "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [277, 198, 291, 210], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [292, 199, 356, 210], "score": 0.94, "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [356, 198, 359, 210], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3, "bbox_fs": [126, 174, 487, 210]}, {"type": "text", "bbox": [135, 209, 261, 220], "lines": [{"bbox": [137, 209, 261, 222], "spans": [{"bbox": [137, 209, 198, 221], "score": 1.0, "content": "Suppose that ", "type": "text"}, {"bbox": [198, 211, 258, 222], "score": 0.93, "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [258, 209, 261, 221], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [137, 209, 261, 222]}, {"type": "text", "bbox": [125, 220, 487, 256], "lines": [{"bbox": [137, 220, 487, 235], "spans": [{"bbox": [137, 220, 194, 235], "score": 1.0, "content": "First assume", "type": "text"}, {"bbox": [195, 223, 236, 234], "score": 0.92, "content": "d(G^{\\prime})=1", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [237, 220, 268, 235], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [268, 223, 308, 234], "score": 0.93, "content": "G^{\\prime}=\\langle c_{2}\\rangle", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [309, 220, 349, 235], "score": 1.0, "content": "and thus", "type": "text"}, {"bbox": [350, 222, 416, 234], "score": 0.92, "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [416, 220, 441, 235], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [442, 223, 487, 234], "score": 0.9, "content": "[a b^{2},c_{2}]=", "type": "inline_equation", "height": 11, "width": 45}], "index": 6}, {"bbox": [126, 234, 486, 246], "spans": [{"bbox": [126, 235, 144, 246], "score": 0.92, "content": "c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [145, 234, 190, 246], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [191, 235, 261, 246], "score": 0.92, "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [262, 234, 323, 246], "score": 1.0, "content": "(cf. Lemma ", "type": "text"}, {"bbox": [323, 236, 329, 243], "score": 0.26, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [329, 234, 402, 246], "score": 1.0, "content": " of [1]). Hence, ", "type": "text"}, {"bbox": [403, 235, 448, 246], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [448, 234, 486, 246], "score": 1.0, "content": ", and so", "type": "text"}], "index": 7}, {"bbox": [126, 244, 438, 259], "spans": [{"bbox": [126, 247, 183, 258], "score": 0.92, "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "type": "inline_equation", "height": 11, "width": 57}, {"bbox": [183, 244, 216, 259], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [216, 247, 288, 258], "score": 0.91, "content": "(N:G^{\\prime})=2^{m-1}", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [288, 244, 325, 259], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [325, 247, 388, 258], "score": 0.91, "content": "(N:N^{\\prime})=2^{m}", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [388, 244, 438, 259], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 8}], "index": 7, "bbox_fs": [126, 220, 487, 259]}, {"type": "text", "bbox": [124, 256, 486, 316], "lines": [{"bbox": [136, 256, 487, 270], "spans": [{"bbox": [136, 256, 225, 270], "score": 1.0, "content": "Next, assume that ", "type": "text"}, {"bbox": [225, 259, 271, 270], "score": 0.93, "content": "d(G^{\\prime})\\;=\\;2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [272, 256, 309, 270], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [309, 258, 385, 270], "score": 0.93, "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [385, 256, 487, 270], "score": 1.0, "content": "by Lemma 1. Notice", "type": "text"}], "index": 9}, {"bbox": [124, 268, 487, 284], "spans": [{"bbox": [124, 268, 149, 284], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [149, 270, 218, 282], "score": 0.92, "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [219, 268, 244, 284], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 270, 314, 282], "score": 0.92, "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [314, 268, 348, 284], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [349, 272, 387, 282], "score": 0.93, "content": "\\eta_{j}~\\in~G_{j}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [388, 268, 408, 284], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [409, 272, 447, 281], "score": 0.91, "content": "j~=~4,5", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [447, 268, 487, 284], "score": 1.0, "content": ". Hence", "type": "text"}], "index": 10}, {"bbox": [126, 280, 487, 295], "spans": [{"bbox": [126, 283, 218, 294], "score": 0.92, "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [218, 280, 252, 295], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [253, 282, 326, 294], "score": 0.92, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [327, 280, 376, 295], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [376, 282, 471, 294], "score": 0.91, "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [471, 280, 487, 295], "score": 1.0, "content": " by", "type": "text"}], "index": 11}, {"bbox": [124, 293, 485, 307], "spans": [{"bbox": [124, 293, 248, 307], "score": 1.0, "content": "Lemma 3. Therefore, by [5], ", "type": "text"}, {"bbox": [248, 295, 286, 304], "score": 0.92, "content": "N^{\\prime}\\supseteq G_{4}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [286, 293, 360, 307], "score": 1.0, "content": ". But notice that", "type": "text"}, {"bbox": [361, 296, 399, 304], "score": 0.94, "content": "N_{3}\\subseteq G_{4}", "type": "inline_equation", "height": 8, "width": 38}, {"bbox": [399, 293, 429, 307], "score": 1.0, "content": ". Thus", "type": "text"}, {"bbox": [430, 295, 485, 306], "score": 0.93, "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "type": "inline_equation", "height": 11, "width": 55}], "index": 12}, {"bbox": [124, 304, 462, 318], "spans": [{"bbox": [124, 304, 157, 318], "score": 1.0, "content": "and so ", "type": "text"}, {"bbox": [157, 307, 214, 317], "score": 0.93, "content": "(G^{\\prime}:N^{\\prime})=4", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [215, 304, 335, 318], "score": 1.0, "content": " which in turn implies that ", "type": "text"}, {"bbox": [335, 307, 408, 317], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+1}", "type": "inline_equation", "height": 10, "width": 73}, {"bbox": [409, 304, 462, 318], "score": 1.0, "content": ", as desired.", "type": "text"}], "index": 13}], "index": 11, "bbox_fs": [124, 256, 487, 318]}, {"type": "text", "bbox": [125, 316, 486, 340], "lines": [{"bbox": [137, 318, 487, 330], "spans": [{"bbox": [137, 318, 208, 330], "score": 1.0, "content": "Finally, assume ", "type": "text"}, {"bbox": [208, 319, 250, 329], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [251, 318, 284, 330], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [284, 319, 343, 329], "score": 0.94, "content": "d(G^{\\prime}/G_{5})=3", "type": "inline_equation", "height": 10, "width": 59}, {"bbox": [344, 318, 487, 330], "score": 1.0, "content": ". Moreover there exists an exact", "type": "text"}], "index": 14}, {"bbox": [125, 331, 167, 343], "spans": [{"bbox": [125, 331, 167, 343], "score": 1.0, "content": "sequence", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [125, 318, 487, 343]}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "lines": [{"bbox": [230, 348, 380, 359], "spans": [{"bbox": [230, 348, 380, 359], "score": 0.91, "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 362, 487, 422], "lines": [{"bbox": [125, 363, 485, 378], "spans": [{"bbox": [125, 363, 168, 378], "score": 1.0, "content": "and thus ", "type": "text"}, {"bbox": [169, 366, 263, 377], "score": 0.91, "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "type": "inline_equation", "height": 11, "width": 94}, {"bbox": [263, 363, 451, 378], "score": 1.0, "content": ". Hence it suffices to prove the result for ", "type": "text"}, {"bbox": [451, 367, 485, 376], "score": 0.92, "content": "G_{5}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 34}], "index": 17}, {"bbox": [126, 376, 487, 390], "spans": [{"bbox": [126, 376, 227, 390], "score": 1.0, "content": "which we now assume. ", "type": "text"}, {"bbox": [227, 378, 311, 389], "score": 0.91, "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "type": "inline_equation", "height": 11, "width": 84}, {"bbox": [311, 376, 462, 390], "score": 1.0, "content": "and so, arguing as above, we have ", "type": "text"}, {"bbox": [462, 378, 487, 388], "score": 0.87, "content": "N^{\\prime}=", "type": "inline_equation", "height": 10, "width": 25}], "index": 18}, {"bbox": [126, 388, 485, 402], "spans": [{"bbox": [126, 389, 287, 401], "score": 0.9, "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [287, 388, 324, 402], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [325, 391, 361, 401], "score": 0.94, "content": "\\eta_{j}\\;\\in\\;G_{j}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [362, 388, 393, 402], "score": 1.0, "content": ". But ", "type": "text"}, {"bbox": [393, 389, 485, 401], "score": 0.91, "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "type": "inline_equation", "height": 12, "width": 92}], "index": 19}, {"bbox": [126, 399, 487, 415], "spans": [{"bbox": [126, 402, 142, 412], "score": 0.92, "content": "\\langle c_{2}^{4}\\rangle", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [143, 399, 201, 415], "score": 1.0, "content": ". Therefore, ", "type": "text"}, {"bbox": [201, 401, 268, 412], "score": 0.93, "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [268, 399, 379, 415], "score": 1.0, "content": ". From this we see that ", "type": "text"}, {"bbox": [379, 402, 442, 412], "score": 0.9, "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [442, 399, 487, 415], "score": 1.0, "content": " and thus", "type": "text"}], "index": 20}, {"bbox": [126, 410, 250, 426], "spans": [{"bbox": [126, 414, 199, 425], "score": 0.92, "content": "(N:N^{\\prime})=2^{m+2}", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [199, 410, 250, 426], "score": 1.0, "content": " as desired.", "type": "text"}], "index": 21}], "index": 19, "bbox_fs": [125, 363, 487, 426]}, {"type": "text", "bbox": [126, 423, 486, 447], "lines": [{"bbox": [136, 424, 487, 437], "spans": [{"bbox": [136, 424, 222, 437], "score": 1.0, "content": "Now suppose that ", "type": "text"}, {"bbox": [222, 425, 290, 437], "score": 0.94, "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [290, 424, 487, 437], "score": 1.0, "content": ". Then the proof is essentially the same as", "type": "text"}], "index": 22}, {"bbox": [124, 434, 355, 450], "spans": [{"bbox": [124, 434, 242, 450], "score": 1.0, "content": "above once we notice that ", "type": "text"}, {"bbox": [243, 437, 312, 448], "score": 0.91, "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [313, 434, 337, 450], "score": 1.0, "content": " mod ", "type": "text"}, {"bbox": [337, 439, 349, 448], "score": 0.91, "content": "G_{5}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 434, 355, 450], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "bbox_fs": [124, 424, 487, 450]}, {"type": "text", "bbox": [137, 447, 281, 459], "lines": [{"bbox": [137, 448, 279, 460], "spans": [{"bbox": [137, 448, 279, 460], "score": 1.0, "content": "This establishes the proposition.", "type": "text"}], "index": 24}], "index": 24, "bbox_fs": [137, 448, 279, 460]}, {"type": "title", "bbox": [213, 471, 397, 484], "lines": [{"bbox": [214, 474, 397, 484], "spans": [{"bbox": [214, 474, 397, 484], "score": 1.0, "content": "3. Number Theoretic Preliminaries", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [125, 489, 486, 526], "lines": [{"bbox": [126, 492, 486, 505], "spans": [{"bbox": [126, 492, 218, 505], "score": 1.0, "content": "Proposition 2. Let ", "type": "text"}, {"bbox": [218, 493, 238, 503], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [238, 492, 486, 505], "score": 1.0, "content": " be a quadratic extension, and assume that the class num-", "type": "text"}], "index": 26}, {"bbox": [126, 502, 487, 517], "spans": [{"bbox": [126, 502, 154, 517], "score": 1.0, "content": "ber of ", "type": "text"}, {"bbox": [154, 505, 160, 513], "score": 0.76, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [160, 502, 166, 517], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [166, 505, 185, 515], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [186, 502, 235, 517], "score": 1.0, "content": ", is odd. If ", "type": "text"}, {"bbox": [236, 505, 245, 513], "score": 0.82, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [245, 502, 487, 517], "score": 1.0, "content": " has an unramified cyclic extension M of order 4, then", "type": "text"}], "index": 27}, {"bbox": [126, 515, 288, 528], "spans": [{"bbox": [126, 516, 147, 527], "score": 0.87, "content": "M/k", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [147, 515, 214, 528], "score": 1.0, "content": " is normal and ", "type": "text"}, {"bbox": [215, 516, 285, 527], "score": 0.93, "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "type": "inline_equation", "height": 11, "width": 70}, {"bbox": [285, 515, 288, 528], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27, "bbox_fs": [126, 492, 487, 528]}, {"type": "text", "bbox": [125, 532, 486, 545], "lines": [{"bbox": [126, 534, 486, 547], "spans": [{"bbox": [126, 534, 329, 547], "score": 1.0, "content": "Proof. R´edei and Reichardt [12] proved this for ", "type": "text"}, {"bbox": [329, 536, 356, 545], "score": 0.92, "content": "k=\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [356, 534, 486, 547], "score": 1.0, "content": "; the general case is analogous.", "type": "text"}], "index": 29}], "index": 29, "bbox_fs": [126, 534, 486, 547]}, {"type": "text", "bbox": [125, 564, 487, 589], "lines": [{"bbox": [137, 565, 486, 579], "spans": [{"bbox": [137, 565, 486, 579], "score": 1.0, "content": "We shall make extensive use of the class number formula for extensions of type", "type": "text"}], "index": 30}, {"bbox": [126, 578, 153, 591], "spans": [{"bbox": [126, 579, 148, 590], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 578, 153, 591], "score": 1.0, "content": ":", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [126, 565, 486, 591]}, {"type": "text", "bbox": [125, 595, 486, 619], "lines": [{"bbox": [125, 597, 486, 609], "spans": [{"bbox": [125, 597, 218, 609], "score": 1.0, "content": "Proposition 3. Let", "type": "text"}, {"bbox": [219, 597, 239, 609], "score": 0.87, "content": "K/k", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [239, 597, 486, 609], "score": 1.0, "content": " be a normal quartic extension with Galois group of type", "type": "text"}], "index": 32}, {"bbox": [126, 610, 437, 621], "spans": [{"bbox": [126, 610, 148, 621], "score": 0.91, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 610, 187, 621], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [188, 610, 198, 621], "score": 0.86, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [198, 610, 202, 621], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [202, 610, 248, 621], "score": 0.77, "content": "(j=1,2,3)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [248, 610, 437, 621], "score": 1.0, "content": ") denote the quadratic subextensions. Then", "type": "text"}], "index": 33}], "index": 32.5, "bbox_fs": [125, 597, 486, 621]}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "lines": [{"bbox": [203, 626, 404, 639], "spans": [{"bbox": [203, 626, 404, 639], "score": 0.9, "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "type": "interline_equation"}], "index": 34}], "index": 34}, {"type": "text", "bbox": [125, 644, 486, 681], "lines": [{"bbox": [127, 646, 487, 658], "spans": [{"bbox": [127, 646, 154, 658], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 647, 255, 658], "score": 0.91, "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "type": "inline_equation", "height": 11, "width": 101}, {"bbox": [256, 646, 370, 658], "score": 1.0, "content": " denotes the unit index of ", "type": "text"}, {"bbox": [371, 647, 390, 658], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [391, 646, 396, 658], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [396, 647, 409, 658], "score": 0.69, "content": "(E_{j}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [410, 646, 487, 658], "score": 1.0, "content": " is the unit group", "type": "text"}], "index": 35}, {"bbox": [126, 658, 487, 670], "spans": [{"bbox": [126, 658, 137, 669], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 659, 147, 670], "score": 0.8, "content": "k_{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [147, 658, 157, 669], "score": 1.0, "content": "), ", "type": "text"}, {"bbox": [158, 659, 163, 667], "score": 0.8, "content": "d", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [164, 658, 319, 669], "score": 1.0, "content": " is the number of infinite primes in", "type": "text"}, {"bbox": [320, 658, 326, 667], "score": 0.74, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [327, 658, 393, 669], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [393, 659, 413, 669], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [413, 658, 418, 669], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [418, 660, 425, 667], "score": 0.44, "content": "\\kappa", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [425, 658, 455, 669], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [455, 659, 462, 667], "score": 0.8, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [463, 658, 487, 669], "score": 1.0, "content": "-rank", "type": "text"}], "index": 36}, {"bbox": [126, 670, 466, 682], "spans": [{"bbox": [126, 670, 202, 682], "score": 1.0, "content": "of the unit group ", "type": "text"}, {"bbox": [202, 671, 215, 681], "score": 0.87, "content": "E_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [215, 670, 230, 682], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [230, 671, 236, 680], "score": 0.77, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [236, 670, 261, 682], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [261, 670, 286, 680], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [286, 670, 344, 682], "score": 1.0, "content": " except when ", "type": "text"}, {"bbox": [344, 670, 403, 682], "score": 0.93, "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [403, 670, 436, 682], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [437, 672, 461, 680], "score": 0.88, "content": "\\upsilon=1", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [462, 670, 466, 682], "score": 1.0, "content": ".", "type": "text"}], "index": 37}], "index": 36, "bbox_fs": [126, 646, 487, 682]}, {"type": "text", "bbox": [126, 687, 192, 700], "lines": [{"bbox": [126, 687, 194, 702], "spans": [{"bbox": [126, 687, 194, 702], "score": 1.0, "content": "Proof. See [10].", "type": "text"}], "index": 38}], "index": 38, "bbox_fs": [126, 687, 194, 702]}]}
|
[{"type": "text", "bbox": [125, 111, 427, 124], "content": "", "index": 0}, {"type": "interline_equation", "bbox": [219, 129, 391, 169], "content": "", "index": 1}, {"type": "text", "bbox": [124, 171, 486, 208], "content": "Proof. Without loss of generality we assume that is metabelian. Let , where mod . Also let and (without loss of generality). Then or .", "index": 2}, {"type": "text", "bbox": [135, 209, 261, 220], "content": "Suppose that .", "index": 3}, {"type": "text", "bbox": [125, 220, 487, 256], "content": "First assume . Then and thus . But for some (cf. Lemma of [1]). Hence, , and so . Since , we get as desired.", "index": 4}, {"type": "text", "bbox": [124, 256, 486, 316], "content": "Next, assume that . Then by Lemma 1. Notice that and where for . Hence and so . But then by Lemma 3. Therefore, by [5], . But notice that . Thus and so which in turn implies that , as desired.", "index": 5}, {"type": "text", "bbox": [125, 316, 486, 340], "content": "Finally, assume . Then . Moreover there exists an exact sequence", "index": 6}, {"type": "interline_equation", "bbox": [230, 348, 380, 359], "content": "", "index": 7}, {"type": "text", "bbox": [124, 362, 487, 422], "content": "and thus . Hence it suffices to prove the result for which we now assume. and so, arguing as above, we have , where . But . Therefore, . From this we see that and thus as desired.", "index": 8}, {"type": "text", "bbox": [126, 423, 486, 447], "content": "Now suppose that . Then the proof is essentially the same as above once we notice that mod .", "index": 9}, {"type": "text", "bbox": [137, 447, 281, 459], "content": "This establishes the proposition.", "index": 10}, {"type": "title", "bbox": [213, 471, 397, 484], "content": "3. Number Theoretic Preliminaries", "index": 11}, {"type": "text", "bbox": [125, 489, 486, 526], "content": "Proposition 2. Let be a quadratic extension, and assume that the class num- ber of , , is odd. If has an unramified cyclic extension M of order 4, then is normal and .", "index": 12}, {"type": "text", "bbox": [125, 532, 486, 545], "content": "Proof. R´edei and Reichardt [12] proved this for ; the general case is analogous.", "index": 13}, {"type": "text", "bbox": [125, 564, 487, 589], "content": "We shall make extensive use of the class number formula for extensions of type :", "index": 14}, {"type": "text", "bbox": [125, 595, 486, 619], "content": "Proposition 3. Let be a normal quartic extension with Galois group of type , and let ) denote the quadratic subextensions. Then", "index": 15}, {"type": "interline_equation", "bbox": [203, 626, 404, 639], "content": "", "index": 16}, {"type": "text", "bbox": [125, 644, 486, 681], "content": "where denotes the unit index of is the unit group of ), is the number of infinite primes in that ramify in , is the -rank of the unit group of , and except when , where .", "index": 17}, {"type": "text", "bbox": [126, 687, 192, 700], "content": "Proof. See [10].", "index": 18}]
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[{"bbox": [126, 174, 485, 187], "content": "Proof. Without loss of generality we assume that is metabelian. Let ,", "parent_index": 2, "line_index": 0}, {"bbox": [126, 186, 487, 199], "content": "where mod . Also let and (without loss", "parent_index": 2, "line_index": 1}, {"bbox": [126, 198, 359, 210], "content": "of generality). Then or .", "parent_index": 2, "line_index": 2}, {"bbox": [137, 209, 261, 222], "content": "Suppose that .", "parent_index": 3, "line_index": 0}, {"bbox": [137, 220, 487, 235], "content": "First assume . Then and thus . But", "parent_index": 4, "line_index": 0}, {"bbox": [126, 234, 486, 246], "content": "for some (cf. Lemma of [1]). Hence, , and so", "parent_index": 4, "line_index": 1}, {"bbox": [126, 244, 438, 259], "content": ". Since , we get as desired.", "parent_index": 4, "line_index": 2}, {"bbox": [136, 256, 487, 270], "content": "Next, assume that . Then by Lemma 1. Notice", "parent_index": 5, "line_index": 0}, {"bbox": [124, 268, 487, 284], "content": "that and where for . Hence", "parent_index": 5, "line_index": 1}, {"bbox": [126, 280, 487, 295], "content": "and so . But then by", "parent_index": 5, "line_index": 2}, {"bbox": [124, 293, 485, 307], "content": "Lemma 3. Therefore, by [5], . But notice that . Thus", "parent_index": 5, "line_index": 3}, {"bbox": [124, 304, 462, 318], "content": "and so which in turn implies that , as desired.", "parent_index": 5, "line_index": 4}, {"bbox": [137, 318, 487, 330], "content": "Finally, assume . Then . Moreover there exists an exact", "parent_index": 6, "line_index": 0}, {"bbox": [125, 331, 167, 343], "content": "sequence", "parent_index": 6, "line_index": 1}, {"bbox": [125, 363, 485, 378], "content": "and thus . Hence it suffices to prove the result for", "parent_index": 8, "line_index": 0}, {"bbox": [126, 376, 487, 390], "content": "which we now assume. and so, arguing as above, we have", "parent_index": 8, "line_index": 1}, {"bbox": [126, 388, 485, 402], "content": ", where . But", "parent_index": 8, "line_index": 2}, {"bbox": [126, 399, 487, 415], "content": ". Therefore, . From this we see that and thus", "parent_index": 8, "line_index": 3}, {"bbox": [126, 410, 250, 426], "content": "as desired.", "parent_index": 8, "line_index": 4}, {"bbox": [136, 424, 487, 437], "content": "Now suppose that . Then the proof is essentially the same as", "parent_index": 9, "line_index": 0}, {"bbox": [124, 434, 355, 450], "content": "above once we notice that mod .", "parent_index": 9, "line_index": 1}, {"bbox": [137, 448, 279, 460], "content": "This establishes the proposition.", "parent_index": 10, "line_index": 0}, {"bbox": [214, 474, 397, 484], "content": "3. Number Theoretic Preliminaries", "parent_index": 11, "line_index": 0}, {"bbox": [126, 492, 486, 505], "content": "Proposition 2. Let be a quadratic extension, and assume that the class num-", "parent_index": 12, "line_index": 0}, {"bbox": [126, 502, 487, 517], "content": "ber of , , is odd. If has an unramified cyclic extension M of order 4, then", "parent_index": 12, "line_index": 1}, {"bbox": [126, 515, 288, 528], "content": "is normal and .", "parent_index": 12, "line_index": 2}, {"bbox": [126, 534, 486, 547], "content": "Proof. R´edei and Reichardt [12] proved this for ; the general case is analogous.", "parent_index": 13, "line_index": 0}, {"bbox": [137, 565, 486, 579], "content": "We shall make extensive use of the class number formula for extensions of type", "parent_index": 14, "line_index": 0}, {"bbox": [126, 578, 153, 591], "content": ":", "parent_index": 14, "line_index": 1}, {"bbox": [125, 597, 486, 609], "content": "Proposition 3. Let be a normal quartic extension with Galois group of type", "parent_index": 15, "line_index": 0}, {"bbox": [126, 610, 437, 621], "content": ", and let ) denote the quadratic subextensions. Then", "parent_index": 15, "line_index": 1}, {"bbox": [127, 646, 487, 658], "content": "where denotes the unit index of is the unit group", "parent_index": 17, "line_index": 0}, {"bbox": [126, 658, 487, 670], "content": "of ), is the number of infinite primes in that ramify in , is the -rank", "parent_index": 17, "line_index": 1}, {"bbox": [126, 670, 466, 682], "content": "of the unit group of , and except when , where .", "parent_index": 17, "line_index": 2}, {"bbox": [126, 687, 194, 702], "content": "Proof. See [10].", "parent_index": 18, "line_index": 0}]
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[]
|
[{"bbox": [219, 129, 391, 169], "content": "(N:N^{\\prime})\\;\\left\\{\\begin{array}{l l}{{=}}&{{2^{m}\\;\\;\\;\\;\\;i f\\,d(G^{\\prime})=1}}\\\\ {{=}}&{{2^{m+1}\\;\\;\\;\\;i f\\,d(G^{\\prime})=2}}\\\\ {{\\geq}}&{{2^{m+2}\\;\\;\\;\\;i f\\,d(G^{\\prime})\\geq3}}\\end{array}\\right..", "parent_index": 1, "subtype": "interline"}, {"bbox": [344, 176, 352, 183], "content": "G", "parent_index": 2, "subtype": "inline"}, {"bbox": [439, 175, 482, 186], "content": "G=\\langle a,b\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [155, 186, 213, 195], "content": "a^{2}\\equiv b^{2^{\\prime\\prime\\prime}}\\equiv1", "parent_index": 2, "subtype": "inline"}, {"bbox": [238, 188, 250, 197], "content": "G_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [295, 187, 345, 198], "content": "H=\\langle b,G^{\\prime}\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [369, 187, 424, 198], "content": "K=\\langle a b,G^{\\prime}\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [217, 199, 276, 210], "content": "N=\\langle a b^{2},G^{\\prime}\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [292, 199, 356, 210], "content": "N=\\langle a,b^{4},G^{\\prime}\\rangle", "parent_index": 2, "subtype": "inline"}, {"bbox": [198, 211, 258, 222], "content": "N=\\left\\langle a b^{2},G^{\\prime}\\right\\rangle", "parent_index": 3, "subtype": "inline"}, {"bbox": [195, 223, 236, 234], "content": "d(G^{\\prime})=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [268, 223, 308, 234], "content": "G^{\\prime}=\\langle c_{2}\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [350, 222, 416, 234], "content": "N^{\\prime}=\\left\\langle[a b^{2},c_{2}]\\right\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [442, 223, 487, 234], "content": "[a b^{2},c_{2}]=", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 235, 144, 246], "content": "c_{2}^{2}\\eta_{4}", "parent_index": 4, "subtype": "inline"}, {"bbox": [191, 235, 261, 246], "content": "\\eta_{4}\\,\\in\\,G_{4}\\,=\\,\\langle c_{2}^{4}\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [323, 236, 329, 243], "content": "^{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [403, 235, 448, 246], "content": "N^{\\prime}\\,=\\,\\left\\langle c_{2}^{2}\\right\\rangle", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 247, 183, 258], "content": "\\left(G^{\\prime}:N^{\\prime}\\right)=2", "parent_index": 4, "subtype": "inline"}, {"bbox": [216, 247, 288, 258], "content": "(N:G^{\\prime})=2^{m-1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [325, 247, 388, 258], "content": "(N:N^{\\prime})=2^{m}", "parent_index": 4, "subtype": "inline"}, {"bbox": [225, 259, 271, 270], "content": "d(G^{\\prime})\\;=\\;2", "parent_index": 5, "subtype": "inline"}, {"bbox": [309, 258, 385, 270], "content": "N\\,=\\,\\langle a b^{2},c_{2},c_{3}\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [149, 270, 218, 282], "content": "[a b^{2},c_{2}]~=~c_{2}^{2}\\eta_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [244, 270, 314, 282], "content": "[a b^{2},c_{3}]\\;=\\;c_{3}^{2}\\eta_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [349, 272, 387, 282], "content": "\\eta_{j}~\\in~G_{j}", "parent_index": 5, "subtype": "inline"}, {"bbox": [409, 272, 447, 281], "content": "j~=~4,5", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 283, 218, 294], "content": "N^{\\prime}=\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},N_{3}\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [253, 282, 326, 294], "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5}\\rangle\\subseteq N^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [376, 282, 471, 294], "content": "N^{\\prime}G_{5}\\supseteq\\langle c_{2}^{4},c_{3}^{2}\\rangle=G_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [248, 295, 286, 304], "content": "N^{\\prime}\\supseteq G_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [361, 296, 399, 304], "content": "N_{3}\\subseteq G_{4}", "parent_index": 5, "subtype": "inline"}, {"bbox": [430, 295, 485, 306], "content": "N^{\\prime}=\\left\\langle c_{2}^{2},c_{3}^{2}\\right\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [157, 307, 214, 317], "content": "(G^{\\prime}:N^{\\prime})=4", "parent_index": 5, "subtype": "inline"}, {"bbox": [335, 307, 408, 317], "content": "(N:N^{\\prime})=2^{m+1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [208, 319, 250, 329], "content": "d(G^{\\prime})\\geq3", "parent_index": 6, "subtype": "inline"}, {"bbox": [284, 319, 343, 329], "content": "d(G^{\\prime}/G_{5})=3", "parent_index": 6, "subtype": "inline"}, {"bbox": [230, 348, 380, 359], "content": "N/N^{\\prime}\\longrightarrow(N/G_{5})/(N/G_{5})^{\\prime}\\longrightarrow1,", "parent_index": 7, "subtype": "interline"}, {"bbox": [169, 366, 263, 377], "content": "\\#N^{\\mathrm{ab}}\\,\\geq\\,\\#(N/G_{5})^{\\mathrm{ab}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [451, 367, 485, 376], "content": "G_{5}\\,=\\,1", "parent_index": 8, "subtype": "inline"}, {"bbox": [227, 378, 311, 389], "content": "N=\\langle a b^{2},c_{2},c_{3},c_{4}\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [462, 378, 487, 388], "content": "N^{\\prime}=", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 389, 287, 401], "content": "\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\eta_{5},c_{4}^{2}\\eta_{6},N_{3}\\rangle\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2},N_{3}\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [325, 391, 361, 401], "content": "\\eta_{j}\\;\\in\\;G_{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [393, 389, 485, 401], "content": "N_{3}\\,=\\,\\langle[a b^{2},c_{2}^{2}\\eta_{4}]\\rangle\\,=", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 402, 142, 412], "content": "\\langle c_{2}^{4}\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [201, 401, 268, 412], "content": "N^{\\prime}\\,=\\,\\langle c_{2}^{2}\\eta_{4},c_{3}^{2}\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [379, 402, 442, 412], "content": "(G^{\\prime}\\,:\\,N^{\\prime})\\,=\\,8", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 414, 199, 425], "content": "(N:N^{\\prime})=2^{m+2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [222, 425, 290, 437], "content": "N\\,=\\,\\langle a,b^{4},G^{\\prime}\\rangle", "parent_index": 9, "subtype": "inline"}, {"bbox": [243, 437, 312, 448], "content": "[a,b^{4}]\\equiv c_{3}{}^{2}c_{2}{}^{-4}", "parent_index": 9, "subtype": "inline"}, {"bbox": [337, 439, 349, 448], "content": "G_{5}", "parent_index": 9, "subtype": "inline"}, {"bbox": [218, 493, 238, 503], "content": "K/k", "parent_index": 12, "subtype": "inline"}, {"bbox": [154, 505, 160, 513], "content": "k", "parent_index": 12, "subtype": "inline"}, {"bbox": [166, 505, 185, 515], "content": "h(k)", "parent_index": 12, "subtype": "inline"}, {"bbox": [236, 505, 245, 513], "content": "K", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 516, 147, 527], "content": "M/k", "parent_index": 12, "subtype": "inline"}, {"bbox": [215, 516, 285, 527], "content": "\\operatorname{Gal}(M/k)\\simeq D_{4}", "parent_index": 12, "subtype": "inline"}, {"bbox": [329, 536, 356, 545], "content": "k=\\mathbb{Q}", "parent_index": 13, "subtype": "inline"}, {"bbox": [126, 579, 148, 590], "content": "(2,2)", "parent_index": 14, "subtype": "inline"}, {"bbox": [219, 597, 239, 609], "content": "K/k", "parent_index": 15, "subtype": "inline"}, {"bbox": [126, 610, 148, 621], "content": "(2,2)", "parent_index": 15, "subtype": "inline"}, {"bbox": [188, 610, 198, 621], "content": "k_{j}", "parent_index": 15, "subtype": "inline"}, {"bbox": [202, 610, 248, 621], "content": "(j=1,2,3)", "parent_index": 15, "subtype": "inline"}, {"bbox": [203, 626, 404, 639], "content": "h(K)=2^{d-\\kappa-2-v}q(K)h(k_{1})h(k_{2})h(k_{3})/h(k)^{2},", "parent_index": 16, "subtype": "interline"}, {"bbox": [154, 647, 255, 658], "content": "q(K)=(E_{K}:E_{1}E_{2}E_{3})", "parent_index": 17, "subtype": "inline"}, {"bbox": [371, 647, 390, 658], "content": "K/k", "parent_index": 17, "subtype": "inline"}, {"bbox": [396, 647, 409, 658], "content": "(E_{j}", "parent_index": 17, "subtype": "inline"}, {"bbox": [137, 659, 147, 670], "content": "k_{j}", "parent_index": 17, "subtype": "inline"}, {"bbox": [158, 659, 163, 667], "content": "d", "parent_index": 17, "subtype": "inline"}, {"bbox": [320, 658, 326, 667], "content": "k", "parent_index": 17, "subtype": "inline"}, {"bbox": [393, 659, 413, 669], "content": "K/k", "parent_index": 17, "subtype": "inline"}, {"bbox": [418, 660, 425, 667], "content": "\\kappa", "parent_index": 17, "subtype": "inline"}, {"bbox": [455, 659, 462, 667], "content": "\\mathbb{Z}", "parent_index": 17, "subtype": "inline"}, {"bbox": [202, 671, 215, 681], "content": "E_{k}", "parent_index": 17, "subtype": "inline"}, {"bbox": [230, 671, 236, 680], "content": "k", "parent_index": 17, "subtype": "inline"}, {"bbox": [261, 670, 286, 680], "content": "\\upsilon=0", "parent_index": 17, "subtype": "inline"}, {"bbox": [344, 670, 403, 682], "content": "K\\subseteq k(\\sqrt{E_{k}}\\,)", "parent_index": 17, "subtype": "inline"}, {"bbox": [437, 672, 461, 680], "content": "\\upsilon=1", "parent_index": 17, "subtype": "inline"}]
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Another important result is the ambiguous class number formula. For cyclic extensions $K/k$ , let $\operatorname{Am}(K/k)$ denote the group of ideal classes in $K$ fixed by $\operatorname{Gal}(K/k)$ , i.e. the ambiguous ideal class group of $K$ , and $\mathrm{{Am}_{2}}$ its 2-Sylow subgroup.
Proposition 4. Let $K/k$ be a cyclic extension of prime degree $p$ ; then the number of ambiguous ideal classes is given by
$$
\#\operatorname{Am}(K/k)=h(k)\,{\frac{p^{t-1}}{(E:H)}},
$$
where $t$ is the number of primes (including those at $\infty$ ) of $k$ that ramify in $K/k$ , $E$ is the unit group of $k$ , and $H$ is its subgroup consisting of norms of elements from $K^{\times}$ . Moreover, $\mathrm{Cl}_{p}(K)$ is trivial if and only if $p\nmid\#\operatorname{Am}(K/k)$ .
Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $\operatorname{Am}(K/k)$ is defined by the exact sequence
$$
1\;\longrightarrow\;\mathrm{Am}(K/k)\;\longrightarrow\;\mathrm{Cl}(K)\;\longrightarrow\;\mathrm{Cl}(K)^{1-\sigma}\;\longrightarrow\;1,
$$
where $\sigma$ generates $\operatorname{Gal}(K/k)$ . Taking $p$ -parts we see that $p\nmid\#\operatorname{Am}(K/k)$ is equivalent to $\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{1-\sigma}$ . By induction we get $\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{(1-\sigma)^{\nu}}$ , but since $(1-\sigma)^{p}\equiv0$ mod $p$ in the group ring $\mathbb{Z}[G]$ , this implies $\operatorname{Cl}_{p}(K)\subseteq\operatorname{Cl}_{p}(K)^{p}$ . But then $\mathrm{Cl}_{p}(K)$ must be trivial. 口
We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $h(k)$ is odd, then it is known that $\#\operatorname{Am}_{2}(K/k)=2^{r}$ where $r=\mathrm{rank}\,\mathrm{Cl}_{2}(K)$ .
We also need a result essentially due to G. Gras [4]:
Proposition 5. Let $K/k$ be a quadratic extension of number fields and assume that $h_{2}(k)=\#\operatorname{Am}_{2}(K/k)=2$ . Then $K/k$ is ramified and
$$
\operatorname{Cl}_{2}(K)\simeq\left\{\!\!\begin{array}{l l}{{(2,2)~o r~\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq3)~}}&{{i f\#\kappa_{K/k}=1,}}\\ {{\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq1)~}}&{{i f\#\kappa_{K/k}=2,}}\end{array}\!\!\right.
$$
where $\kappa_{K/k}$ denotes the set of ideal classes of $k$ that become principal (capitulate) in $K$ .
Proof. We first notice that $K/k$ is ramified. If the extension were unramified, then $K$ would be the 2-class field of $k$ , and since $\mathrm{Cl_{2}}(k)$ is cyclic, it would follow that $\mathrm{Cl}_{2}(K)=1$ , contrary to assumption.
Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $K/k$ be a cyclic extension of prime power order $p^{r}$ , and let $\sigma$ be a generator of $G\,=\,\operatorname{Gal}(K/k)$ . For any $p$ -group $M$ on which $G$ acts we put $M_{i}\,=\,\{m\,\in\,M\,:\,m^{(1-\sigma)^{i}}\,=\,1\}$ . Moreover, let $\nu$ be the algebraic norm, that is, exponentiation by $1+\sigma+\sigma^{2}+...+\sigma^{p^{\intercal}-1}$ . Then [4, Cor. 4.3] reads
Lemma 4. Suppose that $M^{\nu}=1$ ; let $n$ be the smallest positive integer such that $M_{n}\,=\,M$ and write $n=a(p-1)+b$ with integers $a\geq0$ and $0\,\leq\,b\,\leq\,p\,-\,2$ . If $\#\,M_{i+1}/M_{i}=p$ for $i=0,1,\dots,n-1$ , then $M\simeq(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}$ .
We claim that if $\kappa_{K/k}~=~2$ , then $M\ =\ \mathrm{Cl}_{2}(K)$ satisfies the assumptions of Lemma 4: in fact, let $j=j_{k\rightarrow K}$ denote the transfer of ideal classes. Then $c^{1+\sigma}=$ $j(N_{K/k}c)$ for any ideal class $c\,\in\,\mathrm{Cl}_{2}(K)$ , hence $M^{\nu}=j(\mathrm{Cl}_{2}(k))\,=\,1$ . Moreover,
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<html><body>
<p data-bbox="124 111 487 149">Another important result is the ambiguous class number formula. For cyclic extensions $K/k$ , let $\operatorname{Am}(K/k)$ denote the group of ideal classes in $K$ fixed by $\operatorname{Gal}(K/k)$ , i.e. the ambiguous ideal class group of $K$ , and $\mathrm{{Am}_{2}}$ its 2-Sylow subgroup. </p>
<p data-bbox="125 154 487 178">Proposition 4. Let $K/k$ be a cyclic extension of prime degree $p$ ; then the number of ambiguous ideal classes is given by </p>
<div class="equation" data-bbox="242 183 367 209">$$
\#\operatorname{Am}(K/k)=h(k)\,{\frac{p^{t-1}}{(E:H)}},
$$</div>
<p data-bbox="125 211 486 248">where $t$ is the number of primes (including those at $\infty$ ) of $k$ that ramify in $K/k$ , $E$ is the unit group of $k$ , and $H$ is its subgroup consisting of norms of elements from $K^{\times}$ . Moreover, $\mathrm{Cl}_{p}(K)$ is trivial if and only if $p\nmid\#\operatorname{Am}(K/k)$ . </p>
<p data-bbox="126 252 487 277">Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $\operatorname{Am}(K/k)$ is defined by the exact sequence </p>
<div class="equation" data-bbox="162 284 447 295">$$
1\;\longrightarrow\;\mathrm{Am}(K/k)\;\longrightarrow\;\mathrm{Cl}(K)\;\longrightarrow\;\mathrm{Cl}(K)^{1-\sigma}\;\longrightarrow\;1,
$$</div>
<p data-bbox="125 298 487 347">where $\sigma$ generates $\operatorname{Gal}(K/k)$ . Taking $p$ -parts we see that $p\nmid\#\operatorname{Am}(K/k)$ is equivalent to $\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{1-\sigma}$ . By induction we get $\operatorname{Cl}_{p}(K)=\operatorname{Cl}_{p}(K)^{(1-\sigma)^{\nu}}$ , but since $(1-\sigma)^{p}\equiv0$ mod $p$ in the group ring $\mathbb{Z}[G]$ , this implies $\operatorname{Cl}_{p}(K)\subseteq\operatorname{Cl}_{p}(K)^{p}$ . But then $\mathrm{Cl}_{p}(K)$ must be trivial. 口 </p>
<p data-bbox="125 353 486 389">We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $h(k)$ is odd, then it is known that $\#\operatorname{Am}_{2}(K/k)=2^{r}$ where $r=\mathrm{rank}\,\mathrm{Cl}_{2}(K)$ . </p>
<p data-bbox="137 393 365 406">We also need a result essentially due to G. Gras [4]: </p>
<p data-bbox="125 411 486 436">Proposition 5. Let $K/k$ be a quadratic extension of number fields and assume that $h_{2}(k)=\#\operatorname{Am}_{2}(K/k)=2$ . Then $K/k$ is ramified and </p>
<div class="equation" data-bbox="189 441 421 473">$$
\operatorname{Cl}_{2}(K)\simeq\left\{\!\!\begin{array}{l l}{{(2,2)~o r~\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq3)~}}&{{i f\#\kappa_{K/k}=1,}}\\ {{\mathbb{Z}/2^{n}\mathbb{Z}~(n\geq1)~}}&{{i f\#\kappa_{K/k}=2,}}\end{array}\!\!\right.
$$</div>
<p data-bbox="126 475 486 499">where $\kappa_{K/k}$ denotes the set of ideal classes of $k$ that become principal (capitulate) in $K$ . </p>
<p data-bbox="125 504 486 541">Proof. We first notice that $K/k$ is ramified. If the extension were unramified, then $K$ would be the 2-class field of $k$ , and since $\mathrm{Cl_{2}}(k)$ is cyclic, it would follow that $\mathrm{Cl}_{2}(K)=1$ , contrary to assumption. </p>
<p data-bbox="125 542 487 615">Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $K/k$ be a cyclic extension of prime power order $p^{r}$ , and let $\sigma$ be a generator of $G\,=\,\operatorname{Gal}(K/k)$ . For any $p$ -group $M$ on which $G$ acts we put $M_{i}\,=\,\{m\,\in\,M\,:\,m^{(1-\sigma)^{i}}\,=\,1\}$ . Moreover, let $\nu$ be the algebraic norm, that is, exponentiation by $1+\sigma+\sigma^{2}+...+\sigma^{p^{\intercal}-1}$ . Then [4, Cor. 4.3] reads </p>
<p data-bbox="124 619 487 658">Lemma 4. Suppose that $M^{\nu}=1$ ; let $n$ be the smallest positive integer such that $M_{n}\,=\,M$ and write $n=a(p-1)+b$ with integers $a\geq0$ and $0\,\leq\,b\,\leq\,p\,-\,2$ . If $\#\,M_{i+1}/M_{i}=p$ for $i=0,1,\dots,n-1$ , then $M\simeq(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}$ . </p>
<p data-bbox="124 662 487 701">We claim that if $\kappa_{K/k}~=~2$ , then $M\ =\ \mathrm{Cl}_{2}(K)$ satisfies the assumptions of Lemma 4: in fact, let $j=j_{k\rightarrow K}$ denote the transfer of ideal classes. Then $c^{1+\sigma}=$ $j(N_{K/k}c)$ for any ideal class $c\,\in\,\mathrm{Cl}_{2}(K)$ , hence $M^{\nu}=j(\mathrm{Cl}_{2}(k))\,=\,1$ . Moreover, </p>
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0003244v1
| 5 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Another important result is the ambiguous class number formula. For cyclic extensions $K/k$ , let $\\operatorname{Am}(K/k)$ denote the group of ideal classes in $K$ fixed by $\\operatorname{Gal}(K/k)$ , i.e. the ambiguous ideal class group of $K$ , and $\\mathrm{{Am}_{2}}$ its 2-Sylow subgroup. ", "page_idx": 5}, {"type": "text", "text": "Proposition 4. Let $K/k$ be a cyclic extension of prime degree $p$ ; then the number of ambiguous ideal classes is given by ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $t$ is the number of primes (including those at $\\infty$ ) of $k$ that ramify in $K/k$ , $E$ is the unit group of $k$ , and $H$ is its subgroup consisting of norms of elements from $K^{\\times}$ . Moreover, $\\mathrm{Cl}_{p}(K)$ is trivial if and only if $p\\nmid\\#\\operatorname{Am}(K/k)$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that $\\operatorname{Am}(K/k)$ is defined by the exact sequence ", "page_idx": 5}, {"type": "equation", "text": "$$\n1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $\\sigma$ generates $\\operatorname{Gal}(K/k)$ . Taking $p$ -parts we see that $p\\nmid\\#\\operatorname{Am}(K/k)$ is equivalent to $\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}$ . By induction we get $\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}$ , but since $(1-\\sigma)^{p}\\equiv0$ mod $p$ in the group ring $\\mathbb{Z}[G]$ , this implies $\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}$ . But then $\\mathrm{Cl}_{p}(K)$ must be trivial. 口 ", "page_idx": 5}, {"type": "text", "text": "We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number $h(k)$ is odd, then it is known that $\\#\\operatorname{Am}_{2}(K/k)=2^{r}$ where $r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)$ . ", "page_idx": 5}, {"type": "text", "text": "We also need a result essentially due to G. Gras [4]: ", "page_idx": 5}, {"type": "text", "text": "Proposition 5. Let $K/k$ be a quadratic extension of number fields and assume that $h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2$ . Then $K/k$ is ramified and ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "where $\\kappa_{K/k}$ denotes the set of ideal classes of $k$ that become principal (capitulate) in $K$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. We first notice that $K/k$ is ramified. If the extension were unramified, then $K$ would be the 2-class field of $k$ , and since $\\mathrm{Cl_{2}}(k)$ is cyclic, it would follow that $\\mathrm{Cl}_{2}(K)=1$ , contrary to assumption. ", "page_idx": 5}, {"type": "text", "text": "Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let $K/k$ be a cyclic extension of prime power order $p^{r}$ , and let $\\sigma$ be a generator of $G\\,=\\,\\operatorname{Gal}(K/k)$ . For any $p$ -group $M$ on which $G$ acts we put $M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}$ . Moreover, let $\\nu$ be the algebraic norm, that is, exponentiation by $1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}$ . Then [4, Cor. 4.3] reads ", "page_idx": 5}, {"type": "text", "text": "Lemma 4. Suppose that $M^{\\nu}=1$ ; let $n$ be the smallest positive integer such that $M_{n}\\,=\\,M$ and write $n=a(p-1)+b$ with integers $a\\geq0$ and $0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2$ . If $\\#\\,M_{i+1}/M_{i}=p$ for $i=0,1,\\dots,n-1$ , then $M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}$ . ", "page_idx": 5}, {"type": "text", "text": "We claim that if $\\kappa_{K/k}~=~2$ , then $M\\ =\\ \\mathrm{Cl}_{2}(K)$ satisfies the assumptions of Lemma 4: in fact, let $j=j_{k\\rightarrow K}$ denote the transfer of ideal classes. Then $c^{1+\\sigma}=$ $j(N_{K/k}c)$ for any ideal class $c\\,\\in\\,\\mathrm{Cl}_{2}(K)$ , hence $M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1$ . Moreover, ", "page_idx": 5}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 111, 487, 149], "lines": [{"bbox": [138, 114, 486, 126], "spans": [{"bbox": [138, 114, 486, 126], "score": 1.0, "content": "Another important result is the ambiguous class number formula. For cyclic", "type": "text"}], "index": 0}, {"bbox": [126, 127, 485, 138], "spans": [{"bbox": [126, 127, 175, 138], "score": 1.0, "content": "extensions ", "type": "text"}, {"bbox": [175, 127, 195, 138], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [195, 127, 218, 138], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [219, 127, 262, 138], "score": 0.92, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [262, 127, 434, 138], "score": 1.0, "content": " denote the group of ideal classes in ", "type": "text"}, {"bbox": [434, 128, 444, 135], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [444, 127, 485, 138], "score": 1.0, "content": " fixed by", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 151], "spans": [{"bbox": [126, 139, 168, 150], "score": 0.89, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [169, 138, 337, 151], "score": 1.0, "content": ", i.e. the ambiguous ideal class group of", "type": "text"}, {"bbox": [338, 140, 347, 147], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [348, 138, 370, 151], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [370, 140, 390, 149], "score": 0.48, "content": "\\mathrm{{Am}_{2}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [391, 138, 486, 151], "score": 1.0, "content": " its 2-Sylow subgroup.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [125, 154, 487, 178], "lines": [{"bbox": [125, 156, 486, 169], "spans": [{"bbox": [125, 156, 218, 169], "score": 1.0, "content": "Proposition 4. Let ", "type": "text"}, {"bbox": [218, 158, 238, 168], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [239, 156, 402, 169], "score": 1.0, "content": " be a cyclic extension of prime degree ", "type": "text"}, {"bbox": [403, 161, 408, 167], "score": 0.86, "content": "p", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [408, 156, 486, 169], "score": 1.0, "content": "; then the number", "type": "text"}], "index": 3}, {"bbox": [126, 168, 291, 181], "spans": [{"bbox": [126, 168, 291, 181], "score": 1.0, "content": "of ambiguous ideal classes is given by", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 211, 486, 248], "lines": [{"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 153, 225], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 215, 158, 222], "score": 0.34, "content": "t", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [158, 213, 350, 225], "score": 1.0, "content": " is the number of primes (including those at ", "type": "text"}, {"bbox": [351, 217, 361, 222], "score": 0.8, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [361, 213, 379, 225], "score": 1.0, "content": ") of ", "type": "text"}, {"bbox": [379, 214, 385, 222], "score": 0.79, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [385, 213, 451, 225], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [451, 214, 471, 225], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [471, 213, 476, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 215, 485, 222], "score": 0.82, "content": "E", "type": "inline_equation", "height": 7, "width": 8}], "index": 6}, {"bbox": [125, 225, 487, 237], "spans": [{"bbox": [125, 225, 213, 237], "score": 1.0, "content": "is the unit group of ", "type": "text"}, {"bbox": [213, 227, 219, 234], "score": 0.85, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [219, 225, 245, 237], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [245, 227, 254, 234], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 225, 487, 237], "score": 1.0, "content": " is its subgroup consisting of norms of elements from", "type": "text"}], "index": 7}, {"bbox": [126, 237, 401, 250], "spans": [{"bbox": [126, 238, 142, 246], "score": 0.89, "content": "K^{\\times}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [142, 237, 196, 250], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [196, 238, 228, 249], "score": 0.94, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [228, 237, 331, 250], "score": 1.0, "content": " is trivial if and only if ", "type": "text"}, {"bbox": [331, 237, 398, 249], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [399, 237, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [126, 252, 487, 277], "lines": [{"bbox": [126, 254, 487, 269], "spans": [{"bbox": [126, 254, 487, 269], "score": 1.0, "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "type": "text"}], "index": 9}, {"bbox": [125, 266, 456, 280], "spans": [{"bbox": [125, 266, 268, 280], "score": 1.0, "content": "(see e.g. Moriya [11]), note that ", "type": "text"}, {"bbox": [268, 268, 311, 279], "score": 0.91, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [312, 266, 456, 280], "score": 1.0, "content": " is defined by the exact sequence", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 298, 487, 347], "lines": [{"bbox": [126, 299, 485, 313], "spans": [{"bbox": [126, 299, 154, 313], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 304, 161, 309], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [162, 299, 208, 313], "score": 1.0, "content": " generates ", "type": "text"}, {"bbox": [208, 301, 251, 312], "score": 0.93, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [252, 299, 291, 313], "score": 1.0, "content": ". Taking ", "type": "text"}, {"bbox": [291, 304, 297, 311], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [297, 299, 378, 313], "score": 1.0, "content": "-parts we see that ", "type": "text"}, {"bbox": [378, 301, 445, 312], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [445, 299, 485, 313], "score": 1.0, "content": " is equiv-", "type": "text"}], "index": 12}, {"bbox": [124, 311, 487, 327], "spans": [{"bbox": [124, 311, 163, 327], "score": 1.0, "content": "alent to ", "type": "text"}, {"bbox": [164, 313, 257, 325], "score": 0.9, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [258, 311, 358, 327], "score": 1.0, "content": ". By induction we get ", "type": "text"}, {"bbox": [359, 313, 463, 325], "score": 0.92, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [463, 311, 487, 327], "score": 1.0, "content": ", but", "type": "text"}], "index": 13}, {"bbox": [125, 324, 485, 338], "spans": [{"bbox": [125, 324, 151, 338], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [151, 325, 206, 336], "score": 0.94, "content": "(1-\\sigma)^{p}\\equiv0", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [207, 324, 230, 338], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [231, 329, 236, 336], "score": 0.84, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [236, 324, 318, 338], "score": 1.0, "content": " in the group ring ", "type": "text"}, {"bbox": [318, 326, 339, 336], "score": 0.93, "content": "\\mathbb{Z}[G]", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [339, 324, 399, 338], "score": 1.0, "content": ", this implies ", "type": "text"}, {"bbox": [399, 326, 482, 336], "score": 0.91, "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "type": "inline_equation", "height": 10, "width": 83}, {"bbox": [483, 324, 485, 338], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [125, 336, 486, 349], "spans": [{"bbox": [125, 336, 168, 349], "score": 1.0, "content": "But then ", "type": "text"}, {"bbox": [168, 338, 200, 348], "score": 0.93, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [200, 336, 271, 349], "score": 1.0, "content": " must be trivial.", "type": "text"}, {"bbox": [476, 338, 486, 347], "score": 0.9896161556243896, "content": "口", "type": "text"}], "index": 15}], "index": 13.5}, {"type": "text", "bbox": [125, 353, 486, 389], "lines": [{"bbox": [125, 355, 486, 367], "spans": [{"bbox": [125, 355, 486, 367], "score": 1.0, "content": "We make one further remark concerning the ambiguous class number formula that", "type": "text"}], "index": 16}, {"bbox": [126, 367, 486, 379], "spans": [{"bbox": [126, 367, 324, 379], "score": 1.0, "content": "will be useful below. If the class number ", "type": "text"}, {"bbox": [324, 368, 343, 379], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [344, 367, 486, 379], "score": 1.0, "content": " is odd, then it is known that", "type": "text"}], "index": 17}, {"bbox": [126, 379, 312, 391], "spans": [{"bbox": [126, 380, 205, 390], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "type": "inline_equation", "height": 10, "width": 79}, {"bbox": [206, 379, 237, 391], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [238, 380, 309, 390], "score": 0.92, "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [309, 379, 312, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17}, {"type": "text", "bbox": [137, 393, 365, 406], "lines": [{"bbox": [137, 395, 364, 408], "spans": [{"bbox": [137, 395, 364, 408], "score": 1.0, "content": "We also need a result essentially due to G. Gras [4]:", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 411, 486, 436], "lines": [{"bbox": [126, 414, 487, 426], "spans": [{"bbox": [126, 414, 219, 426], "score": 1.0, "content": "Proposition 5. Let ", "type": "text"}, {"bbox": [220, 415, 240, 425], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [240, 414, 487, 426], "score": 1.0, "content": " be a quadratic extension of number fields and assume", "type": "text"}], "index": 20}, {"bbox": [126, 426, 382, 437], "spans": [{"bbox": [126, 426, 146, 437], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 427, 259, 437], "score": 0.93, "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 113}, {"bbox": [259, 426, 291, 437], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [292, 426, 312, 437], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [312, 426, 382, 437], "score": 1.0, "content": " is ramified and", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [126, 475, 486, 499], "lines": [{"bbox": [125, 475, 487, 492], "spans": [{"bbox": [125, 475, 154, 492], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 481, 176, 489], "score": 0.88, "content": "\\kappa_{K/k}", "type": "inline_equation", "height": 8, "width": 22}, {"bbox": [176, 475, 329, 492], "score": 1.0, "content": " denotes the set of ideal classes of ", "type": "text"}, {"bbox": [329, 478, 335, 486], "score": 0.81, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [335, 475, 487, 492], "score": 1.0, "content": " that become principal (capitulate)", "type": "text"}], "index": 23}, {"bbox": [126, 489, 152, 500], "spans": [{"bbox": [126, 489, 137, 500], "score": 1.0, "content": "in", "type": "text"}, {"bbox": [138, 491, 148, 498], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [148, 489, 152, 500], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [125, 504, 486, 541], "lines": [{"bbox": [127, 507, 485, 519], "spans": [{"bbox": [127, 507, 245, 519], "score": 1.0, "content": "Proof. We first notice that ", "type": "text"}, {"bbox": [245, 508, 264, 518], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [264, 507, 485, 519], "score": 1.0, "content": " is ramified. If the extension were unramified, then", "type": "text"}], "index": 25}, {"bbox": [126, 518, 486, 531], "spans": [{"bbox": [126, 520, 135, 528], "score": 0.9, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [136, 518, 266, 531], "score": 1.0, "content": " would be the 2-class field of ", "type": "text"}, {"bbox": [267, 520, 272, 528], "score": 0.87, "content": "k", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [273, 518, 323, 531], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [324, 520, 352, 530], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [352, 518, 486, 531], "score": 1.0, "content": " is cyclic, it would follow that", "type": "text"}], "index": 26}, {"bbox": [126, 530, 286, 545], "spans": [{"bbox": [126, 532, 176, 542], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)=1", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [176, 530, 286, 545], "score": 1.0, "content": ", contrary to assumption.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [125, 542, 487, 615], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Before we start with the rest of the proof, we cite the results of Gras that", "type": "text"}], "index": 28}, {"bbox": [126, 555, 486, 568], "spans": [{"bbox": [126, 555, 486, 568], "score": 1.0, "content": "we need (we could also give a slightly longer direct proof without referring to", "type": "text"}], "index": 29}, {"bbox": [125, 565, 484, 580], "spans": [{"bbox": [125, 565, 204, 580], "score": 1.0, "content": "his results). Let ", "type": "text"}, {"bbox": [204, 568, 224, 578], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [224, 565, 425, 580], "score": 1.0, "content": " be a cyclic extension of prime power order ", "type": "text"}, {"bbox": [425, 568, 435, 577], "score": 0.92, "content": "p^{r}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 565, 478, 580], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [478, 571, 484, 576], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}], "index": 30}, {"bbox": [124, 578, 486, 592], "spans": [{"bbox": [124, 578, 207, 592], "score": 1.0, "content": "be a generator of ", "type": "text"}, {"bbox": [207, 579, 275, 590], "score": 0.91, "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [275, 578, 323, 592], "score": 1.0, "content": ". For any ", "type": "text"}, {"bbox": [323, 583, 329, 589], "score": 0.87, "content": "p", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [329, 578, 361, 592], "score": 1.0, "content": "-group ", "type": "text"}, {"bbox": [361, 580, 372, 587], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [372, 578, 420, 592], "score": 1.0, "content": " on which ", "type": "text"}, {"bbox": [420, 580, 428, 588], "score": 0.9, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [429, 578, 486, 592], "score": 1.0, "content": " acts we put", "type": "text"}], "index": 31}, {"bbox": [126, 589, 487, 605], "spans": [{"bbox": [126, 590, 265, 604], "score": 0.9, "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "type": "inline_equation", "height": 14, "width": 139}, {"bbox": [266, 589, 337, 605], "score": 1.0, "content": ". Moreover, let ", "type": "text"}, {"bbox": [338, 596, 344, 601], "score": 0.86, "content": "\\nu", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [344, 589, 487, 605], "score": 1.0, "content": " be the algebraic norm, that is,", "type": "text"}], "index": 32}, {"bbox": [124, 602, 429, 618], "spans": [{"bbox": [124, 602, 207, 618], "score": 1.0, "content": "exponentiation by ", "type": "text"}, {"bbox": [207, 604, 314, 614], "score": 0.9, "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "type": "inline_equation", "height": 10, "width": 107}, {"bbox": [315, 602, 429, 618], "score": 1.0, "content": ". Then [4, Cor. 4.3] reads", "type": "text"}], "index": 33}], "index": 30.5}, {"type": "text", "bbox": [124, 619, 487, 658], "lines": [{"bbox": [124, 622, 487, 635], "spans": [{"bbox": [124, 622, 239, 635], "score": 1.0, "content": "Lemma 4. Suppose that ", "type": "text"}, {"bbox": [239, 623, 275, 632], "score": 0.85, "content": "M^{\\nu}=1", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [276, 622, 296, 635], "score": 1.0, "content": "; let ", "type": "text"}, {"bbox": [297, 627, 303, 631], "score": 0.71, "content": "n", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [303, 622, 487, 635], "score": 1.0, "content": " be the smallest positive integer such that", "type": "text"}], "index": 34}, {"bbox": [126, 634, 487, 647], "spans": [{"bbox": [126, 635, 168, 645], "score": 0.86, "content": "M_{n}\\,=\\,M", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 634, 216, 647], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [217, 635, 292, 646], "score": 0.92, "content": "n=a(p-1)+b", "type": "inline_equation", "height": 11, "width": 75}, {"bbox": [293, 634, 355, 647], "score": 1.0, "content": " with integers ", "type": "text"}, {"bbox": [356, 636, 381, 644], "score": 0.89, "content": "a\\geq0", "type": "inline_equation", "height": 8, "width": 25}, {"bbox": [382, 634, 405, 647], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 636, 469, 645], "score": 0.91, "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "type": "inline_equation", "height": 9, "width": 64}, {"bbox": [469, 634, 487, 647], "score": 1.0, "content": ". If", "type": "text"}], "index": 35}, {"bbox": [125, 645, 472, 660], "spans": [{"bbox": [125, 646, 195, 658], "score": 0.91, "content": "\\#\\,M_{i+1}/M_{i}=p", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [196, 645, 213, 660], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [214, 648, 293, 657], "score": 0.87, "content": "i=0,1,\\dots,n-1", "type": "inline_equation", "height": 9, "width": 79}, {"bbox": [293, 645, 320, 660], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [321, 646, 468, 658], "score": 0.93, "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "type": "inline_equation", "height": 12, "width": 147}, {"bbox": [468, 645, 472, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 35}, {"type": "text", "bbox": [124, 662, 487, 701], "lines": [{"bbox": [137, 663, 487, 677], "spans": [{"bbox": [137, 663, 218, 677], "score": 1.0, "content": "We claim that if ", "type": "text"}, {"bbox": [218, 665, 264, 677], "score": 0.89, "content": "\\kappa_{K/k}~=~2", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 663, 295, 677], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [296, 666, 357, 676], "score": 0.93, "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [357, 663, 487, 677], "score": 1.0, "content": " satisfies the assumptions of", "type": "text"}], "index": 37}, {"bbox": [123, 676, 486, 690], "spans": [{"bbox": [123, 676, 222, 690], "score": 1.0, "content": "Lemma 4: in fact, let", "type": "text"}, {"bbox": [223, 678, 266, 688], "score": 0.91, "content": "j=j_{k\\rightarrow K}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [266, 676, 454, 690], "score": 1.0, "content": " denote the transfer of ideal classes. Then ", "type": "text"}, {"bbox": [454, 677, 486, 687], "score": 0.87, "content": "c^{1+\\sigma}=", "type": "inline_equation", "height": 10, "width": 32}], "index": 38}, {"bbox": [126, 689, 486, 702], "spans": [{"bbox": [126, 690, 167, 702], "score": 0.92, "content": "j(N_{K/k}c)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [167, 690, 254, 702], "score": 1.0, "content": " for any ideal class ", "type": "text"}, {"bbox": [254, 689, 305, 701], "score": 0.92, "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [305, 690, 339, 702], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [340, 690, 432, 701], "score": 0.92, "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [433, 690, 486, 702], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 39}], "index": 38}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [124, 90, 131, 99], "lines": [{"bbox": [126, 93, 132, 101], "spans": [{"bbox": [126, 93, 132, 101], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 487, 149], "lines": [{"bbox": [138, 114, 486, 126], "spans": [{"bbox": [138, 114, 486, 126], "score": 1.0, "content": "Another important result is the ambiguous class number formula. For cyclic", "type": "text"}], "index": 0}, {"bbox": [126, 127, 485, 138], "spans": [{"bbox": [126, 127, 175, 138], "score": 1.0, "content": "extensions ", "type": "text"}, {"bbox": [175, 127, 195, 138], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [195, 127, 218, 138], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [219, 127, 262, 138], "score": 0.92, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [262, 127, 434, 138], "score": 1.0, "content": " denote the group of ideal classes in ", "type": "text"}, {"bbox": [434, 128, 444, 135], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [444, 127, 485, 138], "score": 1.0, "content": " fixed by", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 151], "spans": [{"bbox": [126, 139, 168, 150], "score": 0.89, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [169, 138, 337, 151], "score": 1.0, "content": ", i.e. the ambiguous ideal class group of", "type": "text"}, {"bbox": [338, 140, 347, 147], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [348, 138, 370, 151], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [370, 140, 390, 149], "score": 0.48, "content": "\\mathrm{{Am}_{2}}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [391, 138, 486, 151], "score": 1.0, "content": " its 2-Sylow subgroup.", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [126, 114, 486, 151]}, {"type": "text", "bbox": [125, 154, 487, 178], "lines": [{"bbox": [125, 156, 486, 169], "spans": [{"bbox": [125, 156, 218, 169], "score": 1.0, "content": "Proposition 4. Let ", "type": "text"}, {"bbox": [218, 158, 238, 168], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [239, 156, 402, 169], "score": 1.0, "content": " be a cyclic extension of prime degree ", "type": "text"}, {"bbox": [403, 161, 408, 167], "score": 0.86, "content": "p", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [408, 156, 486, 169], "score": 1.0, "content": "; then the number", "type": "text"}], "index": 3}, {"bbox": [126, 168, 291, 181], "spans": [{"bbox": [126, 168, 291, 181], "score": 1.0, "content": "of ambiguous ideal classes is given by", "type": "text"}], "index": 4}], "index": 3.5, "bbox_fs": [125, 156, 486, 181]}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "lines": [{"bbox": [242, 183, 367, 209], "spans": [{"bbox": [242, 183, 367, 209], "score": 0.94, "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [125, 211, 486, 248], "lines": [{"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 153, 225], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 215, 158, 222], "score": 0.34, "content": "t", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [158, 213, 350, 225], "score": 1.0, "content": " is the number of primes (including those at ", "type": "text"}, {"bbox": [351, 217, 361, 222], "score": 0.8, "content": "\\infty", "type": "inline_equation", "height": 5, "width": 10}, {"bbox": [361, 213, 379, 225], "score": 1.0, "content": ") of ", "type": "text"}, {"bbox": [379, 214, 385, 222], "score": 0.79, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [385, 213, 451, 225], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [451, 214, 471, 225], "score": 0.89, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [471, 213, 476, 225], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [477, 215, 485, 222], "score": 0.82, "content": "E", "type": "inline_equation", "height": 7, "width": 8}], "index": 6}, {"bbox": [125, 225, 487, 237], "spans": [{"bbox": [125, 225, 213, 237], "score": 1.0, "content": "is the unit group of ", "type": "text"}, {"bbox": [213, 227, 219, 234], "score": 0.85, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [219, 225, 245, 237], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [245, 227, 254, 234], "score": 0.85, "content": "H", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [255, 225, 487, 237], "score": 1.0, "content": " is its subgroup consisting of norms of elements from", "type": "text"}], "index": 7}, {"bbox": [126, 237, 401, 250], "spans": [{"bbox": [126, 238, 142, 246], "score": 0.89, "content": "K^{\\times}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [142, 237, 196, 250], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [196, 238, 228, 249], "score": 0.94, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [228, 237, 331, 250], "score": 1.0, "content": " is trivial if and only if ", "type": "text"}, {"bbox": [331, 237, 398, 249], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [399, 237, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7, "bbox_fs": [125, 213, 487, 250]}, {"type": "text", "bbox": [126, 252, 487, 277], "lines": [{"bbox": [126, 254, 487, 269], "spans": [{"bbox": [126, 254, 487, 269], "score": 1.0, "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "type": "text"}], "index": 9}, {"bbox": [125, 266, 456, 280], "spans": [{"bbox": [125, 266, 268, 280], "score": 1.0, "content": "(see e.g. Moriya [11]), note that ", "type": "text"}, {"bbox": [268, 268, 311, 279], "score": 0.91, "content": "\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [312, 266, 456, 280], "score": 1.0, "content": " is defined by the exact sequence", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [125, 254, 487, 280]}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "lines": [{"bbox": [162, 284, 447, 295], "spans": [{"bbox": [162, 284, 447, 295], "score": 0.86, "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 298, 487, 347], "lines": [{"bbox": [126, 299, 485, 313], "spans": [{"bbox": [126, 299, 154, 313], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 304, 161, 309], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [162, 299, 208, 313], "score": 1.0, "content": " generates ", "type": "text"}, {"bbox": [208, 301, 251, 312], "score": 0.93, "content": "\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [252, 299, 291, 313], "score": 1.0, "content": ". Taking ", "type": "text"}, {"bbox": [291, 304, 297, 311], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [297, 299, 378, 313], "score": 1.0, "content": "-parts we see that ", "type": "text"}, {"bbox": [378, 301, 445, 312], "score": 0.9, "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [445, 299, 485, 313], "score": 1.0, "content": " is equiv-", "type": "text"}], "index": 12}, {"bbox": [124, 311, 487, 327], "spans": [{"bbox": [124, 311, 163, 327], "score": 1.0, "content": "alent to ", "type": "text"}, {"bbox": [164, 313, 257, 325], "score": 0.9, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [258, 311, 358, 327], "score": 1.0, "content": ". By induction we get ", "type": "text"}, {"bbox": [359, 313, 463, 325], "score": 0.92, "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [463, 311, 487, 327], "score": 1.0, "content": ", but", "type": "text"}], "index": 13}, {"bbox": [125, 324, 485, 338], "spans": [{"bbox": [125, 324, 151, 338], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [151, 325, 206, 336], "score": 0.94, "content": "(1-\\sigma)^{p}\\equiv0", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [207, 324, 230, 338], "score": 1.0, "content": " mod", "type": "text"}, {"bbox": [231, 329, 236, 336], "score": 0.84, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [236, 324, 318, 338], "score": 1.0, "content": " in the group ring ", "type": "text"}, {"bbox": [318, 326, 339, 336], "score": 0.93, "content": "\\mathbb{Z}[G]", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [339, 324, 399, 338], "score": 1.0, "content": ", this implies ", "type": "text"}, {"bbox": [399, 326, 482, 336], "score": 0.91, "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "type": "inline_equation", "height": 10, "width": 83}, {"bbox": [483, 324, 485, 338], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [125, 336, 486, 349], "spans": [{"bbox": [125, 336, 168, 349], "score": 1.0, "content": "But then ", "type": "text"}, {"bbox": [168, 338, 200, 348], "score": 0.93, "content": "\\mathrm{Cl}_{p}(K)", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [200, 336, 271, 349], "score": 1.0, "content": " must be trivial.", "type": "text"}, {"bbox": [476, 338, 486, 347], "score": 0.9896161556243896, "content": "口", "type": "text"}], "index": 15}], "index": 13.5, "bbox_fs": [124, 299, 487, 349]}, {"type": "text", "bbox": [125, 353, 486, 389], "lines": [{"bbox": [125, 355, 486, 367], "spans": [{"bbox": [125, 355, 486, 367], "score": 1.0, "content": "We make one further remark concerning the ambiguous class number formula that", "type": "text"}], "index": 16}, {"bbox": [126, 367, 486, 379], "spans": [{"bbox": [126, 367, 324, 379], "score": 1.0, "content": "will be useful below. If the class number ", "type": "text"}, {"bbox": [324, 368, 343, 379], "score": 0.92, "content": "h(k)", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [344, 367, 486, 379], "score": 1.0, "content": " is odd, then it is known that", "type": "text"}], "index": 17}, {"bbox": [126, 379, 312, 391], "spans": [{"bbox": [126, 380, 205, 390], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "type": "inline_equation", "height": 10, "width": 79}, {"bbox": [206, 379, 237, 391], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [238, 380, 309, 390], "score": 0.92, "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [309, 379, 312, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17, "bbox_fs": [125, 355, 486, 391]}, {"type": "text", "bbox": [137, 393, 365, 406], "lines": [{"bbox": [137, 395, 364, 408], "spans": [{"bbox": [137, 395, 364, 408], "score": 1.0, "content": "We also need a result essentially due to G. Gras [4]:", "type": "text"}], "index": 19}], "index": 19, "bbox_fs": [137, 395, 364, 408]}, {"type": "text", "bbox": [125, 411, 486, 436], "lines": [{"bbox": [126, 414, 487, 426], "spans": [{"bbox": [126, 414, 219, 426], "score": 1.0, "content": "Proposition 5. Let ", "type": "text"}, {"bbox": [220, 415, 240, 425], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [240, 414, 487, 426], "score": 1.0, "content": " be a quadratic extension of number fields and assume", "type": "text"}], "index": 20}, {"bbox": [126, 426, 382, 437], "spans": [{"bbox": [126, 426, 146, 437], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [146, 427, 259, 437], "score": 0.93, "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 113}, {"bbox": [259, 426, 291, 437], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [292, 426, 312, 437], "score": 0.88, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [312, 426, 382, 437], "score": 1.0, "content": " is ramified and", "type": "text"}], "index": 21}], "index": 20.5, "bbox_fs": [126, 414, 487, 437]}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "lines": [{"bbox": [189, 441, 421, 473], "spans": [{"bbox": [189, 441, 421, 473], "score": 0.92, "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [126, 475, 486, 499], "lines": [{"bbox": [125, 475, 487, 492], "spans": [{"bbox": [125, 475, 154, 492], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 481, 176, 489], "score": 0.88, "content": "\\kappa_{K/k}", "type": "inline_equation", "height": 8, "width": 22}, {"bbox": [176, 475, 329, 492], "score": 1.0, "content": " denotes the set of ideal classes of ", "type": "text"}, {"bbox": [329, 478, 335, 486], "score": 0.81, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [335, 475, 487, 492], "score": 1.0, "content": " that become principal (capitulate)", "type": "text"}], "index": 23}, {"bbox": [126, 489, 152, 500], "spans": [{"bbox": [126, 489, 137, 500], "score": 1.0, "content": "in", "type": "text"}, {"bbox": [138, 491, 148, 498], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [148, 489, 152, 500], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [125, 475, 487, 500]}, {"type": "text", "bbox": [125, 504, 486, 541], "lines": [{"bbox": [127, 507, 485, 519], "spans": [{"bbox": [127, 507, 245, 519], "score": 1.0, "content": "Proof. We first notice that ", "type": "text"}, {"bbox": [245, 508, 264, 518], "score": 0.92, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [264, 507, 485, 519], "score": 1.0, "content": " is ramified. If the extension were unramified, then", "type": "text"}], "index": 25}, {"bbox": [126, 518, 486, 531], "spans": [{"bbox": [126, 520, 135, 528], "score": 0.9, "content": "K", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [136, 518, 266, 531], "score": 1.0, "content": " would be the 2-class field of ", "type": "text"}, {"bbox": [267, 520, 272, 528], "score": 0.87, "content": "k", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [273, 518, 323, 531], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [324, 520, 352, 530], "score": 0.93, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [352, 518, 486, 531], "score": 1.0, "content": " is cyclic, it would follow that", "type": "text"}], "index": 26}, {"bbox": [126, 530, 286, 545], "spans": [{"bbox": [126, 532, 176, 542], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)=1", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [176, 530, 286, 545], "score": 1.0, "content": ", contrary to assumption.", "type": "text"}], "index": 27}], "index": 26, "bbox_fs": [126, 507, 486, 545]}, {"type": "text", "bbox": [125, 542, 487, 615], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Before we start with the rest of the proof, we cite the results of Gras that", "type": "text"}], "index": 28}, {"bbox": [126, 555, 486, 568], "spans": [{"bbox": [126, 555, 486, 568], "score": 1.0, "content": "we need (we could also give a slightly longer direct proof without referring to", "type": "text"}], "index": 29}, {"bbox": [125, 565, 484, 580], "spans": [{"bbox": [125, 565, 204, 580], "score": 1.0, "content": "his results). Let ", "type": "text"}, {"bbox": [204, 568, 224, 578], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [224, 565, 425, 580], "score": 1.0, "content": " be a cyclic extension of prime power order ", "type": "text"}, {"bbox": [425, 568, 435, 577], "score": 0.92, "content": "p^{r}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 565, 478, 580], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [478, 571, 484, 576], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 6}], "index": 30}, {"bbox": [124, 578, 486, 592], "spans": [{"bbox": [124, 578, 207, 592], "score": 1.0, "content": "be a generator of ", "type": "text"}, {"bbox": [207, 579, 275, 590], "score": 0.91, "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [275, 578, 323, 592], "score": 1.0, "content": ". For any ", "type": "text"}, {"bbox": [323, 583, 329, 589], "score": 0.87, "content": "p", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [329, 578, 361, 592], "score": 1.0, "content": "-group ", "type": "text"}, {"bbox": [361, 580, 372, 587], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [372, 578, 420, 592], "score": 1.0, "content": " on which ", "type": "text"}, {"bbox": [420, 580, 428, 588], "score": 0.9, "content": "G", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [429, 578, 486, 592], "score": 1.0, "content": " acts we put", "type": "text"}], "index": 31}, {"bbox": [126, 589, 487, 605], "spans": [{"bbox": [126, 590, 265, 604], "score": 0.9, "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "type": "inline_equation", "height": 14, "width": 139}, {"bbox": [266, 589, 337, 605], "score": 1.0, "content": ". Moreover, let ", "type": "text"}, {"bbox": [338, 596, 344, 601], "score": 0.86, "content": "\\nu", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [344, 589, 487, 605], "score": 1.0, "content": " be the algebraic norm, that is,", "type": "text"}], "index": 32}, {"bbox": [124, 602, 429, 618], "spans": [{"bbox": [124, 602, 207, 618], "score": 1.0, "content": "exponentiation by ", "type": "text"}, {"bbox": [207, 604, 314, 614], "score": 0.9, "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "type": "inline_equation", "height": 10, "width": 107}, {"bbox": [315, 602, 429, 618], "score": 1.0, "content": ". Then [4, Cor. 4.3] reads", "type": "text"}], "index": 33}], "index": 30.5, "bbox_fs": [124, 542, 487, 618]}, {"type": "text", "bbox": [124, 619, 487, 658], "lines": [{"bbox": [124, 622, 487, 635], "spans": [{"bbox": [124, 622, 239, 635], "score": 1.0, "content": "Lemma 4. Suppose that ", "type": "text"}, {"bbox": [239, 623, 275, 632], "score": 0.85, "content": "M^{\\nu}=1", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [276, 622, 296, 635], "score": 1.0, "content": "; let ", "type": "text"}, {"bbox": [297, 627, 303, 631], "score": 0.71, "content": "n", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [303, 622, 487, 635], "score": 1.0, "content": " be the smallest positive integer such that", "type": "text"}], "index": 34}, {"bbox": [126, 634, 487, 647], "spans": [{"bbox": [126, 635, 168, 645], "score": 0.86, "content": "M_{n}\\,=\\,M", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 634, 216, 647], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [217, 635, 292, 646], "score": 0.92, "content": "n=a(p-1)+b", "type": "inline_equation", "height": 11, "width": 75}, {"bbox": [293, 634, 355, 647], "score": 1.0, "content": " with integers ", "type": "text"}, {"bbox": [356, 636, 381, 644], "score": 0.89, "content": "a\\geq0", "type": "inline_equation", "height": 8, "width": 25}, {"bbox": [382, 634, 405, 647], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 636, 469, 645], "score": 0.91, "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "type": "inline_equation", "height": 9, "width": 64}, {"bbox": [469, 634, 487, 647], "score": 1.0, "content": ". If", "type": "text"}], "index": 35}, {"bbox": [125, 645, 472, 660], "spans": [{"bbox": [125, 646, 195, 658], "score": 0.91, "content": "\\#\\,M_{i+1}/M_{i}=p", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [196, 645, 213, 660], "score": 1.0, "content": " for", "type": "text"}, {"bbox": [214, 648, 293, 657], "score": 0.87, "content": "i=0,1,\\dots,n-1", "type": "inline_equation", "height": 9, "width": 79}, {"bbox": [293, 645, 320, 660], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [321, 646, 468, 658], "score": 0.93, "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "type": "inline_equation", "height": 12, "width": 147}, {"bbox": [468, 645, 472, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 35, "bbox_fs": [124, 622, 487, 660]}, {"type": "text", "bbox": [124, 662, 487, 701], "lines": [{"bbox": [137, 663, 487, 677], "spans": [{"bbox": [137, 663, 218, 677], "score": 1.0, "content": "We claim that if ", "type": "text"}, {"bbox": [218, 665, 264, 677], "score": 0.89, "content": "\\kappa_{K/k}~=~2", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 663, 295, 677], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [296, 666, 357, 676], "score": 0.93, "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [357, 663, 487, 677], "score": 1.0, "content": " satisfies the assumptions of", "type": "text"}], "index": 37}, {"bbox": [123, 676, 486, 690], "spans": [{"bbox": [123, 676, 222, 690], "score": 1.0, "content": "Lemma 4: in fact, let", "type": "text"}, {"bbox": [223, 678, 266, 688], "score": 0.91, "content": "j=j_{k\\rightarrow K}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [266, 676, 454, 690], "score": 1.0, "content": " denote the transfer of ideal classes. Then ", "type": "text"}, {"bbox": [454, 677, 486, 687], "score": 0.87, "content": "c^{1+\\sigma}=", "type": "inline_equation", "height": 10, "width": 32}], "index": 38}, {"bbox": [126, 689, 486, 702], "spans": [{"bbox": [126, 690, 167, 702], "score": 0.92, "content": "j(N_{K/k}c)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [167, 690, 254, 702], "score": 1.0, "content": " for any ideal class ", "type": "text"}, {"bbox": [254, 689, 305, 701], "score": 0.92, "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [305, 690, 339, 702], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [340, 690, 432, 701], "score": 0.92, "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [433, 690, 486, 702], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 39}], "index": 38, "bbox_fs": [123, 663, 487, 702]}]}
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[{"type": "text", "bbox": [124, 111, 487, 149], "content": "Another important result is the ambiguous class number formula. For cyclic extensions , let denote the group of ideal classes in fixed by , i.e. the ambiguous ideal class group of , and its 2-Sylow subgroup.", "index": 0}, {"type": "text", "bbox": [125, 154, 487, 178], "content": "Proposition 4. Let be a cyclic extension of prime degree ; then the number of ambiguous ideal classes is given by", "index": 1}, {"type": "interline_equation", "bbox": [242, 183, 367, 209], "content": "", "index": 2}, {"type": "text", "bbox": [125, 211, 486, 248], "content": "where is the number of primes (including those at ) of that ramify in , is the unit group of , and is its subgroup consisting of norms of elements from . Moreover, is trivial if and only if .", "index": 3}, {"type": "text", "bbox": [126, 252, 487, 277], "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion (see e.g. Moriya [11]), note that is defined by the exact sequence", "index": 4}, {"type": "interline_equation", "bbox": [162, 284, 447, 295], "content": "", "index": 5}, {"type": "text", "bbox": [125, 298, 487, 347], "content": "where generates . Taking -parts we see that is equiv- alent to . By induction we get , but since mod in the group ring , this implies . But then must be trivial. 口", "index": 6}, {"type": "text", "bbox": [125, 353, 486, 389], "content": "We make one further remark concerning the ambiguous class number formula that will be useful below. If the class number is odd, then it is known that where .", "index": 7}, {"type": "text", "bbox": [137, 393, 365, 406], "content": "We also need a result essentially due to G. Gras [4]:", "index": 8}, {"type": "text", "bbox": [125, 411, 486, 436], "content": "Proposition 5. Let be a quadratic extension of number fields and assume that . Then is ramified and", "index": 9}, {"type": "interline_equation", "bbox": [189, 441, 421, 473], "content": "", "index": 10}, {"type": "text", "bbox": [126, 475, 486, 499], "content": "where denotes the set of ideal classes of that become principal (capitulate) in .", "index": 11}, {"type": "text", "bbox": [125, 504, 486, 541], "content": "Proof. We first notice that is ramified. If the extension were unramified, then would be the 2-class field of , and since is cyclic, it would follow that , contrary to assumption.", "index": 12}, {"type": "text", "bbox": [125, 542, 487, 615], "content": "Before we start with the rest of the proof, we cite the results of Gras that we need (we could also give a slightly longer direct proof without referring to his results). Let be a cyclic extension of prime power order , and let be a generator of . For any -group on which acts we put . Moreover, let be the algebraic norm, that is, exponentiation by . Then [4, Cor. 4.3] reads", "index": 13}, {"type": "text", "bbox": [124, 619, 487, 658], "content": "Lemma 4. Suppose that ; let be the smallest positive integer such that and write with integers and . If for , then .", "index": 14}, {"type": "text", "bbox": [124, 662, 487, 701], "content": "We claim that if , then satisfies the assumptions of Lemma 4: in fact, let denote the transfer of ideal classes. Then for any ideal class , hence . Moreover,", "index": 15}]
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[{"bbox": [138, 114, 486, 126], "content": "Another important result is the ambiguous class number formula. For cyclic", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 485, 138], "content": "extensions , let denote the group of ideal classes in fixed by", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 486, 151], "content": ", i.e. the ambiguous ideal class group of , and its 2-Sylow subgroup.", "parent_index": 0, "line_index": 2}, {"bbox": [125, 156, 486, 169], "content": "Proposition 4. Let be a cyclic extension of prime degree ; then the number", "parent_index": 1, "line_index": 0}, {"bbox": [126, 168, 291, 181], "content": "of ambiguous ideal classes is given by", "parent_index": 1, "line_index": 1}, {"bbox": [126, 213, 485, 225], "content": "where is the number of primes (including those at ) of that ramify in ,", "parent_index": 3, "line_index": 0}, {"bbox": [125, 225, 487, 237], "content": "is the unit group of , and is its subgroup consisting of norms of elements from", "parent_index": 3, "line_index": 1}, {"bbox": [126, 237, 401, 250], "content": ". Moreover, is trivial if and only if .", "parent_index": 3, "line_index": 2}, {"bbox": [126, 254, 487, 269], "content": "Proof. See Lang [9, part II] for the formula. For a proof of the second assertion", "parent_index": 4, "line_index": 0}, {"bbox": [125, 266, 456, 280], "content": "(see e.g. Moriya [11]), note that is defined by the exact sequence", "parent_index": 4, "line_index": 1}, {"bbox": [126, 299, 485, 313], "content": "where generates . Taking -parts we see that is equiv-", "parent_index": 6, "line_index": 0}, {"bbox": [124, 311, 487, 327], "content": "alent to . By induction we get , but", "parent_index": 6, "line_index": 1}, {"bbox": [125, 324, 485, 338], "content": "since mod in the group ring , this implies .", "parent_index": 6, "line_index": 2}, {"bbox": [125, 336, 486, 349], "content": "But then must be trivial. 口", "parent_index": 6, "line_index": 3}, {"bbox": [125, 355, 486, 367], "content": "We make one further remark concerning the ambiguous class number formula that", "parent_index": 7, "line_index": 0}, {"bbox": [126, 367, 486, 379], "content": "will be useful below. If the class number is odd, then it is known that", "parent_index": 7, "line_index": 1}, {"bbox": [126, 379, 312, 391], "content": "where .", "parent_index": 7, "line_index": 2}, {"bbox": [137, 395, 364, 408], "content": "We also need a result essentially due to G. Gras [4]:", "parent_index": 8, "line_index": 0}, {"bbox": [126, 414, 487, 426], "content": "Proposition 5. Let be a quadratic extension of number fields and assume", "parent_index": 9, "line_index": 0}, {"bbox": [126, 426, 382, 437], "content": "that . Then is ramified and", "parent_index": 9, "line_index": 1}, {"bbox": [125, 475, 487, 492], "content": "where denotes the set of ideal classes of that become principal (capitulate)", "parent_index": 11, "line_index": 0}, {"bbox": [126, 489, 152, 500], "content": "in .", "parent_index": 11, "line_index": 1}, {"bbox": [127, 507, 485, 519], "content": "Proof. We first notice that is ramified. If the extension were unramified, then", "parent_index": 12, "line_index": 0}, {"bbox": [126, 518, 486, 531], "content": "would be the 2-class field of , and since is cyclic, it would follow that", "parent_index": 12, "line_index": 1}, {"bbox": [126, 530, 286, 545], "content": ", contrary to assumption.", "parent_index": 12, "line_index": 2}, {"bbox": [137, 542, 486, 555], "content": "Before we start with the rest of the proof, we cite the results of Gras that", "parent_index": 13, "line_index": 0}, {"bbox": [126, 555, 486, 568], "content": "we need (we could also give a slightly longer direct proof without referring to", "parent_index": 13, "line_index": 1}, {"bbox": [125, 565, 484, 580], "content": "his results). Let be a cyclic extension of prime power order , and let", "parent_index": 13, "line_index": 2}, {"bbox": [124, 578, 486, 592], "content": "be a generator of . For any -group on which acts we put", "parent_index": 13, "line_index": 3}, {"bbox": [126, 589, 487, 605], "content": ". Moreover, let be the algebraic norm, that is,", "parent_index": 13, "line_index": 4}, {"bbox": [124, 602, 429, 618], "content": "exponentiation by . Then [4, Cor. 4.3] reads", "parent_index": 13, "line_index": 5}, {"bbox": [124, 622, 487, 635], "content": "Lemma 4. Suppose that ; let be the smallest positive integer such that", "parent_index": 14, "line_index": 0}, {"bbox": [126, 634, 487, 647], "content": "and write with integers and . If", "parent_index": 14, "line_index": 1}, {"bbox": [125, 645, 472, 660], "content": "for , then .", "parent_index": 14, "line_index": 2}, {"bbox": [137, 663, 487, 677], "content": "We claim that if , then satisfies the assumptions of", "parent_index": 15, "line_index": 0}, {"bbox": [123, 676, 486, 690], "content": "Lemma 4: in fact, let denote the transfer of ideal classes. Then", "parent_index": 15, "line_index": 1}, {"bbox": [126, 689, 486, 702], "content": "for any ideal class , hence . Moreover,", "parent_index": 15, "line_index": 2}]
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[]
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[{"bbox": [175, 127, 195, 138], "content": "K/k", "parent_index": 0, "subtype": "inline"}, {"bbox": [219, 127, 262, 138], "content": "\\operatorname{Am}(K/k)", "parent_index": 0, "subtype": "inline"}, {"bbox": [434, 128, 444, 135], "content": "K", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 139, 168, 150], "content": "\\operatorname{Gal}(K/k)", "parent_index": 0, "subtype": "inline"}, {"bbox": [338, 140, 347, 147], "content": "K", "parent_index": 0, "subtype": "inline"}, {"bbox": [370, 140, 390, 149], "content": "\\mathrm{{Am}_{2}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [218, 158, 238, 168], "content": "K/k", "parent_index": 1, "subtype": "inline"}, {"bbox": [403, 161, 408, 167], "content": "p", "parent_index": 1, "subtype": "inline"}, {"bbox": [242, 183, 367, 209], "content": "\\#\\operatorname{Am}(K/k)=h(k)\\,{\\frac{p^{t-1}}{(E:H)}},", "parent_index": 2, "subtype": "interline"}, {"bbox": [154, 215, 158, 222], "content": "t", "parent_index": 3, "subtype": "inline"}, {"bbox": [351, 217, 361, 222], "content": "\\infty", "parent_index": 3, "subtype": "inline"}, {"bbox": [379, 214, 385, 222], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [451, 214, 471, 225], "content": "K/k", "parent_index": 3, "subtype": "inline"}, {"bbox": [477, 215, 485, 222], "content": "E", "parent_index": 3, "subtype": "inline"}, {"bbox": [213, 227, 219, 234], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [245, 227, 254, 234], "content": "H", "parent_index": 3, "subtype": "inline"}, {"bbox": [126, 238, 142, 246], "content": "K^{\\times}", "parent_index": 3, "subtype": "inline"}, {"bbox": [196, 238, 228, 249], "content": "\\mathrm{Cl}_{p}(K)", "parent_index": 3, "subtype": "inline"}, {"bbox": [331, 237, 398, 249], "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "parent_index": 3, "subtype": "inline"}, {"bbox": [268, 268, 311, 279], "content": "\\operatorname{Am}(K/k)", "parent_index": 4, "subtype": "inline"}, {"bbox": [162, 284, 447, 295], "content": "1\\;\\longrightarrow\\;\\mathrm{Am}(K/k)\\;\\longrightarrow\\;\\mathrm{Cl}(K)\\;\\longrightarrow\\;\\mathrm{Cl}(K)^{1-\\sigma}\\;\\longrightarrow\\;1,", "parent_index": 5, "subtype": "interline"}, {"bbox": [155, 304, 161, 309], "content": "\\sigma", "parent_index": 6, "subtype": "inline"}, {"bbox": [208, 301, 251, 312], "content": "\\operatorname{Gal}(K/k)", "parent_index": 6, "subtype": "inline"}, {"bbox": [291, 304, 297, 311], "content": "p", "parent_index": 6, "subtype": "inline"}, {"bbox": [378, 301, 445, 312], "content": "p\\nmid\\#\\operatorname{Am}(K/k)", "parent_index": 6, "subtype": "inline"}, {"bbox": [164, 313, 257, 325], "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{1-\\sigma}", "parent_index": 6, "subtype": "inline"}, {"bbox": [359, 313, 463, 325], "content": "\\operatorname{Cl}_{p}(K)=\\operatorname{Cl}_{p}(K)^{(1-\\sigma)^{\\nu}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [151, 325, 206, 336], "content": "(1-\\sigma)^{p}\\equiv0", "parent_index": 6, "subtype": "inline"}, {"bbox": [231, 329, 236, 336], "content": "p", "parent_index": 6, "subtype": "inline"}, {"bbox": [318, 326, 339, 336], "content": "\\mathbb{Z}[G]", "parent_index": 6, "subtype": "inline"}, {"bbox": [399, 326, 482, 336], "content": "\\operatorname{Cl}_{p}(K)\\subseteq\\operatorname{Cl}_{p}(K)^{p}", "parent_index": 6, "subtype": "inline"}, {"bbox": [168, 338, 200, 348], "content": "\\mathrm{Cl}_{p}(K)", "parent_index": 6, "subtype": "inline"}, {"bbox": [324, 368, 343, 379], "content": "h(k)", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 380, 205, 390], "content": "\\#\\operatorname{Am}_{2}(K/k)=2^{r}", "parent_index": 7, "subtype": "inline"}, {"bbox": [238, 380, 309, 390], "content": "r=\\mathrm{rank}\\,\\mathrm{Cl}_{2}(K)", "parent_index": 7, "subtype": "inline"}, {"bbox": [220, 415, 240, 425], "content": "K/k", "parent_index": 9, "subtype": "inline"}, {"bbox": [146, 427, 259, 437], "content": "h_{2}(k)=\\#\\operatorname{Am}_{2}(K/k)=2", "parent_index": 9, "subtype": "inline"}, {"bbox": [292, 426, 312, 437], "content": "K/k", "parent_index": 9, "subtype": "inline"}, {"bbox": [189, 441, 421, 473], "content": "\\operatorname{Cl}_{2}(K)\\simeq\\left\\{\\!\\!\\begin{array}{l l}{{(2,2)~o r~\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq3)~}}&{{i f\\#\\kappa_{K/k}=1,}}\\\\ {{\\mathbb{Z}/2^{n}\\mathbb{Z}~(n\\geq1)~}}&{{i f\\#\\kappa_{K/k}=2,}}\\end{array}\\!\\!\\right.", "parent_index": 10, "subtype": "interline"}, {"bbox": [154, 481, 176, 489], "content": "\\kappa_{K/k}", "parent_index": 11, "subtype": "inline"}, {"bbox": [329, 478, 335, 486], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [138, 491, 148, 498], "content": "K", "parent_index": 11, "subtype": "inline"}, {"bbox": [245, 508, 264, 518], "content": "K/k", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 520, 135, 528], "content": "K", "parent_index": 12, "subtype": "inline"}, {"bbox": [267, 520, 272, 528], "content": "k", "parent_index": 12, "subtype": "inline"}, {"bbox": [324, 520, 352, 530], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 532, 176, 542], "content": "\\mathrm{Cl}_{2}(K)=1", "parent_index": 12, "subtype": "inline"}, {"bbox": [204, 568, 224, 578], "content": "K/k", "parent_index": 13, "subtype": "inline"}, {"bbox": [425, 568, 435, 577], "content": "p^{r}", "parent_index": 13, "subtype": "inline"}, {"bbox": [478, 571, 484, 576], "content": "\\sigma", "parent_index": 13, "subtype": "inline"}, {"bbox": [207, 579, 275, 590], "content": "G\\,=\\,\\operatorname{Gal}(K/k)", "parent_index": 13, "subtype": "inline"}, {"bbox": [323, 583, 329, 589], "content": "p", "parent_index": 13, "subtype": "inline"}, {"bbox": [361, 580, 372, 587], "content": "M", "parent_index": 13, "subtype": "inline"}, {"bbox": [420, 580, 428, 588], "content": "G", "parent_index": 13, "subtype": "inline"}, {"bbox": [126, 590, 265, 604], "content": "M_{i}\\,=\\,\\{m\\,\\in\\,M\\,:\\,m^{(1-\\sigma)^{i}}\\,=\\,1\\}", "parent_index": 13, "subtype": "inline"}, {"bbox": [338, 596, 344, 601], "content": "\\nu", "parent_index": 13, "subtype": "inline"}, {"bbox": [207, 604, 314, 614], "content": "1+\\sigma+\\sigma^{2}+...+\\sigma^{p^{\\intercal}-1}", "parent_index": 13, "subtype": "inline"}, {"bbox": [239, 623, 275, 632], "content": "M^{\\nu}=1", "parent_index": 14, "subtype": "inline"}, {"bbox": [297, 627, 303, 631], "content": "n", "parent_index": 14, "subtype": "inline"}, {"bbox": [126, 635, 168, 645], "content": "M_{n}\\,=\\,M", "parent_index": 14, "subtype": "inline"}, {"bbox": [217, 635, 292, 646], "content": "n=a(p-1)+b", "parent_index": 14, "subtype": "inline"}, {"bbox": [356, 636, 381, 644], "content": "a\\geq0", "parent_index": 14, "subtype": "inline"}, {"bbox": [405, 636, 469, 645], "content": "0\\,\\leq\\,b\\,\\leq\\,p\\,-\\,2", "parent_index": 14, "subtype": "inline"}, {"bbox": [125, 646, 195, 658], "content": "\\#\\,M_{i+1}/M_{i}=p", "parent_index": 14, "subtype": "inline"}, {"bbox": [214, 648, 293, 657], "content": "i=0,1,\\dots,n-1", "parent_index": 14, "subtype": "inline"}, {"bbox": [321, 646, 468, 658], "content": "M\\simeq(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}", "parent_index": 14, "subtype": "inline"}, {"bbox": [218, 665, 264, 677], "content": "\\kappa_{K/k}~=~2", "parent_index": 15, "subtype": "inline"}, {"bbox": [296, 666, 357, 676], "content": "M\\ =\\ \\mathrm{Cl}_{2}(K)", "parent_index": 15, "subtype": "inline"}, {"bbox": [223, 678, 266, 688], "content": "j=j_{k\\rightarrow K}", "parent_index": 15, "subtype": "inline"}, {"bbox": [454, 677, 486, 687], "content": "c^{1+\\sigma}=", "parent_index": 15, "subtype": "inline"}, {"bbox": [126, 690, 167, 702], "content": "j(N_{K/k}c)", "parent_index": 15, "subtype": "inline"}, {"bbox": [254, 689, 305, 701], "content": "c\\,\\in\\,\\mathrm{Cl}_{2}(K)", "parent_index": 15, "subtype": "inline"}, {"bbox": [340, 690, 432, 701], "content": "M^{\\nu}=j(\\mathrm{Cl}_{2}(k))\\,=\\,1", "parent_index": 15, "subtype": "inline"}]
|
[]
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$M_{1}\;=\;\mathrm{Am}_{2}(K/k)$ in our case, hence $M_{1}/M_{0}$ has order 2. Since the orders of $M_{i+1}/M_{i}$ decrease towards $^{1}$ as $i$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $:M_{i+1}/M_{i}=2$ for all $i<n$ . Since $a=n$ and $b=0$ when $p=2$ , Lemma 4 now implies that $\mathrm{Cl}_{2}(K)\simeq\mathbb{Z}/2^{n}\mathbb{Z}$ , that is, the 2-class group is cyclic.
The second result of Gras that we need is [4, Prop. 4.3]
Lemma 5. Suppose that $M^{\nu}\ne1$ but assume the other conditions in Lemma 4. Then $n\geq2$ and
$$
M\simeq\left\{\!\!\begin{array}{l l}{(\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n<p;}\\ {(\mathbb{Z}/p\mathbb{Z})^{p}\ \mathrm{or}\ (\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n=p;}\\ {(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}}&{\mathrm{if}\ n>p.}\end{array}\right.
$$
If $\kappa_{K/k}=1$ , then this lemma shows that $\mathrm{Cl}_{2}(K)$ is either cyclic of order $\geq4$ or of type $(2,2)$ . (Notice that the hypothesis of the lemma is satisfied since $K/k$ is ramified implying that the norm $N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$ is onto; and so the argument above this lemma applies.) It remains to show that the case $\mathrm{Cl}_{2}(K)\simeq$ $\mathbb{Z}/4\mathbb{Z}$ cannot occur here.
Now assume that $\operatorname{Cl}_{2}(K)~=~\langle{\cal C}\rangle~\simeq~\mathbb{Z}/4\mathbb{Z}$ ; since $K/k$ is ramified, the norm $N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$ is onto, and using $\kappa_{K/k}=1$ once more we find $C^{1+\sigma}=c$ , where $c$ is the nontrivial ideal class from $\mathrm{Cl_{2}}(k)$ . On the other hand, $c\in\mathrm{Cl}_{2}(k)$ still has order 2 in $\mathrm{Cl}_{2}(K)$ , hence we must also have $C^{2}=C^{1+\sigma}$ . But this implies that $C^{\sigma}=C$ , i.e. that each ideal class in $K$ is ambiguous, contradicting our assumption that $\#\operatorname{Am}_{2}(K/k)=2$ . 口
# 4. Arithmetic of some Dihedral Extensions
In this section we study the arithmetic of some dihedral extensions $L/\mathbb{Q}$ , that is, normal extensions $L$ of $\mathbb{Q}$ with Galois group $\operatorname{Gal}(L/\mathbb{Q})\simeq D_{4}$ , the dihedral group of order 8. Hence $D_{4}$ may be presented as $\langle\tau,\sigma|\tau^{2}=\sigma^{4}=1,\tau\sigma\tau=\sigma^{-1}\rangle$ . Now consider the following diagrams (Galois correspondence):
In this situation, we let $q_{1}=(E_{L}:E_{1}E_{1}^{\prime}E_{K})$ and $q_{2}=(E_{L}:E_{2}E_{2}^{\prime}E_{K})$ denote the unit indices of the bicyclic extensions $L/k_{1}$ and $L/k_{2}$ , where $E_{i}$ and $E_{i}^{\prime}$ are the unit groups in $K_{i}$ and $K_{i}^{\prime}$ respectively. Finally, let $\kappa_{i}$ denote the kernel of the transfer of ideal classes $j_{k_{i}\to K_{i}}:\mathrm{Cl}_{2}(k_{i})\longrightarrow\mathrm{Cl}_{2}(K_{i})$ for $i=1,2$ .
The following remark will be used several times: if $K_{1}\;=\;k_{1}(\sqrt{\alpha}\,)$ for some $\alpha\in k_{1}$ , then $k_{2}=\mathbb{Q}({\sqrt{a}}\,)$ , where $a=\alpha\alpha^{\prime}$ is the norm of $\alpha$ . To see this, let $\gamma=\sqrt{\alpha}$ ; then $\gamma^{\tau}\,=\,\gamma$ , since $\gamma\ \in\ K_{1}$ . Clearly $\gamma^{1+\sigma}\,=\,\sqrt{a}\,\in\,K$ and hence fixed by $\sigma^{2}$ . Furthermore,
$$
(\gamma^{1+\sigma})^{\sigma\tau}=\gamma^{\sigma\tau+\sigma^{2}\tau}=\gamma^{\tau\sigma^{3}+\tau\sigma^{2}}=(\gamma^{\tau})^{\sigma^{3}+\sigma^{2}}=\gamma^{\sigma^{3}+\sigma^{2}}=\gamma^{(1+\sigma)\sigma^{2}}=\gamma^{1+\sigma},
$$
implying that ${\sqrt{a}}\ \in\ k_{2}$ . Finally notice that ${\sqrt{a}}\ \not\in\ \mathbb{Q}$ , since otherwise $\sqrt{\alpha^{\prime}}\,=$ ${\sqrt{a}}/{\sqrt{\alpha}}\in K_{1}$ implying that $K_{1}/\mathbb{Q}$ is normal, which is not the case.
|
<html><body>
<p data-bbox="125 111 487 160">$M_{1}\;=\;\mathrm{Am}_{2}(K/k)$ in our case, hence $M_{1}/M_{0}$ has order 2. Since the orders of $M_{i+1}/M_{i}$ decrease towards $^{1}$ as $i$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $:M_{i+1}/M_{i}=2$ for all $i<n$ . Since $a=n$ and $b=0$ when $p=2$ , Lemma 4 now implies that $\mathrm{Cl}_{2}(K)\simeq\mathbb{Z}/2^{n}\mathbb{Z}$ , that is, the 2-class group is cyclic. </p>
<p data-bbox="136 160 380 172">The second result of Gras that we need is [4, Prop. 4.3] </p>
<p data-bbox="124 177 486 201">Lemma 5. Suppose that $M^{\nu}\ne1$ but assume the other conditions in Lemma 4. Then $n\geq2$ and </p>
<div class="equation" data-bbox="191 205 420 251">$$
M\simeq\left\{\!\!\begin{array}{l l}{(\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n<p;}\\ {(\mathbb{Z}/p\mathbb{Z})^{p}\ \mathrm{or}\ (\mathbb{Z}/p^{2}\mathbb{Z})\times(\mathbb{Z}/p\mathbb{Z})^{n-2}}&{\mathrm{if}\ n=p;}\\ {(\mathbb{Z}/p^{a+1}\mathbb{Z})^{b}\times(\mathbb{Z}/p^{a}\mathbb{Z})^{p-1-b}}&{\mathrm{if}\ n>p.}\end{array}\right.
$$</div>
<p data-bbox="124 253 487 314">If $\kappa_{K/k}=1$ , then this lemma shows that $\mathrm{Cl}_{2}(K)$ is either cyclic of order $\geq4$ or of type $(2,2)$ . (Notice that the hypothesis of the lemma is satisfied since $K/k$ is ramified implying that the norm $N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$ is onto; and so the argument above this lemma applies.) It remains to show that the case $\mathrm{Cl}_{2}(K)\simeq$ $\mathbb{Z}/4\mathbb{Z}$ cannot occur here. </p>
<p data-bbox="125 314 486 387">Now assume that $\operatorname{Cl}_{2}(K)~=~\langle{\cal C}\rangle~\simeq~\mathbb{Z}/4\mathbb{Z}$ ; since $K/k$ is ramified, the norm $N_{K/k}:\operatorname{Cl}_{2}(K)\longrightarrow\operatorname{Cl}_{2}(k)$ is onto, and using $\kappa_{K/k}=1$ once more we find $C^{1+\sigma}=c$ , where $c$ is the nontrivial ideal class from $\mathrm{Cl_{2}}(k)$ . On the other hand, $c\in\mathrm{Cl}_{2}(k)$ still has order 2 in $\mathrm{Cl}_{2}(K)$ , hence we must also have $C^{2}=C^{1+\sigma}$ . But this implies that $C^{\sigma}=C$ , i.e. that each ideal class in $K$ is ambiguous, contradicting our assumption that $\#\operatorname{Am}_{2}(K/k)=2$ . 口 </p>
<h1 data-bbox="193 393 418 406">4. Arithmetic of some Dihedral Extensions </h1>
<p data-bbox="124 412 486 460">In this section we study the arithmetic of some dihedral extensions $L/\mathbb{Q}$ , that is, normal extensions $L$ of $\mathbb{Q}$ with Galois group $\operatorname{Gal}(L/\mathbb{Q})\simeq D_{4}$ , the dihedral group of order 8. Hence $D_{4}$ may be presented as $\langle\tau,\sigma|\tau^{2}=\sigma^{4}=1,\tau\sigma\tau=\sigma^{-1}\rangle$ . Now consider the following diagrams (Galois correspondence): </p>
<div class="image" data-bbox="131 468 478 554"><img data-bbox="131 468 478 554"/></div>
<p data-bbox="125 556 487 605">In this situation, we let $q_{1}=(E_{L}:E_{1}E_{1}^{\prime}E_{K})$ and $q_{2}=(E_{L}:E_{2}E_{2}^{\prime}E_{K})$ denote the unit indices of the bicyclic extensions $L/k_{1}$ and $L/k_{2}$ , where $E_{i}$ and $E_{i}^{\prime}$ are the unit groups in $K_{i}$ and $K_{i}^{\prime}$ respectively. Finally, let $\kappa_{i}$ denote the kernel of the transfer of ideal classes $j_{k_{i}\to K_{i}}:\mathrm{Cl}_{2}(k_{i})\longrightarrow\mathrm{Cl}_{2}(K_{i})$ for $i=1,2$ . </p>
<p data-bbox="125 606 486 653">The following remark will be used several times: if $K_{1}\;=\;k_{1}(\sqrt{\alpha}\,)$ for some $\alpha\in k_{1}$ , then $k_{2}=\mathbb{Q}({\sqrt{a}}\,)$ , where $a=\alpha\alpha^{\prime}$ is the norm of $\alpha$ . To see this, let $\gamma=\sqrt{\alpha}$ ; then $\gamma^{\tau}\,=\,\gamma$ , since $\gamma\ \in\ K_{1}$ . Clearly $\gamma^{1+\sigma}\,=\,\sqrt{a}\,\in\,K$ and hence fixed by $\sigma^{2}$ . Furthermore, </p>
<div class="equation" data-bbox="142 658 466 672">$$
(\gamma^{1+\sigma})^{\sigma\tau}=\gamma^{\sigma\tau+\sigma^{2}\tau}=\gamma^{\tau\sigma^{3}+\tau\sigma^{2}}=(\gamma^{\tau})^{\sigma^{3}+\sigma^{2}}=\gamma^{\sigma^{3}+\sigma^{2}}=\gamma^{(1+\sigma)\sigma^{2}}=\gamma^{1+\sigma},
$$</div>
<p data-bbox="124 674 486 700">implying that ${\sqrt{a}}\ \in\ k_{2}$ . Finally notice that ${\sqrt{a}}\ \not\in\ \mathbb{Q}$ , since otherwise $\sqrt{\alpha^{\prime}}\,=$ ${\sqrt{a}}/{\sqrt{\alpha}}\in K_{1}$ implying that $K_{1}/\mathbb{Q}$ is normal, which is not the case. </p>
</body></html>
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0003244v1
| 6 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "$M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)$ in our case, hence $M_{1}/M_{0}$ has order 2. Since the orders of $M_{i+1}/M_{i}$ decrease towards $^{1}$ as $i$ grows (Gras [4, Prop. 4.1.ii)]), we conclude that # $:M_{i+1}/M_{i}=2$ for all $i<n$ . Since $a=n$ and $b=0$ when $p=2$ , Lemma 4 now implies that $\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}$ , that is, the 2-class group is cyclic. ", "page_idx": 6}, {"type": "text", "text": "The second result of Gras that we need is [4, Prop. 4.3] ", "page_idx": 6}, {"type": "text", "text": "Lemma 5. Suppose that $M^{\\nu}\\ne1$ but assume the other conditions in Lemma 4. Then $n\\geq2$ and ", "page_idx": 6}, {"type": "equation", "text": "$$\nM\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "If $\\kappa_{K/k}=1$ , then this lemma shows that $\\mathrm{Cl}_{2}(K)$ is either cyclic of order $\\geq4$ or of type $(2,2)$ . (Notice that the hypothesis of the lemma is satisfied since $K/k$ is ramified implying that the norm $N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$ is onto; and so the argument above this lemma applies.) It remains to show that the case $\\mathrm{Cl}_{2}(K)\\simeq$ $\\mathbb{Z}/4\\mathbb{Z}$ cannot occur here. ", "page_idx": 6}, {"type": "text", "text": "Now assume that $\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}$ ; since $K/k$ is ramified, the norm $N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)$ is onto, and using $\\kappa_{K/k}=1$ once more we find $C^{1+\\sigma}=c$ , where $c$ is the nontrivial ideal class from $\\mathrm{Cl_{2}}(k)$ . On the other hand, $c\\in\\mathrm{Cl}_{2}(k)$ still has order 2 in $\\mathrm{Cl}_{2}(K)$ , hence we must also have $C^{2}=C^{1+\\sigma}$ . But this implies that $C^{\\sigma}=C$ , i.e. that each ideal class in $K$ is ambiguous, contradicting our assumption that $\\#\\operatorname{Am}_{2}(K/k)=2$ . 口 ", "page_idx": 6}, {"type": "text", "text": "4. Arithmetic of some Dihedral Extensions ", "text_level": 1, "page_idx": 6}, {"type": "text", "text": "In this section we study the arithmetic of some dihedral extensions $L/\\mathbb{Q}$ , that is, normal extensions $L$ of $\\mathbb{Q}$ with Galois group $\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}$ , the dihedral group of order 8. Hence $D_{4}$ may be presented as $\\langle\\tau,\\sigma|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle$ . Now consider the following diagrams (Galois correspondence): ", "page_idx": 6}, {"type": "image", "img_path": "images/110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg", "img_caption": [], "img_footnote": [], "page_idx": 6}, {"type": "text", "text": "In this situation, we let $q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})$ and $q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})$ denote the unit indices of the bicyclic extensions $L/k_{1}$ and $L/k_{2}$ , where $E_{i}$ and $E_{i}^{\\prime}$ are the unit groups in $K_{i}$ and $K_{i}^{\\prime}$ respectively. Finally, let $\\kappa_{i}$ denote the kernel of the transfer of ideal classes $j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})$ for $i=1,2$ . ", "page_idx": 6}, {"type": "text", "text": "The following remark will be used several times: if $K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)$ for some $\\alpha\\in k_{1}$ , then $k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)$ , where $a=\\alpha\\alpha^{\\prime}$ is the norm of $\\alpha$ . To see this, let $\\gamma=\\sqrt{\\alpha}$ ; then $\\gamma^{\\tau}\\,=\\,\\gamma$ , since $\\gamma\\ \\in\\ K_{1}$ . Clearly $\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K$ and hence fixed by $\\sigma^{2}$ . Furthermore, ", "page_idx": 6}, {"type": "equation", "text": "$$\n(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "implying that ${\\sqrt{a}}\\ \\in\\ k_{2}$ . Finally notice that ${\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}$ , since otherwise $\\sqrt{\\alpha^{\\prime}}\\,=$ ${\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}$ implying that $K_{1}/\\mathbb{Q}$ is normal, which is not the case. ", "page_idx": 6}]
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 487, 160], "lines": [{"bbox": [126, 114, 487, 126], "spans": [{"bbox": [126, 115, 206, 126], "score": 0.92, "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [207, 114, 297, 126], "score": 1.0, "content": " in our case, hence ", "type": "text"}, {"bbox": [298, 115, 330, 126], "score": 0.94, "content": "M_{1}/M_{0}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [331, 114, 487, 126], "score": 1.0, "content": " has order 2. Since the orders of", "type": "text"}], "index": 0}, {"bbox": [126, 126, 486, 138], "spans": [{"bbox": [126, 128, 167, 138], "score": 0.93, "content": "M_{i+1}/M_{i}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [167, 126, 246, 138], "score": 1.0, "content": " decrease towards ", "type": "text"}, {"bbox": [246, 128, 252, 135], "score": 0.35, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [252, 126, 267, 138], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [267, 128, 271, 135], "score": 0.85, "content": "i", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [271, 126, 486, 138], "score": 1.0, "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 133, 150], "score": 1.0, "content": "#", "type": "text"}, {"bbox": [133, 139, 196, 150], "score": 0.9, "content": ":M_{i+1}/M_{i}=2", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [196, 138, 229, 150], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [230, 140, 254, 147], "score": 0.91, "content": "i<n", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [254, 138, 288, 150], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [288, 142, 314, 147], "score": 0.88, "content": "a=n", "type": "inline_equation", "height": 5, "width": 26}, {"bbox": [315, 138, 337, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 140, 361, 147], "score": 0.91, "content": "b=0", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [362, 138, 391, 150], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [392, 140, 416, 149], "score": 0.92, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [416, 138, 486, 150], "score": 1.0, "content": ", Lemma 4 now", "type": "text"}], "index": 2}, {"bbox": [126, 150, 410, 162], "spans": [{"bbox": [126, 150, 181, 162], "score": 1.0, "content": "implies that ", "type": "text"}, {"bbox": [181, 151, 255, 162], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 74}, {"bbox": [255, 150, 410, 162], "score": 1.0, "content": ", that is, the 2-class group is cyclic.", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "text", "bbox": [136, 160, 380, 172], "lines": [{"bbox": [137, 162, 381, 173], "spans": [{"bbox": [137, 162, 381, 173], "score": 1.0, "content": "The second result of Gras that we need is [4, Prop. 4.3]", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 177, 486, 201], "lines": [{"bbox": [125, 180, 486, 192], "spans": [{"bbox": [125, 180, 240, 192], "score": 1.0, "content": "Lemma 5. Suppose that ", "type": "text"}, {"bbox": [240, 181, 277, 191], "score": 0.92, "content": "M^{\\nu}\\ne1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [278, 180, 486, 192], "score": 1.0, "content": " but assume the other conditions in Lemma 4.", "type": "text"}], "index": 5}, {"bbox": [127, 192, 199, 204], "spans": [{"bbox": [127, 192, 151, 204], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 194, 176, 203], "score": 0.91, "content": "n\\geq2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [177, 192, 199, 204], "score": 1.0, "content": " and", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [124, 253, 487, 314], "lines": [{"bbox": [136, 255, 485, 269], "spans": [{"bbox": [136, 255, 148, 269], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 258, 191, 268], "score": 0.92, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [191, 255, 325, 269], "score": 1.0, "content": ", then this lemma shows that ", "type": "text"}, {"bbox": [326, 257, 357, 267], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [358, 255, 468, 269], "score": 1.0, "content": " is either cyclic of order ", "type": "text"}, {"bbox": [468, 258, 485, 266], "score": 0.84, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 17}], "index": 8}, {"bbox": [125, 268, 485, 280], "spans": [{"bbox": [125, 268, 173, 280], "score": 1.0, "content": "or of type ", "type": "text"}, {"bbox": [173, 269, 196, 280], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [196, 268, 465, 280], "score": 1.0, "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "type": "text"}, {"bbox": [465, 269, 485, 280], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}], "index": 9}, {"bbox": [124, 279, 487, 294], "spans": [{"bbox": [124, 279, 282, 294], "score": 1.0, "content": "is ramified implying that the norm", "type": "text"}, {"bbox": [283, 281, 398, 292], "score": 0.9, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 115}, {"bbox": [399, 279, 487, 294], "score": 1.0, "content": " is onto; and so the", "type": "text"}], "index": 10}, {"bbox": [125, 292, 486, 304], "spans": [{"bbox": [125, 292, 442, 304], "score": 1.0, "content": "argument above this lemma applies.) It remains to show that the case ", "type": "text"}, {"bbox": [442, 293, 486, 303], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K)\\simeq", "type": "inline_equation", "height": 10, "width": 44}], "index": 11}, {"bbox": [126, 303, 234, 317], "spans": [{"bbox": [126, 305, 150, 316], "score": 0.94, "content": "\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 303, 234, 317], "score": 1.0, "content": " cannot occur here.", "type": "text"}], "index": 12}], "index": 10}, {"type": "text", "bbox": [125, 314, 486, 387], "lines": [{"bbox": [135, 315, 487, 329], "spans": [{"bbox": [135, 315, 221, 329], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [221, 317, 330, 327], "score": 0.91, "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 10, "width": 109}, {"bbox": [330, 315, 363, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [363, 317, 383, 327], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [384, 315, 487, 329], "score": 1.0, "content": " is ramified, the norm", "type": "text"}], "index": 13}, {"bbox": [126, 327, 487, 342], "spans": [{"bbox": [126, 329, 239, 340], "score": 0.91, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 113}, {"bbox": [239, 327, 319, 342], "score": 1.0, "content": " is onto, and using ", "type": "text"}, {"bbox": [319, 330, 359, 340], "score": 0.93, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [360, 327, 441, 342], "score": 1.0, "content": " once more we find ", "type": "text"}, {"bbox": [441, 328, 482, 337], "score": 0.92, "content": "C^{1+\\sigma}=c", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [483, 327, 487, 342], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 339, 486, 352], "spans": [{"bbox": [126, 339, 154, 352], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 344, 159, 349], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [159, 339, 301, 352], "score": 1.0, "content": " is the nontrivial ideal class from ", "type": "text"}, {"bbox": [302, 341, 329, 351], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [330, 339, 421, 352], "score": 1.0, "content": ". On the other hand, ", "type": "text"}, {"bbox": [421, 341, 466, 351], "score": 0.93, "content": "c\\in\\mathrm{Cl}_{2}(k)", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [466, 339, 486, 352], "score": 1.0, "content": " still", "type": "text"}], "index": 15}, {"bbox": [124, 350, 487, 365], "spans": [{"bbox": [124, 350, 189, 365], "score": 1.0, "content": "has order 2 in ", "type": "text"}, {"bbox": [190, 353, 221, 363], "score": 0.91, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [222, 350, 337, 365], "score": 1.0, "content": ", hence we must also have ", "type": "text"}, {"bbox": [337, 352, 386, 361], "score": 0.93, "content": "C^{2}=C^{1+\\sigma}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [387, 350, 487, 365], "score": 1.0, "content": ". But this implies that", "type": "text"}], "index": 16}, {"bbox": [126, 363, 485, 376], "spans": [{"bbox": [126, 366, 160, 373], "score": 0.91, "content": "C^{\\sigma}=C", "type": "inline_equation", "height": 7, "width": 34}, {"bbox": [161, 363, 284, 376], "score": 1.0, "content": ", i.e. that each ideal class in ", "type": "text"}, {"bbox": [284, 366, 293, 373], "score": 0.92, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [294, 363, 485, 376], "score": 1.0, "content": " is ambiguous, contradicting our assumption", "type": "text"}], "index": 17}, {"bbox": [126, 376, 487, 388], "spans": [{"bbox": [126, 376, 148, 388], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 377, 223, 387], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [223, 376, 227, 388], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 376, 487, 386], "score": 0.9919366836547852, "content": "口", "type": "text"}], "index": 18}], "index": 15.5}, {"type": "title", "bbox": [193, 393, 418, 406], "lines": [{"bbox": [193, 396, 418, 407], "spans": [{"bbox": [193, 396, 418, 407], "score": 1.0, "content": "4. Arithmetic of some Dihedral Extensions", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [124, 412, 486, 460], "lines": [{"bbox": [137, 413, 485, 426], "spans": [{"bbox": [137, 413, 428, 426], "score": 1.0, "content": "In this section we study the arithmetic of some dihedral extensions", "type": "text"}, {"bbox": [429, 415, 449, 425], "score": 0.94, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [449, 413, 485, 426], "score": 1.0, "content": ", that is,", "type": "text"}], "index": 20}, {"bbox": [125, 425, 487, 439], "spans": [{"bbox": [125, 425, 208, 439], "score": 1.0, "content": "normal extensions ", "type": "text"}, {"bbox": [208, 428, 216, 435], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [216, 425, 231, 439], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [231, 428, 239, 437], "score": 0.91, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [240, 425, 325, 439], "score": 1.0, "content": " with Galois group ", "type": "text"}, {"bbox": [326, 427, 396, 437], "score": 0.93, "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [397, 425, 487, 439], "score": 1.0, "content": ", the dihedral group", "type": "text"}], "index": 21}, {"bbox": [125, 437, 487, 451], "spans": [{"bbox": [125, 437, 208, 451], "score": 1.0, "content": "of order 8. Hence ", "type": "text"}, {"bbox": [209, 440, 222, 448], "score": 0.91, "content": "D_{4}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [222, 437, 320, 451], "score": 1.0, "content": " may be presented as ", "type": "text"}, {"bbox": [320, 438, 455, 450], "score": 0.91, "content": "\\langle\\tau,\\sigma|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "type": "inline_equation", "height": 12, "width": 135}, {"bbox": [456, 437, 487, 451], "score": 1.0, "content": ". Now", "type": "text"}], "index": 22}, {"bbox": [126, 450, 375, 462], "spans": [{"bbox": [126, 450, 375, 462], "score": 1.0, "content": "consider the following diagrams (Galois correspondence):", "type": "text"}], "index": 23}], "index": 21.5}, {"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}, {"type": "text", "bbox": [125, 556, 487, 605], "lines": [{"bbox": [137, 559, 486, 571], "spans": [{"bbox": [137, 559, 244, 571], "score": 1.0, "content": "In this situation, we let ", "type": "text"}, {"bbox": [245, 560, 337, 570], "score": 0.91, "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [338, 559, 360, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 560, 452, 571], "score": 0.91, "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [453, 559, 486, 571], "score": 1.0, "content": " denote", "type": "text"}], "index": 27}, {"bbox": [126, 571, 486, 584], "spans": [{"bbox": [126, 571, 316, 584], "score": 1.0, "content": "the unit indices of the bicyclic extensions ", "type": "text"}, {"bbox": [316, 572, 338, 583], "score": 0.93, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [338, 571, 362, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 572, 384, 583], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [384, 571, 420, 584], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [421, 573, 432, 582], "score": 0.91, "content": "E_{i}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [432, 571, 456, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [456, 572, 467, 583], "score": 0.92, "content": "E_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [468, 571, 486, 584], "score": 1.0, "content": " are", "type": "text"}], "index": 28}, {"bbox": [126, 583, 486, 596], "spans": [{"bbox": [126, 583, 208, 596], "score": 1.0, "content": "the unit groups in ", "type": "text"}, {"bbox": [208, 585, 221, 594], "score": 0.92, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 583, 243, 596], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 584, 255, 595], "score": 0.92, "content": "K_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [256, 583, 367, 596], "score": 1.0, "content": " respectively. Finally, let ", "type": "text"}, {"bbox": [367, 587, 376, 594], "score": 0.9, "content": "\\kappa_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [377, 583, 486, 596], "score": 1.0, "content": " denote the kernel of the", "type": "text"}], "index": 29}, {"bbox": [124, 594, 408, 608], "spans": [{"bbox": [124, 594, 230, 608], "score": 1.0, "content": "transfer of ideal classes ", "type": "text"}, {"bbox": [230, 596, 354, 606], "score": 0.91, "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "type": "inline_equation", "height": 10, "width": 124}, {"bbox": [354, 594, 372, 608], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [372, 597, 404, 606], "score": 0.92, "content": "i=1,2", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [404, 594, 408, 608], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 28.5}, {"type": "text", "bbox": [125, 606, 486, 653], "lines": [{"bbox": [137, 605, 487, 619], "spans": [{"bbox": [137, 605, 376, 619], "score": 1.0, "content": "The following remark will be used several times: if ", "type": "text"}, {"bbox": [377, 608, 442, 618], "score": 0.94, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 10, "width": 65}, {"bbox": [442, 605, 487, 619], "score": 1.0, "content": " for some", "type": "text"}], "index": 31}, {"bbox": [126, 618, 485, 632], "spans": [{"bbox": [126, 621, 154, 630], "score": 0.92, "content": "\\alpha\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [155, 618, 182, 632], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [182, 619, 236, 631], "score": 0.94, "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [236, 618, 270, 632], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 620, 304, 628], "score": 0.93, "content": "a=\\alpha\\alpha^{\\prime}", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 618, 369, 632], "score": 1.0, "content": " is the norm of ", "type": "text"}, {"bbox": [369, 623, 376, 628], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [376, 618, 447, 632], "score": 1.0, "content": ". To see this, let", "type": "text"}, {"bbox": [448, 619, 482, 630], "score": 0.92, "content": "\\gamma=\\sqrt{\\alpha}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [482, 618, 485, 632], "score": 1.0, "content": ";", "type": "text"}], "index": 32}, {"bbox": [124, 630, 487, 644], "spans": [{"bbox": [124, 630, 149, 644], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [150, 633, 184, 642], "score": 0.93, "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [184, 630, 216, 644], "score": 1.0, "content": ", since ", "type": "text"}, {"bbox": [217, 633, 252, 642], "score": 0.93, "content": "\\gamma\\ \\in\\ K_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [253, 630, 299, 644], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [299, 631, 377, 642], "score": 0.94, "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [377, 630, 471, 644], "score": 1.0, "content": " and hence fixed by ", "type": "text"}, {"bbox": [471, 631, 482, 640], "score": 0.9, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 630, 487, 644], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [125, 642, 185, 655], "spans": [{"bbox": [125, 642, 185, 655], "score": 1.0, "content": "Furthermore,", "type": "text"}], "index": 34}], "index": 32.5}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [124, 674, 486, 700], "lines": [{"bbox": [125, 676, 486, 689], "spans": [{"bbox": [125, 677, 191, 689], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [192, 678, 233, 689], "score": 0.93, "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [233, 677, 334, 689], "score": 1.0, "content": ". Finally notice that ", "type": "text"}, {"bbox": [335, 678, 374, 689], "score": 0.94, "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [375, 677, 453, 689], "score": 1.0, "content": ", since otherwise", "type": "text"}, {"bbox": [453, 676, 486, 688], "score": 0.88, "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 36}, {"bbox": [126, 689, 422, 702], "spans": [{"bbox": [126, 690, 185, 701], "score": 0.94, "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 689, 250, 702], "score": 1.0, "content": " implying that ", "type": "text"}, {"bbox": [250, 690, 276, 701], "score": 0.94, "content": "K_{1}/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [277, 689, 422, 702], "score": 1.0, "content": " is normal, which is not the case.", "type": "text"}], "index": 37}], "index": 36.5}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}], "discarded_blocks": [{"type": "discarded", "bbox": [237, 90, 374, 100], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 90, 486, 99], "lines": [{"bbox": [481, 94, 485, 100], "spans": [{"bbox": [481, 94, 485, 100], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 487, 160], "lines": [{"bbox": [126, 114, 487, 126], "spans": [{"bbox": [126, 115, 206, 126], "score": 0.92, "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [207, 114, 297, 126], "score": 1.0, "content": " in our case, hence ", "type": "text"}, {"bbox": [298, 115, 330, 126], "score": 0.94, "content": "M_{1}/M_{0}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [331, 114, 487, 126], "score": 1.0, "content": " has order 2. Since the orders of", "type": "text"}], "index": 0}, {"bbox": [126, 126, 486, 138], "spans": [{"bbox": [126, 128, 167, 138], "score": 0.93, "content": "M_{i+1}/M_{i}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [167, 126, 246, 138], "score": 1.0, "content": " decrease towards ", "type": "text"}, {"bbox": [246, 128, 252, 135], "score": 0.35, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [252, 126, 267, 138], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [267, 128, 271, 135], "score": 0.85, "content": "i", "type": "inline_equation", "height": 7, "width": 4}, {"bbox": [271, 126, 486, 138], "score": 1.0, "content": " grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 486, 150], "spans": [{"bbox": [126, 138, 133, 150], "score": 1.0, "content": "#", "type": "text"}, {"bbox": [133, 139, 196, 150], "score": 0.9, "content": ":M_{i+1}/M_{i}=2", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [196, 138, 229, 150], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [230, 140, 254, 147], "score": 0.91, "content": "i<n", "type": "inline_equation", "height": 7, "width": 24}, {"bbox": [254, 138, 288, 150], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [288, 142, 314, 147], "score": 0.88, "content": "a=n", "type": "inline_equation", "height": 5, "width": 26}, {"bbox": [315, 138, 337, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 140, 361, 147], "score": 0.91, "content": "b=0", "type": "inline_equation", "height": 7, "width": 23}, {"bbox": [362, 138, 391, 150], "score": 1.0, "content": " when ", "type": "text"}, {"bbox": [392, 140, 416, 149], "score": 0.92, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [416, 138, 486, 150], "score": 1.0, "content": ", Lemma 4 now", "type": "text"}], "index": 2}, {"bbox": [126, 150, 410, 162], "spans": [{"bbox": [126, 150, 181, 162], "score": 1.0, "content": "implies that ", "type": "text"}, {"bbox": [181, 151, 255, 162], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 74}, {"bbox": [255, 150, 410, 162], "score": 1.0, "content": ", that is, the 2-class group is cyclic.", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [126, 114, 487, 162]}, {"type": "text", "bbox": [136, 160, 380, 172], "lines": [{"bbox": [137, 162, 381, 173], "spans": [{"bbox": [137, 162, 381, 173], "score": 1.0, "content": "The second result of Gras that we need is [4, Prop. 4.3]", "type": "text"}], "index": 4}], "index": 4, "bbox_fs": [137, 162, 381, 173]}, {"type": "text", "bbox": [124, 177, 486, 201], "lines": [{"bbox": [125, 180, 486, 192], "spans": [{"bbox": [125, 180, 240, 192], "score": 1.0, "content": "Lemma 5. Suppose that ", "type": "text"}, {"bbox": [240, 181, 277, 191], "score": 0.92, "content": "M^{\\nu}\\ne1", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [278, 180, 486, 192], "score": 1.0, "content": " but assume the other conditions in Lemma 4.", "type": "text"}], "index": 5}, {"bbox": [127, 192, 199, 204], "spans": [{"bbox": [127, 192, 151, 204], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 194, 176, 203], "score": 0.91, "content": "n\\geq2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [177, 192, 199, 204], "score": 1.0, "content": " and", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [125, 180, 486, 204]}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "lines": [{"bbox": [191, 205, 420, 251], "spans": [{"bbox": [191, 205, 420, 251], "score": 0.92, "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [124, 253, 487, 314], "lines": [{"bbox": [136, 255, 485, 269], "spans": [{"bbox": [136, 255, 148, 269], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 258, 191, 268], "score": 0.92, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [191, 255, 325, 269], "score": 1.0, "content": ", then this lemma shows that ", "type": "text"}, {"bbox": [326, 257, 357, 267], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [358, 255, 468, 269], "score": 1.0, "content": " is either cyclic of order ", "type": "text"}, {"bbox": [468, 258, 485, 266], "score": 0.84, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 17}], "index": 8}, {"bbox": [125, 268, 485, 280], "spans": [{"bbox": [125, 268, 173, 280], "score": 1.0, "content": "or of type ", "type": "text"}, {"bbox": [173, 269, 196, 280], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [196, 268, 465, 280], "score": 1.0, "content": ". (Notice that the hypothesis of the lemma is satisfied since ", "type": "text"}, {"bbox": [465, 269, 485, 280], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 11, "width": 20}], "index": 9}, {"bbox": [124, 279, 487, 294], "spans": [{"bbox": [124, 279, 282, 294], "score": 1.0, "content": "is ramified implying that the norm", "type": "text"}, {"bbox": [283, 281, 398, 292], "score": 0.9, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 115}, {"bbox": [399, 279, 487, 294], "score": 1.0, "content": " is onto; and so the", "type": "text"}], "index": 10}, {"bbox": [125, 292, 486, 304], "spans": [{"bbox": [125, 292, 442, 304], "score": 1.0, "content": "argument above this lemma applies.) It remains to show that the case ", "type": "text"}, {"bbox": [442, 293, 486, 303], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K)\\simeq", "type": "inline_equation", "height": 10, "width": 44}], "index": 11}, {"bbox": [126, 303, 234, 317], "spans": [{"bbox": [126, 305, 150, 316], "score": 0.94, "content": "\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 303, 234, 317], "score": 1.0, "content": " cannot occur here.", "type": "text"}], "index": 12}], "index": 10, "bbox_fs": [124, 255, 487, 317]}, {"type": "text", "bbox": [125, 314, 486, 387], "lines": [{"bbox": [135, 315, 487, 329], "spans": [{"bbox": [135, 315, 221, 329], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [221, 317, 330, 327], "score": 0.91, "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "type": "inline_equation", "height": 10, "width": 109}, {"bbox": [330, 315, 363, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [363, 317, 383, 327], "score": 0.93, "content": "K/k", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [384, 315, 487, 329], "score": 1.0, "content": " is ramified, the norm", "type": "text"}], "index": 13}, {"bbox": [126, 327, 487, 342], "spans": [{"bbox": [126, 329, 239, 340], "score": 0.91, "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "type": "inline_equation", "height": 11, "width": 113}, {"bbox": [239, 327, 319, 342], "score": 1.0, "content": " is onto, and using ", "type": "text"}, {"bbox": [319, 330, 359, 340], "score": 0.93, "content": "\\kappa_{K/k}=1", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [360, 327, 441, 342], "score": 1.0, "content": " once more we find ", "type": "text"}, {"bbox": [441, 328, 482, 337], "score": 0.92, "content": "C^{1+\\sigma}=c", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [483, 327, 487, 342], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 339, 486, 352], "spans": [{"bbox": [126, 339, 154, 352], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [154, 344, 159, 349], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [159, 339, 301, 352], "score": 1.0, "content": " is the nontrivial ideal class from ", "type": "text"}, {"bbox": [302, 341, 329, 351], "score": 0.92, "content": "\\mathrm{Cl_{2}}(k)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [330, 339, 421, 352], "score": 1.0, "content": ". On the other hand, ", "type": "text"}, {"bbox": [421, 341, 466, 351], "score": 0.93, "content": "c\\in\\mathrm{Cl}_{2}(k)", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [466, 339, 486, 352], "score": 1.0, "content": " still", "type": "text"}], "index": 15}, {"bbox": [124, 350, 487, 365], "spans": [{"bbox": [124, 350, 189, 365], "score": 1.0, "content": "has order 2 in ", "type": "text"}, {"bbox": [190, 353, 221, 363], "score": 0.91, "content": "\\mathrm{Cl}_{2}(K)", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [222, 350, 337, 365], "score": 1.0, "content": ", hence we must also have ", "type": "text"}, {"bbox": [337, 352, 386, 361], "score": 0.93, "content": "C^{2}=C^{1+\\sigma}", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [387, 350, 487, 365], "score": 1.0, "content": ". But this implies that", "type": "text"}], "index": 16}, {"bbox": [126, 363, 485, 376], "spans": [{"bbox": [126, 366, 160, 373], "score": 0.91, "content": "C^{\\sigma}=C", "type": "inline_equation", "height": 7, "width": 34}, {"bbox": [161, 363, 284, 376], "score": 1.0, "content": ", i.e. that each ideal class in ", "type": "text"}, {"bbox": [284, 366, 293, 373], "score": 0.92, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [294, 363, 485, 376], "score": 1.0, "content": " is ambiguous, contradicting our assumption", "type": "text"}], "index": 17}, {"bbox": [126, 376, 487, 388], "spans": [{"bbox": [126, 376, 148, 388], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 377, 223, 387], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [223, 376, 227, 388], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [475, 376, 487, 386], "score": 0.9919366836547852, "content": "口", "type": "text"}], "index": 18}], "index": 15.5, "bbox_fs": [124, 315, 487, 388]}, {"type": "title", "bbox": [193, 393, 418, 406], "lines": [{"bbox": [193, 396, 418, 407], "spans": [{"bbox": [193, 396, 418, 407], "score": 1.0, "content": "4. Arithmetic of some Dihedral Extensions", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [124, 412, 486, 460], "lines": [{"bbox": [137, 413, 485, 426], "spans": [{"bbox": [137, 413, 428, 426], "score": 1.0, "content": "In this section we study the arithmetic of some dihedral extensions", "type": "text"}, {"bbox": [429, 415, 449, 425], "score": 0.94, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 20}, {"bbox": [449, 413, 485, 426], "score": 1.0, "content": ", that is,", "type": "text"}], "index": 20}, {"bbox": [125, 425, 487, 439], "spans": [{"bbox": [125, 425, 208, 439], "score": 1.0, "content": "normal extensions ", "type": "text"}, {"bbox": [208, 428, 216, 435], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [216, 425, 231, 439], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [231, 428, 239, 437], "score": 0.91, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [240, 425, 325, 439], "score": 1.0, "content": " with Galois group ", "type": "text"}, {"bbox": [326, 427, 396, 437], "score": 0.93, "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [397, 425, 487, 439], "score": 1.0, "content": ", the dihedral group", "type": "text"}], "index": 21}, {"bbox": [125, 437, 487, 451], "spans": [{"bbox": [125, 437, 208, 451], "score": 1.0, "content": "of order 8. Hence ", "type": "text"}, {"bbox": [209, 440, 222, 448], "score": 0.91, "content": "D_{4}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [222, 437, 320, 451], "score": 1.0, "content": " may be presented as ", "type": "text"}, {"bbox": [320, 438, 455, 450], "score": 0.91, "content": "\\langle\\tau,\\sigma|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "type": "inline_equation", "height": 12, "width": 135}, {"bbox": [456, 437, 487, 451], "score": 1.0, "content": ". Now", "type": "text"}], "index": 22}, {"bbox": [126, 450, 375, 462], "spans": [{"bbox": [126, 450, 375, 462], "score": 1.0, "content": "consider the following diagrams (Galois correspondence):", "type": "text"}], "index": 23}], "index": 21.5, "bbox_fs": [125, 413, 487, 462]}, {"type": "image", "bbox": [131, 468, 478, 554], "blocks": [{"type": "image_body", "bbox": [131, 468, 478, 554], "group_id": 0, "lines": [{"bbox": [131, 468, 478, 554], "spans": [{"bbox": [131, 468, 478, 554], "score": 0.921, "type": "image", "image_path": "110ff188220c8bb5d817468ee0d73bccaf61d8e5373f09de7f29c9cd995490d3.jpg"}]}], "index": 25, "virtual_lines": [{"bbox": [131, 468, 478, 496.6666666666667], "spans": [], "index": 24}, {"bbox": [131, 496.6666666666667, 478, 525.3333333333334], "spans": [], "index": 25}, {"bbox": [131, 525.3333333333334, 478, 554.0], "spans": [], "index": 26}]}], "index": 25}, {"type": "text", "bbox": [125, 556, 487, 605], "lines": [{"bbox": [137, 559, 486, 571], "spans": [{"bbox": [137, 559, 244, 571], "score": 1.0, "content": "In this situation, we let ", "type": "text"}, {"bbox": [245, 560, 337, 570], "score": 0.91, "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "type": "inline_equation", "height": 10, "width": 92}, {"bbox": [338, 559, 360, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 560, 452, 571], "score": 0.91, "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [453, 559, 486, 571], "score": 1.0, "content": " denote", "type": "text"}], "index": 27}, {"bbox": [126, 571, 486, 584], "spans": [{"bbox": [126, 571, 316, 584], "score": 1.0, "content": "the unit indices of the bicyclic extensions ", "type": "text"}, {"bbox": [316, 572, 338, 583], "score": 0.93, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [338, 571, 362, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 572, 384, 583], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [384, 571, 420, 584], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [421, 573, 432, 582], "score": 0.91, "content": "E_{i}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [432, 571, 456, 584], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [456, 572, 467, 583], "score": 0.92, "content": "E_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [468, 571, 486, 584], "score": 1.0, "content": " are", "type": "text"}], "index": 28}, {"bbox": [126, 583, 486, 596], "spans": [{"bbox": [126, 583, 208, 596], "score": 1.0, "content": "the unit groups in ", "type": "text"}, {"bbox": [208, 585, 221, 594], "score": 0.92, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [221, 583, 243, 596], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 584, 255, 595], "score": 0.92, "content": "K_{i}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [256, 583, 367, 596], "score": 1.0, "content": " respectively. Finally, let ", "type": "text"}, {"bbox": [367, 587, 376, 594], "score": 0.9, "content": "\\kappa_{i}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [377, 583, 486, 596], "score": 1.0, "content": " denote the kernel of the", "type": "text"}], "index": 29}, {"bbox": [124, 594, 408, 608], "spans": [{"bbox": [124, 594, 230, 608], "score": 1.0, "content": "transfer of ideal classes ", "type": "text"}, {"bbox": [230, 596, 354, 606], "score": 0.91, "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "type": "inline_equation", "height": 10, "width": 124}, {"bbox": [354, 594, 372, 608], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [372, 597, 404, 606], "score": 0.92, "content": "i=1,2", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [404, 594, 408, 608], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 28.5, "bbox_fs": [124, 559, 486, 608]}, {"type": "text", "bbox": [125, 606, 486, 653], "lines": [{"bbox": [137, 605, 487, 619], "spans": [{"bbox": [137, 605, 376, 619], "score": 1.0, "content": "The following remark will be used several times: if ", "type": "text"}, {"bbox": [377, 608, 442, 618], "score": 0.94, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 10, "width": 65}, {"bbox": [442, 605, 487, 619], "score": 1.0, "content": " for some", "type": "text"}], "index": 31}, {"bbox": [126, 618, 485, 632], "spans": [{"bbox": [126, 621, 154, 630], "score": 0.92, "content": "\\alpha\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [155, 618, 182, 632], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [182, 619, 236, 631], "score": 0.94, "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [236, 618, 270, 632], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 620, 304, 628], "score": 0.93, "content": "a=\\alpha\\alpha^{\\prime}", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 618, 369, 632], "score": 1.0, "content": " is the norm of ", "type": "text"}, {"bbox": [369, 623, 376, 628], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [376, 618, 447, 632], "score": 1.0, "content": ". To see this, let", "type": "text"}, {"bbox": [448, 619, 482, 630], "score": 0.92, "content": "\\gamma=\\sqrt{\\alpha}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [482, 618, 485, 632], "score": 1.0, "content": ";", "type": "text"}], "index": 32}, {"bbox": [124, 630, 487, 644], "spans": [{"bbox": [124, 630, 149, 644], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [150, 633, 184, 642], "score": 0.93, "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [184, 630, 216, 644], "score": 1.0, "content": ", since ", "type": "text"}, {"bbox": [217, 633, 252, 642], "score": 0.93, "content": "\\gamma\\ \\in\\ K_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [253, 630, 299, 644], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [299, 631, 377, 642], "score": 0.94, "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "type": "inline_equation", "height": 11, "width": 78}, {"bbox": [377, 630, 471, 644], "score": 1.0, "content": " and hence fixed by ", "type": "text"}, {"bbox": [471, 631, 482, 640], "score": 0.9, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [482, 630, 487, 644], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [125, 642, 185, 655], "spans": [{"bbox": [125, 642, 185, 655], "score": 1.0, "content": "Furthermore,", "type": "text"}], "index": 34}], "index": 32.5, "bbox_fs": [124, 605, 487, 655]}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "lines": [{"bbox": [142, 658, 466, 672], "spans": [{"bbox": [142, 658, 466, 672], "score": 0.88, "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "type": "interline_equation"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [124, 674, 486, 700], "lines": [{"bbox": [125, 676, 486, 689], "spans": [{"bbox": [125, 677, 191, 689], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [192, 678, 233, 689], "score": 0.93, "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [233, 677, 334, 689], "score": 1.0, "content": ". Finally notice that ", "type": "text"}, {"bbox": [335, 678, 374, 689], "score": 0.94, "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [375, 677, 453, 689], "score": 1.0, "content": ", since otherwise", "type": "text"}, {"bbox": [453, 676, 486, 688], "score": 0.88, "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 36}, {"bbox": [126, 689, 422, 702], "spans": [{"bbox": [126, 690, 185, 701], "score": 0.94, "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 689, 250, 702], "score": 1.0, "content": " implying that ", "type": "text"}, {"bbox": [250, 690, 276, 701], "score": 0.94, "content": "K_{1}/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [277, 689, 422, 702], "score": 1.0, "content": " is normal, which is not the case.", "type": "text"}], "index": 37}], "index": 36.5, "bbox_fs": [125, 676, 486, 702]}]}
|
[{"type": "text", "bbox": [125, 111, 487, 160], "content": "in our case, hence has order 2. Since the orders of decrease towards as grows (Gras [4, Prop. 4.1.ii)]), we conclude that # for all . Since and when , Lemma 4 now implies that , that is, the 2-class group is cyclic.", "index": 0}, {"type": "text", "bbox": [136, 160, 380, 172], "content": "The second result of Gras that we need is [4, Prop. 4.3]", "index": 1}, {"type": "text", "bbox": [124, 177, 486, 201], "content": "Lemma 5. Suppose that but assume the other conditions in Lemma 4. Then and", "index": 2}, {"type": "interline_equation", "bbox": [191, 205, 420, 251], "content": "", "index": 3}, {"type": "text", "bbox": [124, 253, 487, 314], "content": "If , then this lemma shows that is either cyclic of order or of type . (Notice that the hypothesis of the lemma is satisfied since is ramified implying that the norm is onto; and so the argument above this lemma applies.) It remains to show that the case cannot occur here.", "index": 4}, {"type": "text", "bbox": [125, 314, 486, 387], "content": "Now assume that ; since is ramified, the norm is onto, and using once more we find , where is the nontrivial ideal class from . On the other hand, still has order 2 in , hence we must also have . But this implies that , i.e. that each ideal class in is ambiguous, contradicting our assumption that . 口", "index": 5}, {"type": "title", "bbox": [193, 393, 418, 406], "content": "4. Arithmetic of some Dihedral Extensions", "index": 6}, {"type": "text", "bbox": [124, 412, 486, 460], "content": "In this section we study the arithmetic of some dihedral extensions , that is, normal extensions of with Galois group , the dihedral group of order 8. Hence may be presented as . Now consider the following diagrams (Galois correspondence):", "index": 7}, {"type": "image", "bbox": [131, 468, 478, 554], "content": "", "index": 8}, {"type": "text", "bbox": [125, 556, 487, 605], "content": "In this situation, we let and denote the unit indices of the bicyclic extensions and , where and are the unit groups in and respectively. Finally, let denote the kernel of the transfer of ideal classes for .", "index": 9}, {"type": "text", "bbox": [125, 606, 486, 653], "content": "The following remark will be used several times: if for some , then , where is the norm of . To see this, let ; then , since . Clearly and hence fixed by . Furthermore,", "index": 10}, {"type": "interline_equation", "bbox": [142, 658, 466, 672], "content": "", "index": 11}, {"type": "text", "bbox": [124, 674, 486, 700], "content": "implying that . Finally notice that , since otherwise implying that is normal, which is not the case.", "index": 12}]
|
[{"bbox": [126, 114, 487, 126], "content": "in our case, hence has order 2. Since the orders of", "parent_index": 0, "line_index": 0}, {"bbox": [126, 126, 486, 138], "content": "decrease towards as grows (Gras [4, Prop. 4.1.ii)]), we conclude that", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 486, 150], "content": "# for all . Since and when , Lemma 4 now", "parent_index": 0, "line_index": 2}, {"bbox": [126, 150, 410, 162], "content": "implies that , that is, the 2-class group is cyclic.", "parent_index": 0, "line_index": 3}, {"bbox": [137, 162, 381, 173], "content": "The second result of Gras that we need is [4, Prop. 4.3]", "parent_index": 1, "line_index": 0}, {"bbox": [125, 180, 486, 192], "content": "Lemma 5. Suppose that but assume the other conditions in Lemma 4.", "parent_index": 2, "line_index": 0}, {"bbox": [127, 192, 199, 204], "content": "Then and", "parent_index": 2, "line_index": 1}, {"bbox": [136, 255, 485, 269], "content": "If , then this lemma shows that is either cyclic of order", "parent_index": 4, "line_index": 0}, {"bbox": [125, 268, 485, 280], "content": "or of type . (Notice that the hypothesis of the lemma is satisfied since", "parent_index": 4, "line_index": 1}, {"bbox": [124, 279, 487, 294], "content": "is ramified implying that the norm is onto; and so the", "parent_index": 4, "line_index": 2}, {"bbox": [125, 292, 486, 304], "content": "argument above this lemma applies.) It remains to show that the case", "parent_index": 4, "line_index": 3}, {"bbox": [126, 303, 234, 317], "content": "cannot occur here.", "parent_index": 4, "line_index": 4}, {"bbox": [135, 315, 487, 329], "content": "Now assume that ; since is ramified, the norm", "parent_index": 5, "line_index": 0}, {"bbox": [126, 327, 487, 342], "content": "is onto, and using once more we find ,", "parent_index": 5, "line_index": 1}, {"bbox": [126, 339, 486, 352], "content": "where is the nontrivial ideal class from . On the other hand, still", "parent_index": 5, "line_index": 2}, {"bbox": [124, 350, 487, 365], "content": "has order 2 in , hence we must also have . But this implies that", "parent_index": 5, "line_index": 3}, {"bbox": [126, 363, 485, 376], "content": ", i.e. that each ideal class in is ambiguous, contradicting our assumption", "parent_index": 5, "line_index": 4}, {"bbox": [126, 376, 487, 388], "content": "that . 口", "parent_index": 5, "line_index": 5}, {"bbox": [193, 396, 418, 407], "content": "4. Arithmetic of some Dihedral Extensions", "parent_index": 6, "line_index": 0}, {"bbox": [137, 413, 485, 426], "content": "In this section we study the arithmetic of some dihedral extensions , that is,", "parent_index": 7, "line_index": 0}, {"bbox": [125, 425, 487, 439], "content": "normal extensions of with Galois group , the dihedral group", "parent_index": 7, "line_index": 1}, {"bbox": [125, 437, 487, 451], "content": "of order 8. Hence may be presented as . Now", "parent_index": 7, "line_index": 2}, {"bbox": [126, 450, 375, 462], "content": "consider the following diagrams (Galois correspondence):", "parent_index": 7, "line_index": 3}, {"bbox": [137, 559, 486, 571], "content": "In this situation, we let and denote", "parent_index": 9, "line_index": 0}, {"bbox": [126, 571, 486, 584], "content": "the unit indices of the bicyclic extensions and , where and are", "parent_index": 9, "line_index": 1}, {"bbox": [126, 583, 486, 596], "content": "the unit groups in and respectively. Finally, let denote the kernel of the", "parent_index": 9, "line_index": 2}, {"bbox": [124, 594, 408, 608], "content": "transfer of ideal classes for .", "parent_index": 9, "line_index": 3}, {"bbox": [137, 605, 487, 619], "content": "The following remark will be used several times: if for some", "parent_index": 10, "line_index": 0}, {"bbox": [126, 618, 485, 632], "content": ", then , where is the norm of . To see this, let ;", "parent_index": 10, "line_index": 1}, {"bbox": [124, 630, 487, 644], "content": "then , since . Clearly and hence fixed by .", "parent_index": 10, "line_index": 2}, {"bbox": [125, 642, 185, 655], "content": "Furthermore,", "parent_index": 10, "line_index": 3}, {"bbox": [125, 676, 486, 689], "content": "implying that . Finally notice that , since otherwise", "parent_index": 12, "line_index": 0}, {"bbox": [126, 689, 422, 702], "content": "implying that is normal, which is not the case.", "parent_index": 12, "line_index": 1}]
|
[{"bbox": [131, 468, 478, 554], "content": "", "parent_index": 8, "subtype": "body"}]
|
[{"bbox": [126, 115, 206, 126], "content": "M_{1}\\;=\\;\\mathrm{Am}_{2}(K/k)", "parent_index": 0, "subtype": "inline"}, {"bbox": [298, 115, 330, 126], "content": "M_{1}/M_{0}", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 128, 167, 138], "content": "M_{i+1}/M_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [246, 128, 252, 135], "content": "^{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [267, 128, 271, 135], "content": "i", "parent_index": 0, "subtype": "inline"}, {"bbox": [133, 139, 196, 150], "content": ":M_{i+1}/M_{i}=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [230, 140, 254, 147], "content": "i<n", "parent_index": 0, "subtype": "inline"}, {"bbox": [288, 142, 314, 147], "content": "a=n", "parent_index": 0, "subtype": "inline"}, {"bbox": [338, 140, 361, 147], "content": "b=0", "parent_index": 0, "subtype": "inline"}, {"bbox": [392, 140, 416, 149], "content": "p=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [181, 151, 255, 162], "content": "\\mathrm{Cl}_{2}(K)\\simeq\\mathbb{Z}/2^{n}\\mathbb{Z}", "parent_index": 0, "subtype": "inline"}, {"bbox": [240, 181, 277, 191], "content": "M^{\\nu}\\ne1", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 194, 176, 203], "content": "n\\geq2", "parent_index": 2, "subtype": "inline"}, {"bbox": [191, 205, 420, 251], "content": "M\\simeq\\left\\{\\!\\!\\begin{array}{l l}{(\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n<p;}\\\\ {(\\mathbb{Z}/p\\mathbb{Z})^{p}\\ \\mathrm{or}\\ (\\mathbb{Z}/p^{2}\\mathbb{Z})\\times(\\mathbb{Z}/p\\mathbb{Z})^{n-2}}&{\\mathrm{if}\\ n=p;}\\\\ {(\\mathbb{Z}/p^{a+1}\\mathbb{Z})^{b}\\times(\\mathbb{Z}/p^{a}\\mathbb{Z})^{p-1-b}}&{\\mathrm{if}\\ n>p.}\\end{array}\\right.", "parent_index": 3, "subtype": "interline"}, {"bbox": [149, 258, 191, 268], "content": "\\kappa_{K/k}=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [326, 257, 357, 267], "content": "\\mathrm{Cl}_{2}(K)", "parent_index": 4, "subtype": "inline"}, {"bbox": [468, 258, 485, 266], "content": "\\geq4", "parent_index": 4, "subtype": "inline"}, {"bbox": [173, 269, 196, 280], "content": "(2,2)", "parent_index": 4, "subtype": "inline"}, {"bbox": [465, 269, 485, 280], "content": "K/k", "parent_index": 4, "subtype": "inline"}, {"bbox": [283, 281, 398, 292], "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "parent_index": 4, "subtype": "inline"}, {"bbox": [442, 293, 486, 303], "content": "\\mathrm{Cl}_{2}(K)\\simeq", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 305, 150, 316], "content": "\\mathbb{Z}/4\\mathbb{Z}", "parent_index": 4, "subtype": "inline"}, {"bbox": [221, 317, 330, 327], "content": "\\operatorname{Cl}_{2}(K)~=~\\langle{\\cal C}\\rangle~\\simeq~\\mathbb{Z}/4\\mathbb{Z}", "parent_index": 5, "subtype": "inline"}, {"bbox": [363, 317, 383, 327], "content": "K/k", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 329, 239, 340], "content": "N_{K/k}:\\operatorname{Cl}_{2}(K)\\longrightarrow\\operatorname{Cl}_{2}(k)", "parent_index": 5, "subtype": "inline"}, {"bbox": [319, 330, 359, 340], "content": "\\kappa_{K/k}=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [441, 328, 482, 337], "content": "C^{1+\\sigma}=c", "parent_index": 5, "subtype": "inline"}, {"bbox": [154, 344, 159, 349], "content": "c", "parent_index": 5, "subtype": "inline"}, {"bbox": [302, 341, 329, 351], "content": "\\mathrm{Cl_{2}}(k)", "parent_index": 5, "subtype": "inline"}, {"bbox": [421, 341, 466, 351], "content": "c\\in\\mathrm{Cl}_{2}(k)", "parent_index": 5, "subtype": "inline"}, {"bbox": [190, 353, 221, 363], "content": "\\mathrm{Cl}_{2}(K)", "parent_index": 5, "subtype": "inline"}, {"bbox": [337, 352, 386, 361], "content": "C^{2}=C^{1+\\sigma}", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 366, 160, 373], "content": "C^{\\sigma}=C", "parent_index": 5, "subtype": "inline"}, {"bbox": [284, 366, 293, 373], "content": "K", "parent_index": 5, "subtype": "inline"}, {"bbox": [148, 377, 223, 387], "content": "\\#\\operatorname{Am}_{2}(K/k)=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [429, 415, 449, 425], "content": "L/\\mathbb{Q}", "parent_index": 7, "subtype": "inline"}, {"bbox": [208, 428, 216, 435], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [231, 428, 239, 437], "content": "\\mathbb{Q}", "parent_index": 7, "subtype": "inline"}, {"bbox": [326, 427, 396, 437], "content": "\\operatorname{Gal}(L/\\mathbb{Q})\\simeq D_{4}", "parent_index": 7, "subtype": "inline"}, {"bbox": [209, 440, 222, 448], "content": "D_{4}", "parent_index": 7, "subtype": "inline"}, {"bbox": [320, 438, 455, 450], "content": "\\langle\\tau,\\sigma|\\tau^{2}=\\sigma^{4}=1,\\tau\\sigma\\tau=\\sigma^{-1}\\rangle", "parent_index": 7, "subtype": "inline"}, {"bbox": [245, 560, 337, 570], "content": "q_{1}=(E_{L}:E_{1}E_{1}^{\\prime}E_{K})", "parent_index": 9, "subtype": "inline"}, {"bbox": [360, 560, 452, 571], "content": "q_{2}=(E_{L}:E_{2}E_{2}^{\\prime}E_{K})", "parent_index": 9, "subtype": "inline"}, {"bbox": [316, 572, 338, 583], "content": "L/k_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [362, 572, 384, 583], "content": "L/k_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [421, 573, 432, 582], "content": "E_{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [456, 572, 467, 583], "content": "E_{i}^{\\prime}", "parent_index": 9, "subtype": "inline"}, {"bbox": [208, 585, 221, 594], "content": "K_{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [243, 584, 255, 595], "content": "K_{i}^{\\prime}", "parent_index": 9, "subtype": "inline"}, {"bbox": [367, 587, 376, 594], "content": "\\kappa_{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [230, 596, 354, 606], "content": "j_{k_{i}\\to K_{i}}:\\mathrm{Cl}_{2}(k_{i})\\longrightarrow\\mathrm{Cl}_{2}(K_{i})", "parent_index": 9, "subtype": "inline"}, {"bbox": [372, 597, 404, 606], "content": "i=1,2", "parent_index": 9, "subtype": "inline"}, {"bbox": [377, 608, 442, 618], "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\alpha}\\,)", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 621, 154, 630], "content": "\\alpha\\in k_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [182, 619, 236, 631], "content": "k_{2}=\\mathbb{Q}({\\sqrt{a}}\\,)", "parent_index": 10, "subtype": "inline"}, {"bbox": [270, 620, 304, 628], "content": "a=\\alpha\\alpha^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [369, 623, 376, 628], "content": "\\alpha", "parent_index": 10, "subtype": "inline"}, {"bbox": [448, 619, 482, 630], "content": "\\gamma=\\sqrt{\\alpha}", "parent_index": 10, "subtype": "inline"}, {"bbox": [150, 633, 184, 642], "content": "\\gamma^{\\tau}\\,=\\,\\gamma", "parent_index": 10, "subtype": "inline"}, {"bbox": [217, 633, 252, 642], "content": "\\gamma\\ \\in\\ K_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [299, 631, 377, 642], "content": "\\gamma^{1+\\sigma}\\,=\\,\\sqrt{a}\\,\\in\\,K", "parent_index": 10, "subtype": "inline"}, {"bbox": [471, 631, 482, 640], "content": "\\sigma^{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [142, 658, 466, 672], "content": "(\\gamma^{1+\\sigma})^{\\sigma\\tau}=\\gamma^{\\sigma\\tau+\\sigma^{2}\\tau}=\\gamma^{\\tau\\sigma^{3}+\\tau\\sigma^{2}}=(\\gamma^{\\tau})^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{\\sigma^{3}+\\sigma^{2}}=\\gamma^{(1+\\sigma)\\sigma^{2}}=\\gamma^{1+\\sigma},", "parent_index": 11, "subtype": "interline"}, {"bbox": [192, 678, 233, 689], "content": "{\\sqrt{a}}\\ \\in\\ k_{2}", "parent_index": 12, "subtype": "inline"}, {"bbox": [335, 678, 374, 689], "content": "{\\sqrt{a}}\\ \\not\\in\\ \\mathbb{Q}", "parent_index": 12, "subtype": "inline"}, {"bbox": [453, 676, 486, 688], "content": "\\sqrt{\\alpha^{\\prime}}\\,=", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 690, 185, 701], "content": "{\\sqrt{a}}/{\\sqrt{\\alpha}}\\in K_{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [250, 690, 276, 701], "content": "K_{1}/\\mathbb{Q}", "parent_index": 12, "subtype": "inline"}]
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[]
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Recall that a quadratic extension $K\,=\,k({\sqrt{\alpha}}\,)$ is called essentially ramified if $\alpha{\mathcal{O}}_{k}$ is not an ideal square. This definition is independent of the choice of $\alpha$ .
Proposition 6. Let $L/\mathbb{Q}$ be a non-CM totally complex dihedral extension not containing $\sqrt{-1}$ , and assume that $L/K_{1}$ and $L/K_{2}$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $K$ has norm $^{-1}$ , then $q_{1}q_{2}=2$ .
Proof. Notice first that $k$ cannot be real (in fact, $K$ is not totally real by assumption, and since $L/k$ is a cyclic quartic extension, no infinite prime can ramify in $K/k$ ); thus exactly one of $k_{1}$ , $k_{2}$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $L/k_{1}$ and $L/k_{2}$ (note that $\upsilon=0$ since both $L/K_{1}$ and $L/K_{2}$ are essentially ramified) we find that $2q_{1}q_{2}$ is a square. If we can prove that $q_{1},q_{2}\leq2$ , then $2q_{1}q_{2}$ is a square between 2 and 8, which implies that we must have $2q_{1}q_{2}=4$ and $q_{1}q_{2}=2$ as claimed.
We start by remarking that if $\zeta\eta$ becomes a square in $L$ , where $\zeta$ is a root of unity in $L$ , then so does one of $\pm\eta$ . This follows from the fact that the only non-trivial roots of unity that can be in $L$ are the sixth roots of unity $\langle\zeta_{6}\rangle$ , and here $\zeta_{6}=-\zeta_{3}^{2}$ .
Now we prove that $q_{1}\leq2$ under the assumptions we made; the claim $q_{2}\leq2$ will then follow by symmetry. Assume first that $k_{1}$ is real and let $\varepsilon$ be the fundamental unit of $k_{1}$ . We claim that $\sqrt{\pm\varepsilon}\notin L$ . Suppose otherwise; then $k_{1}(\sqrt{\pm\varepsilon})$ is one of $K_{1}$ , $K_{1}^{\prime}$ or $K$ . If $k_{1}(\sqrt{\pm\varepsilon}\,)=K_{1}$ , then $K_{1}^{\prime}=k_{1}(\sqrt{\pm\varepsilon^{\prime}}\,)$ and $K=k_{1}\big(\sqrt{\varepsilon\varepsilon^{\prime}}\big)$ . (Here and below $x^{\prime}\,=\,x^{\sigma}$ .) This however cannot occur since by assumption $\varepsilon\varepsilon^{\prime}\,=\,-1$ implying that $\sqrt{-1}\in L$ , a contradiction. Similarly, if $k_{1}({\sqrt{\pm\varepsilon}}\,)\,=\,K$ , then again $\sqrt{-1}\in L$ .
Thus $\sqrt{\pm\varepsilon}~\notin~L$ , and $E_{1}~=~\langle-1,\varepsilon,\eta\rangle$ for some unit $\eta\ \in\ E_{1}$ . Suppose that $\sqrt{u\eta}\in L$ for some unit $u\in k_{1}$ . Then $L=K_{1}(\sqrt{u\eta}\,)$ , contradicting our assumption that $L/K_{1}$ is essentially ramified. The same argument shows that $\sqrt{u\eta^{\prime}}\notin L$ , hence either $E_{L}\,=\,\langle\zeta,\varepsilon,\eta,\eta^{\prime}\rangle$ and $q_{1}=1$ or $E_{L}=\langle\zeta,\varepsilon,\eta,\sqrt{u\eta\eta^{\prime}}\rangle$ for some unit $u\in k_{1}$ and $q_{1}=2$ . Here $\zeta$ is a root of unity generating the torsion subgroup $W_{L}$ of $E_{L}$ .
Next consider the case where $k_{1}$ is complex, and let $\varepsilon$ denote the fundamental unit of $k_{2}$ . Then $\pm\varepsilon$ stays fundamental in $L$ by the argument above.
Let $\eta$ be a fundamental unit in $K_{1}$ . If $\pm\eta$ became a square in $L$ , then clearly $L/K_{1}$ could not be essentially ramified. Thus if we have $q_{1}\geq4$ , then $\pm\varepsilon\eta=\alpha^{2}$ is a square in $L$ . Applying $\tau$ to this relation we find that $-1=\varepsilon\varepsilon^{\prime}$ is a square in $L$ , contradicting the assumption that $L$ does not contain $\sqrt{-1}$ . 口
Proposition 7. Suppose that $q_{2}=1$ . Then $K_{2}/k_{2}$ is essentially ramified if and only if $\kappa_{2}=1$ ; if $K_{2}/k_{2}$ is not essentially ramified, then $\kappa_{2}=\langle[6]\rangle$ , where $K_{2}=$ $k_{2}(\sqrt{\beta}\,)$ and $(\beta)=\mathfrak{b}^{2}$ .
Proof. First notice that if $K_{2}/k_{2}$ is not essentially ramified, then $\kappa_{2}\neq1$ : in fact, in this case we have $(\beta)\;=\;6^{2}$ , and if we had $\,\kappa_{2}\,=\,1$ , then $\mathfrak{b}$ would have to be principal, say ${\mathfrak{b}}=(\gamma)$ . This implies that $\beta\,=\,\varepsilon\gamma^{2}$ for some unit $\varepsilon\in k_{2}$ , which in view of $q_{2}=1$ implies that $\varepsilon$ must be a square. But then $\beta$ would be a square, and this is impossible.
Conversely, suppose $\kappa_{2}\neq1$ . Let $\mathfrak{a}$ be a nonprincipal ideal in $k_{2}$ of absolute norm $a$ , and assume that ${\mathfrak{a}}=(\alpha)$ in $K_{2}$ . Then $\alpha^{1-\sigma^{2}}=\eta$ for some unit $\eta\in E_{2}$ , and similarly $\alpha^{\sigma-\sigma^{3}}\,=\,\eta^{\prime}$ , where $\eta^{\prime}$ is a unit in $E_{2}^{\prime}$ . But then $\eta\eta^{\prime}\,=\,\alpha^{1+\sigma-\sigma^{2}-\sigma^{3}}\,\stackrel{\cdot2}{=}$ $N_{L/k}\alpha\,=\,\pm N_{L/k}\mathfrak{a}\,=\,\pm a^{2}\,\stackrel{?}{=}\,\pm1$ in $L^{\times}$ , where $\underline{{\underline{{2}}}}$ means equal up to a square in $L^{\times}$ . Thus $\pm\eta\eta^{\prime}$ is a square in $L$ , so our assumption that $q_{2}=1$ implies that $\pm\eta\eta^{\prime}$ must be a square in $k_{2}$ . The same argument show that $\pm\eta/\eta^{\prime}$ is a square in $k_{2}$ , hence we find $\eta\in k_{2}$ . Thus $\alpha^{1-\sigma^{2}}$ is fixed by $\sigma^{2}$ and so $\beta:=\alpha^{2}\in k_{2}$ . This gives $K_{2}=k_{2}(\sqrt{\beta}\,)$ , hence $K_{2}/k_{2}$ is not essentially ramified, and moreover, $a\sim{\mathfrak{b}}$ . 冏口
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<p data-bbox="125 111 486 136">Recall that a quadratic extension $K\,=\,k({\sqrt{\alpha}}\,)$ is called essentially ramified if $\alpha{\mathcal{O}}_{k}$ is not an ideal square. This definition is independent of the choice of $\alpha$ . </p>
<p data-bbox="125 142 486 179">Proposition 6. Let $L/\mathbb{Q}$ be a non-CM totally complex dihedral extension not containing $\sqrt{-1}$ , and assume that $L/K_{1}$ and $L/K_{2}$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $K$ has norm $^{-1}$ , then $q_{1}q_{2}=2$ . </p>
<p data-bbox="125 185 486 269">Proof. Notice first that $k$ cannot be real (in fact, $K$ is not totally real by assumption, and since $L/k$ is a cyclic quartic extension, no infinite prime can ramify in $K/k$ ); thus exactly one of $k_{1}$ , $k_{2}$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $L/k_{1}$ and $L/k_{2}$ (note that $\upsilon=0$ since both $L/K_{1}$ and $L/K_{2}$ are essentially ramified) we find that $2q_{1}q_{2}$ is a square. If we can prove that $q_{1},q_{2}\leq2$ , then $2q_{1}q_{2}$ is a square between 2 and 8, which implies that we must have $2q_{1}q_{2}=4$ and $q_{1}q_{2}=2$ as claimed. </p>
<p data-bbox="125 270 486 306">We start by remarking that if $\zeta\eta$ becomes a square in $L$ , where $\zeta$ is a root of unity in $L$ , then so does one of $\pm\eta$ . This follows from the fact that the only non-trivial roots of unity that can be in $L$ are the sixth roots of unity $\langle\zeta_{6}\rangle$ , and here $\zeta_{6}=-\zeta_{3}^{2}$ . </p>
<p data-bbox="125 306 486 390">Now we prove that $q_{1}\leq2$ under the assumptions we made; the claim $q_{2}\leq2$ will then follow by symmetry. Assume first that $k_{1}$ is real and let $\varepsilon$ be the fundamental unit of $k_{1}$ . We claim that $\sqrt{\pm\varepsilon}\notin L$ . Suppose otherwise; then $k_{1}(\sqrt{\pm\varepsilon})$ is one of $K_{1}$ , $K_{1}^{\prime}$ or $K$ . If $k_{1}(\sqrt{\pm\varepsilon}\,)=K_{1}$ , then $K_{1}^{\prime}=k_{1}(\sqrt{\pm\varepsilon^{\prime}}\,)$ and $K=k_{1}\big(\sqrt{\varepsilon\varepsilon^{\prime}}\big)$ . (Here and below $x^{\prime}\,=\,x^{\sigma}$ .) This however cannot occur since by assumption $\varepsilon\varepsilon^{\prime}\,=\,-1$ implying that $\sqrt{-1}\in L$ , a contradiction. Similarly, if $k_{1}({\sqrt{\pm\varepsilon}}\,)\,=\,K$ , then again $\sqrt{-1}\in L$ . </p>
<p data-bbox="125 390 486 450">Thus $\sqrt{\pm\varepsilon}~\notin~L$ , and $E_{1}~=~\langle-1,\varepsilon,\eta\rangle$ for some unit $\eta\ \in\ E_{1}$ . Suppose that $\sqrt{u\eta}\in L$ for some unit $u\in k_{1}$ . Then $L=K_{1}(\sqrt{u\eta}\,)$ , contradicting our assumption that $L/K_{1}$ is essentially ramified. The same argument shows that $\sqrt{u\eta^{\prime}}\notin L$ , hence either $E_{L}\,=\,\langle\zeta,\varepsilon,\eta,\eta^{\prime}\rangle$ and $q_{1}=1$ or $E_{L}=\langle\zeta,\varepsilon,\eta,\sqrt{u\eta\eta^{\prime}}\rangle$ for some unit $u\in k_{1}$ and $q_{1}=2$ . Here $\zeta$ is a root of unity generating the torsion subgroup $W_{L}$ of $E_{L}$ . </p>
<p data-bbox="125 451 486 474">Next consider the case where $k_{1}$ is complex, and let $\varepsilon$ denote the fundamental unit of $k_{2}$ . Then $\pm\varepsilon$ stays fundamental in $L$ by the argument above. </p>
<p data-bbox="125 475 486 522">Let $\eta$ be a fundamental unit in $K_{1}$ . If $\pm\eta$ became a square in $L$ , then clearly $L/K_{1}$ could not be essentially ramified. Thus if we have $q_{1}\geq4$ , then $\pm\varepsilon\eta=\alpha^{2}$ is a square in $L$ . Applying $\tau$ to this relation we find that $-1=\varepsilon\varepsilon^{\prime}$ is a square in $L$ , contradicting the assumption that $L$ does not contain $\sqrt{-1}$ . 口 </p>
<p data-bbox="125 530 486 567">Proposition 7. Suppose that $q_{2}=1$ . Then $K_{2}/k_{2}$ is essentially ramified if and only if $\kappa_{2}=1$ ; if $K_{2}/k_{2}$ is not essentially ramified, then $\kappa_{2}=\langle[6]\rangle$ , where $K_{2}=$ $k_{2}(\sqrt{\beta}\,)$ and $(\beta)=\mathfrak{b}^{2}$ . </p>
<p data-bbox="125 573 486 632">Proof. First notice that if $K_{2}/k_{2}$ is not essentially ramified, then $\kappa_{2}\neq1$ : in fact, in this case we have $(\beta)\;=\;6^{2}$ , and if we had $\,\kappa_{2}\,=\,1$ , then $\mathfrak{b}$ would have to be principal, say ${\mathfrak{b}}=(\gamma)$ . This implies that $\beta\,=\,\varepsilon\gamma^{2}$ for some unit $\varepsilon\in k_{2}$ , which in view of $q_{2}=1$ implies that $\varepsilon$ must be a square. But then $\beta$ would be a square, and this is impossible. </p>
<p data-bbox="124 633 486 700">Conversely, suppose $\kappa_{2}\neq1$ . Let $\mathfrak{a}$ be a nonprincipal ideal in $k_{2}$ of absolute norm $a$ , and assume that ${\mathfrak{a}}=(\alpha)$ in $K_{2}$ . Then $\alpha^{1-\sigma^{2}}=\eta$ for some unit $\eta\in E_{2}$ , and similarly $\alpha^{\sigma-\sigma^{3}}\,=\,\eta^{\prime}$ , where $\eta^{\prime}$ is a unit in $E_{2}^{\prime}$ . But then $\eta\eta^{\prime}\,=\,\alpha^{1+\sigma-\sigma^{2}-\sigma^{3}}\,\stackrel{\cdot2}{=}$ $N_{L/k}\alpha\,=\,\pm N_{L/k}\mathfrak{a}\,=\,\pm a^{2}\,\stackrel{?}{=}\,\pm1$ in $L^{\times}$ , where $\underline{{\underline{{2}}}}$ means equal up to a square in $L^{\times}$ . Thus $\pm\eta\eta^{\prime}$ is a square in $L$ , so our assumption that $q_{2}=1$ implies that $\pm\eta\eta^{\prime}$ must be a square in $k_{2}$ . The same argument show that $\pm\eta/\eta^{\prime}$ is a square in $k_{2}$ , hence we find $\eta\in k_{2}$ . Thus $\alpha^{1-\sigma^{2}}$ is fixed by $\sigma^{2}$ and so $\beta:=\alpha^{2}\in k_{2}$ . This gives $K_{2}=k_{2}(\sqrt{\beta}\,)$ , hence $K_{2}/k_{2}$ is not essentially ramified, and moreover, $a\sim{\mathfrak{b}}$ . 冏口 </p>
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0003244v1
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[{"type": "text", "text": "Recall that a quadratic extension $K\\,=\\,k({\\sqrt{\\alpha}}\\,)$ is called essentially ramified if $\\alpha{\\mathcal{O}}_{k}$ is not an ideal square. This definition is independent of the choice of $\\alpha$ . ", "page_idx": 7}, {"type": "text", "text": "Proposition 6. Let $L/\\mathbb{Q}$ be a non-CM totally complex dihedral extension not containing $\\sqrt{-1}$ , and assume that $L/K_{1}$ and $L/K_{2}$ are essentially ramified. If the fundamental unit of the real quadratic subfield of $K$ has norm $^{-1}$ , then $q_{1}q_{2}=2$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. Notice first that $k$ cannot be real (in fact, $K$ is not totally real by assumption, and since $L/k$ is a cyclic quartic extension, no infinite prime can ramify in $K/k$ ); thus exactly one of $k_{1}$ , $k_{2}$ is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for $L/k_{1}$ and $L/k_{2}$ (note that $\\upsilon=0$ since both $L/K_{1}$ and $L/K_{2}$ are essentially ramified) we find that $2q_{1}q_{2}$ is a square. If we can prove that $q_{1},q_{2}\\leq2$ , then $2q_{1}q_{2}$ is a square between 2 and 8, which implies that we must have $2q_{1}q_{2}=4$ and $q_{1}q_{2}=2$ as claimed. ", "page_idx": 7}, {"type": "text", "text": "We start by remarking that if $\\zeta\\eta$ becomes a square in $L$ , where $\\zeta$ is a root of unity in $L$ , then so does one of $\\pm\\eta$ . This follows from the fact that the only non-trivial roots of unity that can be in $L$ are the sixth roots of unity $\\langle\\zeta_{6}\\rangle$ , and here $\\zeta_{6}=-\\zeta_{3}^{2}$ . ", "page_idx": 7}, {"type": "text", "text": "Now we prove that $q_{1}\\leq2$ under the assumptions we made; the claim $q_{2}\\leq2$ will then follow by symmetry. Assume first that $k_{1}$ is real and let $\\varepsilon$ be the fundamental unit of $k_{1}$ . We claim that $\\sqrt{\\pm\\varepsilon}\\notin L$ . Suppose otherwise; then $k_{1}(\\sqrt{\\pm\\varepsilon})$ is one of $K_{1}$ , $K_{1}^{\\prime}$ or $K$ . If $k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}$ , then $K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)$ and $K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)$ . (Here and below $x^{\\prime}\\,=\\,x^{\\sigma}$ .) This however cannot occur since by assumption $\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1$ implying that $\\sqrt{-1}\\in L$ , a contradiction. Similarly, if $k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K$ , then again $\\sqrt{-1}\\in L$ . ", "page_idx": 7}, {"type": "text", "text": "Thus $\\sqrt{\\pm\\varepsilon}~\\notin~L$ , and $E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle$ for some unit $\\eta\\ \\in\\ E_{1}$ . Suppose that $\\sqrt{u\\eta}\\in L$ for some unit $u\\in k_{1}$ . Then $L=K_{1}(\\sqrt{u\\eta}\\,)$ , contradicting our assumption that $L/K_{1}$ is essentially ramified. The same argument shows that $\\sqrt{u\\eta^{\\prime}}\\notin L$ , hence either $E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle$ and $q_{1}=1$ or $E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle$ for some unit $u\\in k_{1}$ and $q_{1}=2$ . Here $\\zeta$ is a root of unity generating the torsion subgroup $W_{L}$ of $E_{L}$ . ", "page_idx": 7}, {"type": "text", "text": "Next consider the case where $k_{1}$ is complex, and let $\\varepsilon$ denote the fundamental unit of $k_{2}$ . Then $\\pm\\varepsilon$ stays fundamental in $L$ by the argument above. ", "page_idx": 7}, {"type": "text", "text": "Let $\\eta$ be a fundamental unit in $K_{1}$ . If $\\pm\\eta$ became a square in $L$ , then clearly $L/K_{1}$ could not be essentially ramified. Thus if we have $q_{1}\\geq4$ , then $\\pm\\varepsilon\\eta=\\alpha^{2}$ is a square in $L$ . Applying $\\tau$ to this relation we find that $-1=\\varepsilon\\varepsilon^{\\prime}$ is a square in $L$ , contradicting the assumption that $L$ does not contain $\\sqrt{-1}$ . 口 ", "page_idx": 7}, {"type": "text", "text": "Proposition 7. Suppose that $q_{2}=1$ . Then $K_{2}/k_{2}$ is essentially ramified if and only if $\\kappa_{2}=1$ ; if $K_{2}/k_{2}$ is not essentially ramified, then $\\kappa_{2}=\\langle[6]\\rangle$ , where $K_{2}=$ $k_{2}(\\sqrt{\\beta}\\,)$ and $(\\beta)=\\mathfrak{b}^{2}$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. First notice that if $K_{2}/k_{2}$ is not essentially ramified, then $\\kappa_{2}\\neq1$ : in fact, in this case we have $(\\beta)\\;=\\;6^{2}$ , and if we had $\\,\\kappa_{2}\\,=\\,1$ , then $\\mathfrak{b}$ would have to be principal, say ${\\mathfrak{b}}=(\\gamma)$ . This implies that $\\beta\\,=\\,\\varepsilon\\gamma^{2}$ for some unit $\\varepsilon\\in k_{2}$ , which in view of $q_{2}=1$ implies that $\\varepsilon$ must be a square. But then $\\beta$ would be a square, and this is impossible. ", "page_idx": 7}, {"type": "text", "text": "Conversely, suppose $\\kappa_{2}\\neq1$ . Let $\\mathfrak{a}$ be a nonprincipal ideal in $k_{2}$ of absolute norm $a$ , and assume that ${\\mathfrak{a}}=(\\alpha)$ in $K_{2}$ . Then $\\alpha^{1-\\sigma^{2}}=\\eta$ for some unit $\\eta\\in E_{2}$ , and similarly $\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}$ , where $\\eta^{\\prime}$ is a unit in $E_{2}^{\\prime}$ . But then $\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}$ $N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1$ in $L^{\\times}$ , where $\\underline{{\\underline{{2}}}}$ means equal up to a square in $L^{\\times}$ . Thus $\\pm\\eta\\eta^{\\prime}$ is a square in $L$ , so our assumption that $q_{2}=1$ implies that $\\pm\\eta\\eta^{\\prime}$ must be a square in $k_{2}$ . The same argument show that $\\pm\\eta/\\eta^{\\prime}$ is a square in $k_{2}$ , hence we find $\\eta\\in k_{2}$ . Thus $\\alpha^{1-\\sigma^{2}}$ is fixed by $\\sigma^{2}$ and so $\\beta:=\\alpha^{2}\\in k_{2}$ . This gives $K_{2}=k_{2}(\\sqrt{\\beta}\\,)$ , hence $K_{2}/k_{2}$ is not essentially ramified, and moreover, $a\\sim{\\mathfrak{b}}$ . 冏口 ", "page_idx": 7}]
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 486, 136], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 291, 126], "score": 1.0, "content": "Recall that a quadratic extension ", "type": "text"}, {"bbox": [291, 115, 346, 126], "score": 0.95, "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [347, 114, 486, 126], "score": 1.0, "content": " is called essentially ramified if", "type": "text"}], "index": 0}, {"bbox": [126, 126, 462, 138], "spans": [{"bbox": [126, 128, 145, 137], "score": 0.92, "content": "\\alpha{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 9, "width": 19}, {"bbox": [145, 126, 452, 138], "score": 1.0, "content": " is not an ideal square. This definition is independent of the choice of ", "type": "text"}, {"bbox": [452, 131, 459, 135], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [459, 126, 462, 138], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [125, 142, 486, 179], "lines": [{"bbox": [126, 145, 487, 158], "spans": [{"bbox": [126, 145, 218, 158], "score": 1.0, "content": "Proposition 6. Let ", "type": "text"}, {"bbox": [218, 146, 239, 156], "score": 0.92, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [239, 145, 487, 158], "score": 1.0, "content": " be a non-CM totally complex dihedral extension not con-", "type": "text"}], "index": 2}, {"bbox": [126, 157, 487, 170], "spans": [{"bbox": [126, 157, 160, 170], "score": 1.0, "content": "taining ", "type": "text"}, {"bbox": [161, 157, 182, 168], "score": 0.91, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [183, 157, 267, 170], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [268, 158, 293, 169], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [293, 157, 317, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 158, 342, 169], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [343, 157, 487, 170], "score": 1.0, "content": " are essentially ramified. If the", "type": "text"}], "index": 3}, {"bbox": [126, 169, 481, 182], "spans": [{"bbox": [126, 169, 341, 182], "score": 1.0, "content": "fundamental unit of the real quadratic subfield of ", "type": "text"}, {"bbox": [342, 171, 351, 178], "score": 0.87, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [352, 169, 398, 182], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [399, 171, 412, 179], "score": 0.85, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [412, 169, 440, 182], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [440, 171, 477, 180], "score": 0.91, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [477, 169, 481, 182], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [125, 185, 486, 269], "lines": [{"bbox": [127, 188, 485, 200], "spans": [{"bbox": [127, 188, 227, 200], "score": 1.0, "content": "Proof. Notice first that ", "type": "text"}, {"bbox": [227, 189, 233, 197], "score": 0.9, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [233, 188, 333, 200], "score": 1.0, "content": " cannot be real (in fact,", "type": "text"}, {"bbox": [334, 190, 343, 197], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [343, 188, 485, 200], "score": 1.0, "content": " is not totally real by assumption,", "type": "text"}], "index": 5}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 170, 212], "score": 1.0, "content": "and since ", "type": "text"}, {"bbox": [170, 201, 188, 211], "score": 0.93, "content": "L/k", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [188, 199, 459, 212], "score": 1.0, "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "type": "text"}, {"bbox": [459, 201, 478, 211], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [479, 199, 486, 212], "score": 1.0, "content": ");", "type": "text"}], "index": 6}, {"bbox": [126, 212, 486, 224], "spans": [{"bbox": [126, 212, 213, 224], "score": 1.0, "content": "thus exactly one of ", "type": "text"}, {"bbox": [213, 213, 223, 222], "score": 0.9, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [223, 212, 228, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [229, 213, 239, 222], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [239, 212, 486, 224], "score": 1.0, "content": " is real, and the other is complex. Multiplying the class", "type": "text"}], "index": 7}, {"bbox": [125, 224, 486, 236], "spans": [{"bbox": [125, 224, 289, 236], "score": 1.0, "content": "number formulas, Proposition 3, for ", "type": "text"}, {"bbox": [289, 225, 311, 235], "score": 0.94, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [311, 224, 334, 236], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 225, 356, 235], "score": 0.94, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [357, 224, 408, 236], "score": 1.0, "content": " (note that ", "type": "text"}, {"bbox": [409, 226, 435, 233], "score": 0.91, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [436, 224, 486, 236], "score": 1.0, "content": " since both", "type": "text"}], "index": 8}, {"bbox": [126, 236, 487, 248], "spans": [{"bbox": [126, 237, 151, 247], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [151, 236, 173, 248], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [173, 237, 198, 247], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [198, 236, 365, 248], "score": 1.0, "content": " are essentially ramified) we find that ", "type": "text"}, {"bbox": [365, 238, 388, 247], "score": 0.91, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [388, 236, 487, 248], "score": 1.0, "content": " is a square. If we can", "type": "text"}], "index": 9}, {"bbox": [125, 248, 486, 260], "spans": [{"bbox": [125, 248, 174, 260], "score": 1.0, "content": "prove that", "type": "text"}, {"bbox": [175, 250, 217, 259], "score": 0.92, "content": "q_{1},q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [218, 248, 246, 260], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [247, 250, 270, 259], "score": 0.92, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [270, 248, 486, 260], "score": 1.0, "content": " is a square between 2 and 8, which implies that", "type": "text"}], "index": 10}, {"bbox": [125, 259, 340, 272], "spans": [{"bbox": [125, 259, 188, 272], "score": 1.0, "content": "we must have ", "type": "text"}, {"bbox": [189, 262, 230, 271], "score": 0.93, "content": "2q_{1}q_{2}=4", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [230, 259, 252, 272], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [252, 262, 288, 270], "score": 0.93, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [289, 259, 340, 272], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 11}], "index": 8}, {"type": "text", "bbox": [125, 270, 486, 306], "lines": [{"bbox": [137, 271, 485, 284], "spans": [{"bbox": [137, 271, 266, 284], "score": 1.0, "content": "We start by remarking that if ", "type": "text"}, {"bbox": [266, 273, 276, 282], "score": 0.92, "content": "\\zeta\\eta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [277, 271, 366, 284], "score": 1.0, "content": " becomes a square in ", "type": "text"}, {"bbox": [366, 273, 374, 280], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [374, 271, 406, 284], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [407, 273, 412, 282], "score": 0.9, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [412, 271, 485, 284], "score": 1.0, "content": " is a root of unity", "type": "text"}], "index": 12}, {"bbox": [126, 284, 486, 295], "spans": [{"bbox": [126, 284, 137, 295], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 285, 145, 293], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [145, 284, 239, 295], "score": 1.0, "content": ", then so does one of", "type": "text"}, {"bbox": [239, 286, 253, 294], "score": 0.9, "content": "\\pm\\eta", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [253, 284, 486, 295], "score": 1.0, "content": ". This follows from the fact that the only non-trivial", "type": "text"}], "index": 13}, {"bbox": [125, 295, 485, 307], "spans": [{"bbox": [125, 295, 251, 307], "score": 1.0, "content": "roots of unity that can be in ", "type": "text"}, {"bbox": [251, 297, 258, 304], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [258, 295, 380, 307], "score": 1.0, "content": " are the sixth roots of unity", "type": "text"}, {"bbox": [380, 297, 397, 307], "score": 0.93, "content": "\\langle\\zeta_{6}\\rangle", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [397, 295, 442, 307], "score": 1.0, "content": ", and here ", "type": "text"}, {"bbox": [442, 296, 482, 307], "score": 0.93, "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [482, 295, 485, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13}, {"type": "text", "bbox": [125, 306, 486, 390], "lines": [{"bbox": [137, 307, 486, 319], "spans": [{"bbox": [137, 307, 222, 319], "score": 1.0, "content": "Now we prove that ", "type": "text"}, {"bbox": [222, 309, 249, 318], "score": 0.92, "content": "q_{1}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [250, 307, 439, 319], "score": 1.0, "content": " under the assumptions we made; the claim ", "type": "text"}, {"bbox": [439, 309, 466, 318], "score": 0.92, "content": "q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [467, 307, 486, 319], "score": 1.0, "content": " will", "type": "text"}], "index": 15}, {"bbox": [125, 320, 487, 330], "spans": [{"bbox": [125, 320, 318, 330], "score": 1.0, "content": "then follow by symmetry. Assume first that ", "type": "text"}, {"bbox": [318, 321, 327, 330], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [328, 320, 392, 330], "score": 1.0, "content": " is real and let ", "type": "text"}, {"bbox": [393, 324, 397, 328], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 4}, {"bbox": [398, 320, 487, 330], "score": 1.0, "content": " be the fundamental", "type": "text"}], "index": 16}, {"bbox": [126, 331, 487, 344], "spans": [{"bbox": [126, 331, 159, 344], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [159, 333, 169, 342], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 331, 244, 344], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [244, 331, 285, 342], "score": 0.93, "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [286, 331, 403, 344], "score": 1.0, "content": ". Suppose otherwise; then ", "type": "text"}, {"bbox": [404, 331, 444, 343], "score": 0.93, "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [444, 331, 487, 344], "score": 1.0, "content": " is one of", "type": "text"}], "index": 17}, {"bbox": [126, 343, 487, 356], "spans": [{"bbox": [126, 346, 139, 354], "score": 0.89, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [139, 343, 145, 356], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [145, 345, 158, 356], "score": 0.91, "content": "K_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [158, 343, 174, 356], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [174, 346, 183, 353], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [184, 343, 201, 356], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [201, 344, 268, 356], "score": 0.94, "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [268, 343, 297, 356], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [297, 344, 367, 356], "score": 0.94, "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [367, 343, 389, 356], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [390, 344, 453, 356], "score": 0.93, "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "type": "inline_equation", "height": 12, "width": 63}, {"bbox": [453, 343, 487, 356], "score": 1.0, "content": ". (Here", "type": "text"}], "index": 18}, {"bbox": [125, 355, 485, 369], "spans": [{"bbox": [125, 355, 175, 369], "score": 1.0, "content": "and below ", "type": "text"}, {"bbox": [175, 357, 212, 365], "score": 0.91, "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "type": "inline_equation", "height": 8, "width": 37}, {"bbox": [212, 355, 443, 369], "score": 1.0, "content": ".) This however cannot occur since by assumption ", "type": "text"}, {"bbox": [443, 357, 485, 366], "score": 0.92, "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "type": "inline_equation", "height": 9, "width": 42}], "index": 19}, {"bbox": [125, 368, 487, 380], "spans": [{"bbox": [125, 368, 189, 380], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [190, 368, 232, 379], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [232, 368, 367, 380], "score": 1.0, "content": ", a contradiction. Similarly, if ", "type": "text"}, {"bbox": [367, 369, 432, 380], "score": 0.94, "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [432, 368, 487, 380], "score": 1.0, "content": ", then again", "type": "text"}], "index": 20}, {"bbox": [126, 380, 170, 392], "spans": [{"bbox": [126, 380, 166, 391], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [167, 380, 170, 392], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18}, {"type": "text", "bbox": [125, 390, 486, 450], "lines": [{"bbox": [137, 390, 486, 405], "spans": [{"bbox": [137, 390, 164, 405], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [164, 392, 210, 403], "score": 0.94, "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [210, 390, 238, 405], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [239, 393, 308, 403], "score": 0.94, "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [309, 390, 379, 405], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [379, 394, 414, 403], "score": 0.93, "content": "\\eta\\ \\in\\ E_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [414, 390, 486, 405], "score": 1.0, "content": ". Suppose that", "type": "text"}], "index": 22}, {"bbox": [126, 404, 485, 417], "spans": [{"bbox": [126, 406, 164, 416], "score": 0.93, "content": "\\sqrt{u\\eta}\\in L", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [165, 404, 228, 417], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [228, 406, 256, 415], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [257, 404, 289, 417], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [289, 405, 351, 416], "score": 0.94, "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [352, 404, 485, 417], "score": 1.0, "content": ", contradicting our assumption", "type": "text"}], "index": 23}, {"bbox": [126, 417, 486, 429], "spans": [{"bbox": [126, 417, 147, 429], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 418, 172, 428], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [172, 417, 413, 429], "score": 1.0, "content": " is essentially ramified. The same argument shows that ", "type": "text"}, {"bbox": [414, 417, 455, 428], "score": 0.94, "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [455, 417, 486, 429], "score": 1.0, "content": ", hence", "type": "text"}], "index": 24}, {"bbox": [126, 428, 484, 442], "spans": [{"bbox": [126, 428, 154, 442], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [155, 430, 227, 440], "score": 0.94, "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [227, 428, 250, 442], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [251, 431, 279, 439], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [280, 428, 295, 442], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [296, 429, 389, 440], "score": 0.93, "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "type": "inline_equation", "height": 11, "width": 93}, {"bbox": [390, 428, 455, 442], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [455, 430, 484, 439], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 29}], "index": 25}, {"bbox": [126, 441, 479, 452], "spans": [{"bbox": [126, 441, 145, 452], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [145, 443, 173, 451], "score": 0.92, "content": "q_{1}=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [173, 441, 203, 452], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [203, 442, 208, 451], "score": 0.88, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [209, 441, 431, 452], "score": 1.0, "content": " is a root of unity generating the torsion subgroup ", "type": "text"}, {"bbox": [431, 442, 446, 451], "score": 0.92, "content": "W_{L}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [447, 441, 461, 452], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [461, 442, 475, 451], "score": 0.91, "content": "E_{L}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [475, 441, 479, 452], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 24}, {"type": "text", "bbox": [125, 451, 486, 474], "lines": [{"bbox": [137, 451, 486, 465], "spans": [{"bbox": [137, 451, 270, 465], "score": 1.0, "content": "Next consider the case where ", "type": "text"}, {"bbox": [271, 454, 280, 463], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 451, 371, 465], "score": 1.0, "content": " is complex, and let ", "type": "text"}, {"bbox": [371, 457, 376, 461], "score": 0.87, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [377, 451, 486, 465], "score": 1.0, "content": " denote the fundamental", "type": "text"}], "index": 27}, {"bbox": [126, 464, 425, 476], "spans": [{"bbox": [126, 464, 158, 476], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [158, 466, 168, 475], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [168, 464, 201, 476], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [201, 466, 214, 474], "score": 0.86, "content": "\\pm\\varepsilon", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [214, 464, 311, 476], "score": 1.0, "content": " stays fundamental in ", "type": "text"}, {"bbox": [311, 466, 318, 473], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [318, 464, 425, 476], "score": 1.0, "content": " by the argument above.", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [125, 475, 486, 522], "lines": [{"bbox": [137, 477, 485, 488], "spans": [{"bbox": [137, 477, 156, 488], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [156, 481, 162, 488], "score": 0.9, "content": "\\eta", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 477, 280, 488], "score": 1.0, "content": " be a fundamental unit in ", "type": "text"}, {"bbox": [280, 478, 294, 487], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [294, 477, 313, 488], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [313, 479, 326, 488], "score": 0.89, "content": "\\pm\\eta", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [327, 477, 419, 488], "score": 1.0, "content": " became a square in ", "type": "text"}, {"bbox": [419, 478, 426, 486], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [426, 477, 485, 488], "score": 1.0, "content": ", then clearly", "type": "text"}], "index": 29}, {"bbox": [126, 488, 487, 501], "spans": [{"bbox": [126, 489, 151, 500], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [151, 488, 375, 501], "score": 1.0, "content": " could not be essentially ramified. Thus if we have ", "type": "text"}, {"bbox": [376, 491, 403, 499], "score": 0.91, "content": "q_{1}\\geq4", "type": "inline_equation", "height": 8, "width": 27}, {"bbox": [404, 488, 432, 501], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [432, 489, 474, 500], "score": 0.93, "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [475, 488, 487, 501], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [125, 500, 487, 513], "spans": [{"bbox": [125, 500, 177, 513], "score": 1.0, "content": "a square in ", "type": "text"}, {"bbox": [178, 502, 185, 509], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [185, 500, 235, 513], "score": 1.0, "content": ". Applying ", "type": "text"}, {"bbox": [236, 505, 241, 509], "score": 0.87, "content": "\\tau", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [242, 500, 370, 513], "score": 1.0, "content": " to this relation we find that ", "type": "text"}, {"bbox": [371, 501, 409, 510], "score": 0.9, "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [410, 500, 475, 513], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [475, 502, 482, 509], "score": 0.87, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [483, 500, 487, 513], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [125, 512, 486, 524], "spans": [{"bbox": [125, 512, 277, 524], "score": 1.0, "content": "contradicting the assumption that", "type": "text"}, {"bbox": [278, 514, 285, 521], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 512, 363, 524], "score": 1.0, "content": " does not contain ", "type": "text"}, {"bbox": [363, 513, 385, 523], "score": 0.92, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [385, 512, 388, 524], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [476, 513, 486, 522], "score": 0.9836314916610718, "content": "口", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "text", "bbox": [125, 530, 486, 567], "lines": [{"bbox": [126, 532, 487, 545], "spans": [{"bbox": [126, 532, 261, 545], "score": 1.0, "content": "Proposition 7. Suppose that ", "type": "text"}, {"bbox": [261, 534, 291, 543], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [292, 532, 327, 545], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [327, 533, 355, 544], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [355, 532, 487, 545], "score": 1.0, "content": " is essentially ramified if and", "type": "text"}], "index": 33}, {"bbox": [127, 543, 486, 558], "spans": [{"bbox": [127, 543, 158, 558], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [158, 546, 189, 555], "score": 0.84, "content": "\\kappa_{2}=1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [190, 543, 205, 558], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [206, 545, 234, 556], "score": 0.9, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [234, 543, 380, 558], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [380, 545, 424, 556], "score": 0.93, "content": "\\kappa_{2}=\\langle[6]\\rangle", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [425, 543, 460, 558], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [460, 545, 486, 555], "score": 0.87, "content": "K_{2}=", "type": "inline_equation", "height": 10, "width": 26}], "index": 34}, {"bbox": [126, 556, 223, 568], "spans": [{"bbox": [126, 556, 160, 568], "score": 0.92, "content": "k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 556, 182, 568], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [182, 556, 219, 568], "score": 0.92, "content": "(\\beta)=\\mathfrak{b}^{2}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [220, 556, 223, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34}, {"type": "text", "bbox": [125, 573, 486, 632], "lines": [{"bbox": [126, 574, 485, 587], "spans": [{"bbox": [126, 574, 243, 587], "score": 1.0, "content": "Proof. First notice that if ", "type": "text"}, {"bbox": [243, 576, 271, 586], "score": 0.93, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [271, 574, 415, 587], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [416, 577, 446, 586], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [446, 574, 485, 587], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 36}, {"bbox": [124, 586, 486, 599], "spans": [{"bbox": [124, 586, 221, 599], "score": 1.0, "content": "in this case we have ", "type": "text"}, {"bbox": [221, 587, 261, 598], "score": 0.93, "content": "(\\beta)\\;=\\;6^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [261, 586, 334, 599], "score": 1.0, "content": ", and if we had ", "type": "text"}, {"bbox": [335, 589, 367, 597], "score": 0.91, "content": "\\,\\kappa_{2}\\,=\\,1", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [367, 586, 397, 599], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [398, 588, 403, 596], "score": 0.83, "content": "\\mathfrak{b}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [403, 586, 486, 599], "score": 1.0, "content": " would have to be", "type": "text"}], "index": 37}, {"bbox": [126, 599, 486, 611], "spans": [{"bbox": [126, 599, 188, 611], "score": 1.0, "content": "principal, say ", "type": "text"}, {"bbox": [189, 600, 222, 610], "score": 0.93, "content": "{\\mathfrak{b}}=(\\gamma)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [222, 599, 309, 611], "score": 1.0, "content": ". This implies that", "type": "text"}, {"bbox": [310, 600, 346, 610], "score": 0.93, "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [347, 599, 412, 611], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [412, 601, 441, 609], "score": 0.92, "content": "\\varepsilon\\in k_{2}", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [441, 599, 486, 611], "score": 1.0, "content": ", which in", "type": "text"}], "index": 38}, {"bbox": [125, 611, 487, 623], "spans": [{"bbox": [125, 611, 159, 623], "score": 1.0, "content": "view of ", "type": "text"}, {"bbox": [160, 613, 187, 622], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [187, 611, 244, 623], "score": 1.0, "content": " implies that", "type": "text"}, {"bbox": [245, 615, 249, 620], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [250, 611, 375, 623], "score": 1.0, "content": " must be a square. But then ", "type": "text"}, {"bbox": [375, 613, 381, 622], "score": 0.91, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [382, 611, 487, 623], "score": 1.0, "content": " would be a square, and", "type": "text"}], "index": 39}, {"bbox": [125, 623, 204, 635], "spans": [{"bbox": [125, 623, 204, 635], "score": 1.0, "content": "this is impossible.", "type": "text"}], "index": 40}], "index": 38}, {"type": "text", "bbox": [124, 633, 486, 700], "lines": [{"bbox": [138, 635, 487, 647], "spans": [{"bbox": [138, 635, 226, 647], "score": 1.0, "content": "Conversely, suppose", "type": "text"}, {"bbox": [227, 636, 255, 646], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 635, 280, 647], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [280, 639, 285, 644], "score": 0.87, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [285, 635, 399, 647], "score": 1.0, "content": " be a nonprincipal ideal in", "type": "text"}, {"bbox": [400, 636, 409, 645], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [410, 635, 487, 647], "score": 1.0, "content": " of absolute norm", "type": "text"}], "index": 41}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 652, 132, 657], "score": 0.85, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [132, 647, 216, 660], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [216, 649, 250, 659], "score": 0.94, "content": "{\\mathfrak{a}}=(\\alpha)", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [251, 647, 266, 660], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [267, 649, 280, 658], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [280, 647, 315, 660], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [316, 646, 363, 659], "score": 0.95, "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [363, 647, 429, 660], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [430, 649, 462, 659], "score": 0.93, "content": "\\eta\\in E_{2}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [462, 647, 486, 660], "score": 1.0, "content": ", and", "type": "text"}], "index": 42}, {"bbox": [123, 658, 485, 676], "spans": [{"bbox": [123, 658, 167, 676], "score": 1.0, "content": "similarly ", "type": "text"}, {"bbox": [168, 661, 219, 673], "score": 0.93, "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [220, 658, 256, 676], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [256, 663, 264, 673], "score": 0.9, "content": "\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [265, 658, 324, 676], "score": 1.0, "content": " is a unit in ", "type": "text"}, {"bbox": [324, 663, 336, 673], "score": 0.92, "content": "E_{2}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [336, 658, 390, 676], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [390, 660, 485, 673], "score": 0.9, "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "type": "inline_equation", "height": 13, "width": 95}], "index": 43}, {"bbox": [125, 674, 487, 690], "spans": [{"bbox": [125, 675, 269, 689], "score": 0.89, "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "type": "inline_equation", "height": 14, "width": 144}, {"bbox": [270, 674, 285, 690], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [285, 677, 299, 686], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [299, 674, 335, 690], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [336, 675, 344, 686], "score": 0.71, "content": "\\underline{{\\underline{{2}}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [344, 674, 487, 690], "score": 1.0, "content": " means equal up to a square in", "type": "text"}], "index": 44}, {"bbox": [126, 689, 484, 702], "spans": [{"bbox": [126, 690, 140, 698], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [140, 689, 172, 702], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [172, 690, 194, 700], "score": 0.92, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [194, 689, 258, 702], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [259, 691, 266, 698], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [266, 689, 376, 702], "score": 1.0, "content": ", so our assumption that ", "type": "text"}, {"bbox": [376, 691, 405, 700], "score": 0.93, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [405, 689, 463, 702], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [464, 690, 484, 700], "score": 0.91, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 20}], "index": 45}], "index": 43}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [125, 91, 131, 99], "lines": [{"bbox": [126, 93, 131, 101], "spans": [{"bbox": [126, 93, 131, 101], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 136], "lines": [{"bbox": [137, 114, 486, 126], "spans": [{"bbox": [137, 114, 291, 126], "score": 1.0, "content": "Recall that a quadratic extension ", "type": "text"}, {"bbox": [291, 115, 346, 126], "score": 0.95, "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [347, 114, 486, 126], "score": 1.0, "content": " is called essentially ramified if", "type": "text"}], "index": 0}, {"bbox": [126, 126, 462, 138], "spans": [{"bbox": [126, 128, 145, 137], "score": 0.92, "content": "\\alpha{\\mathcal{O}}_{k}", "type": "inline_equation", "height": 9, "width": 19}, {"bbox": [145, 126, 452, 138], "score": 1.0, "content": " is not an ideal square. This definition is independent of the choice of ", "type": "text"}, {"bbox": [452, 131, 459, 135], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 7}, {"bbox": [459, 126, 462, 138], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [126, 114, 486, 138]}, {"type": "text", "bbox": [125, 142, 486, 179], "lines": [{"bbox": [126, 145, 487, 158], "spans": [{"bbox": [126, 145, 218, 158], "score": 1.0, "content": "Proposition 6. Let ", "type": "text"}, {"bbox": [218, 146, 239, 156], "score": 0.92, "content": "L/\\mathbb{Q}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [239, 145, 487, 158], "score": 1.0, "content": " be a non-CM totally complex dihedral extension not con-", "type": "text"}], "index": 2}, {"bbox": [126, 157, 487, 170], "spans": [{"bbox": [126, 157, 160, 170], "score": 1.0, "content": "taining ", "type": "text"}, {"bbox": [161, 157, 182, 168], "score": 0.91, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [183, 157, 267, 170], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [268, 158, 293, 169], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [293, 157, 317, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 158, 342, 169], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [343, 157, 487, 170], "score": 1.0, "content": " are essentially ramified. If the", "type": "text"}], "index": 3}, {"bbox": [126, 169, 481, 182], "spans": [{"bbox": [126, 169, 341, 182], "score": 1.0, "content": "fundamental unit of the real quadratic subfield of ", "type": "text"}, {"bbox": [342, 171, 351, 178], "score": 0.87, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [352, 169, 398, 182], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [399, 171, 412, 179], "score": 0.85, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [412, 169, 440, 182], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [440, 171, 477, 180], "score": 0.91, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [477, 169, 481, 182], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3, "bbox_fs": [126, 145, 487, 182]}, {"type": "text", "bbox": [125, 185, 486, 269], "lines": [{"bbox": [127, 188, 485, 200], "spans": [{"bbox": [127, 188, 227, 200], "score": 1.0, "content": "Proof. Notice first that ", "type": "text"}, {"bbox": [227, 189, 233, 197], "score": 0.9, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [233, 188, 333, 200], "score": 1.0, "content": " cannot be real (in fact,", "type": "text"}, {"bbox": [334, 190, 343, 197], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [343, 188, 485, 200], "score": 1.0, "content": " is not totally real by assumption,", "type": "text"}], "index": 5}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 170, 212], "score": 1.0, "content": "and since ", "type": "text"}, {"bbox": [170, 201, 188, 211], "score": 0.93, "content": "L/k", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [188, 199, 459, 212], "score": 1.0, "content": " is a cyclic quartic extension, no infinite prime can ramify in ", "type": "text"}, {"bbox": [459, 201, 478, 211], "score": 0.91, "content": "K/k", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [479, 199, 486, 212], "score": 1.0, "content": ");", "type": "text"}], "index": 6}, {"bbox": [126, 212, 486, 224], "spans": [{"bbox": [126, 212, 213, 224], "score": 1.0, "content": "thus exactly one of ", "type": "text"}, {"bbox": [213, 213, 223, 222], "score": 0.9, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [223, 212, 228, 224], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [229, 213, 239, 222], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [239, 212, 486, 224], "score": 1.0, "content": " is real, and the other is complex. Multiplying the class", "type": "text"}], "index": 7}, {"bbox": [125, 224, 486, 236], "spans": [{"bbox": [125, 224, 289, 236], "score": 1.0, "content": "number formulas, Proposition 3, for ", "type": "text"}, {"bbox": [289, 225, 311, 235], "score": 0.94, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [311, 224, 334, 236], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 225, 356, 235], "score": 0.94, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [357, 224, 408, 236], "score": 1.0, "content": " (note that ", "type": "text"}, {"bbox": [409, 226, 435, 233], "score": 0.91, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [436, 224, 486, 236], "score": 1.0, "content": " since both", "type": "text"}], "index": 8}, {"bbox": [126, 236, 487, 248], "spans": [{"bbox": [126, 237, 151, 247], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [151, 236, 173, 248], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [173, 237, 198, 247], "score": 0.93, "content": "L/K_{2}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [198, 236, 365, 248], "score": 1.0, "content": " are essentially ramified) we find that ", "type": "text"}, {"bbox": [365, 238, 388, 247], "score": 0.91, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [388, 236, 487, 248], "score": 1.0, "content": " is a square. If we can", "type": "text"}], "index": 9}, {"bbox": [125, 248, 486, 260], "spans": [{"bbox": [125, 248, 174, 260], "score": 1.0, "content": "prove that", "type": "text"}, {"bbox": [175, 250, 217, 259], "score": 0.92, "content": "q_{1},q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 42}, {"bbox": [218, 248, 246, 260], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [247, 250, 270, 259], "score": 0.92, "content": "2q_{1}q_{2}", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [270, 248, 486, 260], "score": 1.0, "content": " is a square between 2 and 8, which implies that", "type": "text"}], "index": 10}, {"bbox": [125, 259, 340, 272], "spans": [{"bbox": [125, 259, 188, 272], "score": 1.0, "content": "we must have ", "type": "text"}, {"bbox": [189, 262, 230, 271], "score": 0.93, "content": "2q_{1}q_{2}=4", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [230, 259, 252, 272], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [252, 262, 288, 270], "score": 0.93, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [289, 259, 340, 272], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 11}], "index": 8, "bbox_fs": [125, 188, 487, 272]}, {"type": "text", "bbox": [125, 270, 486, 306], "lines": [{"bbox": [137, 271, 485, 284], "spans": [{"bbox": [137, 271, 266, 284], "score": 1.0, "content": "We start by remarking that if ", "type": "text"}, {"bbox": [266, 273, 276, 282], "score": 0.92, "content": "\\zeta\\eta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [277, 271, 366, 284], "score": 1.0, "content": " becomes a square in ", "type": "text"}, {"bbox": [366, 273, 374, 280], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [374, 271, 406, 284], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [407, 273, 412, 282], "score": 0.9, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [412, 271, 485, 284], "score": 1.0, "content": " is a root of unity", "type": "text"}], "index": 12}, {"bbox": [126, 284, 486, 295], "spans": [{"bbox": [126, 284, 137, 295], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 285, 145, 293], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [145, 284, 239, 295], "score": 1.0, "content": ", then so does one of", "type": "text"}, {"bbox": [239, 286, 253, 294], "score": 0.9, "content": "\\pm\\eta", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [253, 284, 486, 295], "score": 1.0, "content": ". This follows from the fact that the only non-trivial", "type": "text"}], "index": 13}, {"bbox": [125, 295, 485, 307], "spans": [{"bbox": [125, 295, 251, 307], "score": 1.0, "content": "roots of unity that can be in ", "type": "text"}, {"bbox": [251, 297, 258, 304], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [258, 295, 380, 307], "score": 1.0, "content": " are the sixth roots of unity", "type": "text"}, {"bbox": [380, 297, 397, 307], "score": 0.93, "content": "\\langle\\zeta_{6}\\rangle", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [397, 295, 442, 307], "score": 1.0, "content": ", and here ", "type": "text"}, {"bbox": [442, 296, 482, 307], "score": 0.93, "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [482, 295, 485, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 13, "bbox_fs": [125, 271, 486, 307]}, {"type": "text", "bbox": [125, 306, 486, 390], "lines": [{"bbox": [137, 307, 486, 319], "spans": [{"bbox": [137, 307, 222, 319], "score": 1.0, "content": "Now we prove that ", "type": "text"}, {"bbox": [222, 309, 249, 318], "score": 0.92, "content": "q_{1}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [250, 307, 439, 319], "score": 1.0, "content": " under the assumptions we made; the claim ", "type": "text"}, {"bbox": [439, 309, 466, 318], "score": 0.92, "content": "q_{2}\\leq2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [467, 307, 486, 319], "score": 1.0, "content": " will", "type": "text"}], "index": 15}, {"bbox": [125, 320, 487, 330], "spans": [{"bbox": [125, 320, 318, 330], "score": 1.0, "content": "then follow by symmetry. Assume first that ", "type": "text"}, {"bbox": [318, 321, 327, 330], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [328, 320, 392, 330], "score": 1.0, "content": " is real and let ", "type": "text"}, {"bbox": [393, 324, 397, 328], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 4}, {"bbox": [398, 320, 487, 330], "score": 1.0, "content": " be the fundamental", "type": "text"}], "index": 16}, {"bbox": [126, 331, 487, 344], "spans": [{"bbox": [126, 331, 159, 344], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [159, 333, 169, 342], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 331, 244, 344], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [244, 331, 285, 342], "score": 0.93, "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [286, 331, 403, 344], "score": 1.0, "content": ". Suppose otherwise; then ", "type": "text"}, {"bbox": [404, 331, 444, 343], "score": 0.93, "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [444, 331, 487, 344], "score": 1.0, "content": " is one of", "type": "text"}], "index": 17}, {"bbox": [126, 343, 487, 356], "spans": [{"bbox": [126, 346, 139, 354], "score": 0.89, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [139, 343, 145, 356], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [145, 345, 158, 356], "score": 0.91, "content": "K_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [158, 343, 174, 356], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [174, 346, 183, 353], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [184, 343, 201, 356], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [201, 344, 268, 356], "score": 0.94, "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [268, 343, 297, 356], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [297, 344, 367, 356], "score": 0.94, "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [367, 343, 389, 356], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [390, 344, 453, 356], "score": 0.93, "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "type": "inline_equation", "height": 12, "width": 63}, {"bbox": [453, 343, 487, 356], "score": 1.0, "content": ". (Here", "type": "text"}], "index": 18}, {"bbox": [125, 355, 485, 369], "spans": [{"bbox": [125, 355, 175, 369], "score": 1.0, "content": "and below ", "type": "text"}, {"bbox": [175, 357, 212, 365], "score": 0.91, "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "type": "inline_equation", "height": 8, "width": 37}, {"bbox": [212, 355, 443, 369], "score": 1.0, "content": ".) This however cannot occur since by assumption ", "type": "text"}, {"bbox": [443, 357, 485, 366], "score": 0.92, "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "type": "inline_equation", "height": 9, "width": 42}], "index": 19}, {"bbox": [125, 368, 487, 380], "spans": [{"bbox": [125, 368, 189, 380], "score": 1.0, "content": "implying that ", "type": "text"}, {"bbox": [190, 368, 232, 379], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [232, 368, 367, 380], "score": 1.0, "content": ", a contradiction. Similarly, if ", "type": "text"}, {"bbox": [367, 369, 432, 380], "score": 0.94, "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [432, 368, 487, 380], "score": 1.0, "content": ", then again", "type": "text"}], "index": 20}, {"bbox": [126, 380, 170, 392], "spans": [{"bbox": [126, 380, 166, 391], "score": 0.92, "content": "\\sqrt{-1}\\in L", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [167, 380, 170, 392], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18, "bbox_fs": [125, 307, 487, 392]}, {"type": "text", "bbox": [125, 390, 486, 450], "lines": [{"bbox": [137, 390, 486, 405], "spans": [{"bbox": [137, 390, 164, 405], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [164, 392, 210, 403], "score": 0.94, "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [210, 390, 238, 405], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [239, 393, 308, 403], "score": 0.94, "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [309, 390, 379, 405], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [379, 394, 414, 403], "score": 0.93, "content": "\\eta\\ \\in\\ E_{1}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [414, 390, 486, 405], "score": 1.0, "content": ". Suppose that", "type": "text"}], "index": 22}, {"bbox": [126, 404, 485, 417], "spans": [{"bbox": [126, 406, 164, 416], "score": 0.93, "content": "\\sqrt{u\\eta}\\in L", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [165, 404, 228, 417], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [228, 406, 256, 415], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [257, 404, 289, 417], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [289, 405, 351, 416], "score": 0.94, "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [352, 404, 485, 417], "score": 1.0, "content": ", contradicting our assumption", "type": "text"}], "index": 23}, {"bbox": [126, 417, 486, 429], "spans": [{"bbox": [126, 417, 147, 429], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 418, 172, 428], "score": 0.94, "content": "L/K_{1}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [172, 417, 413, 429], "score": 1.0, "content": " is essentially ramified. The same argument shows that ", "type": "text"}, {"bbox": [414, 417, 455, 428], "score": 0.94, "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [455, 417, 486, 429], "score": 1.0, "content": ", hence", "type": "text"}], "index": 24}, {"bbox": [126, 428, 484, 442], "spans": [{"bbox": [126, 428, 154, 442], "score": 1.0, "content": "either ", "type": "text"}, {"bbox": [155, 430, 227, 440], "score": 0.94, "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [227, 428, 250, 442], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [251, 431, 279, 439], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [280, 428, 295, 442], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [296, 429, 389, 440], "score": 0.93, "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "type": "inline_equation", "height": 11, "width": 93}, {"bbox": [390, 428, 455, 442], "score": 1.0, "content": "for some unit ", "type": "text"}, {"bbox": [455, 430, 484, 439], "score": 0.92, "content": "u\\in k_{1}", "type": "inline_equation", "height": 9, "width": 29}], "index": 25}, {"bbox": [126, 441, 479, 452], "spans": [{"bbox": [126, 441, 145, 452], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [145, 443, 173, 451], "score": 0.92, "content": "q_{1}=2", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [173, 441, 203, 452], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [203, 442, 208, 451], "score": 0.88, "content": "\\zeta", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [209, 441, 431, 452], "score": 1.0, "content": " is a root of unity generating the torsion subgroup ", "type": "text"}, {"bbox": [431, 442, 446, 451], "score": 0.92, "content": "W_{L}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [447, 441, 461, 452], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [461, 442, 475, 451], "score": 0.91, "content": "E_{L}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [475, 441, 479, 452], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 24, "bbox_fs": [126, 390, 486, 452]}, {"type": "text", "bbox": [125, 451, 486, 474], "lines": [{"bbox": [137, 451, 486, 465], "spans": [{"bbox": [137, 451, 270, 465], "score": 1.0, "content": "Next consider the case where ", "type": "text"}, {"bbox": [271, 454, 280, 463], "score": 0.92, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 451, 371, 465], "score": 1.0, "content": " is complex, and let ", "type": "text"}, {"bbox": [371, 457, 376, 461], "score": 0.87, "content": "\\varepsilon", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [377, 451, 486, 465], "score": 1.0, "content": " denote the fundamental", "type": "text"}], "index": 27}, {"bbox": [126, 464, 425, 476], "spans": [{"bbox": [126, 464, 158, 476], "score": 1.0, "content": "unit of ", "type": "text"}, {"bbox": [158, 466, 168, 475], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [168, 464, 201, 476], "score": 1.0, "content": ". Then", "type": "text"}, {"bbox": [201, 466, 214, 474], "score": 0.86, "content": "\\pm\\varepsilon", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [214, 464, 311, 476], "score": 1.0, "content": " stays fundamental in ", "type": "text"}, {"bbox": [311, 466, 318, 473], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [318, 464, 425, 476], "score": 1.0, "content": " by the argument above.", "type": "text"}], "index": 28}], "index": 27.5, "bbox_fs": [126, 451, 486, 476]}, {"type": "text", "bbox": [125, 475, 486, 522], "lines": [{"bbox": [137, 477, 485, 488], "spans": [{"bbox": [137, 477, 156, 488], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [156, 481, 162, 488], "score": 0.9, "content": "\\eta", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [162, 477, 280, 488], "score": 1.0, "content": " be a fundamental unit in ", "type": "text"}, {"bbox": [280, 478, 294, 487], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [294, 477, 313, 488], "score": 1.0, "content": ". If", "type": "text"}, {"bbox": [313, 479, 326, 488], "score": 0.89, "content": "\\pm\\eta", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [327, 477, 419, 488], "score": 1.0, "content": " became a square in ", "type": "text"}, {"bbox": [419, 478, 426, 486], "score": 0.89, "content": "L", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [426, 477, 485, 488], "score": 1.0, "content": ", then clearly", "type": "text"}], "index": 29}, {"bbox": [126, 488, 487, 501], "spans": [{"bbox": [126, 489, 151, 500], "score": 0.93, "content": "L/K_{1}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [151, 488, 375, 501], "score": 1.0, "content": " could not be essentially ramified. Thus if we have ", "type": "text"}, {"bbox": [376, 491, 403, 499], "score": 0.91, "content": "q_{1}\\geq4", "type": "inline_equation", "height": 8, "width": 27}, {"bbox": [404, 488, 432, 501], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [432, 489, 474, 500], "score": 0.93, "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [475, 488, 487, 501], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [125, 500, 487, 513], "spans": [{"bbox": [125, 500, 177, 513], "score": 1.0, "content": "a square in ", "type": "text"}, {"bbox": [178, 502, 185, 509], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [185, 500, 235, 513], "score": 1.0, "content": ". Applying ", "type": "text"}, {"bbox": [236, 505, 241, 509], "score": 0.87, "content": "\\tau", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [242, 500, 370, 513], "score": 1.0, "content": " to this relation we find that ", "type": "text"}, {"bbox": [371, 501, 409, 510], "score": 0.9, "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [410, 500, 475, 513], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [475, 502, 482, 509], "score": 0.87, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [483, 500, 487, 513], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [125, 512, 486, 524], "spans": [{"bbox": [125, 512, 277, 524], "score": 1.0, "content": "contradicting the assumption that", "type": "text"}, {"bbox": [278, 514, 285, 521], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [286, 512, 363, 524], "score": 1.0, "content": " does not contain ", "type": "text"}, {"bbox": [363, 513, 385, 523], "score": 0.92, "content": "\\sqrt{-1}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [385, 512, 388, 524], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [476, 513, 486, 522], "score": 0.9836314916610718, "content": "口", "type": "text"}], "index": 32}], "index": 30.5, "bbox_fs": [125, 477, 487, 524]}, {"type": "text", "bbox": [125, 530, 486, 567], "lines": [{"bbox": [126, 532, 487, 545], "spans": [{"bbox": [126, 532, 261, 545], "score": 1.0, "content": "Proposition 7. Suppose that ", "type": "text"}, {"bbox": [261, 534, 291, 543], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [292, 532, 327, 545], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [327, 533, 355, 544], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [355, 532, 487, 545], "score": 1.0, "content": " is essentially ramified if and", "type": "text"}], "index": 33}, {"bbox": [127, 543, 486, 558], "spans": [{"bbox": [127, 543, 158, 558], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [158, 546, 189, 555], "score": 0.84, "content": "\\kappa_{2}=1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [190, 543, 205, 558], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [206, 545, 234, 556], "score": 0.9, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [234, 543, 380, 558], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [380, 545, 424, 556], "score": 0.93, "content": "\\kappa_{2}=\\langle[6]\\rangle", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [425, 543, 460, 558], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [460, 545, 486, 555], "score": 0.87, "content": "K_{2}=", "type": "inline_equation", "height": 10, "width": 26}], "index": 34}, {"bbox": [126, 556, 223, 568], "spans": [{"bbox": [126, 556, 160, 568], "score": 0.92, "content": "k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [160, 556, 182, 568], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [182, 556, 219, 568], "score": 0.92, "content": "(\\beta)=\\mathfrak{b}^{2}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [220, 556, 223, 568], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 34, "bbox_fs": [126, 532, 487, 568]}, {"type": "text", "bbox": [125, 573, 486, 632], "lines": [{"bbox": [126, 574, 485, 587], "spans": [{"bbox": [126, 574, 243, 587], "score": 1.0, "content": "Proof. First notice that if ", "type": "text"}, {"bbox": [243, 576, 271, 586], "score": 0.93, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [271, 574, 415, 587], "score": 1.0, "content": " is not essentially ramified, then ", "type": "text"}, {"bbox": [416, 577, 446, 586], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [446, 574, 485, 587], "score": 1.0, "content": ": in fact,", "type": "text"}], "index": 36}, {"bbox": [124, 586, 486, 599], "spans": [{"bbox": [124, 586, 221, 599], "score": 1.0, "content": "in this case we have ", "type": "text"}, {"bbox": [221, 587, 261, 598], "score": 0.93, "content": "(\\beta)\\;=\\;6^{2}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [261, 586, 334, 599], "score": 1.0, "content": ", and if we had ", "type": "text"}, {"bbox": [335, 589, 367, 597], "score": 0.91, "content": "\\,\\kappa_{2}\\,=\\,1", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [367, 586, 397, 599], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [398, 588, 403, 596], "score": 0.83, "content": "\\mathfrak{b}", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [403, 586, 486, 599], "score": 1.0, "content": " would have to be", "type": "text"}], "index": 37}, {"bbox": [126, 599, 486, 611], "spans": [{"bbox": [126, 599, 188, 611], "score": 1.0, "content": "principal, say ", "type": "text"}, {"bbox": [189, 600, 222, 610], "score": 0.93, "content": "{\\mathfrak{b}}=(\\gamma)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [222, 599, 309, 611], "score": 1.0, "content": ". This implies that", "type": "text"}, {"bbox": [310, 600, 346, 610], "score": 0.93, "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [347, 599, 412, 611], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [412, 601, 441, 609], "score": 0.92, "content": "\\varepsilon\\in k_{2}", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [441, 599, 486, 611], "score": 1.0, "content": ", which in", "type": "text"}], "index": 38}, {"bbox": [125, 611, 487, 623], "spans": [{"bbox": [125, 611, 159, 623], "score": 1.0, "content": "view of ", "type": "text"}, {"bbox": [160, 613, 187, 622], "score": 0.92, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [187, 611, 244, 623], "score": 1.0, "content": " implies that", "type": "text"}, {"bbox": [245, 615, 249, 620], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [250, 611, 375, 623], "score": 1.0, "content": " must be a square. But then ", "type": "text"}, {"bbox": [375, 613, 381, 622], "score": 0.91, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [382, 611, 487, 623], "score": 1.0, "content": " would be a square, and", "type": "text"}], "index": 39}, {"bbox": [125, 623, 204, 635], "spans": [{"bbox": [125, 623, 204, 635], "score": 1.0, "content": "this is impossible.", "type": "text"}], "index": 40}], "index": 38, "bbox_fs": [124, 574, 487, 635]}, {"type": "text", "bbox": [124, 633, 486, 700], "lines": [{"bbox": [138, 635, 487, 647], "spans": [{"bbox": [138, 635, 226, 647], "score": 1.0, "content": "Conversely, suppose", "type": "text"}, {"bbox": [227, 636, 255, 646], "score": 0.92, "content": "\\kappa_{2}\\neq1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 635, 280, 647], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [280, 639, 285, 644], "score": 0.87, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [285, 635, 399, 647], "score": 1.0, "content": " be a nonprincipal ideal in", "type": "text"}, {"bbox": [400, 636, 409, 645], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [410, 635, 487, 647], "score": 1.0, "content": " of absolute norm", "type": "text"}], "index": 41}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 652, 132, 657], "score": 0.85, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [132, 647, 216, 660], "score": 1.0, "content": ", and assume that ", "type": "text"}, {"bbox": [216, 649, 250, 659], "score": 0.94, "content": "{\\mathfrak{a}}=(\\alpha)", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [251, 647, 266, 660], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [267, 649, 280, 658], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [280, 647, 315, 660], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [316, 646, 363, 659], "score": 0.95, "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "type": "inline_equation", "height": 13, "width": 47}, {"bbox": [363, 647, 429, 660], "score": 1.0, "content": " for some unit ", "type": "text"}, {"bbox": [430, 649, 462, 659], "score": 0.93, "content": "\\eta\\in E_{2}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [462, 647, 486, 660], "score": 1.0, "content": ", and", "type": "text"}], "index": 42}, {"bbox": [123, 658, 485, 676], "spans": [{"bbox": [123, 658, 167, 676], "score": 1.0, "content": "similarly ", "type": "text"}, {"bbox": [168, 661, 219, 673], "score": 0.93, "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [220, 658, 256, 676], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [256, 663, 264, 673], "score": 0.9, "content": "\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [265, 658, 324, 676], "score": 1.0, "content": " is a unit in ", "type": "text"}, {"bbox": [324, 663, 336, 673], "score": 0.92, "content": "E_{2}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [336, 658, 390, 676], "score": 1.0, "content": ". But then ", "type": "text"}, {"bbox": [390, 660, 485, 673], "score": 0.9, "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "type": "inline_equation", "height": 13, "width": 95}], "index": 43}, {"bbox": [125, 674, 487, 690], "spans": [{"bbox": [125, 675, 269, 689], "score": 0.89, "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "type": "inline_equation", "height": 14, "width": 144}, {"bbox": [270, 674, 285, 690], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [285, 677, 299, 686], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [299, 674, 335, 690], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [336, 675, 344, 686], "score": 0.71, "content": "\\underline{{\\underline{{2}}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [344, 674, 487, 690], "score": 1.0, "content": " means equal up to a square in", "type": "text"}], "index": 44}, {"bbox": [126, 689, 484, 702], "spans": [{"bbox": [126, 690, 140, 698], "score": 0.89, "content": "L^{\\times}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [140, 689, 172, 702], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [172, 690, 194, 700], "score": 0.92, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [194, 689, 258, 702], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [259, 691, 266, 698], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [266, 689, 376, 702], "score": 1.0, "content": ", so our assumption that ", "type": "text"}, {"bbox": [376, 691, 405, 700], "score": 0.93, "content": "q_{2}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [405, 689, 463, 702], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [464, 690, 484, 700], "score": 0.91, "content": "\\pm\\eta\\eta^{\\prime}", "type": "inline_equation", "height": 10, "width": 20}], "index": 45}, {"bbox": [124, 114, 486, 127], "spans": [{"bbox": [124, 114, 218, 127], "score": 1.0, "content": "must be a square in ", "type": "text", "cross_page": true}, {"bbox": [219, 116, 229, 125], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [229, 114, 377, 127], "score": 1.0, "content": ". The same argument show that ", "type": "text", "cross_page": true}, {"bbox": [378, 115, 404, 126], "score": 0.94, "content": "\\pm\\eta/\\eta^{\\prime}", "type": "inline_equation", "height": 11, "width": 26, "cross_page": true}, {"bbox": [405, 114, 472, 127], "score": 1.0, "content": " is a square in ", "type": "text", "cross_page": true}, {"bbox": [472, 116, 482, 125], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [482, 114, 486, 127], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [124, 126, 487, 140], "spans": [{"bbox": [124, 126, 189, 140], "score": 1.0, "content": "hence we find ", "type": "text", "cross_page": true}, {"bbox": [189, 129, 217, 139], "score": 0.93, "content": "\\eta\\in k_{2}", "type": "inline_equation", "height": 10, "width": 28, "cross_page": true}, {"bbox": [218, 126, 250, 140], "score": 1.0, "content": ". Thus ", "type": "text", "cross_page": true}, {"bbox": [251, 126, 276, 137], "score": 0.93, "content": "\\alpha^{1-\\sigma^{2}}", "type": "inline_equation", "height": 11, "width": 25, "cross_page": true}, {"bbox": [277, 126, 329, 140], "score": 1.0, "content": " is fixed by ", "type": "text", "cross_page": true}, {"bbox": [329, 128, 340, 137], "score": 0.91, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11, "cross_page": true}, {"bbox": [340, 126, 375, 140], "score": 1.0, "content": " and so ", "type": "text", "cross_page": true}, {"bbox": [375, 128, 433, 139], "score": 0.93, "content": "\\beta:=\\alpha^{2}\\in k_{2}", "type": "inline_equation", "height": 11, "width": 58, "cross_page": true}, {"bbox": [433, 126, 487, 140], "score": 1.0, "content": ". This gives", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [126, 139, 487, 151], "spans": [{"bbox": [126, 140, 186, 151], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 60, "cross_page": true}, {"bbox": [186, 139, 219, 151], "score": 1.0, "content": ", hence ", "type": "text", "cross_page": true}, {"bbox": [219, 141, 247, 151], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28, "cross_page": true}, {"bbox": [248, 139, 432, 151], "score": 1.0, "content": " is not essentially ramified, and moreover, ", "type": "text", "cross_page": true}, {"bbox": [433, 141, 456, 149], "score": 0.91, "content": "a\\sim{\\mathfrak{b}}", "type": "inline_equation", "height": 8, "width": 23, "cross_page": true}, {"bbox": [456, 139, 462, 151], "score": 1.0, "content": ".", "type": "text", "cross_page": true}, {"bbox": [465, 140, 487, 150], "score": 0.9668625593185425, "content": "冏口", "type": "text", "cross_page": true}], "index": 2}], "index": 43, "bbox_fs": [123, 635, 487, 702]}]}
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[{"type": "text", "bbox": [125, 111, 486, 136], "content": "Recall that a quadratic extension is called essentially ramified if is not an ideal square. This definition is independent of the choice of .", "index": 0}, {"type": "text", "bbox": [125, 142, 486, 179], "content": "Proposition 6. Let be a non-CM totally complex dihedral extension not con- taining , and assume that and are essentially ramified. If the fundamental unit of the real quadratic subfield of has norm , then .", "index": 1}, {"type": "text", "bbox": [125, 185, 486, 269], "content": "Proof. Notice first that cannot be real (in fact, is not totally real by assumption, and since is a cyclic quartic extension, no infinite prime can ramify in ); thus exactly one of , is real, and the other is complex. Multiplying the class number formulas, Proposition 3, for and (note that since both and are essentially ramified) we find that is a square. If we can prove that , then is a square between 2 and 8, which implies that we must have and as claimed.", "index": 2}, {"type": "text", "bbox": [125, 270, 486, 306], "content": "We start by remarking that if becomes a square in , where is a root of unity in , then so does one of . This follows from the fact that the only non-trivial roots of unity that can be in are the sixth roots of unity , and here .", "index": 3}, {"type": "text", "bbox": [125, 306, 486, 390], "content": "Now we prove that under the assumptions we made; the claim will then follow by symmetry. Assume first that is real and let be the fundamental unit of . We claim that . Suppose otherwise; then is one of , or . If , then and . (Here and below .) This however cannot occur since by assumption implying that , a contradiction. Similarly, if , then again .", "index": 4}, {"type": "text", "bbox": [125, 390, 486, 450], "content": "Thus , and for some unit . Suppose that for some unit . Then , contradicting our assumption that is essentially ramified. The same argument shows that , hence either and or for some unit and . Here is a root of unity generating the torsion subgroup of .", "index": 5}, {"type": "text", "bbox": [125, 451, 486, 474], "content": "Next consider the case where is complex, and let denote the fundamental unit of . Then stays fundamental in by the argument above.", "index": 6}, {"type": "text", "bbox": [125, 475, 486, 522], "content": "Let be a fundamental unit in . If became a square in , then clearly could not be essentially ramified. Thus if we have , then is a square in . Applying to this relation we find that is a square in , contradicting the assumption that does not contain . 口", "index": 7}, {"type": "text", "bbox": [125, 530, 486, 567], "content": "Proposition 7. Suppose that . Then is essentially ramified if and only if ; if is not essentially ramified, then , where and .", "index": 8}, {"type": "text", "bbox": [125, 573, 486, 632], "content": "Proof. First notice that if is not essentially ramified, then : in fact, in this case we have , and if we had , then would have to be principal, say . This implies that for some unit , which in view of implies that must be a square. But then would be a square, and this is impossible.", "index": 9}, {"type": "text", "bbox": [124, 633, 486, 700], "content": "Conversely, suppose . Let be a nonprincipal ideal in of absolute norm , and assume that in . Then for some unit , and similarly , where is a unit in . But then in , where means equal up to a square in . Thus is a square in , so our assumption that implies that must be a square in . The same argument show that is a square in , hence we find . Thus is fixed by and so . This gives , hence is not essentially ramified, and moreover, . 冏口", "index": 10}]
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[{"bbox": [137, 114, 486, 126], "content": "Recall that a quadratic extension is called essentially ramified if", "parent_index": 0, "line_index": 0}, {"bbox": [126, 126, 462, 138], "content": "is not an ideal square. This definition is independent of the choice of .", "parent_index": 0, "line_index": 1}, {"bbox": [126, 145, 487, 158], "content": "Proposition 6. Let be a non-CM totally complex dihedral extension not con-", "parent_index": 1, "line_index": 0}, {"bbox": [126, 157, 487, 170], "content": "taining , and assume that and are essentially ramified. If the", "parent_index": 1, "line_index": 1}, {"bbox": [126, 169, 481, 182], "content": "fundamental unit of the real quadratic subfield of has norm , then .", "parent_index": 1, "line_index": 2}, {"bbox": [127, 188, 485, 200], "content": "Proof. Notice first that cannot be real (in fact, is not totally real by assumption,", "parent_index": 2, "line_index": 0}, {"bbox": [126, 199, 486, 212], "content": "and since is a cyclic quartic extension, no infinite prime can ramify in );", "parent_index": 2, "line_index": 1}, {"bbox": [126, 212, 486, 224], "content": "thus exactly one of , is real, and the other is complex. Multiplying the class", "parent_index": 2, "line_index": 2}, {"bbox": [125, 224, 486, 236], "content": "number formulas, Proposition 3, for and (note that since both", "parent_index": 2, "line_index": 3}, {"bbox": [126, 236, 487, 248], "content": "and are essentially ramified) we find that is a square. If we can", "parent_index": 2, "line_index": 4}, {"bbox": [125, 248, 486, 260], "content": "prove that , then is a square between 2 and 8, which implies that", "parent_index": 2, "line_index": 5}, {"bbox": [125, 259, 340, 272], "content": "we must have and as claimed.", "parent_index": 2, "line_index": 6}, {"bbox": [137, 271, 485, 284], "content": "We start by remarking that if becomes a square in , where is a root of unity", "parent_index": 3, "line_index": 0}, {"bbox": [126, 284, 486, 295], "content": "in , then so does one of . This follows from the fact that the only non-trivial", "parent_index": 3, "line_index": 1}, {"bbox": [125, 295, 485, 307], "content": "roots of unity that can be in are the sixth roots of unity , and here .", "parent_index": 3, "line_index": 2}, {"bbox": [137, 307, 486, 319], "content": "Now we prove that under the assumptions we made; the claim will", "parent_index": 4, "line_index": 0}, {"bbox": [125, 320, 487, 330], "content": "then follow by symmetry. Assume first that is real and let be the fundamental", "parent_index": 4, "line_index": 1}, {"bbox": [126, 331, 487, 344], "content": "unit of . We claim that . Suppose otherwise; then is one of", "parent_index": 4, "line_index": 2}, {"bbox": [126, 343, 487, 356], "content": ", or . If , then and . (Here", "parent_index": 4, "line_index": 3}, {"bbox": [125, 355, 485, 369], "content": "and below .) This however cannot occur since by assumption", "parent_index": 4, "line_index": 4}, {"bbox": [125, 368, 487, 380], "content": "implying that , a contradiction. Similarly, if , then again", "parent_index": 4, "line_index": 5}, {"bbox": [126, 380, 170, 392], "content": ".", "parent_index": 4, "line_index": 6}, {"bbox": [137, 390, 486, 405], "content": "Thus , and for some unit . Suppose that", "parent_index": 5, "line_index": 0}, {"bbox": [126, 404, 485, 417], "content": "for some unit . Then , contradicting our assumption", "parent_index": 5, "line_index": 1}, {"bbox": [126, 417, 486, 429], "content": "that is essentially ramified. The same argument shows that , hence", "parent_index": 5, "line_index": 2}, {"bbox": [126, 428, 484, 442], "content": "either and or for some unit", "parent_index": 5, "line_index": 3}, {"bbox": [126, 441, 479, 452], "content": "and . Here is a root of unity generating the torsion subgroup of .", "parent_index": 5, "line_index": 4}, {"bbox": [137, 451, 486, 465], "content": "Next consider the case where is complex, and let denote the fundamental", "parent_index": 6, "line_index": 0}, {"bbox": [126, 464, 425, 476], "content": "unit of . Then stays fundamental in by the argument above.", "parent_index": 6, "line_index": 1}, {"bbox": [137, 477, 485, 488], "content": "Let be a fundamental unit in . If became a square in , then clearly", "parent_index": 7, "line_index": 0}, {"bbox": [126, 488, 487, 501], "content": "could not be essentially ramified. Thus if we have , then is", "parent_index": 7, "line_index": 1}, {"bbox": [125, 500, 487, 513], "content": "a square in . Applying to this relation we find that is a square in ,", "parent_index": 7, "line_index": 2}, {"bbox": [125, 512, 486, 524], "content": "contradicting the assumption that does not contain . 口", "parent_index": 7, "line_index": 3}, {"bbox": [126, 532, 487, 545], "content": "Proposition 7. Suppose that . Then is essentially ramified if and", "parent_index": 8, "line_index": 0}, {"bbox": [127, 543, 486, 558], "content": "only if ; if is not essentially ramified, then , where", "parent_index": 8, "line_index": 1}, {"bbox": [126, 556, 223, 568], "content": "and .", "parent_index": 8, "line_index": 2}, {"bbox": [126, 574, 485, 587], "content": "Proof. First notice that if is not essentially ramified, then : in fact,", "parent_index": 9, "line_index": 0}, {"bbox": [124, 586, 486, 599], "content": "in this case we have , and if we had , then would have to be", "parent_index": 9, "line_index": 1}, {"bbox": [126, 599, 486, 611], "content": "principal, say . This implies that for some unit , which in", "parent_index": 9, "line_index": 2}, {"bbox": [125, 611, 487, 623], "content": "view of implies that must be a square. But then would be a square, and", "parent_index": 9, "line_index": 3}, {"bbox": [125, 623, 204, 635], "content": "this is impossible.", "parent_index": 9, "line_index": 4}, {"bbox": [138, 635, 487, 647], "content": "Conversely, suppose . Let be a nonprincipal ideal in of absolute norm", "parent_index": 10, "line_index": 0}, {"bbox": [126, 646, 486, 660], "content": ", and assume that in . Then for some unit , and", "parent_index": 10, "line_index": 1}, {"bbox": [123, 658, 485, 676], "content": "similarly , where is a unit in . But then", "parent_index": 10, "line_index": 2}, {"bbox": [125, 674, 487, 690], "content": "in , where means equal up to a square in", "parent_index": 10, "line_index": 3}, {"bbox": [126, 689, 484, 702], "content": ". Thus is a square in , so our assumption that implies that", "parent_index": 10, "line_index": 4}, {"bbox": [124, 114, 486, 127], "content": "must be a square in . The same argument show that is a square in ,", "parent_index": 10, "line_index": 5}, {"bbox": [124, 126, 487, 140], "content": "hence we find . Thus is fixed by and so . This gives", "parent_index": 10, "line_index": 6}, {"bbox": [126, 139, 487, 151], "content": ", hence is not essentially ramified, and moreover, . 冏口", "parent_index": 10, "line_index": 7}]
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[{"bbox": [291, 115, 346, 126], "content": "K\\,=\\,k({\\sqrt{\\alpha}}\\,)", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 128, 145, 137], "content": "\\alpha{\\mathcal{O}}_{k}", "parent_index": 0, "subtype": "inline"}, {"bbox": [452, 131, 459, 135], "content": "\\alpha", "parent_index": 0, "subtype": "inline"}, {"bbox": [218, 146, 239, 156], "content": "L/\\mathbb{Q}", "parent_index": 1, "subtype": "inline"}, {"bbox": [161, 157, 182, 168], "content": "\\sqrt{-1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [268, 158, 293, 169], "content": "L/K_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [317, 158, 342, 169], "content": "L/K_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [342, 171, 351, 178], "content": "K", "parent_index": 1, "subtype": "inline"}, {"bbox": [399, 171, 412, 179], "content": "^{-1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [440, 171, 477, 180], "content": "q_{1}q_{2}=2", "parent_index": 1, "subtype": "inline"}, {"bbox": [227, 189, 233, 197], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [334, 190, 343, 197], "content": "K", "parent_index": 2, "subtype": "inline"}, {"bbox": [170, 201, 188, 211], "content": "L/k", "parent_index": 2, "subtype": "inline"}, {"bbox": [459, 201, 478, 211], "content": "K/k", "parent_index": 2, "subtype": "inline"}, {"bbox": [213, 213, 223, 222], "content": "k_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [229, 213, 239, 222], "content": "k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [289, 225, 311, 235], "content": "L/k_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [334, 225, 356, 235], "content": "L/k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [409, 226, 435, 233], "content": "\\upsilon=0", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 237, 151, 247], "content": "L/K_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [173, 237, 198, 247], "content": "L/K_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [365, 238, 388, 247], "content": "2q_{1}q_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [175, 250, 217, 259], "content": "q_{1},q_{2}\\leq2", "parent_index": 2, "subtype": "inline"}, {"bbox": [247, 250, 270, 259], "content": "2q_{1}q_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [189, 262, 230, 271], "content": "2q_{1}q_{2}=4", "parent_index": 2, "subtype": "inline"}, {"bbox": [252, 262, 288, 270], "content": "q_{1}q_{2}=2", "parent_index": 2, "subtype": "inline"}, {"bbox": [266, 273, 276, 282], "content": "\\zeta\\eta", "parent_index": 3, "subtype": "inline"}, {"bbox": [366, 273, 374, 280], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [407, 273, 412, 282], "content": "\\zeta", "parent_index": 3, "subtype": "inline"}, {"bbox": [138, 285, 145, 293], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [239, 286, 253, 294], "content": "\\pm\\eta", "parent_index": 3, "subtype": "inline"}, {"bbox": [251, 297, 258, 304], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [380, 297, 397, 307], "content": "\\langle\\zeta_{6}\\rangle", "parent_index": 3, "subtype": "inline"}, {"bbox": [442, 296, 482, 307], "content": "\\zeta_{6}=-\\zeta_{3}^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [222, 309, 249, 318], "content": "q_{1}\\leq2", "parent_index": 4, "subtype": "inline"}, {"bbox": [439, 309, 466, 318], "content": "q_{2}\\leq2", "parent_index": 4, "subtype": "inline"}, {"bbox": [318, 321, 327, 330], "content": "k_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [393, 324, 397, 328], "content": "\\varepsilon", "parent_index": 4, "subtype": "inline"}, {"bbox": [159, 333, 169, 342], "content": "k_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [244, 331, 285, 342], "content": "\\sqrt{\\pm\\varepsilon}\\notin L", "parent_index": 4, "subtype": "inline"}, {"bbox": [404, 331, 444, 343], "content": "k_{1}(\\sqrt{\\pm\\varepsilon})", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 346, 139, 354], "content": "K_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [145, 345, 158, 356], "content": "K_{1}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [174, 346, 183, 353], "content": "K", "parent_index": 4, "subtype": "inline"}, {"bbox": [201, 344, 268, 356], "content": "k_{1}(\\sqrt{\\pm\\varepsilon}\\,)=K_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [297, 344, 367, 356], "content": "K_{1}^{\\prime}=k_{1}(\\sqrt{\\pm\\varepsilon^{\\prime}}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [390, 344, 453, 356], "content": "K=k_{1}\\big(\\sqrt{\\varepsilon\\varepsilon^{\\prime}}\\big)", "parent_index": 4, "subtype": "inline"}, {"bbox": [175, 357, 212, 365], "content": "x^{\\prime}\\,=\\,x^{\\sigma}", "parent_index": 4, "subtype": "inline"}, {"bbox": [443, 357, 485, 366], "content": "\\varepsilon\\varepsilon^{\\prime}\\,=\\,-1", "parent_index": 4, "subtype": "inline"}, {"bbox": [190, 368, 232, 379], "content": "\\sqrt{-1}\\in L", "parent_index": 4, "subtype": "inline"}, {"bbox": [367, 369, 432, 380], "content": "k_{1}({\\sqrt{\\pm\\varepsilon}}\\,)\\,=\\,K", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 380, 166, 391], "content": "\\sqrt{-1}\\in L", "parent_index": 4, "subtype": "inline"}, {"bbox": [164, 392, 210, 403], "content": "\\sqrt{\\pm\\varepsilon}~\\notin~L", "parent_index": 5, "subtype": "inline"}, {"bbox": [239, 393, 308, 403], "content": "E_{1}~=~\\langle-1,\\varepsilon,\\eta\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [379, 394, 414, 403], "content": "\\eta\\ \\in\\ E_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 406, 164, 416], "content": "\\sqrt{u\\eta}\\in L", "parent_index": 5, "subtype": "inline"}, {"bbox": [228, 406, 256, 415], "content": "u\\in k_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [289, 405, 351, 416], "content": "L=K_{1}(\\sqrt{u\\eta}\\,)", "parent_index": 5, "subtype": "inline"}, {"bbox": [147, 418, 172, 428], "content": "L/K_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [414, 417, 455, 428], "content": "\\sqrt{u\\eta^{\\prime}}\\notin L", "parent_index": 5, "subtype": "inline"}, {"bbox": [155, 430, 227, 440], "content": "E_{L}\\,=\\,\\langle\\zeta,\\varepsilon,\\eta,\\eta^{\\prime}\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [251, 431, 279, 439], "content": "q_{1}=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [296, 429, 389, 440], "content": "E_{L}=\\langle\\zeta,\\varepsilon,\\eta,\\sqrt{u\\eta\\eta^{\\prime}}\\rangle", "parent_index": 5, "subtype": "inline"}, {"bbox": [455, 430, 484, 439], "content": "u\\in k_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [145, 443, 173, 451], "content": "q_{1}=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [203, 442, 208, 451], "content": "\\zeta", "parent_index": 5, "subtype": "inline"}, {"bbox": [431, 442, 446, 451], "content": "W_{L}", "parent_index": 5, "subtype": "inline"}, {"bbox": [461, 442, 475, 451], "content": "E_{L}", "parent_index": 5, "subtype": "inline"}, {"bbox": [271, 454, 280, 463], "content": "k_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [371, 457, 376, 461], "content": "\\varepsilon", "parent_index": 6, "subtype": "inline"}, {"bbox": [158, 466, 168, 475], "content": "k_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [201, 466, 214, 474], "content": "\\pm\\varepsilon", "parent_index": 6, "subtype": "inline"}, {"bbox": [311, 466, 318, 473], "content": "L", "parent_index": 6, "subtype": "inline"}, {"bbox": [156, 481, 162, 488], "content": "\\eta", "parent_index": 7, "subtype": "inline"}, {"bbox": [280, 478, 294, 487], "content": "K_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [313, 479, 326, 488], "content": "\\pm\\eta", "parent_index": 7, "subtype": "inline"}, {"bbox": [419, 478, 426, 486], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 489, 151, 500], "content": "L/K_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [376, 491, 403, 499], "content": "q_{1}\\geq4", "parent_index": 7, "subtype": "inline"}, {"bbox": [432, 489, 474, 500], "content": "\\pm\\varepsilon\\eta=\\alpha^{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [178, 502, 185, 509], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [236, 505, 241, 509], "content": "\\tau", "parent_index": 7, "subtype": "inline"}, {"bbox": [371, 501, 409, 510], "content": "-1=\\varepsilon\\varepsilon^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [475, 502, 482, 509], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [278, 514, 285, 521], "content": "L", "parent_index": 7, "subtype": "inline"}, {"bbox": [363, 513, 385, 523], "content": "\\sqrt{-1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [261, 534, 291, 543], "content": "q_{2}=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [327, 533, 355, 544], "content": "K_{2}/k_{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [158, 546, 189, 555], "content": "\\kappa_{2}=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [206, 545, 234, 556], "content": "K_{2}/k_{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [380, 545, 424, 556], "content": "\\kappa_{2}=\\langle[6]\\rangle", "parent_index": 8, "subtype": "inline"}, {"bbox": [460, 545, 486, 555], "content": "K_{2}=", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 556, 160, 568], "content": "k_{2}(\\sqrt{\\beta}\\,)", "parent_index": 8, "subtype": "inline"}, {"bbox": [182, 556, 219, 568], "content": "(\\beta)=\\mathfrak{b}^{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [243, 576, 271, 586], "content": "K_{2}/k_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [416, 577, 446, 586], "content": "\\kappa_{2}\\neq1", "parent_index": 9, "subtype": "inline"}, {"bbox": [221, 587, 261, 598], "content": "(\\beta)\\;=\\;6^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [335, 589, 367, 597], "content": "\\,\\kappa_{2}\\,=\\,1", "parent_index": 9, "subtype": "inline"}, {"bbox": [398, 588, 403, 596], "content": "\\mathfrak{b}", "parent_index": 9, "subtype": "inline"}, {"bbox": [189, 600, 222, 610], "content": "{\\mathfrak{b}}=(\\gamma)", "parent_index": 9, "subtype": "inline"}, {"bbox": [310, 600, 346, 610], "content": "\\beta\\,=\\,\\varepsilon\\gamma^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [412, 601, 441, 609], "content": "\\varepsilon\\in k_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [160, 613, 187, 622], "content": "q_{2}=1", "parent_index": 9, "subtype": "inline"}, {"bbox": [245, 615, 249, 620], "content": "\\varepsilon", "parent_index": 9, "subtype": "inline"}, {"bbox": [375, 613, 381, 622], "content": "\\beta", "parent_index": 9, "subtype": "inline"}, {"bbox": [227, 636, 255, 646], "content": "\\kappa_{2}\\neq1", "parent_index": 10, "subtype": "inline"}, {"bbox": [280, 639, 285, 644], "content": "\\mathfrak{a}", "parent_index": 10, "subtype": "inline"}, {"bbox": [400, 636, 409, 645], "content": "k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 652, 132, 657], "content": "a", "parent_index": 10, "subtype": "inline"}, {"bbox": [216, 649, 250, 659], "content": "{\\mathfrak{a}}=(\\alpha)", "parent_index": 10, "subtype": "inline"}, {"bbox": [267, 649, 280, 658], "content": "K_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [316, 646, 363, 659], "content": "\\alpha^{1-\\sigma^{2}}=\\eta", "parent_index": 10, "subtype": "inline"}, {"bbox": [430, 649, 462, 659], "content": "\\eta\\in E_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [168, 661, 219, 673], "content": "\\alpha^{\\sigma-\\sigma^{3}}\\,=\\,\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [256, 663, 264, 673], "content": "\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [324, 663, 336, 673], "content": "E_{2}^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [390, 660, 485, 673], "content": "\\eta\\eta^{\\prime}\\,=\\,\\alpha^{1+\\sigma-\\sigma^{2}-\\sigma^{3}}\\,\\stackrel{\\cdot2}{=}", "parent_index": 10, "subtype": "inline"}, {"bbox": [125, 675, 269, 689], "content": "N_{L/k}\\alpha\\,=\\,\\pm N_{L/k}\\mathfrak{a}\\,=\\,\\pm a^{2}\\,\\stackrel{?}{=}\\,\\pm1", "parent_index": 10, "subtype": "inline"}, {"bbox": [285, 677, 299, 686], "content": "L^{\\times}", "parent_index": 10, "subtype": "inline"}, {"bbox": [336, 675, 344, 686], "content": "\\underline{{\\underline{{2}}}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 690, 140, 698], "content": "L^{\\times}", "parent_index": 10, "subtype": "inline"}, {"bbox": [172, 690, 194, 700], "content": "\\pm\\eta\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [259, 691, 266, 698], "content": "L", "parent_index": 10, "subtype": "inline"}, {"bbox": [376, 691, 405, 700], "content": "q_{2}=1", "parent_index": 10, "subtype": "inline"}, {"bbox": [464, 690, 484, 700], "content": "\\pm\\eta\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [219, 116, 229, 125], "content": "k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [378, 115, 404, 126], "content": "\\pm\\eta/\\eta^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [472, 116, 482, 125], "content": "k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [189, 129, 217, 139], "content": "\\eta\\in k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [251, 126, 276, 137], "content": "\\alpha^{1-\\sigma^{2}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [329, 128, 340, 137], "content": "\\sigma^{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [375, 128, 433, 139], "content": "\\beta:=\\alpha^{2}\\in k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 140, 186, 151], "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "parent_index": 10, "subtype": "inline"}, {"bbox": [219, 141, 247, 151], "content": "K_{2}/k_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [433, 141, 456, 149], "content": "a\\sim{\\mathfrak{b}}", "parent_index": 10, "subtype": "inline"}]
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From now on assume that $k$ is one of the imaginary quadratic fields of type A) or $\mathrm{B}$ ) as explained in the Introduction. Let
Then there exist two unramified cyclic quartic extensions of $k$ which are $D_{4}$ over $\mathbb{Q}$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the $C_{4}$ -factorization $d=d_{1}d_{2}\cdot d_{3}$ implies that unramified cyclic quartic extensions of $k\,=\,\mathbb{Q}({\sqrt{d}}\,)$ are constructed by choosing a “primitive” solution $\left(x,y,z\right)$ of $d_{1}d_{2}X^{2}+d_{3}Y^{2}\,=\,Z^{2}$ and putting $L=k(\sqrt{d_{1}d_{2}},\sqrt{\alpha}\,)$ with $\alpha=z+x\sqrt{d_{1}d_{2}}$ (primitive here means that $\alpha$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $\widetilde{L}=k(\sqrt{d_{1}d_{2}},\sqrt{d_{1}\alpha}\,)$ . If we put $\beta\,=\,{\textstyle\frac{1}{2}}(z+y\sqrt{d_{3}}\,)$ , then it is an elementary exerc i se to show that $\alpha\beta$ is a square in $L$ , hence we also have $L=k(\sqrt{d_{3}},\sqrt{\beta}\,)$ etc. If $d_{3}=-4$ , then it is easy to see that we may choose $\beta$ as the fundamental unit of $k_{2}$ ; if $d_{3}\neq-4$ , then genus theory says that a) the class number $h$ of $k_{2}$ is twice an odd number $u$ ; and b) the prime ideal ${\mathfrak{p}}_{3}$ above $d_{3}$ in $k_{2}$ is in the principal genus, so ${\mathfrak{p}}_{3}^{u}=(\pi_{3})$ is principal. Again it can be checked that $\beta=\pm\pi_{3}$ for a suitable choice of the sign.
Example. Consider the case $d\,=\,-31\cdot5\cdot8$ ; here $\pi_{3}\,=\,\pm(3+2{\sqrt{10}}\,)$ , and the positive sign is correct since $3\,{+}\,2{\sqrt{10}}\equiv(1\,{+}\,{\sqrt{10}}\,)^{2}$ mod 4 is primary. The minimal polynomial of $\sqrt{\pi_{3}}$ is $f(x)=x^{4}-6x^{2}-31$ : compare Table 1.
The fields $K_{2}=k_{2}(\sqrt{\alpha}\,)$ and $\tilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$ will play a dominant role in the proof below; they are both contai n ed in $M=F({\sqrt{\alpha}}\,)$ for $F=k_{2}(\sqrt{d_{2}}\,)$ , and it is the ambiguous class group $\mathrm{Am}(M/F)$ that contains the information we are interested in.
Lemma 6. The field $F$ has odd class number (even in the strict sense), and we have $\#\operatorname{Am}(M/F)\mid$ 2. In particular, $\mathrm{Cl_{2}}(M)$ is cyclic (though possibly trivial).
Proof. The class group in the strict sense of $k_{2}$ is cyclic of order 2 by R´edei’s theory [12] (since $(d_{2}/p_{3})=(d_{3}/p_{2})=-1$ in case A) and $(d_{1}/p_{2})=(d_{2}/p_{1})=-1$ in case B)). Since $F$ is the Hilbert class field of $k_{2}$ in the strict sense, its class number in the strict sense is odd.
Next we apply the ambiguous class number formula. In case A), $F$ is complex, and exactly the two primes above $d_{3}$ ramify in $M/F$ . Note that $M\,=\,F({\sqrt{\alpha}}\,)$ with $\alpha$ primary of norm $d_{3}y^{2}$ ; there are four primes above $d_{3}$ in $F$ , and exactly two of them divide $\alpha$ to an odd power, so $t\ =\ 2$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $\#\operatorname{Am}_{2}(M/F)=2/(E:H)\leq2$ , and $\mathrm{Cl_{2}}(M)$ is cyclic.
In case B), however, $F$ is real; since $\alpha\,\in\,k_{2}$ has norm $d_{3}y^{2}\,<\,0$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $M/F$ . As in case A), there are two finite primes above $d_{3}$ that ramify in $M/F$ , so we get $\#\operatorname{Am}_{2}(M/F)=8/(E:H)$ . Since $F$ has odd class number in the strict sense, $F$ has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has $\mathbb{Z}$ -rank 2, i.e. $(E:H)\geq4$ by consideration of the infinite primes alone. In particular, $\#\operatorname{Am}_{2}(M/F)\leq2$ in case B). 口
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<p data-bbox="125 157 486 181">From now on assume that $k$ is one of the imaginary quadratic fields of type A) or $\mathrm{B}$ ) as explained in the Introduction. Let </p>
<p data-bbox="124 214 486 385">Then there exist two unramified cyclic quartic extensions of $k$ which are $D_{4}$ over $\mathbb{Q}$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the $C_{4}$ -factorization $d=d_{1}d_{2}\cdot d_{3}$ implies that unramified cyclic quartic extensions of $k\,=\,\mathbb{Q}({\sqrt{d}}\,)$ are constructed by choosing a “primitive” solution $\left(x,y,z\right)$ of $d_{1}d_{2}X^{2}+d_{3}Y^{2}\,=\,Z^{2}$ and putting $L=k(\sqrt{d_{1}d_{2}},\sqrt{\alpha}\,)$ with $\alpha=z+x\sqrt{d_{1}d_{2}}$ (primitive here means that $\alpha$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $\widetilde{L}=k(\sqrt{d_{1}d_{2}},\sqrt{d_{1}\alpha}\,)$ . If we put $\beta\,=\,{\textstyle\frac{1}{2}}(z+y\sqrt{d_{3}}\,)$ , then it is an elementary exerc i se to show that $\alpha\beta$ is a square in $L$ , hence we also have $L=k(\sqrt{d_{3}},\sqrt{\beta}\,)$ etc. If $d_{3}=-4$ , then it is easy to see that we may choose $\beta$ as the fundamental unit of $k_{2}$ ; if $d_{3}\neq-4$ , then genus theory says that a) the class number $h$ of $k_{2}$ is twice an odd number $u$ ; and b) the prime ideal ${\mathfrak{p}}_{3}$ above $d_{3}$ in $k_{2}$ is in the principal genus, so ${\mathfrak{p}}_{3}^{u}=(\pi_{3})$ is principal. Again it can be checked that $\beta=\pm\pi_{3}$ for a suitable choice of the sign. </p>
<p data-bbox="125 389 486 427">Example. Consider the case $d\,=\,-31\cdot5\cdot8$ ; here $\pi_{3}\,=\,\pm(3+2{\sqrt{10}}\,)$ , and the positive sign is correct since $3\,{+}\,2{\sqrt{10}}\equiv(1\,{+}\,{\sqrt{10}}\,)^{2}$ mod 4 is primary. The minimal polynomial of $\sqrt{\pi_{3}}$ is $f(x)=x^{4}-6x^{2}-31$ : compare Table 1. </p>
<p data-bbox="125 433 486 483">The fields $K_{2}=k_{2}(\sqrt{\alpha}\,)$ and $\tilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$ will play a dominant role in the proof below; they are both contai n ed in $M=F({\sqrt{\alpha}}\,)$ for $F=k_{2}(\sqrt{d_{2}}\,)$ , and it is the ambiguous class group $\mathrm{Am}(M/F)$ that contains the information we are interested in. </p>
<p data-bbox="125 489 486 513">Lemma 6. The field $F$ has odd class number (even in the strict sense), and we have $\#\operatorname{Am}(M/F)\mid$ 2. In particular, $\mathrm{Cl_{2}}(M)$ is cyclic (though possibly trivial). </p>
<p data-bbox="126 519 486 567">Proof. The class group in the strict sense of $k_{2}$ is cyclic of order 2 by R´edei’s theory [12] (since $(d_{2}/p_{3})=(d_{3}/p_{2})=-1$ in case A) and $(d_{1}/p_{2})=(d_{2}/p_{1})=-1$ in case B)). Since $F$ is the Hilbert class field of $k_{2}$ in the strict sense, its class number in the strict sense is odd. </p>
<p data-bbox="125 568 486 640">Next we apply the ambiguous class number formula. In case A), $F$ is complex, and exactly the two primes above $d_{3}$ ramify in $M/F$ . Note that $M\,=\,F({\sqrt{\alpha}}\,)$ with $\alpha$ primary of norm $d_{3}y^{2}$ ; there are four primes above $d_{3}$ in $F$ , and exactly two of them divide $\alpha$ to an odd power, so $t\ =\ 2$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $\#\operatorname{Am}_{2}(M/F)=2/(E:H)\leq2$ , and $\mathrm{Cl_{2}}(M)$ is cyclic. </p>
<p data-bbox="125 640 486 700">In case B), however, $F$ is real; since $\alpha\,\in\,k_{2}$ has norm $d_{3}y^{2}\,<\,0$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $M/F$ . As in case A), there are two finite primes above $d_{3}$ that ramify in $M/F$ , so we get $\#\operatorname{Am}_{2}(M/F)=8/(E:H)$ . Since $F$ has odd class number in the strict sense, $F$ has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has $\mathbb{Z}$ -rank 2, i.e. $(E:H)\geq4$ by consideration of the infinite primes alone. In particular, $\#\operatorname{Am}_{2}(M/F)\leq2$ in case B). 口 </p>
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0003244v1
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[{"type": "text", "text": "", "page_idx": 8}, {"type": "text", "text": "From now on assume that $k$ is one of the imaginary quadratic fields of type A) or $\\mathrm{B}$ ) as explained in the Introduction. Let ", "page_idx": 8}, {"type": "text", "text": "Then there exist two unramified cyclic quartic extensions of $k$ which are $D_{4}$ over $\\mathbb{Q}$ (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the $C_{4}$ -factorization $d=d_{1}d_{2}\\cdot d_{3}$ implies that unramified cyclic quartic extensions of $k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)$ are constructed by choosing a “primitive” solution $\\left(x,y,z\\right)$ of $d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}$ and putting $L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)$ with $\\alpha=z+x\\sqrt{d_{1}d_{2}}$ (primitive here means that $\\alpha$ should not be divisible by rational integers); the other unramified cyclic quartic extension is then $\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)$ . If we put $\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)$ , then it is an elementary exerc i se to show that $\\alpha\\beta$ is a square in $L$ , hence we also have $L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)$ etc. If $d_{3}=-4$ , then it is easy to see that we may choose $\\beta$ as the fundamental unit of $k_{2}$ ; if $d_{3}\\neq-4$ , then genus theory says that a) the class number $h$ of $k_{2}$ is twice an odd number $u$ ; and b) the prime ideal ${\\mathfrak{p}}_{3}$ above $d_{3}$ in $k_{2}$ is in the principal genus, so ${\\mathfrak{p}}_{3}^{u}=(\\pi_{3})$ is principal. Again it can be checked that $\\beta=\\pm\\pi_{3}$ for a suitable choice of the sign. ", "page_idx": 8}, {"type": "text", "text": "Example. Consider the case $d\\,=\\,-31\\cdot5\\cdot8$ ; here $\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)$ , and the positive sign is correct since $3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}$ mod 4 is primary. The minimal polynomial of $\\sqrt{\\pi_{3}}$ is $f(x)=x^{4}-6x^{2}-31$ : compare Table 1. ", "page_idx": 8}, {"type": "text", "text": "The fields $K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$ and $\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$ will play a dominant role in the proof below; they are both contai n ed in $M=F({\\sqrt{\\alpha}}\\,)$ for $F=k_{2}(\\sqrt{d_{2}}\\,)$ , and it is the ambiguous class group $\\mathrm{Am}(M/F)$ that contains the information we are interested in. ", "page_idx": 8}, {"type": "text", "text": "Lemma 6. The field $F$ has odd class number (even in the strict sense), and we have $\\#\\operatorname{Am}(M/F)\\mid$ 2. In particular, $\\mathrm{Cl_{2}}(M)$ is cyclic (though possibly trivial). ", "page_idx": 8}, {"type": "text", "text": "Proof. The class group in the strict sense of $k_{2}$ is cyclic of order 2 by R´edei’s theory [12] (since $(d_{2}/p_{3})=(d_{3}/p_{2})=-1$ in case A) and $(d_{1}/p_{2})=(d_{2}/p_{1})=-1$ in case B)). Since $F$ is the Hilbert class field of $k_{2}$ in the strict sense, its class number in the strict sense is odd. ", "page_idx": 8}, {"type": "text", "text": "Next we apply the ambiguous class number formula. In case A), $F$ is complex, and exactly the two primes above $d_{3}$ ramify in $M/F$ . Note that $M\\,=\\,F({\\sqrt{\\alpha}}\\,)$ with $\\alpha$ primary of norm $d_{3}y^{2}$ ; there are four primes above $d_{3}$ in $F$ , and exactly two of them divide $\\alpha$ to an odd power, so $t\\ =\\ 2$ by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, $\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2$ , and $\\mathrm{Cl_{2}}(M)$ is cyclic. ", "page_idx": 8}, {"type": "text", "text": "In case B), however, $F$ is real; since $\\alpha\\,\\in\\,k_{2}$ has norm $d_{3}y^{2}\\,<\\,0$ , it has mixed signature, hence there are exactly two infinite primes that ramify in $M/F$ . As in case A), there are two finite primes above $d_{3}$ that ramify in $M/F$ , so we get $\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)$ . Since $F$ has odd class number in the strict sense, $F$ has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has $\\mathbb{Z}$ -rank 2, i.e. $(E:H)\\geq4$ by consideration of the infinite primes alone. In particular, $\\#\\operatorname{Am}_{2}(M/F)\\leq2$ in case B). 口 ", "page_idx": 8}]
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"poly": [894.0, 1397.0, 1303.0, 1397.0, 1303.0, 1432.0, 894.0, 1432.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [665.0, 259.0, 1035.0, 259.0, 1035.0, 280.0, 665.0, 280.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1336.0, 261.0, 1350.0, 261.0, 1350.0, 282.0, 1336.0, 282.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 8, "height": 2200, "width": 1700}}
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{"preproc_blocks": [{"type": "text", "bbox": [125, 112, 486, 150], "lines": [{"bbox": [124, 114, 486, 127], "spans": [{"bbox": [124, 114, 218, 127], "score": 1.0, "content": "must be a square in ", "type": "text"}, {"bbox": [219, 116, 229, 125], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [229, 114, 377, 127], "score": 1.0, "content": ". The same argument show that ", "type": "text"}, {"bbox": [378, 115, 404, 126], "score": 0.94, "content": "\\pm\\eta/\\eta^{\\prime}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [405, 114, 472, 127], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [472, 116, 482, 125], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [482, 114, 486, 127], "score": 1.0, "content": ",", "type": "text"}], "index": 0}, {"bbox": [124, 126, 487, 140], "spans": [{"bbox": [124, 126, 189, 140], "score": 1.0, "content": "hence we find ", "type": "text"}, {"bbox": [189, 129, 217, 139], "score": 0.93, "content": "\\eta\\in k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [218, 126, 250, 140], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [251, 126, 276, 137], "score": 0.93, "content": "\\alpha^{1-\\sigma^{2}}", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [277, 126, 329, 140], "score": 1.0, "content": " is fixed by ", "type": "text"}, {"bbox": [329, 128, 340, 137], "score": 0.91, "content": "\\sigma^{2}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [340, 126, 375, 140], "score": 1.0, "content": " and so ", "type": "text"}, {"bbox": [375, 128, 433, 139], "score": 0.93, "content": "\\beta:=\\alpha^{2}\\in k_{2}", "type": "inline_equation", "height": 11, "width": 58}, {"bbox": [433, 126, 487, 140], "score": 1.0, "content": ". This gives", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 151], "spans": [{"bbox": [126, 140, 186, 151], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [186, 139, 219, 151], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [219, 141, 247, 151], "score": 0.94, "content": "K_{2}/k_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [248, 139, 432, 151], "score": 1.0, "content": " is not essentially ramified, and moreover, ", "type": "text"}, {"bbox": [433, 141, 456, 149], "score": 0.91, "content": "a\\sim{\\mathfrak{b}}", "type": "inline_equation", "height": 8, "width": 23}, {"bbox": [456, 139, 462, 151], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [465, 140, 487, 150], "score": 0.9668625593185425, "content": "冏口", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [125, 157, 486, 181], "lines": [{"bbox": [137, 159, 484, 172], "spans": [{"bbox": [137, 159, 255, 172], "score": 1.0, "content": "From now on assume that ", "type": "text"}, {"bbox": [255, 162, 261, 169], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [261, 159, 484, 172], "score": 1.0, "content": " is one of the imaginary quadratic fields of type A)", "type": "text"}], "index": 3}, {"bbox": [126, 172, 316, 183], "spans": [{"bbox": [126, 172, 137, 183], "score": 1.0, "content": "or", "type": "text"}, {"bbox": [138, 173, 145, 181], "score": 0.43, "content": "\\mathrm{B}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [146, 172, 316, 183], "score": 1.0, "content": ") as explained in the Introduction. Let", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [124, 214, 486, 385], "lines": [{"bbox": [138, 217, 484, 228], "spans": [{"bbox": [138, 217, 414, 228], "score": 1.0, "content": "Then there exist two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [414, 218, 420, 225], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [420, 217, 471, 228], "score": 1.0, "content": " which are ", "type": "text"}, {"bbox": [472, 218, 484, 227], "score": 0.92, "content": "D_{4}", "type": "inline_equation", "height": 9, "width": 12}], "index": 5}, {"bbox": [126, 229, 486, 240], "spans": [{"bbox": [126, 229, 148, 240], "score": 1.0, "content": "over ", "type": "text"}, {"bbox": [149, 230, 157, 239], "score": 0.9, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [157, 229, 486, 240], "score": 1.0, "content": " (see Proposition 2). Let us say a few words about their construction.", "type": "text"}], "index": 6}, {"bbox": [126, 240, 484, 252], "spans": [{"bbox": [126, 240, 360, 252], "score": 1.0, "content": "Consider e.g. case B); by R´edei’s theory (see [12]), the", "type": "text"}, {"bbox": [361, 242, 372, 250], "score": 0.91, "content": "C_{4}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [373, 240, 433, 252], "score": 1.0, "content": "-factorization ", "type": "text"}, {"bbox": [433, 242, 484, 250], "score": 0.93, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 51}], "index": 7}, {"bbox": [125, 252, 486, 264], "spans": [{"bbox": [125, 253, 359, 264], "score": 1.0, "content": "implies that unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [359, 252, 412, 264], "score": 0.94, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [413, 253, 486, 264], "score": 1.0, "content": " are constructed", "type": "text"}], "index": 8}, {"bbox": [124, 263, 487, 278], "spans": [{"bbox": [124, 263, 282, 278], "score": 1.0, "content": "by choosing a “primitive” solution ", "type": "text"}, {"bbox": [283, 266, 316, 276], "score": 0.93, "content": "\\left(x,y,z\\right)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [316, 263, 331, 278], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [332, 265, 428, 275], "score": 0.92, "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "type": "inline_equation", "height": 10, "width": 96}, {"bbox": [429, 263, 487, 278], "score": 1.0, "content": " and putting", "type": "text"}], "index": 9}, {"bbox": [126, 276, 487, 289], "spans": [{"bbox": [126, 277, 208, 288], "score": 0.94, "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [209, 276, 234, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [234, 277, 305, 288], "score": 0.93, "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [305, 276, 428, 289], "score": 1.0, "content": " (primitive here means that ", "type": "text"}, {"bbox": [429, 281, 435, 286], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [436, 276, 487, 289], "score": 1.0, "content": " should not", "type": "text"}], "index": 10}, {"bbox": [125, 288, 487, 301], "spans": [{"bbox": [125, 288, 487, 301], "score": 1.0, "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "type": "text"}], "index": 11}, {"bbox": [126, 300, 486, 315], "spans": [{"bbox": [126, 300, 149, 315], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [149, 301, 241, 313], "score": 0.92, "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [242, 300, 294, 315], "score": 1.0, "content": ". If we put ", "type": "text"}, {"bbox": [294, 302, 372, 314], "score": 0.96, "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [372, 300, 486, 315], "score": 1.0, "content": ", then it is an elementary", "type": "text"}], "index": 12}, {"bbox": [126, 315, 486, 326], "spans": [{"bbox": [126, 315, 220, 326], "score": 1.0, "content": "exerc i se to show that ", "type": "text"}, {"bbox": [221, 316, 233, 326], "score": 0.92, "content": "\\alpha\\beta", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [234, 315, 296, 326], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [297, 317, 304, 324], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 315, 394, 326], "score": 1.0, "content": ", hence we also have ", "type": "text"}, {"bbox": [394, 315, 466, 326], "score": 0.94, "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [466, 315, 486, 326], "score": 1.0, "content": " etc.", "type": "text"}], "index": 13}, {"bbox": [125, 326, 487, 339], "spans": [{"bbox": [125, 326, 136, 339], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [136, 329, 172, 337], "score": 0.94, "content": "d_{3}=-4", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [172, 326, 359, 339], "score": 1.0, "content": ", then it is easy to see that we may choose ", "type": "text"}, {"bbox": [360, 329, 366, 338], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [366, 326, 487, 339], "score": 1.0, "content": " as the fundamental unit of", "type": "text"}], "index": 14}, {"bbox": [126, 338, 487, 351], "spans": [{"bbox": [126, 340, 136, 349], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 338, 150, 351], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [151, 340, 187, 349], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [187, 338, 405, 351], "score": 1.0, "content": ", then genus theory says that a) the class number ", "type": "text"}, {"bbox": [405, 340, 411, 348], "score": 0.91, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [411, 338, 425, 351], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [425, 340, 435, 349], "score": 0.92, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 338, 487, 351], "score": 1.0, "content": " is twice an", "type": "text"}], "index": 15}, {"bbox": [126, 352, 486, 363], "spans": [{"bbox": [126, 352, 181, 363], "score": 1.0, "content": "odd number ", "type": "text"}, {"bbox": [181, 355, 187, 360], "score": 0.88, "content": "u", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [187, 352, 291, 363], "score": 1.0, "content": "; and b) the prime ideal ", "type": "text"}, {"bbox": [291, 354, 301, 362], "score": 0.89, "content": "{\\mathfrak{p}}_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [301, 352, 331, 363], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [331, 353, 341, 361], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [341, 352, 354, 363], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [355, 353, 365, 361], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [365, 352, 486, 363], "score": 1.0, "content": " is in the principal genus, so", "type": "text"}], "index": 16}, {"bbox": [126, 363, 486, 375], "spans": [{"bbox": [126, 364, 168, 374], "score": 0.93, "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 363, 356, 375], "score": 1.0, "content": " is principal. Again it can be checked that ", "type": "text"}, {"bbox": [356, 364, 394, 374], "score": 0.93, "content": "\\beta=\\pm\\pi_{3}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [394, 363, 486, 375], "score": 1.0, "content": " for a suitable choice", "type": "text"}], "index": 17}, {"bbox": [125, 374, 175, 388], "spans": [{"bbox": [125, 374, 175, 388], "score": 1.0, "content": "of the sign.", "type": "text"}], "index": 18}], "index": 11.5}, {"type": "text", "bbox": [125, 389, 486, 427], "lines": [{"bbox": [126, 392, 486, 404], "spans": [{"bbox": [126, 392, 262, 404], "score": 1.0, "content": "Example. Consider the case ", "type": "text"}, {"bbox": [262, 394, 329, 402], "score": 0.9, "content": "d\\,=\\,-31\\cdot5\\cdot8", "type": "inline_equation", "height": 8, "width": 67}, {"bbox": [329, 392, 357, 404], "score": 1.0, "content": "; here ", "type": "text"}, {"bbox": [358, 392, 443, 404], "score": 0.92, "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [444, 392, 486, 404], "score": 1.0, "content": ", and the", "type": "text"}], "index": 19}, {"bbox": [126, 404, 486, 417], "spans": [{"bbox": [126, 404, 248, 417], "score": 1.0, "content": "positive sign is correct since ", "type": "text"}, {"bbox": [248, 405, 346, 416], "score": 0.94, "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "type": "inline_equation", "height": 11, "width": 98}, {"bbox": [346, 404, 486, 417], "score": 1.0, "content": " mod 4 is primary. The minimal", "type": "text"}], "index": 20}, {"bbox": [126, 416, 397, 429], "spans": [{"bbox": [126, 416, 189, 429], "score": 1.0, "content": "polynomial of ", "type": "text"}, {"bbox": [189, 418, 208, 429], "score": 0.93, "content": "\\sqrt{\\pi_{3}}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [208, 416, 220, 429], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [221, 417, 313, 428], "score": 0.92, "content": "f(x)=x^{4}-6x^{2}-31", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [313, 416, 397, 429], "score": 1.0, "content": ": compare Table 1.", "type": "text"}], "index": 21}], "index": 20}, {"type": "text", "bbox": [125, 433, 486, 483], "lines": [{"bbox": [137, 435, 486, 449], "spans": [{"bbox": [137, 435, 184, 449], "score": 1.0, "content": "The fields ", "type": "text"}, {"bbox": [185, 437, 246, 448], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [246, 435, 269, 449], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [269, 435, 340, 448], "score": 0.92, "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [340, 435, 486, 449], "score": 1.0, "content": " will play a dominant role in the", "type": "text"}], "index": 22}, {"bbox": [125, 448, 486, 460], "spans": [{"bbox": [125, 449, 297, 460], "score": 1.0, "content": "proof below; they are both contai n ed in ", "type": "text"}, {"bbox": [297, 449, 353, 460], "score": 0.94, "content": "M=F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [353, 449, 370, 460], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [370, 448, 428, 460], "score": 0.94, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [429, 449, 486, 460], "score": 1.0, "content": ", and it is the", "type": "text"}], "index": 23}, {"bbox": [126, 460, 487, 474], "spans": [{"bbox": [126, 460, 227, 474], "score": 1.0, "content": "ambiguous class group ", "type": "text"}, {"bbox": [228, 461, 275, 472], "score": 0.91, "content": "\\mathrm{Am}(M/F)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [275, 460, 487, 474], "score": 1.0, "content": " that contains the information we are interested", "type": "text"}], "index": 24}, {"bbox": [126, 473, 138, 484], "spans": [{"bbox": [126, 473, 138, 484], "score": 1.0, "content": "in.", "type": "text"}], "index": 25}], "index": 23.5}, {"type": "text", "bbox": [125, 489, 486, 513], "lines": [{"bbox": [124, 491, 487, 503], "spans": [{"bbox": [124, 491, 223, 503], "score": 1.0, "content": "Lemma 6. The field ", "type": "text"}, {"bbox": [223, 493, 232, 500], "score": 0.87, "content": "F", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [232, 491, 487, 503], "score": 1.0, "content": " has odd class number (even in the strict sense), and we", "type": "text"}], "index": 26}, {"bbox": [125, 502, 469, 515], "spans": [{"bbox": [125, 502, 149, 515], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [149, 504, 213, 514], "score": 0.84, "content": "\\#\\operatorname{Am}(M/F)\\mid", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [213, 502, 287, 515], "score": 1.0, "content": "2. In particular, ", "type": "text"}, {"bbox": [288, 504, 321, 514], "score": 0.92, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [321, 502, 469, 515], "score": 1.0, "content": " is cyclic (though possibly trivial).", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "text", "bbox": [126, 519, 486, 567], "lines": [{"bbox": [125, 521, 486, 535], "spans": [{"bbox": [125, 521, 316, 535], "score": 1.0, "content": "Proof. The class group in the strict sense of ", "type": "text"}, {"bbox": [317, 524, 326, 532], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [326, 521, 486, 535], "score": 1.0, "content": " is cyclic of order 2 by R´edei’s theory", "type": "text"}], "index": 28}, {"bbox": [126, 533, 486, 547], "spans": [{"bbox": [126, 533, 173, 547], "score": 1.0, "content": "[12] (since ", "type": "text"}, {"bbox": [173, 535, 277, 545], "score": 0.92, "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [278, 533, 347, 547], "score": 1.0, "content": " in case A) and ", "type": "text"}, {"bbox": [348, 535, 452, 545], "score": 0.92, "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [452, 533, 486, 547], "score": 1.0, "content": " in case", "type": "text"}], "index": 29}, {"bbox": [125, 546, 486, 558], "spans": [{"bbox": [125, 546, 173, 558], "score": 1.0, "content": "B)). Since ", "type": "text"}, {"bbox": [173, 547, 181, 555], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [182, 546, 304, 558], "score": 1.0, "content": " is the Hilbert class field of ", "type": "text"}, {"bbox": [304, 547, 314, 556], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [314, 546, 486, 558], "score": 1.0, "content": " in the strict sense, its class number in", "type": "text"}], "index": 30}, {"bbox": [125, 558, 225, 570], "spans": [{"bbox": [125, 558, 225, 570], "score": 1.0, "content": "the strict sense is odd.", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [125, 568, 486, 640], "lines": [{"bbox": [137, 569, 484, 581], "spans": [{"bbox": [137, 569, 424, 581], "score": 1.0, "content": "Next we apply the ambiguous class number formula. In case A), ", "type": "text"}, {"bbox": [425, 571, 433, 578], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [433, 569, 484, 581], "score": 1.0, "content": " is complex,", "type": "text"}], "index": 32}, {"bbox": [126, 581, 485, 594], "spans": [{"bbox": [126, 581, 283, 594], "score": 1.0, "content": "and exactly the two primes above ", "type": "text"}, {"bbox": [283, 583, 293, 592], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [293, 581, 342, 594], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [343, 583, 366, 593], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [366, 581, 424, 594], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [425, 582, 485, 593], "score": 0.94, "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 60}], "index": 33}, {"bbox": [126, 594, 486, 606], "spans": [{"bbox": [126, 594, 149, 606], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [150, 598, 156, 603], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [156, 594, 237, 606], "score": 1.0, "content": " primary of norm ", "type": "text"}, {"bbox": [238, 594, 257, 604], "score": 0.93, "content": "d_{3}y^{2}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [258, 594, 392, 606], "score": 1.0, "content": "; there are four primes above ", "type": "text"}, {"bbox": [392, 595, 402, 604], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [402, 594, 418, 606], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [418, 595, 426, 603], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [427, 594, 486, 606], "score": 1.0, "content": ", and exactly", "type": "text"}], "index": 34}, {"bbox": [126, 606, 487, 618], "spans": [{"bbox": [126, 606, 216, 618], "score": 1.0, "content": "two of them divide ", "type": "text"}, {"bbox": [217, 610, 223, 614], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [224, 606, 324, 618], "score": 1.0, "content": " to an odd power, so ", "type": "text"}, {"bbox": [325, 608, 352, 614], "score": 0.91, "content": "t\\ =\\ 2", "type": "inline_equation", "height": 6, "width": 27}, {"bbox": [352, 606, 487, 618], "score": 1.0, "content": " by the decomposition law in", "type": "text"}], "index": 35}, {"bbox": [125, 617, 487, 630], "spans": [{"bbox": [125, 617, 487, 630], "score": 1.0, "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "type": "text"}], "index": 36}, {"bbox": [125, 629, 360, 641], "spans": [{"bbox": [125, 630, 262, 641], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "type": "inline_equation", "height": 11, "width": 137}, {"bbox": [262, 629, 286, 641], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [287, 630, 320, 641], "score": 0.86, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [320, 629, 360, 641], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 37}], "index": 34.5}, {"type": "text", "bbox": [125, 640, 486, 700], "lines": [{"bbox": [137, 641, 486, 653], "spans": [{"bbox": [137, 641, 232, 653], "score": 1.0, "content": "In case B), however, ", "type": "text"}, {"bbox": [232, 643, 240, 650], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [241, 641, 303, 653], "score": 1.0, "content": " is real; since ", "type": "text"}, {"bbox": [303, 643, 333, 652], "score": 0.93, "content": "\\alpha\\,\\in\\,k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [334, 641, 382, 653], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [383, 642, 423, 652], "score": 0.94, "content": "d_{3}y^{2}\\,<\\,0", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [423, 641, 486, 653], "score": 1.0, "content": ", it has mixed", "type": "text"}], "index": 38}, {"bbox": [125, 653, 486, 665], "spans": [{"bbox": [125, 653, 439, 665], "score": 1.0, "content": "signature, hence there are exactly two infinite primes that ramify in ", "type": "text"}, {"bbox": [439, 654, 462, 665], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [463, 653, 486, 665], "score": 1.0, "content": ". As", "type": "text"}], "index": 39}, {"bbox": [125, 664, 486, 678], "spans": [{"bbox": [125, 664, 331, 678], "score": 1.0, "content": "in case A), there are two finite primes above ", "type": "text"}, {"bbox": [331, 667, 340, 676], "score": 0.92, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [341, 664, 412, 678], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [412, 666, 435, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [436, 664, 486, 678], "score": 1.0, "content": ", so we get", "type": "text"}], "index": 40}, {"bbox": [126, 678, 485, 689], "spans": [{"bbox": [126, 678, 246, 689], "score": 0.91, "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "type": "inline_equation", "height": 11, "width": 120}, {"bbox": [247, 678, 281, 689], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [281, 679, 289, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [290, 678, 476, 689], "score": 1.0, "content": " has odd class number in the strict sense, ", "type": "text"}, {"bbox": [477, 679, 485, 686], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}], "index": 41}, {"bbox": [126, 690, 486, 702], "spans": [{"bbox": [126, 690, 486, 702], "score": 1.0, "content": "has units of independent signs. This implies that the group of units that are positive", "type": "text"}], "index": 42}], "index": 40}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 99], "lines": [{"bbox": [480, 93, 486, 101], "spans": [{"bbox": [480, 93, 486, 101], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 112, 486, 150], "lines": [], "index": 1, "bbox_fs": [124, 114, 487, 151], "lines_deleted": true}, {"type": "text", "bbox": [125, 157, 486, 181], "lines": [{"bbox": [137, 159, 484, 172], "spans": [{"bbox": [137, 159, 255, 172], "score": 1.0, "content": "From now on assume that ", "type": "text"}, {"bbox": [255, 162, 261, 169], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [261, 159, 484, 172], "score": 1.0, "content": " is one of the imaginary quadratic fields of type A)", "type": "text"}], "index": 3}, {"bbox": [126, 172, 316, 183], "spans": [{"bbox": [126, 172, 137, 183], "score": 1.0, "content": "or", "type": "text"}, {"bbox": [138, 173, 145, 181], "score": 0.43, "content": "\\mathrm{B}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [146, 172, 316, 183], "score": 1.0, "content": ") as explained in the Introduction. Let", "type": "text"}], "index": 4}], "index": 3.5, "bbox_fs": [126, 159, 484, 183]}, {"type": "text", "bbox": [124, 214, 486, 385], "lines": [{"bbox": [138, 217, 484, 228], "spans": [{"bbox": [138, 217, 414, 228], "score": 1.0, "content": "Then there exist two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [414, 218, 420, 225], "score": 0.89, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [420, 217, 471, 228], "score": 1.0, "content": " which are ", "type": "text"}, {"bbox": [472, 218, 484, 227], "score": 0.92, "content": "D_{4}", "type": "inline_equation", "height": 9, "width": 12}], "index": 5}, {"bbox": [126, 229, 486, 240], "spans": [{"bbox": [126, 229, 148, 240], "score": 1.0, "content": "over ", "type": "text"}, {"bbox": [149, 230, 157, 239], "score": 0.9, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [157, 229, 486, 240], "score": 1.0, "content": " (see Proposition 2). Let us say a few words about their construction.", "type": "text"}], "index": 6}, {"bbox": [126, 240, 484, 252], "spans": [{"bbox": [126, 240, 360, 252], "score": 1.0, "content": "Consider e.g. case B); by R´edei’s theory (see [12]), the", "type": "text"}, {"bbox": [361, 242, 372, 250], "score": 0.91, "content": "C_{4}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [373, 240, 433, 252], "score": 1.0, "content": "-factorization ", "type": "text"}, {"bbox": [433, 242, 484, 250], "score": 0.93, "content": "d=d_{1}d_{2}\\cdot d_{3}", "type": "inline_equation", "height": 8, "width": 51}], "index": 7}, {"bbox": [125, 252, 486, 264], "spans": [{"bbox": [125, 253, 359, 264], "score": 1.0, "content": "implies that unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [359, 252, 412, 264], "score": 0.94, "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [413, 253, 486, 264], "score": 1.0, "content": " are constructed", "type": "text"}], "index": 8}, {"bbox": [124, 263, 487, 278], "spans": [{"bbox": [124, 263, 282, 278], "score": 1.0, "content": "by choosing a “primitive” solution ", "type": "text"}, {"bbox": [283, 266, 316, 276], "score": 0.93, "content": "\\left(x,y,z\\right)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [316, 263, 331, 278], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [332, 265, 428, 275], "score": 0.92, "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "type": "inline_equation", "height": 10, "width": 96}, {"bbox": [429, 263, 487, 278], "score": 1.0, "content": " and putting", "type": "text"}], "index": 9}, {"bbox": [126, 276, 487, 289], "spans": [{"bbox": [126, 277, 208, 288], "score": 0.94, "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [209, 276, 234, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [234, 277, 305, 288], "score": 0.93, "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [305, 276, 428, 289], "score": 1.0, "content": " (primitive here means that ", "type": "text"}, {"bbox": [429, 281, 435, 286], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [436, 276, 487, 289], "score": 1.0, "content": " should not", "type": "text"}], "index": 10}, {"bbox": [125, 288, 487, 301], "spans": [{"bbox": [125, 288, 487, 301], "score": 1.0, "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "type": "text"}], "index": 11}, {"bbox": [126, 300, 486, 315], "spans": [{"bbox": [126, 300, 149, 315], "score": 1.0, "content": "then", "type": "text"}, {"bbox": [149, 301, 241, 313], "score": 0.92, "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [242, 300, 294, 315], "score": 1.0, "content": ". If we put ", "type": "text"}, {"bbox": [294, 302, 372, 314], "score": 0.96, "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [372, 300, 486, 315], "score": 1.0, "content": ", then it is an elementary", "type": "text"}], "index": 12}, {"bbox": [126, 315, 486, 326], "spans": [{"bbox": [126, 315, 220, 326], "score": 1.0, "content": "exerc i se to show that ", "type": "text"}, {"bbox": [221, 316, 233, 326], "score": 0.92, "content": "\\alpha\\beta", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [234, 315, 296, 326], "score": 1.0, "content": " is a square in ", "type": "text"}, {"bbox": [297, 317, 304, 324], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 315, 394, 326], "score": 1.0, "content": ", hence we also have ", "type": "text"}, {"bbox": [394, 315, 466, 326], "score": 0.94, "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [466, 315, 486, 326], "score": 1.0, "content": " etc.", "type": "text"}], "index": 13}, {"bbox": [125, 326, 487, 339], "spans": [{"bbox": [125, 326, 136, 339], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [136, 329, 172, 337], "score": 0.94, "content": "d_{3}=-4", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [172, 326, 359, 339], "score": 1.0, "content": ", then it is easy to see that we may choose ", "type": "text"}, {"bbox": [360, 329, 366, 338], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [366, 326, 487, 339], "score": 1.0, "content": " as the fundamental unit of", "type": "text"}], "index": 14}, {"bbox": [126, 338, 487, 351], "spans": [{"bbox": [126, 340, 136, 349], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 338, 150, 351], "score": 1.0, "content": "; if ", "type": "text"}, {"bbox": [151, 340, 187, 349], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [187, 338, 405, 351], "score": 1.0, "content": ", then genus theory says that a) the class number ", "type": "text"}, {"bbox": [405, 340, 411, 348], "score": 0.91, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [411, 338, 425, 351], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [425, 340, 435, 349], "score": 0.92, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [435, 338, 487, 351], "score": 1.0, "content": " is twice an", "type": "text"}], "index": 15}, {"bbox": [126, 352, 486, 363], "spans": [{"bbox": [126, 352, 181, 363], "score": 1.0, "content": "odd number ", "type": "text"}, {"bbox": [181, 355, 187, 360], "score": 0.88, "content": "u", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [187, 352, 291, 363], "score": 1.0, "content": "; and b) the prime ideal ", "type": "text"}, {"bbox": [291, 354, 301, 362], "score": 0.89, "content": "{\\mathfrak{p}}_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [301, 352, 331, 363], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [331, 353, 341, 361], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [341, 352, 354, 363], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [355, 353, 365, 361], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [365, 352, 486, 363], "score": 1.0, "content": " is in the principal genus, so", "type": "text"}], "index": 16}, {"bbox": [126, 363, 486, 375], "spans": [{"bbox": [126, 364, 168, 374], "score": 0.93, "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [168, 363, 356, 375], "score": 1.0, "content": " is principal. Again it can be checked that ", "type": "text"}, {"bbox": [356, 364, 394, 374], "score": 0.93, "content": "\\beta=\\pm\\pi_{3}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [394, 363, 486, 375], "score": 1.0, "content": " for a suitable choice", "type": "text"}], "index": 17}, {"bbox": [125, 374, 175, 388], "spans": [{"bbox": [125, 374, 175, 388], "score": 1.0, "content": "of the sign.", "type": "text"}], "index": 18}], "index": 11.5, "bbox_fs": [124, 217, 487, 388]}, {"type": "text", "bbox": [125, 389, 486, 427], "lines": [{"bbox": [126, 392, 486, 404], "spans": [{"bbox": [126, 392, 262, 404], "score": 1.0, "content": "Example. Consider the case ", "type": "text"}, {"bbox": [262, 394, 329, 402], "score": 0.9, "content": "d\\,=\\,-31\\cdot5\\cdot8", "type": "inline_equation", "height": 8, "width": 67}, {"bbox": [329, 392, 357, 404], "score": 1.0, "content": "; here ", "type": "text"}, {"bbox": [358, 392, 443, 404], "score": 0.92, "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [444, 392, 486, 404], "score": 1.0, "content": ", and the", "type": "text"}], "index": 19}, {"bbox": [126, 404, 486, 417], "spans": [{"bbox": [126, 404, 248, 417], "score": 1.0, "content": "positive sign is correct since ", "type": "text"}, {"bbox": [248, 405, 346, 416], "score": 0.94, "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "type": "inline_equation", "height": 11, "width": 98}, {"bbox": [346, 404, 486, 417], "score": 1.0, "content": " mod 4 is primary. The minimal", "type": "text"}], "index": 20}, {"bbox": [126, 416, 397, 429], "spans": [{"bbox": [126, 416, 189, 429], "score": 1.0, "content": "polynomial of ", "type": "text"}, {"bbox": [189, 418, 208, 429], "score": 0.93, "content": "\\sqrt{\\pi_{3}}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [208, 416, 220, 429], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [221, 417, 313, 428], "score": 0.92, "content": "f(x)=x^{4}-6x^{2}-31", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [313, 416, 397, 429], "score": 1.0, "content": ": compare Table 1.", "type": "text"}], "index": 21}], "index": 20, "bbox_fs": [126, 392, 486, 429]}, {"type": "text", "bbox": [125, 433, 486, 483], "lines": [{"bbox": [137, 435, 486, 449], "spans": [{"bbox": [137, 435, 184, 449], "score": 1.0, "content": "The fields ", "type": "text"}, {"bbox": [185, 437, 246, 448], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [246, 435, 269, 449], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [269, 435, 340, 448], "score": 0.92, "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [340, 435, 486, 449], "score": 1.0, "content": " will play a dominant role in the", "type": "text"}], "index": 22}, {"bbox": [125, 448, 486, 460], "spans": [{"bbox": [125, 449, 297, 460], "score": 1.0, "content": "proof below; they are both contai n ed in ", "type": "text"}, {"bbox": [297, 449, 353, 460], "score": 0.94, "content": "M=F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [353, 449, 370, 460], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [370, 448, 428, 460], "score": 0.94, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [429, 449, 486, 460], "score": 1.0, "content": ", and it is the", "type": "text"}], "index": 23}, {"bbox": [126, 460, 487, 474], "spans": [{"bbox": [126, 460, 227, 474], "score": 1.0, "content": "ambiguous class group ", "type": "text"}, {"bbox": [228, 461, 275, 472], "score": 0.91, "content": "\\mathrm{Am}(M/F)", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [275, 460, 487, 474], "score": 1.0, "content": " that contains the information we are interested", "type": "text"}], "index": 24}, {"bbox": [126, 473, 138, 484], "spans": [{"bbox": [126, 473, 138, 484], "score": 1.0, "content": "in.", "type": "text"}], "index": 25}], "index": 23.5, "bbox_fs": [125, 435, 487, 484]}, {"type": "text", "bbox": [125, 489, 486, 513], "lines": [{"bbox": [124, 491, 487, 503], "spans": [{"bbox": [124, 491, 223, 503], "score": 1.0, "content": "Lemma 6. The field ", "type": "text"}, {"bbox": [223, 493, 232, 500], "score": 0.87, "content": "F", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [232, 491, 487, 503], "score": 1.0, "content": " has odd class number (even in the strict sense), and we", "type": "text"}], "index": 26}, {"bbox": [125, 502, 469, 515], "spans": [{"bbox": [125, 502, 149, 515], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [149, 504, 213, 514], "score": 0.84, "content": "\\#\\operatorname{Am}(M/F)\\mid", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [213, 502, 287, 515], "score": 1.0, "content": "2. In particular, ", "type": "text"}, {"bbox": [288, 504, 321, 514], "score": 0.92, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [321, 502, 469, 515], "score": 1.0, "content": " is cyclic (though possibly trivial).", "type": "text"}], "index": 27}], "index": 26.5, "bbox_fs": [124, 491, 487, 515]}, {"type": "text", "bbox": [126, 519, 486, 567], "lines": [{"bbox": [125, 521, 486, 535], "spans": [{"bbox": [125, 521, 316, 535], "score": 1.0, "content": "Proof. The class group in the strict sense of ", "type": "text"}, {"bbox": [317, 524, 326, 532], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [326, 521, 486, 535], "score": 1.0, "content": " is cyclic of order 2 by R´edei’s theory", "type": "text"}], "index": 28}, {"bbox": [126, 533, 486, 547], "spans": [{"bbox": [126, 533, 173, 547], "score": 1.0, "content": "[12] (since ", "type": "text"}, {"bbox": [173, 535, 277, 545], "score": 0.92, "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [278, 533, 347, 547], "score": 1.0, "content": " in case A) and ", "type": "text"}, {"bbox": [348, 535, 452, 545], "score": 0.92, "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "type": "inline_equation", "height": 10, "width": 104}, {"bbox": [452, 533, 486, 547], "score": 1.0, "content": " in case", "type": "text"}], "index": 29}, {"bbox": [125, 546, 486, 558], "spans": [{"bbox": [125, 546, 173, 558], "score": 1.0, "content": "B)). Since ", "type": "text"}, {"bbox": [173, 547, 181, 555], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [182, 546, 304, 558], "score": 1.0, "content": " is the Hilbert class field of ", "type": "text"}, {"bbox": [304, 547, 314, 556], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [314, 546, 486, 558], "score": 1.0, "content": " in the strict sense, its class number in", "type": "text"}], "index": 30}, {"bbox": [125, 558, 225, 570], "spans": [{"bbox": [125, 558, 225, 570], "score": 1.0, "content": "the strict sense is odd.", "type": "text"}], "index": 31}], "index": 29.5, "bbox_fs": [125, 521, 486, 570]}, {"type": "text", "bbox": [125, 568, 486, 640], "lines": [{"bbox": [137, 569, 484, 581], "spans": [{"bbox": [137, 569, 424, 581], "score": 1.0, "content": "Next we apply the ambiguous class number formula. In case A), ", "type": "text"}, {"bbox": [425, 571, 433, 578], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [433, 569, 484, 581], "score": 1.0, "content": " is complex,", "type": "text"}], "index": 32}, {"bbox": [126, 581, 485, 594], "spans": [{"bbox": [126, 581, 283, 594], "score": 1.0, "content": "and exactly the two primes above ", "type": "text"}, {"bbox": [283, 583, 293, 592], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [293, 581, 342, 594], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [343, 583, 366, 593], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [366, 581, 424, 594], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [425, 582, 485, 593], "score": 0.94, "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "type": "inline_equation", "height": 11, "width": 60}], "index": 33}, {"bbox": [126, 594, 486, 606], "spans": [{"bbox": [126, 594, 149, 606], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [150, 598, 156, 603], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [156, 594, 237, 606], "score": 1.0, "content": " primary of norm ", "type": "text"}, {"bbox": [238, 594, 257, 604], "score": 0.93, "content": "d_{3}y^{2}", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [258, 594, 392, 606], "score": 1.0, "content": "; there are four primes above ", "type": "text"}, {"bbox": [392, 595, 402, 604], "score": 0.91, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [402, 594, 418, 606], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [418, 595, 426, 603], "score": 0.9, "content": "F", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [427, 594, 486, 606], "score": 1.0, "content": ", and exactly", "type": "text"}], "index": 34}, {"bbox": [126, 606, 487, 618], "spans": [{"bbox": [126, 606, 216, 618], "score": 1.0, "content": "two of them divide ", "type": "text"}, {"bbox": [217, 610, 223, 614], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [224, 606, 324, 618], "score": 1.0, "content": " to an odd power, so ", "type": "text"}, {"bbox": [325, 608, 352, 614], "score": 0.91, "content": "t\\ =\\ 2", "type": "inline_equation", "height": 6, "width": 27}, {"bbox": [352, 606, 487, 618], "score": 1.0, "content": " by the decomposition law in", "type": "text"}], "index": 35}, {"bbox": [125, 617, 487, 630], "spans": [{"bbox": [125, 617, 487, 630], "score": 1.0, "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "type": "text"}], "index": 36}, {"bbox": [125, 629, 360, 641], "spans": [{"bbox": [125, 630, 262, 641], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "type": "inline_equation", "height": 11, "width": 137}, {"bbox": [262, 629, 286, 641], "score": 1.0, "content": ", and", "type": "text"}, {"bbox": [287, 630, 320, 641], "score": 0.86, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [320, 629, 360, 641], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 37}], "index": 34.5, "bbox_fs": [125, 569, 487, 641]}, {"type": "text", "bbox": [125, 640, 486, 700], "lines": [{"bbox": [137, 641, 486, 653], "spans": [{"bbox": [137, 641, 232, 653], "score": 1.0, "content": "In case B), however, ", "type": "text"}, {"bbox": [232, 643, 240, 650], "score": 0.89, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [241, 641, 303, 653], "score": 1.0, "content": " is real; since ", "type": "text"}, {"bbox": [303, 643, 333, 652], "score": 0.93, "content": "\\alpha\\,\\in\\,k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [334, 641, 382, 653], "score": 1.0, "content": " has norm ", "type": "text"}, {"bbox": [383, 642, 423, 652], "score": 0.94, "content": "d_{3}y^{2}\\,<\\,0", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [423, 641, 486, 653], "score": 1.0, "content": ", it has mixed", "type": "text"}], "index": 38}, {"bbox": [125, 653, 486, 665], "spans": [{"bbox": [125, 653, 439, 665], "score": 1.0, "content": "signature, hence there are exactly two infinite primes that ramify in ", "type": "text"}, {"bbox": [439, 654, 462, 665], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [463, 653, 486, 665], "score": 1.0, "content": ". As", "type": "text"}], "index": 39}, {"bbox": [125, 664, 486, 678], "spans": [{"bbox": [125, 664, 331, 678], "score": 1.0, "content": "in case A), there are two finite primes above ", "type": "text"}, {"bbox": [331, 667, 340, 676], "score": 0.92, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [341, 664, 412, 678], "score": 1.0, "content": " that ramify in ", "type": "text"}, {"bbox": [412, 666, 435, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [436, 664, 486, 678], "score": 1.0, "content": ", so we get", "type": "text"}], "index": 40}, {"bbox": [126, 678, 485, 689], "spans": [{"bbox": [126, 678, 246, 689], "score": 0.91, "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "type": "inline_equation", "height": 11, "width": 120}, {"bbox": [247, 678, 281, 689], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [281, 679, 289, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [290, 678, 476, 689], "score": 1.0, "content": " has odd class number in the strict sense, ", "type": "text"}, {"bbox": [477, 679, 485, 686], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}], "index": 41}, {"bbox": [126, 690, 486, 702], "spans": [{"bbox": [126, 690, 486, 702], "score": 1.0, "content": "has units of independent signs. This implies that the group of units that are positive", "type": "text"}], "index": 42}, {"bbox": [125, 114, 486, 126], "spans": [{"bbox": [125, 114, 297, 126], "score": 1.0, "content": "at the two ramified infinite primes has ", "type": "text", "cross_page": true}, {"bbox": [297, 116, 304, 123], "score": 0.9, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 7, "width": 7, "cross_page": true}, {"bbox": [304, 114, 358, 126], "score": 1.0, "content": "-rank 2, i.e. ", "type": "text", "cross_page": true}, {"bbox": [358, 115, 410, 126], "score": 0.92, "content": "(E:H)\\geq4", "type": "inline_equation", "height": 11, "width": 52, "cross_page": true}, {"bbox": [410, 114, 486, 126], "score": 1.0, "content": " by consideration", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [125, 127, 486, 138], "spans": [{"bbox": [125, 127, 311, 138], "score": 1.0, "content": "of the infinite primes alone. In particular, ", "type": "text", "cross_page": true}, {"bbox": [311, 127, 391, 138], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "type": "inline_equation", "height": 11, "width": 80, "cross_page": true}, {"bbox": [391, 127, 441, 138], "score": 1.0, "content": " in case B).", "type": "text", "cross_page": true}, {"bbox": [476, 127, 486, 137], "score": 0.9776784181594849, "content": "口", "type": "text", "cross_page": true}], "index": 1}], "index": 40, "bbox_fs": [125, 641, 486, 702]}]}
|
[{"type": "text", "bbox": [125, 112, 486, 150], "content": "", "index": 0}, {"type": "text", "bbox": [125, 157, 486, 181], "content": "From now on assume that is one of the imaginary quadratic fields of type A) or ) as explained in the Introduction. Let", "index": 1}, {"type": "text", "bbox": [124, 214, 486, 385], "content": "Then there exist two unramified cyclic quartic extensions of which are over (see Proposition 2). Let us say a few words about their construction. Consider e.g. case B); by R´edei’s theory (see [12]), the -factorization implies that unramified cyclic quartic extensions of are constructed by choosing a “primitive” solution of and putting with (primitive here means that should not be divisible by rational integers); the other unramified cyclic quartic extension is then . If we put , then it is an elementary exerc i se to show that is a square in , hence we also have etc. If , then it is easy to see that we may choose as the fundamental unit of ; if , then genus theory says that a) the class number of is twice an odd number ; and b) the prime ideal above in is in the principal genus, so is principal. Again it can be checked that for a suitable choice of the sign.", "index": 2}, {"type": "text", "bbox": [125, 389, 486, 427], "content": "Example. Consider the case ; here , and the positive sign is correct since mod 4 is primary. The minimal polynomial of is : compare Table 1.", "index": 3}, {"type": "text", "bbox": [125, 433, 486, 483], "content": "The fields and will play a dominant role in the proof below; they are both contai n ed in for , and it is the ambiguous class group that contains the information we are interested in.", "index": 4}, {"type": "text", "bbox": [125, 489, 486, 513], "content": "Lemma 6. The field has odd class number (even in the strict sense), and we have 2. In particular, is cyclic (though possibly trivial).", "index": 5}, {"type": "text", "bbox": [126, 519, 486, 567], "content": "Proof. The class group in the strict sense of is cyclic of order 2 by R´edei’s theory [12] (since in case A) and in case B)). Since is the Hilbert class field of in the strict sense, its class number in the strict sense is odd.", "index": 6}, {"type": "text", "bbox": [125, 568, 486, 640], "content": "Next we apply the ambiguous class number formula. In case A), is complex, and exactly the two primes above ramify in . Note that with primary of norm ; there are four primes above in , and exactly two of them divide to an odd power, so by the decomposition law in quadratic Kummer extensions. By Proposition 4 and the remarks following it, , and is cyclic.", "index": 7}, {"type": "text", "bbox": [125, 640, 486, 700], "content": "In case B), however, is real; since has norm , it has mixed signature, hence there are exactly two infinite primes that ramify in . As in case A), there are two finite primes above that ramify in , so we get . Since has odd class number in the strict sense, has units of independent signs. This implies that the group of units that are positive at the two ramified infinite primes has -rank 2, i.e. by consideration of the infinite primes alone. In particular, in case B). 口", "index": 8}]
|
[{"bbox": [137, 159, 484, 172], "content": "From now on assume that is one of the imaginary quadratic fields of type A)", "parent_index": 1, "line_index": 0}, {"bbox": [126, 172, 316, 183], "content": "or ) as explained in the Introduction. Let", "parent_index": 1, "line_index": 1}, {"bbox": [138, 217, 484, 228], "content": "Then there exist two unramified cyclic quartic extensions of which are", "parent_index": 2, "line_index": 0}, {"bbox": [126, 229, 486, 240], "content": "over (see Proposition 2). Let us say a few words about their construction.", "parent_index": 2, "line_index": 1}, {"bbox": [126, 240, 484, 252], "content": "Consider e.g. case B); by R´edei’s theory (see [12]), the -factorization", "parent_index": 2, "line_index": 2}, {"bbox": [125, 252, 486, 264], "content": "implies that unramified cyclic quartic extensions of are constructed", "parent_index": 2, "line_index": 3}, {"bbox": [124, 263, 487, 278], "content": "by choosing a “primitive” solution of and putting", "parent_index": 2, "line_index": 4}, {"bbox": [126, 276, 487, 289], "content": "with (primitive here means that should not", "parent_index": 2, "line_index": 5}, {"bbox": [125, 288, 487, 301], "content": "be divisible by rational integers); the other unramified cyclic quartic extension is", "parent_index": 2, "line_index": 6}, {"bbox": [126, 300, 486, 315], "content": "then . If we put , then it is an elementary", "parent_index": 2, "line_index": 7}, {"bbox": [126, 315, 486, 326], "content": "exerc i se to show that is a square in , hence we also have etc.", "parent_index": 2, "line_index": 8}, {"bbox": [125, 326, 487, 339], "content": "If , then it is easy to see that we may choose as the fundamental unit of", "parent_index": 2, "line_index": 9}, {"bbox": [126, 338, 487, 351], "content": "; if , then genus theory says that a) the class number of is twice an", "parent_index": 2, "line_index": 10}, {"bbox": [126, 352, 486, 363], "content": "odd number ; and b) the prime ideal above in is in the principal genus, so", "parent_index": 2, "line_index": 11}, {"bbox": [126, 363, 486, 375], "content": "is principal. Again it can be checked that for a suitable choice", "parent_index": 2, "line_index": 12}, {"bbox": [125, 374, 175, 388], "content": "of the sign.", "parent_index": 2, "line_index": 13}, {"bbox": [126, 392, 486, 404], "content": "Example. Consider the case ; here , and the", "parent_index": 3, "line_index": 0}, {"bbox": [126, 404, 486, 417], "content": "positive sign is correct since mod 4 is primary. The minimal", "parent_index": 3, "line_index": 1}, {"bbox": [126, 416, 397, 429], "content": "polynomial of is : compare Table 1.", "parent_index": 3, "line_index": 2}, {"bbox": [137, 435, 486, 449], "content": "The fields and will play a dominant role in the", "parent_index": 4, "line_index": 0}, {"bbox": [125, 448, 486, 460], "content": "proof below; they are both contai n ed in for , and it is the", "parent_index": 4, "line_index": 1}, {"bbox": [126, 460, 487, 474], "content": "ambiguous class group that contains the information we are interested", "parent_index": 4, "line_index": 2}, {"bbox": [126, 473, 138, 484], "content": "in.", "parent_index": 4, "line_index": 3}, {"bbox": [124, 491, 487, 503], "content": "Lemma 6. The field has odd class number (even in the strict sense), and we", "parent_index": 5, "line_index": 0}, {"bbox": [125, 502, 469, 515], "content": "have 2. In particular, is cyclic (though possibly trivial).", "parent_index": 5, "line_index": 1}, {"bbox": [125, 521, 486, 535], "content": "Proof. The class group in the strict sense of is cyclic of order 2 by R´edei’s theory", "parent_index": 6, "line_index": 0}, {"bbox": [126, 533, 486, 547], "content": "[12] (since in case A) and in case", "parent_index": 6, "line_index": 1}, {"bbox": [125, 546, 486, 558], "content": "B)). Since is the Hilbert class field of in the strict sense, its class number in", "parent_index": 6, "line_index": 2}, {"bbox": [125, 558, 225, 570], "content": "the strict sense is odd.", "parent_index": 6, "line_index": 3}, {"bbox": [137, 569, 484, 581], "content": "Next we apply the ambiguous class number formula. In case A), is complex,", "parent_index": 7, "line_index": 0}, {"bbox": [126, 581, 485, 594], "content": "and exactly the two primes above ramify in . Note that", "parent_index": 7, "line_index": 1}, {"bbox": [126, 594, 486, 606], "content": "with primary of norm ; there are four primes above in , and exactly", "parent_index": 7, "line_index": 2}, {"bbox": [126, 606, 487, 618], "content": "two of them divide to an odd power, so by the decomposition law in", "parent_index": 7, "line_index": 3}, {"bbox": [125, 617, 487, 630], "content": "quadratic Kummer extensions. By Proposition 4 and the remarks following it,", "parent_index": 7, "line_index": 4}, {"bbox": [125, 629, 360, 641], "content": ", and is cyclic.", "parent_index": 7, "line_index": 5}, {"bbox": [137, 641, 486, 653], "content": "In case B), however, is real; since has norm , it has mixed", "parent_index": 8, "line_index": 0}, {"bbox": [125, 653, 486, 665], "content": "signature, hence there are exactly two infinite primes that ramify in . As", "parent_index": 8, "line_index": 1}, {"bbox": [125, 664, 486, 678], "content": "in case A), there are two finite primes above that ramify in , so we get", "parent_index": 8, "line_index": 2}, {"bbox": [126, 678, 485, 689], "content": ". Since has odd class number in the strict sense,", "parent_index": 8, "line_index": 3}, {"bbox": [126, 690, 486, 702], "content": "has units of independent signs. This implies that the group of units that are positive", "parent_index": 8, "line_index": 4}, {"bbox": [125, 114, 486, 126], "content": "at the two ramified infinite primes has -rank 2, i.e. by consideration", "parent_index": 8, "line_index": 5}, {"bbox": [125, 127, 486, 138], "content": "of the infinite primes alone. In particular, in case B). 口", "parent_index": 8, "line_index": 6}]
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[]
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[{"bbox": [255, 162, 261, 169], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 173, 145, 181], "content": "\\mathrm{B}", "parent_index": 1, "subtype": "inline"}, {"bbox": [414, 218, 420, 225], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [472, 218, 484, 227], "content": "D_{4}", "parent_index": 2, "subtype": "inline"}, {"bbox": [149, 230, 157, 239], "content": "\\mathbb{Q}", "parent_index": 2, "subtype": "inline"}, {"bbox": [361, 242, 372, 250], "content": "C_{4}", "parent_index": 2, "subtype": "inline"}, {"bbox": [433, 242, 484, 250], "content": "d=d_{1}d_{2}\\cdot d_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [359, 252, 412, 264], "content": "k\\,=\\,\\mathbb{Q}({\\sqrt{d}}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [283, 266, 316, 276], "content": "\\left(x,y,z\\right)", "parent_index": 2, "subtype": "inline"}, {"bbox": [332, 265, 428, 275], "content": "d_{1}d_{2}X^{2}+d_{3}Y^{2}\\,=\\,Z^{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 277, 208, 288], "content": "L=k(\\sqrt{d_{1}d_{2}},\\sqrt{\\alpha}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [234, 277, 305, 288], "content": "\\alpha=z+x\\sqrt{d_{1}d_{2}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [429, 281, 435, 286], "content": "\\alpha", "parent_index": 2, "subtype": "inline"}, {"bbox": [149, 301, 241, 313], "content": "\\widetilde{L}=k(\\sqrt{d_{1}d_{2}},\\sqrt{d_{1}\\alpha}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [294, 302, 372, 314], "content": "\\beta\\,=\\,{\\textstyle\\frac{1}{2}}(z+y\\sqrt{d_{3}}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [221, 316, 233, 326], "content": "\\alpha\\beta", "parent_index": 2, "subtype": "inline"}, {"bbox": [297, 317, 304, 324], "content": "L", "parent_index": 2, "subtype": "inline"}, {"bbox": [394, 315, 466, 326], "content": "L=k(\\sqrt{d_{3}},\\sqrt{\\beta}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [136, 329, 172, 337], "content": "d_{3}=-4", "parent_index": 2, "subtype": "inline"}, {"bbox": [360, 329, 366, 338], "content": "\\beta", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 340, 136, 349], "content": "k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [151, 340, 187, 349], "content": "d_{3}\\neq-4", "parent_index": 2, "subtype": "inline"}, {"bbox": [405, 340, 411, 348], "content": "h", "parent_index": 2, "subtype": "inline"}, {"bbox": [425, 340, 435, 349], "content": "k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [181, 355, 187, 360], "content": "u", "parent_index": 2, "subtype": "inline"}, {"bbox": [291, 354, 301, 362], "content": "{\\mathfrak{p}}_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [331, 353, 341, 361], "content": "d_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [355, 353, 365, 361], "content": "k_{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 364, 168, 374], "content": "{\\mathfrak{p}}_{3}^{u}=(\\pi_{3})", "parent_index": 2, "subtype": "inline"}, {"bbox": [356, 364, 394, 374], "content": "\\beta=\\pm\\pi_{3}", "parent_index": 2, "subtype": "inline"}, {"bbox": [262, 394, 329, 402], "content": "d\\,=\\,-31\\cdot5\\cdot8", "parent_index": 3, "subtype": "inline"}, {"bbox": [358, 392, 443, 404], "content": "\\pi_{3}\\,=\\,\\pm(3+2{\\sqrt{10}}\\,)", "parent_index": 3, "subtype": "inline"}, {"bbox": [248, 405, 346, 416], "content": "3\\,{+}\\,2{\\sqrt{10}}\\equiv(1\\,{+}\\,{\\sqrt{10}}\\,)^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [189, 418, 208, 429], "content": "\\sqrt{\\pi_{3}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [221, 417, 313, 428], "content": "f(x)=x^{4}-6x^{2}-31", "parent_index": 3, "subtype": "inline"}, {"bbox": [185, 437, 246, 448], "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [269, 435, 340, 448], "content": "\\tilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "parent_index": 4, "subtype": "inline"}, {"bbox": [297, 449, 353, 460], "content": "M=F({\\sqrt{\\alpha}}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [370, 448, 428, 460], "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [228, 461, 275, 472], "content": "\\mathrm{Am}(M/F)", "parent_index": 4, "subtype": "inline"}, {"bbox": [223, 493, 232, 500], "content": "F", "parent_index": 5, "subtype": "inline"}, {"bbox": [149, 504, 213, 514], "content": "\\#\\operatorname{Am}(M/F)\\mid", "parent_index": 5, "subtype": "inline"}, {"bbox": [288, 504, 321, 514], "content": "\\mathrm{Cl_{2}}(M)", "parent_index": 5, "subtype": "inline"}, {"bbox": [317, 524, 326, 532], "content": "k_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [173, 535, 277, 545], "content": "(d_{2}/p_{3})=(d_{3}/p_{2})=-1", "parent_index": 6, "subtype": "inline"}, {"bbox": [348, 535, 452, 545], "content": "(d_{1}/p_{2})=(d_{2}/p_{1})=-1", "parent_index": 6, "subtype": "inline"}, {"bbox": [173, 547, 181, 555], "content": "F", "parent_index": 6, "subtype": "inline"}, {"bbox": [304, 547, 314, 556], "content": "k_{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [425, 571, 433, 578], "content": "F", "parent_index": 7, "subtype": "inline"}, {"bbox": [283, 583, 293, 592], "content": "d_{3}", "parent_index": 7, "subtype": "inline"}, {"bbox": [343, 583, 366, 593], "content": "M/F", "parent_index": 7, "subtype": "inline"}, {"bbox": [425, 582, 485, 593], "content": "M\\,=\\,F({\\sqrt{\\alpha}}\\,)", "parent_index": 7, "subtype": "inline"}, {"bbox": [150, 598, 156, 603], "content": "\\alpha", "parent_index": 7, "subtype": "inline"}, {"bbox": [238, 594, 257, 604], "content": "d_{3}y^{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [392, 595, 402, 604], "content": "d_{3}", "parent_index": 7, "subtype": "inline"}, {"bbox": [418, 595, 426, 603], "content": "F", "parent_index": 7, "subtype": "inline"}, {"bbox": [217, 610, 223, 614], "content": "\\alpha", "parent_index": 7, "subtype": "inline"}, {"bbox": [325, 608, 352, 614], "content": "t\\ =\\ 2", "parent_index": 7, "subtype": "inline"}, {"bbox": [125, 630, 262, 641], "content": "\\#\\operatorname{Am}_{2}(M/F)=2/(E:H)\\leq2", "parent_index": 7, "subtype": "inline"}, {"bbox": [287, 630, 320, 641], "content": "\\mathrm{Cl_{2}}(M)", "parent_index": 7, "subtype": "inline"}, {"bbox": [232, 643, 240, 650], "content": "F", "parent_index": 8, "subtype": "inline"}, {"bbox": [303, 643, 333, 652], "content": "\\alpha\\,\\in\\,k_{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [383, 642, 423, 652], "content": "d_{3}y^{2}\\,<\\,0", "parent_index": 8, "subtype": "inline"}, {"bbox": [439, 654, 462, 665], "content": "M/F", "parent_index": 8, "subtype": "inline"}, {"bbox": [331, 667, 340, 676], "content": "d_{3}", "parent_index": 8, "subtype": "inline"}, {"bbox": [412, 666, 435, 677], "content": "M/F", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 678, 246, 689], "content": "\\#\\operatorname{Am}_{2}(M/F)=8/(E:H)", "parent_index": 8, "subtype": "inline"}, {"bbox": [281, 679, 289, 686], "content": "F", "parent_index": 8, "subtype": "inline"}, {"bbox": [477, 679, 485, 686], "content": "F", "parent_index": 8, "subtype": "inline"}, {"bbox": [297, 116, 304, 123], "content": "\\mathbb{Z}", "parent_index": 8, "subtype": "inline"}, {"bbox": [358, 115, 410, 126], "content": "(E:H)\\geq4", "parent_index": 8, "subtype": "inline"}, {"bbox": [311, 127, 391, 138], "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "parent_index": 8, "subtype": "inline"}]
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[]
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Next we derive some relations between the class groups of $K_{2}$ and ${\tilde{K}}_{2}$ ; these relations will allow us to use each of them as our field $K$ in Theorem 1.
Proposition 8. Let $L$ and $\widetilde{L}$ be the two unramified cyclic quartic extensions of $k$ , and let $K_{2}$ and ${\widetilde{K}}_{2}$ be two qu a dratic extensions of $k_{2}$ in $L$ and $\widetilde{L}$ , respectively, which are not normal o ver $\mathbb{Q}$ .
a) We have 4 $|\mathit{\Omega}_{h}(K_{2})$ if and only if $4\mid h(\widetilde{K}_{2})$ ;
b) $I f\,4\mid h(K_{2})$ , then one of $\mathrm{Cl}_{2}(K_{2})$ or $\mathrm{Cl}_{2}(\widetilde{K}_{2})$ has type $(2,2)$ , whereas the other is cyclic of order $\geq4$ .
Proof. Notice that the prime dividing $\mathrm{disc}(k_{1})$ splits in $k_{2}$ . Throughout this proof, let $\mathfrak{p}$ be one of the primes of $k_{2}$ dividing $\mathrm{disc}(k_{1})$ .
If we write $K_{2}=k_{2}(\sqrt{\alpha}\,)$ for some $\alpha\in k_{2}$ , then $\widetilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$ . In fact, $K_{2}$ and ${\tilde{K}}_{2}$ are the only extensions $F/k_{2}$ of $k_{2}$ with the p roperties
1. $F/k_{2}$ is a quadratic extension unramified outside $\mathfrak{p}$ ;
Therefore it suffices to observe that if $k_{2}(\sqrt{\alpha}\,)$ has these properties, then so does $k_{2}(\sqrt{d_{2}\alpha}\,)$ . But this is elementary.
In particular, the compositum $M\,=\,K_{2}\tilde{K}_{2}\,=\,k_{2}(\sqrt{d_{2}},\sqrt{\alpha}\,)$ is an extension of type $(2,2)$ over $k_{2}$ with subextensions $K_{2}$ , $\widetilde{K}_{2}$ and $F=k_{2}(\sqrt{d_{2}}\,)$ . Clearly $F$ is the unramified quadratic extension of $k_{2}$ , so bo t h $M/K_{2}$ and $M/\widetilde{K}_{2}$ are unramified. If $K_{2}$ had 2-class number 2, then $M$ would have odd class numb e r, and $M$ would also be the 2-class field of ${\tilde{K}}_{2}$ . Thus $2\parallel h(K_{2})$ implies that $2\parallel h(\widetilde{K}_{2})$ . This proves part a) of the proposition.
Before we go on, we give a Hasse diagram for the fields occurring in this proof:
Now assume that $4\,\mid\,h(K_{2})$ . Since $\mathrm{Cl_{2}}(M)$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $N/M$ , and the uniqueness implies at once that $N/k_{2}$ is normal. Hence $G\,=\,\operatorname{Gal}(N/k_{2})$ is a group of order 8 containing a subgroup of type $(2,2)\,\simeq\,\mathrm{Gal}(N/F)$ : in fact, if $\operatorname{Gal}(N/F)$ were cyclic, then the primes ramifying in $M/F$ would also ramify in $N/M$ contradicting the fact that $N/M$ is unramified. There are three groups satisfying these conditions: $G=(2,4)$ , $G=(2,2,2)$ and $G=D_{4}$ . We claim that $G$ is non-abelian; once we have proved this, it follows that exactly one of the groups $\mathrm{Gal}(N/K_{2})$ and $\mathrm{Gal}(N/\widetilde{K}_{2})$ is cyclic, and that the other is not, which is what we want to prove.
So assume that $G$ is abelian. Then $M/F$ is ramified at two finite primes $\mathfrak{q}$ and ${\mathfrak{q}}^{\prime}$ of $F$ dividing $\mathfrak{p}$ (in $k_{2}$ ); if $F_{1}$ and $F_{2}$ denote the quadratic subextensions of $N/F$ different from $M$ then $F_{1}/F$ and $F_{2}/F$ must be ramified at a finite prime (since
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<html><body>
<p data-bbox="125 145 486 169">Next we derive some relations between the class groups of $K_{2}$ and ${\tilde{K}}_{2}$ ; these relations will allow us to use each of them as our field $K$ in Theorem 1. </p>
<p data-bbox="124 174 486 212">Proposition 8. Let $L$ and $\widetilde{L}$ be the two unramified cyclic quartic extensions of $k$ , and let $K_{2}$ and ${\widetilde{K}}_{2}$ be two qu a dratic extensions of $k_{2}$ in $L$ and $\widetilde{L}$ , respectively, which are not normal o ver $\mathbb{Q}$ . </p>
<p data-bbox="133 214 487 253">a) We have 4 $|\mathit{\Omega}_{h}(K_{2})$ if and only if $4\mid h(\widetilde{K}_{2})$ ;
b) $I f\,4\mid h(K_{2})$ , then one of $\mathrm{Cl}_{2}(K_{2})$ or $\mathrm{Cl}_{2}(\widetilde{K}_{2})$ has type $(2,2)$ , whereas the other is cyclic of order $\geq4$ . </p>
<p data-bbox="125 259 486 284">Proof. Notice that the prime dividing $\mathrm{disc}(k_{1})$ splits in $k_{2}$ . Throughout this proof, let $\mathfrak{p}$ be one of the primes of $k_{2}$ dividing $\mathrm{disc}(k_{1})$ . </p>
<p data-bbox="125 285 486 311">If we write $K_{2}=k_{2}(\sqrt{\alpha}\,)$ for some $\alpha\in k_{2}$ , then $\widetilde{K}_{2}=k_{2}\big(\sqrt{d_{2}\alpha}\,\big)$ . In fact, $K_{2}$ and ${\tilde{K}}_{2}$ are the only extensions $F/k_{2}$ of $k_{2}$ with the p roperties </p>
<p data-bbox="124 311 363 322">1. $F/k_{2}$ is a quadratic extension unramified outside $\mathfrak{p}$ ; </p>
<p data-bbox="124 334 487 358">Therefore it suffices to observe that if $k_{2}(\sqrt{\alpha}\,)$ has these properties, then so does $k_{2}(\sqrt{d_{2}\alpha}\,)$ . But this is elementary. </p>
<p data-bbox="124 358 487 434">In particular, the compositum $M\,=\,K_{2}\tilde{K}_{2}\,=\,k_{2}(\sqrt{d_{2}},\sqrt{\alpha}\,)$ is an extension of type $(2,2)$ over $k_{2}$ with subextensions $K_{2}$ , $\widetilde{K}_{2}$ and $F=k_{2}(\sqrt{d_{2}}\,)$ . Clearly $F$ is the unramified quadratic extension of $k_{2}$ , so bo t h $M/K_{2}$ and $M/\widetilde{K}_{2}$ are unramified. If $K_{2}$ had 2-class number 2, then $M$ would have odd class numb e r, and $M$ would also be the 2-class field of ${\tilde{K}}_{2}$ . Thus $2\parallel h(K_{2})$ implies that $2\parallel h(\widetilde{K}_{2})$ . This proves part a) of the proposition. </p>
<p data-bbox="132 434 485 447">Before we go on, we give a Hasse diagram for the fields occurring in this proof: </p>
<div class="image" data-bbox="253 460 376 545"><img data-bbox="253 460 376 545"/></div>
<p data-bbox="124 554 487 663">Now assume that $4\,\mid\,h(K_{2})$ . Since $\mathrm{Cl_{2}}(M)$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $N/M$ , and the uniqueness implies at once that $N/k_{2}$ is normal. Hence $G\,=\,\operatorname{Gal}(N/k_{2})$ is a group of order 8 containing a subgroup of type $(2,2)\,\simeq\,\mathrm{Gal}(N/F)$ : in fact, if $\operatorname{Gal}(N/F)$ were cyclic, then the primes ramifying in $M/F$ would also ramify in $N/M$ contradicting the fact that $N/M$ is unramified. There are three groups satisfying these conditions: $G=(2,4)$ , $G=(2,2,2)$ and $G=D_{4}$ . We claim that $G$ is non-abelian; once we have proved this, it follows that exactly one of the groups $\mathrm{Gal}(N/K_{2})$ and $\mathrm{Gal}(N/\widetilde{K}_{2})$ is cyclic, and that the other is not, which is what we want to prove. </p>
<p data-bbox="124 663 486 700">So assume that $G$ is abelian. Then $M/F$ is ramified at two finite primes $\mathfrak{q}$ and ${\mathfrak{q}}^{\prime}$ of $F$ dividing $\mathfrak{p}$ (in $k_{2}$ ); if $F_{1}$ and $F_{2}$ denote the quadratic subextensions of $N/F$ different from $M$ then $F_{1}/F$ and $F_{2}/F$ must be ramified at a finite prime (since </p>
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0003244v1
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[{"type": "text", "text": "", "page_idx": 9}, {"type": "text", "text": "Next we derive some relations between the class groups of $K_{2}$ and ${\\tilde{K}}_{2}$ ; these relations will allow us to use each of them as our field $K$ in Theorem 1. ", "page_idx": 9}, {"type": "text", "text": "Proposition 8. Let $L$ and $\\widetilde{L}$ be the two unramified cyclic quartic extensions of $k$ , and let $K_{2}$ and ${\\widetilde{K}}_{2}$ be two qu a dratic extensions of $k_{2}$ in $L$ and $\\widetilde{L}$ , respectively, which are not normal o ver $\\mathbb{Q}$ . ", "page_idx": 9}, {"type": "text", "text": "a) We have 4 $|\\mathit{\\Omega}_{h}(K_{2})$ if and only if $4\\mid h(\\widetilde{K}_{2})$ ; \nb) $I f\\,4\\mid h(K_{2})$ , then one of $\\mathrm{Cl}_{2}(K_{2})$ or $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$ has type $(2,2)$ , whereas the other is cyclic of order $\\geq4$ . ", "page_idx": 9}, {"type": "text", "text": "Proof. Notice that the prime dividing $\\mathrm{disc}(k_{1})$ splits in $k_{2}$ . Throughout this proof, let $\\mathfrak{p}$ be one of the primes of $k_{2}$ dividing $\\mathrm{disc}(k_{1})$ . ", "page_idx": 9}, {"type": "text", "text": "If we write $K_{2}=k_{2}(\\sqrt{\\alpha}\\,)$ for some $\\alpha\\in k_{2}$ , then $\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)$ . In fact, $K_{2}$ and ${\\tilde{K}}_{2}$ are the only extensions $F/k_{2}$ of $k_{2}$ with the p roperties ", "page_idx": 9}, {"type": "text", "text": "1. $F/k_{2}$ is a quadratic extension unramified outside $\\mathfrak{p}$ ; ", "page_idx": 9}, {"type": "text", "text": "Therefore it suffices to observe that if $k_{2}(\\sqrt{\\alpha}\\,)$ has these properties, then so does $k_{2}(\\sqrt{d_{2}\\alpha}\\,)$ . But this is elementary. ", "page_idx": 9}, {"type": "text", "text": "In particular, the compositum $M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)$ is an extension of type $(2,2)$ over $k_{2}$ with subextensions $K_{2}$ , $\\widetilde{K}_{2}$ and $F=k_{2}(\\sqrt{d_{2}}\\,)$ . Clearly $F$ is the unramified quadratic extension of $k_{2}$ , so bo t h $M/K_{2}$ and $M/\\widetilde{K}_{2}$ are unramified. If $K_{2}$ had 2-class number 2, then $M$ would have odd class numb e r, and $M$ would also be the 2-class field of ${\\tilde{K}}_{2}$ . Thus $2\\parallel h(K_{2})$ implies that $2\\parallel h(\\widetilde{K}_{2})$ . This proves part a) of the proposition. ", "page_idx": 9}, {"type": "text", "text": "Before we go on, we give a Hasse diagram for the fields occurring in this proof: ", "page_idx": 9}, {"type": "image", "img_path": "images/d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg", "img_caption": [], "img_footnote": [], "page_idx": 9}, {"type": "text", "text": "Now assume that $4\\,\\mid\\,h(K_{2})$ . Since $\\mathrm{Cl_{2}}(M)$ is cyclic by Lemma 6, there is a unique quadratic unramified extension $N/M$ , and the uniqueness implies at once that $N/k_{2}$ is normal. Hence $G\\,=\\,\\operatorname{Gal}(N/k_{2})$ is a group of order 8 containing a subgroup of type $(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)$ : in fact, if $\\operatorname{Gal}(N/F)$ were cyclic, then the primes ramifying in $M/F$ would also ramify in $N/M$ contradicting the fact that $N/M$ is unramified. There are three groups satisfying these conditions: $G=(2,4)$ , $G=(2,2,2)$ and $G=D_{4}$ . We claim that $G$ is non-abelian; once we have proved this, it follows that exactly one of the groups $\\mathrm{Gal}(N/K_{2})$ and $\\mathrm{Gal}(N/\\widetilde{K}_{2})$ is cyclic, and that the other is not, which is what we want to prove. ", "page_idx": 9}, {"type": "text", "text": "So assume that $G$ is abelian. Then $M/F$ is ramified at two finite primes $\\mathfrak{q}$ and ${\\mathfrak{q}}^{\\prime}$ of $F$ dividing $\\mathfrak{p}$ (in $k_{2}$ ); if $F_{1}$ and $F_{2}$ denote the quadratic subextensions of $N/F$ different from $M$ then $F_{1}/F$ and $F_{2}/F$ must be ramified at a finite prime (since ", "page_idx": 9}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 111, 487, 137], "lines": [{"bbox": [125, 114, 486, 126], "spans": [{"bbox": [125, 114, 297, 126], "score": 1.0, "content": "at the two ramified infinite primes has ", "type": "text"}, {"bbox": [297, 116, 304, 123], "score": 0.9, "content": "\\mathbb{Z}", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [304, 114, 358, 126], "score": 1.0, "content": "-rank 2, i.e. ", "type": "text"}, {"bbox": [358, 115, 410, 126], "score": 0.92, "content": "(E:H)\\geq4", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [410, 114, 486, 126], "score": 1.0, "content": " by consideration", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 138], "spans": [{"bbox": [125, 127, 311, 138], "score": 1.0, "content": "of the infinite primes alone. In particular, ", "type": "text"}, {"bbox": [311, 127, 391, 138], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(M/F)\\leq2", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [391, 127, 441, 138], "score": 1.0, "content": " in case B).", "type": "text"}, {"bbox": [476, 127, 486, 137], "score": 0.9776784181594849, "content": "口", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [125, 145, 486, 169], "lines": [{"bbox": [136, 146, 487, 159], "spans": [{"bbox": [136, 147, 404, 159], "score": 1.0, "content": "Next we derive some relations between the class groups of ", "type": "text"}, {"bbox": [404, 149, 417, 158], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [417, 147, 441, 159], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [442, 146, 455, 158], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [455, 147, 487, 159], "score": 1.0, "content": "; these", "type": "text"}], "index": 2}, {"bbox": [125, 159, 439, 170], "spans": [{"bbox": [125, 159, 363, 170], "score": 1.0, "content": "relations will allow us to use each of them as our field", "type": "text"}, {"bbox": [364, 161, 373, 168], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [374, 159, 439, 170], "score": 1.0, "content": " in Theorem 1.", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [124, 174, 486, 212], "lines": [{"bbox": [126, 177, 485, 190], "spans": [{"bbox": [126, 177, 218, 190], "score": 1.0, "content": "Proposition 8. Let ", "type": "text"}, {"bbox": [218, 180, 226, 187], "score": 0.76, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 177, 248, 190], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 177, 255, 187], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [256, 177, 476, 190], "score": 1.0, "content": " be the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [476, 180, 482, 187], "score": 0.83, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [482, 177, 485, 190], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [126, 189, 486, 203], "spans": [{"bbox": [126, 190, 158, 203], "score": 1.0, "content": "and let ", "type": "text"}, {"bbox": [158, 192, 171, 201], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [172, 190, 192, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [193, 189, 206, 201], "score": 0.92, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [206, 190, 339, 203], "score": 1.0, "content": " be two qu a dratic extensions of", "type": "text"}, {"bbox": [340, 192, 349, 201], "score": 0.81, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [350, 190, 363, 203], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [364, 191, 371, 200], "score": 0.62, "content": "L", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [372, 190, 392, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [393, 189, 400, 200], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [400, 190, 486, 203], "score": 1.0, "content": ", respectively, which", "type": "text"}], "index": 5}, {"bbox": [126, 203, 228, 214], "spans": [{"bbox": [126, 203, 216, 214], "score": 1.0, "content": "are not normal o ver ", "type": "text"}, {"bbox": [217, 204, 225, 213], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [225, 203, 228, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [133, 214, 487, 253], "lines": [{"bbox": [136, 216, 340, 229], "spans": [{"bbox": [136, 217, 199, 229], "score": 1.0, "content": "a) We have 4", "type": "text"}, {"bbox": [199, 219, 232, 229], "score": 0.85, "content": "|\\mathit{\\Omega}_{h}(K_{2})", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [232, 217, 294, 229], "score": 1.0, "content": " if and only if", "type": "text"}, {"bbox": [295, 216, 336, 229], "score": 0.89, "content": "4\\mid h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [336, 217, 340, 229], "score": 1.0, "content": ";", "type": "text"}], "index": 7}, {"bbox": [135, 230, 486, 243], "spans": [{"bbox": [135, 231, 153, 243], "score": 1.0, "content": "b) ", "type": "text"}, {"bbox": [154, 232, 200, 243], "score": 0.67, "content": "I f\\,4\\mid h(K_{2})", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 231, 256, 243], "score": 1.0, "content": ", then one of", "type": "text"}, {"bbox": [257, 232, 292, 243], "score": 0.9, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [292, 231, 306, 243], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [307, 230, 342, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [343, 231, 382, 243], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [382, 232, 405, 243], "score": 0.84, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [405, 231, 486, 243], "score": 1.0, "content": ", whereas the other", "type": "text"}], "index": 8}, {"bbox": [150, 243, 246, 254], "spans": [{"bbox": [150, 243, 226, 254], "score": 1.0, "content": "is cyclic of order", "type": "text"}, {"bbox": [227, 245, 243, 253], "score": 0.83, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [243, 243, 246, 254], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [125, 259, 486, 284], "lines": [{"bbox": [126, 262, 486, 274], "spans": [{"bbox": [126, 262, 293, 274], "score": 1.0, "content": "Proof. Notice that the prime dividing ", "type": "text"}, {"bbox": [293, 262, 327, 273], "score": 0.82, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [327, 262, 367, 274], "score": 1.0, "content": " splits in ", "type": "text"}, {"bbox": [368, 263, 378, 272], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [378, 262, 486, 274], "score": 1.0, "content": ". Throughout this proof,", "type": "text"}], "index": 10}, {"bbox": [125, 273, 342, 286], "spans": [{"bbox": [125, 273, 140, 286], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 277, 146, 285], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [146, 273, 252, 286], "score": 1.0, "content": " be one of the primes of ", "type": "text"}, {"bbox": [252, 275, 262, 284], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [262, 273, 304, 286], "score": 1.0, "content": " dividing ", "type": "text"}, {"bbox": [304, 275, 338, 285], "score": 0.85, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [339, 273, 342, 286], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "text", "bbox": [125, 285, 486, 311], "lines": [{"bbox": [136, 285, 484, 299], "spans": [{"bbox": [136, 286, 189, 299], "score": 1.0, "content": "If we write ", "type": "text"}, {"bbox": [189, 287, 251, 298], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [252, 286, 296, 299], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [296, 288, 326, 297], "score": 0.92, "content": "\\alpha\\in k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [326, 286, 355, 299], "score": 1.0, "content": ", then", "type": "text"}, {"bbox": [356, 285, 427, 298], "score": 0.93, "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [427, 286, 471, 299], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [471, 289, 484, 297], "score": 0.91, "content": "K_{2}", "type": "inline_equation", "height": 8, "width": 13}], "index": 12}, {"bbox": [126, 299, 399, 312], "spans": [{"bbox": [126, 300, 145, 312], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 299, 158, 311], "score": 0.93, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [159, 300, 265, 312], "score": 1.0, "content": " are the only extensions ", "type": "text"}, {"bbox": [265, 301, 287, 312], "score": 0.93, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [287, 300, 301, 312], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [302, 302, 311, 311], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [312, 300, 399, 312], "score": 1.0, "content": " with the p roperties", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [124, 311, 363, 322], "lines": [{"bbox": [126, 311, 363, 325], "spans": [{"bbox": [126, 311, 137, 325], "score": 1.0, "content": "1. ", "type": "text"}, {"bbox": [138, 313, 160, 324], "score": 0.91, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [160, 311, 354, 325], "score": 1.0, "content": " is a quadratic extension unramified outside ", "type": "text"}, {"bbox": [354, 316, 360, 323], "score": 0.84, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [360, 311, 363, 325], "score": 1.0, "content": ";", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [124, 334, 487, 358], "lines": [{"bbox": [125, 335, 486, 348], "spans": [{"bbox": [125, 335, 297, 348], "score": 1.0, "content": "Therefore it suffices to observe that if ", "type": "text"}, {"bbox": [297, 336, 331, 347], "score": 0.94, "content": "k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [331, 335, 486, 348], "score": 1.0, "content": " has these properties, then so does", "type": "text"}], "index": 15}, {"bbox": [126, 347, 277, 360], "spans": [{"bbox": [126, 348, 169, 360], "score": 0.94, "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [170, 347, 277, 360], "score": 1.0, "content": ". But this is elementary.", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "text", "bbox": [124, 358, 487, 434], "lines": [{"bbox": [137, 360, 488, 374], "spans": [{"bbox": [137, 360, 276, 374], "score": 1.0, "content": "In particular, the compositum ", "type": "text"}, {"bbox": [276, 360, 402, 372], "score": 0.94, "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 126}, {"bbox": [402, 360, 488, 374], "score": 1.0, "content": " is an extension of", "type": "text"}], "index": 17}, {"bbox": [125, 373, 486, 387], "spans": [{"bbox": [125, 374, 148, 387], "score": 1.0, "content": "type ", "type": "text"}, {"bbox": [149, 375, 171, 386], "score": 0.93, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [171, 374, 195, 387], "score": 1.0, "content": " over ", "type": "text"}, {"bbox": [195, 376, 205, 385], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 374, 294, 387], "score": 1.0, "content": " with subextensions ", "type": "text"}, {"bbox": [294, 376, 307, 385], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [308, 374, 313, 387], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [313, 373, 326, 385], "score": 0.92, "content": "\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 374, 348, 387], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [349, 374, 408, 385], "score": 0.92, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [408, 374, 449, 387], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [450, 376, 457, 383], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 374, 486, 387], "score": 1.0, "content": " is the", "type": "text"}], "index": 18}, {"bbox": [126, 387, 487, 400], "spans": [{"bbox": [126, 388, 275, 400], "score": 1.0, "content": "unramified quadratic extension of ", "type": "text"}, {"bbox": [275, 389, 285, 398], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [285, 388, 326, 400], "score": 1.0, "content": ", so bo t h ", "type": "text"}, {"bbox": [326, 389, 354, 399], "score": 0.93, "content": "M/K_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [354, 388, 376, 400], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [376, 387, 405, 399], "score": 0.94, "content": "M/\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [405, 388, 487, 400], "score": 1.0, "content": " are unramified. If", "type": "text"}], "index": 19}, {"bbox": [126, 399, 486, 411], "spans": [{"bbox": [126, 401, 139, 410], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 399, 261, 411], "score": 1.0, "content": " had 2-class number 2, then ", "type": "text"}, {"bbox": [261, 401, 272, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [272, 399, 425, 411], "score": 1.0, "content": " would have odd class numb e r, and ", "type": "text"}, {"bbox": [426, 401, 437, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [437, 399, 486, 411], "score": 1.0, "content": " would also", "type": "text"}], "index": 20}, {"bbox": [124, 411, 486, 425], "spans": [{"bbox": [124, 411, 219, 425], "score": 1.0, "content": "be the 2-class field of", "type": "text"}, {"bbox": [220, 411, 233, 423], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [233, 411, 264, 425], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [265, 413, 307, 424], "score": 0.95, "content": "2\\parallel h(K_{2})", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [307, 411, 364, 425], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [364, 411, 406, 424], "score": 0.94, "content": "2\\parallel h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [407, 411, 486, 425], "score": 1.0, "content": ". This proves part", "type": "text"}], "index": 21}, {"bbox": [125, 424, 220, 436], "spans": [{"bbox": [125, 424, 220, 436], "score": 1.0, "content": "a) of the proposition.", "type": "text"}], "index": 22}], "index": 19.5}, {"type": "text", "bbox": [132, 434, 485, 447], "lines": [{"bbox": [137, 436, 484, 449], "spans": [{"bbox": [137, 436, 484, 449], "score": 1.0, "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "type": "text"}], "index": 23}], "index": 23}, {"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}, {"type": "text", "bbox": [124, 554, 487, 663], "lines": [{"bbox": [137, 556, 487, 569], "spans": [{"bbox": [137, 556, 220, 569], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [220, 558, 263, 568], "score": 0.93, "content": "4\\,\\mid\\,h(K_{2})", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [264, 556, 300, 569], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [301, 558, 334, 568], "score": 0.91, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [334, 556, 487, 569], "score": 1.0, "content": " is cyclic by Lemma 6, there is a", "type": "text"}], "index": 26}, {"bbox": [126, 569, 486, 580], "spans": [{"bbox": [126, 569, 299, 580], "score": 1.0, "content": "unique quadratic unramified extension ", "type": "text"}, {"bbox": [299, 569, 323, 580], "score": 0.91, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [323, 569, 486, 580], "score": 1.0, "content": ", and the uniqueness implies at once", "type": "text"}], "index": 27}, {"bbox": [125, 579, 487, 593], "spans": [{"bbox": [125, 579, 148, 593], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 581, 171, 592], "score": 0.93, "content": "N/k_{2}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [172, 579, 257, 593], "score": 1.0, "content": " is normal. Hence ", "type": "text"}, {"bbox": [258, 581, 329, 592], "score": 0.93, "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [329, 579, 487, 593], "score": 1.0, "content": " is a group of order 8 containing a", "type": "text"}], "index": 28}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 205, 604], "score": 1.0, "content": "subgroup of type ", "type": "text"}, {"bbox": [206, 593, 288, 604], "score": 0.93, "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [288, 592, 343, 604], "score": 1.0, "content": ": in fact, if ", "type": "text"}, {"bbox": [343, 593, 388, 604], "score": 0.92, "content": "\\operatorname{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [388, 592, 486, 604], "score": 1.0, "content": " were cyclic, then the", "type": "text"}], "index": 29}, {"bbox": [126, 605, 486, 616], "spans": [{"bbox": [126, 605, 217, 616], "score": 1.0, "content": "primes ramifying in ", "type": "text"}, {"bbox": [217, 605, 240, 616], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [240, 605, 339, 616], "score": 1.0, "content": " would also ramify in ", "type": "text"}, {"bbox": [339, 605, 363, 616], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [363, 605, 486, 616], "score": 1.0, "content": " contradicting the fact that", "type": "text"}], "index": 30}, {"bbox": [126, 616, 485, 628], "spans": [{"bbox": [126, 617, 150, 628], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 616, 438, 628], "score": 1.0, "content": " is unramified. There are three groups satisfying these conditions: ", "type": "text"}, {"bbox": [439, 617, 482, 628], "score": 0.94, "content": "G=(2,4)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [483, 616, 485, 628], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [126, 628, 486, 640], "spans": [{"bbox": [126, 629, 180, 640], "score": 0.94, "content": "G=(2,2,2)", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [181, 628, 203, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [204, 630, 239, 639], "score": 0.93, "content": "G=D_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [239, 628, 314, 640], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [314, 630, 322, 637], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [322, 628, 486, 640], "score": 1.0, "content": " is non-abelian; once we have proved", "type": "text"}], "index": 32}, {"bbox": [125, 640, 485, 654], "spans": [{"bbox": [125, 641, 324, 654], "score": 1.0, "content": "this, it follows that exactly one of the groups ", "type": "text"}, {"bbox": [324, 642, 373, 653], "score": 0.91, "content": "\\mathrm{Gal}(N/K_{2})", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [374, 641, 395, 654], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [396, 640, 445, 653], "score": 0.92, "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [445, 641, 485, 654], "score": 1.0, "content": " is cyclic,", "type": "text"}], "index": 33}, {"bbox": [125, 653, 383, 665], "spans": [{"bbox": [125, 653, 383, 665], "score": 1.0, "content": "and that the other is not, which is what we want to prove.", "type": "text"}], "index": 34}], "index": 30}, {"type": "text", "bbox": [124, 663, 486, 700], "lines": [{"bbox": [137, 665, 487, 678], "spans": [{"bbox": [137, 665, 208, 678], "score": 1.0, "content": "So assume that ", "type": "text"}, {"bbox": [208, 667, 216, 674], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [217, 665, 295, 678], "score": 1.0, "content": " is abelian. Then ", "type": "text"}, {"bbox": [295, 666, 318, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [318, 665, 460, 678], "score": 1.0, "content": " is ramified at two finite primes ", "type": "text"}, {"bbox": [460, 669, 465, 676], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [466, 665, 487, 678], "score": 1.0, "content": " and", "type": "text"}], "index": 35}, {"bbox": [126, 677, 485, 690], "spans": [{"bbox": [126, 678, 134, 688], "score": 0.9, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [134, 677, 147, 690], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [148, 679, 156, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [156, 677, 196, 690], "score": 1.0, "content": " dividing", "type": "text"}, {"bbox": [197, 681, 202, 688], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [203, 677, 220, 690], "score": 1.0, "content": " (in ", "type": "text"}, {"bbox": [221, 679, 230, 687], "score": 0.86, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [231, 677, 248, 690], "score": 1.0, "content": "); if", "type": "text"}, {"bbox": [249, 679, 259, 687], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [260, 677, 282, 690], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [282, 679, 293, 687], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [293, 677, 463, 690], "score": 1.0, "content": " denote the quadratic subextensions of ", "type": "text"}, {"bbox": [464, 678, 485, 689], "score": 0.94, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}], "index": 36}, {"bbox": [125, 689, 486, 702], "spans": [{"bbox": [125, 689, 190, 702], "score": 1.0, "content": "different from ", "type": "text"}, {"bbox": [190, 691, 201, 698], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [201, 689, 228, 702], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [228, 690, 252, 701], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [253, 689, 275, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [276, 690, 300, 701], "score": 0.94, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [300, 689, 486, 702], "score": 1.0, "content": " must be ramified at a finite prime (since", "type": "text"}], "index": 37}], "index": 36}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [126, 91, 135, 99], "lines": [{"bbox": [125, 92, 136, 102], "spans": [{"bbox": [125, 92, 136, 102], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 111, 487, 137], "lines": [], "index": 0.5, "bbox_fs": [125, 114, 486, 138], "lines_deleted": true}, {"type": "text", "bbox": [125, 145, 486, 169], "lines": [{"bbox": [136, 146, 487, 159], "spans": [{"bbox": [136, 147, 404, 159], "score": 1.0, "content": "Next we derive some relations between the class groups of ", "type": "text"}, {"bbox": [404, 149, 417, 158], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [417, 147, 441, 159], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [442, 146, 455, 158], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [455, 147, 487, 159], "score": 1.0, "content": "; these", "type": "text"}], "index": 2}, {"bbox": [125, 159, 439, 170], "spans": [{"bbox": [125, 159, 363, 170], "score": 1.0, "content": "relations will allow us to use each of them as our field", "type": "text"}, {"bbox": [364, 161, 373, 168], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [374, 159, 439, 170], "score": 1.0, "content": " in Theorem 1.", "type": "text"}], "index": 3}], "index": 2.5, "bbox_fs": [125, 146, 487, 170]}, {"type": "text", "bbox": [124, 174, 486, 212], "lines": [{"bbox": [126, 177, 485, 190], "spans": [{"bbox": [126, 177, 218, 190], "score": 1.0, "content": "Proposition 8. Let ", "type": "text"}, {"bbox": [218, 180, 226, 187], "score": 0.76, "content": "L", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [226, 177, 248, 190], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [248, 177, 255, 187], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [256, 177, 476, 190], "score": 1.0, "content": " be the two unramified cyclic quartic extensions of ", "type": "text"}, {"bbox": [476, 180, 482, 187], "score": 0.83, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [482, 177, 485, 190], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [126, 189, 486, 203], "spans": [{"bbox": [126, 190, 158, 203], "score": 1.0, "content": "and let ", "type": "text"}, {"bbox": [158, 192, 171, 201], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [172, 190, 192, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [193, 189, 206, 201], "score": 0.92, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [206, 190, 339, 203], "score": 1.0, "content": " be two qu a dratic extensions of", "type": "text"}, {"bbox": [340, 192, 349, 201], "score": 0.81, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [350, 190, 363, 203], "score": 1.0, "content": " in", "type": "text"}, {"bbox": [364, 191, 371, 200], "score": 0.62, "content": "L", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [372, 190, 392, 203], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [393, 189, 400, 200], "score": 0.86, "content": "\\widetilde{L}", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [400, 190, 486, 203], "score": 1.0, "content": ", respectively, which", "type": "text"}], "index": 5}, {"bbox": [126, 203, 228, 214], "spans": [{"bbox": [126, 203, 216, 214], "score": 1.0, "content": "are not normal o ver ", "type": "text"}, {"bbox": [217, 204, 225, 213], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [225, 203, 228, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5, "bbox_fs": [126, 177, 486, 214]}, {"type": "list", "bbox": [133, 214, 487, 253], "lines": [{"bbox": [136, 216, 340, 229], "spans": [{"bbox": [136, 217, 199, 229], "score": 1.0, "content": "a) We have 4", "type": "text"}, {"bbox": [199, 219, 232, 229], "score": 0.85, "content": "|\\mathit{\\Omega}_{h}(K_{2})", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [232, 217, 294, 229], "score": 1.0, "content": " if and only if", "type": "text"}, {"bbox": [295, 216, 336, 229], "score": 0.89, "content": "4\\mid h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [336, 217, 340, 229], "score": 1.0, "content": ";", "type": "text"}], "index": 7, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [135, 230, 486, 243], "spans": [{"bbox": [135, 231, 153, 243], "score": 1.0, "content": "b) ", "type": "text"}, {"bbox": [154, 232, 200, 243], "score": 0.67, "content": "I f\\,4\\mid h(K_{2})", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 231, 256, 243], "score": 1.0, "content": ", then one of", "type": "text"}, {"bbox": [257, 232, 292, 243], "score": 0.9, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [292, 231, 306, 243], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [307, 230, 342, 243], "score": 0.92, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [343, 231, 382, 243], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [382, 232, 405, 243], "score": 0.84, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [405, 231, 486, 243], "score": 1.0, "content": ", whereas the other", "type": "text"}], "index": 8, "is_list_start_line": true}, {"bbox": [150, 243, 246, 254], "spans": [{"bbox": [150, 243, 226, 254], "score": 1.0, "content": "is cyclic of order", "type": "text"}, {"bbox": [227, 245, 243, 253], "score": 0.83, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [243, 243, 246, 254], "score": 1.0, "content": ".", "type": "text"}], "index": 9, "is_list_end_line": true}], "index": 8, "bbox_fs": [135, 216, 486, 254]}, {"type": "text", "bbox": [125, 259, 486, 284], "lines": [{"bbox": [126, 262, 486, 274], "spans": [{"bbox": [126, 262, 293, 274], "score": 1.0, "content": "Proof. Notice that the prime dividing ", "type": "text"}, {"bbox": [293, 262, 327, 273], "score": 0.82, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [327, 262, 367, 274], "score": 1.0, "content": " splits in ", "type": "text"}, {"bbox": [368, 263, 378, 272], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [378, 262, 486, 274], "score": 1.0, "content": ". Throughout this proof,", "type": "text"}], "index": 10}, {"bbox": [125, 273, 342, 286], "spans": [{"bbox": [125, 273, 140, 286], "score": 1.0, "content": "let ", "type": "text"}, {"bbox": [140, 277, 146, 285], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [146, 273, 252, 286], "score": 1.0, "content": " be one of the primes of ", "type": "text"}, {"bbox": [252, 275, 262, 284], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [262, 273, 304, 286], "score": 1.0, "content": " dividing ", "type": "text"}, {"bbox": [304, 275, 338, 285], "score": 0.85, "content": "\\mathrm{disc}(k_{1})", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [339, 273, 342, 286], "score": 1.0, "content": ".", "type": "text"}], "index": 11}], "index": 10.5, "bbox_fs": [125, 262, 486, 286]}, {"type": "text", "bbox": [125, 285, 486, 311], "lines": [{"bbox": [136, 285, 484, 299], "spans": [{"bbox": [136, 286, 189, 299], "score": 1.0, "content": "If we write ", "type": "text"}, {"bbox": [189, 287, 251, 298], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [252, 286, 296, 299], "score": 1.0, "content": " for some ", "type": "text"}, {"bbox": [296, 288, 326, 297], "score": 0.92, "content": "\\alpha\\in k_{2}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [326, 286, 355, 299], "score": 1.0, "content": ", then", "type": "text"}, {"bbox": [356, 285, 427, 298], "score": 0.93, "content": "\\widetilde{K}_{2}=k_{2}\\big(\\sqrt{d_{2}\\alpha}\\,\\big)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [427, 286, 471, 299], "score": 1.0, "content": ". In fact, ", "type": "text"}, {"bbox": [471, 289, 484, 297], "score": 0.91, "content": "K_{2}", "type": "inline_equation", "height": 8, "width": 13}], "index": 12}, {"bbox": [126, 299, 399, 312], "spans": [{"bbox": [126, 300, 145, 312], "score": 1.0, "content": "and", "type": "text"}, {"bbox": [145, 299, 158, 311], "score": 0.93, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [159, 300, 265, 312], "score": 1.0, "content": " are the only extensions ", "type": "text"}, {"bbox": [265, 301, 287, 312], "score": 0.93, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [287, 300, 301, 312], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [302, 302, 311, 311], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [312, 300, 399, 312], "score": 1.0, "content": " with the p roperties", "type": "text"}], "index": 13}], "index": 12.5, "bbox_fs": [126, 285, 484, 312]}, {"type": "text", "bbox": [124, 311, 363, 322], "lines": [{"bbox": [126, 311, 363, 325], "spans": [{"bbox": [126, 311, 137, 325], "score": 1.0, "content": "1. ", "type": "text"}, {"bbox": [138, 313, 160, 324], "score": 0.91, "content": "F/k_{2}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [160, 311, 354, 325], "score": 1.0, "content": " is a quadratic extension unramified outside ", "type": "text"}, {"bbox": [354, 316, 360, 323], "score": 0.84, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [360, 311, 363, 325], "score": 1.0, "content": ";", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [126, 311, 363, 325]}, {"type": "text", "bbox": [124, 334, 487, 358], "lines": [{"bbox": [125, 335, 486, 348], "spans": [{"bbox": [125, 335, 297, 348], "score": 1.0, "content": "Therefore it suffices to observe that if ", "type": "text"}, {"bbox": [297, 336, 331, 347], "score": 0.94, "content": "k_{2}(\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [331, 335, 486, 348], "score": 1.0, "content": " has these properties, then so does", "type": "text"}], "index": 15}, {"bbox": [126, 347, 277, 360], "spans": [{"bbox": [126, 348, 169, 360], "score": 0.94, "content": "k_{2}(\\sqrt{d_{2}\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [170, 347, 277, 360], "score": 1.0, "content": ". But this is elementary.", "type": "text"}], "index": 16}], "index": 15.5, "bbox_fs": [125, 335, 486, 360]}, {"type": "text", "bbox": [124, 358, 487, 434], "lines": [{"bbox": [137, 360, 488, 374], "spans": [{"bbox": [137, 360, 276, 374], "score": 1.0, "content": "In particular, the compositum ", "type": "text"}, {"bbox": [276, 360, 402, 372], "score": 0.94, "content": "M\\,=\\,K_{2}\\tilde{K}_{2}\\,=\\,k_{2}(\\sqrt{d_{2}},\\sqrt{\\alpha}\\,)", "type": "inline_equation", "height": 12, "width": 126}, {"bbox": [402, 360, 488, 374], "score": 1.0, "content": " is an extension of", "type": "text"}], "index": 17}, {"bbox": [125, 373, 486, 387], "spans": [{"bbox": [125, 374, 148, 387], "score": 1.0, "content": "type ", "type": "text"}, {"bbox": [149, 375, 171, 386], "score": 0.93, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [171, 374, 195, 387], "score": 1.0, "content": " over ", "type": "text"}, {"bbox": [195, 376, 205, 385], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 374, 294, 387], "score": 1.0, "content": " with subextensions ", "type": "text"}, {"bbox": [294, 376, 307, 385], "score": 0.89, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [308, 374, 313, 387], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [313, 373, 326, 385], "score": 0.92, "content": "\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 374, 348, 387], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [349, 374, 408, 385], "score": 0.92, "content": "F=k_{2}(\\sqrt{d_{2}}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [408, 374, 449, 387], "score": 1.0, "content": ". Clearly ", "type": "text"}, {"bbox": [450, 376, 457, 383], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 374, 486, 387], "score": 1.0, "content": " is the", "type": "text"}], "index": 18}, {"bbox": [126, 387, 487, 400], "spans": [{"bbox": [126, 388, 275, 400], "score": 1.0, "content": "unramified quadratic extension of ", "type": "text"}, {"bbox": [275, 389, 285, 398], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [285, 388, 326, 400], "score": 1.0, "content": ", so bo t h ", "type": "text"}, {"bbox": [326, 389, 354, 399], "score": 0.93, "content": "M/K_{2}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [354, 388, 376, 400], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [376, 387, 405, 399], "score": 0.94, "content": "M/\\widetilde{K}_{2}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [405, 388, 487, 400], "score": 1.0, "content": " are unramified. If", "type": "text"}], "index": 19}, {"bbox": [126, 399, 486, 411], "spans": [{"bbox": [126, 401, 139, 410], "score": 0.92, "content": "K_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [139, 399, 261, 411], "score": 1.0, "content": " had 2-class number 2, then ", "type": "text"}, {"bbox": [261, 401, 272, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [272, 399, 425, 411], "score": 1.0, "content": " would have odd class numb e r, and ", "type": "text"}, {"bbox": [426, 401, 437, 408], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [437, 399, 486, 411], "score": 1.0, "content": " would also", "type": "text"}], "index": 20}, {"bbox": [124, 411, 486, 425], "spans": [{"bbox": [124, 411, 219, 425], "score": 1.0, "content": "be the 2-class field of", "type": "text"}, {"bbox": [220, 411, 233, 423], "score": 0.92, "content": "{\\tilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [233, 411, 264, 425], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [265, 413, 307, 424], "score": 0.95, "content": "2\\parallel h(K_{2})", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [307, 411, 364, 425], "score": 1.0, "content": " implies that ", "type": "text"}, {"bbox": [364, 411, 406, 424], "score": 0.94, "content": "2\\parallel h(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [407, 411, 486, 425], "score": 1.0, "content": ". This proves part", "type": "text"}], "index": 21}, {"bbox": [125, 424, 220, 436], "spans": [{"bbox": [125, 424, 220, 436], "score": 1.0, "content": "a) of the proposition.", "type": "text"}], "index": 22}], "index": 19.5, "bbox_fs": [124, 360, 488, 436]}, {"type": "text", "bbox": [132, 434, 485, 447], "lines": [{"bbox": [137, 436, 484, 449], "spans": [{"bbox": [137, 436, 484, 449], "score": 1.0, "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "type": "text"}], "index": 23}], "index": 23, "bbox_fs": [137, 436, 484, 449]}, {"type": "image", "bbox": [253, 460, 376, 545], "blocks": [{"type": "image_body", "bbox": [253, 460, 376, 545], "group_id": 0, "lines": [{"bbox": [253, 460, 376, 545], "spans": [{"bbox": [253, 460, 376, 545], "score": 0.944, "type": "image", "image_path": "d372d949890fcb556f1018db84ad61f8f2a0dc1ea6a962a89ead2aeec6283a9b.jpg"}]}], "index": 24.5, "virtual_lines": [{"bbox": [253, 460, 376, 502.5], "spans": [], "index": 24}, {"bbox": [253, 502.5, 376, 545.0], "spans": [], "index": 25}]}], "index": 24.5}, {"type": "text", "bbox": [124, 554, 487, 663], "lines": [{"bbox": [137, 556, 487, 569], "spans": [{"bbox": [137, 556, 220, 569], "score": 1.0, "content": "Now assume that ", "type": "text"}, {"bbox": [220, 558, 263, 568], "score": 0.93, "content": "4\\,\\mid\\,h(K_{2})", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [264, 556, 300, 569], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [301, 558, 334, 568], "score": 0.91, "content": "\\mathrm{Cl_{2}}(M)", "type": "inline_equation", "height": 10, "width": 33}, {"bbox": [334, 556, 487, 569], "score": 1.0, "content": " is cyclic by Lemma 6, there is a", "type": "text"}], "index": 26}, {"bbox": [126, 569, 486, 580], "spans": [{"bbox": [126, 569, 299, 580], "score": 1.0, "content": "unique quadratic unramified extension ", "type": "text"}, {"bbox": [299, 569, 323, 580], "score": 0.91, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [323, 569, 486, 580], "score": 1.0, "content": ", and the uniqueness implies at once", "type": "text"}], "index": 27}, {"bbox": [125, 579, 487, 593], "spans": [{"bbox": [125, 579, 148, 593], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 581, 171, 592], "score": 0.93, "content": "N/k_{2}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [172, 579, 257, 593], "score": 1.0, "content": " is normal. Hence ", "type": "text"}, {"bbox": [258, 581, 329, 592], "score": 0.93, "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [329, 579, 487, 593], "score": 1.0, "content": " is a group of order 8 containing a", "type": "text"}], "index": 28}, {"bbox": [126, 592, 486, 604], "spans": [{"bbox": [126, 592, 205, 604], "score": 1.0, "content": "subgroup of type ", "type": "text"}, {"bbox": [206, 593, 288, 604], "score": 0.93, "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 82}, {"bbox": [288, 592, 343, 604], "score": 1.0, "content": ": in fact, if ", "type": "text"}, {"bbox": [343, 593, 388, 604], "score": 0.92, "content": "\\operatorname{Gal}(N/F)", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [388, 592, 486, 604], "score": 1.0, "content": " were cyclic, then the", "type": "text"}], "index": 29}, {"bbox": [126, 605, 486, 616], "spans": [{"bbox": [126, 605, 217, 616], "score": 1.0, "content": "primes ramifying in ", "type": "text"}, {"bbox": [217, 605, 240, 616], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [240, 605, 339, 616], "score": 1.0, "content": " would also ramify in ", "type": "text"}, {"bbox": [339, 605, 363, 616], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [363, 605, 486, 616], "score": 1.0, "content": " contradicting the fact that", "type": "text"}], "index": 30}, {"bbox": [126, 616, 485, 628], "spans": [{"bbox": [126, 617, 150, 628], "score": 0.94, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [150, 616, 438, 628], "score": 1.0, "content": " is unramified. There are three groups satisfying these conditions: ", "type": "text"}, {"bbox": [439, 617, 482, 628], "score": 0.94, "content": "G=(2,4)", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [483, 616, 485, 628], "score": 1.0, "content": ",", "type": "text"}], "index": 31}, {"bbox": [126, 628, 486, 640], "spans": [{"bbox": [126, 629, 180, 640], "score": 0.94, "content": "G=(2,2,2)", "type": "inline_equation", "height": 11, "width": 54}, {"bbox": [181, 628, 203, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [204, 630, 239, 639], "score": 0.93, "content": "G=D_{4}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [239, 628, 314, 640], "score": 1.0, "content": ". We claim that ", "type": "text"}, {"bbox": [314, 630, 322, 637], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [322, 628, 486, 640], "score": 1.0, "content": " is non-abelian; once we have proved", "type": "text"}], "index": 32}, {"bbox": [125, 640, 485, 654], "spans": [{"bbox": [125, 641, 324, 654], "score": 1.0, "content": "this, it follows that exactly one of the groups ", "type": "text"}, {"bbox": [324, 642, 373, 653], "score": 0.91, "content": "\\mathrm{Gal}(N/K_{2})", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [374, 641, 395, 654], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [396, 640, 445, 653], "score": 0.92, "content": "\\mathrm{Gal}(N/\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [445, 641, 485, 654], "score": 1.0, "content": " is cyclic,", "type": "text"}], "index": 33}, {"bbox": [125, 653, 383, 665], "spans": [{"bbox": [125, 653, 383, 665], "score": 1.0, "content": "and that the other is not, which is what we want to prove.", "type": "text"}], "index": 34}], "index": 30, "bbox_fs": [125, 556, 487, 665]}, {"type": "text", "bbox": [124, 663, 486, 700], "lines": [{"bbox": [137, 665, 487, 678], "spans": [{"bbox": [137, 665, 208, 678], "score": 1.0, "content": "So assume that ", "type": "text"}, {"bbox": [208, 667, 216, 674], "score": 0.91, "content": "G", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [217, 665, 295, 678], "score": 1.0, "content": " is abelian. Then ", "type": "text"}, {"bbox": [295, 666, 318, 677], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [318, 665, 460, 678], "score": 1.0, "content": " is ramified at two finite primes ", "type": "text"}, {"bbox": [460, 669, 465, 676], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [466, 665, 487, 678], "score": 1.0, "content": " and", "type": "text"}], "index": 35}, {"bbox": [126, 677, 485, 690], "spans": [{"bbox": [126, 678, 134, 688], "score": 0.9, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [134, 677, 147, 690], "score": 1.0, "content": " of", "type": "text"}, {"bbox": [148, 679, 156, 686], "score": 0.91, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [156, 677, 196, 690], "score": 1.0, "content": " dividing", "type": "text"}, {"bbox": [197, 681, 202, 688], "score": 0.85, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [203, 677, 220, 690], "score": 1.0, "content": " (in ", "type": "text"}, {"bbox": [221, 679, 230, 687], "score": 0.86, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [231, 677, 248, 690], "score": 1.0, "content": "); if", "type": "text"}, {"bbox": [249, 679, 259, 687], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [260, 677, 282, 690], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [282, 679, 293, 687], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [293, 677, 463, 690], "score": 1.0, "content": " denote the quadratic subextensions of ", "type": "text"}, {"bbox": [464, 678, 485, 689], "score": 0.94, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}], "index": 36}, {"bbox": [125, 689, 486, 702], "spans": [{"bbox": [125, 689, 190, 702], "score": 1.0, "content": "different from ", "type": "text"}, {"bbox": [190, 691, 201, 698], "score": 0.9, "content": "M", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [201, 689, 228, 702], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [228, 690, 252, 701], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [253, 689, 275, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [276, 690, 300, 701], "score": 0.94, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [300, 689, 486, 702], "score": 1.0, "content": " must be ramified at a finite prime (since", "type": "text"}], "index": 37}], "index": 36, "bbox_fs": [125, 665, 487, 702]}]}
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[{"type": "text", "bbox": [124, 111, 487, 137], "content": "", "index": 0}, {"type": "text", "bbox": [125, 145, 486, 169], "content": "Next we derive some relations between the class groups of and ; these relations will allow us to use each of them as our field in Theorem 1.", "index": 1}, {"type": "text", "bbox": [124, 174, 486, 212], "content": "Proposition 8. Let and be the two unramified cyclic quartic extensions of , and let and be two qu a dratic extensions of in and , respectively, which are not normal o ver .", "index": 2}, {"type": "list", "bbox": [133, 214, 487, 253], "content": "", "index": 3}, {"type": "text", "bbox": [125, 259, 486, 284], "content": "Proof. Notice that the prime dividing splits in . Throughout this proof, let be one of the primes of dividing .", "index": 4}, {"type": "text", "bbox": [125, 285, 486, 311], "content": "If we write for some , then . In fact, and are the only extensions of with the p roperties", "index": 5}, {"type": "text", "bbox": [124, 311, 363, 322], "content": "1. is a quadratic extension unramified outside ;", "index": 6}, {"type": "text", "bbox": [124, 334, 487, 358], "content": "Therefore it suffices to observe that if has these properties, then so does . But this is elementary.", "index": 7}, {"type": "text", "bbox": [124, 358, 487, 434], "content": "In particular, the compositum is an extension of type over with subextensions , and . Clearly is the unramified quadratic extension of , so bo t h and are unramified. If had 2-class number 2, then would have odd class numb e r, and would also be the 2-class field of . Thus implies that . This proves part a) of the proposition.", "index": 8}, {"type": "text", "bbox": [132, 434, 485, 447], "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "index": 9}, {"type": "image", "bbox": [253, 460, 376, 545], "content": "", "index": 10}, {"type": "text", "bbox": [124, 554, 487, 663], "content": "Now assume that . Since is cyclic by Lemma 6, there is a unique quadratic unramified extension , and the uniqueness implies at once that is normal. Hence is a group of order 8 containing a subgroup of type : in fact, if were cyclic, then the primes ramifying in would also ramify in contradicting the fact that is unramified. There are three groups satisfying these conditions: , and . We claim that is non-abelian; once we have proved this, it follows that exactly one of the groups and is cyclic, and that the other is not, which is what we want to prove.", "index": 11}, {"type": "text", "bbox": [124, 663, 486, 700], "content": "So assume that is abelian. Then is ramified at two finite primes and of dividing (in ); if and denote the quadratic subextensions of different from then and must be ramified at a finite prime (since", "index": 12}]
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[{"bbox": [136, 146, 487, 159], "content": "Next we derive some relations between the class groups of and ; these", "parent_index": 1, "line_index": 0}, {"bbox": [125, 159, 439, 170], "content": "relations will allow us to use each of them as our field in Theorem 1.", "parent_index": 1, "line_index": 1}, {"bbox": [126, 177, 485, 190], "content": "Proposition 8. Let and be the two unramified cyclic quartic extensions of ,", "parent_index": 2, "line_index": 0}, {"bbox": [126, 189, 486, 203], "content": "and let and be two qu a dratic extensions of in and , respectively, which", "parent_index": 2, "line_index": 1}, {"bbox": [126, 203, 228, 214], "content": "are not normal o ver .", "parent_index": 2, "line_index": 2}, {"bbox": [136, 216, 340, 229], "content": "a) We have 4 if and only if ;", "parent_index": 3, "line_index": 0}, {"bbox": [135, 230, 486, 243], "content": "b) , then one of or has type , whereas the other", "parent_index": 3, "line_index": 1}, {"bbox": [150, 243, 246, 254], "content": "is cyclic of order .", "parent_index": 3, "line_index": 2}, {"bbox": [126, 262, 486, 274], "content": "Proof. Notice that the prime dividing splits in . Throughout this proof,", "parent_index": 4, "line_index": 0}, {"bbox": [125, 273, 342, 286], "content": "let be one of the primes of dividing .", "parent_index": 4, "line_index": 1}, {"bbox": [136, 285, 484, 299], "content": "If we write for some , then . In fact,", "parent_index": 5, "line_index": 0}, {"bbox": [126, 299, 399, 312], "content": "and are the only extensions of with the p roperties", "parent_index": 5, "line_index": 1}, {"bbox": [126, 311, 363, 325], "content": "1. is a quadratic extension unramified outside ;", "parent_index": 6, "line_index": 0}, {"bbox": [125, 335, 486, 348], "content": "Therefore it suffices to observe that if has these properties, then so does", "parent_index": 7, "line_index": 0}, {"bbox": [126, 347, 277, 360], "content": ". But this is elementary.", "parent_index": 7, "line_index": 1}, {"bbox": [137, 360, 488, 374], "content": "In particular, the compositum is an extension of", "parent_index": 8, "line_index": 0}, {"bbox": [125, 373, 486, 387], "content": "type over with subextensions , and . Clearly is the", "parent_index": 8, "line_index": 1}, {"bbox": [126, 387, 487, 400], "content": "unramified quadratic extension of , so bo t h and are unramified. If", "parent_index": 8, "line_index": 2}, {"bbox": [126, 399, 486, 411], "content": "had 2-class number 2, then would have odd class numb e r, and would also", "parent_index": 8, "line_index": 3}, {"bbox": [124, 411, 486, 425], "content": "be the 2-class field of . Thus implies that . This proves part", "parent_index": 8, "line_index": 4}, {"bbox": [125, 424, 220, 436], "content": "a) of the proposition.", "parent_index": 8, "line_index": 5}, {"bbox": [137, 436, 484, 449], "content": "Before we go on, we give a Hasse diagram for the fields occurring in this proof:", "parent_index": 9, "line_index": 0}, {"bbox": [137, 556, 487, 569], "content": "Now assume that . Since is cyclic by Lemma 6, there is a", "parent_index": 11, "line_index": 0}, {"bbox": [126, 569, 486, 580], "content": "unique quadratic unramified extension , and the uniqueness implies at once", "parent_index": 11, "line_index": 1}, {"bbox": [125, 579, 487, 593], "content": "that is normal. Hence is a group of order 8 containing a", "parent_index": 11, "line_index": 2}, {"bbox": [126, 592, 486, 604], "content": "subgroup of type : in fact, if were cyclic, then the", "parent_index": 11, "line_index": 3}, {"bbox": [126, 605, 486, 616], "content": "primes ramifying in would also ramify in contradicting the fact that", "parent_index": 11, "line_index": 4}, {"bbox": [126, 616, 485, 628], "content": "is unramified. There are three groups satisfying these conditions: ,", "parent_index": 11, "line_index": 5}, {"bbox": [126, 628, 486, 640], "content": "and . We claim that is non-abelian; once we have proved", "parent_index": 11, "line_index": 6}, {"bbox": [125, 640, 485, 654], "content": "this, it follows that exactly one of the groups and is cyclic,", "parent_index": 11, "line_index": 7}, {"bbox": [125, 653, 383, 665], "content": "and that the other is not, which is what we want to prove.", "parent_index": 11, "line_index": 8}, {"bbox": [137, 665, 487, 678], "content": "So assume that is abelian. Then is ramified at two finite primes and", "parent_index": 12, "line_index": 0}, {"bbox": [126, 677, 485, 690], "content": "of dividing (in ); if and denote the quadratic subextensions of", "parent_index": 12, "line_index": 1}, {"bbox": [125, 689, 486, 702], "content": "different from then and must be ramified at a finite prime (since", "parent_index": 12, "line_index": 2}]
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[{"bbox": [253, 460, 376, 545], "content": "", "parent_index": 10, "subtype": "body"}]
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[265, 413, 307, 424], "content": "2\\parallel h(K_{2})", "parent_index": 8, "subtype": "inline"}, {"bbox": [364, 411, 406, 424], "content": "2\\parallel h(\\widetilde{K}_{2})", "parent_index": 8, "subtype": "inline"}, {"bbox": [220, 558, 263, 568], "content": "4\\,\\mid\\,h(K_{2})", "parent_index": 11, "subtype": "inline"}, {"bbox": [301, 558, 334, 568], "content": "\\mathrm{Cl_{2}}(M)", "parent_index": 11, "subtype": "inline"}, {"bbox": [299, 569, 323, 580], "content": "N/M", "parent_index": 11, "subtype": "inline"}, {"bbox": [148, 581, 171, 592], "content": "N/k_{2}", "parent_index": 11, "subtype": "inline"}, {"bbox": [258, 581, 329, 592], "content": "G\\,=\\,\\operatorname{Gal}(N/k_{2})", "parent_index": 11, "subtype": "inline"}, {"bbox": [206, 593, 288, 604], "content": "(2,2)\\,\\simeq\\,\\mathrm{Gal}(N/F)", "parent_index": 11, "subtype": "inline"}, {"bbox": [343, 593, 388, 604], "content": "\\operatorname{Gal}(N/F)", "parent_index": 11, "subtype": "inline"}, {"bbox": [217, 605, 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{"bbox": [276, 690, 300, 701], "content": "F_{2}/F", "parent_index": 12, "subtype": "inline"}]
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[]
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$F$ has odd class number in the strict sense: see Lemma 6); since both $F_{1}$ and $F_{2}$ are normal (even abelian) over $k_{2}$ , ramification at $\mathfrak{q}$ implies ramification at the conjugated ideal ${\mathfrak{q}}^{\prime}$ . Hence both $\mathfrak{q}$ and ${\mathfrak{q}}^{\prime}$ ramify in $F_{1}/F$ and $F_{2}/F$ , and since they also ramify in $M/F$ , they must ramify completely in $N/F$ , again contradicting the fact that $N/M$ is unramified.
We have proved that $\mathrm{Cl_{2}}(K_{2})$ and $\mathrm{Cl}_{2}(\widetilde{K}_{2})$ contain subgroups of type (4) and $(2,2)$ , respectively. Now we wish to apply P roposition 5. But we have to compute $\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})$ . Since the class number of ${\widetilde{K}}_{2}$ is even, it is sufficient to show that $\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})\,\leq\,2$ . In case A), there is e xactly one ramified prime (it divides $d_{1}$ ), hen c e $\#\operatorname{Am}_{2}({\widetilde{K}}_{2}/k_{2})\,=\,2/(E:H)\,\le\,2$ . In case B), there are two ramified primes (one is infin i te, the other divides $d_{3}$ ), hence $\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})=4/(E:H)$ ; but $^{-1}$ is not a norm residue at the ramified infinite prime, h e nce $(E:H)\ge2$ and $\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})\leq2$ as claimed.
Now P roposition 5 implies that $\mathrm{Cl}_{2}(K_{2})$ is cyclic of order $\geq4$ , and that $\mathrm{Cl}_{2}(\widetilde{K}_{2})\simeq$ $(2,2)$ . This concludes our proof. 口
Proposition 9. Assume that $k$ is one of the imaginary quadratic fields of type $A)$ or $B$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $k$ . Let $L$ be one of them, and write
Then $\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$ unless possibly when $d_{3}=-4$ in case $B$ ).
Proof. Observe that $\upsilon=0$ in case A) and B); Kuroda’s class number formulas for $L/k_{1}$ and $L/k_{2}$ gives
$$
h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}
$$
in case A) and
$$
h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}
$$
in case B). Multiplying them together and plugging in the class number formula for $K/\mathbb{Q}$ yields
$$
h_{2}(L)^{2}=\frac{q_{1}\,q_{2}}{8}\frac{h_{2}(K_{1})^{2}\,h_{2}(K_{2})^{2}\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\,h_{2}(k_{2})^{2}}.
$$
Now $h_{2}(k_{1})=1$ , $h_{2}(k_{2})=2$ and $q_{1}q_{2}=2$ (by Proposition 6), and taking the square root we find $\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$ as claimed. 口
# 5. Classification
In this section we apply the results obtained in the last few sections to give a proof for Theorem 1.
Proof of Theorem 1. Let $L$ be one of the two cyclic quartic unramified extensions of $k$ , and let $N$ be the subgroup of $\operatorname{Gal}(k^{2}/k)$ fixing $L$ . Then $N$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities:
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<html><body>
<p data-bbox="125 111 486 172">$F$ has odd class number in the strict sense: see Lemma 6); since both $F_{1}$ and $F_{2}$ are normal (even abelian) over $k_{2}$ , ramification at $\mathfrak{q}$ implies ramification at the conjugated ideal ${\mathfrak{q}}^{\prime}$ . Hence both $\mathfrak{q}$ and ${\mathfrak{q}}^{\prime}$ ramify in $F_{1}/F$ and $F_{2}/F$ , and since they also ramify in $M/F$ , they must ramify completely in $N/F$ , again contradicting the fact that $N/M$ is unramified. </p>
<p data-bbox="125 173 486 275">We have proved that $\mathrm{Cl_{2}}(K_{2})$ and $\mathrm{Cl}_{2}(\widetilde{K}_{2})$ contain subgroups of type (4) and $(2,2)$ , respectively. Now we wish to apply P roposition 5. But we have to compute $\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})$ . Since the class number of ${\widetilde{K}}_{2}$ is even, it is sufficient to show that $\#\operatorname{Am}_{2}(\widetilde{K}_{2}/k_{2})\,\leq\,2$ . In case A), there is e xactly one ramified prime (it divides $d_{1}$ ), hen c e $\#\operatorname{Am}_{2}({\widetilde{K}}_{2}/k_{2})\,=\,2/(E:H)\,\le\,2$ . In case B), there are two ramified primes (one is infin i te, the other divides $d_{3}$ ), hence $\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})=4/(E:H)$ ; but $^{-1}$ is not a norm residue at the ramified infinite prime, h e nce $(E:H)\ge2$ and $\#\operatorname{Am}_{2}(\tilde{K}_{2}/k_{2})\leq2$ as claimed. </p>
<p data-bbox="125 276 486 301">Now P roposition 5 implies that $\mathrm{Cl}_{2}(K_{2})$ is cyclic of order $\geq4$ , and that $\mathrm{Cl}_{2}(\widetilde{K}_{2})\simeq$ $(2,2)$ . This concludes our proof. 口 </p>
<p data-bbox="125 310 487 346">Proposition 9. Assume that $k$ is one of the imaginary quadratic fields of type $A)$ or $B$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $k$ . Let $L$ be one of them, and write </p>
<p data-bbox="126 380 465 394">Then $\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$ unless possibly when $d_{3}=-4$ in case $B$ ). </p>
<p data-bbox="125 401 486 426">Proof. Observe that $\upsilon=0$ in case A) and B); Kuroda’s class number formulas for $L/k_{1}$ and $L/k_{2}$ gives </p>
<div class="equation" data-bbox="199 433 412 459">$$
h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}
$$</div>
<p data-bbox="124 463 191 475">in case A) and </p>
<div class="equation" data-bbox="199 484 412 509">$$
h_{2}(L)\ =\ \frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\ =\ \frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}
$$</div>
<p data-bbox="125 514 486 538">in case B). Multiplying them together and plugging in the class number formula for $K/\mathbb{Q}$ yields </p>
<div class="equation" data-bbox="219 542 392 568">$$
h_{2}(L)^{2}=\frac{q_{1}\,q_{2}}{8}\frac{h_{2}(K_{1})^{2}\,h_{2}(K_{2})^{2}\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\,h_{2}(k_{2})^{2}}.
$$</div>
<p data-bbox="124 570 487 595">Now $h_{2}(k_{1})=1$ , $h_{2}(k_{2})=2$ and $q_{1}q_{2}=2$ (by Proposition 6), and taking the square root we find $\begin{array}{r}{h_{2}(L)=\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\end{array}$ as claimed. 口 </p>
<h1 data-bbox="261 610 349 623">5. Classification </h1>
<p data-bbox="125 629 486 653">In this section we apply the results obtained in the last few sections to give a proof for Theorem 1. </p>
<p data-bbox="124 662 486 700">Proof of Theorem 1. Let $L$ be one of the two cyclic quartic unramified extensions of $k$ , and let $N$ be the subgroup of $\operatorname{Gal}(k^{2}/k)$ fixing $L$ . Then $N$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities: </p>
</body></html>
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0003244v1
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[{"type": "text", "text": "$F$ has odd class number in the strict sense: see Lemma 6); since both $F_{1}$ and $F_{2}$ are normal (even abelian) over $k_{2}$ , ramification at $\\mathfrak{q}$ implies ramification at the conjugated ideal ${\\mathfrak{q}}^{\\prime}$ . Hence both $\\mathfrak{q}$ and ${\\mathfrak{q}}^{\\prime}$ ramify in $F_{1}/F$ and $F_{2}/F$ , and since they also ramify in $M/F$ , they must ramify completely in $N/F$ , again contradicting the fact that $N/M$ is unramified. ", "page_idx": 10}, {"type": "text", "text": "We have proved that $\\mathrm{Cl_{2}}(K_{2})$ and $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})$ contain subgroups of type (4) and $(2,2)$ , respectively. Now we wish to apply P roposition 5. But we have to compute $\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})$ . Since the class number of ${\\widetilde{K}}_{2}$ is even, it is sufficient to show that $\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2$ . In case A), there is e xactly one ramified prime (it divides $d_{1}$ ), hen c e $\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2$ . In case B), there are two ramified primes (one is infin i te, the other divides $d_{3}$ ), hence $\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)$ ; but $^{-1}$ is not a norm residue at the ramified infinite prime, h e nce $(E:H)\\ge2$ and $\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2$ as claimed. ", "page_idx": 10}, {"type": "text", "text": "Now P roposition 5 implies that $\\mathrm{Cl}_{2}(K_{2})$ is cyclic of order $\\geq4$ , and that $\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq$ $(2,2)$ . This concludes our proof. 口 ", "page_idx": 10}, {"type": "text", "text": "Proposition 9. Assume that $k$ is one of the imaginary quadratic fields of type $A)$ or $B$ ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of $k$ . Let $L$ be one of them, and write ", "page_idx": 10}, {"type": "text", "text": "Then $\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$ unless possibly when $d_{3}=-4$ in case $B$ ). ", "page_idx": 10}, {"type": "text", "text": "Proof. Observe that $\\upsilon=0$ in case A) and B); Kuroda’s class number formulas for $L/k_{1}$ and $L/k_{2}$ gives ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "in case A) and ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "in case B). Multiplying them together and plugging in the class number formula for $K/\\mathbb{Q}$ yields ", "page_idx": 10}, {"type": "equation", "text": "$$\nh_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "Now $h_{2}(k_{1})=1$ , $h_{2}(k_{2})=2$ and $q_{1}q_{2}=2$ (by Proposition 6), and taking the square root we find $\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}$ as claimed. 口 ", "page_idx": 10}, {"type": "text", "text": "5. Classification ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "In this section we apply the results obtained in the last few sections to give a proof for Theorem 1. ", "page_idx": 10}, {"type": "text", "text": "Proof of Theorem 1. Let $L$ be one of the two cyclic quartic unramified extensions of $k$ , and let $N$ be the subgroup of $\\operatorname{Gal}(k^{2}/k)$ fixing $L$ . Then $N$ satisfies the assumptions of Proposition 1, thus there are only the following possibilities: ", "page_idx": 10}]
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{"preproc_blocks": [{"type": "text", "bbox": [125, 111, 486, 172], "lines": [{"bbox": [126, 114, 484, 127], "spans": [{"bbox": [126, 116, 134, 123], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [134, 114, 439, 127], "score": 1.0, "content": " has odd class number in the strict sense: see Lemma 6); since both ", "type": "text"}, {"bbox": [439, 116, 450, 125], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [450, 114, 473, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 116, 484, 125], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 9, "width": 10}], "index": 0}, {"bbox": [126, 127, 486, 139], "spans": [{"bbox": [126, 127, 268, 139], "score": 1.0, "content": "are normal (even abelian) over ", "type": "text"}, {"bbox": [269, 128, 279, 137], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [279, 127, 356, 139], "score": 1.0, "content": ", ramification at ", "type": "text"}, {"bbox": [356, 130, 361, 137], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [362, 127, 486, 139], "score": 1.0, "content": " implies ramification at the", "type": "text"}], "index": 1}, {"bbox": [126, 138, 485, 150], "spans": [{"bbox": [126, 138, 199, 150], "score": 1.0, "content": "conjugated ideal", "type": "text"}, {"bbox": [200, 139, 208, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [208, 138, 267, 150], "score": 1.0, "content": ". Hence both ", "type": "text"}, {"bbox": [267, 142, 272, 149], "score": 0.88, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 138, 294, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 139, 302, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [302, 138, 347, 150], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [347, 139, 371, 149], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [371, 138, 393, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [393, 139, 417, 150], "score": 0.93, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [417, 138, 485, 150], "score": 1.0, "content": ", and since they", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 163], "spans": [{"bbox": [125, 150, 188, 163], "score": 1.0, "content": "also ramify in ", "type": "text"}, {"bbox": [189, 151, 212, 162], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [212, 150, 357, 163], "score": 1.0, "content": ", they must ramify completely in ", "type": "text"}, {"bbox": [357, 151, 378, 162], "score": 0.93, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [379, 150, 486, 163], "score": 1.0, "content": ", again contradicting the", "type": "text"}], "index": 3}, {"bbox": [126, 162, 255, 174], "spans": [{"bbox": [126, 162, 167, 174], "score": 1.0, "content": "fact that ", "type": "text"}, {"bbox": [167, 163, 191, 174], "score": 0.93, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [191, 162, 255, 174], "score": 1.0, "content": " is unramified.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [125, 173, 486, 275], "lines": [{"bbox": [137, 174, 487, 188], "spans": [{"bbox": [137, 174, 234, 188], "score": 1.0, "content": "We have proved that ", "type": "text"}, {"bbox": [235, 176, 270, 187], "score": 0.94, "content": "\\mathrm{Cl_{2}}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [270, 174, 294, 188], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 174, 329, 187], "score": 0.93, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [330, 174, 487, 188], "score": 1.0, "content": " contain subgroups of type (4) and", "type": "text"}], "index": 5}, {"bbox": [126, 187, 486, 200], "spans": [{"bbox": [126, 188, 148, 199], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 187, 486, 200], "score": 1.0, "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "type": "text"}], "index": 6}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 192, 212], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [192, 200, 314, 212], "score": 1.0, "content": ". Since the class number of", "type": "text"}, {"bbox": [314, 199, 327, 211], "score": 0.93, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 200, 486, 212], "score": 1.0, "content": " is even, it is sufficient to show that", "type": "text"}], "index": 7}, {"bbox": [126, 212, 487, 227], "spans": [{"bbox": [126, 212, 214, 225], "score": 0.9, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [214, 212, 487, 227], "score": 1.0, "content": ". In case A), there is e xactly one ramified prime (it divides", "type": "text"}], "index": 8}, {"bbox": [126, 226, 486, 238], "spans": [{"bbox": [126, 228, 136, 237], "score": 0.47, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 226, 174, 238], "score": 1.0, "content": "), hen c e ", "type": "text"}, {"bbox": [175, 226, 324, 238], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "type": "inline_equation", "height": 12, "width": 149}, {"bbox": [324, 226, 486, 238], "score": 1.0, "content": ". In case B), there are two ramified", "type": "text"}], "index": 9}, {"bbox": [126, 239, 486, 252], "spans": [{"bbox": [126, 240, 307, 252], "score": 1.0, "content": "primes (one is infin i te, the other divides ", "type": "text"}, {"bbox": [308, 242, 317, 251], "score": 0.89, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [318, 240, 356, 252], "score": 1.0, "content": "), hence ", "type": "text"}, {"bbox": [356, 239, 483, 252], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "type": "inline_equation", "height": 13, "width": 127}, {"bbox": [483, 240, 486, 252], "score": 1.0, "content": ";", "type": "text"}], "index": 10}, {"bbox": [125, 251, 487, 264], "spans": [{"bbox": [125, 251, 144, 264], "score": 1.0, "content": "but", "type": "text"}, {"bbox": [144, 254, 157, 262], "score": 0.89, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [157, 251, 414, 264], "score": 1.0, "content": " is not a norm residue at the ramified infinite prime, h e nce ", "type": "text"}, {"bbox": [414, 253, 466, 263], "score": 0.92, "content": "(E:H)\\ge2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [466, 251, 487, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 11}, {"bbox": [126, 264, 262, 277], "spans": [{"bbox": [126, 264, 210, 277], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [210, 265, 262, 277], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 12}], "index": 8.5}, {"type": "text", "bbox": [125, 276, 486, 301], "lines": [{"bbox": [137, 277, 487, 290], "spans": [{"bbox": [137, 278, 272, 290], "score": 1.0, "content": "Now P roposition 5 implies that ", "type": "text"}, {"bbox": [272, 280, 308, 290], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [308, 278, 380, 290], "score": 1.0, "content": " is cyclic of order", "type": "text"}, {"bbox": [380, 281, 396, 289], "score": 0.7, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [396, 278, 439, 290], "score": 1.0, "content": ", and that", "type": "text"}, {"bbox": [439, 277, 487, 290], "score": 0.91, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "type": "inline_equation", "height": 13, "width": 48}], "index": 13}, {"bbox": [126, 290, 487, 302], "spans": [{"bbox": [126, 291, 148, 302], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 290, 267, 302], "score": 1.0, "content": ". This concludes our proof.", "type": "text"}, {"bbox": [476, 291, 487, 301], "score": 0.9864116907119751, "content": "口", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "text", "bbox": [125, 310, 487, 346], "lines": [{"bbox": [125, 311, 486, 326], "spans": [{"bbox": [125, 311, 261, 326], "score": 1.0, "content": "Proposition 9. Assume that ", "type": "text"}, {"bbox": [261, 314, 268, 322], "score": 0.75, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [268, 311, 486, 326], "score": 1.0, "content": " is one of the imaginary quadratic fields of type", "type": "text"}], "index": 15}, {"bbox": [126, 324, 486, 337], "spans": [{"bbox": [126, 325, 138, 336], "score": 0.28, "content": "A)", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [139, 324, 154, 337], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [154, 326, 162, 334], "score": 0.77, "content": "B", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [163, 324, 486, 337], "score": 1.0, "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "type": "text"}], "index": 16}, {"bbox": [126, 337, 374, 348], "spans": [{"bbox": [126, 337, 219, 348], "score": 1.0, "content": "quartic extensions of ", "type": "text"}, {"bbox": [219, 338, 225, 345], "score": 0.81, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [226, 337, 249, 348], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [250, 338, 257, 345], "score": 0.8, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [257, 337, 374, 348], "score": 1.0, "content": " be one of them, and write", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [126, 380, 465, 394], "lines": [{"bbox": [127, 382, 464, 396], "spans": [{"bbox": [127, 382, 151, 396], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 383, 282, 396], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 13, "width": 130}, {"bbox": [282, 382, 376, 396], "score": 1.0, "content": " unless possibly when ", "type": "text"}, {"bbox": [377, 385, 413, 394], "score": 0.9, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [413, 382, 450, 396], "score": 1.0, "content": " in case ", "type": "text"}, {"bbox": [450, 385, 457, 392], "score": 0.77, "content": "B", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 382, 464, 396], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [125, 401, 486, 426], "lines": [{"bbox": [127, 403, 486, 415], "spans": [{"bbox": [127, 403, 217, 415], "score": 1.0, "content": "Proof. Observe that ", "type": "text"}, {"bbox": [217, 405, 242, 412], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 25}, {"bbox": [242, 403, 486, 415], "score": 1.0, "content": " in case A) and B); Kuroda’s class number formulas for", "type": "text"}], "index": 19}, {"bbox": [126, 415, 218, 428], "spans": [{"bbox": [126, 416, 147, 426], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [148, 415, 169, 428], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [170, 416, 192, 426], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [192, 415, 218, 428], "score": 1.0, "content": " gives", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 463, 191, 475], "lines": [{"bbox": [124, 465, 191, 477], "spans": [{"bbox": [124, 465, 191, 477], "score": 1.0, "content": "in case A) and", "type": "text"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [125, 514, 486, 538], "lines": [{"bbox": [125, 516, 486, 529], "spans": [{"bbox": [125, 516, 486, 529], "score": 1.0, "content": "in case B). Multiplying them together and plugging in the class number formula", "type": "text"}], "index": 24}, {"bbox": [126, 528, 192, 541], "spans": [{"bbox": [126, 528, 141, 541], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [141, 529, 163, 540], "score": 0.93, "content": "K/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [163, 528, 192, 541], "score": 1.0, "content": " yields", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 570, 487, 595], "lines": [{"bbox": [125, 572, 487, 586], "spans": [{"bbox": [125, 572, 147, 586], "score": 1.0, "content": "Now", "type": "text"}, {"bbox": [148, 573, 194, 584], "score": 0.93, "content": "h_{2}(k_{1})=1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [195, 572, 199, 586], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 573, 246, 584], "score": 0.92, "content": "h_{2}(k_{2})=2", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [246, 572, 267, 586], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [267, 574, 303, 583], "score": 0.92, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [304, 572, 487, 586], "score": 1.0, "content": " (by Proposition 6), and taking the square", "type": "text"}], "index": 27}, {"bbox": [126, 585, 486, 596], "spans": [{"bbox": [126, 585, 182, 596], "score": 1.0, "content": "root we find ", "type": "text"}, {"bbox": [182, 585, 312, 596], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 11, "width": 130}, {"bbox": [312, 585, 364, 596], "score": 1.0, "content": " as claimed.", "type": "text"}, {"bbox": [475, 585, 486, 595], "score": 0.9940622448921204, "content": "口", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "title", "bbox": [261, 610, 349, 623], "lines": [{"bbox": [262, 612, 349, 625], "spans": [{"bbox": [262, 612, 349, 625], "score": 1.0, "content": "5. Classification", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 629, 486, 653], "lines": [{"bbox": [136, 631, 487, 643], "spans": [{"bbox": [136, 631, 487, 643], "score": 1.0, "content": "In this section we apply the results obtained in the last few sections to give a", "type": "text"}], "index": 30}, {"bbox": [124, 644, 218, 654], "spans": [{"bbox": [124, 644, 218, 654], "score": 1.0, "content": "proof for Theorem 1.", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [124, 662, 486, 700], "lines": [{"bbox": [126, 665, 487, 677], "spans": [{"bbox": [126, 665, 236, 677], "score": 1.0, "content": "Proof of Theorem 1. Let ", "type": "text"}, {"bbox": [237, 667, 244, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [244, 665, 487, 677], "score": 1.0, "content": " be one of the two cyclic quartic unramified extensions", "type": "text"}], "index": 32}, {"bbox": [126, 677, 486, 689], "spans": [{"bbox": [126, 677, 138, 689], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 679, 145, 686], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [145, 677, 189, 689], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [189, 679, 199, 686], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [199, 677, 294, 689], "score": 1.0, "content": " be the subgroup of ", "type": "text"}, {"bbox": [295, 677, 339, 689], "score": 0.93, "content": "\\operatorname{Gal}(k^{2}/k)", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [339, 677, 372, 689], "score": 1.0, "content": " fixing ", "type": "text"}, {"bbox": [372, 679, 379, 686], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [380, 677, 418, 689], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [419, 679, 428, 686], "score": 0.91, "content": "N", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [428, 677, 486, 689], "score": 1.0, "content": " satisfies the", "type": "text"}], "index": 33}, {"bbox": [126, 690, 458, 702], "spans": [{"bbox": [126, 690, 458, 702], "score": 1.0, "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "type": "text"}], "index": 34}], "index": 33}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 91, 485, 99], "lines": [{"bbox": [476, 93, 486, 101], "spans": [{"bbox": [476, 93, 486, 101], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 111, 486, 172], "lines": [{"bbox": [126, 114, 484, 127], "spans": [{"bbox": [126, 116, 134, 123], "score": 0.9, "content": "F", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [134, 114, 439, 127], "score": 1.0, "content": " has odd class number in the strict sense: see Lemma 6); since both ", "type": "text"}, {"bbox": [439, 116, 450, 125], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [450, 114, 473, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 116, 484, 125], "score": 0.92, "content": "F_{2}", "type": "inline_equation", "height": 9, "width": 10}], "index": 0}, {"bbox": [126, 127, 486, 139], "spans": [{"bbox": [126, 127, 268, 139], "score": 1.0, "content": "are normal (even abelian) over ", "type": "text"}, {"bbox": [269, 128, 279, 137], "score": 0.91, "content": "k_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [279, 127, 356, 139], "score": 1.0, "content": ", ramification at ", "type": "text"}, {"bbox": [356, 130, 361, 137], "score": 0.87, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [362, 127, 486, 139], "score": 1.0, "content": " implies ramification at the", "type": "text"}], "index": 1}, {"bbox": [126, 138, 485, 150], "spans": [{"bbox": [126, 138, 199, 150], "score": 1.0, "content": "conjugated ideal", "type": "text"}, {"bbox": [200, 139, 208, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [208, 138, 267, 150], "score": 1.0, "content": ". Hence both ", "type": "text"}, {"bbox": [267, 142, 272, 149], "score": 0.88, "content": "\\mathfrak{q}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 138, 294, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 139, 302, 149], "score": 0.91, "content": "{\\mathfrak{q}}^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [302, 138, 347, 150], "score": 1.0, "content": " ramify in ", "type": "text"}, {"bbox": [347, 139, 371, 149], "score": 0.94, "content": "F_{1}/F", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [371, 138, 393, 150], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [393, 139, 417, 150], "score": 0.93, "content": "F_{2}/F", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [417, 138, 485, 150], "score": 1.0, "content": ", and since they", "type": "text"}], "index": 2}, {"bbox": [125, 150, 486, 163], "spans": [{"bbox": [125, 150, 188, 163], "score": 1.0, "content": "also ramify in ", "type": "text"}, {"bbox": [189, 151, 212, 162], "score": 0.93, "content": "M/F", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [212, 150, 357, 163], "score": 1.0, "content": ", they must ramify completely in ", "type": "text"}, {"bbox": [357, 151, 378, 162], "score": 0.93, "content": "N/F", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [379, 150, 486, 163], "score": 1.0, "content": ", again contradicting the", "type": "text"}], "index": 3}, {"bbox": [126, 162, 255, 174], "spans": [{"bbox": [126, 162, 167, 174], "score": 1.0, "content": "fact that ", "type": "text"}, {"bbox": [167, 163, 191, 174], "score": 0.93, "content": "N/M", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [191, 162, 255, 174], "score": 1.0, "content": " is unramified.", "type": "text"}], "index": 4}], "index": 2, "bbox_fs": [125, 114, 486, 174]}, {"type": "text", "bbox": [125, 173, 486, 275], "lines": [{"bbox": [137, 174, 487, 188], "spans": [{"bbox": [137, 174, 234, 188], "score": 1.0, "content": "We have proved that ", "type": "text"}, {"bbox": [235, 176, 270, 187], "score": 0.94, "content": "\\mathrm{Cl_{2}}(K_{2})", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [270, 174, 294, 188], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 174, 329, 187], "score": 0.93, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [330, 174, 487, 188], "score": 1.0, "content": " contain subgroups of type (4) and", "type": "text"}], "index": 5}, {"bbox": [126, 187, 486, 200], "spans": [{"bbox": [126, 188, 148, 199], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 187, 486, 200], "score": 1.0, "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "type": "text"}], "index": 6}, {"bbox": [126, 199, 486, 212], "spans": [{"bbox": [126, 199, 192, 212], "score": 0.92, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [192, 200, 314, 212], "score": 1.0, "content": ". Since the class number of", "type": "text"}, {"bbox": [314, 199, 327, 211], "score": 0.93, "content": "{\\widetilde{K}}_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [327, 200, 486, 212], "score": 1.0, "content": " is even, it is sufficient to show that", "type": "text"}], "index": 7}, {"bbox": [126, 212, 487, 227], "spans": [{"bbox": [126, 212, 214, 225], "score": 0.9, "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "type": "inline_equation", "height": 13, "width": 88}, {"bbox": [214, 212, 487, 227], "score": 1.0, "content": ". In case A), there is e xactly one ramified prime (it divides", "type": "text"}], "index": 8}, {"bbox": [126, 226, 486, 238], "spans": [{"bbox": [126, 228, 136, 237], "score": 0.47, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [136, 226, 174, 238], "score": 1.0, "content": "), hen c e ", "type": "text"}, {"bbox": [175, 226, 324, 238], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "type": "inline_equation", "height": 12, "width": 149}, {"bbox": [324, 226, 486, 238], "score": 1.0, "content": ". In case B), there are two ramified", "type": "text"}], "index": 9}, {"bbox": [126, 239, 486, 252], "spans": [{"bbox": [126, 240, 307, 252], "score": 1.0, "content": "primes (one is infin i te, the other divides ", "type": "text"}, {"bbox": [308, 242, 317, 251], "score": 0.89, "content": "d_{3}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [318, 240, 356, 252], "score": 1.0, "content": "), hence ", "type": "text"}, {"bbox": [356, 239, 483, 252], "score": 0.94, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "type": "inline_equation", "height": 13, "width": 127}, {"bbox": [483, 240, 486, 252], "score": 1.0, "content": ";", "type": "text"}], "index": 10}, {"bbox": [125, 251, 487, 264], "spans": [{"bbox": [125, 251, 144, 264], "score": 1.0, "content": "but", "type": "text"}, {"bbox": [144, 254, 157, 262], "score": 0.89, "content": "^{-1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [157, 251, 414, 264], "score": 1.0, "content": " is not a norm residue at the ramified infinite prime, h e nce ", "type": "text"}, {"bbox": [414, 253, 466, 263], "score": 0.92, "content": "(E:H)\\ge2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [466, 251, 487, 264], "score": 1.0, "content": " and", "type": "text"}], "index": 11}, {"bbox": [126, 264, 262, 277], "spans": [{"bbox": [126, 264, 210, 277], "score": 0.93, "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [210, 265, 262, 277], "score": 1.0, "content": " as claimed.", "type": "text"}], "index": 12}], "index": 8.5, "bbox_fs": [125, 174, 487, 277]}, {"type": "text", "bbox": [125, 276, 486, 301], "lines": [{"bbox": [137, 277, 487, 290], "spans": [{"bbox": [137, 278, 272, 290], "score": 1.0, "content": "Now P roposition 5 implies that ", "type": "text"}, {"bbox": [272, 280, 308, 290], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [308, 278, 380, 290], "score": 1.0, "content": " is cyclic of order", "type": "text"}, {"bbox": [380, 281, 396, 289], "score": 0.7, "content": "\\geq4", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [396, 278, 439, 290], "score": 1.0, "content": ", and that", "type": "text"}, {"bbox": [439, 277, 487, 290], "score": 0.91, "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "type": "inline_equation", "height": 13, "width": 48}], "index": 13}, {"bbox": [126, 290, 487, 302], "spans": [{"bbox": [126, 291, 148, 302], "score": 0.92, "content": "(2,2)", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [149, 290, 267, 302], "score": 1.0, "content": ". This concludes our proof.", "type": "text"}, {"bbox": [476, 291, 487, 301], "score": 0.9864116907119751, "content": "口", "type": "text"}], "index": 14}], "index": 13.5, "bbox_fs": [126, 277, 487, 302]}, {"type": "text", "bbox": [125, 310, 487, 346], "lines": [{"bbox": [125, 311, 486, 326], "spans": [{"bbox": [125, 311, 261, 326], "score": 1.0, "content": "Proposition 9. Assume that ", "type": "text"}, {"bbox": [261, 314, 268, 322], "score": 0.75, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [268, 311, 486, 326], "score": 1.0, "content": " is one of the imaginary quadratic fields of type", "type": "text"}], "index": 15}, {"bbox": [126, 324, 486, 337], "spans": [{"bbox": [126, 325, 138, 336], "score": 0.28, "content": "A)", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [139, 324, 154, 337], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [154, 326, 162, 334], "score": 0.77, "content": "B", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [163, 324, 486, 337], "score": 1.0, "content": ") as explained in the Introduction. Then there exist two unramified cyclic", "type": "text"}], "index": 16}, {"bbox": [126, 337, 374, 348], "spans": [{"bbox": [126, 337, 219, 348], "score": 1.0, "content": "quartic extensions of ", "type": "text"}, {"bbox": [219, 338, 225, 345], "score": 0.81, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [226, 337, 249, 348], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [250, 338, 257, 345], "score": 0.8, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [257, 337, 374, 348], "score": 1.0, "content": " be one of them, and write", "type": "text"}], "index": 17}], "index": 16, "bbox_fs": [125, 311, 486, 348]}, {"type": "text", "bbox": [126, 380, 465, 394], "lines": [{"bbox": [127, 382, 464, 396], "spans": [{"bbox": [127, 382, 151, 396], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [152, 383, 282, 396], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 13, "width": 130}, {"bbox": [282, 382, 376, 396], "score": 1.0, "content": " unless possibly when ", "type": "text"}, {"bbox": [377, 385, 413, 394], "score": 0.9, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [413, 382, 450, 396], "score": 1.0, "content": " in case ", "type": "text"}, {"bbox": [450, 385, 457, 392], "score": 0.77, "content": "B", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [458, 382, 464, 396], "score": 1.0, "content": ").", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [127, 382, 464, 396]}, {"type": "text", "bbox": [125, 401, 486, 426], "lines": [{"bbox": [127, 403, 486, 415], "spans": [{"bbox": [127, 403, 217, 415], "score": 1.0, "content": "Proof. Observe that ", "type": "text"}, {"bbox": [217, 405, 242, 412], "score": 0.88, "content": "\\upsilon=0", "type": "inline_equation", "height": 7, "width": 25}, {"bbox": [242, 403, 486, 415], "score": 1.0, "content": " in case A) and B); Kuroda’s class number formulas for", "type": "text"}], "index": 19}, {"bbox": [126, 415, 218, 428], "spans": [{"bbox": [126, 416, 147, 426], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 10, "width": 21}, {"bbox": [148, 415, 169, 428], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [170, 416, 192, 426], "score": 0.93, "content": "L/k_{2}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [192, 415, 218, 428], "score": 1.0, "content": " gives", "type": "text"}], "index": 20}], "index": 19.5, "bbox_fs": [126, 403, 486, 428]}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "lines": [{"bbox": [199, 433, 412, 459], "spans": [{"bbox": [199, 433, 412, 459], "score": 0.91, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 463, 191, 475], "lines": [{"bbox": [124, 465, 191, 477], "spans": [{"bbox": [124, 465, 191, 477], "score": 1.0, "content": "in case A) and", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [124, 465, 191, 477]}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "lines": [{"bbox": [199, 484, 412, 509], "spans": [{"bbox": [199, 484, 412, 509], "score": 0.9, "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [125, 514, 486, 538], "lines": [{"bbox": [125, 516, 486, 529], "spans": [{"bbox": [125, 516, 486, 529], "score": 1.0, "content": "in case B). Multiplying them together and plugging in the class number formula", "type": "text"}], "index": 24}, {"bbox": [126, 528, 192, 541], "spans": [{"bbox": [126, 528, 141, 541], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [141, 529, 163, 540], "score": 0.93, "content": "K/\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [163, 528, 192, 541], "score": 1.0, "content": " yields", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [125, 516, 486, 541]}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "lines": [{"bbox": [219, 542, 392, 568], "spans": [{"bbox": [219, 542, 392, 568], "score": 0.94, "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 570, 487, 595], "lines": [{"bbox": [125, 572, 487, 586], "spans": [{"bbox": [125, 572, 147, 586], "score": 1.0, "content": "Now", "type": "text"}, {"bbox": [148, 573, 194, 584], "score": 0.93, "content": "h_{2}(k_{1})=1", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [195, 572, 199, 586], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 573, 246, 584], "score": 0.92, "content": "h_{2}(k_{2})=2", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [246, 572, 267, 586], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [267, 574, 303, 583], "score": 0.92, "content": "q_{1}q_{2}=2", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [304, 572, 487, 586], "score": 1.0, "content": " (by Proposition 6), and taking the square", "type": "text"}], "index": 27}, {"bbox": [126, 585, 486, 596], "spans": [{"bbox": [126, 585, 182, 596], "score": 1.0, "content": "root we find ", "type": "text"}, {"bbox": [182, 585, 312, 596], "score": 0.91, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "type": "inline_equation", "height": 11, "width": 130}, {"bbox": [312, 585, 364, 596], "score": 1.0, "content": " as claimed.", "type": "text"}, {"bbox": [475, 585, 486, 595], "score": 0.9940622448921204, "content": "口", "type": "text"}], "index": 28}], "index": 27.5, "bbox_fs": [125, 572, 487, 596]}, {"type": "title", "bbox": [261, 610, 349, 623], "lines": [{"bbox": [262, 612, 349, 625], "spans": [{"bbox": [262, 612, 349, 625], "score": 1.0, "content": "5. Classification", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 629, 486, 653], "lines": [{"bbox": [136, 631, 487, 643], "spans": [{"bbox": [136, 631, 487, 643], "score": 1.0, "content": "In this section we apply the results obtained in the last few sections to give a", "type": "text"}], "index": 30}, {"bbox": [124, 644, 218, 654], "spans": [{"bbox": [124, 644, 218, 654], "score": 1.0, "content": "proof for Theorem 1.", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [124, 631, 487, 654]}, {"type": "text", "bbox": [124, 662, 486, 700], "lines": [{"bbox": [126, 665, 487, 677], "spans": [{"bbox": [126, 665, 236, 677], "score": 1.0, "content": "Proof of Theorem 1. Let ", "type": "text"}, {"bbox": [237, 667, 244, 674], "score": 0.91, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [244, 665, 487, 677], "score": 1.0, "content": " be one of the two cyclic quartic unramified extensions", "type": "text"}], "index": 32}, {"bbox": [126, 677, 486, 689], "spans": [{"bbox": [126, 677, 138, 689], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 679, 145, 686], "score": 0.88, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [145, 677, 189, 689], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [189, 679, 199, 686], "score": 0.9, "content": "N", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [199, 677, 294, 689], "score": 1.0, "content": " be the subgroup of ", "type": "text"}, {"bbox": [295, 677, 339, 689], "score": 0.93, "content": "\\operatorname{Gal}(k^{2}/k)", "type": "inline_equation", "height": 12, "width": 44}, {"bbox": [339, 677, 372, 689], "score": 1.0, "content": " fixing ", "type": "text"}, {"bbox": [372, 679, 379, 686], "score": 0.89, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [380, 677, 418, 689], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [419, 679, 428, 686], "score": 0.91, "content": "N", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [428, 677, 486, 689], "score": 1.0, "content": " satisfies the", "type": "text"}], "index": 33}, {"bbox": [126, 690, 458, 702], "spans": [{"bbox": [126, 690, 458, 702], "score": 1.0, "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "type": "text"}], "index": 34}], "index": 33, "bbox_fs": [126, 665, 487, 702]}]}
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[{"type": "text", "bbox": [125, 111, 486, 172], "content": "has odd class number in the strict sense: see Lemma 6); since both and are normal (even abelian) over , ramification at implies ramification at the conjugated ideal . Hence both and ramify in and , and since they also ramify in , they must ramify completely in , again contradicting the fact that is unramified.", "index": 0}, {"type": "text", "bbox": [125, 173, 486, 275], "content": "We have proved that and contain subgroups of type (4) and , respectively. Now we wish to apply P roposition 5. But we have to compute . Since the class number of is even, it is sufficient to show that . In case A), there is e xactly one ramified prime (it divides ), hen c e . In case B), there are two ramified primes (one is infin i te, the other divides ), hence ; but is not a norm residue at the ramified infinite prime, h e nce and as claimed.", "index": 1}, {"type": "text", "bbox": [125, 276, 486, 301], "content": "Now P roposition 5 implies that is cyclic of order , and that . This concludes our proof. 口", "index": 2}, {"type": "text", "bbox": [125, 310, 487, 346], "content": "Proposition 9. Assume that is one of the imaginary quadratic fields of type or ) as explained in the Introduction. Then there exist two unramified cyclic quartic extensions of . Let be one of them, and write", "index": 3}, {"type": "text", "bbox": [126, 380, 465, 394], "content": "Then unless possibly when in case ).", "index": 4}, {"type": "text", "bbox": [125, 401, 486, 426], "content": "Proof. Observe that in case A) and B); Kuroda’s class number formulas for and gives", "index": 5}, {"type": "interline_equation", "bbox": [199, 433, 412, 459], "content": "", "index": 6}, {"type": "text", "bbox": [124, 463, 191, 475], "content": "in case A) and", "index": 7}, {"type": "interline_equation", "bbox": [199, 484, 412, 509], "content": "", "index": 8}, {"type": "text", "bbox": [125, 514, 486, 538], "content": "in case B). Multiplying them together and plugging in the class number formula for yields", "index": 9}, {"type": "interline_equation", "bbox": [219, 542, 392, 568], "content": "", "index": 10}, {"type": "text", "bbox": [124, 570, 487, 595], "content": "Now , and (by Proposition 6), and taking the square root we find as claimed. 口", "index": 11}, {"type": "title", "bbox": [261, 610, 349, 623], "content": "5. Classification", "index": 12}, {"type": "text", "bbox": [125, 629, 486, 653], "content": "In this section we apply the results obtained in the last few sections to give a proof for Theorem 1.", "index": 13}, {"type": "text", "bbox": [124, 662, 486, 700], "content": "Proof of Theorem 1. Let be one of the two cyclic quartic unramified extensions of , and let be the subgroup of fixing . Then satisfies the assumptions of Proposition 1, thus there are only the following possibilities:", "index": 14}]
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[{"bbox": [126, 114, 484, 127], "content": "has odd class number in the strict sense: see Lemma 6); since both and", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 486, 139], "content": "are normal (even abelian) over , ramification at implies ramification at the", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 485, 150], "content": "conjugated ideal . Hence both and ramify in and , and since they", "parent_index": 0, "line_index": 2}, {"bbox": [125, 150, 486, 163], "content": "also ramify in , they must ramify completely in , again contradicting the", "parent_index": 0, "line_index": 3}, {"bbox": [126, 162, 255, 174], "content": "fact that is unramified.", "parent_index": 0, "line_index": 4}, {"bbox": [137, 174, 487, 188], "content": "We have proved that and contain subgroups of type (4) and", "parent_index": 1, "line_index": 0}, {"bbox": [126, 187, 486, 200], "content": ", respectively. Now we wish to apply P roposition 5. But we have to compute", "parent_index": 1, "line_index": 1}, {"bbox": [126, 199, 486, 212], "content": ". Since the class number of is even, it is sufficient to show that", "parent_index": 1, "line_index": 2}, {"bbox": [126, 212, 487, 227], "content": ". In case A), there is e xactly one ramified prime (it divides", "parent_index": 1, "line_index": 3}, {"bbox": [126, 226, 486, 238], "content": "), hen c e . In case B), there are two ramified", "parent_index": 1, "line_index": 4}, {"bbox": [126, 239, 486, 252], "content": "primes (one is infin i te, the other divides ), hence ;", "parent_index": 1, "line_index": 5}, {"bbox": [125, 251, 487, 264], "content": "but is not a norm residue at the ramified infinite prime, h e nce and", "parent_index": 1, "line_index": 6}, {"bbox": [126, 264, 262, 277], "content": "as claimed.", "parent_index": 1, "line_index": 7}, {"bbox": [137, 277, 487, 290], "content": "Now P roposition 5 implies that is cyclic of order , and that", "parent_index": 2, "line_index": 0}, {"bbox": [126, 290, 487, 302], "content": ". This concludes our proof. 口", "parent_index": 2, "line_index": 1}, {"bbox": [125, 311, 486, 326], "content": "Proposition 9. Assume that is one of the imaginary quadratic fields of type", "parent_index": 3, "line_index": 0}, {"bbox": [126, 324, 486, 337], "content": "or ) as explained in the Introduction. Then there exist two unramified cyclic", "parent_index": 3, "line_index": 1}, {"bbox": [126, 337, 374, 348], "content": "quartic extensions of . Let be one of them, and write", "parent_index": 3, "line_index": 2}, {"bbox": [127, 382, 464, 396], "content": "Then unless possibly when in case ).", "parent_index": 4, "line_index": 0}, {"bbox": [127, 403, 486, 415], "content": "Proof. Observe that in case A) and B); Kuroda’s class number formulas for", "parent_index": 5, "line_index": 0}, {"bbox": [126, 415, 218, 428], "content": "and gives", "parent_index": 5, "line_index": 1}, {"bbox": [124, 465, 191, 477], "content": "in case A) and", "parent_index": 7, "line_index": 0}, {"bbox": [125, 516, 486, 529], "content": "in case B). Multiplying them together and plugging in the class number formula", "parent_index": 9, "line_index": 0}, {"bbox": [126, 528, 192, 541], "content": "for yields", "parent_index": 9, "line_index": 1}, {"bbox": [125, 572, 487, 586], "content": "Now , and (by Proposition 6), and taking the square", "parent_index": 11, "line_index": 0}, {"bbox": [126, 585, 486, 596], "content": "root we find as claimed. 口", "parent_index": 11, "line_index": 1}, {"bbox": [262, 612, 349, 625], "content": "5. Classification", "parent_index": 12, "line_index": 0}, {"bbox": [136, 631, 487, 643], "content": "In this section we apply the results obtained in the last few sections to give a", "parent_index": 13, "line_index": 0}, {"bbox": [124, 644, 218, 654], "content": "proof for Theorem 1.", "parent_index": 13, "line_index": 1}, {"bbox": [126, 665, 487, 677], "content": "Proof of Theorem 1. Let be one of the two cyclic quartic unramified extensions", "parent_index": 14, "line_index": 0}, {"bbox": [126, 677, 486, 689], "content": "of , and let be the subgroup of fixing . Then satisfies the", "parent_index": 14, "line_index": 1}, {"bbox": [126, 690, 458, 702], "content": "assumptions of Proposition 1, thus there are only the following possibilities:", "parent_index": 14, "line_index": 2}]
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[]
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[{"bbox": [126, 116, 134, 123], "content": "F", "parent_index": 0, "subtype": "inline"}, {"bbox": [439, 116, 450, 125], "content": "F_{1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [474, 116, 484, 125], "content": "F_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [269, 128, 279, 137], "content": "k_{2}", "parent_index": 0, "subtype": "inline"}, {"bbox": [356, 130, 361, 137], "content": "\\mathfrak{q}", "parent_index": 0, "subtype": "inline"}, {"bbox": [200, 139, 208, 149], "content": "{\\mathfrak{q}}^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [267, 142, 272, 149], "content": "\\mathfrak{q}", "parent_index": 0, "subtype": "inline"}, {"bbox": [294, 139, 302, 149], "content": "{\\mathfrak{q}}^{\\prime}", "parent_index": 0, "subtype": "inline"}, {"bbox": [347, 139, 371, 149], "content": "F_{1}/F", "parent_index": 0, "subtype": "inline"}, {"bbox": [393, 139, 417, 150], "content": "F_{2}/F", "parent_index": 0, "subtype": "inline"}, {"bbox": [189, 151, 212, 162], "content": "M/F", "parent_index": 0, "subtype": "inline"}, {"bbox": [357, 151, 378, 162], "content": "N/F", "parent_index": 0, "subtype": "inline"}, {"bbox": [167, 163, 191, 174], "content": "N/M", "parent_index": 0, "subtype": "inline"}, {"bbox": [235, 176, 270, 187], "content": "\\mathrm{Cl_{2}}(K_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [294, 174, 329, 187], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 188, 148, 199], "content": "(2,2)", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 199, 192, 212], "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [314, 199, 327, 211], "content": "{\\widetilde{K}}_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 212, 214, 225], "content": "\\#\\operatorname{Am}_{2}(\\widetilde{K}_{2}/k_{2})\\,\\leq\\,2", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 228, 136, 237], "content": "d_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [175, 226, 324, 238], "content": "\\#\\operatorname{Am}_{2}({\\widetilde{K}}_{2}/k_{2})\\,=\\,2/(E:H)\\,\\le\\,2", "parent_index": 1, "subtype": "inline"}, {"bbox": [308, 242, 317, 251], "content": "d_{3}", "parent_index": 1, "subtype": "inline"}, {"bbox": [356, 239, 483, 252], "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})=4/(E:H)", "parent_index": 1, "subtype": "inline"}, {"bbox": [144, 254, 157, 262], "content": "^{-1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [414, 253, 466, 263], "content": "(E:H)\\ge2", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 264, 210, 277], "content": "\\#\\operatorname{Am}_{2}(\\tilde{K}_{2}/k_{2})\\leq2", "parent_index": 1, "subtype": "inline"}, {"bbox": [272, 280, 308, 290], "content": "\\mathrm{Cl}_{2}(K_{2})", "parent_index": 2, "subtype": "inline"}, {"bbox": [380, 281, 396, 289], "content": "\\geq4", "parent_index": 2, "subtype": "inline"}, {"bbox": [439, 277, 487, 290], "content": "\\mathrm{Cl}_{2}(\\widetilde{K}_{2})\\simeq", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 291, 148, 302], "content": "(2,2)", "parent_index": 2, "subtype": "inline"}, {"bbox": [261, 314, 268, 322], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [126, 325, 138, 336], "content": "A)", "parent_index": 3, "subtype": "inline"}, {"bbox": [154, 326, 162, 334], "content": "B", "parent_index": 3, "subtype": "inline"}, {"bbox": [219, 338, 225, 345], "content": "k", "parent_index": 3, "subtype": "inline"}, {"bbox": [250, 338, 257, 345], "content": "L", "parent_index": 3, "subtype": "inline"}, {"bbox": [152, 383, 282, 396], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "parent_index": 4, "subtype": "inline"}, {"bbox": [377, 385, 413, 394], "content": "d_{3}=-4", "parent_index": 4, "subtype": "inline"}, {"bbox": [450, 385, 457, 392], "content": "B", "parent_index": 4, "subtype": "inline"}, {"bbox": [217, 405, 242, 412], "content": "\\upsilon=0", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 416, 147, 426], "content": "L/k_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [170, 416, 192, 426], "content": "L/k_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [199, 433, 412, 459], "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{2h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{4h_{2}(k_{2})^{2}}", "parent_index": 6, "subtype": "interline"}, {"bbox": [199, 484, 412, 509], "content": "h_{2}(L)\\ =\\ \\frac{q_{1}h_{2}(K_{1})^{2}h_{2}(K)}{4h_{2}(k_{1})^{2}}\\ =\\ \\frac{q_{2}h_{2}(K_{2})^{2}h_{2}(K)}{2h_{2}(k_{2})^{2}}", "parent_index": 8, "subtype": "interline"}, {"bbox": [141, 529, 163, 540], "content": "K/\\mathbb{Q}", "parent_index": 9, "subtype": "inline"}, {"bbox": [219, 542, 392, 568], "content": "h_{2}(L)^{2}=\\frac{q_{1}\\,q_{2}}{8}\\frac{h_{2}(K_{1})^{2}\\,h_{2}(K_{2})^{2}\\,h_{2}(k)^{2}}{h_{2}(k_{1})^{2}\\,h_{2}(k_{2})^{2}}.", "parent_index": 10, "subtype": "interline"}, {"bbox": [148, 573, 194, 584], "content": "h_{2}(k_{1})=1", "parent_index": 11, "subtype": "inline"}, {"bbox": [199, 573, 246, 584], "content": "h_{2}(k_{2})=2", "parent_index": 11, "subtype": "inline"}, {"bbox": [267, 574, 303, 583], "content": "q_{1}q_{2}=2", "parent_index": 11, "subtype": "inline"}, {"bbox": [182, 585, 312, 596], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{4}h_{2}(k)h_{2}(K_{1})h_{2}(K_{2})}\\end{array}", "parent_index": 11, "subtype": "inline"}, {"bbox": [237, 667, 244, 674], "content": "L", "parent_index": 14, "subtype": "inline"}, {"bbox": [139, 679, 145, 686], "content": "k", "parent_index": 14, "subtype": "inline"}, {"bbox": [189, 679, 199, 686], "content": "N", "parent_index": 14, "subtype": "inline"}, {"bbox": [295, 677, 339, 689], "content": "\\operatorname{Gal}(k^{2}/k)", "parent_index": 14, "subtype": "inline"}, {"bbox": [372, 679, 379, 686], "content": "L", "parent_index": 14, "subtype": "inline"}, {"bbox": [419, 679, 428, 686], "content": "N", "parent_index": 14, "subtype": "inline"}]
|
[]
|
|
<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>
Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $d_{3}=-4$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $d(G^{\prime})\geq3$ if one of the class numbers $h_{2}(K_{1})$ or $h_{2}(K_{2})$ is at least 8. Therefore it suffices to examine the cases $h_{2}(K_{2})=2$ and $h_{2}(K_{2})=4$ (recall from above that $h_{2}(K_{2})$ is always even).
We start by considering case A); it is sufficient to show that $h_{2}(K_{1})h_{2}(K_{2})\neq4$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8.
a) If $h_{2}(K_{2})\,=\,2$ , then $\#\kappa_{2}\,=\,2$ by Proposition 5, hence $q_{2}\,=\,2$ by Proposition 7 and then $q_{1}=1$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $h_{2}(K_{1})=1$ and $h_{2}(L)=2^{m}$ .
It can be shown using the ambiguous class number formula that $\mathrm{Cl}_{2}(K_{1})$ is trivial if and only if $\varepsilon_{1}$ is a quadratic nonresidue modulo the prime ideal over $d_{2}$ in $k_{1}$ ; by Scholz’s reciprocity law, this is equivalent to $(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$ , and this agrees with the criterion given in [1].
b) If $h_{2}(K_{2})=4$ , we may assume that $\mathrm{Cl}_{2}(K_{2})=(4)$ from Proposition 8.b). Then $\#\kappa_{2}=2$ by Proposition 5, $q_{2}=2$ by Proposition 7 and $q_{1}=1$ by Proposition 6. Using the class number formula we get $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+2}$ .
Thus in both cases we have $h_{2}(K_{1})h_{2}(K_{2})\neq4$ , and by the table at the beginning of this proof this implies that rank $\mathrm{Cl}_{2}(k^{1})\neq2$ in case A).
Next we consider case B); here we have to distinguish between $d_{3}\neq-4$ (case $B_{1}$ ) and $d_{3}=-4$ (case $B_{2}$ ).
Let us start with case $B_{1}$ ). a) If $h_{2}(K_{2})=2$ , then $\#\kappa_{2}=2$ , $q_{2}=2$ and $q_{1}=1$ as above. The class number formula gives $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+1}$ . b) If $\mathrm{Cl}_{2}(K_{2})=(4)$ (which we may assume without loss of generality by Proposition 8.b)) then $\#\kappa_{2}\,=\,2$ , $q_{2}\,=\,2$ and $q_{1}\,=\,1$ , again exactly as above. This implies $h_{2}(K_{1})=4$ and $h_{2}(L)=2^{m+3}$ .
Here we apply Kuroda’s class number formula (see [10]) to $L/k_{1}$ , and since $h_{2}(k_{1})=$ $^{1}$ and $h_{2}(K_{1})=h_{2}(K_{1}^{\prime})$ , we get $\begin{array}{r}{h_{2}(L)=\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\end{array}$ . From $K_{2}=k_{2}(\sqrt{\varepsilon}\,)$ (for a suitable choice of $L$ ; the other possibility is ${\tilde{K}}_{2}=k_{2}({\sqrt{d_{2}\varepsilon}}\,))$ , where $\varepsilon$ is the fundamental unit of $k_{2}$ , we deduce that the uni t $\varepsilon$ , which still is fundamental in $k$ , becomes a square in $L$ , and this implies that $q_{1}\geq2$ . Moreover, we have $K_{1}\,=\,k_{1}(\sqrt{\pi\lambda})$ , where $\pi,\lambda\,\equiv\,1$ mod 4 are prime factors of $d_{1}$ and $d_{2}$ in $k_{1}\,=\,\mathbb{Q}(i)$ , respectively. This shows that $K_{1}$ has even class number, because $K_{1}(\sqrt{\pi}\,)/K_{1}$ is easily seen to be unramified.
Thus $2\mid q_{1},\,2\mid h_{2}(K_{1})$ , and so we find that $h_{2}(L)$ is divisible by $2^{m}\cdot2\cdot4=2^{m+3}$ . In particular, we always have $d(G^{\prime})\geq3$ in this case.
This concludes the proof.
|
<html><body>
<div class="table" data-bbox="228 110 383 167"><table data-bbox="228 110 383 167"><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></div>
<p data-bbox="124 187 486 248">Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $d_{3}=-4$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $d(G^{\prime})\geq3$ if one of the class numbers $h_{2}(K_{1})$ or $h_{2}(K_{2})$ is at least 8. Therefore it suffices to examine the cases $h_{2}(K_{2})=2$ and $h_{2}(K_{2})=4$ (recall from above that $h_{2}(K_{2})$ is always even). </p>
<p data-bbox="124 263 485 298">We start by considering case A); it is sufficient to show that $h_{2}(K_{1})h_{2}(K_{2})\neq4$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8. </p>
<p data-bbox="124 299 486 334">a) If $h_{2}(K_{2})\,=\,2$ , then $\#\kappa_{2}\,=\,2$ by Proposition 5, hence $q_{2}\,=\,2$ by Proposition 7 and then $q_{1}=1$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $h_{2}(K_{1})=1$ and $h_{2}(L)=2^{m}$ . </p>
<p data-bbox="125 335 486 382">It can be shown using the ambiguous class number formula that $\mathrm{Cl}_{2}(K_{1})$ is trivial if and only if $\varepsilon_{1}$ is a quadratic nonresidue modulo the prime ideal over $d_{2}$ in $k_{1}$ ; by Scholz’s reciprocity law, this is equivalent to $(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$ , and this agrees with the criterion given in [1]. </p>
<p data-bbox="125 382 485 418">b) If $h_{2}(K_{2})=4$ , we may assume that $\mathrm{Cl}_{2}(K_{2})=(4)$ from Proposition 8.b). Then $\#\kappa_{2}=2$ by Proposition 5, $q_{2}=2$ by Proposition 7 and $q_{1}=1$ by Proposition 6. Using the class number formula we get $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+2}$ . </p>
<p data-bbox="124 418 486 442">Thus in both cases we have $h_{2}(K_{1})h_{2}(K_{2})\neq4$ , and by the table at the beginning of this proof this implies that rank $\mathrm{Cl}_{2}(k^{1})\neq2$ in case A). </p>
<p data-bbox="127 457 486 481">Next we consider case B); here we have to distinguish between $d_{3}\neq-4$ (case $B_{1}$ ) and $d_{3}=-4$ (case $B_{2}$ ). </p>
<p data-bbox="124 484 487 553">Let us start with case $B_{1}$ ). a) If $h_{2}(K_{2})=2$ , then $\#\kappa_{2}=2$ , $q_{2}=2$ and $q_{1}=1$ as above. The class number formula gives $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+1}$ . b) If $\mathrm{Cl}_{2}(K_{2})=(4)$ (which we may assume without loss of generality by Proposition 8.b)) then $\#\kappa_{2}\,=\,2$ , $q_{2}\,=\,2$ and $q_{1}\,=\,1$ , again exactly as above. This implies $h_{2}(K_{1})=4$ and $h_{2}(L)=2^{m+3}$ . </p>
<p data-bbox="125 564 486 663">Here we apply Kuroda’s class number formula (see [10]) to $L/k_{1}$ , and since $h_{2}(k_{1})=$ $^{1}$ and $h_{2}(K_{1})=h_{2}(K_{1}^{\prime})$ , we get $\begin{array}{r}{h_{2}(L)=\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\end{array}$ . From $K_{2}=k_{2}(\sqrt{\varepsilon}\,)$ (for a suitable choice of $L$ ; the other possibility is ${\tilde{K}}_{2}=k_{2}({\sqrt{d_{2}\varepsilon}}\,))$ , where $\varepsilon$ is the fundamental unit of $k_{2}$ , we deduce that the uni t $\varepsilon$ , which still is fundamental in $k$ , becomes a square in $L$ , and this implies that $q_{1}\geq2$ . Moreover, we have $K_{1}\,=\,k_{1}(\sqrt{\pi\lambda})$ , where $\pi,\lambda\,\equiv\,1$ mod 4 are prime factors of $d_{1}$ and $d_{2}$ in $k_{1}\,=\,\mathbb{Q}(i)$ , respectively. This shows that $K_{1}$ has even class number, because $K_{1}(\sqrt{\pi}\,)/K_{1}$ is easily seen to be unramified. </p>
<p data-bbox="125 663 485 687">Thus $2\mid q_{1},\,2\mid h_{2}(K_{1})$ , and so we find that $h_{2}(L)$ is divisible by $2^{m}\cdot2\cdot4=2^{m+3}$ . In particular, we always have $d(G^{\prime})\geq3$ in this case. </p>
<p data-bbox="137 688 249 699">This concludes the proof. </p>
</body></html>
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0003244v1
| 11 | 612 | 792 | 1,275 | 1,650 |
[{"type": "table", "img_path": "images/295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg", "table_caption": [], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>\n\n", "page_idx": 11}, {"type": "text", "text": "Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if $d_{3}=-4$ in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have $d(G^{\\prime})\\geq3$ if one of the class numbers $h_{2}(K_{1})$ or $h_{2}(K_{2})$ is at least 8. Therefore it suffices to examine the cases $h_{2}(K_{2})=2$ and $h_{2}(K_{2})=4$ (recall from above that $h_{2}(K_{2})$ is always even). ", "page_idx": 11}, {"type": "text", "text": "We start by considering case A); it is sufficient to show that $h_{2}(K_{1})h_{2}(K_{2})\\neq4$ . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8. ", "page_idx": 11}, {"type": "text", "text": "a) If $h_{2}(K_{2})\\,=\\,2$ , then $\\#\\kappa_{2}\\,=\\,2$ by Proposition 5, hence $q_{2}\\,=\\,2$ by Proposition 7 and then $q_{1}=1$ by Proposition 6. The class number formulas in the proof of Proposition 9 now give $h_{2}(K_{1})=1$ and $h_{2}(L)=2^{m}$ . ", "page_idx": 11}, {"type": "text", "text": "It can be shown using the ambiguous class number formula that $\\mathrm{Cl}_{2}(K_{1})$ is trivial if and only if $\\varepsilon_{1}$ is a quadratic nonresidue modulo the prime ideal over $d_{2}$ in $k_{1}$ ; by Scholz’s reciprocity law, this is equivalent to $(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1$ , and this agrees with the criterion given in [1]. ", "page_idx": 11}, {"type": "text", "text": "b) If $h_{2}(K_{2})=4$ , we may assume that $\\mathrm{Cl}_{2}(K_{2})=(4)$ from Proposition 8.b). Then $\\#\\kappa_{2}=2$ by Proposition 5, $q_{2}=2$ by Proposition 7 and $q_{1}=1$ by Proposition 6. Using the class number formula we get $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+2}$ . ", "page_idx": 11}, {"type": "text", "text": "Thus in both cases we have $h_{2}(K_{1})h_{2}(K_{2})\\neq4$ , and by the table at the beginning of this proof this implies that rank $\\mathrm{Cl}_{2}(k^{1})\\neq2$ in case A). ", "page_idx": 11}, {"type": "text", "text": "Next we consider case B); here we have to distinguish between $d_{3}\\neq-4$ (case $B_{1}$ ) and $d_{3}=-4$ (case $B_{2}$ ). ", "page_idx": 11}, {"type": "text", "text": "Let us start with case $B_{1}$ ). a) If $h_{2}(K_{2})=2$ , then $\\#\\kappa_{2}=2$ , $q_{2}=2$ and $q_{1}=1$ as above. The class number formula gives $h_{2}(K_{1})=2$ and $h_{2}(L)=2^{m+1}$ . b) If $\\mathrm{Cl}_{2}(K_{2})=(4)$ (which we may assume without loss of generality by Proposition 8.b)) then $\\#\\kappa_{2}\\,=\\,2$ , $q_{2}\\,=\\,2$ and $q_{1}\\,=\\,1$ , again exactly as above. This implies $h_{2}(K_{1})=4$ and $h_{2}(L)=2^{m+3}$ . ", "page_idx": 11}, {"type": "text", "text": "Here we apply Kuroda’s class number formula (see [10]) to $L/k_{1}$ , and since $h_{2}(k_{1})=$ $^{1}$ and $h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})$ , we get $\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}$ . From $K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)$ (for a suitable choice of $L$ ; the other possibility is ${\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))$ , where $\\varepsilon$ is the fundamental unit of $k_{2}$ , we deduce that the uni t $\\varepsilon$ , which still is fundamental in $k$ , becomes a square in $L$ , and this implies that $q_{1}\\geq2$ . Moreover, we have $K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})$ , where $\\pi,\\lambda\\,\\equiv\\,1$ mod 4 are prime factors of $d_{1}$ and $d_{2}$ in $k_{1}\\,=\\,\\mathbb{Q}(i)$ , respectively. This shows that $K_{1}$ has even class number, because $K_{1}(\\sqrt{\\pi}\\,)/K_{1}$ is easily seen to be unramified. ", "page_idx": 11}, {"type": "text", "text": "Thus $2\\mid q_{1},\\,2\\mid h_{2}(K_{1})$ , and so we find that $h_{2}(L)$ is divisible by $2^{m}\\cdot2\\cdot4=2^{m+3}$ . In particular, we always have $d(G^{\\prime})\\geq3$ in this case. ", "page_idx": 11}, {"type": "text", "text": "This concludes the proof. ", "page_idx": 11}]
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{"preproc_blocks": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}, {"type": "text", "bbox": [124, 187, 486, 248], "lines": [{"bbox": [125, 189, 486, 201], "spans": [{"bbox": [125, 189, 486, 201], "score": 1.0, "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "type": "text"}], "index": 5}, {"bbox": [125, 201, 487, 213], "spans": [{"bbox": [125, 201, 194, 213], "score": 1.0, "content": "claim to hold if ", "type": "text"}, {"bbox": [194, 203, 230, 212], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [230, 201, 487, 213], "score": 1.0, "content": " in case B)) is a consequence of the class number formula of", "type": "text"}], "index": 6}, {"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 290, 225], "score": 1.0, "content": "Proposition 9. In particular, we have ", "type": "text"}, {"bbox": [290, 214, 332, 225], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [332, 213, 453, 225], "score": 1.0, "content": " if one of the class numbers ", "type": "text"}, {"bbox": [454, 214, 485, 225], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 31}], "index": 7}, {"bbox": [126, 225, 486, 237], "spans": [{"bbox": [126, 225, 138, 237], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [138, 226, 169, 237], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [170, 225, 415, 237], "score": 1.0, "content": " is at least 8. Therefore it suffices to examine the cases ", "type": "text"}, {"bbox": [415, 226, 465, 237], "score": 0.94, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [466, 225, 486, 237], "score": 1.0, "content": " and", "type": "text"}], "index": 8}, {"bbox": [126, 237, 385, 250], "spans": [{"bbox": [126, 238, 175, 249], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 237, 282, 250], "score": 1.0, "content": " (recall from above that ", "type": "text"}, {"bbox": [282, 238, 313, 249], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [313, 237, 385, 250], "score": 1.0, "content": " is always even).", "type": "text"}], "index": 9}], "index": 7}, {"type": "text", "bbox": [124, 263, 485, 298], "lines": [{"bbox": [137, 263, 486, 278], "spans": [{"bbox": [137, 263, 401, 278], "score": 1.0, "content": "We start by considering case A); it is sufficient to show that ", "type": "text"}, {"bbox": [402, 266, 482, 276], "score": 0.94, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [483, 263, 486, 278], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 275, 486, 290], "spans": [{"bbox": [125, 275, 486, 290], "score": 1.0, "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "type": "text"}], "index": 11}, {"bbox": [126, 290, 135, 299], "spans": [{"bbox": [126, 290, 135, 299], "score": 1.0, "content": "8.", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [124, 299, 486, 334], "lines": [{"bbox": [125, 300, 487, 313], "spans": [{"bbox": [125, 300, 149, 313], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [150, 302, 202, 312], "score": 0.92, "content": "h_{2}(K_{2})\\,=\\,2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [202, 300, 232, 313], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [232, 302, 272, 311], "score": 0.92, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [272, 300, 385, 313], "score": 1.0, "content": " by Proposition 5, hence ", "type": "text"}, {"bbox": [385, 303, 416, 312], "score": 0.92, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [416, 300, 487, 313], "score": 1.0, "content": " by Proposition", "type": "text"}], "index": 13}, {"bbox": [124, 311, 487, 326], "spans": [{"bbox": [124, 311, 178, 326], "score": 1.0, "content": "7 and then ", "type": "text"}, {"bbox": [178, 315, 208, 323], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [209, 311, 487, 326], "score": 1.0, "content": " by Proposition 6. The class number formulas in the proof of", "type": "text"}], "index": 14}, {"bbox": [125, 324, 356, 336], "spans": [{"bbox": [125, 324, 228, 336], "score": 1.0, "content": "Proposition 9 now give", "type": "text"}, {"bbox": [229, 325, 279, 336], "score": 0.94, "content": "h_{2}(K_{1})=1", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [279, 324, 300, 336], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [301, 325, 352, 336], "score": 0.94, "content": "h_{2}(L)=2^{m}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [352, 324, 356, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [125, 335, 486, 382], "lines": [{"bbox": [137, 336, 486, 348], "spans": [{"bbox": [137, 336, 411, 348], "score": 1.0, "content": "It can be shown using the ambiguous class number formula that", "type": "text"}, {"bbox": [412, 338, 447, 348], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 336, 486, 348], "score": 1.0, "content": " is trivial", "type": "text"}], "index": 16}, {"bbox": [125, 348, 485, 361], "spans": [{"bbox": [125, 348, 184, 361], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [185, 353, 194, 359], "score": 0.89, "content": "\\varepsilon_{1}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [194, 348, 434, 361], "score": 1.0, "content": " is a quadratic nonresidue modulo the prime ideal over ", "type": "text"}, {"bbox": [434, 350, 444, 359], "score": 0.92, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [444, 348, 458, 361], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [459, 350, 468, 359], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [469, 348, 485, 361], "score": 1.0, "content": "; by", "type": "text"}], "index": 17}, {"bbox": [126, 361, 486, 373], "spans": [{"bbox": [126, 361, 322, 373], "score": 1.0, "content": "Scholz’s reciprocity law, this is equivalent to ", "type": "text"}, {"bbox": [322, 361, 414, 372], "score": 0.93, "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [414, 361, 486, 373], "score": 1.0, "content": ", and this agrees", "type": "text"}], "index": 18}, {"bbox": [126, 372, 258, 384], "spans": [{"bbox": [126, 372, 258, 384], "score": 1.0, "content": "with the criterion given in [1].", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "text", "bbox": [125, 382, 485, 418], "lines": [{"bbox": [125, 383, 486, 397], "spans": [{"bbox": [125, 383, 148, 397], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [149, 385, 198, 396], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [199, 383, 295, 397], "score": 1.0, "content": ", we may assume that", "type": "text"}, {"bbox": [296, 385, 358, 396], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [358, 383, 486, 397], "score": 1.0, "content": " from Proposition 8.b). Then", "type": "text"}], "index": 20}, {"bbox": [126, 397, 486, 409], "spans": [{"bbox": [126, 398, 164, 407], "score": 0.92, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [164, 397, 247, 409], "score": 1.0, "content": " by Proposition 5, ", "type": "text"}, {"bbox": [247, 398, 276, 407], "score": 0.93, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [276, 397, 376, 409], "score": 1.0, "content": " by Proposition 7 and ", "type": "text"}, {"bbox": [376, 398, 405, 407], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [406, 397, 486, 409], "score": 1.0, "content": " by Proposition 6.", "type": "text"}], "index": 21}, {"bbox": [124, 407, 435, 421], "spans": [{"bbox": [124, 407, 298, 421], "score": 1.0, "content": "Using the class number formula we get ", "type": "text"}, {"bbox": [299, 409, 348, 420], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [348, 407, 370, 421], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 408, 431, 420], "score": 0.93, "content": "h_{2}(L)=2^{m+2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [432, 407, 435, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [124, 418, 486, 442], "lines": [{"bbox": [137, 419, 486, 433], "spans": [{"bbox": [137, 419, 256, 433], "score": 1.0, "content": "Thus in both cases we have", "type": "text"}, {"bbox": [257, 421, 337, 432], "score": 0.93, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [337, 419, 486, 433], "score": 1.0, "content": ", and by the table at the beginning", "type": "text"}], "index": 23}, {"bbox": [126, 432, 380, 444], "spans": [{"bbox": [126, 432, 278, 444], "score": 1.0, "content": "of this proof this implies that rank", "type": "text"}, {"bbox": [279, 433, 329, 443], "score": 0.89, "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [330, 432, 380, 444], "score": 1.0, "content": " in case A).", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [127, 457, 486, 481], "lines": [{"bbox": [136, 459, 487, 472], "spans": [{"bbox": [136, 459, 421, 472], "score": 1.0, "content": "Next we consider case B); here we have to distinguish between ", "type": "text"}, {"bbox": [421, 461, 460, 471], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [460, 459, 487, 472], "score": 1.0, "content": " (case", "type": "text"}], "index": 25}, {"bbox": [126, 471, 248, 484], "spans": [{"bbox": [126, 473, 138, 482], "score": 0.82, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [138, 471, 164, 484], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [164, 473, 201, 482], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [201, 471, 228, 484], "score": 1.0, "content": " (case ", "type": "text"}, {"bbox": [228, 473, 241, 482], "score": 0.88, "content": "B_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [241, 471, 248, 484], "score": 1.0, "content": ").", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [124, 484, 487, 553], "lines": [{"bbox": [137, 483, 256, 496], "spans": [{"bbox": [137, 483, 236, 496], "score": 1.0, "content": "Let us start with case ", "type": "text"}, {"bbox": [236, 485, 248, 494], "score": 0.9, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [249, 483, 256, 496], "score": 1.0, "content": ").", "type": "text"}], "index": 27}, {"bbox": [126, 495, 486, 509], "spans": [{"bbox": [126, 495, 149, 509], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [149, 496, 200, 507], "score": 0.93, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [200, 495, 230, 509], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [230, 497, 268, 506], "score": 0.91, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [269, 495, 275, 509], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [275, 497, 304, 506], "score": 0.91, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [304, 495, 327, 509], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 497, 357, 506], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [357, 495, 486, 509], "score": 1.0, "content": " as above. The class number", "type": "text"}], "index": 28}, {"bbox": [123, 507, 324, 520], "spans": [{"bbox": [123, 507, 187, 520], "score": 1.0, "content": "formula gives ", "type": "text"}, {"bbox": [187, 509, 236, 519], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [236, 507, 258, 520], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [259, 508, 320, 519], "score": 0.93, "content": "h_{2}(L)=2^{m+1}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [320, 507, 324, 520], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 518, 486, 533], "spans": [{"bbox": [125, 518, 147, 533], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [147, 520, 208, 531], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [209, 518, 486, 533], "score": 1.0, "content": " (which we may assume without loss of generality by Proposition", "type": "text"}], "index": 30}, {"bbox": [125, 531, 485, 544], "spans": [{"bbox": [125, 531, 175, 544], "score": 1.0, "content": "8.b)) then ", "type": "text"}, {"bbox": [175, 533, 216, 542], "score": 0.91, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [217, 531, 223, 544], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [223, 533, 254, 542], "score": 0.91, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [255, 531, 279, 544], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [280, 533, 311, 542], "score": 0.93, "content": "q_{1}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [311, 531, 485, 544], "score": 1.0, "content": ", again exactly as above. This implies", "type": "text"}], "index": 31}, {"bbox": [126, 542, 263, 556], "spans": [{"bbox": [126, 544, 175, 555], "score": 0.94, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 542, 198, 556], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 544, 259, 555], "score": 0.92, "content": "h_{2}(L)=2^{m+3}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [259, 542, 263, 556], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 29.5}, {"type": "text", "bbox": [125, 564, 486, 663], "lines": [{"bbox": [125, 566, 487, 580], "spans": [{"bbox": [125, 566, 377, 580], "score": 1.0, "content": "Here we apply Kuroda’s class number formula (see [10]) to ", "type": "text"}, {"bbox": [378, 568, 399, 579], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [399, 566, 446, 580], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [447, 568, 487, 579], "score": 0.91, "content": "h_{2}(k_{1})=", "type": "inline_equation", "height": 11, "width": 40}], "index": 33}, {"bbox": [126, 579, 486, 592], "spans": [{"bbox": [126, 581, 131, 588], "score": 0.43, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [132, 579, 154, 592], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [154, 580, 231, 591], "score": 0.93, "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "type": "inline_equation", "height": 11, "width": 77}, {"bbox": [231, 579, 268, 592], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [269, 579, 453, 591], "score": 0.93, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "type": "inline_equation", "height": 12, "width": 184}, {"bbox": [454, 579, 486, 592], "score": 1.0, "content": ". From", "type": "text"}], "index": 34}, {"bbox": [126, 592, 485, 605], "spans": [{"bbox": [126, 594, 185, 605], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 594, 294, 605], "score": 1.0, "content": " (for a suitable choice of ", "type": "text"}, {"bbox": [295, 595, 302, 602], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [302, 594, 409, 605], "score": 1.0, "content": "; the other possibility is", "type": "text"}, {"bbox": [409, 592, 482, 605], "score": 0.92, "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [482, 594, 485, 605], "score": 1.0, "content": ",", "type": "text"}], "index": 35}, {"bbox": [126, 604, 487, 617], "spans": [{"bbox": [126, 604, 155, 617], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 609, 160, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [161, 604, 286, 617], "score": 1.0, "content": " is the fundamental unit of ", "type": "text"}, {"bbox": [286, 607, 297, 615], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [297, 604, 416, 617], "score": 1.0, "content": ", we deduce that the uni t ", "type": "text"}, {"bbox": [416, 609, 421, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [421, 604, 487, 617], "score": 1.0, "content": ", which still is", "type": "text"}], "index": 36}, {"bbox": [124, 617, 487, 630], "spans": [{"bbox": [124, 617, 195, 630], "score": 1.0, "content": "fundamental in ", "type": "text"}, {"bbox": [195, 619, 201, 627], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [201, 617, 298, 630], "score": 1.0, "content": ", becomes a square in ", "type": "text"}, {"bbox": [299, 619, 306, 626], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [306, 617, 406, 630], "score": 1.0, "content": ", and this implies that ", "type": "text"}, {"bbox": [406, 619, 434, 628], "score": 0.91, "content": "q_{1}\\geq2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [434, 617, 487, 630], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 37}, {"bbox": [125, 628, 484, 642], "spans": [{"bbox": [125, 628, 165, 642], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [166, 629, 236, 641], "score": 0.94, "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [236, 628, 273, 642], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [273, 631, 312, 640], "score": 0.73, "content": "\\pi,\\lambda\\,\\equiv\\,1", "type": "inline_equation", "height": 9, "width": 39}, {"bbox": [313, 628, 440, 642], "score": 1.0, "content": " mod 4 are prime factors of ", "type": "text"}, {"bbox": [440, 631, 450, 640], "score": 0.91, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [450, 628, 474, 642], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [475, 631, 484, 640], "score": 0.91, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 9}], "index": 38}, {"bbox": [126, 642, 486, 653], "spans": [{"bbox": [126, 642, 138, 653], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 642, 184, 653], "score": 0.92, "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [184, 642, 326, 653], "score": 1.0, "content": ", respectively. This shows that ", "type": "text"}, {"bbox": [327, 643, 340, 652], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [340, 642, 486, 653], "score": 1.0, "content": " has even class number, because", "type": "text"}], "index": 39}, {"bbox": [126, 653, 319, 665], "spans": [{"bbox": [126, 654, 181, 665], "score": 0.93, "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [181, 653, 319, 665], "score": 1.0, "content": " is easily seen to be unramified.", "type": "text"}], "index": 40}], "index": 36.5}, {"type": "text", "bbox": [125, 663, 485, 687], "lines": [{"bbox": [136, 662, 486, 679], "spans": [{"bbox": [136, 662, 162, 679], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [163, 666, 235, 677], "score": 0.91, "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [235, 662, 325, 679], "score": 1.0, "content": ", and so we find that ", "type": "text"}, {"bbox": [325, 666, 350, 677], "score": 0.94, "content": "h_{2}(L)", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [350, 662, 413, 679], "score": 1.0, "content": " is divisible by", "type": "text"}, {"bbox": [414, 666, 482, 674], "score": 0.92, "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "type": "inline_equation", "height": 8, "width": 68}, {"bbox": [482, 662, 486, 679], "score": 1.0, "content": ".", "type": "text"}], "index": 41}, {"bbox": [126, 678, 354, 689], "spans": [{"bbox": [126, 678, 256, 689], "score": 1.0, "content": "In particular, we always have ", "type": "text"}, {"bbox": [257, 678, 299, 689], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [299, 678, 354, 689], "score": 1.0, "content": " in this case.", "type": "text"}], "index": 42}], "index": 41.5}, {"type": "text", "bbox": [137, 688, 249, 699], "lines": [{"bbox": [137, 689, 249, 700], "spans": [{"bbox": [137, 689, 249, 700], "score": 1.0, "content": "This concludes the proof.", "type": "text"}], "index": 43}], "index": 43}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [476, 687, 486, 698], "lines": [{"bbox": [476, 690, 486, 700], "spans": [{"bbox": [476, 690, 486, 700], "score": 0.9837851524353027, "content": "口", "type": "text"}]}]}, {"type": "discarded", "bbox": [125, 91, 135, 99], "lines": [{"bbox": [125, 92, 136, 102], "spans": [{"bbox": [125, 92, 136, 102], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "table", "bbox": [228, 110, 383, 167], "blocks": [{"type": "table_body", "bbox": [228, 110, 383, 167], "group_id": 0, "lines": [{"bbox": [228, 110, 383, 167], "spans": [{"bbox": [228, 110, 383, 167], "score": 0.961, "html": "<html><body><table><tr><td>d(G')</td><td>h2(L)</td><td>h2(K1)h2(K2)</td></tr><tr><td>1</td><td>2m</td><td>2</td></tr><tr><td>2</td><td>2m+1</td><td>4</td></tr><tr><td>>3</td><td>≥2m+2</td><td>M8</td></tr></table></body></html>", "type": "table", "image_path": "295f4e323014a6c679e39dff1a665cb1d71c61c9d44701d37ba45e8e808d410a.jpg"}]}], "index": 2, "virtual_lines": [{"bbox": [228, 110, 383, 123], "spans": [], "index": 0}, {"bbox": [228, 123, 383, 136], "spans": [], "index": 1}, {"bbox": [228, 136, 383, 149], "spans": [], "index": 2}, {"bbox": [228, 149, 383, 162], "spans": [], "index": 3}, {"bbox": [228, 162, 383, 175], "spans": [], "index": 4}]}], "index": 2}, {"type": "text", "bbox": [124, 187, 486, 248], "lines": [{"bbox": [125, 189, 486, 201], "spans": [{"bbox": [125, 189, 486, 201], "score": 1.0, "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "type": "text"}], "index": 5}, {"bbox": [125, 201, 487, 213], "spans": [{"bbox": [125, 201, 194, 213], "score": 1.0, "content": "claim to hold if ", "type": "text"}, {"bbox": [194, 203, 230, 212], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [230, 201, 487, 213], "score": 1.0, "content": " in case B)) is a consequence of the class number formula of", "type": "text"}], "index": 6}, {"bbox": [126, 213, 485, 225], "spans": [{"bbox": [126, 213, 290, 225], "score": 1.0, "content": "Proposition 9. In particular, we have ", "type": "text"}, {"bbox": [290, 214, 332, 225], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [332, 213, 453, 225], "score": 1.0, "content": " if one of the class numbers ", "type": "text"}, {"bbox": [454, 214, 485, 225], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 31}], "index": 7}, {"bbox": [126, 225, 486, 237], "spans": [{"bbox": [126, 225, 138, 237], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [138, 226, 169, 237], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [170, 225, 415, 237], "score": 1.0, "content": " is at least 8. Therefore it suffices to examine the cases ", "type": "text"}, {"bbox": [415, 226, 465, 237], "score": 0.94, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [466, 225, 486, 237], "score": 1.0, "content": " and", "type": "text"}], "index": 8}, {"bbox": [126, 237, 385, 250], "spans": [{"bbox": [126, 238, 175, 249], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 237, 282, 250], "score": 1.0, "content": " (recall from above that ", "type": "text"}, {"bbox": [282, 238, 313, 249], "score": 0.94, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [313, 237, 385, 250], "score": 1.0, "content": " is always even).", "type": "text"}], "index": 9}], "index": 7, "bbox_fs": [125, 189, 487, 250]}, {"type": "text", "bbox": [124, 263, 485, 298], "lines": [{"bbox": [137, 263, 486, 278], "spans": [{"bbox": [137, 263, 401, 278], "score": 1.0, "content": "We start by considering case A); it is sufficient to show that ", "type": "text"}, {"bbox": [402, 266, 482, 276], "score": 0.94, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [483, 263, 486, 278], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 275, 486, 290], "spans": [{"bbox": [125, 275, 486, 290], "score": 1.0, "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "type": "text"}], "index": 11}, {"bbox": [126, 290, 135, 299], "spans": [{"bbox": [126, 290, 135, 299], "score": 1.0, "content": "8.", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [125, 263, 486, 299]}, {"type": "text", "bbox": [124, 299, 486, 334], "lines": [{"bbox": [125, 300, 487, 313], "spans": [{"bbox": [125, 300, 149, 313], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [150, 302, 202, 312], "score": 0.92, "content": "h_{2}(K_{2})\\,=\\,2", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [202, 300, 232, 313], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [232, 302, 272, 311], "score": 0.92, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [272, 300, 385, 313], "score": 1.0, "content": " by Proposition 5, hence ", "type": "text"}, {"bbox": [385, 303, 416, 312], "score": 0.92, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [416, 300, 487, 313], "score": 1.0, "content": " by Proposition", "type": "text"}], "index": 13}, {"bbox": [124, 311, 487, 326], "spans": [{"bbox": [124, 311, 178, 326], "score": 1.0, "content": "7 and then ", "type": "text"}, {"bbox": [178, 315, 208, 323], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [209, 311, 487, 326], "score": 1.0, "content": " by Proposition 6. The class number formulas in the proof of", "type": "text"}], "index": 14}, {"bbox": [125, 324, 356, 336], "spans": [{"bbox": [125, 324, 228, 336], "score": 1.0, "content": "Proposition 9 now give", "type": "text"}, {"bbox": [229, 325, 279, 336], "score": 0.94, "content": "h_{2}(K_{1})=1", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [279, 324, 300, 336], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [301, 325, 352, 336], "score": 0.94, "content": "h_{2}(L)=2^{m}", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [352, 324, 356, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14, "bbox_fs": [124, 300, 487, 336]}, {"type": "text", "bbox": [125, 335, 486, 382], "lines": [{"bbox": [137, 336, 486, 348], "spans": [{"bbox": [137, 336, 411, 348], "score": 1.0, "content": "It can be shown using the ambiguous class number formula that", "type": "text"}, {"bbox": [412, 338, 447, 348], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [447, 336, 486, 348], "score": 1.0, "content": " is trivial", "type": "text"}], "index": 16}, {"bbox": [125, 348, 485, 361], "spans": [{"bbox": [125, 348, 184, 361], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [185, 353, 194, 359], "score": 0.89, "content": "\\varepsilon_{1}", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [194, 348, 434, 361], "score": 1.0, "content": " is a quadratic nonresidue modulo the prime ideal over ", "type": "text"}, {"bbox": [434, 350, 444, 359], "score": 0.92, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [444, 348, 458, 361], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [459, 350, 468, 359], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [469, 348, 485, 361], "score": 1.0, "content": "; by", "type": "text"}], "index": 17}, {"bbox": [126, 361, 486, 373], "spans": [{"bbox": [126, 361, 322, 373], "score": 1.0, "content": "Scholz’s reciprocity law, this is equivalent to ", "type": "text"}, {"bbox": [322, 361, 414, 372], "score": 0.93, "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "type": "inline_equation", "height": 11, "width": 92}, {"bbox": [414, 361, 486, 373], "score": 1.0, "content": ", and this agrees", "type": "text"}], "index": 18}, {"bbox": [126, 372, 258, 384], "spans": [{"bbox": [126, 372, 258, 384], "score": 1.0, "content": "with the criterion given in [1].", "type": "text"}], "index": 19}], "index": 17.5, "bbox_fs": [125, 336, 486, 384]}, {"type": "text", "bbox": [125, 382, 485, 418], "lines": [{"bbox": [125, 383, 486, 397], "spans": [{"bbox": [125, 383, 148, 397], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [149, 385, 198, 396], "score": 0.93, "content": "h_{2}(K_{2})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [199, 383, 295, 397], "score": 1.0, "content": ", we may assume that", "type": "text"}, {"bbox": [296, 385, 358, 396], "score": 0.94, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [358, 383, 486, 397], "score": 1.0, "content": " from Proposition 8.b). Then", "type": "text"}], "index": 20}, {"bbox": [126, 397, 486, 409], "spans": [{"bbox": [126, 398, 164, 407], "score": 0.92, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [164, 397, 247, 409], "score": 1.0, "content": " by Proposition 5, ", "type": "text"}, {"bbox": [247, 398, 276, 407], "score": 0.93, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [276, 397, 376, 409], "score": 1.0, "content": " by Proposition 7 and ", "type": "text"}, {"bbox": [376, 398, 405, 407], "score": 0.92, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [406, 397, 486, 409], "score": 1.0, "content": " by Proposition 6.", "type": "text"}], "index": 21}, {"bbox": [124, 407, 435, 421], "spans": [{"bbox": [124, 407, 298, 421], "score": 1.0, "content": "Using the class number formula we get ", "type": "text"}, {"bbox": [299, 409, 348, 420], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [348, 407, 370, 421], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 408, 431, 420], "score": 0.93, "content": "h_{2}(L)=2^{m+2}", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [432, 407, 435, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21, "bbox_fs": [124, 383, 486, 421]}, {"type": "text", "bbox": [124, 418, 486, 442], "lines": [{"bbox": [137, 419, 486, 433], "spans": [{"bbox": [137, 419, 256, 433], "score": 1.0, "content": "Thus in both cases we have", "type": "text"}, {"bbox": [257, 421, 337, 432], "score": 0.93, "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "type": "inline_equation", "height": 11, "width": 80}, {"bbox": [337, 419, 486, 433], "score": 1.0, "content": ", and by the table at the beginning", "type": "text"}], "index": 23}, {"bbox": [126, 432, 380, 444], "spans": [{"bbox": [126, 432, 278, 444], "score": 1.0, "content": "of this proof this implies that rank", "type": "text"}, {"bbox": [279, 433, 329, 443], "score": 0.89, "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [330, 432, 380, 444], "score": 1.0, "content": " in case A).", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [126, 419, 486, 444]}, {"type": "text", "bbox": [127, 457, 486, 481], "lines": [{"bbox": [136, 459, 487, 472], "spans": [{"bbox": [136, 459, 421, 472], "score": 1.0, "content": "Next we consider case B); here we have to distinguish between ", "type": "text"}, {"bbox": [421, 461, 460, 471], "score": 0.93, "content": "d_{3}\\neq-4", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [460, 459, 487, 472], "score": 1.0, "content": " (case", "type": "text"}], "index": 25}, {"bbox": [126, 471, 248, 484], "spans": [{"bbox": [126, 473, 138, 482], "score": 0.82, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [138, 471, 164, 484], "score": 1.0, "content": ") and ", "type": "text"}, {"bbox": [164, 473, 201, 482], "score": 0.92, "content": "d_{3}=-4", "type": "inline_equation", "height": 9, "width": 37}, {"bbox": [201, 471, 228, 484], "score": 1.0, "content": " (case ", "type": "text"}, {"bbox": [228, 473, 241, 482], "score": 0.88, "content": "B_{2}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [241, 471, 248, 484], "score": 1.0, "content": ").", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [126, 459, 487, 484]}, {"type": "text", "bbox": [124, 484, 487, 553], "lines": [{"bbox": [137, 483, 256, 496], "spans": [{"bbox": [137, 483, 236, 496], "score": 1.0, "content": "Let us start with case ", "type": "text"}, {"bbox": [236, 485, 248, 494], "score": 0.9, "content": "B_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [249, 483, 256, 496], "score": 1.0, "content": ").", "type": "text"}], "index": 27}, {"bbox": [126, 495, 486, 509], "spans": [{"bbox": [126, 495, 149, 509], "score": 1.0, "content": "a) If ", "type": "text"}, {"bbox": [149, 496, 200, 507], "score": 0.93, "content": "h_{2}(K_{2})=2", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [200, 495, 230, 509], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [230, 497, 268, 506], "score": 0.91, "content": "\\#\\kappa_{2}=2", "type": "inline_equation", "height": 9, "width": 38}, {"bbox": [269, 495, 275, 509], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [275, 497, 304, 506], "score": 0.91, "content": "q_{2}=2", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [304, 495, 327, 509], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [328, 497, 357, 506], "score": 0.93, "content": "q_{1}=1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [357, 495, 486, 509], "score": 1.0, "content": " as above. The class number", "type": "text"}], "index": 28}, {"bbox": [123, 507, 324, 520], "spans": [{"bbox": [123, 507, 187, 520], "score": 1.0, "content": "formula gives ", "type": "text"}, {"bbox": [187, 509, 236, 519], "score": 0.94, "content": "h_{2}(K_{1})=2", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [236, 507, 258, 520], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [259, 508, 320, 519], "score": 0.93, "content": "h_{2}(L)=2^{m+1}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [320, 507, 324, 520], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 518, 486, 533], "spans": [{"bbox": [125, 518, 147, 533], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [147, 520, 208, 531], "score": 0.92, "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [209, 518, 486, 533], "score": 1.0, "content": " (which we may assume without loss of generality by Proposition", "type": "text"}], "index": 30}, {"bbox": [125, 531, 485, 544], "spans": [{"bbox": [125, 531, 175, 544], "score": 1.0, "content": "8.b)) then ", "type": "text"}, {"bbox": [175, 533, 216, 542], "score": 0.91, "content": "\\#\\kappa_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [217, 531, 223, 544], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [223, 533, 254, 542], "score": 0.91, "content": "q_{2}\\,=\\,2", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [255, 531, 279, 544], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [280, 533, 311, 542], "score": 0.93, "content": "q_{1}\\,=\\,1", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [311, 531, 485, 544], "score": 1.0, "content": ", again exactly as above. This implies", "type": "text"}], "index": 31}, {"bbox": [126, 542, 263, 556], "spans": [{"bbox": [126, 544, 175, 555], "score": 0.94, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [176, 542, 198, 556], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 544, 259, 555], "score": 0.92, "content": "h_{2}(L)=2^{m+3}", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [259, 542, 263, 556], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 29.5, "bbox_fs": [123, 483, 486, 556]}, {"type": "text", "bbox": [125, 564, 486, 663], "lines": [{"bbox": [125, 566, 487, 580], "spans": [{"bbox": [125, 566, 377, 580], "score": 1.0, "content": "Here we apply Kuroda’s class number formula (see [10]) to ", "type": "text"}, {"bbox": [378, 568, 399, 579], "score": 0.92, "content": "L/k_{1}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [399, 566, 446, 580], "score": 1.0, "content": ", and since ", "type": "text"}, {"bbox": [447, 568, 487, 579], "score": 0.91, "content": "h_{2}(k_{1})=", "type": "inline_equation", "height": 11, "width": 40}], "index": 33}, {"bbox": [126, 579, 486, 592], "spans": [{"bbox": [126, 581, 131, 588], "score": 0.43, "content": "^{1}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [132, 579, 154, 592], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [154, 580, 231, 591], "score": 0.93, "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "type": "inline_equation", "height": 11, "width": 77}, {"bbox": [231, 579, 268, 592], "score": 1.0, "content": ", we get ", "type": "text"}, {"bbox": [269, 579, 453, 591], "score": 0.93, "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "type": "inline_equation", "height": 12, "width": 184}, {"bbox": [454, 579, 486, 592], "score": 1.0, "content": ". From", "type": "text"}], "index": 34}, {"bbox": [126, 592, 485, 605], "spans": [{"bbox": [126, 594, 185, 605], "score": 0.94, "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "type": "inline_equation", "height": 11, "width": 59}, {"bbox": [185, 594, 294, 605], "score": 1.0, "content": " (for a suitable choice of ", "type": "text"}, {"bbox": [295, 595, 302, 602], "score": 0.9, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [302, 594, 409, 605], "score": 1.0, "content": "; the other possibility is", "type": "text"}, {"bbox": [409, 592, 482, 605], "score": 0.92, "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [482, 594, 485, 605], "score": 1.0, "content": ",", "type": "text"}], "index": 35}, {"bbox": [126, 604, 487, 617], "spans": [{"bbox": [126, 604, 155, 617], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [155, 609, 160, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [161, 604, 286, 617], "score": 1.0, "content": " is the fundamental unit of ", "type": "text"}, {"bbox": [286, 607, 297, 615], "score": 0.9, "content": "k_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [297, 604, 416, 617], "score": 1.0, "content": ", we deduce that the uni t ", "type": "text"}, {"bbox": [416, 609, 421, 614], "score": 0.88, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [421, 604, 487, 617], "score": 1.0, "content": ", which still is", "type": "text"}], "index": 36}, {"bbox": [124, 617, 487, 630], "spans": [{"bbox": [124, 617, 195, 630], "score": 1.0, "content": "fundamental in ", "type": "text"}, {"bbox": [195, 619, 201, 627], "score": 0.88, "content": "k", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [201, 617, 298, 630], "score": 1.0, "content": ", becomes a square in ", "type": "text"}, {"bbox": [299, 619, 306, 626], "score": 0.88, "content": "L", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [306, 617, 406, 630], "score": 1.0, "content": ", and this implies that ", "type": "text"}, {"bbox": [406, 619, 434, 628], "score": 0.91, "content": "q_{1}\\geq2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [434, 617, 487, 630], "score": 1.0, "content": ". Moreover,", "type": "text"}], "index": 37}, {"bbox": [125, 628, 484, 642], "spans": [{"bbox": [125, 628, 165, 642], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [166, 629, 236, 641], "score": 0.94, "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [236, 628, 273, 642], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [273, 631, 312, 640], "score": 0.73, "content": "\\pi,\\lambda\\,\\equiv\\,1", "type": "inline_equation", "height": 9, "width": 39}, {"bbox": [313, 628, 440, 642], "score": 1.0, "content": " mod 4 are prime factors of ", "type": "text"}, {"bbox": [440, 631, 450, 640], "score": 0.91, "content": "d_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [450, 628, 474, 642], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [475, 631, 484, 640], "score": 0.91, "content": "d_{2}", "type": "inline_equation", "height": 9, "width": 9}], "index": 38}, {"bbox": [126, 642, 486, 653], "spans": [{"bbox": [126, 642, 138, 653], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [138, 642, 184, 653], "score": 0.92, "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [184, 642, 326, 653], "score": 1.0, "content": ", respectively. This shows that ", "type": "text"}, {"bbox": [327, 643, 340, 652], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [340, 642, 486, 653], "score": 1.0, "content": " has even class number, because", "type": "text"}], "index": 39}, {"bbox": [126, 653, 319, 665], "spans": [{"bbox": [126, 654, 181, 665], "score": 0.93, "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [181, 653, 319, 665], "score": 1.0, "content": " is easily seen to be unramified.", "type": "text"}], "index": 40}], "index": 36.5, "bbox_fs": [124, 566, 487, 665]}, {"type": "text", "bbox": [125, 663, 485, 687], "lines": [{"bbox": [136, 662, 486, 679], "spans": [{"bbox": [136, 662, 162, 679], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [163, 666, 235, 677], "score": 0.91, "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 72}, {"bbox": [235, 662, 325, 679], "score": 1.0, "content": ", and so we find that ", "type": "text"}, {"bbox": [325, 666, 350, 677], "score": 0.94, "content": "h_{2}(L)", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [350, 662, 413, 679], "score": 1.0, "content": " is divisible by", "type": "text"}, {"bbox": [414, 666, 482, 674], "score": 0.92, "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "type": "inline_equation", "height": 8, "width": 68}, {"bbox": [482, 662, 486, 679], "score": 1.0, "content": ".", "type": "text"}], "index": 41}, {"bbox": [126, 678, 354, 689], "spans": [{"bbox": [126, 678, 256, 689], "score": 1.0, "content": "In particular, we always have ", "type": "text"}, {"bbox": [257, 678, 299, 689], "score": 0.94, "content": "d(G^{\\prime})\\geq3", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [299, 678, 354, 689], "score": 1.0, "content": " in this case.", "type": "text"}], "index": 42}], "index": 41.5, "bbox_fs": [126, 662, 486, 689]}, {"type": "text", "bbox": [137, 688, 249, 699], "lines": [{"bbox": [137, 689, 249, 700], "spans": [{"bbox": [137, 689, 249, 700], "score": 1.0, "content": "This concludes the proof.", "type": "text"}], "index": 43}], "index": 43, "bbox_fs": [137, 689, 249, 700]}]}
|
[{"type": "table", "bbox": [228, 110, 383, 167], "content": "", "index": 0}, {"type": "text", "bbox": [124, 187, 486, 248], "content": "Here, the first two columns follow from Proposition 1, the last (which we do not claim to hold if in case B)) is a consequence of the class number formula of Proposition 9. In particular, we have if one of the class numbers or is at least 8. Therefore it suffices to examine the cases and (recall from above that is always even).", "index": 1}, {"type": "text", "bbox": [124, 263, 485, 298], "content": "We start by considering case A); it is sufficient to show that . We now apply Proposition 5; notice that we may do so by the proof of Proposition 8.", "index": 2}, {"type": "text", "bbox": [124, 299, 486, 334], "content": "a) If , then by Proposition 5, hence by Proposition 7 and then by Proposition 6. The class number formulas in the proof of Proposition 9 now give and .", "index": 3}, {"type": "text", "bbox": [125, 335, 486, 382], "content": "It can be shown using the ambiguous class number formula that is trivial if and only if is a quadratic nonresidue modulo the prime ideal over in ; by Scholz’s reciprocity law, this is equivalent to , and this agrees with the criterion given in [1].", "index": 4}, {"type": "text", "bbox": [125, 382, 485, 418], "content": "b) If , we may assume that from Proposition 8.b). Then by Proposition 5, by Proposition 7 and by Proposition 6. Using the class number formula we get and .", "index": 5}, {"type": "text", "bbox": [124, 418, 486, 442], "content": "Thus in both cases we have , and by the table at the beginning of this proof this implies that rank in case A).", "index": 6}, {"type": "text", "bbox": [127, 457, 486, 481], "content": "Next we consider case B); here we have to distinguish between (case ) and (case ).", "index": 7}, {"type": "text", "bbox": [124, 484, 487, 553], "content": "Let us start with case ). a) If , then , and as above. The class number formula gives and . b) If (which we may assume without loss of generality by Proposition 8.b)) then , and , again exactly as above. This implies and .", "index": 8}, {"type": "text", "bbox": [125, 564, 486, 663], "content": "Here we apply Kuroda’s class number formula (see [10]) to , and since and , we get . From (for a suitable choice of ; the other possibility is , where is the fundamental unit of , we deduce that the uni t , which still is fundamental in , becomes a square in , and this implies that . Moreover, we have , where mod 4 are prime factors of and in , respectively. This shows that has even class number, because is easily seen to be unramified.", "index": 9}, {"type": "text", "bbox": [125, 663, 485, 687], "content": "Thus , and so we find that is divisible by . In particular, we always have in this case.", "index": 10}, {"type": "text", "bbox": [137, 688, 249, 699], "content": "This concludes the proof.", "index": 11}]
|
[{"bbox": [125, 189, 486, 201], "content": "Here, the first two columns follow from Proposition 1, the last (which we do not", "parent_index": 1, "line_index": 0}, {"bbox": [125, 201, 487, 213], "content": "claim to hold if in case B)) is a consequence of the class number formula of", "parent_index": 1, "line_index": 1}, {"bbox": [126, 213, 485, 225], "content": "Proposition 9. In particular, we have if one of the class numbers", "parent_index": 1, "line_index": 2}, {"bbox": [126, 225, 486, 237], "content": "or is at least 8. Therefore it suffices to examine the cases and", "parent_index": 1, "line_index": 3}, {"bbox": [126, 237, 385, 250], "content": "(recall from above that is always even).", "parent_index": 1, "line_index": 4}, {"bbox": [137, 263, 486, 278], "content": "We start by considering case A); it is sufficient to show that .", "parent_index": 2, "line_index": 0}, {"bbox": [125, 275, 486, 290], "content": "We now apply Proposition 5; notice that we may do so by the proof of Proposition", "parent_index": 2, "line_index": 1}, {"bbox": [126, 290, 135, 299], "content": "8.", "parent_index": 2, "line_index": 2}, {"bbox": [125, 300, 487, 313], "content": "a) If , then by Proposition 5, hence by Proposition", "parent_index": 3, "line_index": 0}, {"bbox": [124, 311, 487, 326], "content": "7 and then by Proposition 6. The class number formulas in the proof of", "parent_index": 3, "line_index": 1}, {"bbox": [125, 324, 356, 336], "content": "Proposition 9 now give and .", "parent_index": 3, "line_index": 2}, {"bbox": [137, 336, 486, 348], "content": "It can be shown using the ambiguous class number formula that is trivial", "parent_index": 4, "line_index": 0}, {"bbox": [125, 348, 485, 361], "content": "if and only if is a quadratic nonresidue modulo the prime ideal over in ; by", "parent_index": 4, "line_index": 1}, {"bbox": [126, 361, 486, 373], "content": "Scholz’s reciprocity law, this is equivalent to , and this agrees", "parent_index": 4, "line_index": 2}, {"bbox": [126, 372, 258, 384], "content": "with the criterion given in [1].", "parent_index": 4, "line_index": 3}, {"bbox": [125, 383, 486, 397], "content": "b) If , we may assume that from Proposition 8.b). Then", "parent_index": 5, "line_index": 0}, {"bbox": [126, 397, 486, 409], "content": "by Proposition 5, by Proposition 7 and by Proposition 6.", "parent_index": 5, "line_index": 1}, {"bbox": [124, 407, 435, 421], "content": "Using the class number formula we get and .", "parent_index": 5, "line_index": 2}, {"bbox": [137, 419, 486, 433], "content": "Thus in both cases we have , and by the table at the beginning", "parent_index": 6, "line_index": 0}, {"bbox": [126, 432, 380, 444], "content": "of this proof this implies that rank in case A).", "parent_index": 6, "line_index": 1}, {"bbox": [136, 459, 487, 472], "content": "Next we consider case B); here we have to distinguish between (case", "parent_index": 7, "line_index": 0}, {"bbox": [126, 471, 248, 484], "content": ") and (case ).", "parent_index": 7, "line_index": 1}, {"bbox": [137, 483, 256, 496], "content": "Let us start with case ).", "parent_index": 8, "line_index": 0}, {"bbox": [126, 495, 486, 509], "content": "a) If , then , and as above. The class number", "parent_index": 8, "line_index": 1}, {"bbox": [123, 507, 324, 520], "content": "formula gives and .", "parent_index": 8, "line_index": 2}, {"bbox": [125, 518, 486, 533], "content": "b) If (which we may assume without loss of generality by Proposition", "parent_index": 8, "line_index": 3}, {"bbox": [125, 531, 485, 544], "content": "8.b)) then , and , again exactly as above. This implies", "parent_index": 8, "line_index": 4}, {"bbox": [126, 542, 263, 556], "content": "and .", "parent_index": 8, "line_index": 5}, {"bbox": [125, 566, 487, 580], "content": "Here we apply Kuroda’s class number formula (see [10]) to , and since", "parent_index": 9, "line_index": 0}, {"bbox": [126, 579, 486, 592], "content": "and , we get . From", "parent_index": 9, "line_index": 1}, {"bbox": [126, 592, 485, 605], "content": "(for a suitable choice of ; the other possibility is ,", "parent_index": 9, "line_index": 2}, {"bbox": [126, 604, 487, 617], "content": "where is the fundamental unit of , we deduce that the uni t , which still is", "parent_index": 9, "line_index": 3}, {"bbox": [124, 617, 487, 630], "content": "fundamental in , becomes a square in , and this implies that . Moreover,", "parent_index": 9, "line_index": 4}, {"bbox": [125, 628, 484, 642], "content": "we have , where mod 4 are prime factors of and", "parent_index": 9, "line_index": 5}, {"bbox": [126, 642, 486, 653], "content": "in , respectively. This shows that has even class number, because", "parent_index": 9, "line_index": 6}, {"bbox": [126, 653, 319, 665], "content": "is easily seen to be unramified.", "parent_index": 9, "line_index": 7}, {"bbox": [136, 662, 486, 679], "content": "Thus , and so we find that is divisible by .", "parent_index": 10, "line_index": 0}, {"bbox": [126, 678, 354, 689], "content": "In particular, we always have in this case.", "parent_index": 10, "line_index": 1}, {"bbox": [137, 689, 249, 700], "content": "This concludes the proof.", "parent_index": 11, "line_index": 0}]
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[]
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[{"bbox": [194, 203, 230, 212], "content": "d_{3}=-4", "parent_index": 1, "subtype": "inline"}, {"bbox": [290, 214, 332, 225], "content": "d(G^{\\prime})\\geq3", "parent_index": 1, "subtype": "inline"}, {"bbox": [454, 214, 485, 225], "content": "h_{2}(K_{1})", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 226, 169, 237], "content": "h_{2}(K_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [415, 226, 465, 237], "content": "h_{2}(K_{2})=2", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 238, 175, 249], "content": "h_{2}(K_{2})=4", "parent_index": 1, "subtype": "inline"}, {"bbox": [282, 238, 313, 249], "content": "h_{2}(K_{2})", "parent_index": 1, "subtype": "inline"}, {"bbox": [402, 266, 482, 276], "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "parent_index": 2, "subtype": "inline"}, {"bbox": [150, 302, 202, 312], "content": "h_{2}(K_{2})\\,=\\,2", "parent_index": 3, "subtype": "inline"}, {"bbox": [232, 302, 272, 311], "content": "\\#\\kappa_{2}\\,=\\,2", "parent_index": 3, "subtype": "inline"}, {"bbox": [385, 303, 416, 312], "content": "q_{2}\\,=\\,2", "parent_index": 3, "subtype": "inline"}, {"bbox": [178, 315, 208, 323], "content": "q_{1}=1", "parent_index": 3, "subtype": "inline"}, {"bbox": [229, 325, 279, 336], "content": "h_{2}(K_{1})=1", "parent_index": 3, "subtype": "inline"}, {"bbox": [301, 325, 352, 336], "content": "h_{2}(L)=2^{m}", "parent_index": 3, "subtype": "inline"}, {"bbox": [412, 338, 447, 348], "content": "\\mathrm{Cl}_{2}(K_{1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [185, 353, 194, 359], "content": "\\varepsilon_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [434, 350, 444, 359], "content": "d_{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [459, 350, 468, 359], "content": "k_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [322, 361, 414, 372], "content": "(d_{1}/d_{2})_{4}(d_{2}/d_{1})_{4}=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [149, 385, 198, 396], "content": "h_{2}(K_{2})=4", "parent_index": 5, "subtype": "inline"}, {"bbox": [296, 385, 358, 396], "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 398, 164, 407], "content": "\\#\\kappa_{2}=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [247, 398, 276, 407], "content": "q_{2}=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [376, 398, 405, 407], "content": "q_{1}=1", "parent_index": 5, "subtype": "inline"}, {"bbox": [299, 409, 348, 420], "content": "h_{2}(K_{1})=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [370, 408, 431, 420], "content": "h_{2}(L)=2^{m+2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [257, 421, 337, 432], "content": "h_{2}(K_{1})h_{2}(K_{2})\\neq4", "parent_index": 6, "subtype": "inline"}, {"bbox": [279, 433, 329, 443], "content": "\\mathrm{Cl}_{2}(k^{1})\\neq2", "parent_index": 6, "subtype": "inline"}, {"bbox": [421, 461, 460, 471], "content": "d_{3}\\neq-4", "parent_index": 7, "subtype": "inline"}, {"bbox": [126, 473, 138, 482], "content": "B_{1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [164, 473, 201, 482], "content": "d_{3}=-4", "parent_index": 7, "subtype": "inline"}, {"bbox": [228, 473, 241, 482], "content": "B_{2}", "parent_index": 7, "subtype": "inline"}, {"bbox": [236, 485, 248, 494], "content": "B_{1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [149, 496, 200, 507], "content": "h_{2}(K_{2})=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [230, 497, 268, 506], "content": "\\#\\kappa_{2}=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [275, 497, 304, 506], "content": "q_{2}=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [328, 497, 357, 506], "content": "q_{1}=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [187, 509, 236, 519], "content": "h_{2}(K_{1})=2", "parent_index": 8, "subtype": "inline"}, {"bbox": [259, 508, 320, 519], "content": "h_{2}(L)=2^{m+1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [147, 520, 208, 531], "content": "\\mathrm{Cl}_{2}(K_{2})=(4)", "parent_index": 8, "subtype": "inline"}, {"bbox": [175, 533, 216, 542], "content": "\\#\\kappa_{2}\\,=\\,2", "parent_index": 8, "subtype": "inline"}, {"bbox": [223, 533, 254, 542], "content": "q_{2}\\,=\\,2", "parent_index": 8, "subtype": "inline"}, {"bbox": [280, 533, 311, 542], "content": "q_{1}\\,=\\,1", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 544, 175, 555], "content": "h_{2}(K_{1})=4", "parent_index": 8, "subtype": "inline"}, {"bbox": [198, 544, 259, 555], "content": "h_{2}(L)=2^{m+3}", "parent_index": 8, "subtype": "inline"}, {"bbox": [378, 568, 399, 579], "content": "L/k_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [447, 568, 487, 579], "content": "h_{2}(k_{1})=", "parent_index": 9, "subtype": "inline"}, {"bbox": [126, 581, 131, 588], "content": "^{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [154, 580, 231, 591], "content": "h_{2}(K_{1})=h_{2}(K_{1}^{\\prime})", "parent_index": 9, "subtype": "inline"}, {"bbox": [269, 579, 453, 591], "content": "\\begin{array}{r}{h_{2}(L)=\\frac{1}{2}q_{1}h_{2}(K_{1})^{2}h_{2}(k)=2^{m}q_{1}h_{2}(K_{1})^{2}}\\end{array}", "parent_index": 9, "subtype": "inline"}, {"bbox": [126, 594, 185, 605], "content": "K_{2}=k_{2}(\\sqrt{\\varepsilon}\\,)", "parent_index": 9, "subtype": "inline"}, {"bbox": [295, 595, 302, 602], "content": "L", "parent_index": 9, "subtype": "inline"}, {"bbox": [409, 592, 482, 605], "content": "{\\tilde{K}}_{2}=k_{2}({\\sqrt{d_{2}\\varepsilon}}\\,))", "parent_index": 9, "subtype": "inline"}, {"bbox": [155, 609, 160, 614], "content": "\\varepsilon", "parent_index": 9, "subtype": "inline"}, {"bbox": [286, 607, 297, 615], "content": "k_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [416, 609, 421, 614], "content": "\\varepsilon", "parent_index": 9, "subtype": "inline"}, {"bbox": [195, 619, 201, 627], "content": "k", "parent_index": 9, "subtype": "inline"}, {"bbox": [299, 619, 306, 626], "content": "L", "parent_index": 9, "subtype": "inline"}, {"bbox": [406, 619, 434, 628], "content": "q_{1}\\geq2", "parent_index": 9, "subtype": "inline"}, {"bbox": [166, 629, 236, 641], "content": "K_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\lambda})", "parent_index": 9, "subtype": "inline"}, {"bbox": [273, 631, 312, 640], "content": "\\pi,\\lambda\\,\\equiv\\,1", "parent_index": 9, "subtype": "inline"}, {"bbox": [440, 631, 450, 640], "content": "d_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [475, 631, 484, 640], "content": "d_{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [138, 642, 184, 653], "content": "k_{1}\\,=\\,\\mathbb{Q}(i)", "parent_index": 9, "subtype": "inline"}, {"bbox": [327, 643, 340, 652], "content": "K_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [126, 654, 181, 665], "content": "K_{1}(\\sqrt{\\pi}\\,)/K_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [163, 666, 235, 677], "content": "2\\mid q_{1},\\,2\\mid h_{2}(K_{1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [325, 666, 350, 677], "content": "h_{2}(L)", "parent_index": 10, "subtype": "inline"}, {"bbox": [414, 666, 482, 674], "content": "2^{m}\\cdot2\\cdot4=2^{m+3}", "parent_index": 10, "subtype": "inline"}, {"bbox": [257, 678, 299, 689], "content": "d(G^{\\prime})\\geq3", "parent_index": 10, "subtype": "inline"}]
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The referee (whom we’d like to thank for a couple of helpful remarks) asked whether $h_{2}(K)=2$ and $h_{2}(K)>2$ infinitely often. Let us show how to prove that both possibilities occur with equal density.
Before we can do this, we have to study the quadratic extensions $K_{1}$ and $\widetilde{K}_{1}$ of $k_{1}$ more closely. We assume that $d_{2}\,=\,p$ and $d_{3}~=~r$ are odd primes in t he following, and then say how to modify the arguments in the case $d_{2}=8$ or $d_{3}=-8$ . The primes $p$ and $r$ split in $k_{1}$ as $p\mathcal{O}_{1}\,=\,\mathfrak{p p}^{\prime}$ and $r\mathcal{O}_{1}\,=\,\mathfrak{r r}^{\prime}$ . Let $h$ denote the odd class number of $k_{1}$ and write ${\mathfrak{p}}^{h}\,=\,(\pi)$ and $\mathfrak{r}^{h}\,=\,(\rho)$ for primary elements $\pi$ and $\rho$ (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields $K$ with odd class number $h$ (see [7]): if ${\mathfrak{a}}^{h}\;=\;\alpha{\mathcal{O}}_{K}$ for an ideal $\mathfrak{p}$ with odd norm, then $\alpha$ can be chosen primary (i.e. congruent to a square mod $4{\cal O}_{K}$ ) if and only if $\mathfrak{a}$ is primary (i.e. $[\varepsilon/{\mathfrak a}]=+1$ for all units $\varepsilon\in{\mathcal{O}}_{K}^{\times}$ , where $[\,\cdot\,/\,\cdot\,]$ denotes the quadratic residue symbol in $K$ )). Let $\big[\cdot\big/\cdot\big]$ denote the quadratic residue symbol in $k_{1}$ . Then $[\pi/\rho][\pi^{\prime}/\rho]=[p/\rho]=(p/r)=-1$ , so we may choose the conjugates in such a way that $[\pi/\rho]=+1$ and $[\pi^{\prime}/\rho]=[\pi/\rho^{\prime}]=-1$ .
Put $K_{1}\;=\;k_{1}(\sqrt{\pi\rho}\,)$ and $\tilde{K}_{1}\,=\,k_{1}(\sqrt{\pi\rho^{\prime}})$ ; we claim that $h_{2}(\tilde{K}_{1})\;=\;2$ . This is equivalent to $h_{2}(\widetilde{L}_{1})\,=\,1$ , where $\tilde{L}_{1}\,=\,k_{1}(\sqrt{\pi},\sqrt{\rho^{\prime}})$ is a quad r atic unramified extension of $\widetilde{K}_{1}$ . Put $\widetilde{F}_{1}=k_{1}(\sqrt{\pi}\,)$ a n d apply the ambiguous class number formula to $\widetilde{F}_{1}/k_{1}$ an d $\widetilde{L}_{1}/\widetilde{F}_{1}$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $\mathrm{Am}(\widetilde{F}_{1}/k_{1})\,=\,\mathrm{Am}(\widetilde{L}_{1}/\widetilde{F}_{1})\,=\,1$ ; note that we have used the assumption that $[\pi/\rho^{\prime}]=-1$ in deducing th a t ${\mathfrak{r}}^{\prime}$ is inert in $\widetilde{F}_{1}/k_{1}$ .
In our proof of Theorem 1 we have seen that there are th e following possibilities when $h_{2}(K_{2})$ | 4:
<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>
In order to decide whether $\widetilde{q}_{2}=1$ or $\widetilde{q}_{2}=2$ , recall that we have $h_{2}(K_{1})=4$ ; thus $\widetilde{K}_{1}$ must be the field with 2 -class nu mber 2, and this implies $h_{2}(\widetilde{L})\,=\,2^{m+2}$ and $\widetilde{q}_{2}=1$ . In particular we see that $4\mid h_{2}(K_{2})$ if and only if $4\mid h_{2}(K_{1})$ as long as $K_{1}=k_{1}(\sqrt{\pi\rho}\,)$ with $[\pi/\rho]=+1$ .
The ambiguous class number formula shows that $\mathrm{Cl}_{2}(K_{1})$ is cyclic, thus 4 | $h_{2}(K_{1})$ if and only if $2\mid h_{2}(L_{1})$ , where $L_{1}=K_{1}(\sqrt{\pi}\,)$ is the quadratic unramified extension of $K_{1}$ . Applying the ambiguous class number formula to $L_{1}/F_{1}$ , where $F_{1}\;=\;k_{1}(\sqrt{\pi}\,)$ , we see that $2\:\:|\:\:h_{2}(L_{1})$ if and only if $(E\,:\,H)\;=\;1$ . Now $E$ is generated by a root of unity (which always is a norm residue at primes dividing $r\,\equiv\,1\;\mathrm{mod}\;4$ ) and a fundamental unit $\varepsilon$ . Therefore $(E\,:\,H)\;=\;1$ if and only if $\{\varepsilon/\mathfrak{R}_{1}\}\,=\,\{\varepsilon/\mathfrak{R}_{2}\}\,=\,+1$ , where $\mathfrak{r}{\mathcal{O}}_{F_{1}}\,=\,\mathfrak{R}_{1}\mathfrak{R}_{2}$ and where $\{\,\cdot\,/\,\cdot\,\}$ denotes the quadratic residue symbol in $F_{1}$ . Since $\{\varepsilon/\mathfrak{R}_{1}\}\{\varepsilon/\mathfrak{R}_{2}\}=[\varepsilon/\mathfrak{r}]=+1$ , we have proved that $4\mid h_{2}(K_{1})$ if and only if the prime ideal $\Re_{1}$ above $\mathfrak{r}$ splits in the quadratic extension $F_{1}(\sqrt{\varepsilon})$ . But if we fix $p$ and $q$ , this happens for exactly half of the values of $r$ satisfying $(p/r)=-1$ , $(q/r)=+1$ .
If $d_{2}=8$ and $p=2$ , then $2\mathcal{O}_{k_{1}}=22^{\prime}$ , and we have to choose $2^{h}=(\pi)$ in such a way that $k_{1}(\sqrt{\pi}\,)/k_{1}$ is unramified outside $\mathfrak{p}$ . The residue symbols $\left[\alpha/2\right]$ are defined as Kronecker symbols via the splitting of $^{2}$ in the quadratic extension $k_{1}(\sqrt{\alpha}\,)/k_{1}$ . With these modifactions, the above arguments remain valid.
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<html><body>
<p data-bbox="125 112 486 148">The referee (whom we’d like to thank for a couple of helpful remarks) asked whether $h_{2}(K)=2$ and $h_{2}(K)>2$ infinitely often. Let us show how to prove that both possibilities occur with equal density. </p>
<p data-bbox="124 149 486 304">Before we can do this, we have to study the quadratic extensions $K_{1}$ and $\widetilde{K}_{1}$ of $k_{1}$ more closely. We assume that $d_{2}\,=\,p$ and $d_{3}~=~r$ are odd primes in t he following, and then say how to modify the arguments in the case $d_{2}=8$ or $d_{3}=-8$ . The primes $p$ and $r$ split in $k_{1}$ as $p\mathcal{O}_{1}\,=\,\mathfrak{p p}^{\prime}$ and $r\mathcal{O}_{1}\,=\,\mathfrak{r r}^{\prime}$ . Let $h$ denote the odd class number of $k_{1}$ and write ${\mathfrak{p}}^{h}\,=\,(\pi)$ and $\mathfrak{r}^{h}\,=\,(\rho)$ for primary elements $\pi$ and $\rho$ (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields $K$ with odd class number $h$ (see [7]): if ${\mathfrak{a}}^{h}\;=\;\alpha{\mathcal{O}}_{K}$ for an ideal $\mathfrak{p}$ with odd norm, then $\alpha$ can be chosen primary (i.e. congruent to a square mod $4{\cal O}_{K}$ ) if and only if $\mathfrak{a}$ is primary (i.e. $[\varepsilon/{\mathfrak a}]=+1$ for all units $\varepsilon\in{\mathcal{O}}_{K}^{\times}$ , where $[\,\cdot\,/\,\cdot\,]$ denotes the quadratic residue symbol in $K$ )). Let $\big[\cdot\big/\cdot\big]$ denote the quadratic residue symbol in $k_{1}$ . Then $[\pi/\rho][\pi^{\prime}/\rho]=[p/\rho]=(p/r)=-1$ , so we may choose the conjugates in such a way that $[\pi/\rho]=+1$ and $[\pi^{\prime}/\rho]=[\pi/\rho^{\prime}]=-1$ . </p>
<p data-bbox="124 306 487 384">Put $K_{1}\;=\;k_{1}(\sqrt{\pi\rho}\,)$ and $\tilde{K}_{1}\,=\,k_{1}(\sqrt{\pi\rho^{\prime}})$ ; we claim that $h_{2}(\tilde{K}_{1})\;=\;2$ . This is equivalent to $h_{2}(\widetilde{L}_{1})\,=\,1$ , where $\tilde{L}_{1}\,=\,k_{1}(\sqrt{\pi},\sqrt{\rho^{\prime}})$ is a quad r atic unramified extension of $\widetilde{K}_{1}$ . Put $\widetilde{F}_{1}=k_{1}(\sqrt{\pi}\,)$ a n d apply the ambiguous class number formula to $\widetilde{F}_{1}/k_{1}$ an d $\widetilde{L}_{1}/\widetilde{F}_{1}$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $\mathrm{Am}(\widetilde{F}_{1}/k_{1})\,=\,\mathrm{Am}(\widetilde{L}_{1}/\widetilde{F}_{1})\,=\,1$ ; note that we have used the assumption that $[\pi/\rho^{\prime}]=-1$ in deducing th a t ${\mathfrak{r}}^{\prime}$ is inert in $\widetilde{F}_{1}/k_{1}$ . </p>
<div class="table" data-bbox="189 412 422 468"><p class="caption" data-bbox="125 385 486 408">In our proof of Theorem 1 we have seen that there are th e following possibilities when $h_{2}(K_{2})$ | 4: </p><table data-bbox="189 412 422 468"><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></div>
<p data-bbox="125 470 486 520">In order to decide whether $\widetilde{q}_{2}=1$ or $\widetilde{q}_{2}=2$ , recall that we have $h_{2}(K_{1})=4$ ; thus $\widetilde{K}_{1}$ must be the field with 2 -class nu mber 2, and this implies $h_{2}(\widetilde{L})\,=\,2^{m+2}$ and $\widetilde{q}_{2}=1$ . In particular we see that $4\mid h_{2}(K_{2})$ if and only if $4\mid h_{2}(K_{1})$ as long as $K_{1}=k_{1}(\sqrt{\pi\rho}\,)$ with $[\pi/\rho]=+1$ . </p>
<p data-bbox="124 520 486 651">The ambiguous class number formula shows that $\mathrm{Cl}_{2}(K_{1})$ is cyclic, thus 4 | $h_{2}(K_{1})$ if and only if $2\mid h_{2}(L_{1})$ , where $L_{1}=K_{1}(\sqrt{\pi}\,)$ is the quadratic unramified extension of $K_{1}$ . Applying the ambiguous class number formula to $L_{1}/F_{1}$ , where $F_{1}\;=\;k_{1}(\sqrt{\pi}\,)$ , we see that $2\:\:|\:\:h_{2}(L_{1})$ if and only if $(E\,:\,H)\;=\;1$ . Now $E$ is generated by a root of unity (which always is a norm residue at primes dividing $r\,\equiv\,1\;\mathrm{mod}\;4$ ) and a fundamental unit $\varepsilon$ . Therefore $(E\,:\,H)\;=\;1$ if and only if $\{\varepsilon/\mathfrak{R}_{1}\}\,=\,\{\varepsilon/\mathfrak{R}_{2}\}\,=\,+1$ , where $\mathfrak{r}{\mathcal{O}}_{F_{1}}\,=\,\mathfrak{R}_{1}\mathfrak{R}_{2}$ and where $\{\,\cdot\,/\,\cdot\,\}$ denotes the quadratic residue symbol in $F_{1}$ . Since $\{\varepsilon/\mathfrak{R}_{1}\}\{\varepsilon/\mathfrak{R}_{2}\}=[\varepsilon/\mathfrak{r}]=+1$ , we have proved that $4\mid h_{2}(K_{1})$ if and only if the prime ideal $\Re_{1}$ above $\mathfrak{r}$ splits in the quadratic extension $F_{1}(\sqrt{\varepsilon})$ . But if we fix $p$ and $q$ , this happens for exactly half of the values of $r$ satisfying $(p/r)=-1$ , $(q/r)=+1$ . </p>
<p data-bbox="125 651 486 699">If $d_{2}=8$ and $p=2$ , then $2\mathcal{O}_{k_{1}}=22^{\prime}$ , and we have to choose $2^{h}=(\pi)$ in such a way that $k_{1}(\sqrt{\pi}\,)/k_{1}$ is unramified outside $\mathfrak{p}$ . The residue symbols $\left[\alpha/2\right]$ are defined as Kronecker symbols via the splitting of $^{2}$ in the quadratic extension $k_{1}(\sqrt{\alpha}\,)/k_{1}$ . With these modifactions, the above arguments remain valid. </p>
</body></html>
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0003244v1
| 12 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "The referee (whom we’d like to thank for a couple of helpful remarks) asked whether $h_{2}(K)=2$ and $h_{2}(K)>2$ infinitely often. Let us show how to prove that both possibilities occur with equal density. ", "page_idx": 12}, {"type": "text", "text": "Before we can do this, we have to study the quadratic extensions $K_{1}$ and $\\widetilde{K}_{1}$ of $k_{1}$ more closely. We assume that $d_{2}\\,=\\,p$ and $d_{3}~=~r$ are odd primes in t he following, and then say how to modify the arguments in the case $d_{2}=8$ or $d_{3}=-8$ . The primes $p$ and $r$ split in $k_{1}$ as $p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}$ and $r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}$ . Let $h$ denote the odd class number of $k_{1}$ and write ${\\mathfrak{p}}^{h}\\,=\\,(\\pi)$ and $\\mathfrak{r}^{h}\\,=\\,(\\rho)$ for primary elements $\\pi$ and $\\rho$ (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields $K$ with odd class number $h$ (see [7]): if ${\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}$ for an ideal $\\mathfrak{p}$ with odd norm, then $\\alpha$ can be chosen primary (i.e. congruent to a square mod $4{\\cal O}_{K}$ ) if and only if $\\mathfrak{a}$ is primary (i.e. $[\\varepsilon/{\\mathfrak a}]=+1$ for all units $\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}$ , where $[\\,\\cdot\\,/\\,\\cdot\\,]$ denotes the quadratic residue symbol in $K$ )). Let $\\big[\\cdot\\big/\\cdot\\big]$ denote the quadratic residue symbol in $k_{1}$ . Then $[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1$ , so we may choose the conjugates in such a way that $[\\pi/\\rho]=+1$ and $[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1$ . ", "page_idx": 12}, {"type": "text", "text": "Put $K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)$ and $\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})$ ; we claim that $h_{2}(\\tilde{K}_{1})\\;=\\;2$ . This is equivalent to $h_{2}(\\widetilde{L}_{1})\\,=\\,1$ , where $\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})$ is a quad r atic unramified extension of $\\widetilde{K}_{1}$ . Put $\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)$ a n d apply the ambiguous class number formula to $\\widetilde{F}_{1}/k_{1}$ an d $\\widetilde{L}_{1}/\\widetilde{F}_{1}$ : since there is only one ramified prime in each of these two ext e nsions, w e fin d $\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1$ ; note that we have used the assumption that $[\\pi/\\rho^{\\prime}]=-1$ in deducing th a t ${\\mathfrak{r}}^{\\prime}$ is inert in $\\widetilde{F}_{1}/k_{1}$ . ", "page_idx": 12}, {"type": "table", "img_path": "images/dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg", "table_caption": ["In our proof of Theorem 1 we have seen that there are th e following possibilities when $h_{2}(K_{2})$ | 4: "], "table_footnote": [], "table_body": "\n\n<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>\n\n", "page_idx": 12}, {"type": "text", "text": "In order to decide whether $\\widetilde{q}_{2}=1$ or $\\widetilde{q}_{2}=2$ , recall that we have $h_{2}(K_{1})=4$ ; thus $\\widetilde{K}_{1}$ must be the field with 2 -class nu mber 2, and this implies $h_{2}(\\widetilde{L})\\,=\\,2^{m+2}$ and $\\widetilde{q}_{2}=1$ . In particular we see that $4\\mid h_{2}(K_{2})$ if and only if $4\\mid h_{2}(K_{1})$ as long as $K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)$ with $[\\pi/\\rho]=+1$ . ", "page_idx": 12}, {"type": "text", "text": "The ambiguous class number formula shows that $\\mathrm{Cl}_{2}(K_{1})$ is cyclic, thus 4 | $h_{2}(K_{1})$ if and only if $2\\mid h_{2}(L_{1})$ , where $L_{1}=K_{1}(\\sqrt{\\pi}\\,)$ is the quadratic unramified extension of $K_{1}$ . Applying the ambiguous class number formula to $L_{1}/F_{1}$ , where $F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)$ , we see that $2\\:\\:|\\:\\:h_{2}(L_{1})$ if and only if $(E\\,:\\,H)\\;=\\;1$ . Now $E$ is generated by a root of unity (which always is a norm residue at primes dividing $r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4$ ) and a fundamental unit $\\varepsilon$ . Therefore $(E\\,:\\,H)\\;=\\;1$ if and only if $\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1$ , where $\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}$ and where $\\{\\,\\cdot\\,/\\,\\cdot\\,\\}$ denotes the quadratic residue symbol in $F_{1}$ . Since $\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1$ , we have proved that $4\\mid h_{2}(K_{1})$ if and only if the prime ideal $\\Re_{1}$ above $\\mathfrak{r}$ splits in the quadratic extension $F_{1}(\\sqrt{\\varepsilon})$ . But if we fix $p$ and $q$ , this happens for exactly half of the values of $r$ satisfying $(p/r)=-1$ , $(q/r)=+1$ . ", "page_idx": 12}, {"type": "text", "text": "If $d_{2}=8$ and $p=2$ , then $2\\mathcal{O}_{k_{1}}=22^{\\prime}$ , and we have to choose $2^{h}=(\\pi)$ in such a way that $k_{1}(\\sqrt{\\pi}\\,)/k_{1}$ is unramified outside $\\mathfrak{p}$ . The residue symbols $\\left[\\alpha/2\\right]$ are defined as Kronecker symbols via the splitting of $^{2}$ in the quadratic extension $k_{1}(\\sqrt{\\alpha}\\,)/k_{1}$ . With these modifactions, the above arguments remain valid. ", "page_idx": 12}]
|
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{"preproc_blocks": [{"type": "text", "bbox": [125, 112, 486, 148], "lines": [{"bbox": [137, 114, 486, 127], "spans": [{"bbox": [137, 114, 486, 127], "score": 1.0, "content": "The referee (whom we’d like to thank for a couple of helpful remarks) asked", "type": "text"}], "index": 0}, {"bbox": [126, 126, 485, 138], "spans": [{"bbox": [126, 126, 164, 138], "score": 1.0, "content": "whether ", "type": "text"}, {"bbox": [164, 127, 209, 138], "score": 0.94, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [210, 126, 232, 138], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 127, 278, 138], "score": 0.94, "content": "h_{2}(K)>2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [278, 126, 485, 138], "score": 1.0, "content": " infinitely often. Let us show how to prove that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 313, 150], "spans": [{"bbox": [126, 138, 313, 150], "score": 1.0, "content": "both possibilities occur with equal density.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [124, 149, 486, 304], "lines": [{"bbox": [137, 150, 484, 163], "spans": [{"bbox": [137, 150, 434, 163], "score": 1.0, "content": "Before we can do this, we have to study the quadratic extensions ", "type": "text"}, {"bbox": [434, 153, 447, 161], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [448, 150, 471, 163], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [471, 150, 484, 161], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}], "index": 3}, {"bbox": [125, 161, 487, 176], "spans": [{"bbox": [125, 161, 138, 176], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 164, 148, 173], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [149, 161, 293, 176], "score": 1.0, "content": " more closely. We assume that ", "type": "text"}, {"bbox": [294, 164, 325, 174], "score": 0.93, "content": "d_{2}\\,=\\,p", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [326, 161, 350, 176], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [350, 164, 382, 173], "score": 0.93, "content": "d_{3}~=~r", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [382, 161, 487, 176], "score": 1.0, "content": " are odd primes in t he", "type": "text"}], "index": 4}, {"bbox": [125, 174, 487, 187], "spans": [{"bbox": [125, 174, 404, 187], "score": 1.0, "content": "following, and then say how to modify the arguments in the case ", "type": "text"}, {"bbox": [404, 176, 432, 185], "score": 0.92, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [433, 174, 446, 187], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [446, 177, 482, 185], "score": 0.93, "content": "d_{3}=-8", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [483, 174, 487, 187], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [126, 187, 487, 199], "spans": [{"bbox": [126, 187, 180, 199], "score": 1.0, "content": "The primes ", "type": "text"}, {"bbox": [181, 191, 186, 198], "score": 0.9, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [186, 187, 210, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 191, 215, 196], "score": 0.88, "content": "r", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [216, 187, 255, 199], "score": 1.0, "content": " split in ", "type": "text"}, {"bbox": [255, 189, 265, 197], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [265, 187, 282, 199], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [282, 188, 330, 198], "score": 0.92, "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [330, 187, 353, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 188, 399, 197], "score": 0.92, "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "type": "inline_equation", "height": 9, "width": 45}, {"bbox": [399, 187, 427, 199], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [427, 189, 433, 196], "score": 0.87, "content": "h", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [434, 187, 487, 199], "score": 1.0, "content": " denote the", "type": "text"}], "index": 6}, {"bbox": [125, 198, 487, 211], "spans": [{"bbox": [125, 198, 221, 211], "score": 1.0, "content": "odd class number of ", "type": "text"}, {"bbox": [221, 200, 231, 209], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [231, 198, 282, 211], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [282, 199, 323, 210], "score": 0.92, "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [324, 198, 348, 211], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 199, 387, 210], "score": 0.92, "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [388, 198, 487, 211], "score": 1.0, "content": " for primary elements", "type": "text"}], "index": 7}, {"bbox": [126, 210, 487, 223], "spans": [{"bbox": [126, 215, 132, 219], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [132, 210, 156, 223], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [157, 215, 162, 222], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [163, 210, 487, 223], "score": 1.0, "content": " (this is can easily be proved directly, but it is also a very special case", "type": "text"}], "index": 8}, {"bbox": [125, 223, 486, 235], "spans": [{"bbox": [125, 223, 432, 235], "score": 1.0, "content": "of Hilbert’s first supplementary law for quadratic reciprocity in fields ", "type": "text"}, {"bbox": [433, 225, 442, 232], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [443, 223, 486, 235], "score": 1.0, "content": " with odd", "type": "text"}], "index": 9}, {"bbox": [125, 234, 485, 247], "spans": [{"bbox": [125, 234, 188, 247], "score": 1.0, "content": "class number ", "type": "text"}, {"bbox": [188, 236, 194, 244], "score": 0.88, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [195, 234, 255, 247], "score": 1.0, "content": " (see [7]): if ", "type": "text"}, {"bbox": [256, 235, 306, 245], "score": 0.92, "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [307, 234, 367, 247], "score": 1.0, "content": " for an ideal ", "type": "text"}, {"bbox": [368, 236, 374, 246], "score": 0.75, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [374, 234, 478, 247], "score": 1.0, "content": " with odd norm, then ", "type": "text"}, {"bbox": [478, 239, 485, 244], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}], "index": 10}, {"bbox": [126, 247, 487, 259], "spans": [{"bbox": [126, 247, 378, 259], "score": 1.0, "content": "can be chosen primary (i.e. congruent to a square mod ", "type": "text"}, {"bbox": [378, 248, 398, 257], "score": 0.89, "content": "4{\\cal O}_{K}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [399, 247, 469, 259], "score": 1.0, "content": ") if and only if ", "type": "text"}, {"bbox": [469, 250, 474, 255], "score": 0.84, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [474, 247, 487, 259], "score": 1.0, "content": " is", "type": "text"}], "index": 11}, {"bbox": [125, 258, 487, 272], "spans": [{"bbox": [125, 258, 186, 272], "score": 1.0, "content": "primary (i.e. ", "type": "text"}, {"bbox": [187, 259, 234, 270], "score": 0.93, "content": "[\\varepsilon/{\\mathfrak a}]=+1", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [235, 258, 293, 272], "score": 1.0, "content": " for all units ", "type": "text"}, {"bbox": [293, 259, 327, 271], "score": 0.93, "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [327, 258, 362, 272], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [363, 259, 386, 270], "score": 0.95, "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [386, 258, 487, 272], "score": 1.0, "content": " denotes the quadratic", "type": "text"}], "index": 12}, {"bbox": [126, 271, 486, 283], "spans": [{"bbox": [126, 271, 205, 283], "score": 1.0, "content": "residue symbol in ", "type": "text"}, {"bbox": [205, 272, 215, 280], "score": 0.86, "content": "K", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [216, 271, 246, 283], "score": 1.0, "content": ")). Let ", "type": "text"}, {"bbox": [247, 272, 270, 282], "score": 0.94, "content": "\\big[\\cdot\\big/\\cdot\\big]", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [270, 271, 445, 283], "score": 1.0, "content": " denote the quadratic residue symbol in ", "type": "text"}, {"bbox": [445, 272, 455, 281], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [455, 271, 486, 283], "score": 1.0, "content": ". Then", "type": "text"}], "index": 13}, {"bbox": [126, 282, 485, 295], "spans": [{"bbox": [126, 284, 271, 294], "score": 0.9, "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "type": "inline_equation", "height": 10, "width": 145}, {"bbox": [271, 282, 485, 295], "score": 1.0, "content": ", so we may choose the conjugates in such a way", "type": "text"}], "index": 14}, {"bbox": [125, 294, 310, 307], "spans": [{"bbox": [125, 294, 147, 307], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 295, 195, 306], "score": 0.93, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [196, 294, 218, 307], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [218, 295, 307, 306], "score": 0.91, "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [307, 294, 310, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 9}, {"type": "text", "bbox": [124, 306, 487, 384], "lines": [{"bbox": [136, 307, 486, 320], "spans": [{"bbox": [136, 307, 158, 320], "score": 1.0, "content": "Put ", "type": "text"}, {"bbox": [159, 309, 228, 320], "score": 0.93, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [229, 307, 253, 320], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [253, 307, 326, 319], "score": 0.91, "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [326, 307, 400, 320], "score": 1.0, "content": "; we claim that ", "type": "text"}, {"bbox": [400, 307, 454, 319], "score": 0.94, "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [455, 307, 486, 320], "score": 1.0, "content": ". This", "type": "text"}], "index": 16}, {"bbox": [125, 321, 486, 334], "spans": [{"bbox": [125, 321, 198, 334], "score": 1.0, "content": "is equivalent to ", "type": "text"}, {"bbox": [198, 321, 249, 333], "score": 0.92, "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [250, 321, 286, 334], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [286, 321, 368, 333], "score": 0.93, "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [369, 321, 486, 334], "score": 1.0, "content": " is a quad r atic unramified", "type": "text"}], "index": 17}, {"bbox": [126, 333, 486, 349], "spans": [{"bbox": [126, 333, 180, 349], "score": 1.0, "content": "extension of", "type": "text"}, {"bbox": [180, 334, 194, 345], "score": 0.9, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [194, 333, 219, 349], "score": 1.0, "content": ". Put ", "type": "text"}, {"bbox": [219, 334, 277, 347], "score": 0.93, "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [278, 333, 486, 349], "score": 1.0, "content": " a n d apply the ambiguous class number formula", "type": "text"}], "index": 18}, {"bbox": [125, 347, 486, 361], "spans": [{"bbox": [125, 347, 138, 361], "score": 1.0, "content": "to", "type": "text"}, {"bbox": [138, 347, 164, 360], "score": 0.95, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [165, 347, 188, 361], "score": 1.0, "content": " an d ", "type": "text"}, {"bbox": [188, 347, 216, 360], "score": 0.94, "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [216, 347, 486, 361], "score": 1.0, "content": ": since there is only one ramified prime in each of these two", "type": "text"}], "index": 19}, {"bbox": [125, 360, 486, 373], "spans": [{"bbox": [125, 361, 214, 373], "score": 1.0, "content": "ext e nsions, w e fin d ", "type": "text"}, {"bbox": [215, 360, 353, 373], "score": 0.94, "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "type": "inline_equation", "height": 13, "width": 138}, {"bbox": [354, 361, 486, 373], "score": 1.0, "content": "; note that we have used the", "type": "text"}], "index": 20}, {"bbox": [126, 374, 415, 386], "spans": [{"bbox": [126, 374, 200, 386], "score": 1.0, "content": "assumption that ", "type": "text"}, {"bbox": [200, 375, 251, 386], "score": 0.94, "content": "[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [251, 374, 329, 386], "score": 1.0, "content": " in deducing th a t ", "type": "text"}, {"bbox": [329, 375, 336, 384], "score": 0.88, "content": "{\\mathfrak{r}}^{\\prime}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [336, 374, 384, 386], "score": 1.0, "content": " is inert in", "type": "text"}, {"bbox": [385, 374, 410, 386], "score": 0.94, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [411, 374, 415, 386], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18.5}, {"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " | 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}, {"type": "text", "bbox": [125, 470, 486, 520], "lines": [{"bbox": [124, 471, 486, 486], "spans": [{"bbox": [124, 471, 245, 486], "score": 1.0, "content": "In order to decide whether", "type": "text"}, {"bbox": [246, 474, 274, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [274, 471, 289, 486], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [289, 474, 317, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=2", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [318, 471, 410, 486], "score": 1.0, "content": ", recall that we have ", "type": "text"}, {"bbox": [410, 474, 460, 484], "score": 0.93, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [460, 471, 486, 486], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [126, 483, 487, 498], "spans": [{"bbox": [126, 485, 139, 496], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [139, 483, 401, 498], "score": 1.0, "content": " must be the field with 2 -class nu mber 2, and this implies ", "type": "text"}, {"bbox": [402, 484, 465, 497], "score": 0.92, "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [465, 483, 487, 498], "score": 1.0, "content": " and", "type": "text"}], "index": 30}, {"bbox": [126, 497, 487, 511], "spans": [{"bbox": [126, 499, 155, 509], "score": 0.92, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [156, 497, 278, 511], "score": 1.0, "content": ". In particular we see that ", "type": "text"}, {"bbox": [278, 499, 325, 509], "score": 0.91, "content": "4\\mid h_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [325, 497, 392, 511], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [392, 499, 437, 509], "score": 0.86, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [437, 497, 487, 511], "score": 1.0, "content": " as long as", "type": "text"}], "index": 31}, {"bbox": [126, 510, 268, 522], "spans": [{"bbox": [126, 511, 191, 522], "score": 0.94, "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [191, 510, 216, 522], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [217, 511, 265, 521], "score": 0.94, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [265, 510, 268, 522], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 30.5}, {"type": "text", "bbox": [124, 520, 486, 651], "lines": [{"bbox": [137, 521, 487, 534], "spans": [{"bbox": [137, 521, 365, 534], "score": 1.0, "content": "The ambiguous class number formula shows that ", "type": "text"}, {"bbox": [365, 523, 400, 533], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [400, 521, 487, 534], "score": 1.0, "content": " is cyclic, thus 4 |", "type": "text"}], "index": 33}, {"bbox": [126, 534, 486, 545], "spans": [{"bbox": [126, 535, 157, 545], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 534, 221, 545], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [221, 535, 265, 545], "score": 0.93, "content": "2\\mid h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [265, 534, 299, 545], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [300, 534, 362, 545], "score": 0.94, "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [362, 534, 486, 545], "score": 1.0, "content": " is the quadratic unramified", "type": "text"}], "index": 34}, {"bbox": [125, 545, 487, 558], "spans": [{"bbox": [125, 545, 182, 558], "score": 1.0, "content": "extension of ", "type": "text"}, {"bbox": [182, 547, 195, 556], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [196, 545, 425, 558], "score": 1.0, "content": ". Applying the ambiguous class number formula to ", "type": "text"}, {"bbox": [425, 547, 452, 557], "score": 0.95, "content": "L_{1}/F_{1}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [453, 545, 487, 558], "score": 1.0, "content": ", where", "type": "text"}], "index": 35}, {"bbox": [126, 558, 487, 570], "spans": [{"bbox": [126, 559, 188, 569], "score": 0.95, "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [189, 558, 254, 570], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [254, 559, 300, 569], "score": 0.89, "content": "2\\:\\:|\\:\\:h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [300, 558, 369, 570], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [369, 559, 430, 569], "score": 0.92, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [430, 558, 465, 570], "score": 1.0, "content": ". Now ", "type": "text"}, {"bbox": [465, 559, 474, 567], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [474, 558, 487, 570], "score": 1.0, "content": " is", "type": "text"}], "index": 36}, {"bbox": [125, 569, 487, 582], "spans": [{"bbox": [125, 569, 487, 582], "score": 1.0, "content": "generated by a root of unity (which always is a norm residue at primes dividing", "type": "text"}], "index": 37}, {"bbox": [126, 582, 485, 593], "spans": [{"bbox": [126, 583, 184, 591], "score": 0.8, "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "type": "inline_equation", "height": 8, "width": 58}, {"bbox": [185, 582, 304, 593], "score": 1.0, "content": ") and a fundamental unit ", "type": "text"}, {"bbox": [304, 586, 310, 591], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [310, 582, 368, 593], "score": 1.0, "content": ". Therefore ", "type": "text"}, {"bbox": [368, 583, 430, 593], "score": 0.91, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [430, 582, 485, 593], "score": 1.0, "content": " if and only", "type": "text"}], "index": 38}, {"bbox": [125, 594, 487, 606], "spans": [{"bbox": [125, 594, 136, 606], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [136, 595, 247, 605], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "type": "inline_equation", "height": 10, "width": 111}, {"bbox": [248, 594, 284, 606], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [284, 595, 348, 605], "score": 0.91, "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [348, 594, 402, 606], "score": 1.0, "content": " and where ", "type": "text"}, {"bbox": [402, 595, 430, 605], "score": 0.93, "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [430, 594, 487, 606], "score": 1.0, "content": " denotes the", "type": "text"}], "index": 39}, {"bbox": [125, 606, 487, 617], "spans": [{"bbox": [125, 606, 247, 617], "score": 1.0, "content": "quadratic residue symbol in ", "type": "text"}, {"bbox": [248, 607, 258, 616], "score": 0.91, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 606, 291, 617], "score": 1.0, "content": ". Since", "type": "text"}, {"bbox": [291, 606, 414, 617], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "type": "inline_equation", "height": 11, "width": 123}, {"bbox": [415, 606, 487, 617], "score": 1.0, "content": ", we have proved", "type": "text"}], "index": 40}, {"bbox": [126, 617, 487, 630], "spans": [{"bbox": [126, 617, 147, 630], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 618, 195, 629], "score": 0.91, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [195, 617, 332, 630], "score": 1.0, "content": " if and only if the prime ideal ", "type": "text"}, {"bbox": [333, 619, 345, 628], "score": 0.91, "content": "\\Re_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [346, 617, 378, 630], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [378, 621, 383, 626], "score": 0.65, "content": "\\mathfrak{r}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [383, 617, 487, 630], "score": 1.0, "content": " splits in the quadratic", "type": "text"}], "index": 41}, {"bbox": [126, 630, 486, 641], "spans": [{"bbox": [126, 630, 169, 641], "score": 1.0, "content": "extension ", "type": "text"}, {"bbox": [170, 630, 203, 641], "score": 0.93, "content": "F_{1}(\\sqrt{\\varepsilon})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [203, 630, 266, 641], "score": 1.0, "content": ". But if we fix ", "type": "text"}, {"bbox": [267, 633, 272, 640], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 630, 293, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 633, 299, 640], "score": 0.88, "content": "q", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [299, 630, 486, 641], "score": 1.0, "content": ", this happens for exactly half of the values", "type": "text"}], "index": 42}, {"bbox": [125, 641, 297, 654], "spans": [{"bbox": [125, 641, 137, 654], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 646, 142, 650], "score": 0.89, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [142, 641, 190, 654], "score": 1.0, "content": " satisfying ", "type": "text"}, {"bbox": [190, 642, 239, 653], "score": 0.92, "content": "(p/r)=-1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [239, 641, 244, 654], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [244, 642, 293, 653], "score": 0.91, "content": "(q/r)=+1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [293, 641, 297, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 38}, {"type": "text", "bbox": [125, 651, 486, 699], "lines": [{"bbox": [136, 652, 487, 666], "spans": [{"bbox": [136, 652, 147, 666], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [148, 655, 176, 664], "score": 0.93, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [176, 652, 198, 666], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 655, 222, 664], "score": 0.9, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [222, 652, 250, 666], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [250, 654, 298, 665], "score": 0.92, "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [299, 652, 405, 666], "score": 1.0, "content": ", and we have to choose ", "type": "text"}, {"bbox": [405, 653, 442, 665], "score": 0.94, "content": "2^{h}=(\\pi)", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [443, 652, 487, 666], "score": 1.0, "content": " in such a", "type": "text"}], "index": 44}, {"bbox": [127, 665, 486, 677], "spans": [{"bbox": [127, 665, 166, 677], "score": 1.0, "content": "way that ", "type": "text"}, {"bbox": [167, 666, 214, 677], "score": 0.94, "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [215, 665, 309, 677], "score": 1.0, "content": " is unramified outside", "type": "text"}, {"bbox": [310, 669, 315, 676], "score": 0.74, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [316, 665, 412, 677], "score": 1.0, "content": ". The residue symbols", "type": "text"}, {"bbox": [413, 666, 435, 677], "score": 0.92, "content": "\\left[\\alpha/2\\right]", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [435, 665, 486, 677], "score": 1.0, "content": " are defined", "type": "text"}], "index": 45}, {"bbox": [125, 677, 486, 689], "spans": [{"bbox": [125, 677, 307, 689], "score": 1.0, "content": "as Kronecker symbols via the splitting of ", "type": "text"}, {"bbox": [308, 681, 313, 686], "score": 0.41, "content": "^{2}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [313, 677, 433, 689], "score": 1.0, "content": " in the quadratic extension ", "type": "text"}, {"bbox": [433, 678, 482, 689], "score": 0.93, "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [482, 677, 486, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 46}, {"bbox": [126, 689, 390, 702], "spans": [{"bbox": [126, 689, 390, 702], "score": 1.0, "content": "With these modifactions, the above arguments remain valid.", "type": "text"}], "index": 47}], "index": 45.5}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [{"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " | 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [238, 90, 373, 99], "lines": [{"bbox": [239, 93, 372, 100], "spans": [{"bbox": [239, 93, 372, 100], "score": 1.0, "content": "IMAGINARY QUADRATIC FIELDS", "type": "text"}]}]}, {"type": "discarded", "bbox": [476, 91, 486, 99], "lines": [{"bbox": [476, 92, 487, 101], "spans": [{"bbox": [476, 92, 487, 101], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 112, 486, 148], "lines": [{"bbox": [137, 114, 486, 127], "spans": [{"bbox": [137, 114, 486, 127], "score": 1.0, "content": "The referee (whom we’d like to thank for a couple of helpful remarks) asked", "type": "text"}], "index": 0}, {"bbox": [126, 126, 485, 138], "spans": [{"bbox": [126, 126, 164, 138], "score": 1.0, "content": "whether ", "type": "text"}, {"bbox": [164, 127, 209, 138], "score": 0.94, "content": "h_{2}(K)=2", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [210, 126, 232, 138], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 127, 278, 138], "score": 0.94, "content": "h_{2}(K)>2", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [278, 126, 485, 138], "score": 1.0, "content": " infinitely often. Let us show how to prove that", "type": "text"}], "index": 1}, {"bbox": [126, 138, 313, 150], "spans": [{"bbox": [126, 138, 313, 150], "score": 1.0, "content": "both possibilities occur with equal density.", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [126, 114, 486, 150]}, {"type": "text", "bbox": [124, 149, 486, 304], "lines": [{"bbox": [137, 150, 484, 163], "spans": [{"bbox": [137, 150, 434, 163], "score": 1.0, "content": "Before we can do this, we have to study the quadratic extensions ", "type": "text"}, {"bbox": [434, 153, 447, 161], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [448, 150, 471, 163], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [471, 150, 484, 161], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}], "index": 3}, {"bbox": [125, 161, 487, 176], "spans": [{"bbox": [125, 161, 138, 176], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 164, 148, 173], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [149, 161, 293, 176], "score": 1.0, "content": " more closely. We assume that ", "type": "text"}, {"bbox": [294, 164, 325, 174], "score": 0.93, "content": "d_{2}\\,=\\,p", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [326, 161, 350, 176], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [350, 164, 382, 173], "score": 0.93, "content": "d_{3}~=~r", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [382, 161, 487, 176], "score": 1.0, "content": " are odd primes in t he", "type": "text"}], "index": 4}, {"bbox": [125, 174, 487, 187], "spans": [{"bbox": [125, 174, 404, 187], "score": 1.0, "content": "following, and then say how to modify the arguments in the case ", "type": "text"}, {"bbox": [404, 176, 432, 185], "score": 0.92, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [433, 174, 446, 187], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [446, 177, 482, 185], "score": 0.93, "content": "d_{3}=-8", "type": "inline_equation", "height": 8, "width": 36}, {"bbox": [483, 174, 487, 187], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [126, 187, 487, 199], "spans": [{"bbox": [126, 187, 180, 199], "score": 1.0, "content": "The primes ", "type": "text"}, {"bbox": [181, 191, 186, 198], "score": 0.9, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [186, 187, 210, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 191, 215, 196], "score": 0.88, "content": "r", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [216, 187, 255, 199], "score": 1.0, "content": " split in ", "type": "text"}, {"bbox": [255, 189, 265, 197], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [265, 187, 282, 199], "score": 1.0, "content": " as ", "type": "text"}, {"bbox": [282, 188, 330, 198], "score": 0.92, "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [330, 187, 353, 199], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 188, 399, 197], "score": 0.92, "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "type": "inline_equation", "height": 9, "width": 45}, {"bbox": [399, 187, 427, 199], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [427, 189, 433, 196], "score": 0.87, "content": "h", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [434, 187, 487, 199], "score": 1.0, "content": " denote the", "type": "text"}], "index": 6}, {"bbox": [125, 198, 487, 211], "spans": [{"bbox": [125, 198, 221, 211], "score": 1.0, "content": "odd class number of ", "type": "text"}, {"bbox": [221, 200, 231, 209], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [231, 198, 282, 211], "score": 1.0, "content": " and write ", "type": "text"}, {"bbox": [282, 199, 323, 210], "score": 0.92, "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [324, 198, 348, 211], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [348, 199, 387, 210], "score": 0.92, "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [388, 198, 487, 211], "score": 1.0, "content": " for primary elements", "type": "text"}], "index": 7}, {"bbox": [126, 210, 487, 223], "spans": [{"bbox": [126, 215, 132, 219], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 4, "width": 6}, {"bbox": [132, 210, 156, 223], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [157, 215, 162, 222], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [163, 210, 487, 223], "score": 1.0, "content": " (this is can easily be proved directly, but it is also a very special case", "type": "text"}], "index": 8}, {"bbox": [125, 223, 486, 235], "spans": [{"bbox": [125, 223, 432, 235], "score": 1.0, "content": "of Hilbert’s first supplementary law for quadratic reciprocity in fields ", "type": "text"}, {"bbox": [433, 225, 442, 232], "score": 0.9, "content": "K", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [443, 223, 486, 235], "score": 1.0, "content": " with odd", "type": "text"}], "index": 9}, {"bbox": [125, 234, 485, 247], "spans": [{"bbox": [125, 234, 188, 247], "score": 1.0, "content": "class number ", "type": "text"}, {"bbox": [188, 236, 194, 244], "score": 0.88, "content": "h", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [195, 234, 255, 247], "score": 1.0, "content": " (see [7]): if ", "type": "text"}, {"bbox": [256, 235, 306, 245], "score": 0.92, "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [307, 234, 367, 247], "score": 1.0, "content": " for an ideal ", "type": "text"}, {"bbox": [368, 236, 374, 246], "score": 0.75, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [374, 234, 478, 247], "score": 1.0, "content": " with odd norm, then ", "type": "text"}, {"bbox": [478, 239, 485, 244], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 7}], "index": 10}, {"bbox": [126, 247, 487, 259], "spans": [{"bbox": [126, 247, 378, 259], "score": 1.0, "content": "can be chosen primary (i.e. congruent to a square mod ", "type": "text"}, {"bbox": [378, 248, 398, 257], "score": 0.89, "content": "4{\\cal O}_{K}", "type": "inline_equation", "height": 9, "width": 20}, {"bbox": [399, 247, 469, 259], "score": 1.0, "content": ") if and only if ", "type": "text"}, {"bbox": [469, 250, 474, 255], "score": 0.84, "content": "\\mathfrak{a}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [474, 247, 487, 259], "score": 1.0, "content": " is", "type": "text"}], "index": 11}, {"bbox": [125, 258, 487, 272], "spans": [{"bbox": [125, 258, 186, 272], "score": 1.0, "content": "primary (i.e. ", "type": "text"}, {"bbox": [187, 259, 234, 270], "score": 0.93, "content": "[\\varepsilon/{\\mathfrak a}]=+1", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [235, 258, 293, 272], "score": 1.0, "content": " for all units ", "type": "text"}, {"bbox": [293, 259, 327, 271], "score": 0.93, "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [327, 258, 362, 272], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [363, 259, 386, 270], "score": 0.95, "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [386, 258, 487, 272], "score": 1.0, "content": " denotes the quadratic", "type": "text"}], "index": 12}, {"bbox": [126, 271, 486, 283], "spans": [{"bbox": [126, 271, 205, 283], "score": 1.0, "content": "residue symbol in ", "type": "text"}, {"bbox": [205, 272, 215, 280], "score": 0.86, "content": "K", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [216, 271, 246, 283], "score": 1.0, "content": ")). Let ", "type": "text"}, {"bbox": [247, 272, 270, 282], "score": 0.94, "content": "\\big[\\cdot\\big/\\cdot\\big]", "type": "inline_equation", "height": 10, "width": 23}, {"bbox": [270, 271, 445, 283], "score": 1.0, "content": " denote the quadratic residue symbol in ", "type": "text"}, {"bbox": [445, 272, 455, 281], "score": 0.91, "content": "k_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [455, 271, 486, 283], "score": 1.0, "content": ". Then", "type": "text"}], "index": 13}, {"bbox": [126, 282, 485, 295], "spans": [{"bbox": [126, 284, 271, 294], "score": 0.9, "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "type": "inline_equation", "height": 10, "width": 145}, {"bbox": [271, 282, 485, 295], "score": 1.0, "content": ", so we may choose the conjugates in such a way", "type": "text"}], "index": 14}, {"bbox": [125, 294, 310, 307], "spans": [{"bbox": [125, 294, 147, 307], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [147, 295, 195, 306], "score": 0.93, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [196, 294, 218, 307], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [218, 295, 307, 306], "score": 0.91, "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 89}, {"bbox": [307, 294, 310, 307], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 9, "bbox_fs": [125, 150, 487, 307]}, {"type": "text", "bbox": [124, 306, 487, 384], "lines": [{"bbox": [136, 307, 486, 320], "spans": [{"bbox": [136, 307, 158, 320], "score": 1.0, "content": "Put ", "type": "text"}, {"bbox": [159, 309, 228, 320], "score": 0.93, "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 69}, {"bbox": [229, 307, 253, 320], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [253, 307, 326, 319], "score": 0.91, "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [326, 307, 400, 320], "score": 1.0, "content": "; we claim that ", "type": "text"}, {"bbox": [400, 307, 454, 319], "score": 0.94, "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [455, 307, 486, 320], "score": 1.0, "content": ". This", "type": "text"}], "index": 16}, {"bbox": [125, 321, 486, 334], "spans": [{"bbox": [125, 321, 198, 334], "score": 1.0, "content": "is equivalent to ", "type": "text"}, {"bbox": [198, 321, 249, 333], "score": 0.92, "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [250, 321, 286, 334], "score": 1.0, "content": ", where", "type": "text"}, {"bbox": [286, 321, 368, 333], "score": 0.93, "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [369, 321, 486, 334], "score": 1.0, "content": " is a quad r atic unramified", "type": "text"}], "index": 17}, {"bbox": [126, 333, 486, 349], "spans": [{"bbox": [126, 333, 180, 349], "score": 1.0, "content": "extension of", "type": "text"}, {"bbox": [180, 334, 194, 345], "score": 0.9, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [194, 333, 219, 349], "score": 1.0, "content": ". Put ", "type": "text"}, {"bbox": [219, 334, 277, 347], "score": 0.93, "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [278, 333, 486, 349], "score": 1.0, "content": " a n d apply the ambiguous class number formula", "type": "text"}], "index": 18}, {"bbox": [125, 347, 486, 361], "spans": [{"bbox": [125, 347, 138, 361], "score": 1.0, "content": "to", "type": "text"}, {"bbox": [138, 347, 164, 360], "score": 0.95, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [165, 347, 188, 361], "score": 1.0, "content": " an d ", "type": "text"}, {"bbox": [188, 347, 216, 360], "score": 0.94, "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [216, 347, 486, 361], "score": 1.0, "content": ": since there is only one ramified prime in each of these two", "type": "text"}], "index": 19}, {"bbox": [125, 360, 486, 373], "spans": [{"bbox": [125, 361, 214, 373], "score": 1.0, "content": "ext e nsions, w e fin d ", "type": "text"}, {"bbox": [215, 360, 353, 373], "score": 0.94, "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "type": "inline_equation", "height": 13, "width": 138}, {"bbox": [354, 361, 486, 373], "score": 1.0, "content": "; note that we have used the", "type": "text"}], "index": 20}, {"bbox": [126, 374, 415, 386], "spans": [{"bbox": [126, 374, 200, 386], "score": 1.0, "content": "assumption that ", "type": "text"}, {"bbox": [200, 375, 251, 386], "score": 0.94, "content": "[\\pi/\\rho^{\\prime}]=-1", "type": "inline_equation", "height": 11, "width": 51}, {"bbox": [251, 374, 329, 386], "score": 1.0, "content": " in deducing th a t ", "type": "text"}, {"bbox": [329, 375, 336, 384], "score": 0.88, "content": "{\\mathfrak{r}}^{\\prime}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [336, 374, 384, 386], "score": 1.0, "content": " is inert in", "type": "text"}, {"bbox": [385, 374, 410, 386], "score": 0.94, "content": "\\widetilde{F}_{1}/k_{1}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [411, 374, 415, 386], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 18.5, "bbox_fs": [125, 307, 486, 386]}, {"type": "table", "bbox": [189, 412, 422, 468], "blocks": [{"type": "table_caption", "bbox": [125, 385, 486, 408], "group_id": 0, "lines": [{"bbox": [137, 386, 485, 399], "spans": [{"bbox": [137, 386, 485, 399], "score": 1.0, "content": "In our proof of Theorem 1 we have seen that there are th e following possibilities", "type": "text"}], "index": 22}, {"bbox": [126, 398, 200, 411], "spans": [{"bbox": [126, 398, 151, 411], "score": 1.0, "content": "when", "type": "text"}, {"bbox": [152, 399, 183, 410], "score": 0.76, "content": "h_{2}(K_{2})", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [183, 398, 200, 411], "score": 1.0, "content": " | 4:", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "table_body", "bbox": [189, 412, 422, 468], "group_id": 0, "lines": [{"bbox": [189, 412, 422, 468], "spans": [{"bbox": [189, 412, 422, 468], "score": 0.975, "html": "<html><body><table><tr><td>q2</td><td>Cl2(K2)</td><td>q1</td><td>h2(K1)</td><td>q2</td><td>Cl2(K2)</td><td>h2(L)</td></tr><tr><td>2</td><td>(2)</td><td>1</td><td>2</td><td>2</td><td>(2)</td><td>2m+1</td></tr><tr><td>2</td><td>(4)</td><td>1</td><td>4</td><td>？</td><td>(2,2)</td><td>2m+3</td></tr></table></body></html>", "type": "table", "image_path": "dfdcf3e7e1dbc3d44e3061767f642a3e9778b62e79babc37d43af68dae85a584.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [189, 412, 422, 425], "spans": [], "index": 24}, {"bbox": [189, 425, 422, 438], "spans": [], "index": 25}, {"bbox": [189, 438, 422, 451], "spans": [], "index": 26}, {"bbox": [189, 451, 422, 464], "spans": [], "index": 27}, {"bbox": [189, 464, 422, 477], "spans": [], "index": 28}]}], "index": 24.25}, {"type": "text", "bbox": [125, 470, 486, 520], "lines": [{"bbox": [124, 471, 486, 486], "spans": [{"bbox": [124, 471, 245, 486], "score": 1.0, "content": "In order to decide whether", "type": "text"}, {"bbox": [246, 474, 274, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [274, 471, 289, 486], "score": 1.0, "content": " or", "type": "text"}, {"bbox": [289, 474, 317, 484], "score": 0.93, "content": "\\widetilde{q}_{2}=2", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [318, 471, 410, 486], "score": 1.0, "content": ", recall that we have ", "type": "text"}, {"bbox": [410, 474, 460, 484], "score": 0.93, "content": "h_{2}(K_{1})=4", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [460, 471, 486, 486], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [126, 483, 487, 498], "spans": [{"bbox": [126, 485, 139, 496], "score": 0.92, "content": "\\widetilde{K}_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [139, 483, 401, 498], "score": 1.0, "content": " must be the field with 2 -class nu mber 2, and this implies ", "type": "text"}, {"bbox": [402, 484, 465, 497], "score": 0.92, "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [465, 483, 487, 498], "score": 1.0, "content": " and", "type": "text"}], "index": 30}, {"bbox": [126, 497, 487, 511], "spans": [{"bbox": [126, 499, 155, 509], "score": 0.92, "content": "\\widetilde{q}_{2}=1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [156, 497, 278, 511], "score": 1.0, "content": ". In particular we see that ", "type": "text"}, {"bbox": [278, 499, 325, 509], "score": 0.91, "content": "4\\mid h_{2}(K_{2})", "type": "inline_equation", "height": 10, "width": 47}, {"bbox": [325, 497, 392, 511], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [392, 499, 437, 509], "score": 0.86, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [437, 497, 487, 511], "score": 1.0, "content": " as long as", "type": "text"}], "index": 31}, {"bbox": [126, 510, 268, 522], "spans": [{"bbox": [126, 511, 191, 522], "score": 0.94, "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [191, 510, 216, 522], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [217, 511, 265, 521], "score": 0.94, "content": "[\\pi/\\rho]=+1", "type": "inline_equation", "height": 10, "width": 48}, {"bbox": [265, 510, 268, 522], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 30.5, "bbox_fs": [124, 471, 487, 522]}, {"type": "text", "bbox": [124, 520, 486, 651], "lines": [{"bbox": [137, 521, 487, 534], "spans": [{"bbox": [137, 521, 365, 534], "score": 1.0, "content": "The ambiguous class number formula shows that ", "type": "text"}, {"bbox": [365, 523, 400, 533], "score": 0.93, "content": "\\mathrm{Cl}_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [400, 521, 487, 534], "score": 1.0, "content": " is cyclic, thus 4 |", "type": "text"}], "index": 33}, {"bbox": [126, 534, 486, 545], "spans": [{"bbox": [126, 535, 157, 545], "score": 0.94, "content": "h_{2}(K_{1})", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 534, 221, 545], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [221, 535, 265, 545], "score": 0.93, "content": "2\\mid h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 44}, {"bbox": [265, 534, 299, 545], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [300, 534, 362, 545], "score": 0.94, "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [362, 534, 486, 545], "score": 1.0, "content": " is the quadratic unramified", "type": "text"}], "index": 34}, {"bbox": [125, 545, 487, 558], "spans": [{"bbox": [125, 545, 182, 558], "score": 1.0, "content": "extension of ", "type": "text"}, {"bbox": [182, 547, 195, 556], "score": 0.92, "content": "K_{1}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [196, 545, 425, 558], "score": 1.0, "content": ". Applying the ambiguous class number formula to ", "type": "text"}, {"bbox": [425, 547, 452, 557], "score": 0.95, "content": "L_{1}/F_{1}", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [453, 545, 487, 558], "score": 1.0, "content": ", where", "type": "text"}], "index": 35}, {"bbox": [126, 558, 487, 570], "spans": [{"bbox": [126, 559, 188, 569], "score": 0.95, "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [189, 558, 254, 570], "score": 1.0, "content": ", we see that ", "type": "text"}, {"bbox": [254, 559, 300, 569], "score": 0.89, "content": "2\\:\\:|\\:\\:h_{2}(L_{1})", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [300, 558, 369, 570], "score": 1.0, "content": " if and only if ", "type": "text"}, {"bbox": [369, 559, 430, 569], "score": 0.92, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [430, 558, 465, 570], "score": 1.0, "content": ". Now ", "type": "text"}, {"bbox": [465, 559, 474, 567], "score": 0.9, "content": "E", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [474, 558, 487, 570], "score": 1.0, "content": " is", "type": "text"}], "index": 36}, {"bbox": [125, 569, 487, 582], "spans": [{"bbox": [125, 569, 487, 582], "score": 1.0, "content": "generated by a root of unity (which always is a norm residue at primes dividing", "type": "text"}], "index": 37}, {"bbox": [126, 582, 485, 593], "spans": [{"bbox": [126, 583, 184, 591], "score": 0.8, "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "type": "inline_equation", "height": 8, "width": 58}, {"bbox": [185, 582, 304, 593], "score": 1.0, "content": ") and a fundamental unit ", "type": "text"}, {"bbox": [304, 586, 310, 591], "score": 0.89, "content": "\\varepsilon", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [310, 582, 368, 593], "score": 1.0, "content": ". Therefore ", "type": "text"}, {"bbox": [368, 583, 430, 593], "score": 0.91, "content": "(E\\,:\\,H)\\;=\\;1", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [430, 582, 485, 593], "score": 1.0, "content": " if and only", "type": "text"}], "index": 38}, {"bbox": [125, 594, 487, 606], "spans": [{"bbox": [125, 594, 136, 606], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [136, 595, 247, 605], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "type": "inline_equation", "height": 10, "width": 111}, {"bbox": [248, 594, 284, 606], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [284, 595, 348, 605], "score": 0.91, "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "type": "inline_equation", "height": 10, "width": 64}, {"bbox": [348, 594, 402, 606], "score": 1.0, "content": " and where ", "type": "text"}, {"bbox": [402, 595, 430, 605], "score": 0.93, "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [430, 594, 487, 606], "score": 1.0, "content": " denotes the", "type": "text"}], "index": 39}, {"bbox": [125, 606, 487, 617], "spans": [{"bbox": [125, 606, 247, 617], "score": 1.0, "content": "quadratic residue symbol in ", "type": "text"}, {"bbox": [248, 607, 258, 616], "score": 0.91, "content": "F_{1}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 606, 291, 617], "score": 1.0, "content": ". Since", "type": "text"}, {"bbox": [291, 606, 414, 617], "score": 0.92, "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "type": "inline_equation", "height": 11, "width": 123}, {"bbox": [415, 606, 487, 617], "score": 1.0, "content": ", we have proved", "type": "text"}], "index": 40}, {"bbox": [126, 617, 487, 630], "spans": [{"bbox": [126, 617, 147, 630], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [148, 618, 195, 629], "score": 0.91, "content": "4\\mid h_{2}(K_{1})", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [195, 617, 332, 630], "score": 1.0, "content": " if and only if the prime ideal ", "type": "text"}, {"bbox": [333, 619, 345, 628], "score": 0.91, "content": "\\Re_{1}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [346, 617, 378, 630], "score": 1.0, "content": " above ", "type": "text"}, {"bbox": [378, 621, 383, 626], "score": 0.65, "content": "\\mathfrak{r}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [383, 617, 487, 630], "score": 1.0, "content": " splits in the quadratic", "type": "text"}], "index": 41}, {"bbox": [126, 630, 486, 641], "spans": [{"bbox": [126, 630, 169, 641], "score": 1.0, "content": "extension ", "type": "text"}, {"bbox": [170, 630, 203, 641], "score": 0.93, "content": "F_{1}(\\sqrt{\\varepsilon})", "type": "inline_equation", "height": 11, "width": 33}, {"bbox": [203, 630, 266, 641], "score": 1.0, "content": ". But if we fix ", "type": "text"}, {"bbox": [267, 633, 272, 640], "score": 0.88, "content": "p", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [272, 630, 293, 641], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [294, 633, 299, 640], "score": 0.88, "content": "q", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [299, 630, 486, 641], "score": 1.0, "content": ", this happens for exactly half of the values", "type": "text"}], "index": 42}, {"bbox": [125, 641, 297, 654], "spans": [{"bbox": [125, 641, 137, 654], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [137, 646, 142, 650], "score": 0.89, "content": "r", "type": "inline_equation", "height": 4, "width": 5}, {"bbox": [142, 641, 190, 654], "score": 1.0, "content": " satisfying ", "type": "text"}, {"bbox": [190, 642, 239, 653], "score": 0.92, "content": "(p/r)=-1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [239, 641, 244, 654], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [244, 642, 293, 653], "score": 0.91, "content": "(q/r)=+1", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [293, 641, 297, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 43}], "index": 38, "bbox_fs": [125, 521, 487, 654]}, {"type": "text", "bbox": [125, 651, 486, 699], "lines": [{"bbox": [136, 652, 487, 666], "spans": [{"bbox": [136, 652, 147, 666], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [148, 655, 176, 664], "score": 0.93, "content": "d_{2}=8", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [176, 652, 198, 666], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [198, 655, 222, 664], "score": 0.9, "content": "p=2", "type": "inline_equation", "height": 9, "width": 24}, {"bbox": [222, 652, 250, 666], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [250, 654, 298, 665], "score": 0.92, "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "type": "inline_equation", "height": 11, "width": 48}, {"bbox": [299, 652, 405, 666], "score": 1.0, "content": ", and we have to choose ", "type": "text"}, {"bbox": [405, 653, 442, 665], "score": 0.94, "content": "2^{h}=(\\pi)", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [443, 652, 487, 666], "score": 1.0, "content": " in such a", "type": "text"}], "index": 44}, {"bbox": [127, 665, 486, 677], "spans": [{"bbox": [127, 665, 166, 677], "score": 1.0, "content": "way that ", "type": "text"}, {"bbox": [167, 666, 214, 677], "score": 0.94, "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [215, 665, 309, 677], "score": 1.0, "content": " is unramified outside", "type": "text"}, {"bbox": [310, 669, 315, 676], "score": 0.74, "content": "\\mathfrak{p}", "type": "inline_equation", "height": 7, "width": 5}, {"bbox": [316, 665, 412, 677], "score": 1.0, "content": ". The residue symbols", "type": "text"}, {"bbox": [413, 666, 435, 677], "score": 0.92, "content": "\\left[\\alpha/2\\right]", "type": "inline_equation", "height": 11, "width": 22}, {"bbox": [435, 665, 486, 677], "score": 1.0, "content": " are defined", "type": "text"}], "index": 45}, {"bbox": [125, 677, 486, 689], "spans": [{"bbox": [125, 677, 307, 689], "score": 1.0, "content": "as Kronecker symbols via the splitting of ", "type": "text"}, {"bbox": [308, 681, 313, 686], "score": 0.41, "content": "^{2}", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [313, 677, 433, 689], "score": 1.0, "content": " in the quadratic extension ", "type": "text"}, {"bbox": [433, 678, 482, 689], "score": 0.93, "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "type": "inline_equation", "height": 11, "width": 49}, {"bbox": [482, 677, 486, 689], "score": 1.0, "content": ".", "type": "text"}], "index": 46}, {"bbox": [126, 689, 390, 702], "spans": [{"bbox": [126, 689, 390, 702], "score": 1.0, "content": "With these modifactions, the above arguments remain valid.", "type": "text"}], "index": 47}], "index": 45.5, "bbox_fs": [125, 652, 487, 702]}]}
|
[{"type": "text", "bbox": [125, 112, 486, 148], "content": "The referee (whom we’d like to thank for a couple of helpful remarks) asked whether and infinitely often. Let us show how to prove that both possibilities occur with equal density.", "index": 0}, {"type": "text", "bbox": [124, 149, 486, 304], "content": "Before we can do this, we have to study the quadratic extensions and of more closely. We assume that and are odd primes in t he following, and then say how to modify the arguments in the case or . The primes and split in as and . Let denote the odd class number of and write and for primary elements and (this is can easily be proved directly, but it is also a very special case of Hilbert’s first supplementary law for quadratic reciprocity in fields with odd class number (see [7]): if for an ideal with odd norm, then can be chosen primary (i.e. congruent to a square mod ) if and only if is primary (i.e. for all units , where denotes the quadratic residue symbol in )). Let denote the quadratic residue symbol in . Then , so we may choose the conjugates in such a way that and .", "index": 1}, {"type": "text", "bbox": [124, 306, 487, 384], "content": "Put and ; we claim that . This is equivalent to , where is a quad r atic unramified extension of . Put a n d apply the ambiguous class number formula to an d : since there is only one ramified prime in each of these two ext e nsions, w e fin d ; note that we have used the assumption that in deducing th a t is inert in .", "index": 2}, {"type": "table", "bbox": [189, 412, 422, 468], "content": "", "index": 3}, {"type": "text", "bbox": [125, 470, 486, 520], "content": "In order to decide whether or , recall that we have ; thus must be the field with 2 -class nu mber 2, and this implies and . In particular we see that if and only if as long as with .", "index": 4}, {"type": "text", "bbox": [124, 520, 486, 651], "content": "The ambiguous class number formula shows that is cyclic, thus 4 | if and only if , where is the quadratic unramified extension of . Applying the ambiguous class number formula to , where , we see that if and only if . Now is generated by a root of unity (which always is a norm residue at primes dividing ) and a fundamental unit . Therefore if and only if , where and where denotes the quadratic residue symbol in . Since , we have proved that if and only if the prime ideal above splits in the quadratic extension . But if we fix and , this happens for exactly half of the values of satisfying , .", "index": 5}, {"type": "text", "bbox": [125, 651, 486, 699], "content": "If and , then , and we have to choose in such a way that is unramified outside . The residue symbols are defined as Kronecker symbols via the splitting of in the quadratic extension . With these modifactions, the above arguments remain valid.", "index": 6}]
|
[{"bbox": [137, 114, 486, 127], "content": "The referee (whom we’d like to thank for a couple of helpful remarks) asked", "parent_index": 0, "line_index": 0}, {"bbox": [126, 126, 485, 138], "content": "whether and infinitely often. Let us show how to prove that", "parent_index": 0, "line_index": 1}, {"bbox": [126, 138, 313, 150], "content": "both possibilities occur with equal density.", "parent_index": 0, "line_index": 2}, {"bbox": [137, 150, 484, 163], "content": "Before we can do this, we have to study the quadratic extensions and", "parent_index": 1, "line_index": 0}, {"bbox": [125, 161, 487, 176], "content": "of more closely. We assume that and are odd primes in t he", "parent_index": 1, "line_index": 1}, {"bbox": [125, 174, 487, 187], "content": "following, and then say how to modify the arguments in the case or .", "parent_index": 1, "line_index": 2}, {"bbox": [126, 187, 487, 199], "content": "The primes and split in as and . Let denote the", "parent_index": 1, "line_index": 3}, {"bbox": [125, 198, 487, 211], "content": "odd class number of and write and for primary elements", "parent_index": 1, "line_index": 4}, {"bbox": [126, 210, 487, 223], "content": "and (this is can easily be proved directly, but it is also a very special case", "parent_index": 1, "line_index": 5}, {"bbox": [125, 223, 486, 235], "content": "of Hilbert’s first supplementary law for quadratic reciprocity in fields with odd", "parent_index": 1, "line_index": 6}, {"bbox": [125, 234, 485, 247], "content": "class number (see [7]): if for an ideal with odd norm, then", "parent_index": 1, "line_index": 7}, {"bbox": [126, 247, 487, 259], "content": "can be chosen primary (i.e. congruent to a square mod ) if and only if is", "parent_index": 1, "line_index": 8}, {"bbox": [125, 258, 487, 272], "content": "primary (i.e. for all units , where denotes the quadratic", "parent_index": 1, "line_index": 9}, {"bbox": [126, 271, 486, 283], "content": "residue symbol in )). Let denote the quadratic residue symbol in . Then", "parent_index": 1, "line_index": 10}, {"bbox": [126, 282, 485, 295], "content": ", so we may choose the conjugates in such a way", "parent_index": 1, "line_index": 11}, {"bbox": [125, 294, 310, 307], "content": "that and .", "parent_index": 1, "line_index": 12}, {"bbox": [136, 307, 486, 320], "content": "Put and ; we claim that . This", "parent_index": 2, "line_index": 0}, {"bbox": [125, 321, 486, 334], "content": "is equivalent to , where is a quad r atic unramified", "parent_index": 2, "line_index": 1}, {"bbox": [126, 333, 486, 349], "content": "extension of . Put a n d apply the ambiguous class number formula", "parent_index": 2, "line_index": 2}, {"bbox": [125, 347, 486, 361], "content": "to an d : since there is only one ramified prime in each of these two", "parent_index": 2, "line_index": 3}, {"bbox": [125, 360, 486, 373], "content": "ext e nsions, w e fin d ; note that we have used the", "parent_index": 2, "line_index": 4}, {"bbox": [126, 374, 415, 386], "content": "assumption that in deducing th a t is inert in .", "parent_index": 2, "line_index": 5}, {"bbox": [124, 471, 486, 486], "content": "In order to decide whether or , recall that we have ; thus", "parent_index": 4, "line_index": 0}, {"bbox": [126, 483, 487, 498], "content": "must be the field with 2 -class nu mber 2, and this implies and", "parent_index": 4, "line_index": 1}, {"bbox": [126, 497, 487, 511], "content": ". In particular we see that if and only if as long as", "parent_index": 4, "line_index": 2}, {"bbox": [126, 510, 268, 522], "content": "with .", "parent_index": 4, "line_index": 3}, {"bbox": [137, 521, 487, 534], "content": "The ambiguous class number formula shows that is cyclic, thus 4 |", "parent_index": 5, "line_index": 0}, {"bbox": [126, 534, 486, 545], "content": "if and only if , where is the quadratic unramified", "parent_index": 5, "line_index": 1}, {"bbox": [125, 545, 487, 558], "content": "extension of . Applying the ambiguous class number formula to , where", "parent_index": 5, "line_index": 2}, {"bbox": [126, 558, 487, 570], "content": ", we see that if and only if . Now is", "parent_index": 5, "line_index": 3}, {"bbox": [125, 569, 487, 582], "content": "generated by a root of unity (which always is a norm residue at primes dividing", "parent_index": 5, "line_index": 4}, {"bbox": [126, 582, 485, 593], "content": ") and a fundamental unit . Therefore if and only", "parent_index": 5, "line_index": 5}, {"bbox": [125, 594, 487, 606], "content": "if , where and where denotes the", "parent_index": 5, "line_index": 6}, {"bbox": [125, 606, 487, 617], "content": "quadratic residue symbol in . Since , we have proved", "parent_index": 5, "line_index": 7}, {"bbox": [126, 617, 487, 630], "content": "that if and only if the prime ideal above splits in the quadratic", "parent_index": 5, "line_index": 8}, {"bbox": [126, 630, 486, 641], "content": "extension . But if we fix and , this happens for exactly half of the values", "parent_index": 5, "line_index": 9}, {"bbox": [125, 641, 297, 654], "content": "of satisfying , .", "parent_index": 5, "line_index": 10}, {"bbox": [136, 652, 487, 666], "content": "If and , then , and we have to choose in such a", "parent_index": 6, "line_index": 0}, {"bbox": [127, 665, 486, 677], "content": "way that is unramified outside . The residue symbols are defined", "parent_index": 6, "line_index": 1}, {"bbox": [125, 677, 486, 689], "content": "as Kronecker symbols via the splitting of in the quadratic extension .", "parent_index": 6, "line_index": 2}, {"bbox": [126, 689, 390, 702], "content": "With these modifactions, the above arguments remain valid.", "parent_index": 6, "line_index": 3}]
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[]
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[{"bbox": [164, 127, 209, 138], "content": "h_{2}(K)=2", "parent_index": 0, "subtype": "inline"}, {"bbox": [232, 127, 278, 138], "content": "h_{2}(K)>2", "parent_index": 0, "subtype": "inline"}, {"bbox": [434, 153, 447, 161], "content": "K_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [471, 150, 484, 161], "content": "\\widetilde{K}_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 164, 148, 173], "content": "k_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [294, 164, 325, 174], "content": "d_{2}\\,=\\,p", "parent_index": 1, "subtype": "inline"}, {"bbox": [350, 164, 382, 173], "content": "d_{3}~=~r", "parent_index": 1, "subtype": "inline"}, {"bbox": [404, 176, 432, 185], "content": "d_{2}=8", "parent_index": 1, "subtype": "inline"}, {"bbox": [446, 177, 482, 185], "content": "d_{3}=-8", "parent_index": 1, "subtype": "inline"}, {"bbox": [181, 191, 186, 198], "content": "p", "parent_index": 1, "subtype": "inline"}, {"bbox": [210, 191, 215, 196], "content": "r", "parent_index": 1, "subtype": "inline"}, {"bbox": [255, 189, 265, 197], "content": "k_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [282, 188, 330, 198], "content": "p\\mathcal{O}_{1}\\,=\\,\\mathfrak{p p}^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [354, 188, 399, 197], "content": "r\\mathcal{O}_{1}\\,=\\,\\mathfrak{r r}^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [427, 189, 433, 196], "content": "h", "parent_index": 1, "subtype": "inline"}, {"bbox": [221, 200, 231, 209], "content": "k_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [282, 199, 323, 210], "content": "{\\mathfrak{p}}^{h}\\,=\\,(\\pi)", "parent_index": 1, "subtype": "inline"}, {"bbox": [348, 199, 387, 210], "content": "\\mathfrak{r}^{h}\\,=\\,(\\rho)", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 215, 132, 219], "content": "\\pi", "parent_index": 1, "subtype": "inline"}, {"bbox": [157, 215, 162, 222], "content": "\\rho", "parent_index": 1, "subtype": "inline"}, {"bbox": [433, 225, 442, 232], "content": "K", "parent_index": 1, "subtype": "inline"}, {"bbox": [188, 236, 194, 244], "content": "h", "parent_index": 1, "subtype": "inline"}, {"bbox": [256, 235, 306, 245], "content": "{\\mathfrak{a}}^{h}\\;=\\;\\alpha{\\mathcal{O}}_{K}", "parent_index": 1, "subtype": "inline"}, {"bbox": [368, 236, 374, 246], "content": "\\mathfrak{p}", "parent_index": 1, "subtype": "inline"}, {"bbox": [478, 239, 485, 244], "content": "\\alpha", "parent_index": 1, "subtype": "inline"}, {"bbox": [378, 248, 398, 257], "content": "4{\\cal O}_{K}", "parent_index": 1, "subtype": "inline"}, {"bbox": [469, 250, 474, 255], "content": "\\mathfrak{a}", "parent_index": 1, "subtype": "inline"}, {"bbox": [187, 259, 234, 270], "content": "[\\varepsilon/{\\mathfrak a}]=+1", "parent_index": 1, "subtype": "inline"}, {"bbox": [293, 259, 327, 271], "content": "\\varepsilon\\in{\\mathcal{O}}_{K}^{\\times}", "parent_index": 1, "subtype": "inline"}, {"bbox": [363, 259, 386, 270], "content": "[\\,\\cdot\\,/\\,\\cdot\\,]", "parent_index": 1, "subtype": "inline"}, {"bbox": [205, 272, 215, 280], "content": "K", "parent_index": 1, "subtype": "inline"}, {"bbox": [247, 272, 270, 282], "content": "\\big[\\cdot\\big/\\cdot\\big]", "parent_index": 1, "subtype": "inline"}, {"bbox": [445, 272, 455, 281], "content": "k_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [126, 284, 271, 294], "content": "[\\pi/\\rho][\\pi^{\\prime}/\\rho]=[p/\\rho]=(p/r)=-1", "parent_index": 1, "subtype": "inline"}, {"bbox": [147, 295, 195, 306], "content": "[\\pi/\\rho]=+1", "parent_index": 1, "subtype": "inline"}, {"bbox": [218, 295, 307, 306], "content": "[\\pi^{\\prime}/\\rho]=[\\pi/\\rho^{\\prime}]=-1", "parent_index": 1, "subtype": "inline"}, {"bbox": [159, 309, 228, 320], "content": "K_{1}\\;=\\;k_{1}(\\sqrt{\\pi\\rho}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [253, 307, 326, 319], "content": "\\tilde{K}_{1}\\,=\\,k_{1}(\\sqrt{\\pi\\rho^{\\prime}})", "parent_index": 2, "subtype": "inline"}, {"bbox": [400, 307, 454, 319], "content": "h_{2}(\\tilde{K}_{1})\\;=\\;2", "parent_index": 2, "subtype": "inline"}, {"bbox": [198, 321, 249, 333], "content": "h_{2}(\\widetilde{L}_{1})\\,=\\,1", "parent_index": 2, "subtype": "inline"}, {"bbox": [286, 321, 368, 333], "content": "\\tilde{L}_{1}\\,=\\,k_{1}(\\sqrt{\\pi},\\sqrt{\\rho^{\\prime}})", "parent_index": 2, "subtype": "inline"}, {"bbox": [180, 334, 194, 345], "content": "\\widetilde{K}_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [219, 334, 277, 347], "content": "\\widetilde{F}_{1}=k_{1}(\\sqrt{\\pi}\\,)", "parent_index": 2, "subtype": "inline"}, {"bbox": [138, 347, 164, 360], "content": "\\widetilde{F}_{1}/k_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [188, 347, 216, 360], "content": "\\widetilde{L}_{1}/\\widetilde{F}_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [215, 360, 353, 373], "content": "\\mathrm{Am}(\\widetilde{F}_{1}/k_{1})\\,=\\,\\mathrm{Am}(\\widetilde{L}_{1}/\\widetilde{F}_{1})\\,=\\,1", "parent_index": 2, "subtype": "inline"}, {"bbox": [200, 375, 251, 386], "content": "[\\pi/\\rho^{\\prime}]=-1", "parent_index": 2, "subtype": "inline"}, {"bbox": [329, 375, 336, 384], "content": "{\\mathfrak{r}}^{\\prime}", "parent_index": 2, "subtype": "inline"}, {"bbox": [385, 374, 410, 386], "content": "\\widetilde{F}_{1}/k_{1}", "parent_index": 2, "subtype": "inline"}, {"bbox": [246, 474, 274, 484], "content": "\\widetilde{q}_{2}=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [289, 474, 317, 484], "content": "\\widetilde{q}_{2}=2", "parent_index": 4, "subtype": "inline"}, {"bbox": [410, 474, 460, 484], "content": "h_{2}(K_{1})=4", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 485, 139, 496], "content": "\\widetilde{K}_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [402, 484, 465, 497], "content": "h_{2}(\\widetilde{L})\\,=\\,2^{m+2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 499, 155, 509], "content": "\\widetilde{q}_{2}=1", "parent_index": 4, "subtype": "inline"}, {"bbox": [278, 499, 325, 509], "content": "4\\mid h_{2}(K_{2})", "parent_index": 4, "subtype": "inline"}, {"bbox": [392, 499, 437, 509], "content": "4\\mid h_{2}(K_{1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 511, 191, 522], "content": "K_{1}=k_{1}(\\sqrt{\\pi\\rho}\\,)", "parent_index": 4, "subtype": "inline"}, {"bbox": [217, 511, 265, 521], "content": "[\\pi/\\rho]=+1", "parent_index": 4, "subtype": "inline"}, {"bbox": [365, 523, 400, 533], "content": "\\mathrm{Cl}_{2}(K_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 535, 157, 545], "content": "h_{2}(K_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [221, 535, 265, 545], "content": "2\\mid h_{2}(L_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [300, 534, 362, 545], "content": "L_{1}=K_{1}(\\sqrt{\\pi}\\,)", "parent_index": 5, "subtype": "inline"}, {"bbox": [182, 547, 195, 556], "content": "K_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [425, 547, 452, 557], "content": "L_{1}/F_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 559, 188, 569], "content": "F_{1}\\;=\\;k_{1}(\\sqrt{\\pi}\\,)", "parent_index": 5, "subtype": "inline"}, {"bbox": [254, 559, 300, 569], "content": "2\\:\\:|\\:\\:h_{2}(L_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [369, 559, 430, 569], "content": "(E\\,:\\,H)\\;=\\;1", "parent_index": 5, "subtype": "inline"}, {"bbox": [465, 559, 474, 567], "content": "E", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 583, 184, 591], "content": "r\\,\\equiv\\,1\\;\\mathrm{mod}\\;4", "parent_index": 5, "subtype": "inline"}, {"bbox": [304, 586, 310, 591], "content": "\\varepsilon", "parent_index": 5, "subtype": "inline"}, {"bbox": [368, 583, 430, 593], "content": "(E\\,:\\,H)\\;=\\;1", "parent_index": 5, "subtype": "inline"}, {"bbox": [136, 595, 247, 605], "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\,=\\,\\{\\varepsilon/\\mathfrak{R}_{2}\\}\\,=\\,+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [284, 595, 348, 605], "content": "\\mathfrak{r}{\\mathcal{O}}_{F_{1}}\\,=\\,\\mathfrak{R}_{1}\\mathfrak{R}_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [402, 595, 430, 605], "content": "\\{\\,\\cdot\\,/\\,\\cdot\\,\\}", "parent_index": 5, "subtype": "inline"}, {"bbox": [248, 607, 258, 616], "content": "F_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [291, 606, 414, 617], "content": "\\{\\varepsilon/\\mathfrak{R}_{1}\\}\\{\\varepsilon/\\mathfrak{R}_{2}\\}=[\\varepsilon/\\mathfrak{r}]=+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [148, 618, 195, 629], "content": "4\\mid h_{2}(K_{1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [333, 619, 345, 628], "content": "\\Re_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [378, 621, 383, 626], "content": "\\mathfrak{r}", "parent_index": 5, "subtype": "inline"}, {"bbox": [170, 630, 203, 641], "content": "F_{1}(\\sqrt{\\varepsilon})", "parent_index": 5, "subtype": "inline"}, {"bbox": [267, 633, 272, 640], "content": "p", "parent_index": 5, "subtype": "inline"}, {"bbox": [294, 633, 299, 640], "content": "q", "parent_index": 5, "subtype": "inline"}, {"bbox": [137, 646, 142, 650], "content": "r", "parent_index": 5, "subtype": "inline"}, {"bbox": [190, 642, 239, 653], "content": "(p/r)=-1", "parent_index": 5, "subtype": "inline"}, {"bbox": [244, 642, 293, 653], "content": "(q/r)=+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [148, 655, 176, 664], "content": "d_{2}=8", "parent_index": 6, "subtype": "inline"}, {"bbox": [198, 655, 222, 664], "content": "p=2", "parent_index": 6, "subtype": "inline"}, {"bbox": [250, 654, 298, 665], "content": "2\\mathcal{O}_{k_{1}}=22^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [405, 653, 442, 665], "content": "2^{h}=(\\pi)", "parent_index": 6, "subtype": "inline"}, {"bbox": [167, 666, 214, 677], "content": "k_{1}(\\sqrt{\\pi}\\,)/k_{1}", "parent_index": 6, "subtype": "inline"}, {"bbox": [310, 669, 315, 676], "content": "\\mathfrak{p}", "parent_index": 6, "subtype": "inline"}, {"bbox": [413, 666, 435, 677], "content": "\\left[\\alpha/2\\right]", "parent_index": 6, "subtype": "inline"}, {"bbox": [308, 681, 313, 686], "content": "^{2}", "parent_index": 6, "subtype": "inline"}, {"bbox": [433, 678, 482, 689], "content": "k_{1}(\\sqrt{\\alpha}\\,)/k_{1}", "parent_index": 6, "subtype": "inline"}]
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# References
[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic $\mathrm{Cl}_{2}(k^{1})$ , J. Number Theory 67 (1997), 229–245.
[2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27.
[3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337.
[4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48.
[5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95.
[6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965.
[7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127.
[8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967.
[9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990.
[10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260.
[11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18.
[12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74.
[13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkorperturm, Arch. Math. 34 (1980), 307–312.
(E. Benjamin) Mathematics Department, Unity College $E$ -mail address: [email protected]
(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany $E$ -mail address: [email protected]
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<html><body>
<h1 data-bbox="276 113 336 123">References </h1>
<p data-bbox="136 129 487 360">[1] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary Quadratic Fields k with Cyclic $\mathrm{Cl}_{2}(k^{1})$ , J. Number Theory 67 (1997), 229–245.
[2] N. Blackburn, On Prime-Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc. 53 (1957), 19–27.
[3] N. Blackburn, On Prime Power Groups with Two Generators, Proc. Cambridge Phil. Soc. 54 (1958), 327–337.
[4] G. Gras, Sur les ℓ-classes d’ide´aux dans les extensions cycliques relatives de degre´ premier ℓ, Ann. Inst. Fourier 23 (1973), 1–48.
[5] P. Hall, A Contribution to the Theory of Groups of Prime Power Order, Proc. London Math. Soc. 36 (1933), 29–95.
[6]H.Hasse,Bericht iber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkorper, Teil I: Reziprozitatsgesetz, Physica Verlag, Wurzburg 1965.
[7] D. Hilbert, Uber die Theorie des relativ-quadratischen Zahlkorpers, Math. Ann. 51 (1899), 1–127.
[8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967.
[9] S. Lang, Cyclotomic Fields. I, II, Springer Verlag 1990.
[10] F. Lemmermeyer, Kuroda’s Class Number Formula, Acta Arith. 66.3 (1994), 245–260.
[11] M. Moriya, Uber die Klassenzahl eines relativzyklischen Zahlkorpers von Primzahlgrad, Jap. J. Math. 10 (1933), 1–18.
[12] L. R´edei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. Reine Angew. Math. 170 (1933), 69-74.
[13] B. Schmithals, Konstruktion imaginarquadratischer Korper mit unendlichem Klassenkorperturm, Arch. Math. 34 (1980), 307–312. </p>
<p data-bbox="137 368 363 388">(E. Benjamin) Mathematics Department, Unity College $E$ -mail address: [email protected] </p>
<p data-bbox="138 395 390 415">(F. Lemmermeyer) Erwin-Rohde-Str. 19, Heidelberg, Germany $E$ -mail address: [email protected] </p>
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0003244v1
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# Special Lagrangian Geometry in irreducible symplectic 4-folds
Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy
# Abstract
Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold $X$ are obtained by complex submanifolds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: indeed all special Lagrangian submanifolds of $X$ turn out to be real analytic.
MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures.
REF.: 75/99/FM/GEO
# 1 Introduction
Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. Indeed, D-branes are special Lagrangian submanifolds equipped with a flat $U(1)$ line bundle. In physical literature, special Lagrangian submanifolds of the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersymmetric cycles or BPS states.
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<html><body>
<h1 data-bbox="123 166 486 215">Special Lagrangian Geometry in irreducible symplectic 4-folds </h1>
<p data-bbox="199 231 411 285">Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy </p>
<h1 data-bbox="281 328 329 341">Abstract </h1>
<p data-bbox="139 348 471 443">Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold $X$ are obtained by complex submanifolds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: indeed all special Lagrangian submanifolds of $X$ turn out to be real analytic. </p>
<p data-bbox="127 455 460 483">MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures. </p>
<p data-bbox="248 495 362 508">REF.: 75/99/FM/GEO </p>
<h1 data-bbox="110 529 246 549">1 Introduction </h1>
<p data-bbox="110 560 500 647">Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. Indeed, D-branes are special Lagrangian submanifolds equipped with a flat $U(1)$ line bundle. In physical literature, special Lagrangian submanifolds of the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersymmetric cycles or BPS states. </p>
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[{"type": "title", "bbox": [123, 166, 486, 215], "content": "Special Lagrangian Geometry in irreducible symplectic 4-folds", "index": 0}, {"type": "text", "bbox": [199, 231, 411, 285], "content": "Alessandro Arsie S.I.S.S.A. - I.S.A.S. Via Beirut 4 - 34013 Trieste, Italy", "index": 1}, {"type": "title", "bbox": [281, 328, 329, 341], "content": "Abstract", "index": 2}, {"type": "text", "bbox": [139, 348, 471, 443], "content": "Having fixed a Kaehler class and the unique corresponding hyper- kaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold are obtained by complex submani- folds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: in- deed all special Lagrangian submanifolds of turn out to be real analytic.", "index": 3}, {"type": "text", "bbox": [127, 455, 460, 483], "content": "MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20 Keywords: special Lagrangian submanifolds, hyperkaehler structures.", "index": 4}, {"type": "text", "bbox": [248, 495, 362, 508], "content": "REF.: 75/99/FM/GEO", "index": 5}, {"type": "title", "bbox": [110, 529, 246, 549], "content": "1 Introduction", "index": 6}, {"type": "text", "bbox": [110, 560, 500, 647], "content": "Under the flourishing research activity on D-branes in string theory, the role of special Lagrangian submanifolds in physics has become more and more relevant (see for example [1]) untill it was eventually conjectured in [11] that they can be considered as the cornerstones of the mirror phenomenon. In- deed, D-branes are special Lagrangian submanifolds equipped with a flat line bundle. In physical literature, special Lagrangian submanifolds of the compactification space are related to physical states which retain part of the vacuum supersymmetry: for this reason they are often called supersym- metric cycles or BPS states.", "index": 7}]
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[{"bbox": [124, 170, 486, 191], "content": "Special Lagrangian Geometry in irreducible", "parent_index": 0, "line_index": 0}, {"bbox": [230, 196, 379, 214], "content": "symplectic 4-folds", "parent_index": 0, "line_index": 1}, {"bbox": [251, 234, 359, 248], "content": "Alessandro Arsie", "parent_index": 1, "line_index": 0}, {"bbox": [244, 252, 363, 266], "content": "S.I.S.S.A. - I.S.A.S.", "parent_index": 1, "line_index": 1}, {"bbox": [199, 268, 409, 286], "content": "Via Beirut 4 - 34013 Trieste, Italy", "parent_index": 1, "line_index": 2}, {"bbox": [280, 330, 330, 342], "content": "Abstract", "parent_index": 2, "line_index": 0}, {"bbox": [155, 351, 470, 363], "content": "Having fixed a Kaehler class and the unique corresponding hyper-", "parent_index": 3, "line_index": 0}, {"bbox": [138, 364, 472, 378], "content": "kaehler metric, we prove that all special Lagrangian submanifolds of", "parent_index": 3, "line_index": 1}, {"bbox": [140, 379, 470, 390], "content": "an irreducible symplectic 4-fold are obtained by complex submani-", "parent_index": 3, "line_index": 2}, {"bbox": [138, 391, 470, 404], "content": "folds via a generalization of the so called hyperkaehler rotation trick;", "parent_index": 3, "line_index": 3}, {"bbox": [139, 405, 470, 417], "content": "thus they retain part of the rigidity of the complex submanifolds: in-", "parent_index": 3, "line_index": 4}, {"bbox": [139, 419, 471, 431], "content": "deed all special Lagrangian submanifolds of turn out to be real", "parent_index": 3, "line_index": 5}, {"bbox": [139, 433, 180, 444], "content": "analytic.", "parent_index": 3, "line_index": 6}, {"bbox": [127, 458, 437, 471], "content": "MSC (1991): Primary: 53C15, Secondary: 53A40, 51P05, 53C20", "parent_index": 4, "line_index": 0}, {"bbox": [127, 473, 460, 486], "content": "Keywords: special Lagrangian submanifolds, hyperkaehler structures.", "parent_index": 4, "line_index": 1}, {"bbox": [248, 497, 360, 509], "content": "REF.: 75/99/FM/GEO", "parent_index": 5, "line_index": 0}, {"bbox": [110, 532, 246, 549], "content": "1 Introduction", "parent_index": 6, "line_index": 0}, {"bbox": [110, 563, 500, 577], "content": "Under the flourishing research activity on D-branes in string theory, the role", "parent_index": 7, "line_index": 0}, {"bbox": [110, 577, 501, 592], "content": "of special Lagrangian submanifolds in physics has become more and more", "parent_index": 7, "line_index": 1}, {"bbox": [110, 592, 500, 605], "content": "relevant (see for example [1]) untill it was eventually conjectured in [11] that", "parent_index": 7, "line_index": 2}, {"bbox": [110, 606, 499, 620], "content": "they can be considered as the cornerstones of the mirror phenomenon. In-", "parent_index": 7, "line_index": 3}, {"bbox": [110, 621, 500, 635], "content": "deed, D-branes are special Lagrangian submanifolds equipped with a flat", "parent_index": 7, "line_index": 4}, {"bbox": [110, 634, 502, 650], "content": "line bundle. In physical literature, special Lagrangian submanifolds of", "parent_index": 7, "line_index": 5}, {"bbox": [111, 128, 501, 141], "content": "the compactification space are related to physical states which retain part of", "parent_index": 7, "line_index": 6}, {"bbox": [109, 142, 498, 157], "content": "the vacuum supersymmetry: for this reason they are often called supersym-", "parent_index": 7, "line_index": 7}, {"bbox": [110, 156, 254, 170], "content": "metric cycles or BPS states.", "parent_index": 7, "line_index": 8}]
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Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds.
We first recall the following:
Definition 1.1: A complex manifold $X$ is called irreducible symplectic if it satisfies the following three conditions:
1) $X$ is compact and Kaehler;
$\boldsymbol{\mathcal{Q}}$ ) $X$ is simply connected;
3) $H^{0}(X,\Omega_{X}^{2})$ is spanned by an everywhere non-degenerate 2-form $\omega$ .
In particular, irreducible symplectic manifolds are special cases of CalabiYau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form $\omega$ ). In dimension 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]).
Finally, let us recall from [4] the following:
Definition 1.2: Let $X$ be a Calabi-Yau n-fold, with Kaehler form $\omega$ and holomorphic nowhere vanishing n-form $\Omega$ . $A$ (real) $\boldsymbol{n}$ -dimensional submanifold $j:\Lambda\hookrightarrow X$ of $X$ is called special Lagrangian if the following two conditions are satisfied:
1) $\Lambda$ is Lagrangian with respect to $\omega$ , i.e. $j^{*}\omega=0$ ;
2) there exists a multiple $\Omega^{\prime}$ of $\Omega$ such that $j^{*}\mathrm{Im}(\Omega^{\prime})\!=\!0_{.}$ ; one can prove (see [4]) that both conditions are equivalent to:
$\mathit{1}$ ’) $j^{*}\mathrm{Re}(\Omega^{\prime})=V o l_{g}(\Lambda)$ .
The condition $1^{\prime}$ ) in the previous definition means that the real part of $\Omega^{\prime}$ restricts to the volume form of $\Lambda$ , induced by the Calabi-Yau Riemannian metric $g$ . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point).
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<p data-bbox="109 169 500 255">Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds. </p>
<p data-bbox="127 255 273 269">We first recall the following: </p>
<p data-bbox="110 270 501 298">Definition 1.1: A complex manifold $X$ is called irreducible symplectic if it satisfies the following three conditions: </p>
<p data-bbox="129 299 283 312">1) $X$ is compact and Kaehler; </p>
<p data-bbox="128 314 262 327">$\boldsymbol{\mathcal{Q}}$ ) $X$ is simply connected; </p>
<p data-bbox="127 327 482 342">3) $H^{0}(X,\Omega_{X}^{2})$ is spanned by an everywhere non-degenerate 2-form $\omega$ . </p>
<p data-bbox="109 343 500 515">In particular, irreducible symplectic manifolds are special cases of CalabiYau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form $\omega$ ). In dimension 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]). </p>
<p data-bbox="126 516 347 529">Finally, let us recall from [4] the following: </p>
<p data-bbox="110 530 500 587">Definition 1.2: Let $X$ be a Calabi-Yau n-fold, with Kaehler form $\omega$ and holomorphic nowhere vanishing n-form $\Omega$ . $A$ (real) $\boldsymbol{n}$ -dimensional submanifold $j:\Lambda\hookrightarrow X$ of $X$ is called special Lagrangian if the following two conditions are satisfied: </p>
<p data-bbox="129 588 388 602">1) $\Lambda$ is Lagrangian with respect to $\omega$ , i.e. $j^{*}\omega=0$ ; </p>
<p data-bbox="109 603 502 631">2) there exists a multiple $\Omega^{\prime}$ of $\Omega$ such that $j^{*}\mathrm{Im}(\Omega^{\prime})\!=\!0_{.}$ ; one can prove (see [4]) that both conditions are equivalent to: </p>
<p data-bbox="127 631 249 645">$\mathit{1}$ ’) $j^{*}\mathrm{Re}(\Omega^{\prime})=V o l_{g}(\Lambda)$ . </p>
<p data-bbox="110 646 500 674">The condition $1^{\prime}$ ) in the previous definition means that the real part of $\Omega^{\prime}$ restricts to the volume form of $\Lambda$ , induced by the Calabi-Yau Riemannian metric $g$ . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point). </p>
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{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [{"bbox": [111, 128, 501, 141], "spans": [{"bbox": [111, 128, 501, 141], "score": 1.0, "content": "the compactification space are related to physical states which retain part of", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 157], "spans": [{"bbox": [109, 142, 498, 157], "score": 1.0, "content": "the vacuum supersymmetry: for this reason they are often called supersym-", "type": "text"}], "index": 1}, {"bbox": [110, 156, 254, 170], "spans": [{"bbox": [110, 156, 254, 170], "score": 1.0, "content": "metric cycles or BPS states.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [109, 169, 500, 255], "lines": [{"bbox": [127, 171, 500, 186], "spans": [{"bbox": [127, 171, 500, 186], "score": 1.0, "content": "Despite their importance, there are very few explicit examples of special", "type": "text"}], "index": 3}, {"bbox": [110, 186, 501, 200], "spans": [{"bbox": [110, 186, 501, 200], "score": 1.0, "content": "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an", "type": "text"}], "index": 4}, {"bbox": [109, 200, 501, 214], "spans": [{"bbox": [109, 200, 501, 214], "score": 1.0, "content": "irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a", "type": "text"}], "index": 5}, {"bbox": [109, 214, 501, 230], "spans": [{"bbox": [109, 214, 501, 230], "score": 1.0, "content": "complete control of the special Lagrangian geometry of its submanifolds, via", "type": "text"}], "index": 6}, {"bbox": [110, 229, 499, 243], "spans": [{"bbox": [110, 229, 499, 243], "score": 1.0, "content": "a sort of hyperkaehler trick; moreover this enables us to prove that special", "type": "text"}], "index": 7}, {"bbox": [109, 243, 498, 258], "spans": [{"bbox": [109, 243, 498, 258], "score": 1.0, "content": "Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "type": "text"}], "index": 8}], "index": 5.5}, {"type": "text", "bbox": [127, 255, 273, 269], "lines": [{"bbox": [127, 256, 272, 271], "spans": [{"bbox": [127, 256, 272, 271], "score": 1.0, "content": "We first recall the following:", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [110, 270, 501, 298], "lines": [{"bbox": [127, 272, 502, 287], "spans": [{"bbox": [127, 272, 320, 287], "score": 1.0, "content": "Definition 1.1: A complex manifold ", "type": "text"}, {"bbox": [320, 274, 331, 282], "score": 0.67, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 272, 502, 287], "score": 1.0, "content": " is called irreducible symplectic if", "type": "text"}], "index": 10}, {"bbox": [111, 288, 321, 300], "spans": [{"bbox": [111, 288, 321, 300], "score": 1.0, "content": "it satisfies the following three conditions:", "type": "text"}], "index": 11}], "index": 10.5}, {"type": "text", "bbox": [129, 299, 283, 312], "lines": [{"bbox": [129, 300, 282, 314], "spans": [{"bbox": [129, 300, 142, 314], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 303, 154, 311], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [154, 300, 282, 314], "score": 1.0, "content": " is compact and Kaehler;", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [128, 314, 262, 327], "lines": [{"bbox": [128, 316, 260, 328], "spans": [{"bbox": [128, 317, 135, 326], "score": 0.64, "content": "\\boldsymbol{\\mathcal{Q}}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [135, 316, 142, 328], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [142, 317, 154, 326], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [154, 316, 260, 328], "score": 1.0, "content": " is simply connected;", "type": "text"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [127, 327, 482, 342], "lines": [{"bbox": [128, 329, 483, 344], "spans": [{"bbox": [128, 329, 142, 344], "score": 1.0, "content": "3) ", "type": "text"}, {"bbox": [142, 330, 199, 343], "score": 0.94, "content": "H^{0}(X,\\Omega_{X}^{2})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [199, 329, 470, 344], "score": 1.0, "content": " is spanned by an everywhere non-degenerate 2-form ", "type": "text"}, {"bbox": [470, 335, 479, 340], "score": 0.41, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [479, 329, 483, 344], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [109, 343, 500, 515], "lines": [{"bbox": [127, 345, 499, 357], "spans": [{"bbox": [127, 345, 499, 357], "score": 1.0, "content": "In particular, irreducible symplectic manifolds are special cases of Calabi-", "type": "text"}], "index": 15}, {"bbox": [110, 359, 500, 372], "spans": [{"bbox": [110, 359, 500, 372], "score": 1.0, "content": "Yau manifolds (the top holomorphic form which trivializes the canonical line", "type": "text"}], "index": 16}, {"bbox": [110, 374, 500, 387], "spans": [{"bbox": [110, 374, 428, 387], "score": 1.0, "content": "bundle is given by a suitable power of the holomorphic 2-form ", "type": "text"}, {"bbox": [429, 378, 437, 384], "score": 0.87, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [437, 374, 500, 387], "score": 1.0, "content": "). In dimen-", "type": "text"}], "index": 17}, {"bbox": [110, 388, 500, 402], "spans": [{"bbox": [110, 388, 500, 402], "score": 1.0, "content": "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "type": "text"}], "index": 18}, {"bbox": [109, 402, 502, 416], "spans": [{"bbox": [109, 402, 502, 416], "score": 1.0, "content": "irreducible symplectic manifolds appear as higher-dimensional analogues of", "type": "text"}], "index": 19}, {"bbox": [110, 417, 500, 431], "spans": [{"bbox": [110, 417, 500, 431], "score": 1.0, "content": "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "type": "text"}], "index": 20}, {"bbox": [109, 432, 500, 444], "spans": [{"bbox": [109, 432, 500, 444], "score": 1.0, "content": "are very few explicit examples of irreducible symplectic manifolds. Indeed", "type": "text"}], "index": 21}, {"bbox": [110, 446, 502, 459], "spans": [{"bbox": [110, 446, 502, 459], "score": 1.0, "content": "almost all known examples turn out to be birational to two standard series of", "type": "text"}], "index": 22}, {"bbox": [110, 461, 500, 474], "spans": [{"bbox": [110, 461, 500, 474], "score": 1.0, "content": "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "type": "text"}], "index": 23}, {"bbox": [110, 475, 499, 488], "spans": [{"bbox": [110, 475, 499, 488], "score": 1.0, "content": "varieties (both series were first studied in [2]), but quite recently O’Grady", "type": "text"}], "index": 24}, {"bbox": [109, 489, 500, 503], "spans": [{"bbox": [109, 489, 500, 503], "score": 1.0, "content": "has constructed irreducible symplectic manifolds which are not birational to", "type": "text"}], "index": 25}, {"bbox": [110, 503, 357, 517], "spans": [{"bbox": [110, 503, 357, 517], "score": 1.0, "content": "any of the elements of the two groups (see [10]).", "type": "text"}], "index": 26}], "index": 20.5}, {"type": "text", "bbox": [126, 516, 347, 529], "lines": [{"bbox": [127, 517, 346, 531], "spans": [{"bbox": [127, 517, 346, 531], "score": 1.0, "content": "Finally, let us recall from [4] the following:", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [110, 530, 500, 587], "lines": [{"bbox": [127, 531, 498, 546], "spans": [{"bbox": [127, 531, 243, 546], "score": 1.0, "content": "Definition 1.2: Let ", "type": "text"}, {"bbox": [243, 534, 254, 542], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [255, 531, 491, 546], "score": 1.0, "content": " be a Calabi-Yau n-fold, with Kaehler form ", "type": "text"}, {"bbox": [491, 537, 498, 542], "score": 0.33, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 7}], "index": 28}, {"bbox": [111, 547, 499, 560], "spans": [{"bbox": [111, 547, 338, 560], "score": 1.0, "content": "and holomorphic nowhere vanishing n-form ", "type": "text"}, {"bbox": [338, 548, 347, 557], "score": 0.45, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 547, 356, 560], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [356, 547, 365, 557], "score": 0.4, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [366, 547, 402, 560], "score": 1.0, "content": " (real) ", "type": "text"}, {"bbox": [402, 551, 409, 557], "score": 0.57, "content": "\\boldsymbol{n}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [410, 547, 499, 560], "score": 1.0, "content": "-dimensional sub-", "type": "text"}], "index": 29}, {"bbox": [111, 561, 500, 575], "spans": [{"bbox": [111, 561, 159, 575], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [159, 563, 219, 573], "score": 0.9, "content": "j:\\Lambda\\hookrightarrow X", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [219, 561, 237, 575], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [238, 563, 249, 571], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [249, 561, 500, 575], "score": 1.0, "content": " is called special Lagrangian if the following two", "type": "text"}], "index": 30}, {"bbox": [111, 576, 232, 589], "spans": [{"bbox": [111, 576, 232, 589], "score": 1.0, "content": "conditions are satisfied:", "type": "text"}], "index": 31}], "index": 29.5}, {"type": "text", "bbox": [129, 588, 388, 602], "lines": [{"bbox": [129, 590, 387, 604], "spans": [{"bbox": [129, 590, 142, 604], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 591, 151, 601], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [152, 590, 305, 604], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [306, 594, 314, 600], "score": 0.67, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [314, 590, 342, 604], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [343, 592, 383, 603], "score": 0.91, "content": "j^{*}\\omega=0", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [383, 590, 387, 604], "score": 1.0, "content": ";", "type": "text"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [109, 603, 502, 631], "lines": [{"bbox": [127, 604, 501, 620], "spans": [{"bbox": [127, 604, 262, 620], "score": 1.0, "content": "2) there exists a multiple ", "type": "text"}, {"bbox": [262, 606, 274, 615], "score": 0.88, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [274, 604, 292, 620], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [293, 606, 302, 615], "score": 0.79, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 604, 357, 620], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [358, 605, 419, 618], "score": 0.67, "content": "j^{*}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [420, 604, 501, 620], "score": 1.0, "content": "; one can prove", "type": "text"}], "index": 33}, {"bbox": [111, 619, 351, 634], "spans": [{"bbox": [111, 619, 351, 634], "score": 1.0, "content": "(see [4]) that both conditions are equivalent to:", "type": "text"}], "index": 34}], "index": 33.5}, {"type": "text", "bbox": [127, 631, 249, 645], "lines": [{"bbox": [128, 631, 249, 648], "spans": [{"bbox": [128, 634, 134, 644], "score": 0.26, "content": "\\mathit{1}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [135, 631, 146, 648], "score": 1.0, "content": "’) ", "type": "text"}, {"bbox": [146, 634, 248, 647], "score": 0.92, "content": "j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [248, 631, 249, 648], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 35}, {"type": "text", "bbox": [110, 646, 500, 674], "lines": [{"bbox": [127, 646, 501, 663], "spans": [{"bbox": [127, 646, 204, 663], "score": 1.0, "content": "The condition ", "type": "text"}, {"bbox": [204, 649, 213, 658], "score": 0.7, "content": "1^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 646, 501, 663], "score": 1.0, "content": ") in the previous definition means that the real part of", "type": "text"}], "index": 36}, {"bbox": [110, 661, 500, 676], "spans": [{"bbox": [110, 663, 121, 672], "score": 0.89, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 661, 282, 676], "score": 1.0, "content": " restricts to the volume form of", "type": "text"}, {"bbox": [283, 664, 291, 672], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [292, 661, 500, 676], "score": 1.0, "content": ", induced by the Calabi-Yau Riemannian", "type": "text"}], "index": 37}], "index": 36.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [300, 692, 309, 705], "spans": [{"bbox": [300, 692, 309, 705], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [], "index": 1, "bbox_fs": [109, 128, 501, 170], "lines_deleted": true}, {"type": "text", "bbox": [109, 169, 500, 255], "lines": [{"bbox": [127, 171, 500, 186], "spans": [{"bbox": [127, 171, 500, 186], "score": 1.0, "content": "Despite their importance, there are very few explicit examples of special", "type": "text"}], "index": 3}, {"bbox": [110, 186, 501, 200], "spans": [{"bbox": [110, 186, 501, 200], "score": 1.0, "content": "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an", "type": "text"}], "index": 4}, {"bbox": [109, 200, 501, 214], "spans": [{"bbox": [109, 200, 501, 214], "score": 1.0, "content": "irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a", "type": "text"}], "index": 5}, {"bbox": [109, 214, 501, 230], "spans": [{"bbox": [109, 214, 501, 230], "score": 1.0, "content": "complete control of the special Lagrangian geometry of its submanifolds, via", "type": "text"}], "index": 6}, {"bbox": [110, 229, 499, 243], "spans": [{"bbox": [110, 229, 499, 243], "score": 1.0, "content": "a sort of hyperkaehler trick; moreover this enables us to prove that special", "type": "text"}], "index": 7}, {"bbox": [109, 243, 498, 258], "spans": [{"bbox": [109, 243, 498, 258], "score": 1.0, "content": "Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "type": "text"}], "index": 8}], "index": 5.5, "bbox_fs": [109, 171, 501, 258]}, {"type": "text", "bbox": [127, 255, 273, 269], "lines": [{"bbox": [127, 256, 272, 271], "spans": [{"bbox": [127, 256, 272, 271], "score": 1.0, "content": "We first recall the following:", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [127, 256, 272, 271]}, {"type": "text", "bbox": [110, 270, 501, 298], "lines": [{"bbox": [127, 272, 502, 287], "spans": [{"bbox": [127, 272, 320, 287], "score": 1.0, "content": "Definition 1.1: A complex manifold ", "type": "text"}, {"bbox": [320, 274, 331, 282], "score": 0.67, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 272, 502, 287], "score": 1.0, "content": " is called irreducible symplectic if", "type": "text"}], "index": 10}, {"bbox": [111, 288, 321, 300], "spans": [{"bbox": [111, 288, 321, 300], "score": 1.0, "content": "it satisfies the following three conditions:", "type": "text"}], "index": 11}], "index": 10.5, "bbox_fs": [111, 272, 502, 300]}, {"type": "text", "bbox": [129, 299, 283, 312], "lines": [{"bbox": [129, 300, 282, 314], "spans": [{"bbox": [129, 300, 142, 314], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 303, 154, 311], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [154, 300, 282, 314], "score": 1.0, "content": " is compact and Kaehler;", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [129, 300, 282, 314]}, {"type": "text", "bbox": [128, 314, 262, 327], "lines": [{"bbox": [128, 316, 260, 328], "spans": [{"bbox": [128, 317, 135, 326], "score": 0.64, "content": "\\boldsymbol{\\mathcal{Q}}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [135, 316, 142, 328], "score": 1.0, "content": ") ", "type": "text"}, {"bbox": [142, 317, 154, 326], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [154, 316, 260, 328], "score": 1.0, "content": " is simply connected;", "type": "text"}], "index": 13}], "index": 13, "bbox_fs": [128, 316, 260, 328]}, {"type": "text", "bbox": [127, 327, 482, 342], "lines": [{"bbox": [128, 329, 483, 344], "spans": [{"bbox": [128, 329, 142, 344], "score": 1.0, "content": "3) ", "type": "text"}, {"bbox": [142, 330, 199, 343], "score": 0.94, "content": "H^{0}(X,\\Omega_{X}^{2})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [199, 329, 470, 344], "score": 1.0, "content": " is spanned by an everywhere non-degenerate 2-form ", "type": "text"}, {"bbox": [470, 335, 479, 340], "score": 0.41, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 9}, {"bbox": [479, 329, 483, 344], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [128, 329, 483, 344]}, {"type": "text", "bbox": [109, 343, 500, 515], "lines": [{"bbox": [127, 345, 499, 357], "spans": [{"bbox": [127, 345, 499, 357], "score": 1.0, "content": "In particular, irreducible symplectic manifolds are special cases of Calabi-", "type": "text"}], "index": 15}, {"bbox": [110, 359, 500, 372], "spans": [{"bbox": [110, 359, 500, 372], "score": 1.0, "content": "Yau manifolds (the top holomorphic form which trivializes the canonical line", "type": "text"}], "index": 16}, {"bbox": [110, 374, 500, 387], "spans": [{"bbox": [110, 374, 428, 387], "score": 1.0, "content": "bundle is given by a suitable power of the holomorphic 2-form ", "type": "text"}, {"bbox": [429, 378, 437, 384], "score": 0.87, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [437, 374, 500, 387], "score": 1.0, "content": "). In dimen-", "type": "text"}], "index": 17}, {"bbox": [110, 388, 500, 402], "spans": [{"bbox": [110, 388, 500, 402], "score": 1.0, "content": "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "type": "text"}], "index": 18}, {"bbox": [109, 402, 502, 416], "spans": [{"bbox": [109, 402, 502, 416], "score": 1.0, "content": "irreducible symplectic manifolds appear as higher-dimensional analogues of", "type": "text"}], "index": 19}, {"bbox": [110, 417, 500, 431], "spans": [{"bbox": [110, 417, 500, 431], "score": 1.0, "content": "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "type": "text"}], "index": 20}, {"bbox": [109, 432, 500, 444], "spans": [{"bbox": [109, 432, 500, 444], "score": 1.0, "content": "are very few explicit examples of irreducible symplectic manifolds. Indeed", "type": "text"}], "index": 21}, {"bbox": [110, 446, 502, 459], "spans": [{"bbox": [110, 446, 502, 459], "score": 1.0, "content": "almost all known examples turn out to be birational to two standard series of", "type": "text"}], "index": 22}, {"bbox": [110, 461, 500, 474], "spans": [{"bbox": [110, 461, 500, 474], "score": 1.0, "content": "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "type": "text"}], "index": 23}, {"bbox": [110, 475, 499, 488], "spans": [{"bbox": [110, 475, 499, 488], "score": 1.0, "content": "varieties (both series were first studied in [2]), but quite recently O’Grady", "type": "text"}], "index": 24}, {"bbox": [109, 489, 500, 503], "spans": [{"bbox": [109, 489, 500, 503], "score": 1.0, "content": "has constructed irreducible symplectic manifolds which are not birational to", "type": "text"}], "index": 25}, {"bbox": [110, 503, 357, 517], "spans": [{"bbox": [110, 503, 357, 517], "score": 1.0, "content": "any of the elements of the two groups (see [10]).", "type": "text"}], "index": 26}], "index": 20.5, "bbox_fs": [109, 345, 502, 517]}, {"type": "text", "bbox": [126, 516, 347, 529], "lines": [{"bbox": [127, 517, 346, 531], "spans": [{"bbox": [127, 517, 346, 531], "score": 1.0, "content": "Finally, let us recall from [4] the following:", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [127, 517, 346, 531]}, {"type": "text", "bbox": [110, 530, 500, 587], "lines": [{"bbox": [127, 531, 498, 546], "spans": [{"bbox": [127, 531, 243, 546], "score": 1.0, "content": "Definition 1.2: Let ", "type": "text"}, {"bbox": [243, 534, 254, 542], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [255, 531, 491, 546], "score": 1.0, "content": " be a Calabi-Yau n-fold, with Kaehler form ", "type": "text"}, {"bbox": [491, 537, 498, 542], "score": 0.33, "content": "\\omega", "type": "inline_equation", "height": 5, "width": 7}], "index": 28}, {"bbox": [111, 547, 499, 560], "spans": [{"bbox": [111, 547, 338, 560], "score": 1.0, "content": "and holomorphic nowhere vanishing n-form ", "type": "text"}, {"bbox": [338, 548, 347, 557], "score": 0.45, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [347, 547, 356, 560], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [356, 547, 365, 557], "score": 0.4, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [366, 547, 402, 560], "score": 1.0, "content": " (real) ", "type": "text"}, {"bbox": [402, 551, 409, 557], "score": 0.57, "content": "\\boldsymbol{n}", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [410, 547, 499, 560], "score": 1.0, "content": "-dimensional sub-", "type": "text"}], "index": 29}, {"bbox": [111, 561, 500, 575], "spans": [{"bbox": [111, 561, 159, 575], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [159, 563, 219, 573], "score": 0.9, "content": "j:\\Lambda\\hookrightarrow X", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [219, 561, 237, 575], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [238, 563, 249, 571], "score": 0.87, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [249, 561, 500, 575], "score": 1.0, "content": " is called special Lagrangian if the following two", "type": "text"}], "index": 30}, {"bbox": [111, 576, 232, 589], "spans": [{"bbox": [111, 576, 232, 589], "score": 1.0, "content": "conditions are satisfied:", "type": "text"}], "index": 31}], "index": 29.5, "bbox_fs": [111, 531, 500, 589]}, {"type": "text", "bbox": [129, 588, 388, 602], "lines": [{"bbox": [129, 590, 387, 604], "spans": [{"bbox": [129, 590, 142, 604], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [142, 591, 151, 601], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [152, 590, 305, 604], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [306, 594, 314, 600], "score": 0.67, "content": "\\omega", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [314, 590, 342, 604], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [343, 592, 383, 603], "score": 0.91, "content": "j^{*}\\omega=0", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [383, 590, 387, 604], "score": 1.0, "content": ";", "type": "text"}], "index": 32}], "index": 32, "bbox_fs": [129, 590, 387, 604]}, {"type": "text", "bbox": [109, 603, 502, 631], "lines": [{"bbox": [127, 604, 501, 620], "spans": [{"bbox": [127, 604, 262, 620], "score": 1.0, "content": "2) there exists a multiple ", "type": "text"}, {"bbox": [262, 606, 274, 615], "score": 0.88, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [274, 604, 292, 620], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [293, 606, 302, 615], "score": 0.79, "content": "\\Omega", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 604, 357, 620], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [358, 605, 419, 618], "score": 0.67, "content": "j^{*}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [420, 604, 501, 620], "score": 1.0, "content": "; one can prove", "type": "text"}], "index": 33}, {"bbox": [111, 619, 351, 634], "spans": [{"bbox": [111, 619, 351, 634], "score": 1.0, "content": "(see [4]) that both conditions are equivalent to:", "type": "text"}], "index": 34}], "index": 33.5, "bbox_fs": [111, 604, 501, 634]}, {"type": "text", "bbox": [127, 631, 249, 645], "lines": [{"bbox": [128, 631, 249, 648], "spans": [{"bbox": [128, 634, 134, 644], "score": 0.26, "content": "\\mathit{1}", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [135, 631, 146, 648], "score": 1.0, "content": "’) ", "type": "text"}, {"bbox": [146, 634, 248, 647], "score": 0.92, "content": "j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [248, 631, 249, 648], "score": 1.0, "content": ".", "type": "text"}], "index": 35}], "index": 35, "bbox_fs": [128, 631, 249, 648]}, {"type": "text", "bbox": [110, 646, 500, 674], "lines": [{"bbox": [127, 646, 501, 663], "spans": [{"bbox": [127, 646, 204, 663], "score": 1.0, "content": "The condition ", "type": "text"}, {"bbox": [204, 649, 213, 658], "score": 0.7, "content": "1^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 646, 501, 663], "score": 1.0, "content": ") in the previous definition means that the real part of", "type": "text"}], "index": 36}, {"bbox": [110, 661, 500, 676], "spans": [{"bbox": [110, 663, 121, 672], "score": 0.89, "content": "\\Omega^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 661, 282, 676], "score": 1.0, "content": " restricts to the volume form of", "type": "text"}, {"bbox": [283, 664, 291, 672], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [292, 661, 500, 676], "score": 1.0, "content": ", induced by the Calabi-Yau Riemannian", "type": "text"}], "index": 37}, {"bbox": [109, 128, 502, 142], "spans": [{"bbox": [109, 128, 147, 142], "score": 1.0, "content": "metric ", "type": "text", "cross_page": true}, {"bbox": [147, 133, 154, 141], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7, "cross_page": true}, {"bbox": [154, 128, 502, 142], "score": 1.0, "content": ". In this way special Lagrangian submanifolds are considered as a", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 142, 483, 156], "spans": [{"bbox": [110, 142, 483, 156], "score": 1.0, "content": "type of calibrated submanifolds (see [4] for further details on this point).", "type": "text", "cross_page": true}], "index": 1}], "index": 36.5, "bbox_fs": [110, 646, 501, 676]}]}
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[{"type": "text", "bbox": [110, 125, 500, 168], "content": "", "index": 0}, {"type": "text", "bbox": [109, 169, 500, 255], "content": "Despite their importance, there are very few explicit examples of special Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a complete control of the special Lagrangian geometry of its submanifolds, via a sort of hyperkaehler trick; moreover this enables us to prove that special Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "index": 1}, {"type": "text", "bbox": [127, 255, 273, 269], "content": "We first recall the following:", "index": 2}, {"type": "text", "bbox": [110, 270, 501, 298], "content": "Definition 1.1: A complex manifold is called irreducible symplectic if it satisfies the following three conditions:", "index": 3}, {"type": "text", "bbox": [129, 299, 283, 312], "content": "1) is compact and Kaehler;", "index": 4}, {"type": "text", "bbox": [128, 314, 262, 327], "content": ") is simply connected;", "index": 5}, {"type": "text", "bbox": [127, 327, 482, 342], "content": "3) is spanned by an everywhere non-degenerate 2-form .", "index": 6}, {"type": "text", "bbox": [109, 343, 500, 515], "content": "In particular, irreducible symplectic manifolds are special cases of Calabi- Yau manifolds (the top holomorphic form which trivializes the canonical line bundle is given by a suitable power of the holomorphic 2-form ). In dimen- sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed irreducible symplectic manifolds appear as higher-dimensional analogues of K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there are very few explicit examples of irreducible symplectic manifolds. Indeed almost all known examples turn out to be birational to two standard series of examples: Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (both series were first studied in [2]), but quite recently O’Grady has constructed irreducible symplectic manifolds which are not birational to any of the elements of the two groups (see [10]).", "index": 7}, {"type": "text", "bbox": [126, 516, 347, 529], "content": "Finally, let us recall from [4] the following:", "index": 8}, {"type": "text", "bbox": [110, 530, 500, 587], "content": "Definition 1.2: Let be a Calabi-Yau n-fold, with Kaehler form and holomorphic nowhere vanishing n-form . (real) -dimensional sub- manifold of is called special Lagrangian if the following two conditions are satisfied:", "index": 9}, {"type": "text", "bbox": [129, 588, 388, 602], "content": "1) is Lagrangian with respect to , i.e. ;", "index": 10}, {"type": "text", "bbox": [109, 603, 502, 631], "content": "2) there exists a multiple of such that ; one can prove (see [4]) that both conditions are equivalent to:", "index": 11}, {"type": "text", "bbox": [127, 631, 249, 645], "content": "’) .", "index": 12}, {"type": "text", "bbox": [110, 646, 500, 674], "content": "The condition ) in the previous definition means that the real part of restricts to the volume form of , induced by the Calabi-Yau Riemannian metric . In this way special Lagrangian submanifolds are considered as a type of calibrated submanifolds (see [4] for further details on this point).", "index": 13}]
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[{"bbox": [127, 171, 500, 186], "content": "Despite their importance, there are very few explicit examples of special", "parent_index": 1, "line_index": 0}, {"bbox": [110, 186, 501, 200], "content": "Lagrangian submanifolds, especially in Calabi-Yau 3-folds. However, in an", "parent_index": 1, "line_index": 1}, {"bbox": [109, 200, 501, 214], "content": "irreducible symplectic 4-fold (realized as a hyperkaehler manifold) we have a", "parent_index": 1, "line_index": 2}, {"bbox": [109, 214, 501, 230], "content": "complete control of the special Lagrangian geometry of its submanifolds, via", "parent_index": 1, "line_index": 3}, {"bbox": [110, 229, 499, 243], "content": "a sort of hyperkaehler trick; moreover this enables us to prove that special", "parent_index": 1, "line_index": 4}, {"bbox": [109, 243, 498, 258], "content": "Lagrangian submanifolds retain part of the rigidity of complex submanifolds.", "parent_index": 1, "line_index": 5}, {"bbox": [127, 256, 272, 271], "content": "We first recall the following:", "parent_index": 2, "line_index": 0}, {"bbox": [127, 272, 502, 287], "content": "Definition 1.1: A complex manifold is called irreducible symplectic if", "parent_index": 3, "line_index": 0}, {"bbox": [111, 288, 321, 300], "content": "it satisfies the following three conditions:", "parent_index": 3, "line_index": 1}, {"bbox": [129, 300, 282, 314], "content": "1) is compact and Kaehler;", "parent_index": 4, "line_index": 0}, {"bbox": [128, 316, 260, 328], "content": ") is simply connected;", "parent_index": 5, "line_index": 0}, {"bbox": [128, 329, 483, 344], "content": "3) is spanned by an everywhere non-degenerate 2-form .", "parent_index": 6, "line_index": 0}, {"bbox": [127, 345, 499, 357], "content": "In particular, irreducible symplectic manifolds are special cases of Calabi-", "parent_index": 7, "line_index": 0}, {"bbox": [110, 359, 500, 372], "content": "Yau manifolds (the top holomorphic form which trivializes the canonical line", "parent_index": 7, "line_index": 1}, {"bbox": [110, 374, 500, 387], "content": "bundle is given by a suitable power of the holomorphic 2-form ). In dimen-", "parent_index": 7, "line_index": 2}, {"bbox": [110, 388, 500, 402], "content": "sion 2, K3 surfaces are the only irreducible symplectic manifolds, and indeed", "parent_index": 7, "line_index": 3}, {"bbox": [109, 402, 502, 416], "content": "irreducible symplectic manifolds appear as higher-dimensional analogues of", "parent_index": 7, "line_index": 4}, {"bbox": [110, 417, 500, 431], "content": "K3 surfaces, as strongly suggested in [5]. Unfortunately, up to now there", "parent_index": 7, "line_index": 5}, {"bbox": [109, 432, 500, 444], "content": "are very few explicit examples of irreducible symplectic manifolds. Indeed", "parent_index": 7, "line_index": 6}, {"bbox": [110, 446, 502, 459], "content": "almost all known examples turn out to be birational to two standard series of", "parent_index": 7, "line_index": 7}, {"bbox": [110, 461, 500, 474], "content": "examples: Hilbert schemes of points on K3 surfaces and generalized Kummer", "parent_index": 7, "line_index": 8}, {"bbox": [110, 475, 499, 488], "content": "varieties (both series were first studied in [2]), but quite recently O’Grady", "parent_index": 7, "line_index": 9}, {"bbox": [109, 489, 500, 503], "content": "has constructed irreducible symplectic manifolds which are not birational to", "parent_index": 7, "line_index": 10}, {"bbox": [110, 503, 357, 517], "content": "any of the elements of the two groups (see [10]).", "parent_index": 7, "line_index": 11}, {"bbox": [127, 517, 346, 531], "content": "Finally, let us recall from [4] the following:", "parent_index": 8, "line_index": 0}, {"bbox": [127, 531, 498, 546], "content": "Definition 1.2: Let be a Calabi-Yau n-fold, with Kaehler form", "parent_index": 9, "line_index": 0}, {"bbox": [111, 547, 499, 560], "content": "and holomorphic nowhere vanishing n-form . (real) -dimensional sub-", "parent_index": 9, "line_index": 1}, {"bbox": [111, 561, 500, 575], "content": "manifold of is called special Lagrangian if the following two", "parent_index": 9, "line_index": 2}, {"bbox": [111, 576, 232, 589], "content": "conditions are satisfied:", "parent_index": 9, "line_index": 3}, {"bbox": [129, 590, 387, 604], "content": "1) is Lagrangian with respect to , i.e. ;", "parent_index": 10, "line_index": 0}, {"bbox": [127, 604, 501, 620], "content": "2) there exists a multiple of such that ; one can prove", "parent_index": 11, "line_index": 0}, {"bbox": [111, 619, 351, 634], "content": "(see [4]) that both conditions are equivalent to:", "parent_index": 11, "line_index": 1}, {"bbox": [128, 631, 249, 648], "content": "’) .", "parent_index": 12, "line_index": 0}, {"bbox": [127, 646, 501, 663], "content": "The condition ) in the previous definition means that the real part of", "parent_index": 13, "line_index": 0}, {"bbox": [110, 661, 500, 676], "content": "restricts to the volume form of , induced by the Calabi-Yau Riemannian", "parent_index": 13, "line_index": 1}, {"bbox": [109, 128, 502, 142], "content": "metric . In this way special Lagrangian submanifolds are considered as a", "parent_index": 13, "line_index": 2}, {"bbox": [110, 142, 483, 156], "content": "type of calibrated submanifolds (see [4] for further details on this point).", "parent_index": 13, "line_index": 3}]
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[{"bbox": [320, 274, 331, 282], "content": "X", "parent_index": 3, "subtype": "inline"}, {"bbox": [142, 303, 154, 311], "content": "X", "parent_index": 4, "subtype": "inline"}, {"bbox": [128, 317, 135, 326], "content": "\\boldsymbol{\\mathcal{Q}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [142, 317, 154, 326], "content": "X", "parent_index": 5, "subtype": "inline"}, {"bbox": [142, 330, 199, 343], "content": "H^{0}(X,\\Omega_{X}^{2})", "parent_index": 6, "subtype": "inline"}, {"bbox": [470, 335, 479, 340], "content": "\\omega", "parent_index": 6, "subtype": "inline"}, {"bbox": [429, 378, 437, 384], "content": "\\omega", "parent_index": 7, "subtype": "inline"}, {"bbox": [243, 534, 254, 542], "content": "X", "parent_index": 9, "subtype": "inline"}, {"bbox": [491, 537, 498, 542], "content": "\\omega", "parent_index": 9, "subtype": "inline"}, {"bbox": [338, 548, 347, 557], "content": "\\Omega", "parent_index": 9, "subtype": "inline"}, {"bbox": [356, 547, 365, 557], "content": "A", "parent_index": 9, "subtype": "inline"}, {"bbox": [402, 551, 409, 557], "content": "\\boldsymbol{n}", "parent_index": 9, "subtype": "inline"}, {"bbox": [159, 563, 219, 573], "content": "j:\\Lambda\\hookrightarrow X", "parent_index": 9, "subtype": "inline"}, {"bbox": [238, 563, 249, 571], "content": "X", "parent_index": 9, "subtype": "inline"}, {"bbox": [142, 591, 151, 601], "content": "\\Lambda", "parent_index": 10, "subtype": "inline"}, {"bbox": [306, 594, 314, 600], "content": "\\omega", "parent_index": 10, "subtype": "inline"}, {"bbox": [343, 592, 383, 603], "content": "j^{*}\\omega=0", "parent_index": 10, "subtype": "inline"}, {"bbox": [262, 606, 274, 615], "content": "\\Omega^{\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [293, 606, 302, 615], "content": "\\Omega", "parent_index": 11, "subtype": "inline"}, {"bbox": [358, 605, 419, 618], "content": "j^{*}\\mathrm{Im}(\\Omega^{\\prime})\\!=\\!0_{.}", "parent_index": 11, "subtype": "inline"}, {"bbox": [128, 634, 134, 644], "content": "\\mathit{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [146, 634, 248, 647], "content": "j^{*}\\mathrm{Re}(\\Omega^{\\prime})=V o l_{g}(\\Lambda)", "parent_index": 12, "subtype": "inline"}, {"bbox": [204, 649, 213, 658], "content": "1^{\\prime}", "parent_index": 13, "subtype": "inline"}, {"bbox": [110, 663, 121, 672], "content": "\\Omega^{\\prime}", "parent_index": 13, "subtype": "inline"}, {"bbox": [283, 664, 291, 672], "content": "\\Lambda", "parent_index": 13, "subtype": "inline"}, {"bbox": [147, 133, 154, 141], "content": "g", "parent_index": 13, "subtype": "inline"}]
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# 2 Characterization of special Lagrangian submanifolds
In this section we will describe all special Lagrangian submanifolds of an irreducible symplectic 4-fold $X$ (having fixed a Kaehler class $[\omega]$ in the Kaehler cone). The key result is the following:
Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures.
Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold $X$ . By Yau’s Theorem this determines a unique hyperkaehler metric $g$ . Choose a hyperkaehler structure $(I,J,K)$ compatible with the metric $g$ (notice that the triple $(I,J,K)$ is not uniquely determined) and consider the associated symplectic structures $\omega_{I}(.,.):=g(I.,.)$ , $\omega_{J}(.,.):=g(J.,.)$ and $\omega_{K}(.,.):=g(K.,.)$ .
Consider a special Lagrangian submanifold $\Lambda$ in the complex structure $K$ (this is not restrictive, since $(I,J,K)$ is not uniquely determined); that is assume that $\Lambda$ is calibrated by the real part of the holomorphic (in the structure $K$ ) 4-form:
$$
\Omega_{K}:=\frac{1}{2!}(\omega_{I}+i\omega_{J})^{2}.
$$
Notice that the real and immaginary part of $\Omega_{K}$ are then given by:
$$
\mathrm{Re}(\Omega_{K})=\frac{1}{2}(\omega_{I}^{2}-\omega_{J}^{2})\quad\mathrm{Im}(\Omega_{K})=\omega_{I}\wedge\omega_{J}.
$$
Obviously, by the property of being special Lagrangian we have that $\Lambda$ is Lagrangian with respect to $\omega_{K}$ . We will prove that having fixed the calibration, if $\Lambda$ is not Lagrangian also with respect to $\omega_{I}$ , then it is necessarily Lagrangian with respect to $\omega_{J}$ . First we work locally and consider $V:=T_{p}\Lambda$ $y\in\Lambda)$ , spanned by $(w_{1},w_{2},w_{3},w_{4})$ . Since $\Lambda$ is assumed not to be Lagrangian with respect to $\omega_{I}$ , we have to deal with two cases.
First case: $V$ is a symplectic vector space for the structure $\omega_{I}$ . In this case we can choose a symplectic basis for $V$ and this can always be chosen to be of the form $v_{1},I v_{1},v_{2},I v_{2}$ . Then $V$ is Lagrangian in the symplectic structure $\omega_{J}$ ; indeed $\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\omega_{K}(v_{1},v_{1})=0$ ; analogously for $\omega_{J}(v_{2},I v_{2})$ ; $\omega_{J}(v_{1},I v_{2})\;=\;g(J v_{1},I v_{2})\;=\;-\omega_{K}(v_{1},v_{2})\;=\;0$ since $v_{1},v_{2}$ belong to a Lagrangian subspace of $\omega_{K}$ , and analogously for $\omega_{J}(v_{2},I v_{1})=-\omega_{K}(v_{2},v_{1})=0$ . Thus $V$ is also Lagrangian for the symplectic structure $\omega_{J}$ .
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<html><body>
<h1 data-bbox="111 175 500 216">2 Characterization of special Lagrangian submanifolds </h1>
<p data-bbox="110 227 500 271">In this section we will describe all special Lagrangian submanifolds of an irreducible symplectic 4-fold $X$ (having fixed a Kaehler class $[\omega]$ in the Kaehler cone). The key result is the following: </p>
<p data-bbox="110 272 500 314">Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures. </p>
<p data-bbox="109 315 500 402">Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold $X$ . By Yau’s Theorem this determines a unique hyperkaehler metric $g$ . Choose a hyperkaehler structure $(I,J,K)$ compatible with the metric $g$ (notice that the triple $(I,J,K)$ is not uniquely determined) and consider the associated symplectic structures $\omega_{I}(.,.):=g(I.,.)$ , $\omega_{J}(.,.):=g(J.,.)$ and $\omega_{K}(.,.):=g(K.,.)$ . </p>
<p data-bbox="109 402 500 459">Consider a special Lagrangian submanifold $\Lambda$ in the complex structure $K$ (this is not restrictive, since $(I,J,K)$ is not uniquely determined); that is assume that $\Lambda$ is calibrated by the real part of the holomorphic (in the structure $K$ ) 4-form: </p>
<div class="equation" data-bbox="250 459 359 486">$$
\Omega_{K}:=\frac{1}{2!}(\omega_{I}+i\omega_{J})^{2}.
$$</div>
<p data-bbox="109 488 456 503">Notice that the real and immaginary part of $\Omega_{K}$ are then given by: </p>
<div class="equation" data-bbox="193 514 416 541">$$
\mathrm{Re}(\Omega_{K})=\frac{1}{2}(\omega_{I}^{2}-\omega_{J}^{2})\quad\mathrm{Im}(\Omega_{K})=\omega_{I}\wedge\omega_{J}.
$$</div>
<p data-bbox="109 543 500 630">Obviously, by the property of being special Lagrangian we have that $\Lambda$ is Lagrangian with respect to $\omega_{K}$ . We will prove that having fixed the calibration, if $\Lambda$ is not Lagrangian also with respect to $\omega_{I}$ , then it is necessarily Lagrangian with respect to $\omega_{J}$ . First we work locally and consider $V:=T_{p}\Lambda$ $y\in\Lambda)$ , spanned by $(w_{1},w_{2},w_{3},w_{4})$ . Since $\Lambda$ is assumed not to be Lagrangian with respect to $\omega_{I}$ , we have to deal with two cases. </p>
<p data-bbox="110 631 500 659">First case: $V$ is a symplectic vector space for the structure $\omega_{I}$ . In this case we can choose a symplectic basis for $V$ and this can always be chosen to be of the form $v_{1},I v_{1},v_{2},I v_{2}$ . Then $V$ is Lagrangian in the symplectic structure $\omega_{J}$ ; indeed $\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\omega_{K}(v_{1},v_{1})=0$ ; analogously for $\omega_{J}(v_{2},I v_{2})$ ; $\omega_{J}(v_{1},I v_{2})\;=\;g(J v_{1},I v_{2})\;=\;-\omega_{K}(v_{1},v_{2})\;=\;0$ since $v_{1},v_{2}$ belong to a Lagrangian subspace of $\omega_{K}$ , and analogously for $\omega_{J}(v_{2},I v_{1})=-\omega_{K}(v_{2},v_{1})=0$ . Thus $V$ is also Lagrangian for the symplectic structure $\omega_{J}$ . </p>
</body></html>
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0001060v1
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In this way special Lagrangian submanifolds are considered as a", "type": "text"}], "index": 0}, {"bbox": [110, 142, 483, 156], "spans": [{"bbox": [110, 142, 483, 156], "score": 1.0, "content": "type of calibrated submanifolds (see [4] for further details on this point).", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "title", "bbox": [111, 175, 500, 216], "lines": [{"bbox": [111, 178, 498, 197], "spans": [{"bbox": [111, 181, 122, 194], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [138, 178, 498, 197], "score": 1.0, "content": "Characterization of special Lagrangian sub-", "type": "text"}], "index": 2}, {"bbox": [140, 201, 221, 218], "spans": [{"bbox": [140, 201, 221, 218], "score": 1.0, "content": "manifolds", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [110, 227, 500, 271], "lines": [{"bbox": [109, 231, 498, 244], "spans": [{"bbox": [109, 231, 498, 244], "score": 1.0, "content": "In this section we will describe all special Lagrangian submanifolds of an ir-", "type": "text"}], "index": 4}, {"bbox": [110, 245, 499, 259], "spans": [{"bbox": [110, 245, 248, 259], "score": 1.0, "content": "reducible symplectic 4-fold ", "type": "text"}, {"bbox": [248, 247, 259, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [259, 245, 409, 259], "score": 1.0, "content": " (having fixed a Kaehler class ", "type": "text"}, {"bbox": [410, 246, 424, 258], "score": 0.88, "content": "[\\omega]", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [424, 245, 499, 259], "score": 1.0, "content": " in the Kaehler", "type": "text"}], "index": 5}, {"bbox": [110, 259, 305, 273], "spans": [{"bbox": [110, 259, 305, 273], "score": 1.0, "content": "cone). The key result is the following:", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [110, 272, 500, 314], "lines": [{"bbox": [127, 273, 501, 287], "spans": [{"bbox": [127, 273, 501, 287], "score": 1.0, "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an", "type": "text"}], "index": 7}, {"bbox": [110, 288, 502, 302], "spans": [{"bbox": [110, 288, 502, 302], "score": 1.0, "content": "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "type": "text"}], "index": 8}, {"bbox": [111, 302, 427, 316], "spans": [{"bbox": [111, 302, 427, 316], "score": 1.0, "content": "Lagrangian with respect to two different symplectic structures.", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [109, 315, 500, 402], "lines": [{"bbox": [127, 317, 499, 330], "spans": [{"bbox": [127, 317, 499, 330], "score": 1.0, "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "type": "text"}], "index": 10}, {"bbox": [110, 332, 500, 345], "spans": [{"bbox": [110, 332, 134, 345], "score": 1.0, "content": "fold ", "type": "text"}, {"bbox": [134, 333, 145, 342], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [146, 332, 500, 345], "score": 1.0, "content": ". By Yau’s Theorem this determines a unique hyperkaehler metric", "type": "text"}], "index": 11}, {"bbox": [110, 346, 499, 361], "spans": [{"bbox": [110, 351, 117, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 346, 299, 361], "score": 1.0, "content": ". Choose a hyperkaehler structure ", "type": "text"}, {"bbox": [299, 347, 342, 360], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [343, 346, 492, 361], "score": 1.0, "content": " compatible with the metric ", "type": "text"}, {"bbox": [493, 351, 499, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}], "index": 12}, {"bbox": [110, 361, 500, 375], "spans": [{"bbox": [110, 361, 232, 375], "score": 1.0, "content": "(notice that the triple ", "type": "text"}, {"bbox": [232, 361, 275, 374], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [276, 361, 500, 375], "score": 1.0, "content": " is not uniquely determined) and consider", "type": "text"}], "index": 13}, {"bbox": [109, 375, 500, 389], "spans": [{"bbox": [109, 375, 298, 389], "score": 1.0, "content": "the associated symplectic structures ", "type": "text"}, {"bbox": [298, 376, 383, 388], "score": 0.84, "content": "\\omega_{I}(.,.):=g(I.,.)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [384, 375, 389, 389], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [390, 376, 476, 388], "score": 0.91, "content": "\\omega_{J}(.,.):=g(J.,.)", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [477, 375, 500, 389], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [110, 388, 208, 404], "spans": [{"bbox": [110, 390, 203, 403], "score": 0.94, "content": "\\omega_{K}(.,.):=g(K.,.)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [203, 388, 208, 404], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5}, {"type": "text", "bbox": [109, 402, 500, 459], "lines": [{"bbox": [127, 403, 500, 418], "spans": [{"bbox": [127, 403, 356, 418], "score": 1.0, "content": "Consider a special Lagrangian submanifold ", "type": "text"}, {"bbox": [357, 405, 365, 414], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [366, 403, 500, 418], "score": 1.0, "content": " in the complex structure", "type": "text"}], "index": 16}, {"bbox": [110, 417, 500, 433], "spans": [{"bbox": [110, 420, 121, 428], "score": 0.89, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [122, 417, 276, 433], "score": 1.0, "content": " (this is not restrictive, since ", "type": "text"}, {"bbox": [277, 419, 320, 432], "score": 0.95, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [320, 417, 500, 433], "score": 1.0, "content": " is not uniquely determined); that", "type": "text"}], "index": 17}, {"bbox": [108, 431, 500, 447], "spans": [{"bbox": [108, 431, 190, 447], "score": 1.0, "content": "is assume that ", "type": "text"}, {"bbox": [190, 434, 199, 443], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 431, 500, 447], "score": 1.0, "content": " is calibrated by the real part of the holomorphic (in the", "type": "text"}], "index": 18}, {"bbox": [109, 446, 218, 461], "spans": [{"bbox": [109, 446, 160, 461], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 449, 172, 458], "score": 0.88, "content": "K", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [172, 446, 218, 461], "score": 1.0, "content": ") 4-form:", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [109, 488, 456, 503], "lines": [{"bbox": [109, 491, 455, 505], "spans": [{"bbox": [109, 491, 341, 505], "score": 1.0, "content": "Notice that the real and immaginary part of ", "type": "text"}, {"bbox": [341, 492, 358, 503], "score": 0.93, "content": "\\Omega_{K}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [358, 491, 455, 505], "score": 1.0, "content": " are then given by:", "type": "text"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [109, 543, 500, 630], "lines": [{"bbox": [127, 545, 501, 560], "spans": [{"bbox": [127, 545, 501, 560], "score": 1.0, "content": "Obviously, by the property of being special Lagrangian we have that", "type": "text"}], "index": 23}, {"bbox": [110, 560, 500, 575], "spans": [{"bbox": [110, 562, 119, 571], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 560, 282, 575], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [283, 565, 299, 573], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [299, 560, 500, 575], "score": 1.0, "content": ". We will prove that having fixed the", "type": "text"}], "index": 24}, {"bbox": [109, 575, 500, 589], "spans": [{"bbox": [109, 575, 185, 589], "score": 1.0, "content": "calibration, if ", "type": "text"}, {"bbox": [185, 577, 194, 586], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [194, 575, 403, 589], "score": 1.0, "content": " is not Lagrangian also with respect to ", "type": "text"}, {"bbox": [404, 580, 416, 587], "score": 0.88, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [417, 575, 500, 589], "score": 1.0, "content": ", then it is nec-", "type": "text"}], "index": 25}, {"bbox": [109, 590, 500, 603], "spans": [{"bbox": [109, 590, 296, 603], "score": 1.0, "content": "essarily Lagrangian with respect to ", "type": "text"}, {"bbox": [297, 594, 310, 602], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [310, 590, 500, 603], "score": 1.0, "content": ". First we work locally and consider", "type": "text"}], "index": 26}, {"bbox": [110, 604, 501, 618], "spans": [{"bbox": [110, 605, 161, 618], "score": 0.88, "content": "V:=T_{p}\\Lambda", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [161, 604, 167, 618], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [168, 605, 203, 617], "score": 0.65, "content": "y\\in\\Lambda)", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [204, 604, 275, 618], "score": 1.0, "content": ", spanned by ", "type": "text"}, {"bbox": [276, 604, 353, 617], "score": 0.91, "content": "(w_{1},w_{2},w_{3},w_{4})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [353, 604, 394, 618], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [394, 605, 403, 614], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [403, 604, 501, 618], "score": 1.0, "content": " is assumed not to", "type": "text"}], "index": 27}, {"bbox": [110, 618, 448, 632], "spans": [{"bbox": [110, 618, 268, 632], "score": 1.0, "content": "be Lagrangian with respect to ", "type": "text"}, {"bbox": [268, 623, 281, 630], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [281, 618, 448, 632], "score": 1.0, "content": ", we have to deal with two cases.", "type": "text"}], "index": 28}], "index": 25.5}, {"type": "text", "bbox": [110, 631, 500, 659], "lines": [{"bbox": [126, 631, 500, 647], "spans": [{"bbox": [126, 631, 187, 647], "score": 1.0, "content": "First case: ", "type": "text"}, {"bbox": [188, 634, 197, 643], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [198, 631, 441, 647], "score": 1.0, "content": " is a symplectic vector space for the structure ", "type": "text"}, {"bbox": [442, 637, 454, 645], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [454, 631, 500, 647], "score": 1.0, "content": ". In this", "type": "text"}], "index": 29}, {"bbox": [109, 647, 499, 661], "spans": [{"bbox": [109, 647, 320, 661], "score": 1.0, "content": "case we can choose a symplectic basis for ", "type": "text"}, {"bbox": [321, 649, 330, 657], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [331, 647, 499, 661], "score": 1.0, "content": " and this can always be chosen to", "type": "text"}], "index": 30}], "index": 29.5}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [300, 692, 309, 705], "spans": [{"bbox": [300, 692, 309, 705], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [], "index": 0.5, "bbox_fs": [109, 128, 502, 156], "lines_deleted": true}, {"type": "title", "bbox": [111, 175, 500, 216], "lines": [{"bbox": [111, 178, 498, 197], "spans": [{"bbox": [111, 181, 122, 194], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [138, 178, 498, 197], "score": 1.0, "content": "Characterization of special Lagrangian sub-", "type": "text"}], "index": 2}, {"bbox": [140, 201, 221, 218], "spans": [{"bbox": [140, 201, 221, 218], "score": 1.0, "content": "manifolds", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [110, 227, 500, 271], "lines": [{"bbox": [109, 231, 498, 244], "spans": [{"bbox": [109, 231, 498, 244], "score": 1.0, "content": "In this section we will describe all special Lagrangian submanifolds of an ir-", "type": "text"}], "index": 4}, {"bbox": [110, 245, 499, 259], "spans": [{"bbox": [110, 245, 248, 259], "score": 1.0, "content": "reducible symplectic 4-fold ", "type": "text"}, {"bbox": [248, 247, 259, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [259, 245, 409, 259], "score": 1.0, "content": " (having fixed a Kaehler class ", "type": "text"}, {"bbox": [410, 246, 424, 258], "score": 0.88, "content": "[\\omega]", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [424, 245, 499, 259], "score": 1.0, "content": " in the Kaehler", "type": "text"}], "index": 5}, {"bbox": [110, 259, 305, 273], "spans": [{"bbox": [110, 259, 305, 273], "score": 1.0, "content": "cone). The key result is the following:", "type": "text"}], "index": 6}], "index": 5, "bbox_fs": [109, 231, 499, 273]}, {"type": "text", "bbox": [110, 272, 500, 314], "lines": [{"bbox": [127, 273, 501, 287], "spans": [{"bbox": [127, 273, 501, 287], "score": 1.0, "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an", "type": "text"}], "index": 7}, {"bbox": [110, 288, 502, 302], "spans": [{"bbox": [110, 288, 502, 302], "score": 1.0, "content": "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "type": "text"}], "index": 8}, {"bbox": [111, 302, 427, 316], "spans": [{"bbox": [111, 302, 427, 316], "score": 1.0, "content": "Lagrangian with respect to two different symplectic structures.", "type": "text"}], "index": 9}], "index": 8, "bbox_fs": [110, 273, 502, 316]}, {"type": "text", "bbox": [109, 315, 500, 402], "lines": [{"bbox": [127, 317, 499, 330], "spans": [{"bbox": [127, 317, 499, 330], "score": 1.0, "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "type": "text"}], "index": 10}, {"bbox": [110, 332, 500, 345], "spans": [{"bbox": [110, 332, 134, 345], "score": 1.0, "content": "fold ", "type": "text"}, {"bbox": [134, 333, 145, 342], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [146, 332, 500, 345], "score": 1.0, "content": ". By Yau’s Theorem this determines a unique hyperkaehler metric", "type": "text"}], "index": 11}, {"bbox": [110, 346, 499, 361], "spans": [{"bbox": [110, 351, 117, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 346, 299, 361], "score": 1.0, "content": ". Choose a hyperkaehler structure ", "type": "text"}, {"bbox": [299, 347, 342, 360], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [343, 346, 492, 361], "score": 1.0, "content": " compatible with the metric ", "type": "text"}, {"bbox": [493, 351, 499, 359], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}], "index": 12}, {"bbox": [110, 361, 500, 375], "spans": [{"bbox": [110, 361, 232, 375], "score": 1.0, "content": "(notice that the triple ", "type": "text"}, {"bbox": [232, 361, 275, 374], "score": 0.94, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [276, 361, 500, 375], "score": 1.0, "content": " is not uniquely determined) and consider", "type": "text"}], "index": 13}, {"bbox": [109, 375, 500, 389], "spans": [{"bbox": [109, 375, 298, 389], "score": 1.0, "content": "the associated symplectic structures ", "type": "text"}, {"bbox": [298, 376, 383, 388], "score": 0.84, "content": "\\omega_{I}(.,.):=g(I.,.)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [384, 375, 389, 389], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [390, 376, 476, 388], "score": 0.91, "content": "\\omega_{J}(.,.):=g(J.,.)", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [477, 375, 500, 389], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [110, 388, 208, 404], "spans": [{"bbox": [110, 390, 203, 403], "score": 0.94, "content": "\\omega_{K}(.,.):=g(K.,.)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [203, 388, 208, 404], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5, "bbox_fs": [109, 317, 500, 404]}, {"type": "text", "bbox": [109, 402, 500, 459], "lines": [{"bbox": [127, 403, 500, 418], "spans": [{"bbox": [127, 403, 356, 418], "score": 1.0, "content": "Consider a special Lagrangian submanifold ", "type": "text"}, {"bbox": [357, 405, 365, 414], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [366, 403, 500, 418], "score": 1.0, "content": " in the complex structure", "type": "text"}], "index": 16}, {"bbox": [110, 417, 500, 433], "spans": [{"bbox": [110, 420, 121, 428], "score": 0.89, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [122, 417, 276, 433], "score": 1.0, "content": " (this is not restrictive, since ", "type": "text"}, {"bbox": [277, 419, 320, 432], "score": 0.95, "content": "(I,J,K)", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [320, 417, 500, 433], "score": 1.0, "content": " is not uniquely determined); that", "type": "text"}], "index": 17}, {"bbox": [108, 431, 500, 447], "spans": [{"bbox": [108, 431, 190, 447], "score": 1.0, "content": "is assume that ", "type": "text"}, {"bbox": [190, 434, 199, 443], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 431, 500, 447], "score": 1.0, "content": " is calibrated by the real part of the holomorphic (in the", "type": "text"}], "index": 18}, {"bbox": [109, 446, 218, 461], "spans": [{"bbox": [109, 446, 160, 461], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 449, 172, 458], "score": 0.88, "content": "K", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [172, 446, 218, 461], "score": 1.0, "content": ") 4-form:", "type": "text"}], "index": 19}], "index": 17.5, "bbox_fs": [108, 403, 500, 461]}, {"type": "interline_equation", "bbox": [250, 459, 359, 486], "lines": [{"bbox": [250, 459, 359, 486], "spans": [{"bbox": [250, 459, 359, 486], "score": 0.95, "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [109, 488, 456, 503], "lines": [{"bbox": [109, 491, 455, 505], "spans": [{"bbox": [109, 491, 341, 505], "score": 1.0, "content": "Notice that the real and immaginary part of ", "type": "text"}, {"bbox": [341, 492, 358, 503], "score": 0.93, "content": "\\Omega_{K}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [358, 491, 455, 505], "score": 1.0, "content": " are then given by:", "type": "text"}], "index": 21}], "index": 21, "bbox_fs": [109, 491, 455, 505]}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "lines": [{"bbox": [193, 514, 416, 541], "spans": [{"bbox": [193, 514, 416, 541], "score": 0.94, "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [109, 543, 500, 630], "lines": [{"bbox": [127, 545, 501, 560], "spans": [{"bbox": [127, 545, 501, 560], "score": 1.0, "content": "Obviously, by the property of being special Lagrangian we have that", "type": "text"}], "index": 23}, {"bbox": [110, 560, 500, 575], "spans": [{"bbox": [110, 562, 119, 571], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 560, 282, 575], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [283, 565, 299, 573], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [299, 560, 500, 575], "score": 1.0, "content": ". We will prove that having fixed the", "type": "text"}], "index": 24}, {"bbox": [109, 575, 500, 589], "spans": [{"bbox": [109, 575, 185, 589], "score": 1.0, "content": "calibration, if ", "type": "text"}, {"bbox": [185, 577, 194, 586], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [194, 575, 403, 589], "score": 1.0, "content": " is not Lagrangian also with respect to ", "type": "text"}, {"bbox": [404, 580, 416, 587], "score": 0.88, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [417, 575, 500, 589], "score": 1.0, "content": ", then it is nec-", "type": "text"}], "index": 25}, {"bbox": [109, 590, 500, 603], "spans": [{"bbox": [109, 590, 296, 603], "score": 1.0, "content": "essarily Lagrangian with respect to ", "type": "text"}, {"bbox": [297, 594, 310, 602], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [310, 590, 500, 603], "score": 1.0, "content": ". First we work locally and consider", "type": "text"}], "index": 26}, {"bbox": [110, 604, 501, 618], "spans": [{"bbox": [110, 605, 161, 618], "score": 0.88, "content": "V:=T_{p}\\Lambda", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [161, 604, 167, 618], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [168, 605, 203, 617], "score": 0.65, "content": "y\\in\\Lambda)", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [204, 604, 275, 618], "score": 1.0, "content": ", spanned by ", "type": "text"}, {"bbox": [276, 604, 353, 617], "score": 0.91, "content": "(w_{1},w_{2},w_{3},w_{4})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [353, 604, 394, 618], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [394, 605, 403, 614], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [403, 604, 501, 618], "score": 1.0, "content": " is assumed not to", "type": "text"}], "index": 27}, {"bbox": [110, 618, 448, 632], "spans": [{"bbox": [110, 618, 268, 632], "score": 1.0, "content": "be Lagrangian with respect to ", "type": "text"}, {"bbox": [268, 623, 281, 630], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [281, 618, 448, 632], "score": 1.0, "content": ", we have to deal with two cases.", "type": "text"}], "index": 28}], "index": 25.5, "bbox_fs": [109, 545, 501, 632]}, {"type": "text", "bbox": [110, 631, 500, 659], "lines": [{"bbox": [126, 631, 500, 647], "spans": [{"bbox": [126, 631, 187, 647], "score": 1.0, "content": "First case: ", "type": "text"}, {"bbox": [188, 634, 197, 643], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [198, 631, 441, 647], "score": 1.0, "content": " is a symplectic vector space for the structure ", "type": "text"}, {"bbox": [442, 637, 454, 645], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [454, 631, 500, 647], "score": 1.0, "content": ". In this", "type": "text"}], "index": 29}, {"bbox": [109, 647, 499, 661], "spans": [{"bbox": [109, 647, 320, 661], "score": 1.0, "content": "case we can choose a symplectic basis for ", "type": "text"}, {"bbox": [321, 649, 330, 657], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [331, 647, 499, 661], "score": 1.0, "content": " and this can always be chosen to", "type": "text"}], "index": 30}, {"bbox": [109, 127, 500, 142], "spans": [{"bbox": [109, 127, 185, 142], "score": 1.0, "content": "be of the form ", "type": "text", "cross_page": true}, {"bbox": [186, 129, 255, 141], "score": 0.93, "content": "v_{1},I v_{1},v_{2},I v_{2}", "type": "inline_equation", "height": 12, "width": 69, "cross_page": true}, {"bbox": [256, 127, 294, 142], "score": 1.0, "content": ". Then ", "type": "text", "cross_page": true}, {"bbox": [294, 129, 303, 138], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 9, "cross_page": true}, {"bbox": [304, 127, 500, 142], "score": 1.0, "content": " is Lagrangian in the symplectic struc-", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [109, 142, 501, 157], "spans": [{"bbox": [109, 142, 134, 157], "score": 1.0, "content": "ture ", "type": "text", "cross_page": true}, {"bbox": [135, 147, 148, 155], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13, "cross_page": true}, {"bbox": [149, 142, 192, 157], "score": 1.0, "content": "; indeed ", "type": "text", "cross_page": true}, {"bbox": [193, 144, 496, 156], "score": 0.94, "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "type": "inline_equation", "height": 12, "width": 303, "cross_page": true}, {"bbox": [497, 142, 501, 157], "score": 1.0, "content": ";", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [109, 156, 498, 171], "spans": [{"bbox": [109, 156, 195, 171], "score": 1.0, "content": "analogously for ", "type": "text", "cross_page": true}, {"bbox": [195, 158, 250, 170], "score": 0.92, "content": "\\omega_{J}(v_{2},I v_{2})", "type": "inline_equation", "height": 12, "width": 55, "cross_page": true}, {"bbox": [250, 156, 257, 171], "score": 1.0, "content": "; ", "type": "text", "cross_page": true}, {"bbox": [257, 158, 498, 170], "score": 0.9, "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "type": "inline_equation", "height": 12, "width": 241, "cross_page": true}], "index": 2}, {"bbox": [108, 171, 501, 186], "spans": [{"bbox": [108, 171, 140, 186], "score": 1.0, "content": "since ", "type": "text", "cross_page": true}, {"bbox": [141, 176, 167, 183], "score": 0.9, "content": "v_{1},v_{2}", "type": "inline_equation", "height": 7, "width": 26, "cross_page": true}, {"bbox": [167, 171, 369, 186], "score": 1.0, "content": " belong to a Lagrangian subspace of ", "type": "text", "cross_page": true}, {"bbox": [369, 176, 385, 183], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16, "cross_page": true}, {"bbox": [386, 171, 501, 186], "score": 1.0, "content": ", and analogously for", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [110, 185, 500, 200], "spans": [{"bbox": [110, 186, 262, 199], "score": 0.93, "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "type": "inline_equation", "height": 13, "width": 152, "cross_page": true}, {"bbox": [262, 185, 299, 200], "score": 1.0, "content": ". Thus ", "type": "text", "cross_page": true}, {"bbox": [299, 187, 309, 196], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [309, 185, 500, 200], "score": 1.0, "content": " is also Lagrangian for the symplectic", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [109, 199, 178, 215], "spans": [{"bbox": [109, 199, 160, 215], "score": 1.0, "content": "structure ", "type": "text", "cross_page": true}, {"bbox": [160, 204, 174, 212], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14, "cross_page": true}, {"bbox": [174, 199, 178, 215], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 5}], "index": 29.5, "bbox_fs": [109, 631, 500, 661]}]}
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[{"type": "text", "bbox": [110, 125, 501, 154], "content": "", "index": 0}, {"type": "title", "bbox": [111, 175, 500, 216], "content": "2 Characterization of special Lagrangian sub- manifolds", "index": 1}, {"type": "text", "bbox": [110, 227, 500, 271], "content": "In this section we will describe all special Lagrangian submanifolds of an ir- reducible symplectic 4-fold (having fixed a Kaehler class in the Kaehler cone). The key result is the following:", "index": 2}, {"type": "text", "bbox": [110, 272, 500, 314], "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is Lagrangian with respect to two different symplectic structures.", "index": 3}, {"type": "text", "bbox": [109, 315, 500, 402], "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4- fold . By Yau’s Theorem this determines a unique hyperkaehler metric . Choose a hyperkaehler structure compatible with the metric (notice that the triple is not uniquely determined) and consider the associated symplectic structures , and .", "index": 4}, {"type": "text", "bbox": [109, 402, 500, 459], "content": "Consider a special Lagrangian submanifold in the complex structure (this is not restrictive, since is not uniquely determined); that is assume that is calibrated by the real part of the holomorphic (in the structure ) 4-form:", "index": 5}, {"type": "interline_equation", "bbox": [250, 459, 359, 486], "content": "", "index": 6}, {"type": "text", "bbox": [109, 488, 456, 503], "content": "Notice that the real and immaginary part of are then given by:", "index": 7}, {"type": "interline_equation", "bbox": [193, 514, 416, 541], "content": "", "index": 8}, {"type": "text", "bbox": [109, 543, 500, 630], "content": "Obviously, by the property of being special Lagrangian we have that is Lagrangian with respect to . We will prove that having fixed the calibration, if is not Lagrangian also with respect to , then it is nec- essarily Lagrangian with respect to . First we work locally and consider , spanned by . Since is assumed not to be Lagrangian with respect to , we have to deal with two cases.", "index": 9}, {"type": "text", "bbox": [110, 631, 500, 659], "content": "First case: is a symplectic vector space for the structure . In this case we can choose a symplectic basis for and this can always be chosen to be of the form . Then is Lagrangian in the symplectic struc- ture ; indeed ; analogously for ; since belong to a Lagrangian subspace of , and analogously for . Thus is also Lagrangian for the symplectic structure .", "index": 10}]
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[{"bbox": [111, 178, 498, 197], "content": "2 Characterization of special Lagrangian sub-", "parent_index": 1, "line_index": 0}, {"bbox": [140, 201, 221, 218], "content": "manifolds", "parent_index": 1, "line_index": 1}, {"bbox": [109, 231, 498, 244], "content": "In this section we will describe all special Lagrangian submanifolds of an ir-", "parent_index": 2, "line_index": 0}, {"bbox": [110, 245, 499, 259], "content": "reducible symplectic 4-fold (having fixed a Kaehler class in the Kaehler", "parent_index": 2, "line_index": 1}, {"bbox": [110, 259, 305, 273], "content": "cone). The key result is the following:", "parent_index": 2, "line_index": 2}, {"bbox": [127, 273, 501, 287], "content": "Theorem 2.1: Every connected special Lagrangian submanifold of an", "parent_index": 3, "line_index": 0}, {"bbox": [110, 288, 502, 302], "content": "irreducible symplectic 4-fold is also bi-Lagrangian, in the sense that it is", "parent_index": 3, "line_index": 1}, {"bbox": [111, 302, 427, 316], "content": "Lagrangian with respect to two different symplectic structures.", "parent_index": 3, "line_index": 2}, {"bbox": [127, 317, 499, 330], "content": "Proof: Let us a fix a Kaehler class on the irreducible symplectic 4-", "parent_index": 4, "line_index": 0}, {"bbox": [110, 332, 500, 345], "content": "fold . By Yau’s Theorem this determines a unique hyperkaehler metric", "parent_index": 4, "line_index": 1}, {"bbox": [110, 346, 499, 361], "content": ". Choose a hyperkaehler structure compatible with the metric", "parent_index": 4, "line_index": 2}, {"bbox": [110, 361, 500, 375], "content": "(notice that the triple is not uniquely determined) and consider", "parent_index": 4, "line_index": 3}, {"bbox": [109, 375, 500, 389], "content": "the associated symplectic structures , and", "parent_index": 4, "line_index": 4}, {"bbox": [110, 388, 208, 404], "content": ".", "parent_index": 4, "line_index": 5}, {"bbox": [127, 403, 500, 418], "content": "Consider a special Lagrangian submanifold in the complex structure", "parent_index": 5, "line_index": 0}, {"bbox": [110, 417, 500, 433], "content": "(this is not restrictive, since is not uniquely determined); that", "parent_index": 5, "line_index": 1}, {"bbox": [108, 431, 500, 447], "content": "is assume that is calibrated by the real part of the holomorphic (in the", "parent_index": 5, "line_index": 2}, {"bbox": [109, 446, 218, 461], "content": "structure ) 4-form:", "parent_index": 5, "line_index": 3}, {"bbox": [109, 491, 455, 505], "content": "Notice that the real and immaginary part of are then given by:", "parent_index": 7, "line_index": 0}, {"bbox": [127, 545, 501, 560], "content": "Obviously, by the property of being special Lagrangian we have that", "parent_index": 9, "line_index": 0}, {"bbox": [110, 560, 500, 575], "content": "is Lagrangian with respect to . We will prove that having fixed the", "parent_index": 9, "line_index": 1}, {"bbox": [109, 575, 500, 589], "content": "calibration, if is not Lagrangian also with respect to , then it is nec-", "parent_index": 9, "line_index": 2}, {"bbox": [109, 590, 500, 603], "content": "essarily Lagrangian with respect to . First we work locally and consider", "parent_index": 9, "line_index": 3}, {"bbox": [110, 604, 501, 618], "content": ", spanned by . Since is assumed not to", "parent_index": 9, "line_index": 4}, {"bbox": [110, 618, 448, 632], "content": "be Lagrangian with respect to , we have to deal with two cases.", "parent_index": 9, "line_index": 5}, {"bbox": [126, 631, 500, 647], "content": "First case: is a symplectic vector space for the structure . In this", "parent_index": 10, "line_index": 0}, {"bbox": [109, 647, 499, 661], "content": "case we can choose a symplectic basis for and this can always be chosen to", "parent_index": 10, "line_index": 1}, {"bbox": [109, 127, 500, 142], "content": "be of the form . Then is Lagrangian in the symplectic struc-", "parent_index": 10, "line_index": 2}, {"bbox": [109, 142, 501, 157], "content": "ture ; indeed ;", "parent_index": 10, "line_index": 3}, {"bbox": [109, 156, 498, 171], "content": "analogously for ;", "parent_index": 10, "line_index": 4}, {"bbox": [108, 171, 501, 186], "content": "since belong to a Lagrangian subspace of , and analogously for", "parent_index": 10, "line_index": 5}, {"bbox": [110, 185, 500, 200], "content": ". Thus is also Lagrangian for the symplectic", "parent_index": 10, "line_index": 6}, {"bbox": [109, 199, 178, 215], "content": "structure .", "parent_index": 10, "line_index": 7}]
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[{"bbox": [248, 247, 259, 255], "content": "X", "parent_index": 2, "subtype": "inline"}, {"bbox": [410, 246, 424, 258], "content": "[\\omega]", "parent_index": 2, "subtype": "inline"}, {"bbox": [134, 333, 145, 342], "content": "X", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 351, 117, 359], "content": "g", "parent_index": 4, "subtype": "inline"}, {"bbox": [299, 347, 342, 360], "content": "(I,J,K)", "parent_index": 4, "subtype": "inline"}, {"bbox": [493, 351, 499, 359], "content": "g", "parent_index": 4, "subtype": "inline"}, {"bbox": [232, 361, 275, 374], "content": "(I,J,K)", "parent_index": 4, "subtype": "inline"}, {"bbox": [298, 376, 383, 388], "content": "\\omega_{I}(.,.):=g(I.,.)", "parent_index": 4, "subtype": "inline"}, {"bbox": [390, 376, 476, 388], "content": "\\omega_{J}(.,.):=g(J.,.)", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 390, 203, 403], "content": "\\omega_{K}(.,.):=g(K.,.)", "parent_index": 4, "subtype": "inline"}, {"bbox": [357, 405, 365, 414], "content": "\\Lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [110, 420, 121, 428], "content": "K", "parent_index": 5, "subtype": "inline"}, {"bbox": [277, 419, 320, 432], "content": "(I,J,K)", "parent_index": 5, "subtype": "inline"}, {"bbox": [190, 434, 199, 443], "content": "\\Lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [160, 449, 172, 458], "content": "K", "parent_index": 5, "subtype": "inline"}, {"bbox": [250, 459, 359, 486], "content": "\\Omega_{K}:=\\frac{1}{2!}(\\omega_{I}+i\\omega_{J})^{2}.", "parent_index": 6, "subtype": "interline"}, {"bbox": [341, 492, 358, 503], "content": "\\Omega_{K}", "parent_index": 7, "subtype": "inline"}, {"bbox": [193, 514, 416, 541], "content": "\\mathrm{Re}(\\Omega_{K})=\\frac{1}{2}(\\omega_{I}^{2}-\\omega_{J}^{2})\\quad\\mathrm{Im}(\\Omega_{K})=\\omega_{I}\\wedge\\omega_{J}.", "parent_index": 8, "subtype": "interline"}, {"bbox": [110, 562, 119, 571], "content": "\\Lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [283, 565, 299, 573], "content": "\\omega_{K}", "parent_index": 9, "subtype": "inline"}, {"bbox": [185, 577, 194, 586], "content": "\\Lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [404, 580, 416, 587], "content": "\\omega_{I}", "parent_index": 9, "subtype": "inline"}, {"bbox": [297, 594, 310, 602], "content": "\\omega_{J}", "parent_index": 9, "subtype": "inline"}, {"bbox": [110, 605, 161, 618], "content": "V:=T_{p}\\Lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [168, 605, 203, 617], "content": "y\\in\\Lambda)", "parent_index": 9, "subtype": "inline"}, {"bbox": [276, 604, 353, 617], "content": "(w_{1},w_{2},w_{3},w_{4})", "parent_index": 9, "subtype": "inline"}, {"bbox": [394, 605, 403, 614], "content": "\\Lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [268, 623, 281, 630], "content": "\\omega_{I}", "parent_index": 9, "subtype": "inline"}, {"bbox": [188, 634, 197, 643], "content": "V", "parent_index": 10, "subtype": "inline"}, {"bbox": [442, 637, 454, 645], "content": "\\omega_{I}", "parent_index": 10, "subtype": "inline"}, {"bbox": [321, 649, 330, 657], "content": "V", "parent_index": 10, "subtype": "inline"}, {"bbox": [186, 129, 255, 141], "content": "v_{1},I v_{1},v_{2},I v_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [294, 129, 303, 138], "content": "V", "parent_index": 10, "subtype": "inline"}, {"bbox": [135, 147, 148, 155], "content": "\\omega_{J}", "parent_index": 10, "subtype": "inline"}, {"bbox": [193, 144, 496, 156], "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "parent_index": 10, "subtype": "inline"}, {"bbox": [195, 158, 250, 170], "content": "\\omega_{J}(v_{2},I v_{2})", "parent_index": 10, "subtype": "inline"}, {"bbox": [257, 158, 498, 170], "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "parent_index": 10, "subtype": "inline"}, {"bbox": [141, 176, 167, 183], "content": "v_{1},v_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [369, 176, 385, 183], "content": "\\omega_{K}", "parent_index": 10, "subtype": "inline"}, {"bbox": [110, 186, 262, 199], "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "parent_index": 10, "subtype": "inline"}, {"bbox": [299, 187, 309, 196], "content": "V", "parent_index": 10, "subtype": "inline"}, {"bbox": [160, 204, 174, 212], "content": "\\omega_{J}", "parent_index": 10, "subtype": "inline"}]
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Second case: $V$ is neither symplectic nor Lagrangian for the structure $\omega_{I}$ . Notice $V$ can not be symplectic with respect to $\omega_{J}$ , otherwise by the first case it would be Lagrangian in the strucutre $\omega_{I}$ ; moreover we can assume that $V$ is not Lagrangian with respect to $\omega_{J}$ , otherwise there is nothing to prove. So in this case $V$ is neither Lagrangian nor symplectic in the structure $\omega_{I}$ and in the structure $\omega_{J}$ . This means that $V$ contains a symplectic 2-plane $\pi$ with respect to $\omega_{I}$ and a symplectic 2-plane $\rho$ with respect to $\omega_{J}$ . Indeed, consider $v_{1}\in V$ ; since $V$ is not Lagrangian in the structure $\omega_{I}$ , there exists $v_{2}\in V$ such that $\omega_{I}(v_{1},v_{2})\neq0$ and this implies that the vector subspace $\pi$ spanned by $(v_{1},v_{2})$ is a symplectic vector space for $\omega_{I}$ , which can not be extended to all $V$ . The same reasoning applies in the structure $\omega_{J}$ .
We prove that this can not happen, since it violates the calibration condition. We have to distinguish three different subcases according to the intersection of $\pi$ with $\rho$ .
First subcase: $\pi$ and $\rho$ have zero intersection. If this happens we can always choose a basis of $V$ of the form $(v_{1},I v_{1},v_{2},J v_{2})$ . Write $\pi$ for the 2- plane spanned by $v_{1},I v_{1}$ and $\rho$ for that spanned by $v_{2},J v_{2}$ , so that $V=\pi\oplus\rho$ . Indeed, since $V$ is not Lagrangian with respect to $\omega_{I}$ , it has to contain a symplectic 2-plane like $\pi$ , and similarly for $\rho$ and $\omega_{J}$ . Moreover, since $V$ is not symplectic with respect to $\omega_{I}$ , it turns out that the symplectic 2-plane $\pi$ can not be completed to a symplectic basis of $V$ , so that $V$ has to contain an isotropic 2-plane for $\omega_{I}$ , which is $\rho$ . The same reasoning (with the roles reversed) applies obviously to the symplectic structure $\omega_{J}$ . Hence, in this case we have:
$$
2\mathrm{Re}(\Omega_{K})|_{V}=(\omega_{I}^{2}-\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\omega_{I}(v_{1},I v_{1})\omega_{I}(v_{2},J v_{2})-
$$
$$
\omega_{I}(v_{1},v_{2})\omega_{I}(I v_{1},J v_{2})+\omega_{I}(v_{1},J v_{2})\omega_{I}(I v_{1},v_{2})-\omega_{J}(v_{1},I v_{1})\omega_{J}(v_{2},J v_{2})+
$$
$$
\omega_{J}(v_{1},v_{2})\omega_{J}(I v_{1},J v_{2})-\omega_{J}(v_{1},J v_{2})\omega_{J}(I v_{2},v_{2})=0,
$$
using the defining relations of $\omega_{I},\omega_{J},\omega_{K}$ , the quaternionic relation $I J=K$ , the invariance of $g$ and the fact that $V$ is Lagrangian with respect to $\omega_{K}$ . So this subcase is not consistent with the calibration property.
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<p data-bbox="109 213 500 371">Second case: $V$ is neither symplectic nor Lagrangian for the structure $\omega_{I}$ . Notice $V$ can not be symplectic with respect to $\omega_{J}$ , otherwise by the first case it would be Lagrangian in the strucutre $\omega_{I}$ ; moreover we can assume that $V$ is not Lagrangian with respect to $\omega_{J}$ , otherwise there is nothing to prove. So in this case $V$ is neither Lagrangian nor symplectic in the structure $\omega_{I}$ and in the structure $\omega_{J}$ . This means that $V$ contains a symplectic 2-plane $\pi$ with respect to $\omega_{I}$ and a symplectic 2-plane $\rho$ with respect to $\omega_{J}$ . Indeed, consider $v_{1}\in V$ ; since $V$ is not Lagrangian in the structure $\omega_{I}$ , there exists $v_{2}\in V$ such that $\omega_{I}(v_{1},v_{2})\neq0$ and this implies that the vector subspace $\pi$ spanned by $(v_{1},v_{2})$ is a symplectic vector space for $\omega_{I}$ , which can not be extended to all $V$ . The same reasoning applies in the structure $\omega_{J}$ . </p>
<p data-bbox="109 372 500 414">We prove that this can not happen, since it violates the calibration condition. We have to distinguish three different subcases according to the intersection of $\pi$ with $\rho$ . </p>
<p data-bbox="109 415 500 559">First subcase: $\pi$ and $\rho$ have zero intersection. If this happens we can always choose a basis of $V$ of the form $(v_{1},I v_{1},v_{2},J v_{2})$ . Write $\pi$ for the 2- plane spanned by $v_{1},I v_{1}$ and $\rho$ for that spanned by $v_{2},J v_{2}$ , so that $V=\pi\oplus\rho$ . Indeed, since $V$ is not Lagrangian with respect to $\omega_{I}$ , it has to contain a symplectic 2-plane like $\pi$ , and similarly for $\rho$ and $\omega_{J}$ . Moreover, since $V$ is not symplectic with respect to $\omega_{I}$ , it turns out that the symplectic 2-plane $\pi$ can not be completed to a symplectic basis of $V$ , so that $V$ has to contain an isotropic 2-plane for $\omega_{I}$ , which is $\rho$ . The same reasoning (with the roles reversed) applies obviously to the symplectic structure $\omega_{J}$ . Hence, in this case we have: </p>
<div class="equation" data-bbox="132 569 479 584">$$
2\mathrm{Re}(\Omega_{K})|_{V}=(\omega_{I}^{2}-\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\omega_{I}(v_{1},I v_{1})\omega_{I}(v_{2},J v_{2})-
$$</div>
<div class="equation" data-bbox="123 594 487 608">$$
\omega_{I}(v_{1},v_{2})\omega_{I}(I v_{1},J v_{2})+\omega_{I}(v_{1},J v_{2})\omega_{I}(I v_{1},v_{2})-\omega_{J}(v_{1},I v_{1})\omega_{J}(v_{2},J v_{2})+
$$</div>
<div class="equation" data-bbox="174 614 434 627">$$
\omega_{J}(v_{1},v_{2})\omega_{J}(I v_{1},J v_{2})-\omega_{J}(v_{1},J v_{2})\omega_{J}(I v_{2},v_{2})=0,
$$</div>
<p data-bbox="110 631 501 674">using the defining relations of $\omega_{I},\omega_{J},\omega_{K}$ , the quaternionic relation $I J=K$ , the invariance of $g$ and the fact that $V$ is Lagrangian with respect to $\omega_{K}$ . So this subcase is not consistent with the calibration property. </p>
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0001060v1
| 3 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "", "page_idx": 3}, {"type": "text", "text": "Second case: $V$ is neither symplectic nor Lagrangian for the structure $\\omega_{I}$ . Notice $V$ can not be symplectic with respect to $\\omega_{J}$ , otherwise by the first case it would be Lagrangian in the strucutre $\\omega_{I}$ ; moreover we can assume that $V$ is not Lagrangian with respect to $\\omega_{J}$ , otherwise there is nothing to prove. So in this case $V$ is neither Lagrangian nor symplectic in the structure $\\omega_{I}$ and in the structure $\\omega_{J}$ . This means that $V$ contains a symplectic 2-plane $\\pi$ with respect to $\\omega_{I}$ and a symplectic 2-plane $\\rho$ with respect to $\\omega_{J}$ . Indeed, consider $v_{1}\\in V$ ; since $V$ is not Lagrangian in the structure $\\omega_{I}$ , there exists $v_{2}\\in V$ such that $\\omega_{I}(v_{1},v_{2})\\neq0$ and this implies that the vector subspace $\\pi$ spanned by $(v_{1},v_{2})$ is a symplectic vector space for $\\omega_{I}$ , which can not be extended to all $V$ . The same reasoning applies in the structure $\\omega_{J}$ . ", "page_idx": 3}, {"type": "text", "text": "We prove that this can not happen, since it violates the calibration condition. We have to distinguish three different subcases according to the intersection of $\\pi$ with $\\rho$ . ", "page_idx": 3}, {"type": "text", "text": "First subcase: $\\pi$ and $\\rho$ have zero intersection. If this happens we can always choose a basis of $V$ of the form $(v_{1},I v_{1},v_{2},J v_{2})$ . Write $\\pi$ for the 2- plane spanned by $v_{1},I v_{1}$ and $\\rho$ for that spanned by $v_{2},J v_{2}$ , so that $V=\\pi\\oplus\\rho$ . Indeed, since $V$ is not Lagrangian with respect to $\\omega_{I}$ , it has to contain a symplectic 2-plane like $\\pi$ , and similarly for $\\rho$ and $\\omega_{J}$ . Moreover, since $V$ is not symplectic with respect to $\\omega_{I}$ , it turns out that the symplectic 2-plane $\\pi$ can not be completed to a symplectic basis of $V$ , so that $V$ has to contain an isotropic 2-plane for $\\omega_{I}$ , which is $\\rho$ . The same reasoning (with the roles reversed) applies obviously to the symplectic structure $\\omega_{J}$ . Hence, in this case we have: ", "page_idx": 3}, {"type": "equation", "text": "$$\n2\\mathrm{Re}(\\Omega_{K})|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-\n$$", "text_format": "latex", "page_idx": 3}, {"type": "equation", "text": "$$\n\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+\n$$", "text_format": "latex", "page_idx": 3}, {"type": "equation", "text": "$$\n\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "using the defining relations of $\\omega_{I},\\omega_{J},\\omega_{K}$ , the quaternionic relation $I J=K$ , the invariance of $g$ and the fact that $V$ is Lagrangian with respect to $\\omega_{K}$ . So this subcase is not consistent with the calibration property. ", "page_idx": 3}]
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{"preproc_blocks": [{"type": "text", "bbox": [109, 124, 500, 212], "lines": [{"bbox": [109, 127, 500, 142], "spans": [{"bbox": [109, 127, 185, 142], "score": 1.0, "content": "be of the form ", "type": "text"}, {"bbox": [186, 129, 255, 141], "score": 0.93, "content": "v_{1},I v_{1},v_{2},I v_{2}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [256, 127, 294, 142], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [294, 129, 303, 138], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [304, 127, 500, 142], "score": 1.0, "content": " is Lagrangian in the symplectic struc-", "type": "text"}], "index": 0}, {"bbox": [109, 142, 501, 157], "spans": [{"bbox": [109, 142, 134, 157], "score": 1.0, "content": "ture ", "type": "text"}, {"bbox": [135, 147, 148, 155], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [149, 142, 192, 157], "score": 1.0, "content": "; indeed ", "type": "text"}, {"bbox": [193, 144, 496, 156], "score": 0.94, "content": "\\omega_{J}(v_{1},I v_{1})=g(J v_{1},I v_{1})=g(I J v_{1},-v_{1})=-\\omega_{K}(v_{1},v_{1})=0", "type": "inline_equation", "height": 12, "width": 303}, {"bbox": [497, 142, 501, 157], "score": 1.0, "content": ";", "type": "text"}], "index": 1}, {"bbox": [109, 156, 498, 171], "spans": [{"bbox": [109, 156, 195, 171], "score": 1.0, "content": "analogously for ", "type": "text"}, {"bbox": [195, 158, 250, 170], "score": 0.92, "content": "\\omega_{J}(v_{2},I v_{2})", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [250, 156, 257, 171], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [257, 158, 498, 170], "score": 0.9, "content": "\\omega_{J}(v_{1},I v_{2})\\;=\\;g(J v_{1},I v_{2})\\;=\\;-\\omega_{K}(v_{1},v_{2})\\;=\\;0", "type": "inline_equation", "height": 12, "width": 241}], "index": 2}, {"bbox": [108, 171, 501, 186], "spans": [{"bbox": [108, 171, 140, 186], "score": 1.0, "content": "since ", "type": "text"}, {"bbox": [141, 176, 167, 183], "score": 0.9, "content": "v_{1},v_{2}", "type": "inline_equation", "height": 7, "width": 26}, {"bbox": [167, 171, 369, 186], "score": 1.0, "content": " belong to a Lagrangian subspace of ", "type": "text"}, {"bbox": [369, 176, 385, 183], "score": 0.89, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [386, 171, 501, 186], "score": 1.0, "content": ", and analogously for", "type": "text"}], "index": 3}, {"bbox": [110, 185, 500, 200], "spans": [{"bbox": [110, 186, 262, 199], "score": 0.93, "content": "\\omega_{J}(v_{2},I v_{1})=-\\omega_{K}(v_{2},v_{1})=0", "type": "inline_equation", "height": 13, "width": 152}, {"bbox": [262, 185, 299, 200], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [299, 187, 309, 196], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [309, 185, 500, 200], "score": 1.0, "content": " is also Lagrangian for the symplectic", "type": "text"}], "index": 4}, {"bbox": [109, 199, 178, 215], "spans": [{"bbox": [109, 199, 160, 215], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [160, 204, 174, 212], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [174, 199, 178, 215], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 2.5}, {"type": "text", "bbox": [109, 213, 500, 371], "lines": [{"bbox": [127, 213, 500, 228], "spans": [{"bbox": [127, 213, 195, 228], "score": 1.0, "content": "Second case: ", "type": "text"}, {"bbox": [196, 216, 205, 225], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [205, 213, 483, 228], "score": 1.0, "content": " is neither symplectic nor Lagrangian for the structure ", "type": "text"}, {"bbox": [483, 219, 496, 226], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [496, 213, 500, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 147, 243], "score": 1.0, "content": "Notice ", "type": "text"}, {"bbox": [148, 231, 158, 240], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [158, 228, 365, 243], "score": 1.0, "content": " can not be symplectic with respect to ", "type": "text"}, {"bbox": [366, 234, 379, 241], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [379, 228, 500, 243], "score": 1.0, "content": ", otherwise by the first", "type": "text"}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 349, 257], "score": 1.0, "content": "case it would be Lagrangian in the strucutre ", "type": "text"}, {"bbox": [349, 248, 362, 255], "score": 0.87, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [362, 244, 500, 257], "score": 1.0, "content": "; moreover we can assume", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 272], "spans": [{"bbox": [110, 257, 136, 272], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 259, 146, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [146, 257, 328, 272], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [328, 262, 342, 270], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [342, 257, 500, 272], "score": 1.0, "content": ", otherwise there is nothing to", "type": "text"}], "index": 9}, {"bbox": [110, 273, 501, 286], "spans": [{"bbox": [110, 273, 220, 286], "score": 1.0, "content": "prove. So in this case ", "type": "text"}, {"bbox": [220, 274, 230, 283], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [230, 273, 501, 286], "score": 1.0, "content": " is neither Lagrangian nor symplectic in the structure", "type": "text"}], "index": 10}, {"bbox": [110, 286, 501, 301], "spans": [{"bbox": [110, 291, 122, 299], "score": 0.83, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 286, 229, 301], "score": 1.0, "content": " and in the structure ", "type": "text"}, {"bbox": [230, 291, 243, 299], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [243, 286, 337, 301], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [337, 288, 347, 297], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 286, 501, 301], "score": 1.0, "content": " contains a symplectic 2-plane", "type": "text"}], "index": 11}, {"bbox": [110, 302, 500, 315], "spans": [{"bbox": [110, 306, 117, 311], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [118, 302, 201, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [202, 306, 214, 313], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [214, 302, 348, 315], "score": 1.0, "content": " and a symplectic 2-plane ", "type": "text"}, {"bbox": [349, 306, 355, 314], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [355, 302, 439, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [440, 306, 453, 313], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [453, 302, 500, 315], "score": 1.0, "content": ". Indeed,", "type": "text"}], "index": 12}, {"bbox": [110, 316, 500, 329], "spans": [{"bbox": [110, 316, 155, 329], "score": 1.0, "content": "consider ", "type": "text"}, {"bbox": [156, 317, 191, 327], "score": 0.93, "content": "v_{1}\\in V", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [192, 316, 227, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [228, 317, 237, 326], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [237, 316, 420, 329], "score": 1.0, "content": " is not Lagrangian in the structure ", "type": "text"}, {"bbox": [421, 320, 433, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 316, 500, 329], "score": 1.0, "content": ", there exists", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 331, 147, 342], "score": 0.93, "content": "v_{2}\\in V", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [148, 330, 205, 344], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [206, 331, 278, 343], "score": 0.94, "content": "\\omega_{I}(v_{1},v_{2})\\neq0", "type": "inline_equation", "height": 12, "width": 72}, {"bbox": [278, 330, 500, 344], "score": 1.0, "content": " and this implies that the vector subspace", "type": "text"}], "index": 14}, {"bbox": [110, 344, 500, 359], "spans": [{"bbox": [110, 349, 117, 355], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 344, 184, 359], "score": 1.0, "content": " spanned by ", "type": "text"}, {"bbox": [185, 345, 219, 358], "score": 0.94, "content": "(v_{1},v_{2})", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [219, 344, 389, 359], "score": 1.0, "content": " is a symplectic vector space for ", "type": "text"}, {"bbox": [389, 349, 402, 357], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [402, 344, 500, 359], "score": 1.0, "content": ", which can not be", "type": "text"}], "index": 15}, {"bbox": [110, 358, 455, 374], "spans": [{"bbox": [110, 358, 190, 374], "score": 1.0, "content": "extended to all ", "type": "text"}, {"bbox": [190, 361, 200, 369], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [200, 358, 437, 374], "score": 1.0, "content": ". The same reasoning applies in the structure ", "type": "text"}, {"bbox": [437, 364, 450, 371], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [451, 358, 455, 374], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11}, {"type": "text", "bbox": [109, 372, 500, 414], "lines": [{"bbox": [127, 373, 499, 388], "spans": [{"bbox": [127, 373, 499, 388], "score": 1.0, "content": "We prove that this can not happen, since it violates the calibration con-", "type": "text"}], "index": 17}, {"bbox": [110, 387, 500, 402], "spans": [{"bbox": [110, 387, 500, 402], "score": 1.0, "content": "dition. We have to distinguish three different subcases according to the", "type": "text"}], "index": 18}, {"bbox": [110, 402, 235, 417], "spans": [{"bbox": [110, 402, 186, 417], "score": 1.0, "content": "intersection of ", "type": "text"}, {"bbox": [186, 407, 194, 412], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [194, 402, 223, 417], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 407, 230, 415], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [230, 402, 235, 417], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [109, 415, 500, 559], "lines": [{"bbox": [127, 416, 500, 430], "spans": [{"bbox": [127, 416, 207, 430], "score": 1.0, "content": "First subcase: ", "type": "text"}, {"bbox": [207, 421, 214, 427], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 416, 243, 430], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 421, 249, 429], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [250, 416, 500, 430], "score": 1.0, "content": " have zero intersection. If this happens we can", "type": "text"}], "index": 20}, {"bbox": [110, 431, 500, 445], "spans": [{"bbox": [110, 431, 239, 445], "score": 1.0, "content": "always choose a basis of ", "type": "text"}, {"bbox": [239, 433, 249, 441], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [249, 431, 315, 445], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [316, 432, 396, 444], "score": 0.92, "content": "(v_{1},I v_{1},v_{2},J v_{2})", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [396, 431, 438, 445], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [438, 436, 446, 441], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [446, 431, 500, 445], "score": 1.0, "content": " for the 2-", "type": "text"}], "index": 21}, {"bbox": [110, 445, 500, 460], "spans": [{"bbox": [110, 445, 200, 460], "score": 1.0, "content": "plane spanned by ", "type": "text"}, {"bbox": [200, 447, 232, 458], "score": 0.94, "content": "v_{1},I v_{1}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [233, 445, 257, 460], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 450, 264, 458], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [264, 445, 368, 460], "score": 1.0, "content": " for that spanned by ", "type": "text"}, {"bbox": [368, 447, 402, 458], "score": 0.92, "content": "v_{2},J v_{2}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 445, 446, 460], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [446, 447, 496, 458], "score": 0.93, "content": "V=\\pi\\oplus\\rho", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [496, 445, 500, 460], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [109, 459, 501, 475], "spans": [{"bbox": [109, 459, 182, 475], "score": 1.0, "content": "Indeed, since ", "type": "text"}, {"bbox": [183, 462, 192, 470], "score": 0.91, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [192, 459, 379, 475], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [379, 465, 392, 472], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [392, 459, 501, 475], "score": 1.0, "content": ", it has to contain a", "type": "text"}], "index": 23}, {"bbox": [109, 475, 501, 488], "spans": [{"bbox": [109, 475, 231, 488], "score": 1.0, "content": "symplectic 2-plane like ", "type": "text"}, {"bbox": [231, 479, 239, 485], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [239, 475, 336, 488], "score": 1.0, "content": ", and similarly for ", "type": "text"}, {"bbox": [336, 479, 342, 487], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [343, 475, 369, 488], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 479, 383, 487], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [383, 475, 477, 488], "score": 1.0, "content": ". Moreover, since ", "type": "text"}, {"bbox": [477, 476, 487, 485], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [487, 475, 501, 488], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [110, 490, 500, 502], "spans": [{"bbox": [110, 490, 272, 502], "score": 1.0, "content": "not symplectic with respect to ", "type": "text"}, {"bbox": [272, 493, 285, 501], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [285, 490, 500, 502], "score": 1.0, "content": ", it turns out that the symplectic 2-plane", "type": "text"}], "index": 25}, {"bbox": [110, 504, 500, 516], "spans": [{"bbox": [110, 508, 117, 514], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 504, 357, 516], "score": 1.0, "content": " can not be completed to a symplectic basis of ", "type": "text"}, {"bbox": [357, 505, 367, 514], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 504, 413, 516], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [413, 505, 423, 514], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [423, 504, 500, 516], "score": 1.0, "content": " has to contain", "type": "text"}], "index": 26}, {"bbox": [110, 518, 500, 531], "spans": [{"bbox": [110, 518, 234, 531], "score": 1.0, "content": "an isotropic 2-plane for ", "type": "text"}, {"bbox": [234, 522, 247, 530], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [247, 518, 300, 531], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [300, 523, 307, 531], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 518, 500, 531], "score": 1.0, "content": ". The same reasoning (with the roles", "type": "text"}], "index": 27}, {"bbox": [110, 533, 500, 546], "spans": [{"bbox": [110, 533, 401, 546], "score": 1.0, "content": "reversed) applies obviously to the symplectic structure ", "type": "text"}, {"bbox": [401, 537, 414, 545], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [415, 533, 500, 546], "score": 1.0, "content": ". Hence, in this", "type": "text"}], "index": 28}, {"bbox": [110, 547, 180, 560], "spans": [{"bbox": [110, 547, 180, 560], "score": 1.0, "content": "case we have:", "type": "text"}], "index": 29}], "index": 24.5}, {"type": "interline_equation", "bbox": [132, 569, 479, 584], "lines": [{"bbox": [132, 569, 479, 584], "spans": [{"bbox": [132, 569, 479, 584], "score": 0.82, "content": "2\\mathrm{Re}(\\Omega_{K})|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "lines": [{"bbox": [123, 594, 487, 608], "spans": [{"bbox": [123, 594, 487, 608], "score": 0.87, "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "lines": [{"bbox": [174, 614, 434, 627], "spans": [{"bbox": [174, 614, 434, 627], "score": 0.87, "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "type": "interline_equation"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [110, 631, 501, 674], "lines": [{"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 266, 648], "score": 1.0, "content": "using the defining relations of ", "type": "text"}, {"bbox": [266, 638, 318, 646], "score": 0.91, "content": "\\omega_{I},\\omega_{J},\\omega_{K}", "type": "inline_equation", "height": 8, "width": 52}, {"bbox": [318, 633, 455, 648], "score": 1.0, "content": ", the quaternionic relation ", "type": "text"}, {"bbox": [455, 635, 496, 644], "score": 0.92, "content": "I J=K", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [496, 633, 500, 648], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [110, 648, 500, 662], "spans": [{"bbox": [110, 648, 198, 662], "score": 1.0, "content": "the invariance of ", "type": "text"}, {"bbox": [198, 653, 204, 660], "score": 0.89, "content": "g", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [204, 648, 297, 662], "score": 1.0, "content": " and the fact that ", "type": "text"}, {"bbox": [297, 649, 307, 658], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 648, 462, 662], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [462, 653, 478, 660], "score": 0.91, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [479, 648, 500, 662], "score": 1.0, "content": ". So", "type": "text"}], "index": 34}, {"bbox": [110, 662, 414, 675], "spans": [{"bbox": [110, 662, 414, 675], "score": 1.0, "content": "this subcase is not consistent with the calibration property.", "type": "text"}], "index": 35}], "index": 34}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [132, 569, 479, 584], "lines": [{"bbox": [132, 569, 479, 584], "spans": [{"bbox": [132, 569, 479, 584], "score": 0.82, "content": "2\\mathrm{Re}(\\Omega_{K})|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "lines": [{"bbox": [123, 594, 487, 608], "spans": [{"bbox": [123, 594, 487, 608], "score": 0.87, "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "lines": [{"bbox": [174, 614, 434, 627], "spans": [{"bbox": [174, 614, 434, 627], "score": 0.87, "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "type": "interline_equation"}], "index": 32}], "index": 32}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 124, 500, 212], "lines": [], "index": 2.5, "bbox_fs": [108, 127, 501, 215], "lines_deleted": true}, {"type": "text", "bbox": [109, 213, 500, 371], "lines": [{"bbox": [127, 213, 500, 228], "spans": [{"bbox": [127, 213, 195, 228], "score": 1.0, "content": "Second case: ", "type": "text"}, {"bbox": [196, 216, 205, 225], "score": 0.88, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [205, 213, 483, 228], "score": 1.0, "content": " is neither symplectic nor Lagrangian for the structure ", "type": "text"}, {"bbox": [483, 219, 496, 226], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [496, 213, 500, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 147, 243], "score": 1.0, "content": "Notice ", "type": "text"}, {"bbox": [148, 231, 158, 240], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [158, 228, 365, 243], "score": 1.0, "content": " can not be symplectic with respect to ", "type": "text"}, {"bbox": [366, 234, 379, 241], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [379, 228, 500, 243], "score": 1.0, "content": ", otherwise by the first", "type": "text"}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 349, 257], "score": 1.0, "content": "case it would be Lagrangian in the strucutre ", "type": "text"}, {"bbox": [349, 248, 362, 255], "score": 0.87, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [362, 244, 500, 257], "score": 1.0, "content": "; moreover we can assume", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 272], "spans": [{"bbox": [110, 257, 136, 272], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 259, 146, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [146, 257, 328, 272], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [328, 262, 342, 270], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [342, 257, 500, 272], "score": 1.0, "content": ", otherwise there is nothing to", "type": "text"}], "index": 9}, {"bbox": [110, 273, 501, 286], "spans": [{"bbox": [110, 273, 220, 286], "score": 1.0, "content": "prove. So in this case ", "type": "text"}, {"bbox": [220, 274, 230, 283], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [230, 273, 501, 286], "score": 1.0, "content": " is neither Lagrangian nor symplectic in the structure", "type": "text"}], "index": 10}, {"bbox": [110, 286, 501, 301], "spans": [{"bbox": [110, 291, 122, 299], "score": 0.83, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 286, 229, 301], "score": 1.0, "content": " and in the structure ", "type": "text"}, {"bbox": [230, 291, 243, 299], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [243, 286, 337, 301], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [337, 288, 347, 297], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 286, 501, 301], "score": 1.0, "content": " contains a symplectic 2-plane", "type": "text"}], "index": 11}, {"bbox": [110, 302, 500, 315], "spans": [{"bbox": [110, 306, 117, 311], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [118, 302, 201, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [202, 306, 214, 313], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [214, 302, 348, 315], "score": 1.0, "content": " and a symplectic 2-plane ", "type": "text"}, {"bbox": [349, 306, 355, 314], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [355, 302, 439, 315], "score": 1.0, "content": " with respect to ", "type": "text"}, {"bbox": [440, 306, 453, 313], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [453, 302, 500, 315], "score": 1.0, "content": ". Indeed,", "type": "text"}], "index": 12}, {"bbox": [110, 316, 500, 329], "spans": [{"bbox": [110, 316, 155, 329], "score": 1.0, "content": "consider ", "type": "text"}, {"bbox": [156, 317, 191, 327], "score": 0.93, "content": "v_{1}\\in V", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [192, 316, 227, 329], "score": 1.0, "content": "; since ", "type": "text"}, {"bbox": [228, 317, 237, 326], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [237, 316, 420, 329], "score": 1.0, "content": " is not Lagrangian in the structure ", "type": "text"}, {"bbox": [421, 320, 433, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 316, 500, 329], "score": 1.0, "content": ", there exists", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 331, 147, 342], "score": 0.93, "content": "v_{2}\\in V", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [148, 330, 205, 344], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [206, 331, 278, 343], "score": 0.94, "content": "\\omega_{I}(v_{1},v_{2})\\neq0", "type": "inline_equation", "height": 12, "width": 72}, {"bbox": [278, 330, 500, 344], "score": 1.0, "content": " and this implies that the vector subspace", "type": "text"}], "index": 14}, {"bbox": [110, 344, 500, 359], "spans": [{"bbox": [110, 349, 117, 355], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 344, 184, 359], "score": 1.0, "content": " spanned by ", "type": "text"}, {"bbox": [185, 345, 219, 358], "score": 0.94, "content": "(v_{1},v_{2})", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [219, 344, 389, 359], "score": 1.0, "content": " is a symplectic vector space for ", "type": "text"}, {"bbox": [389, 349, 402, 357], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [402, 344, 500, 359], "score": 1.0, "content": ", which can not be", "type": "text"}], "index": 15}, {"bbox": [110, 358, 455, 374], "spans": [{"bbox": [110, 358, 190, 374], "score": 1.0, "content": "extended to all ", "type": "text"}, {"bbox": [190, 361, 200, 369], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [200, 358, 437, 374], "score": 1.0, "content": ". The same reasoning applies in the structure ", "type": "text"}, {"bbox": [437, 364, 450, 371], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [451, 358, 455, 374], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11, "bbox_fs": [109, 213, 501, 374]}, {"type": "text", "bbox": [109, 372, 500, 414], "lines": [{"bbox": [127, 373, 499, 388], "spans": [{"bbox": [127, 373, 499, 388], "score": 1.0, "content": "We prove that this can not happen, since it violates the calibration con-", "type": "text"}], "index": 17}, {"bbox": [110, 387, 500, 402], "spans": [{"bbox": [110, 387, 500, 402], "score": 1.0, "content": "dition. We have to distinguish three different subcases according to the", "type": "text"}], "index": 18}, {"bbox": [110, 402, 235, 417], "spans": [{"bbox": [110, 402, 186, 417], "score": 1.0, "content": "intersection of ", "type": "text"}, {"bbox": [186, 407, 194, 412], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [194, 402, 223, 417], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 407, 230, 415], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [230, 402, 235, 417], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [110, 373, 500, 417]}, {"type": "text", "bbox": [109, 415, 500, 559], "lines": [{"bbox": [127, 416, 500, 430], "spans": [{"bbox": [127, 416, 207, 430], "score": 1.0, "content": "First subcase: ", "type": "text"}, {"bbox": [207, 421, 214, 427], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [214, 416, 243, 430], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [243, 421, 249, 429], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [250, 416, 500, 430], "score": 1.0, "content": " have zero intersection. If this happens we can", "type": "text"}], "index": 20}, {"bbox": [110, 431, 500, 445], "spans": [{"bbox": [110, 431, 239, 445], "score": 1.0, "content": "always choose a basis of ", "type": "text"}, {"bbox": [239, 433, 249, 441], "score": 0.9, "content": "V", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [249, 431, 315, 445], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [316, 432, 396, 444], "score": 0.92, "content": "(v_{1},I v_{1},v_{2},J v_{2})", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [396, 431, 438, 445], "score": 1.0, "content": ". Write ", "type": "text"}, {"bbox": [438, 436, 446, 441], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [446, 431, 500, 445], "score": 1.0, "content": " for the 2-", "type": "text"}], "index": 21}, {"bbox": [110, 445, 500, 460], "spans": [{"bbox": [110, 445, 200, 460], "score": 1.0, "content": "plane spanned by ", "type": "text"}, {"bbox": [200, 447, 232, 458], "score": 0.94, "content": "v_{1},I v_{1}", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [233, 445, 257, 460], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 450, 264, 458], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [264, 445, 368, 460], "score": 1.0, "content": " for that spanned by ", "type": "text"}, {"bbox": [368, 447, 402, 458], "score": 0.92, "content": "v_{2},J v_{2}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 445, 446, 460], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [446, 447, 496, 458], "score": 0.93, "content": "V=\\pi\\oplus\\rho", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [496, 445, 500, 460], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [109, 459, 501, 475], "spans": [{"bbox": [109, 459, 182, 475], "score": 1.0, "content": "Indeed, since ", "type": "text"}, {"bbox": [183, 462, 192, 470], "score": 0.91, "content": "V", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [192, 459, 379, 475], "score": 1.0, "content": " is not Lagrangian with respect to ", "type": "text"}, {"bbox": [379, 465, 392, 472], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [392, 459, 501, 475], "score": 1.0, "content": ", it has to contain a", "type": "text"}], "index": 23}, {"bbox": [109, 475, 501, 488], "spans": [{"bbox": [109, 475, 231, 488], "score": 1.0, "content": "symplectic 2-plane like ", "type": "text"}, {"bbox": [231, 479, 239, 485], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [239, 475, 336, 488], "score": 1.0, "content": ", and similarly for ", "type": "text"}, {"bbox": [336, 479, 342, 487], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [343, 475, 369, 488], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [370, 479, 383, 487], "score": 0.88, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [383, 475, 477, 488], "score": 1.0, "content": ". Moreover, since ", "type": "text"}, {"bbox": [477, 476, 487, 485], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [487, 475, 501, 488], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [110, 490, 500, 502], "spans": [{"bbox": [110, 490, 272, 502], "score": 1.0, "content": "not symplectic with respect to ", "type": "text"}, {"bbox": [272, 493, 285, 501], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [285, 490, 500, 502], "score": 1.0, "content": ", it turns out that the symplectic 2-plane", "type": "text"}], "index": 25}, {"bbox": [110, 504, 500, 516], "spans": [{"bbox": [110, 508, 117, 514], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 504, 357, 516], "score": 1.0, "content": " can not be completed to a symplectic basis of ", "type": "text"}, {"bbox": [357, 505, 367, 514], "score": 0.91, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 504, 413, 516], "score": 1.0, "content": ", so that ", "type": "text"}, {"bbox": [413, 505, 423, 514], "score": 0.89, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [423, 504, 500, 516], "score": 1.0, "content": " has to contain", "type": "text"}], "index": 26}, {"bbox": [110, 518, 500, 531], "spans": [{"bbox": [110, 518, 234, 531], "score": 1.0, "content": "an isotropic 2-plane for ", "type": "text"}, {"bbox": [234, 522, 247, 530], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [247, 518, 300, 531], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [300, 523, 307, 531], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 518, 500, 531], "score": 1.0, "content": ". The same reasoning (with the roles", "type": "text"}], "index": 27}, {"bbox": [110, 533, 500, 546], "spans": [{"bbox": [110, 533, 401, 546], "score": 1.0, "content": "reversed) applies obviously to the symplectic structure ", "type": "text"}, {"bbox": [401, 537, 414, 545], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [415, 533, 500, 546], "score": 1.0, "content": ". Hence, in this", "type": "text"}], "index": 28}, {"bbox": [110, 547, 180, 560], "spans": [{"bbox": [110, 547, 180, 560], "score": 1.0, "content": "case we have:", "type": "text"}], "index": 29}], "index": 24.5, "bbox_fs": [109, 416, 501, 560]}, {"type": "interline_equation", "bbox": [132, 569, 479, 584], "lines": [{"bbox": [132, 569, 479, 584], "spans": [{"bbox": [132, 569, 479, 584], "score": 0.82, "content": "2\\mathrm{Re}(\\Omega_{K})|_{V}=(\\omega_{I}^{2}-\\omega_{J}^{2})(v_{1},I v_{1},v_{2},J v_{2})=\\omega_{I}(v_{1},I v_{1})\\omega_{I}(v_{2},J v_{2})-", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "lines": [{"bbox": [123, 594, 487, 608], "spans": [{"bbox": [123, 594, 487, 608], "score": 0.87, "content": "\\omega_{I}(v_{1},v_{2})\\omega_{I}(I v_{1},J v_{2})+\\omega_{I}(v_{1},J v_{2})\\omega_{I}(I v_{1},v_{2})-\\omega_{J}(v_{1},I v_{1})\\omega_{J}(v_{2},J v_{2})+", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "lines": [{"bbox": [174, 614, 434, 627], "spans": [{"bbox": [174, 614, 434, 627], "score": 0.87, "content": "\\omega_{J}(v_{1},v_{2})\\omega_{J}(I v_{1},J v_{2})-\\omega_{J}(v_{1},J v_{2})\\omega_{J}(I v_{2},v_{2})=0,", "type": "interline_equation"}], "index": 32}], "index": 32}, {"type": "text", "bbox": [110, 631, 501, 674], "lines": [{"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 266, 648], "score": 1.0, "content": "using the defining relations of ", "type": "text"}, {"bbox": [266, 638, 318, 646], "score": 0.91, "content": "\\omega_{I},\\omega_{J},\\omega_{K}", "type": "inline_equation", "height": 8, "width": 52}, {"bbox": [318, 633, 455, 648], "score": 1.0, "content": ", the quaternionic relation ", "type": "text"}, {"bbox": [455, 635, 496, 644], "score": 0.92, "content": "I J=K", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [496, 633, 500, 648], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [110, 648, 500, 662], "spans": [{"bbox": [110, 648, 198, 662], "score": 1.0, "content": "the invariance of ", "type": "text"}, {"bbox": [198, 653, 204, 660], "score": 0.89, "content": "g", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [204, 648, 297, 662], "score": 1.0, "content": " and the fact that ", "type": "text"}, {"bbox": [297, 649, 307, 658], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 648, 462, 662], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [462, 653, 478, 660], "score": 0.91, "content": "\\omega_{K}", "type": "inline_equation", "height": 7, "width": 16}, {"bbox": [479, 648, 500, 662], "score": 1.0, "content": ". So", "type": "text"}], "index": 34}, {"bbox": [110, 662, 414, 675], "spans": [{"bbox": [110, 662, 414, 675], "score": 1.0, "content": "this subcase is not consistent with the calibration property.", "type": "text"}], "index": 35}], "index": 34, "bbox_fs": [109, 633, 500, 675]}]}
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[{"type": "text", "bbox": [109, 124, 500, 212], "content": "", "index": 0}, {"type": "text", "bbox": [109, 213, 500, 371], "content": "Second case: is neither symplectic nor Lagrangian for the structure . Notice can not be symplectic with respect to , otherwise by the first case it would be Lagrangian in the strucutre ; moreover we can assume that is not Lagrangian with respect to , otherwise there is nothing to prove. So in this case is neither Lagrangian nor symplectic in the structure and in the structure . This means that contains a symplectic 2-plane with respect to and a symplectic 2-plane with respect to . Indeed, consider ; since is not Lagrangian in the structure , there exists such that and this implies that the vector subspace spanned by is a symplectic vector space for , which can not be extended to all . The same reasoning applies in the structure .", "index": 1}, {"type": "text", "bbox": [109, 372, 500, 414], "content": "We prove that this can not happen, since it violates the calibration con- dition. We have to distinguish three different subcases according to the intersection of with .", "index": 2}, {"type": "text", "bbox": [109, 415, 500, 559], "content": "First subcase: and have zero intersection. If this happens we can always choose a basis of of the form . Write for the 2- plane spanned by and for that spanned by , so that . Indeed, since is not Lagrangian with respect to , it has to contain a symplectic 2-plane like , and similarly for and . Moreover, since is not symplectic with respect to , it turns out that the symplectic 2-plane can not be completed to a symplectic basis of , so that has to contain an isotropic 2-plane for , which is . The same reasoning (with the roles reversed) applies obviously to the symplectic structure . Hence, in this case we have:", "index": 3}, {"type": "interline_equation", "bbox": [132, 569, 479, 584], "content": "", "index": 4}, {"type": "interline_equation", "bbox": [123, 594, 487, 608], "content": "", "index": 5}, {"type": "interline_equation", "bbox": [174, 614, 434, 627], "content": "", "index": 6}, {"type": "text", "bbox": [110, 631, 501, 674], "content": "using the defining relations of , the quaternionic relation , the invariance of and the fact that is Lagrangian with respect to . So this subcase is not consistent with the calibration property.", "index": 7}]
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[{"bbox": [127, 213, 500, 228], "content": "Second case: is neither symplectic nor Lagrangian for the structure .", "parent_index": 1, "line_index": 0}, {"bbox": [109, 228, 500, 243], "content": "Notice can not be symplectic with respect to , otherwise by the first", "parent_index": 1, "line_index": 1}, {"bbox": [110, 244, 500, 257], "content": "case it would be Lagrangian in the strucutre ; moreover we can assume", "parent_index": 1, "line_index": 2}, {"bbox": [110, 257, 500, 272], "content": "that is not Lagrangian with respect to , otherwise there is nothing to", "parent_index": 1, "line_index": 3}, {"bbox": [110, 273, 501, 286], "content": "prove. So in this case is neither Lagrangian nor symplectic in the structure", "parent_index": 1, "line_index": 4}, {"bbox": [110, 286, 501, 301], "content": "and in the structure . This means that contains a symplectic 2-plane", "parent_index": 1, "line_index": 5}, {"bbox": [110, 302, 500, 315], "content": "with respect to and a symplectic 2-plane with respect to . Indeed,", "parent_index": 1, "line_index": 6}, {"bbox": [110, 316, 500, 329], "content": "consider ; since is not Lagrangian in the structure , there exists", "parent_index": 1, "line_index": 7}, {"bbox": [110, 330, 500, 344], "content": "such that and this implies that the vector subspace", "parent_index": 1, "line_index": 8}, {"bbox": [110, 344, 500, 359], "content": "spanned by is a symplectic vector space for , which can not be", "parent_index": 1, "line_index": 9}, {"bbox": [110, 358, 455, 374], "content": "extended to all . The same reasoning applies in the structure .", "parent_index": 1, "line_index": 10}, {"bbox": [127, 373, 499, 388], "content": "We prove that this can not happen, since it violates the calibration con-", "parent_index": 2, "line_index": 0}, {"bbox": [110, 387, 500, 402], "content": "dition. We have to distinguish three different subcases according to the", "parent_index": 2, "line_index": 1}, {"bbox": [110, 402, 235, 417], "content": "intersection of with .", "parent_index": 2, "line_index": 2}, {"bbox": [127, 416, 500, 430], "content": "First subcase: and have zero intersection. If this happens we can", "parent_index": 3, "line_index": 0}, {"bbox": [110, 431, 500, 445], "content": "always choose a basis of of the form . Write for the 2-", "parent_index": 3, "line_index": 1}, {"bbox": [110, 445, 500, 460], "content": "plane spanned by and for that spanned by , so that .", "parent_index": 3, "line_index": 2}, {"bbox": [109, 459, 501, 475], "content": "Indeed, since is not Lagrangian with respect to , it has to contain a", "parent_index": 3, "line_index": 3}, {"bbox": [109, 475, 501, 488], "content": "symplectic 2-plane like , and similarly for and . Moreover, since is", "parent_index": 3, "line_index": 4}, {"bbox": [110, 490, 500, 502], "content": "not symplectic with respect to , it turns out that the symplectic 2-plane", "parent_index": 3, "line_index": 5}, {"bbox": [110, 504, 500, 516], "content": "can not be completed to a symplectic basis of , so that has to contain", "parent_index": 3, "line_index": 6}, {"bbox": [110, 518, 500, 531], "content": "an isotropic 2-plane for , which is . The same reasoning (with the roles", "parent_index": 3, "line_index": 7}, {"bbox": [110, 533, 500, 546], "content": "reversed) applies obviously to the symplectic structure . Hence, in this", "parent_index": 3, "line_index": 8}, {"bbox": [110, 547, 180, 560], "content": "case we have:", "parent_index": 3, "line_index": 9}, {"bbox": [109, 633, 500, 648], "content": "using the defining relations of , the quaternionic relation ,", "parent_index": 7, "line_index": 0}, {"bbox": [110, 648, 500, 662], "content": "the invariance of and the fact that is Lagrangian with respect to . So", "parent_index": 7, "line_index": 1}, {"bbox": [110, 662, 414, 675], "content": "this subcase is not consistent with the calibration property.", "parent_index": 7, "line_index": 2}]
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[]
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[]
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Second subcase: $\pi$ and $\rho$ have a 1-dimensional intersection spanned by a vector $v_{1}$ . In this case we can choose a basis of $V$ of the form $(v_{1},I v_{1},J v_{1},w)$ ( $\pi$ is spanned by $(v_{1},I v_{1})$ , while $\rho$ is spanned by $\left(v_{1},J v_{1}\right))$ . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration.
Third subcase: Finally $\pi{=}\rho$ can not clearly happen, since otherwise one can choose a basis of $\pi$ equal to $(v_{1},I v_{1})$ , but then, in this basis $\omega_{J}$ is identically vanishing, contrary to the assumption that $\rho~=~\pi$ is a symplectic 2-plane also for $\omega_{J}$ .
Since the second case can never happen $V$ has to be Lagrangian also with respect to $\omega_{J}$ .
Up to now, we have worked only locally; to conclude the proof it is necessary to show that if $T_{p}\Lambda$ is Lagrangian with respect to $\omega_{J}$ , then it can not be possible that $T_{q}\Lambda$ is Lagrangian with respect to $\omega_{I}$ , for a different $q\in\Lambda$ . Notice that any tangent space to $\Lambda$ can not be Lagrangian with respect to both $\omega_{I}$ and $\omega_{J}$ , otherwise it would violates the calibration condition. Consider now the following smooth sections of $\Lambda^{2}T^{*}\Lambda$ :
$$
\begin{array}{c c c c}{{\alpha_{I,J}:\ \Lambda}}&{{\to}}&{{\Lambda^{2}\,T^{*}\Lambda}}\\ {{p}}&{{\mapsto}}&{{\omega_{I,J}|T_{p}\Lambda}}\end{array}
$$
and the zero section $s_{0}:\Lambda\to\Lambda^{2}\,T^{*}\Lambda$ . Obviously, $s_{0}(\Lambda)$ is closed in $\Lambda^{2}T^{*}\Lambda$ , and by the previous reasoning $\Lambda$ can be decomposed as $\Lambda\,=\,\alpha_{I}^{-1}(s_{0}(\Lambda))\cup$ $\alpha_{J}^{-1}(s_{0}(\Lambda))$ , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since $\Lambda$ is connected, and this implies that one of the two closed subset is empty, so $\Lambda$ is bi-Lagrangian. 口
The previous theorem is important in view of the following:
Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $X$ can be realized as $a$ complex submanifold, via hyperkaehler rotation of the complex structure of $X$ .
Proof: Let $\Lambda$ be a special Lagrangian submanifold of $X$ in the complex structure $K$ . Then by definition $\mathrm{Re}(\Omega_{K})_{|\Lambda}=\mathrm{Vol}_{g}(\Lambda)$ , but by the previous theorem, since $\omega_{J}|_{\Lambda}=0$ this means:
$$
\mathrm{Vol}_{g}(\Lambda)=\frac{1}{2}\int_{\Lambda}\omega_{I}^{2}.
$$
By Wirtinger’s theorem, since $\Lambda$ is assumed to be compact and without border, condition (3) is equivalent to say that $\Lambda$ is a complex submanifold of
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<html><body>
<p data-bbox="109 125 500 197">Second subcase: $\pi$ and $\rho$ have a 1-dimensional intersection spanned by a vector $v_{1}$ . In this case we can choose a basis of $V$ of the form $(v_{1},I v_{1},J v_{1},w)$ ( $\pi$ is spanned by $(v_{1},I v_{1})$ , while $\rho$ is spanned by $\left(v_{1},J v_{1}\right))$ . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration. </p>
<p data-bbox="109 198 500 255">Third subcase: Finally $\pi{=}\rho$ can not clearly happen, since otherwise one can choose a basis of $\pi$ equal to $(v_{1},I v_{1})$ , but then, in this basis $\omega_{J}$ is identically vanishing, contrary to the assumption that $\rho~=~\pi$ is a symplectic 2-plane also for $\omega_{J}$ . </p>
<p data-bbox="109 255 499 284">Since the second case can never happen $V$ has to be Lagrangian also with respect to $\omega_{J}$ . </p>
<p data-bbox="109 284 500 371">Up to now, we have worked only locally; to conclude the proof it is necessary to show that if $T_{p}\Lambda$ is Lagrangian with respect to $\omega_{J}$ , then it can not be possible that $T_{q}\Lambda$ is Lagrangian with respect to $\omega_{I}$ , for a different $q\in\Lambda$ . Notice that any tangent space to $\Lambda$ can not be Lagrangian with respect to both $\omega_{I}$ and $\omega_{J}$ , otherwise it would violates the calibration condition. Consider now the following smooth sections of $\Lambda^{2}T^{*}\Lambda$ : </p>
<div class="equation" data-bbox="244 377 365 411">$$
\begin{array}{c c c c}{{\alpha_{I,J}:\ \Lambda}}&{{\to}}&{{\Lambda^{2}\,T^{*}\Lambda}}\\ {{p}}&{{\mapsto}}&{{\omega_{I,J}|T_{p}\Lambda}}\end{array}
$$</div>
<p data-bbox="109 413 501 487">and the zero section $s_{0}:\Lambda\to\Lambda^{2}\,T^{*}\Lambda$ . Obviously, $s_{0}(\Lambda)$ is closed in $\Lambda^{2}T^{*}\Lambda$ , and by the previous reasoning $\Lambda$ can be decomposed as $\Lambda\,=\,\alpha_{I}^{-1}(s_{0}(\Lambda))\cup$ $\alpha_{J}^{-1}(s_{0}(\Lambda))$ , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since $\Lambda$ is connected, and this implies that one of the two closed subset is empty, so $\Lambda$ is bi-Lagrangian. 口 </p>
<p data-bbox="126 492 435 506">The previous theorem is important in view of the following: </p>
<p data-bbox="110 507 501 563">Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $X$ can be realized as $a$ complex submanifold, via hyperkaehler rotation of the complex structure of $X$ . </p>
<p data-bbox="109 564 500 607">Proof: Let $\Lambda$ be a special Lagrangian submanifold of $X$ in the complex structure $K$ . Then by definition $\mathrm{Re}(\Omega_{K})_{|\Lambda}=\mathrm{Vol}_{g}(\Lambda)$ , but by the previous theorem, since $\omega_{J}|_{\Lambda}=0$ this means: </p>
<div class="equation" data-bbox="257 614 353 642">$$
\mathrm{Vol}_{g}(\Lambda)=\frac{1}{2}\int_{\Lambda}\omega_{I}^{2}.
$$</div>
<p data-bbox="110 645 501 675">By Wirtinger’s theorem, since $\Lambda$ is assumed to be compact and without border, condition (3) is equivalent to say that $\Lambda$ is a complex submanifold of </p>
</body></html>
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0001060v1
| 4 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Second subcase: $\\pi$ and $\\rho$ have a 1-dimensional intersection spanned by a vector $v_{1}$ . In this case we can choose a basis of $V$ of the form $(v_{1},I v_{1},J v_{1},w)$ ( $\\pi$ is spanned by $(v_{1},I v_{1})$ , while $\\rho$ is spanned by $\\left(v_{1},J v_{1}\\right))$ . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration. ", "page_idx": 4}, {"type": "text", "text": "Third subcase: Finally $\\pi{=}\\rho$ can not clearly happen, since otherwise one can choose a basis of $\\pi$ equal to $(v_{1},I v_{1})$ , but then, in this basis $\\omega_{J}$ is identically vanishing, contrary to the assumption that $\\rho~=~\\pi$ is a symplectic 2-plane also for $\\omega_{J}$ . ", "page_idx": 4}, {"type": "text", "text": "Since the second case can never happen $V$ has to be Lagrangian also with respect to $\\omega_{J}$ . ", "page_idx": 4}, {"type": "text", "text": "Up to now, we have worked only locally; to conclude the proof it is necessary to show that if $T_{p}\\Lambda$ is Lagrangian with respect to $\\omega_{J}$ , then it can not be possible that $T_{q}\\Lambda$ is Lagrangian with respect to $\\omega_{I}$ , for a different $q\\in\\Lambda$ . Notice that any tangent space to $\\Lambda$ can not be Lagrangian with respect to both $\\omega_{I}$ and $\\omega_{J}$ , otherwise it would violates the calibration condition. Consider now the following smooth sections of $\\Lambda^{2}T^{*}\\Lambda$ : ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{*}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}|T_{p}\\Lambda}}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "and the zero section $s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{*}\\Lambda$ . Obviously, $s_{0}(\\Lambda)$ is closed in $\\Lambda^{2}T^{*}\\Lambda$ , and by the previous reasoning $\\Lambda$ can be decomposed as $\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup$ $\\alpha_{J}^{-1}(s_{0}(\\Lambda))$ , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since $\\Lambda$ is connected, and this implies that one of the two closed subset is empty, so $\\Lambda$ is bi-Lagrangian. 口 ", "page_idx": 4}, {"type": "text", "text": "The previous theorem is important in view of the following: ", "page_idx": 4}, {"type": "text", "text": "Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $X$ can be realized as $a$ complex submanifold, via hyperkaehler rotation of the complex structure of $X$ . ", "page_idx": 4}, {"type": "text", "text": "Proof: Let $\\Lambda$ be a special Lagrangian submanifold of $X$ in the complex structure $K$ . Then by definition $\\mathrm{Re}(\\Omega_{K})_{|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)$ , but by the previous theorem, since $\\omega_{J}|_{\\Lambda}=0$ this means: ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "By Wirtinger’s theorem, since $\\Lambda$ is assumed to be compact and without border, condition (3) is equivalent to say that $\\Lambda$ is a complex submanifold of ", "page_idx": 4}]
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In this case we can choose a basis of ", "type": "text"}, {"bbox": [348, 144, 358, 153], "score": 0.87, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 142, 420, 156], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [420, 143, 499, 155], "score": 0.92, "content": "(v_{1},I v_{1},J v_{1},w)", "type": "inline_equation", "height": 12, "width": 79}], "index": 1}, {"bbox": [110, 157, 500, 171], "spans": [{"bbox": [110, 157, 114, 171], "score": 1.0, "content": "(", "type": "text"}, {"bbox": [115, 162, 122, 167], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [122, 157, 203, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [203, 158, 244, 170], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 157, 284, 171], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [284, 162, 290, 169], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [291, 157, 371, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [372, 158, 418, 170], "score": 0.92, "content": "\\left(v_{1},J v_{1}\\right))", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [418, 157, 500, 171], "score": 1.0, "content": ". Again by the", "type": "text"}], "index": 2}, {"bbox": [110, 172, 499, 185], "spans": [{"bbox": [110, 172, 499, 185], "score": 1.0, "content": "same computation of the previous subcase one shows that this configuration", "type": "text"}], "index": 3}, {"bbox": [109, 186, 307, 198], "spans": [{"bbox": [109, 186, 307, 198], "score": 1.0, "content": "is not compatible with the calibration.", "type": "text"}], "index": 4}], "index": 2}, {"type": "text", "bbox": [109, 198, 500, 255], "lines": [{"bbox": [127, 199, 500, 215], "spans": [{"bbox": [127, 199, 248, 215], "score": 1.0, "content": "Third subcase: Finally ", "type": "text"}, {"bbox": [248, 204, 271, 213], "score": 0.88, "content": "\\pi{=}\\rho", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [271, 199, 500, 215], "score": 1.0, "content": " can not clearly happen, since otherwise one", "type": "text"}], "index": 5}, {"bbox": [110, 214, 499, 228], "spans": [{"bbox": [110, 214, 221, 228], "score": 1.0, "content": "can choose a basis of ", "type": "text"}, {"bbox": [221, 219, 228, 225], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [229, 214, 278, 228], "score": 1.0, "content": " equal to ", "type": "text"}, {"bbox": [278, 215, 320, 228], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [320, 214, 444, 228], "score": 1.0, "content": ", but then, in this basis ", "type": "text"}, {"bbox": [444, 219, 457, 227], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [458, 214, 499, 228], "score": 1.0, "content": " is iden-", "type": "text"}], "index": 6}, {"bbox": [109, 227, 500, 245], "spans": [{"bbox": [109, 227, 380, 245], "score": 1.0, "content": "tically vanishing, contrary to the assumption that ", "type": "text"}, {"bbox": [380, 234, 415, 242], "score": 0.91, "content": "\\rho~=~\\pi", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [415, 227, 500, 245], "score": 1.0, "content": " is a symplectic", "type": "text"}], "index": 7}, {"bbox": [110, 241, 211, 259], "spans": [{"bbox": [110, 241, 192, 259], "score": 1.0, "content": "2-plane also for ", "type": "text"}, {"bbox": [192, 248, 206, 256], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [206, 241, 211, 259], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5}, {"type": "text", "bbox": [109, 255, 499, 284], "lines": [{"bbox": [126, 257, 499, 271], "spans": [{"bbox": [126, 257, 330, 271], "score": 1.0, "content": "Since the second case can never happen ", "type": "text"}, {"bbox": [331, 259, 340, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [340, 257, 499, 271], "score": 1.0, "content": " has to be Lagrangian also with", "type": "text"}], "index": 9}, {"bbox": [109, 271, 182, 287], "spans": [{"bbox": [109, 271, 164, 287], "score": 1.0, "content": "respect to ", "type": "text"}, {"bbox": [164, 277, 178, 285], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [178, 271, 182, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [109, 284, 500, 371], "lines": [{"bbox": [128, 286, 501, 301], "spans": [{"bbox": [128, 286, 501, 301], "score": 1.0, "content": "Up to now, we have worked only locally; to conclude the proof it is", "type": "text"}], "index": 11}, {"bbox": [110, 302, 501, 315], "spans": [{"bbox": [110, 302, 235, 315], "score": 1.0, "content": "necessary to show that if ", "type": "text"}, {"bbox": [235, 303, 255, 315], "score": 0.93, "content": "T_{p}\\Lambda", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [256, 302, 406, 315], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [406, 306, 420, 313], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [420, 302, 501, 315], "score": 1.0, "content": ", then it can not", "type": "text"}], "index": 12}, {"bbox": [110, 315, 500, 330], "spans": [{"bbox": [110, 315, 195, 330], "score": 1.0, "content": "be possible that ", "type": "text"}, {"bbox": [195, 317, 216, 329], "score": 0.94, "content": "T_{q}\\Lambda", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [216, 315, 373, 330], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [374, 320, 386, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [386, 315, 466, 330], "score": 1.0, "content": ", for a different ", "type": "text"}, {"bbox": [467, 317, 496, 328], "score": 0.93, "content": "q\\in\\Lambda", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [496, 315, 500, 330], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [109, 330, 500, 344], "spans": [{"bbox": [109, 330, 275, 344], "score": 1.0, "content": "Notice that any tangent space to ", "type": "text"}, {"bbox": [275, 331, 284, 340], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 330, 500, 344], "score": 1.0, "content": " can not be Lagrangian with respect to both", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 349, 122, 357], "score": 0.91, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 345, 150, 358], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [150, 349, 164, 357], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [164, 345, 500, 358], "score": 1.0, "content": ", otherwise it would violates the calibration condition. Consider", "type": "text"}], "index": 15}, {"bbox": [109, 359, 344, 372], "spans": [{"bbox": [109, 360, 302, 372], "score": 1.0, "content": "now the following smooth sections of", "type": "text"}, {"bbox": [302, 359, 339, 371], "score": 0.93, "content": "\\Lambda^{2}T^{*}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [339, 360, 344, 372], "score": 1.0, "content": ":", "type": "text"}], "index": 16}], "index": 13.5}, {"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{*}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [109, 413, 501, 487], "lines": [{"bbox": [109, 416, 500, 432], "spans": [{"bbox": [109, 416, 217, 432], "score": 1.0, "content": "and the zero section ", "type": "text"}, {"bbox": [217, 417, 301, 429], "score": 0.94, "content": "s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{*}\\Lambda", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [302, 416, 367, 432], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [367, 418, 395, 430], "score": 0.95, "content": "s_{0}(\\Lambda)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [395, 416, 458, 432], "score": 1.0, "content": " is closed in", "type": "text"}, {"bbox": [459, 417, 496, 429], "score": 0.92, "content": "\\Lambda^{2}T^{*}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [497, 416, 500, 432], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 431, 500, 446], "spans": [{"bbox": [110, 431, 272, 446], "score": 1.0, "content": "and by the previous reasoning ", "type": "text"}, {"bbox": [272, 433, 281, 442], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [281, 431, 405, 446], "score": 1.0, "content": " can be decomposed as ", "type": "text"}, {"bbox": [405, 432, 500, 445], "score": 0.93, "content": "\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup", "type": "inline_equation", "height": 13, "width": 95}], "index": 19}, {"bbox": [110, 444, 501, 461], "spans": [{"bbox": [110, 446, 166, 460], "score": 0.97, "content": "\\alpha_{J}^{-1}(s_{0}(\\Lambda))", "type": "inline_equation", "height": 14, "width": 56}, {"bbox": [166, 444, 501, 461], "score": 1.0, "content": ", that is as the disjoint union of two proper closed subsets. But", "type": "text"}], "index": 20}, {"bbox": [110, 461, 501, 474], "spans": [{"bbox": [110, 461, 272, 474], "score": 1.0, "content": "this is clearly impossible, since ", "type": "text"}, {"bbox": [273, 462, 281, 471], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [281, 461, 501, 474], "score": 1.0, "content": " is connected, and this implies that one of", "type": "text"}], "index": 21}, {"bbox": [109, 474, 502, 489], "spans": [{"bbox": [109, 474, 286, 489], "score": 1.0, "content": "the two closed subset is empty, so ", "type": "text"}, {"bbox": [287, 477, 295, 485], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [296, 474, 385, 489], "score": 1.0, "content": " is bi-Lagrangian.", "type": "text"}, {"bbox": [491, 477, 502, 487], "score": 0.9912325739860535, "content": "口", "type": "text"}], "index": 22}], "index": 20}, {"type": "text", "bbox": [126, 492, 435, 506], "lines": [{"bbox": [127, 493, 434, 509], "spans": [{"bbox": [127, 493, 434, 509], "score": 1.0, "content": "The previous theorem is important in view of the following:", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [110, 507, 501, 563], "lines": [{"bbox": [126, 507, 501, 524], "spans": [{"bbox": [126, 507, 501, 524], "score": 1.0, "content": "Corollary 2.1: Every (connected, compact and without border) special", "type": "text"}], "index": 24}, {"bbox": [111, 523, 500, 537], "spans": [{"bbox": [111, 523, 379, 537], "score": 1.0, "content": "Lagrangian submanifold Λ of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [380, 524, 391, 533], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [391, 523, 493, 537], "score": 1.0, "content": " can be realized as ", "type": "text"}, {"bbox": [493, 528, 500, 533], "score": 0.26, "content": "a", "type": "inline_equation", "height": 5, "width": 7}], "index": 25}, {"bbox": [110, 537, 502, 552], "spans": [{"bbox": [110, 537, 502, 552], "score": 1.0, "content": "complex submanifold, via hyperkaehler rotation of the complex structure of", "type": "text"}], "index": 26}, {"bbox": [110, 552, 126, 565], "spans": [{"bbox": [110, 554, 121, 563], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 552, 126, 565], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "text", "bbox": [109, 564, 500, 607], "lines": [{"bbox": [127, 566, 498, 580], "spans": [{"bbox": [127, 566, 190, 580], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [190, 568, 199, 577], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 566, 407, 580], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [408, 568, 419, 577], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [419, 566, 498, 580], "score": 1.0, "content": " in the complex", "type": "text"}], "index": 28}, {"bbox": [109, 581, 500, 596], "spans": [{"bbox": [109, 581, 160, 596], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [161, 582, 172, 591], "score": 0.91, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [172, 581, 282, 596], "score": 1.0, "content": ". Then by definition ", "type": "text"}, {"bbox": [283, 582, 388, 595], "score": 0.93, "content": "\\mathrm{Re}(\\Omega_{K})_{|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [389, 581, 500, 596], "score": 1.0, "content": ", but by the previous", "type": "text"}], "index": 29}, {"bbox": [110, 595, 296, 610], "spans": [{"bbox": [110, 595, 187, 610], "score": 1.0, "content": "theorem, since ", "type": "text"}, {"bbox": [187, 596, 232, 608], "score": 0.95, "content": "\\omega_{J}|_{\\Lambda}=0", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [233, 595, 296, 610], "score": 1.0, "content": " this means:", "type": "text"}], "index": 30}], "index": 29}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "lines": [{"bbox": [257, 614, 353, 642], "spans": [{"bbox": [257, 614, 353, 642], "score": 0.95, "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [110, 645, 501, 675], "lines": [{"bbox": [111, 648, 500, 661], "spans": [{"bbox": [111, 648, 275, 661], "score": 1.0, "content": "By Wirtinger’s theorem, since ", "type": "text"}, {"bbox": [275, 649, 284, 658], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 648, 500, 661], "score": 1.0, "content": " is assumed to be compact and without", "type": "text"}], "index": 32}, {"bbox": [110, 662, 501, 675], "spans": [{"bbox": [110, 662, 345, 675], "score": 1.0, "content": "border, condition (3) is equivalent to say that ", "type": "text"}, {"bbox": [345, 664, 354, 672], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [354, 662, 501, 675], "score": 1.0, "content": " is a complex submanifold of", "type": "text"}], "index": 33}], "index": 32.5}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{*}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "lines": [{"bbox": [257, 614, 353, 642], "spans": [{"bbox": [257, 614, 353, 642], "score": 0.95, "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "type": "interline_equation"}], "index": 31}], "index": 31}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 309, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "5", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 500, 197], "lines": [{"bbox": [127, 127, 501, 142], "spans": [{"bbox": [127, 127, 214, 142], "score": 1.0, "content": "Second subcase: ", "type": "text"}, {"bbox": [214, 133, 222, 138], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [222, 127, 248, 142], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 133, 254, 141], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [255, 127, 501, 142], "score": 1.0, "content": " have a 1-dimensional intersection spanned by a", "type": "text"}], "index": 0}, {"bbox": [110, 142, 499, 156], "spans": [{"bbox": [110, 142, 144, 156], "score": 1.0, "content": "vector ", "type": "text"}, {"bbox": [145, 147, 155, 154], "score": 0.91, "content": "v_{1}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [155, 142, 348, 156], "score": 1.0, "content": ". In this case we can choose a basis of ", "type": "text"}, {"bbox": [348, 144, 358, 153], "score": 0.87, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [358, 142, 420, 156], "score": 1.0, "content": " of the form ", "type": "text"}, {"bbox": [420, 143, 499, 155], "score": 0.92, "content": "(v_{1},I v_{1},J v_{1},w)", "type": "inline_equation", "height": 12, "width": 79}], "index": 1}, {"bbox": [110, 157, 500, 171], "spans": [{"bbox": [110, 157, 114, 171], "score": 1.0, "content": "(", "type": "text"}, {"bbox": [115, 162, 122, 167], "score": 0.86, "content": "\\pi", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [122, 157, 203, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [203, 158, 244, 170], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 157, 284, 171], "score": 1.0, "content": ", while ", "type": "text"}, {"bbox": [284, 162, 290, 169], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [291, 157, 371, 171], "score": 1.0, "content": " is spanned by ", "type": "text"}, {"bbox": [372, 158, 418, 170], "score": 0.92, "content": "\\left(v_{1},J v_{1}\\right))", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [418, 157, 500, 171], "score": 1.0, "content": ". Again by the", "type": "text"}], "index": 2}, {"bbox": [110, 172, 499, 185], "spans": [{"bbox": [110, 172, 499, 185], "score": 1.0, "content": "same computation of the previous subcase one shows that this configuration", "type": "text"}], "index": 3}, {"bbox": [109, 186, 307, 198], "spans": [{"bbox": [109, 186, 307, 198], "score": 1.0, "content": "is not compatible with the calibration.", "type": "text"}], "index": 4}], "index": 2, "bbox_fs": [109, 127, 501, 198]}, {"type": "text", "bbox": [109, 198, 500, 255], "lines": [{"bbox": [127, 199, 500, 215], "spans": [{"bbox": [127, 199, 248, 215], "score": 1.0, "content": "Third subcase: Finally ", "type": "text"}, {"bbox": [248, 204, 271, 213], "score": 0.88, "content": "\\pi{=}\\rho", "type": "inline_equation", "height": 9, "width": 23}, {"bbox": [271, 199, 500, 215], "score": 1.0, "content": " can not clearly happen, since otherwise one", "type": "text"}], "index": 5}, {"bbox": [110, 214, 499, 228], "spans": [{"bbox": [110, 214, 221, 228], "score": 1.0, "content": "can choose a basis of ", "type": "text"}, {"bbox": [221, 219, 228, 225], "score": 0.89, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [229, 214, 278, 228], "score": 1.0, "content": " equal to ", "type": "text"}, {"bbox": [278, 215, 320, 228], "score": 0.94, "content": "(v_{1},I v_{1})", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [320, 214, 444, 228], "score": 1.0, "content": ", but then, in this basis ", "type": "text"}, {"bbox": [444, 219, 457, 227], "score": 0.91, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [458, 214, 499, 228], "score": 1.0, "content": " is iden-", "type": "text"}], "index": 6}, {"bbox": [109, 227, 500, 245], "spans": [{"bbox": [109, 227, 380, 245], "score": 1.0, "content": "tically vanishing, contrary to the assumption that ", "type": "text"}, {"bbox": [380, 234, 415, 242], "score": 0.91, "content": "\\rho~=~\\pi", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [415, 227, 500, 245], "score": 1.0, "content": " is a symplectic", "type": "text"}], "index": 7}, {"bbox": [110, 241, 211, 259], "spans": [{"bbox": [110, 241, 192, 259], "score": 1.0, "content": "2-plane also for ", "type": "text"}, {"bbox": [192, 248, 206, 256], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [206, 241, 211, 259], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5, "bbox_fs": [109, 199, 500, 259]}, {"type": "text", "bbox": [109, 255, 499, 284], "lines": [{"bbox": [126, 257, 499, 271], "spans": [{"bbox": [126, 257, 330, 271], "score": 1.0, "content": "Since the second case can never happen ", "type": "text"}, {"bbox": [331, 259, 340, 268], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [340, 257, 499, 271], "score": 1.0, "content": " has to be Lagrangian also with", "type": "text"}], "index": 9}, {"bbox": [109, 271, 182, 287], "spans": [{"bbox": [109, 271, 164, 287], "score": 1.0, "content": "respect to ", "type": "text"}, {"bbox": [164, 277, 178, 285], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [178, 271, 182, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [109, 257, 499, 287]}, {"type": "text", "bbox": [109, 284, 500, 371], "lines": [{"bbox": [128, 286, 501, 301], "spans": [{"bbox": [128, 286, 501, 301], "score": 1.0, "content": "Up to now, we have worked only locally; to conclude the proof it is", "type": "text"}], "index": 11}, {"bbox": [110, 302, 501, 315], "spans": [{"bbox": [110, 302, 235, 315], "score": 1.0, "content": "necessary to show that if ", "type": "text"}, {"bbox": [235, 303, 255, 315], "score": 0.93, "content": "T_{p}\\Lambda", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [256, 302, 406, 315], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [406, 306, 420, 313], "score": 0.89, "content": "\\omega_{J}", "type": "inline_equation", "height": 7, "width": 14}, {"bbox": [420, 302, 501, 315], "score": 1.0, "content": ", then it can not", "type": "text"}], "index": 12}, {"bbox": [110, 315, 500, 330], "spans": [{"bbox": [110, 315, 195, 330], "score": 1.0, "content": "be possible that ", "type": "text"}, {"bbox": [195, 317, 216, 329], "score": 0.94, "content": "T_{q}\\Lambda", "type": "inline_equation", "height": 12, "width": 21}, {"bbox": [216, 315, 373, 330], "score": 1.0, "content": " is Lagrangian with respect to ", "type": "text"}, {"bbox": [374, 320, 386, 327], "score": 0.89, "content": "\\omega_{I}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [386, 315, 466, 330], "score": 1.0, "content": ", for a different ", "type": "text"}, {"bbox": [467, 317, 496, 328], "score": 0.93, "content": "q\\in\\Lambda", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [496, 315, 500, 330], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [109, 330, 500, 344], "spans": [{"bbox": [109, 330, 275, 344], "score": 1.0, "content": "Notice that any tangent space to ", "type": "text"}, {"bbox": [275, 331, 284, 340], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 330, 500, 344], "score": 1.0, "content": " can not be Lagrangian with respect to both", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 349, 122, 357], "score": 0.91, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [123, 345, 150, 358], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [150, 349, 164, 357], "score": 0.9, "content": "\\omega_{J}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [164, 345, 500, 358], "score": 1.0, "content": ", otherwise it would violates the calibration condition. Consider", "type": "text"}], "index": 15}, {"bbox": [109, 359, 344, 372], "spans": [{"bbox": [109, 360, 302, 372], "score": 1.0, "content": "now the following smooth sections of", "type": "text"}, {"bbox": [302, 359, 339, 371], "score": 0.93, "content": "\\Lambda^{2}T^{*}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [339, 360, 344, 372], "score": 1.0, "content": ":", "type": "text"}], "index": 16}], "index": 13.5, "bbox_fs": [109, 286, 501, 372]}, {"type": "interline_equation", "bbox": [244, 377, 365, 411], "lines": [{"bbox": [244, 377, 365, 411], "spans": [{"bbox": [244, 377, 365, 411], "score": 0.91, "content": "\\begin{array}{c c c c}{{\\alpha_{I,J}:\\ \\Lambda}}&{{\\to}}&{{\\Lambda^{2}\\,T^{*}\\Lambda}}\\\\ {{p}}&{{\\mapsto}}&{{\\omega_{I,J}|T_{p}\\Lambda}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [109, 413, 501, 487], "lines": [{"bbox": [109, 416, 500, 432], "spans": [{"bbox": [109, 416, 217, 432], "score": 1.0, "content": "and the zero section ", "type": "text"}, {"bbox": [217, 417, 301, 429], "score": 0.94, "content": "s_{0}:\\Lambda\\to\\Lambda^{2}\\,T^{*}\\Lambda", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [302, 416, 367, 432], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [367, 418, 395, 430], "score": 0.95, "content": "s_{0}(\\Lambda)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [395, 416, 458, 432], "score": 1.0, "content": " is closed in", "type": "text"}, {"bbox": [459, 417, 496, 429], "score": 0.92, "content": "\\Lambda^{2}T^{*}\\Lambda", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [497, 416, 500, 432], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 431, 500, 446], "spans": [{"bbox": [110, 431, 272, 446], "score": 1.0, "content": "and by the previous reasoning ", "type": "text"}, {"bbox": [272, 433, 281, 442], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [281, 431, 405, 446], "score": 1.0, "content": " can be decomposed as ", "type": "text"}, {"bbox": [405, 432, 500, 445], "score": 0.93, "content": "\\Lambda\\,=\\,\\alpha_{I}^{-1}(s_{0}(\\Lambda))\\cup", "type": "inline_equation", "height": 13, "width": 95}], "index": 19}, {"bbox": [110, 444, 501, 461], "spans": [{"bbox": [110, 446, 166, 460], "score": 0.97, "content": "\\alpha_{J}^{-1}(s_{0}(\\Lambda))", "type": "inline_equation", "height": 14, "width": 56}, {"bbox": [166, 444, 501, 461], "score": 1.0, "content": ", that is as the disjoint union of two proper closed subsets. But", "type": "text"}], "index": 20}, {"bbox": [110, 461, 501, 474], "spans": [{"bbox": [110, 461, 272, 474], "score": 1.0, "content": "this is clearly impossible, since ", "type": "text"}, {"bbox": [273, 462, 281, 471], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [281, 461, 501, 474], "score": 1.0, "content": " is connected, and this implies that one of", "type": "text"}], "index": 21}, {"bbox": [109, 474, 502, 489], "spans": [{"bbox": [109, 474, 286, 489], "score": 1.0, "content": "the two closed subset is empty, so ", "type": "text"}, {"bbox": [287, 477, 295, 485], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [296, 474, 385, 489], "score": 1.0, "content": " is bi-Lagrangian.", "type": "text"}, {"bbox": [491, 477, 502, 487], "score": 0.9912325739860535, "content": "口", "type": "text"}], "index": 22}], "index": 20, "bbox_fs": [109, 416, 502, 489]}, {"type": "text", "bbox": [126, 492, 435, 506], "lines": [{"bbox": [127, 493, 434, 509], "spans": [{"bbox": [127, 493, 434, 509], "score": 1.0, "content": "The previous theorem is important in view of the following:", "type": "text"}], "index": 23}], "index": 23, "bbox_fs": [127, 493, 434, 509]}, {"type": "text", "bbox": [110, 507, 501, 563], "lines": [{"bbox": [126, 507, 501, 524], "spans": [{"bbox": [126, 507, 501, 524], "score": 1.0, "content": "Corollary 2.1: Every (connected, compact and without border) special", "type": "text"}], "index": 24}, {"bbox": [111, 523, 500, 537], "spans": [{"bbox": [111, 523, 379, 537], "score": 1.0, "content": "Lagrangian submanifold Λ of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [380, 524, 391, 533], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [391, 523, 493, 537], "score": 1.0, "content": " can be realized as ", "type": "text"}, {"bbox": [493, 528, 500, 533], "score": 0.26, "content": "a", "type": "inline_equation", "height": 5, "width": 7}], "index": 25}, {"bbox": [110, 537, 502, 552], "spans": [{"bbox": [110, 537, 502, 552], "score": 1.0, "content": "complex submanifold, via hyperkaehler rotation of the complex structure of", "type": "text"}], "index": 26}, {"bbox": [110, 552, 126, 565], "spans": [{"bbox": [110, 554, 121, 563], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 552, 126, 565], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 25.5, "bbox_fs": [110, 507, 502, 565]}, {"type": "text", "bbox": [109, 564, 500, 607], "lines": [{"bbox": [127, 566, 498, 580], "spans": [{"bbox": [127, 566, 190, 580], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [190, 568, 199, 577], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [199, 566, 407, 580], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [408, 568, 419, 577], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [419, 566, 498, 580], "score": 1.0, "content": " in the complex", "type": "text"}], "index": 28}, {"bbox": [109, 581, 500, 596], "spans": [{"bbox": [109, 581, 160, 596], "score": 1.0, "content": "structure ", "type": "text"}, {"bbox": [161, 582, 172, 591], "score": 0.91, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [172, 581, 282, 596], "score": 1.0, "content": ". Then by definition ", "type": "text"}, {"bbox": [283, 582, 388, 595], "score": 0.93, "content": "\\mathrm{Re}(\\Omega_{K})_{|\\Lambda}=\\mathrm{Vol}_{g}(\\Lambda)", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [389, 581, 500, 596], "score": 1.0, "content": ", but by the previous", "type": "text"}], "index": 29}, {"bbox": [110, 595, 296, 610], "spans": [{"bbox": [110, 595, 187, 610], "score": 1.0, "content": "theorem, since ", "type": "text"}, {"bbox": [187, 596, 232, 608], "score": 0.95, "content": "\\omega_{J}|_{\\Lambda}=0", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [233, 595, 296, 610], "score": 1.0, "content": " this means:", "type": "text"}], "index": 30}], "index": 29, "bbox_fs": [109, 566, 500, 610]}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "lines": [{"bbox": [257, 614, 353, 642], "spans": [{"bbox": [257, 614, 353, 642], "score": 0.95, "content": "\\mathrm{Vol}_{g}(\\Lambda)=\\frac{1}{2}\\int_{\\Lambda}\\omega_{I}^{2}.", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "text", "bbox": [110, 645, 501, 675], "lines": [{"bbox": [111, 648, 500, 661], "spans": [{"bbox": [111, 648, 275, 661], "score": 1.0, "content": "By Wirtinger’s theorem, since ", "type": "text"}, {"bbox": [275, 649, 284, 658], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [284, 648, 500, 661], "score": 1.0, "content": " is assumed to be compact and without", "type": "text"}], "index": 32}, {"bbox": [110, 662, 501, 675], "spans": [{"bbox": [110, 662, 345, 675], "score": 1.0, "content": "border, condition (3) is equivalent to say that ", "type": "text"}, {"bbox": [345, 664, 354, 672], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [354, 662, 501, 675], "score": 1.0, "content": " is a complex submanifold of", "type": "text"}], "index": 33}], "index": 32.5, "bbox_fs": [110, 648, 501, 675]}]}
|
[{"type": "text", "bbox": [109, 125, 500, 197], "content": "Second subcase: and have a 1-dimensional intersection spanned by a vector . In this case we can choose a basis of of the form ( is spanned by , while is spanned by . Again by the same computation of the previous subcase one shows that this configuration is not compatible with the calibration.", "index": 0}, {"type": "text", "bbox": [109, 198, 500, 255], "content": "Third subcase: Finally can not clearly happen, since otherwise one can choose a basis of equal to , but then, in this basis is iden- tically vanishing, contrary to the assumption that is a symplectic 2-plane also for .", "index": 1}, {"type": "text", "bbox": [109, 255, 499, 284], "content": "Since the second case can never happen has to be Lagrangian also with respect to .", "index": 2}, {"type": "text", "bbox": [109, 284, 500, 371], "content": "Up to now, we have worked only locally; to conclude the proof it is necessary to show that if is Lagrangian with respect to , then it can not be possible that is Lagrangian with respect to , for a different . Notice that any tangent space to can not be Lagrangian with respect to both and , otherwise it would violates the calibration condition. Consider now the following smooth sections of :", "index": 3}, {"type": "interline_equation", "bbox": [244, 377, 365, 411], "content": "", "index": 4}, {"type": "text", "bbox": [109, 413, 501, 487], "content": "and the zero section . Obviously, is closed in , and by the previous reasoning can be decomposed as , that is as the disjoint union of two proper closed subsets. But this is clearly impossible, since is connected, and this implies that one of the two closed subset is empty, so is bi-Lagrangian. 口", "index": 5}, {"type": "text", "bbox": [126, 492, 435, 506], "content": "The previous theorem is important in view of the following:", "index": 6}, {"type": "text", "bbox": [110, 507, 501, 563], "content": "Corollary 2.1: Every (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold can be realized as complex submanifold, via hyperkaehler rotation of the complex structure of .", "index": 7}, {"type": "text", "bbox": [109, 564, 500, 607], "content": "Proof: Let be a special Lagrangian submanifold of in the complex structure . Then by definition , but by the previous theorem, since this means:", "index": 8}, {"type": "interline_equation", "bbox": [257, 614, 353, 642], "content": "", "index": 9}, {"type": "text", "bbox": [110, 645, 501, 675], "content": "By Wirtinger’s theorem, since is assumed to be compact and without border, condition (3) is equivalent to say that is a complex submanifold of", "index": 10}]
|
[{"bbox": [127, 127, 501, 142], "content": "Second subcase: and have a 1-dimensional intersection spanned by a", "parent_index": 0, "line_index": 0}, {"bbox": [110, 142, 499, 156], "content": "vector . In this case we can choose a basis of of the form", "parent_index": 0, "line_index": 1}, {"bbox": [110, 157, 500, 171], "content": "( is spanned by , while is spanned by . Again by the", "parent_index": 0, "line_index": 2}, {"bbox": [110, 172, 499, 185], "content": "same computation of the previous subcase one shows that this configuration", "parent_index": 0, "line_index": 3}, {"bbox": [109, 186, 307, 198], "content": "is not compatible with the calibration.", "parent_index": 0, "line_index": 4}, {"bbox": [127, 199, 500, 215], "content": "Third subcase: Finally can not clearly happen, since otherwise one", "parent_index": 1, "line_index": 0}, {"bbox": [110, 214, 499, 228], "content": "can choose a basis of equal to , but then, in this basis is iden-", "parent_index": 1, "line_index": 1}, {"bbox": [109, 227, 500, 245], "content": "tically vanishing, contrary to the assumption that is a symplectic", "parent_index": 1, "line_index": 2}, {"bbox": [110, 241, 211, 259], "content": "2-plane also for .", "parent_index": 1, "line_index": 3}, {"bbox": [126, 257, 499, 271], "content": "Since the second case can never happen has to be Lagrangian also with", "parent_index": 2, "line_index": 0}, {"bbox": [109, 271, 182, 287], "content": "respect to .", "parent_index": 2, "line_index": 1}, {"bbox": [128, 286, 501, 301], "content": "Up to now, we have worked only locally; to conclude the proof it is", "parent_index": 3, "line_index": 0}, {"bbox": [110, 302, 501, 315], "content": "necessary to show that if is Lagrangian with respect to , then it can not", "parent_index": 3, "line_index": 1}, {"bbox": [110, 315, 500, 330], "content": "be possible that is Lagrangian with respect to , for a different .", "parent_index": 3, "line_index": 2}, {"bbox": [109, 330, 500, 344], "content": "Notice that any tangent space to can not be Lagrangian with respect to both", "parent_index": 3, "line_index": 3}, {"bbox": [110, 345, 500, 358], "content": "and , otherwise it would violates the calibration condition. Consider", "parent_index": 3, "line_index": 4}, {"bbox": [109, 359, 344, 372], "content": "now the following smooth sections of :", "parent_index": 3, "line_index": 5}, {"bbox": [109, 416, 500, 432], "content": "and the zero section . Obviously, is closed in ,", "parent_index": 5, "line_index": 0}, {"bbox": [110, 431, 500, 446], "content": "and by the previous reasoning can be decomposed as", "parent_index": 5, "line_index": 1}, {"bbox": [110, 444, 501, 461], "content": ", that is as the disjoint union of two proper closed subsets. But", "parent_index": 5, "line_index": 2}, {"bbox": [110, 461, 501, 474], "content": "this is clearly impossible, since is connected, and this implies that one of", "parent_index": 5, "line_index": 3}, {"bbox": [109, 474, 502, 489], "content": "the two closed subset is empty, so is bi-Lagrangian. 口", "parent_index": 5, "line_index": 4}, {"bbox": [127, 493, 434, 509], "content": "The previous theorem is important in view of the following:", "parent_index": 6, "line_index": 0}, {"bbox": [126, 507, 501, 524], "content": "Corollary 2.1: Every (connected, compact and without border) special", "parent_index": 7, "line_index": 0}, {"bbox": [111, 523, 500, 537], "content": "Lagrangian submanifold Λ of a hyperkaehler 4-fold can be realized as", "parent_index": 7, "line_index": 1}, {"bbox": [110, 537, 502, 552], "content": "complex submanifold, via hyperkaehler rotation of the complex structure of", "parent_index": 7, "line_index": 2}, {"bbox": [110, 552, 126, 565], "content": ".", "parent_index": 7, "line_index": 3}, {"bbox": [127, 566, 498, 580], "content": "Proof: Let be a special Lagrangian submanifold of in the complex", "parent_index": 8, "line_index": 0}, {"bbox": [109, 581, 500, 596], "content": "structure . Then by definition , but by the previous", "parent_index": 8, "line_index": 1}, {"bbox": [110, 595, 296, 610], "content": "theorem, since this means:", "parent_index": 8, "line_index": 2}, {"bbox": [111, 648, 500, 661], "content": "By Wirtinger’s theorem, since is assumed to be compact and without", "parent_index": 10, "line_index": 0}, {"bbox": [110, 662, 501, 675], "content": "border, condition (3) is equivalent to say that is a complex submanifold of", "parent_index": 10, "line_index": 1}]
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$X$ , in the complex structure $I$ , that is performing a hyperkaehler rotation. Notice that in the complex structure $I$ , $\Lambda$ is still a Lagrangian submanifold with respect to $\omega_{K}$ and $\omega_{I}$ , so it is Lagrangian with respect to the holomorphic (in the structure $I$ ) 2-form $\Omega_{I}:=\omega_{J}+i\omega_{K}$ . 口
Collecting the results so far proved, we can show that special Lagrangian submanifolds of $X$ are particularly rigid:
Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $X$ is real analytic.
Proof: Let $\Lambda$ be a special Lagrangian submanifold of $X$ , having fixed some complex structure on $X$ , let us say $K$ ; then, by Corollary 2.1 there exists a new complex structure, let us say $I$ , in which $\Lambda$ is holomorphic, that is, it is locally given by:
$$
f_{1}(z_{1},\ldots,z_{4})=0\quad\mathrm{and}\quad f_{2}(z_{1},\ldots,z_{4})=0.
$$
Now observe that coming back to the original complex structure $K$ , we induce an analytic change of coordinates from the holomorphic coordinates $z^{i}$ $\begin{array}{r}{\int\!\frac{\partial}{\partial z^{i}}\,=\,i\frac{\partial}{\partial z^{i}}\rangle}\end{array}$ to new holomorphic coordinates $w^{i}$ ( $\begin{array}{r}{K\frac{\partial}{\partial w^{i}}\,=\,i\frac{\partial}{\partial w^{i}}\Big)}\end{array}$ i ∂∂wi ) such that locally:
$$
z^{i}=c_{1}w^{i}+c_{2}\bar{w}^{i}\;\;\;\;\bar{z}^{i}=d_{1}w^{i}+d_{2}\bar{w}^{i},
$$
for some complex constants $c_{j},d_{j}$ . Thus in the complex structure $K$ the special Lagrangian submanifold $\Lambda$ is given by $f_{j}(c_{1}w^{i}\!+\!c_{2}\bar{w}^{i},d_{1}w^{i}\!+\!d_{2}\bar{w}^{i})=0$ which is again the zero locus of a set of functions analytic in $w^{i},\bar{w}^{i}$ . 口
Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds $S$ of $X$ ; in particular we have an action of the hyperkaehler rotation on the structure sheaf $O_{S}$ (here, as always, we identify $O_{S}$ with its direct image $j_{*}O_{S}$ , where $j\,:\,S\,\rightarrow\,X$ is the holomorphic embedding). We are thus led to give the following:
Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyperkaehler 4-fold $X$ (in the complex structure $K$ ). Then we define the special Lagrangian structure sheaf ${\mathcal{L}}_{\Lambda}$ as the sheaf obtained by the action of the hyperkaehler rotation on the structure sheaf $O_{\Lambda}$ of $\Lambda$ , as a complex Lagrangian submanifold of $X$ , (in the structure $I$ ).
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<html><body>
<p data-bbox="110 125 501 183">$X$ , in the complex structure $I$ , that is performing a hyperkaehler rotation. Notice that in the complex structure $I$ , $\Lambda$ is still a Lagrangian submanifold with respect to $\omega_{K}$ and $\omega_{I}$ , so it is Lagrangian with respect to the holomorphic (in the structure $I$ ) 2-form $\Omega_{I}:=\omega_{J}+i\omega_{K}$ . 口 </p>
<p data-bbox="109 189 500 218">Collecting the results so far proved, we can show that special Lagrangian submanifolds of $X$ are particularly rigid: </p>
<p data-bbox="110 219 502 246">Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $X$ is real analytic. </p>
<p data-bbox="109 247 500 305">Proof: Let $\Lambda$ be a special Lagrangian submanifold of $X$ , having fixed some complex structure on $X$ , let us say $K$ ; then, by Corollary 2.1 there exists a new complex structure, let us say $I$ , in which $\Lambda$ is holomorphic, that is, it is locally given by: </p>
<div class="equation" data-bbox="194 319 415 333">$$
f_{1}(z_{1},\ldots,z_{4})=0\quad\mathrm{and}\quad f_{2}(z_{1},\ldots,z_{4})=0.
$$</div>
<p data-bbox="109 342 500 398">Now observe that coming back to the original complex structure $K$ , we induce an analytic change of coordinates from the holomorphic coordinates $z^{i}$ $\begin{array}{r}{\int\!\frac{\partial}{\partial z^{i}}\,=\,i\frac{\partial}{\partial z^{i}}\rangle}\end{array}$ to new holomorphic coordinates $w^{i}$ ( $\begin{array}{r}{K\frac{\partial}{\partial w^{i}}\,=\,i\frac{\partial}{\partial w^{i}}\Big)}\end{array}$ i ∂∂wi ) such that locally: </p>
<div class="equation" data-bbox="212 401 396 416">$$
z^{i}=c_{1}w^{i}+c_{2}\bar{w}^{i}\;\;\;\;\bar{z}^{i}=d_{1}w^{i}+d_{2}\bar{w}^{i},
$$</div>
<p data-bbox="109 421 501 465">for some complex constants $c_{j},d_{j}$ . Thus in the complex structure $K$ the special Lagrangian submanifold $\Lambda$ is given by $f_{j}(c_{1}w^{i}\!+\!c_{2}\bar{w}^{i},d_{1}w^{i}\!+\!d_{2}\bar{w}^{i})=0$ which is again the zero locus of a set of functions analytic in $w^{i},\bar{w}^{i}$ . 口 </p>
<p data-bbox="109 471 500 557">Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds $S$ of $X$ ; in particular we have an action of the hyperkaehler rotation on the structure sheaf $O_{S}$ (here, as always, we identify $O_{S}$ with its direct image $j_{*}O_{S}$ , where $j\,:\,S\,\rightarrow\,X$ is the holomorphic embedding). We are thus led to give the following: </p>
<p data-bbox="110 558 501 631">Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyperkaehler 4-fold $X$ (in the complex structure $K$ ). Then we define the special Lagrangian structure sheaf ${\mathcal{L}}_{\Lambda}$ as the sheaf obtained by the action of the hyperkaehler rotation on the structure sheaf $O_{\Lambda}$ of $\Lambda$ , as a complex Lagrangian submanifold of $X$ , (in the structure $I$ ). </p>
</body></html>
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0001060v1
| 5 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "$X$ , in the complex structure $I$ , that is performing a hyperkaehler rotation. Notice that in the complex structure $I$ , $\\Lambda$ is still a Lagrangian submanifold with respect to $\\omega_{K}$ and $\\omega_{I}$ , so it is Lagrangian with respect to the holomorphic (in the structure $I$ ) 2-form $\\Omega_{I}:=\\omega_{J}+i\\omega_{K}$ . 口 ", "page_idx": 5}, {"type": "text", "text": "Collecting the results so far proved, we can show that special Lagrangian submanifolds of $X$ are particularly rigid: ", "page_idx": 5}, {"type": "text", "text": "Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold $X$ is real analytic. ", "page_idx": 5}, {"type": "text", "text": "Proof: Let $\\Lambda$ be a special Lagrangian submanifold of $X$ , having fixed some complex structure on $X$ , let us say $K$ ; then, by Corollary 2.1 there exists a new complex structure, let us say $I$ , in which $\\Lambda$ is holomorphic, that is, it is locally given by: ", "page_idx": 5}, {"type": "equation", "text": "$$\nf_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Now observe that coming back to the original complex structure $K$ , we induce an analytic change of coordinates from the holomorphic coordinates $z^{i}$ $\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}$ to new holomorphic coordinates $w^{i}$ ( $\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}$ i ∂∂wi ) such that locally: ", "page_idx": 5}, {"type": "equation", "text": "$$\nz^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "for some complex constants $c_{j},d_{j}$ . Thus in the complex structure $K$ the special Lagrangian submanifold $\\Lambda$ is given by $f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0$ which is again the zero locus of a set of functions analytic in $w^{i},\\bar{w}^{i}$ . 口 ", "page_idx": 5}, {"type": "text", "text": "Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds $S$ of $X$ ; in particular we have an action of the hyperkaehler rotation on the structure sheaf $O_{S}$ (here, as always, we identify $O_{S}$ with its direct image $j_{*}O_{S}$ , where $j\\,:\\,S\\,\\rightarrow\\,X$ is the holomorphic embedding). We are thus led to give the following: ", "page_idx": 5}, {"type": "text", "text": "Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyperkaehler 4-fold $X$ (in the complex structure $K$ ). Then we define the special Lagrangian structure sheaf ${\\mathcal{L}}_{\\Lambda}$ as the sheaf obtained by the action of the hyperkaehler rotation on the structure sheaf $O_{\\Lambda}$ of $\\Lambda$ , as a complex Lagrangian submanifold of $X$ , (in the structure $I$ ). ", "page_idx": 5}]
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{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 501, 183], "lines": [{"bbox": [110, 128, 499, 142], "spans": [{"bbox": [110, 129, 121, 138], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [121, 128, 261, 142], "score": 1.0, "content": ", in the complex structure ", "type": "text"}, {"bbox": [261, 129, 268, 138], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [268, 128, 499, 142], "score": 1.0, "content": ", that is performing a hyperkaehler rotation.", "type": "text"}], "index": 0}, {"bbox": [109, 142, 500, 156], "spans": [{"bbox": [109, 142, 303, 156], "score": 1.0, "content": "Notice that in the complex structure ", "type": "text"}, {"bbox": [303, 144, 310, 153], "score": 0.87, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [311, 142, 317, 156], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [317, 144, 326, 153], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [326, 142, 500, 156], "score": 1.0, "content": " is still a Lagrangian submanifold", "type": "text"}], "index": 1}, {"bbox": [109, 157, 500, 171], "spans": [{"bbox": [109, 157, 187, 171], "score": 1.0, "content": "with respect to ", "type": "text"}, {"bbox": [187, 161, 203, 169], "score": 0.91, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [203, 157, 226, 171], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [227, 161, 240, 169], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [240, 157, 500, 171], "score": 1.0, "content": ", so it is Lagrangian with respect to the holomorphic", "type": "text"}], "index": 2}, {"bbox": [110, 170, 501, 186], "spans": [{"bbox": [110, 170, 198, 186], "score": 1.0, "content": "(in the structure ", "type": "text"}, {"bbox": [199, 173, 205, 182], "score": 0.87, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [205, 170, 250, 186], "score": 1.0, "content": ") 2-form ", "type": "text"}, {"bbox": [250, 173, 330, 183], "score": 0.93, "content": "\\Omega_{I}:=\\omega_{J}+i\\omega_{K}", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [330, 170, 335, 186], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [491, 174, 501, 183], "score": 0.9884846806526184, "content": "口", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "text", "bbox": [109, 189, 500, 218], "lines": [{"bbox": [127, 190, 500, 207], "spans": [{"bbox": [127, 190, 500, 207], "score": 1.0, "content": "Collecting the results so far proved, we can show that special Lagrangian", "type": "text"}], "index": 4}, {"bbox": [110, 206, 318, 219], "spans": [{"bbox": [110, 206, 194, 219], "score": 1.0, "content": "submanifolds of ", "type": "text"}, {"bbox": [194, 208, 205, 217], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [205, 206, 318, 219], "score": 1.0, "content": " are particularly rigid:", "type": "text"}], "index": 5}], "index": 4.5}, {"type": "text", "bbox": [110, 219, 502, 246], "lines": [{"bbox": [126, 219, 501, 237], "spans": [{"bbox": [126, 219, 501, 237], "score": 1.0, "content": "Proposition 2.1: Any (connected, compact and without border) special", "type": "text"}], "index": 6}, {"bbox": [111, 236, 466, 250], "spans": [{"bbox": [111, 236, 372, 250], "score": 1.0, "content": "Lagrangian submanifold Λ of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [372, 237, 383, 245], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [384, 236, 466, 250], "score": 1.0, "content": " is real analytic.", "type": "text"}], "index": 7}], "index": 6.5}, {"type": "text", "bbox": [109, 247, 500, 305], "lines": [{"bbox": [126, 248, 500, 263], "spans": [{"bbox": [126, 248, 192, 263], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [192, 251, 201, 259], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [201, 248, 416, 263], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [416, 251, 428, 260], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [428, 248, 500, 263], "score": 1.0, "content": ", having fixed", "type": "text"}], "index": 8}, {"bbox": [109, 264, 500, 278], "spans": [{"bbox": [109, 264, 256, 278], "score": 1.0, "content": "some complex structure on ", "type": "text"}, {"bbox": [256, 265, 267, 274], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [268, 264, 331, 278], "score": 1.0, "content": ", let us say ", "type": "text"}, {"bbox": [331, 265, 342, 274], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [343, 264, 500, 278], "score": 1.0, "content": "; then, by Corollary 2.1 there", "type": "text"}], "index": 9}, {"bbox": [110, 279, 499, 291], "spans": [{"bbox": [110, 279, 325, 291], "score": 1.0, "content": "exists a new complex structure, let us say ", "type": "text"}, {"bbox": [325, 280, 331, 289], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [332, 279, 384, 291], "score": 1.0, "content": ", in which ", "type": "text"}, {"bbox": [384, 280, 393, 289], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [393, 279, 499, 291], "score": 1.0, "content": " is holomorphic, that", "type": "text"}], "index": 10}, {"bbox": [110, 294, 232, 306], "spans": [{"bbox": [110, 294, 232, 306], "score": 1.0, "content": "is, it is locally given by:", "type": "text"}], "index": 11}], "index": 9.5}, {"type": "interline_equation", "bbox": [194, 319, 415, 333], "lines": [{"bbox": [194, 319, 415, 333], "spans": [{"bbox": [194, 319, 415, 333], "score": 0.88, "content": "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [109, 342, 500, 398], "lines": [{"bbox": [109, 344, 499, 359], "spans": [{"bbox": [109, 344, 448, 359], "score": 1.0, "content": "Now observe that coming back to the original complex structure ", "type": "text"}, {"bbox": [449, 347, 460, 356], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [460, 344, 499, 359], "score": 1.0, "content": ", we in-", "type": "text"}], "index": 13}, {"bbox": [109, 359, 499, 374], "spans": [{"bbox": [109, 359, 489, 374], "score": 1.0, "content": "duce an analytic change of coordinates from the holomorphic coordinates ", "type": "text"}, {"bbox": [489, 360, 499, 370], "score": 0.91, "content": "z^{i}", "type": "inline_equation", "height": 10, "width": 10}], "index": 14}, {"bbox": [113, 369, 502, 395], "spans": [{"bbox": [113, 374, 175, 389], "score": 0.9, "content": "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}", "type": "inline_equation", "height": 15, "width": 62}, {"bbox": [176, 369, 352, 395], "score": 1.0, "content": " to new holomorphic coordinates ", "type": "text"}, {"bbox": [352, 375, 365, 385], "score": 0.9, "content": "w^{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [365, 369, 373, 395], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [374, 374, 444, 389], "score": 0.92, "content": "\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}", "type": "inline_equation", "height": 15, "width": 70}, {"bbox": [421, 372, 502, 391], "score": 1.0, "content": "i ∂∂wi ) such that", "type": "text"}], "index": 15}, {"bbox": [109, 389, 148, 402], "spans": [{"bbox": [109, 389, 148, 402], "score": 1.0, "content": "locally:", "type": "text"}], "index": 16}], "index": 14.5}, {"type": "interline_equation", "bbox": [212, 401, 396, 416], "lines": [{"bbox": [212, 401, 396, 416], "spans": [{"bbox": [212, 401, 396, 416], "score": 0.92, "content": "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [109, 421, 501, 465], "lines": [{"bbox": [109, 424, 501, 439], "spans": [{"bbox": [109, 424, 261, 439], "score": 1.0, "content": "for some complex constants ", "type": "text"}, {"bbox": [261, 426, 286, 438], "score": 0.94, "content": "c_{j},d_{j}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [287, 424, 466, 439], "score": 1.0, "content": ". Thus in the complex structure ", "type": "text"}, {"bbox": [466, 426, 477, 434], "score": 0.88, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [478, 424, 501, 439], "score": 1.0, "content": " the", "type": "text"}], "index": 18}, {"bbox": [109, 439, 499, 453], "spans": [{"bbox": [109, 439, 273, 453], "score": 1.0, "content": "special Lagrangian submanifold ", "type": "text"}, {"bbox": [273, 440, 281, 449], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [282, 439, 340, 453], "score": 1.0, "content": " is given by ", "type": "text"}, {"bbox": [341, 439, 499, 452], "score": 0.92, "content": "f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "type": "inline_equation", "height": 13, "width": 158}], "index": 19}, {"bbox": [110, 453, 501, 467], "spans": [{"bbox": [110, 453, 424, 467], "score": 1.0, "content": "which is again the zero locus of a set of functions analytic in ", "type": "text"}, {"bbox": [424, 453, 454, 466], "score": 0.81, "content": "w^{i},\\bar{w}^{i}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [454, 453, 458, 467], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [492, 456, 501, 465], "score": 0.9820061922073364, "content": "口", "type": "text"}], "index": 20}], "index": 19}, {"type": "text", "bbox": [109, 471, 500, 557], "lines": [{"bbox": [128, 474, 500, 487], "spans": [{"bbox": [128, 474, 500, 487], "score": 1.0, "content": "Quite naturally, the action of the hyperkaehler rotation can be extended", "type": "text"}], "index": 21}, {"bbox": [109, 488, 500, 501], "spans": [{"bbox": [109, 488, 459, 501], "score": 1.0, "content": "also to the holomorphic functions defined on complex submanifolds ", "type": "text"}, {"bbox": [460, 489, 468, 498], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [468, 488, 484, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [484, 489, 496, 498], "score": 0.88, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [496, 488, 500, 501], "score": 1.0, "content": ";", "type": "text"}], "index": 22}, {"bbox": [109, 503, 501, 516], "spans": [{"bbox": [109, 503, 501, 516], "score": 1.0, "content": "in particular we have an action of the hyperkaehler rotation on the structure", "type": "text"}], "index": 23}, {"bbox": [109, 516, 500, 531], "spans": [{"bbox": [109, 516, 140, 531], "score": 1.0, "content": "sheaf ", "type": "text"}, {"bbox": [140, 518, 155, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [156, 516, 307, 531], "score": 1.0, "content": " (here, as always, we identify ", "type": "text"}, {"bbox": [307, 518, 323, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [323, 516, 436, 531], "score": 1.0, "content": " with its direct image ", "type": "text"}, {"bbox": [436, 518, 462, 529], "score": 0.94, "content": "j_{*}O_{S}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [462, 516, 500, 531], "score": 1.0, "content": ", where", "type": "text"}], "index": 24}, {"bbox": [110, 531, 500, 545], "spans": [{"bbox": [110, 533, 173, 544], "score": 0.92, "content": "j\\,:\\,S\\,\\rightarrow\\,X", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [173, 531, 500, 545], "score": 1.0, "content": " is the holomorphic embedding). We are thus led to give the", "type": "text"}], "index": 25}, {"bbox": [109, 545, 160, 560], "spans": [{"bbox": [109, 545, 160, 560], "score": 1.0, "content": "following:", "type": "text"}], "index": 26}], "index": 23.5}, {"type": "text", "bbox": [110, 558, 501, 631], "lines": [{"bbox": [127, 559, 500, 574], "spans": [{"bbox": [127, 559, 500, 574], "score": 1.0, "content": "Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper-", "type": "text"}], "index": 27}, {"bbox": [109, 574, 501, 589], "spans": [{"bbox": [109, 574, 184, 589], "score": 1.0, "content": "kaehler 4-fold ", "type": "text"}, {"bbox": [185, 577, 195, 585], "score": 0.82, "content": "X", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [196, 574, 335, 589], "score": 1.0, "content": " (in the complex structure ", "type": "text"}, {"bbox": [335, 576, 346, 585], "score": 0.83, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [347, 574, 501, 589], "score": 1.0, "content": "). Then we define the special", "type": "text"}], "index": 28}, {"bbox": [111, 589, 499, 604], "spans": [{"bbox": [111, 589, 249, 604], "score": 1.0, "content": "Lagrangian structure sheaf ", "type": "text"}, {"bbox": [249, 591, 264, 601], "score": 0.88, "content": "{\\mathcal{L}}_{\\Lambda}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [265, 589, 499, 604], "score": 1.0, "content": " as the sheaf obtained by the action of the hy-", "type": "text"}], "index": 29}, {"bbox": [110, 604, 500, 617], "spans": [{"bbox": [110, 604, 324, 617], "score": 1.0, "content": "perkaehler rotation on the structure sheaf ", "type": "text"}, {"bbox": [324, 605, 340, 616], "score": 0.86, "content": "O_{\\Lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [340, 604, 357, 617], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [357, 605, 366, 614], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [366, 604, 500, 617], "score": 1.0, "content": ", as a complex Lagrangian", "type": "text"}], "index": 30}, {"bbox": [110, 618, 310, 632], "spans": [{"bbox": [110, 618, 188, 632], "score": 1.0, "content": "submanifold of ", "type": "text"}, {"bbox": [189, 620, 200, 628], "score": 0.85, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [200, 618, 295, 632], "score": 1.0, "content": ", (in the structure ", "type": "text"}, {"bbox": [295, 619, 301, 628], "score": 0.75, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [302, 618, 310, 632], "score": 1.0, "content": ").", "type": "text"}], "index": 31}], "index": 29}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [194, 319, 415, 333], "lines": [{"bbox": [194, 319, 415, 333], "spans": [{"bbox": [194, 319, 415, 333], "score": 0.88, "content": 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"content": ", in the complex structure ", "type": "text"}, {"bbox": [261, 129, 268, 138], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [268, 128, 499, 142], "score": 1.0, "content": ", that is performing a hyperkaehler rotation.", "type": "text"}], "index": 0}, {"bbox": [109, 142, 500, 156], "spans": [{"bbox": [109, 142, 303, 156], "score": 1.0, "content": "Notice that in the complex structure ", "type": "text"}, {"bbox": [303, 144, 310, 153], "score": 0.87, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [311, 142, 317, 156], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [317, 144, 326, 153], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [326, 142, 500, 156], "score": 1.0, "content": " is still a Lagrangian submanifold", "type": "text"}], "index": 1}, {"bbox": [109, 157, 500, 171], "spans": [{"bbox": [109, 157, 187, 171], "score": 1.0, "content": "with respect to ", "type": "text"}, {"bbox": [187, 161, 203, 169], "score": 0.91, "content": "\\omega_{K}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [203, 157, 226, 171], "score": 1.0, "content": " and", "type": "text"}, {"bbox": [227, 161, 240, 169], "score": 0.9, "content": "\\omega_{I}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [240, 157, 500, 171], "score": 1.0, "content": ", so it is Lagrangian with respect to the holomorphic", "type": "text"}], "index": 2}, {"bbox": [110, 170, 501, 186], "spans": [{"bbox": [110, 170, 198, 186], "score": 1.0, "content": "(in the structure ", "type": "text"}, {"bbox": [199, 173, 205, 182], "score": 0.87, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [205, 170, 250, 186], "score": 1.0, "content": ") 2-form ", "type": "text"}, {"bbox": [250, 173, 330, 183], "score": 0.93, "content": "\\Omega_{I}:=\\omega_{J}+i\\omega_{K}", "type": "inline_equation", "height": 10, "width": 80}, {"bbox": [330, 170, 335, 186], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [491, 174, 501, 183], "score": 0.9884846806526184, "content": "口", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [109, 128, 501, 186]}, {"type": "text", "bbox": [109, 189, 500, 218], "lines": [{"bbox": [127, 190, 500, 207], "spans": [{"bbox": [127, 190, 500, 207], "score": 1.0, "content": "Collecting the results so far proved, we can show that special Lagrangian", "type": "text"}], "index": 4}, {"bbox": [110, 206, 318, 219], "spans": [{"bbox": [110, 206, 194, 219], "score": 1.0, "content": "submanifolds of ", "type": "text"}, {"bbox": [194, 208, 205, 217], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [205, 206, 318, 219], "score": 1.0, "content": " are particularly rigid:", "type": "text"}], "index": 5}], "index": 4.5, "bbox_fs": [110, 190, 500, 219]}, {"type": "text", "bbox": [110, 219, 502, 246], "lines": [{"bbox": [126, 219, 501, 237], "spans": [{"bbox": [126, 219, 501, 237], "score": 1.0, "content": "Proposition 2.1: Any (connected, compact and without border) special", "type": "text"}], "index": 6}, {"bbox": [111, 236, 466, 250], "spans": [{"bbox": [111, 236, 372, 250], "score": 1.0, "content": "Lagrangian submanifold Λ of a hyperkaehler 4-fold ", "type": "text"}, {"bbox": [372, 237, 383, 245], "score": 0.89, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [384, 236, 466, 250], "score": 1.0, "content": " is real analytic.", "type": "text"}], "index": 7}], "index": 6.5, "bbox_fs": [111, 219, 501, 250]}, {"type": "text", "bbox": [109, 247, 500, 305], "lines": [{"bbox": [126, 248, 500, 263], "spans": [{"bbox": [126, 248, 192, 263], "score": 1.0, "content": "Proof: Let ", "type": "text"}, {"bbox": [192, 251, 201, 259], "score": 0.88, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [201, 248, 416, 263], "score": 1.0, "content": " be a special Lagrangian submanifold of ", "type": "text"}, {"bbox": [416, 251, 428, 260], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [428, 248, 500, 263], "score": 1.0, "content": ", having fixed", "type": "text"}], "index": 8}, {"bbox": [109, 264, 500, 278], "spans": [{"bbox": [109, 264, 256, 278], "score": 1.0, "content": "some complex structure on ", "type": "text"}, {"bbox": [256, 265, 267, 274], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [268, 264, 331, 278], "score": 1.0, "content": ", let us say ", "type": "text"}, {"bbox": [331, 265, 342, 274], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [343, 264, 500, 278], "score": 1.0, "content": "; then, by Corollary 2.1 there", "type": "text"}], "index": 9}, {"bbox": [110, 279, 499, 291], "spans": [{"bbox": [110, 279, 325, 291], "score": 1.0, "content": "exists a new complex structure, let us say ", "type": "text"}, {"bbox": [325, 280, 331, 289], "score": 0.89, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [332, 279, 384, 291], "score": 1.0, "content": ", in which ", "type": "text"}, {"bbox": [384, 280, 393, 289], "score": 0.91, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [393, 279, 499, 291], "score": 1.0, "content": " is holomorphic, that", "type": "text"}], "index": 10}, {"bbox": [110, 294, 232, 306], "spans": [{"bbox": [110, 294, 232, 306], "score": 1.0, "content": "is, it is locally given by:", "type": "text"}], "index": 11}], "index": 9.5, "bbox_fs": [109, 248, 500, 306]}, {"type": "interline_equation", "bbox": [194, 319, 415, 333], "lines": [{"bbox": [194, 319, 415, 333], "spans": [{"bbox": [194, 319, 415, 333], "score": 0.88, "content": "f_{1}(z_{1},\\ldots,z_{4})=0\\quad\\mathrm{and}\\quad f_{2}(z_{1},\\ldots,z_{4})=0.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [109, 342, 500, 398], "lines": [{"bbox": [109, 344, 499, 359], "spans": [{"bbox": [109, 344, 448, 359], "score": 1.0, "content": "Now observe that coming back to the original complex structure ", "type": "text"}, {"bbox": [449, 347, 460, 356], "score": 0.89, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [460, 344, 499, 359], "score": 1.0, "content": ", we in-", "type": "text"}], "index": 13}, {"bbox": [109, 359, 499, 374], "spans": [{"bbox": [109, 359, 489, 374], "score": 1.0, "content": "duce an analytic change of coordinates from the holomorphic coordinates ", "type": "text"}, {"bbox": [489, 360, 499, 370], "score": 0.91, "content": "z^{i}", "type": "inline_equation", "height": 10, "width": 10}], "index": 14}, {"bbox": [113, 369, 502, 395], "spans": [{"bbox": [113, 374, 175, 389], "score": 0.9, "content": "\\begin{array}{r}{\\int\\!\\frac{\\partial}{\\partial z^{i}}\\,=\\,i\\frac{\\partial}{\\partial z^{i}}\\rangle}\\end{array}", "type": "inline_equation", "height": 15, "width": 62}, {"bbox": [176, 369, 352, 395], "score": 1.0, "content": " to new holomorphic coordinates ", "type": "text"}, {"bbox": [352, 375, 365, 385], "score": 0.9, "content": "w^{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [365, 369, 373, 395], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [374, 374, 444, 389], "score": 0.92, "content": "\\begin{array}{r}{K\\frac{\\partial}{\\partial w^{i}}\\,=\\,i\\frac{\\partial}{\\partial w^{i}}\\Big)}\\end{array}", "type": "inline_equation", "height": 15, "width": 70}, {"bbox": [421, 372, 502, 391], "score": 1.0, "content": "i ∂∂wi ) such that", "type": "text"}], "index": 15}, {"bbox": [109, 389, 148, 402], "spans": [{"bbox": [109, 389, 148, 402], "score": 1.0, "content": "locally:", "type": "text"}], "index": 16}], "index": 14.5, "bbox_fs": [109, 344, 502, 402]}, {"type": "interline_equation", "bbox": [212, 401, 396, 416], "lines": [{"bbox": [212, 401, 396, 416], "spans": [{"bbox": [212, 401, 396, 416], "score": 0.92, "content": "z^{i}=c_{1}w^{i}+c_{2}\\bar{w}^{i}\\;\\;\\;\\;\\bar{z}^{i}=d_{1}w^{i}+d_{2}\\bar{w}^{i},", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [109, 421, 501, 465], "lines": [{"bbox": [109, 424, 501, 439], "spans": [{"bbox": [109, 424, 261, 439], "score": 1.0, "content": "for some complex constants ", "type": "text"}, {"bbox": [261, 426, 286, 438], "score": 0.94, "content": "c_{j},d_{j}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [287, 424, 466, 439], "score": 1.0, "content": ". Thus in the complex structure ", "type": "text"}, {"bbox": [466, 426, 477, 434], "score": 0.88, "content": "K", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [478, 424, 501, 439], "score": 1.0, "content": " the", "type": "text"}], "index": 18}, {"bbox": [109, 439, 499, 453], "spans": [{"bbox": [109, 439, 273, 453], "score": 1.0, "content": "special Lagrangian submanifold ", "type": "text"}, {"bbox": [273, 440, 281, 449], "score": 0.9, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [282, 439, 340, 453], "score": 1.0, "content": " is given by ", "type": "text"}, {"bbox": [341, 439, 499, 452], "score": 0.92, "content": "f_{j}(c_{1}w^{i}\\!+\\!c_{2}\\bar{w}^{i},d_{1}w^{i}\\!+\\!d_{2}\\bar{w}^{i})=0", "type": "inline_equation", "height": 13, "width": 158}], "index": 19}, {"bbox": [110, 453, 501, 467], "spans": [{"bbox": [110, 453, 424, 467], "score": 1.0, "content": "which is again the zero locus of a set of functions analytic in ", "type": "text"}, {"bbox": [424, 453, 454, 466], "score": 0.81, "content": "w^{i},\\bar{w}^{i}", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [454, 453, 458, 467], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [492, 456, 501, 465], "score": 0.9820061922073364, "content": "口", "type": "text"}], "index": 20}], "index": 19, "bbox_fs": [109, 424, 501, 467]}, {"type": "text", "bbox": [109, 471, 500, 557], "lines": [{"bbox": [128, 474, 500, 487], "spans": [{"bbox": [128, 474, 500, 487], "score": 1.0, "content": "Quite naturally, the action of the hyperkaehler rotation can be extended", "type": "text"}], "index": 21}, {"bbox": [109, 488, 500, 501], "spans": [{"bbox": [109, 488, 459, 501], "score": 1.0, "content": "also to the holomorphic functions defined on complex submanifolds ", "type": "text"}, {"bbox": [460, 489, 468, 498], "score": 0.89, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [468, 488, 484, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [484, 489, 496, 498], "score": 0.88, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [496, 488, 500, 501], "score": 1.0, "content": ";", "type": "text"}], "index": 22}, {"bbox": [109, 503, 501, 516], "spans": [{"bbox": [109, 503, 501, 516], "score": 1.0, "content": "in particular we have an action of the hyperkaehler rotation on the structure", "type": "text"}], "index": 23}, {"bbox": [109, 516, 500, 531], "spans": [{"bbox": [109, 516, 140, 531], "score": 1.0, "content": "sheaf ", "type": "text"}, {"bbox": [140, 518, 155, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [156, 516, 307, 531], "score": 1.0, "content": " (here, as always, we identify ", "type": "text"}, {"bbox": [307, 518, 323, 529], "score": 0.92, "content": "O_{S}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [323, 516, 436, 531], "score": 1.0, "content": " with its direct image ", "type": "text"}, {"bbox": [436, 518, 462, 529], "score": 0.94, "content": "j_{*}O_{S}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [462, 516, 500, 531], "score": 1.0, "content": ", where", "type": "text"}], "index": 24}, {"bbox": [110, 531, 500, 545], "spans": [{"bbox": [110, 533, 173, 544], "score": 0.92, "content": "j\\,:\\,S\\,\\rightarrow\\,X", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [173, 531, 500, 545], "score": 1.0, "content": " is the holomorphic embedding). We are thus led to give the", "type": "text"}], "index": 25}, {"bbox": [109, 545, 160, 560], "spans": [{"bbox": [109, 545, 160, 560], "score": 1.0, "content": "following:", "type": "text"}], "index": 26}], "index": 23.5, "bbox_fs": [109, 474, 501, 560]}, {"type": "text", "bbox": [110, 558, 501, 631], "lines": [{"bbox": [127, 559, 500, 574], "spans": [{"bbox": [127, 559, 500, 574], "score": 1.0, "content": "Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper-", "type": "text"}], "index": 27}, {"bbox": [109, 574, 501, 589], "spans": [{"bbox": [109, 574, 184, 589], "score": 1.0, "content": "kaehler 4-fold ", "type": "text"}, {"bbox": [185, 577, 195, 585], "score": 0.82, "content": "X", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [196, 574, 335, 589], "score": 1.0, "content": " (in the complex structure ", "type": "text"}, {"bbox": [335, 576, 346, 585], "score": 0.83, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [347, 574, 501, 589], "score": 1.0, "content": "). Then we define the special", "type": "text"}], "index": 28}, {"bbox": [111, 589, 499, 604], "spans": [{"bbox": [111, 589, 249, 604], "score": 1.0, "content": "Lagrangian structure sheaf ", "type": "text"}, {"bbox": [249, 591, 264, 601], "score": 0.88, "content": "{\\mathcal{L}}_{\\Lambda}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [265, 589, 499, 604], "score": 1.0, "content": " as the sheaf obtained by the action of the hy-", "type": "text"}], "index": 29}, {"bbox": [110, 604, 500, 617], "spans": [{"bbox": [110, 604, 324, 617], "score": 1.0, "content": "perkaehler rotation on the structure sheaf ", "type": "text"}, {"bbox": [324, 605, 340, 616], "score": 0.86, "content": "O_{\\Lambda}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [340, 604, 357, 617], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [357, 605, 366, 614], "score": 0.61, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [366, 604, 500, 617], "score": 1.0, "content": ", as a complex Lagrangian", "type": "text"}], "index": 30}, {"bbox": [110, 618, 310, 632], "spans": [{"bbox": [110, 618, 188, 632], "score": 1.0, "content": "submanifold of ", "type": "text"}, {"bbox": [189, 620, 200, 628], "score": 0.85, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [200, 618, 295, 632], "score": 1.0, "content": ", (in the structure ", "type": "text"}, {"bbox": [295, 619, 301, 628], "score": 0.75, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [302, 618, 310, 632], "score": 1.0, "content": ").", "type": "text"}], "index": 31}], "index": 29, "bbox_fs": [109, 559, 501, 632]}]}
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[{"type": "text", "bbox": [110, 125, 501, 183], "content": ", in the complex structure , that is performing a hyperkaehler rotation. Notice that in the complex structure , is still a Lagrangian submanifold with respect to and , so it is Lagrangian with respect to the holomorphic (in the structure ) 2-form . 口", "index": 0}, {"type": "text", "bbox": [109, 189, 500, 218], "content": "Collecting the results so far proved, we can show that special Lagrangian submanifolds of are particularly rigid:", "index": 1}, {"type": "text", "bbox": [110, 219, 502, 246], "content": "Proposition 2.1: Any (connected, compact and without border) special Lagrangian submanifold Λ of a hyperkaehler 4-fold is real analytic.", "index": 2}, {"type": "text", "bbox": [109, 247, 500, 305], "content": "Proof: Let be a special Lagrangian submanifold of , having fixed some complex structure on , let us say ; then, by Corollary 2.1 there exists a new complex structure, let us say , in which is holomorphic, that is, it is locally given by:", "index": 3}, {"type": "interline_equation", "bbox": [194, 319, 415, 333], "content": "", "index": 4}, {"type": "text", "bbox": [109, 342, 500, 398], "content": "Now observe that coming back to the original complex structure , we in- duce an analytic change of coordinates from the holomorphic coordinates to new holomorphic coordinates ( i ∂∂wi ) such that locally:", "index": 5}, {"type": "interline_equation", "bbox": [212, 401, 396, 416], "content": "", "index": 6}, {"type": "text", "bbox": [109, 421, 501, 465], "content": "for some complex constants . Thus in the complex structure the special Lagrangian submanifold is given by which is again the zero locus of a set of functions analytic in . 口", "index": 7}, {"type": "text", "bbox": [109, 471, 500, 557], "content": "Quite naturally, the action of the hyperkaehler rotation can be extended also to the holomorphic functions defined on complex submanifolds of ; in particular we have an action of the hyperkaehler rotation on the structure sheaf (here, as always, we identify with its direct image , where is the holomorphic embedding). We are thus led to give the following:", "index": 8}, {"type": "text", "bbox": [110, 558, 501, 631], "content": "Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper- kaehler 4-fold (in the complex structure ). Then we define the special Lagrangian structure sheaf as the sheaf obtained by the action of the hy- perkaehler rotation on the structure sheaf of , as a complex Lagrangian submanifold of , (in the structure ).", "index": 9}]
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[{"bbox": [110, 128, 499, 142], "content": ", in the complex structure , that is performing a hyperkaehler rotation.", "parent_index": 0, "line_index": 0}, {"bbox": [109, 142, 500, 156], "content": "Notice that in the complex structure , is still a Lagrangian submanifold", "parent_index": 0, "line_index": 1}, {"bbox": [109, 157, 500, 171], "content": "with respect to and , so it is Lagrangian with respect to the holomorphic", "parent_index": 0, "line_index": 2}, {"bbox": [110, 170, 501, 186], "content": "(in the structure ) 2-form . 口", "parent_index": 0, "line_index": 3}, {"bbox": [127, 190, 500, 207], "content": "Collecting the results so far proved, we can show that special Lagrangian", "parent_index": 1, "line_index": 0}, {"bbox": [110, 206, 318, 219], "content": "submanifolds of are particularly rigid:", "parent_index": 1, "line_index": 1}, {"bbox": [126, 219, 501, 237], "content": "Proposition 2.1: Any (connected, compact and without border) special", "parent_index": 2, "line_index": 0}, {"bbox": [111, 236, 466, 250], "content": "Lagrangian submanifold Λ of a hyperkaehler 4-fold is real analytic.", "parent_index": 2, "line_index": 1}, {"bbox": [126, 248, 500, 263], "content": "Proof: Let be a special Lagrangian submanifold of , having fixed", "parent_index": 3, "line_index": 0}, {"bbox": [109, 264, 500, 278], "content": "some complex structure on , let us say ; then, by Corollary 2.1 there", "parent_index": 3, "line_index": 1}, {"bbox": [110, 279, 499, 291], "content": "exists a new complex structure, let us say , in which is holomorphic, that", "parent_index": 3, "line_index": 2}, {"bbox": [110, 294, 232, 306], "content": "is, it is locally given by:", "parent_index": 3, "line_index": 3}, {"bbox": [109, 344, 499, 359], "content": "Now observe that coming back to the original complex structure , we in-", "parent_index": 5, "line_index": 0}, {"bbox": [109, 359, 499, 374], "content": "duce an analytic change of coordinates from the holomorphic coordinates", "parent_index": 5, "line_index": 1}, {"bbox": [113, 369, 502, 395], "content": "to new holomorphic coordinates ( i ∂∂wi ) such that", "parent_index": 5, "line_index": 2}, {"bbox": [109, 389, 148, 402], "content": "locally:", "parent_index": 5, "line_index": 3}, {"bbox": [109, 424, 501, 439], "content": "for some complex constants . Thus in the complex structure the", "parent_index": 7, "line_index": 0}, {"bbox": [109, 439, 499, 453], "content": "special Lagrangian submanifold is given by", "parent_index": 7, "line_index": 1}, {"bbox": [110, 453, 501, 467], "content": "which is again the zero locus of a set of functions analytic in . 口", "parent_index": 7, "line_index": 2}, {"bbox": [128, 474, 500, 487], "content": "Quite naturally, the action of the hyperkaehler rotation can be extended", "parent_index": 8, "line_index": 0}, {"bbox": [109, 488, 500, 501], "content": "also to the holomorphic functions defined on complex submanifolds of ;", "parent_index": 8, "line_index": 1}, {"bbox": [109, 503, 501, 516], "content": "in particular we have an action of the hyperkaehler rotation on the structure", "parent_index": 8, "line_index": 2}, {"bbox": [109, 516, 500, 531], "content": "sheaf (here, as always, we identify with its direct image , where", "parent_index": 8, "line_index": 3}, {"bbox": [110, 531, 500, 545], "content": "is the holomorphic embedding). We are thus led to give the", "parent_index": 8, "line_index": 4}, {"bbox": [109, 545, 160, 560], "content": "following:", "parent_index": 8, "line_index": 5}, {"bbox": [127, 559, 500, 574], "content": "Definition 2.2: Let Λ be a special Lagrangian submanifold of a hyper-", "parent_index": 9, "line_index": 0}, {"bbox": [109, 574, 501, 589], "content": "kaehler 4-fold (in the complex structure ). Then we define the special", "parent_index": 9, "line_index": 1}, {"bbox": [111, 589, 499, 604], "content": "Lagrangian structure sheaf as the sheaf obtained by the action of the hy-", "parent_index": 9, "line_index": 2}, {"bbox": [110, 604, 500, 617], "content": "perkaehler rotation on the structure sheaf of , as a complex Lagrangian", "parent_index": 9, "line_index": 3}, {"bbox": [110, 618, 310, 632], "content": "submanifold of , (in the structure ).", "parent_index": 9, "line_index": 4}]
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# 3 Concluding remarks
It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case.
Another observation is related to singular Lagrangian submanifolds: indeed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of Lagrangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in $C^{n}$ which are only smooth, but not real analytic.
The discussion about singular Lagrangian submanifolds leads us to comment on the mirror symmetry construction suggested in [11]. Indeed, according to the recipe of [11], any Calabi-Yau $X$ , admitting a mirror $\hat{X}$ , has a peculiar fibre space structure: on a physical ground it is argued that $X$ can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irreducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure $f:X\to B$ of a projective irreducible symplectic manifold $X$ , with projective base $B$ , the generic fibre $f^{-1}(b)$ is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form $\Omega$ ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and $f$ is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special Lagrangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian surfaces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions.
Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover $\pi:S\to P^{2}$ of the projective plane, ramified along a smooth sextic $C\hookrightarrow P^{2}$ ( $S$ is then realized as a K3 surface). Since any line in $P^{2}$ will intersect generically the sextic $C$ in six distinct point, we have that the covering $\pi:S\to P^{2}$ determines a (flat) family of hyperelliptic curves over the dual projective plane $f:\mathcal{X}\rightarrow P^{2}$ . Then the Altmann-Kleiman compactification of the relative Jacobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over $P^{2}$ , and in fact all fibres are Lagrangian Abelian varieties.
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<h1 data-bbox="109 121 311 141">3 Concluding remarks </h1>
<p data-bbox="109 151 500 194">It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case. </p>
<p data-bbox="109 196 500 324">Another observation is related to singular Lagrangian submanifolds: indeed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of Lagrangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in $C^{n}$ which are only smooth, but not real analytic. </p>
<p data-bbox="109 325 500 657">The discussion about singular Lagrangian submanifolds leads us to comment on the mirror symmetry construction suggested in [11]. Indeed, according to the recipe of [11], any Calabi-Yau $X$ , admitting a mirror $\hat{X}$ , has a peculiar fibre space structure: on a physical ground it is argued that $X$ can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irreducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure $f:X\to B$ of a projective irreducible symplectic manifold $X$ , with projective base $B$ , the generic fibre $f^{-1}(b)$ is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form $\Omega$ ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and $f$ is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special Lagrangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian surfaces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions. </p>
<p data-bbox="126 658 500 671">Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover $\pi:S\to P^{2}$ of the projective plane, ramified along a smooth sextic $C\hookrightarrow P^{2}$ ( $S$ is then realized as a K3 surface). Since any line in $P^{2}$ will intersect generically the sextic $C$ in six distinct point, we have that the covering $\pi:S\to P^{2}$ determines a (flat) family of hyperelliptic curves over the dual projective plane $f:\mathcal{X}\rightarrow P^{2}$ . Then the Altmann-Kleiman compactification of the relative Jacobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over $P^{2}$ , and in fact all fibres are Lagrangian Abelian varieties. </p>
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{"preproc_blocks": [{"type": "title", "bbox": [109, 121, 311, 141], "lines": [{"bbox": [110, 123, 311, 142], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [138, 123, 311, 142], "score": 1.0, "content": "Concluding remarks", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [109, 151, 500, 194], "lines": [{"bbox": [109, 154, 500, 168], "spans": [{"bbox": [109, 154, 500, 168], "score": 1.0, "content": "It is important to remark that all previous results are true also for special", "type": "text"}], "index": 1}, {"bbox": [110, 168, 499, 182], "spans": [{"bbox": [110, 168, 499, 182], "score": 1.0, "content": "Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial", "type": "text"}], "index": 2}, {"bbox": [109, 183, 175, 196], "spans": [{"bbox": [109, 183, 175, 196], "score": 1.0, "content": "in that case.", "type": "text"}], "index": 3}], "index": 2}, {"type": "text", "bbox": [109, 196, 500, 324], "lines": [{"bbox": [127, 198, 498, 210], "spans": [{"bbox": [127, 198, 498, 210], "score": 1.0, "content": "Another observation is related to singular Lagrangian submanifolds: in-", "type": "text"}], "index": 4}, {"bbox": [110, 212, 499, 226], "spans": [{"bbox": [110, 212, 499, 226], "score": 1.0, "content": "deed, by the previous results, it turns out that we can also give examples", "type": "text"}], "index": 5}, {"bbox": [110, 227, 499, 240], "spans": [{"bbox": [110, 227, 499, 240], "score": 1.0, "content": "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "type": "text"}], "index": 6}, {"bbox": [109, 241, 500, 254], "spans": [{"bbox": [109, 241, 500, 254], "score": 1.0, "content": "grangian complex subvarieties. On the other hand, contrary to the case of the", "type": "text"}], "index": 7}, {"bbox": [109, 255, 500, 270], "spans": [{"bbox": [109, 255, 500, 270], "score": 1.0, "content": "corresponding submanifolds, we can not expect that all special Lagrangian", "type": "text"}], "index": 8}, {"bbox": [109, 270, 500, 284], "spans": [{"bbox": [109, 270, 500, 284], "score": 1.0, "content": "subvarieties are obtained in this way, and consequently we can not expect", "type": "text"}], "index": 9}, {"bbox": [109, 284, 500, 298], "spans": [{"bbox": [109, 284, 500, 298], "score": 1.0, "content": "that all special Lagrangian subvarieties are real analytic. Indeed, there are", "type": "text"}], "index": 10}, {"bbox": [109, 298, 498, 312], "spans": [{"bbox": [109, 298, 483, 312], "score": 1.0, "content": "examples (compare [4]) of singular special Lagrangian submanifold in ", "type": "text"}, {"bbox": [484, 300, 498, 309], "score": 0.91, "content": "C^{n}", "type": "inline_equation", "height": 9, "width": 14}], "index": 11}, {"bbox": [110, 312, 342, 326], "spans": [{"bbox": [110, 312, 342, 326], "score": 1.0, "content": "which are only smooth, but not real analytic.", "type": "text"}], "index": 12}], "index": 8}, {"type": "text", "bbox": [109, 325, 500, 657], "lines": [{"bbox": [127, 327, 500, 341], "spans": [{"bbox": [127, 327, 500, 341], "score": 1.0, "content": "The discussion about singular Lagrangian submanifolds leads us to com-", "type": "text"}], "index": 13}, {"bbox": [109, 342, 501, 357], "spans": [{"bbox": [109, 342, 501, 357], "score": 1.0, "content": "ment on the mirror symmetry construction suggested in [11]. Indeed, ac-", "type": "text"}], "index": 14}, {"bbox": [110, 355, 500, 369], "spans": [{"bbox": [110, 357, 344, 369], "score": 1.0, "content": "cording to the recipe of [11], any Calabi-Yau ", "type": "text"}, {"bbox": [344, 358, 355, 366], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [356, 357, 463, 369], "score": 1.0, "content": ", admitting a mirror", "type": "text"}, {"bbox": [463, 355, 474, 366], "score": 0.91, "content": "\\hat{X}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [475, 357, 500, 369], "score": 1.0, "content": ", has", "type": "text"}], "index": 15}, {"bbox": [110, 371, 499, 385], "spans": [{"bbox": [110, 371, 487, 385], "score": 1.0, "content": "a peculiar fibre space structure: on a physical ground it is argued that ", "type": "text"}, {"bbox": [488, 372, 499, 381], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}], "index": 16}, {"bbox": [109, 385, 500, 399], "spans": [{"bbox": [109, 385, 500, 399], "score": 1.0, "content": "can be realized as the total space of a fibration in special Lagrangian tori.", "type": "text"}], "index": 17}, {"bbox": [110, 399, 500, 414], "spans": [{"bbox": [110, 399, 500, 414], "score": 1.0, "content": "Unfortunately, there are very few examples of such realization: in particular,", "type": "text"}], "index": 18}, {"bbox": [110, 415, 500, 427], "spans": [{"bbox": [110, 415, 500, 427], "score": 1.0, "content": "as far as we know, there is only one (partial) example for Calabi-Yau 3-folds", "type": "text"}], "index": 19}, {"bbox": [110, 428, 500, 442], "spans": [{"bbox": [110, 428, 500, 442], "score": 1.0, "content": "of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-", "type": "text"}], "index": 20}, {"bbox": [109, 442, 500, 457], "spans": [{"bbox": [109, 442, 500, 457], "score": 1.0, "content": "ducible symplectic projective manifolds the situation is completely different.", "type": "text"}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre", "type": "text"}], "index": 22}, {"bbox": [110, 472, 500, 486], "spans": [{"bbox": [110, 472, 192, 486], "score": 1.0, "content": "space structure ", "type": "text"}, {"bbox": [193, 473, 253, 484], "score": 0.93, "content": "f:X\\to B", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [254, 472, 500, 486], "score": 1.0, "content": " of a projective irreducible symplectic manifold", "type": "text"}], "index": 23}, {"bbox": [110, 486, 500, 501], "spans": [{"bbox": [110, 488, 121, 497], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [121, 486, 234, 501], "score": 1.0, "content": ", with projective base ", "type": "text"}, {"bbox": [234, 488, 244, 497], "score": 0.9, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [244, 486, 335, 501], "score": 1.0, "content": ", the generic fibre ", "type": "text"}, {"bbox": [336, 487, 369, 500], "score": 0.94, "content": "f^{-1}(b)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [369, 486, 500, 501], "score": 1.0, "content": " is an Abelian variety (up", "type": "text"}], "index": 24}, {"bbox": [110, 502, 499, 514], "spans": [{"bbox": [110, 502, 499, 514], "score": 1.0, "content": "to finite unramified cover), and it is also Lagrangian with respect to the non", "type": "text"}], "index": 25}, {"bbox": [110, 515, 501, 529], "spans": [{"bbox": [110, 515, 275, 529], "score": 1.0, "content": "degenerate holomorphic 2-form ", "type": "text"}, {"bbox": [275, 517, 284, 525], "score": 0.88, "content": "\\Omega", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [284, 515, 501, 529], "score": 1.0, "content": "; moreover, in the case of 4-folds one can", "type": "text"}], "index": 26}, {"bbox": [109, 530, 500, 543], "spans": [{"bbox": [109, 530, 390, 543], "score": 1.0, "content": "prove that the generic fibre is an Abelian surface and ", "type": "text"}, {"bbox": [390, 531, 397, 542], "score": 0.91, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [398, 530, 500, 543], "score": 1.0, "content": " is equidimensional,", "type": "text"}], "index": 27}, {"bbox": [110, 544, 499, 558], "spans": [{"bbox": [110, 544, 499, 558], "score": 1.0, "content": "(i.e. all irreducible components of the fibres have the same dimension). By", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 572], "spans": [{"bbox": [109, 558, 500, 572], "score": 1.0, "content": "Corollary 2.1 it turns out that this fibre space structure can also be realized", "type": "text"}], "index": 29}, {"bbox": [110, 573, 499, 587], "spans": [{"bbox": [110, 573, 499, 587], "score": 1.0, "content": "as a special Lagrangian torus fibration; moreover, in this case all special La-", "type": "text"}], "index": 30}, {"bbox": [109, 587, 500, 601], "spans": [{"bbox": [109, 587, 500, 601], "score": 1.0, "content": "grangian fibres, even the singular ones, are analytic, since they are obtained", "type": "text"}], "index": 31}, {"bbox": [110, 603, 499, 615], "spans": [{"bbox": [110, 603, 499, 615], "score": 1.0, "content": "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "type": "text"}], "index": 32}, {"bbox": [109, 616, 500, 630], "spans": [{"bbox": [109, 616, 500, 630], "score": 1.0, "content": "faces. So, in these cases, we have special Lagrangian torus fibration in which", "type": "text"}], "index": 33}, {"bbox": [110, 631, 501, 645], "spans": [{"bbox": [110, 631, 501, 645], "score": 1.0, "content": "all fibres are analytic: one can hope to understand the degeneration types of", "type": "text"}], "index": 34}, {"bbox": [110, 645, 450, 659], "spans": [{"bbox": [110, 645, 450, 659], "score": 1.0, "content": "singular special Lagrangian tori, moving from these constructions.", "type": "text"}], "index": 35}], "index": 24}, {"type": "text", "bbox": [126, 658, 500, 671], "lines": [{"bbox": [127, 660, 500, 673], "spans": [{"bbox": [127, 660, 500, 673], "score": 1.0, "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "type": "text"}], "index": 36}], "index": 36}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 309, 702], "lines": [{"bbox": [300, 692, 310, 705], "spans": [{"bbox": [300, 692, 310, 705], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [109, 121, 311, 141], "lines": [{"bbox": [110, 123, 311, 142], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [138, 123, 311, 142], "score": 1.0, "content": "Concluding remarks", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [109, 151, 500, 194], "lines": [{"bbox": [109, 154, 500, 168], "spans": [{"bbox": [109, 154, 500, 168], "score": 1.0, "content": "It is important to remark that all previous results are true also for special", "type": "text"}], "index": 1}, {"bbox": [110, 168, 499, 182], "spans": [{"bbox": [110, 168, 499, 182], "score": 1.0, "content": "Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial", "type": "text"}], "index": 2}, {"bbox": [109, 183, 175, 196], "spans": [{"bbox": [109, 183, 175, 196], "score": 1.0, "content": "in that case.", "type": "text"}], "index": 3}], "index": 2, "bbox_fs": [109, 154, 500, 196]}, {"type": "text", "bbox": [109, 196, 500, 324], "lines": [{"bbox": [127, 198, 498, 210], "spans": [{"bbox": [127, 198, 498, 210], "score": 1.0, "content": "Another observation is related to singular Lagrangian submanifolds: in-", "type": "text"}], "index": 4}, {"bbox": [110, 212, 499, 226], "spans": [{"bbox": [110, 212, 499, 226], "score": 1.0, "content": "deed, by the previous results, it turns out that we can also give examples", "type": "text"}], "index": 5}, {"bbox": [110, 227, 499, 240], "spans": [{"bbox": [110, 227, 499, 240], "score": 1.0, "content": "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "type": "text"}], "index": 6}, {"bbox": [109, 241, 500, 254], "spans": [{"bbox": [109, 241, 500, 254], "score": 1.0, "content": "grangian complex subvarieties. On the other hand, contrary to the case of the", "type": "text"}], "index": 7}, {"bbox": [109, 255, 500, 270], "spans": [{"bbox": [109, 255, 500, 270], "score": 1.0, "content": "corresponding submanifolds, we can not expect that all special Lagrangian", "type": "text"}], "index": 8}, {"bbox": [109, 270, 500, 284], "spans": [{"bbox": [109, 270, 500, 284], "score": 1.0, "content": "subvarieties are obtained in this way, and consequently we can not expect", "type": "text"}], "index": 9}, {"bbox": [109, 284, 500, 298], "spans": [{"bbox": [109, 284, 500, 298], "score": 1.0, "content": "that all special Lagrangian subvarieties are real analytic. Indeed, there are", "type": "text"}], "index": 10}, {"bbox": [109, 298, 498, 312], "spans": [{"bbox": [109, 298, 483, 312], "score": 1.0, "content": "examples (compare [4]) of singular special Lagrangian submanifold in ", "type": "text"}, {"bbox": [484, 300, 498, 309], "score": 0.91, "content": "C^{n}", "type": "inline_equation", "height": 9, "width": 14}], "index": 11}, {"bbox": [110, 312, 342, 326], "spans": [{"bbox": [110, 312, 342, 326], "score": 1.0, "content": "which are only smooth, but not real analytic.", "type": "text"}], "index": 12}], "index": 8, "bbox_fs": [109, 198, 500, 326]}, {"type": "text", "bbox": [109, 325, 500, 657], "lines": [{"bbox": [127, 327, 500, 341], "spans": [{"bbox": [127, 327, 500, 341], "score": 1.0, "content": "The discussion about singular Lagrangian submanifolds leads us to com-", "type": "text"}], "index": 13}, {"bbox": [109, 342, 501, 357], "spans": [{"bbox": [109, 342, 501, 357], "score": 1.0, "content": "ment on the mirror symmetry construction suggested in [11]. Indeed, ac-", "type": "text"}], "index": 14}, {"bbox": [110, 355, 500, 369], "spans": [{"bbox": [110, 357, 344, 369], "score": 1.0, "content": "cording to the recipe of [11], any Calabi-Yau ", "type": "text"}, {"bbox": [344, 358, 355, 366], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [356, 357, 463, 369], "score": 1.0, "content": ", admitting a mirror", "type": "text"}, {"bbox": [463, 355, 474, 366], "score": 0.91, "content": "\\hat{X}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [475, 357, 500, 369], "score": 1.0, "content": ", has", "type": "text"}], "index": 15}, {"bbox": [110, 371, 499, 385], "spans": [{"bbox": [110, 371, 487, 385], "score": 1.0, "content": "a peculiar fibre space structure: on a physical ground it is argued that ", "type": "text"}, {"bbox": [488, 372, 499, 381], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}], "index": 16}, {"bbox": [109, 385, 500, 399], "spans": [{"bbox": [109, 385, 500, 399], "score": 1.0, "content": "can be realized as the total space of a fibration in special Lagrangian tori.", "type": "text"}], "index": 17}, {"bbox": [110, 399, 500, 414], "spans": [{"bbox": [110, 399, 500, 414], "score": 1.0, "content": "Unfortunately, there are very few examples of such realization: in particular,", "type": "text"}], "index": 18}, {"bbox": [110, 415, 500, 427], "spans": [{"bbox": [110, 415, 500, 427], "score": 1.0, "content": "as far as we know, there is only one (partial) example for Calabi-Yau 3-folds", "type": "text"}], "index": 19}, {"bbox": [110, 428, 500, 442], "spans": [{"bbox": [110, 428, 500, 442], "score": 1.0, "content": "of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-", "type": "text"}], "index": 20}, {"bbox": [109, 442, 500, 457], "spans": [{"bbox": [109, 442, 500, 457], "score": 1.0, "content": "ducible symplectic projective manifolds the situation is completely different.", "type": "text"}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre", "type": "text"}], "index": 22}, {"bbox": [110, 472, 500, 486], "spans": [{"bbox": [110, 472, 192, 486], "score": 1.0, "content": "space structure ", "type": "text"}, {"bbox": [193, 473, 253, 484], "score": 0.93, "content": "f:X\\to B", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [254, 472, 500, 486], "score": 1.0, "content": " of a projective irreducible symplectic manifold", "type": "text"}], "index": 23}, {"bbox": [110, 486, 500, 501], "spans": [{"bbox": [110, 488, 121, 497], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [121, 486, 234, 501], "score": 1.0, "content": ", with projective base ", "type": "text"}, {"bbox": [234, 488, 244, 497], "score": 0.9, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [244, 486, 335, 501], "score": 1.0, "content": ", the generic fibre ", "type": "text"}, {"bbox": [336, 487, 369, 500], "score": 0.94, "content": "f^{-1}(b)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [369, 486, 500, 501], "score": 1.0, "content": " is an Abelian variety (up", "type": "text"}], "index": 24}, {"bbox": [110, 502, 499, 514], "spans": [{"bbox": [110, 502, 499, 514], "score": 1.0, "content": "to finite unramified cover), and it is also Lagrangian with respect to the non", "type": "text"}], "index": 25}, {"bbox": [110, 515, 501, 529], "spans": [{"bbox": [110, 515, 275, 529], "score": 1.0, "content": "degenerate holomorphic 2-form ", "type": "text"}, {"bbox": [275, 517, 284, 525], "score": 0.88, "content": "\\Omega", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [284, 515, 501, 529], "score": 1.0, "content": "; moreover, in the case of 4-folds one can", "type": "text"}], "index": 26}, {"bbox": [109, 530, 500, 543], "spans": [{"bbox": [109, 530, 390, 543], "score": 1.0, "content": "prove that the generic fibre is an Abelian surface and ", "type": "text"}, {"bbox": [390, 531, 397, 542], "score": 0.91, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [398, 530, 500, 543], "score": 1.0, "content": " is equidimensional,", "type": "text"}], "index": 27}, {"bbox": [110, 544, 499, 558], "spans": [{"bbox": [110, 544, 499, 558], "score": 1.0, "content": "(i.e. all irreducible components of the fibres have the same dimension). By", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 572], "spans": [{"bbox": [109, 558, 500, 572], "score": 1.0, "content": "Corollary 2.1 it turns out that this fibre space structure can also be realized", "type": "text"}], "index": 29}, {"bbox": [110, 573, 499, 587], "spans": [{"bbox": [110, 573, 499, 587], "score": 1.0, "content": "as a special Lagrangian torus fibration; moreover, in this case all special La-", "type": "text"}], "index": 30}, {"bbox": [109, 587, 500, 601], "spans": [{"bbox": [109, 587, 500, 601], "score": 1.0, "content": "grangian fibres, even the singular ones, are analytic, since they are obtained", "type": "text"}], "index": 31}, {"bbox": [110, 603, 499, 615], "spans": [{"bbox": [110, 603, 499, 615], "score": 1.0, "content": "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "type": "text"}], "index": 32}, {"bbox": [109, 616, 500, 630], "spans": [{"bbox": [109, 616, 500, 630], "score": 1.0, "content": "faces. So, in these cases, we have special Lagrangian torus fibration in which", "type": "text"}], "index": 33}, {"bbox": [110, 631, 501, 645], "spans": [{"bbox": [110, 631, 501, 645], "score": 1.0, "content": "all fibres are analytic: one can hope to understand the degeneration types of", "type": "text"}], "index": 34}, {"bbox": [110, 645, 450, 659], "spans": [{"bbox": [110, 645, 450, 659], "score": 1.0, "content": "singular special Lagrangian tori, moving from these constructions.", "type": "text"}], "index": 35}], "index": 24, "bbox_fs": [109, 327, 501, 659]}, {"type": "text", "bbox": [126, 658, 500, 671], "lines": [{"bbox": [127, 660, 500, 673], "spans": [{"bbox": [127, 660, 500, 673], "score": 1.0, "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "type": "text"}], "index": 36}, {"bbox": [109, 128, 500, 142], "spans": [{"bbox": [109, 128, 500, 142], "score": 1.0, "content": "a projective base have been constructed by Markuschevich in [6] and [7]. One", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [109, 141, 499, 156], "spans": [{"bbox": [109, 141, 436, 156], "score": 1.0, "content": "of this constructions is the following: consider a double cover ", "type": "text", "cross_page": true}, {"bbox": [436, 143, 499, 153], "score": 0.91, "content": "\\pi:S\\to P^{2}", "type": "inline_equation", "height": 10, "width": 63, "cross_page": true}], "index": 1}, {"bbox": [109, 156, 500, 171], "spans": [{"bbox": [109, 156, 397, 171], "score": 1.0, "content": "of the projective plane, ramified along a smooth sextic ", "type": "text", "cross_page": true}, {"bbox": [398, 157, 443, 167], "score": 0.93, "content": "C\\hookrightarrow P^{2}", "type": "inline_equation", "height": 10, "width": 45, "cross_page": true}, {"bbox": [443, 156, 451, 171], "score": 1.0, "content": " (", "type": "text", "cross_page": true}, {"bbox": [451, 158, 459, 167], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [460, 156, 500, 171], "score": 1.0, "content": " is then", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [109, 171, 500, 185], "spans": [{"bbox": [109, 171, 336, 185], "score": 1.0, "content": "realized as a K3 surface). Since any line in ", "type": "text", "cross_page": true}, {"bbox": [336, 172, 350, 181], "score": 0.92, "content": "P^{2}", "type": "inline_equation", "height": 9, "width": 14, "cross_page": true}, {"bbox": [350, 171, 500, 185], "score": 1.0, "content": " will intersect generically the", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [110, 185, 500, 199], "spans": [{"bbox": [110, 185, 143, 199], "score": 1.0, "content": "sextic ", "type": "text", "cross_page": true}, {"bbox": [144, 187, 153, 196], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9, "cross_page": true}, {"bbox": [153, 185, 404, 199], "score": 1.0, "content": " in six distinct point, we have that the covering ", "type": "text", "cross_page": true}, {"bbox": [404, 186, 465, 196], "score": 0.93, "content": "\\pi:S\\to P^{2}", "type": "inline_equation", "height": 10, "width": 61, "cross_page": true}, {"bbox": [465, 185, 500, 199], "score": 1.0, "content": " deter-", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [110, 200, 500, 214], "spans": [{"bbox": [110, 200, 500, 214], "score": 1.0, "content": "mines a (flat) family of hyperelliptic curves over the dual projective plane", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [110, 213, 500, 228], "spans": [{"bbox": [110, 215, 170, 227], "score": 0.93, "content": "f:\\mathcal{X}\\rightarrow P^{2}", "type": "inline_equation", "height": 12, "width": 60, "cross_page": true}, {"bbox": [170, 213, 500, 228], "score": 1.0, "content": ". Then the Altmann-Kleiman compactification of the relative Ja-", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "cobian of the family turns out to be a simplectic projective irreducible 4-folds,", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [109, 243, 472, 257], "spans": [{"bbox": [109, 243, 173, 257], "score": 1.0, "content": "fibered over ", "type": "text", "cross_page": true}, {"bbox": [174, 244, 188, 254], "score": 0.91, "content": "P^{2}", "type": "inline_equation", "height": 10, "width": 14, "cross_page": true}, {"bbox": [188, 243, 472, 257], "score": 1.0, "content": ", and in fact all fibres are Lagrangian Abelian varieties.", "type": "text", "cross_page": true}], "index": 8}], "index": 36, "bbox_fs": [127, 660, 500, 673]}]}
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[{"type": "title", "bbox": [109, 121, 311, 141], "content": "3 Concluding remarks", "index": 0}, {"type": "text", "bbox": [109, 151, 500, 194], "content": "It is important to remark that all previous results are true also for special Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial in that case.", "index": 1}, {"type": "text", "bbox": [109, 196, 500, 324], "content": "Another observation is related to singular Lagrangian submanifolds: in- deed, by the previous results, it turns out that we can also give examples of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La- grangian complex subvarieties. On the other hand, contrary to the case of the corresponding submanifolds, we can not expect that all special Lagrangian subvarieties are obtained in this way, and consequently we can not expect that all special Lagrangian subvarieties are real analytic. Indeed, there are examples (compare [4]) of singular special Lagrangian submanifold in which are only smooth, but not real analytic.", "index": 2}, {"type": "text", "bbox": [109, 325, 500, 657], "content": "The discussion about singular Lagrangian submanifolds leads us to com- ment on the mirror symmetry construction suggested in [11]. Indeed, ac- cording to the recipe of [11], any Calabi-Yau , admitting a mirror , has a peculiar fibre space structure: on a physical ground it is argued that can be realized as the total space of a fibration in special Lagrangian tori. Unfortunately, there are very few examples of such realization: in particular, as far as we know, there is only one (partial) example for Calabi-Yau 3-folds of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre- ducible symplectic projective manifolds the situation is completely different. Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre space structure of a projective irreducible symplectic manifold , with projective base , the generic fibre is an Abelian variety (up to finite unramified cover), and it is also Lagrangian with respect to the non degenerate holomorphic 2-form ; moreover, in the case of 4-folds one can prove that the generic fibre is an Abelian surface and is equidimensional, (i.e. all irreducible components of the fibres have the same dimension). By Corollary 2.1 it turns out that this fibre space structure can also be realized as a special Lagrangian torus fibration; moreover, in this case all special La- grangian fibres, even the singular ones, are analytic, since they are obtained by performing a hyperkaehler rotation starting from Lagrangian Abelian sur- faces. So, in these cases, we have special Lagrangian torus fibration in which all fibres are analytic: one can hope to understand the degeneration types of singular special Lagrangian tori, moving from these constructions.", "index": 3}, {"type": "text", "bbox": [126, 658, 500, 671], "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over a projective base have been constructed by Markuschevich in [6] and [7]. One of this constructions is the following: consider a double cover of the projective plane, ramified along a smooth sextic ( is then realized as a K3 surface). Since any line in will intersect generically the sextic in six distinct point, we have that the covering deter- mines a (flat) family of hyperelliptic curves over the dual projective plane . Then the Altmann-Kleiman compactification of the relative Ja- cobian of the family turns out to be a simplectic projective irreducible 4-folds, fibered over , and in fact all fibres are Lagrangian Abelian varieties.", "index": 4}]
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[{"bbox": [110, 123, 311, 142], "content": "3 Concluding remarks", "parent_index": 0, "line_index": 0}, {"bbox": [109, 154, 500, 168], "content": "It is important to remark that all previous results are true also for special", "parent_index": 1, "line_index": 0}, {"bbox": [110, 168, 499, 182], "content": "Lagrangian submanifolds of K3 surfaces, but their proof is completely trivial", "parent_index": 1, "line_index": 1}, {"bbox": [109, 183, 175, 196], "content": "in that case.", "parent_index": 1, "line_index": 2}, {"bbox": [127, 198, 498, 210], "content": "Another observation is related to singular Lagrangian submanifolds: in-", "parent_index": 2, "line_index": 0}, {"bbox": [110, 212, 499, 226], "content": "deed, by the previous results, it turns out that we can also give examples", "parent_index": 2, "line_index": 1}, {"bbox": [110, 227, 499, 240], "content": "of special Lagrangian subvarieties, obtained via hyperkaehler rotation of La-", "parent_index": 2, "line_index": 2}, {"bbox": [109, 241, 500, 254], "content": "grangian complex subvarieties. On the other hand, contrary to the case of the", "parent_index": 2, "line_index": 3}, {"bbox": [109, 255, 500, 270], "content": "corresponding submanifolds, we can not expect that all special Lagrangian", "parent_index": 2, "line_index": 4}, {"bbox": [109, 270, 500, 284], "content": "subvarieties are obtained in this way, and consequently we can not expect", "parent_index": 2, "line_index": 5}, {"bbox": [109, 284, 500, 298], "content": "that all special Lagrangian subvarieties are real analytic. Indeed, there are", "parent_index": 2, "line_index": 6}, {"bbox": [109, 298, 498, 312], "content": "examples (compare [4]) of singular special Lagrangian submanifold in", "parent_index": 2, "line_index": 7}, {"bbox": [110, 312, 342, 326], "content": "which are only smooth, but not real analytic.", "parent_index": 2, "line_index": 8}, {"bbox": [127, 327, 500, 341], "content": "The discussion about singular Lagrangian submanifolds leads us to com-", "parent_index": 3, "line_index": 0}, {"bbox": [109, 342, 501, 357], "content": "ment on the mirror symmetry construction suggested in [11]. Indeed, ac-", "parent_index": 3, "line_index": 1}, {"bbox": [110, 355, 500, 369], "content": "cording to the recipe of [11], any Calabi-Yau , admitting a mirror , has", "parent_index": 3, "line_index": 2}, {"bbox": [110, 371, 499, 385], "content": "a peculiar fibre space structure: on a physical ground it is argued that", "parent_index": 3, "line_index": 3}, {"bbox": [109, 385, 500, 399], "content": "can be realized as the total space of a fibration in special Lagrangian tori.", "parent_index": 3, "line_index": 4}, {"bbox": [110, 399, 500, 414], "content": "Unfortunately, there are very few examples of such realization: in particular,", "parent_index": 3, "line_index": 5}, {"bbox": [110, 415, 500, 427], "content": "as far as we know, there is only one (partial) example for Calabi-Yau 3-folds", "parent_index": 3, "line_index": 6}, {"bbox": [110, 428, 500, 442], "content": "of the so called Borcea-Voisin type (see [3]). Instead, in the case of irre-", "parent_index": 3, "line_index": 7}, {"bbox": [109, 442, 500, 457], "content": "ducible symplectic projective manifolds the situation is completely different.", "parent_index": 3, "line_index": 8}, {"bbox": [109, 457, 500, 471], "content": "Indeed, a recent result of Matsushita (see [8] and [9]) shows that for any fibre", "parent_index": 3, "line_index": 9}, {"bbox": [110, 472, 500, 486], "content": "space structure of a projective irreducible symplectic manifold", "parent_index": 3, "line_index": 10}, {"bbox": [110, 486, 500, 501], "content": ", with projective base , the generic fibre is an Abelian variety (up", "parent_index": 3, "line_index": 11}, {"bbox": [110, 502, 499, 514], "content": "to finite unramified cover), and it is also Lagrangian with respect to the non", "parent_index": 3, "line_index": 12}, {"bbox": [110, 515, 501, 529], "content": "degenerate holomorphic 2-form ; moreover, in the case of 4-folds one can", "parent_index": 3, "line_index": 13}, {"bbox": [109, 530, 500, 543], "content": "prove that the generic fibre is an Abelian surface and is equidimensional,", "parent_index": 3, "line_index": 14}, {"bbox": [110, 544, 499, 558], "content": "(i.e. all irreducible components of the fibres have the same dimension). By", "parent_index": 3, "line_index": 15}, {"bbox": [109, 558, 500, 572], "content": "Corollary 2.1 it turns out that this fibre space structure can also be realized", "parent_index": 3, "line_index": 16}, {"bbox": [110, 573, 499, 587], "content": "as a special Lagrangian torus fibration; moreover, in this case all special La-", "parent_index": 3, "line_index": 17}, {"bbox": [109, 587, 500, 601], "content": "grangian fibres, even the singular ones, are analytic, since they are obtained", "parent_index": 3, "line_index": 18}, {"bbox": [110, 603, 499, 615], "content": "by performing a hyperkaehler rotation starting from Lagrangian Abelian sur-", "parent_index": 3, "line_index": 19}, {"bbox": [109, 616, 500, 630], "content": "faces. So, in these cases, we have special Lagrangian torus fibration in which", "parent_index": 3, "line_index": 20}, {"bbox": [110, 631, 501, 645], "content": "all fibres are analytic: one can hope to understand the degeneration types of", "parent_index": 3, "line_index": 21}, {"bbox": [110, 645, 450, 659], "content": "singular special Lagrangian tori, moving from these constructions.", "parent_index": 3, "line_index": 22}, {"bbox": [127, 660, 500, 673], "content": "Explicit examples of projective irreducible symplectic 4-folds, fibered over", "parent_index": 4, "line_index": 0}, {"bbox": [109, 128, 500, 142], "content": "a projective base have been constructed by Markuschevich in [6] and [7]. One", "parent_index": 4, "line_index": 1}, {"bbox": [109, 141, 499, 156], "content": "of this constructions is the following: consider a double cover", "parent_index": 4, "line_index": 2}, {"bbox": [109, 156, 500, 171], "content": "of the projective plane, ramified along a smooth sextic ( is then", "parent_index": 4, "line_index": 3}, {"bbox": [109, 171, 500, 185], "content": "realized as a K3 surface). Since any line in will intersect generically the", "parent_index": 4, "line_index": 4}, {"bbox": [110, 185, 500, 199], "content": "sextic in six distinct point, we have that the covering deter-", "parent_index": 4, "line_index": 5}, {"bbox": [110, 200, 500, 214], "content": "mines a (flat) family of hyperelliptic curves over the dual projective plane", "parent_index": 4, "line_index": 6}, {"bbox": [110, 213, 500, 228], "content": ". Then the Altmann-Kleiman compactification of the relative Ja-", "parent_index": 4, "line_index": 7}, {"bbox": [110, 230, 500, 243], "content": "cobian of the family turns out to be a simplectic projective irreducible 4-folds,", "parent_index": 4, "line_index": 8}, {"bbox": [109, 243, 472, 257], "content": "fibered over , and in fact all fibres are Lagrangian Abelian varieties.", "parent_index": 4, "line_index": 9}]
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[]
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[{"bbox": [484, 300, 498, 309], "content": "C^{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [344, 358, 355, 366], "content": "X", "parent_index": 3, "subtype": "inline"}, {"bbox": [463, 355, 474, 366], "content": "\\hat{X}", "parent_index": 3, "subtype": "inline"}, {"bbox": [488, 372, 499, 381], "content": "X", "parent_index": 3, "subtype": "inline"}, {"bbox": [193, 473, 253, 484], "content": "f:X\\to B", "parent_index": 3, "subtype": "inline"}, {"bbox": [110, 488, 121, 497], "content": "X", "parent_index": 3, "subtype": "inline"}, {"bbox": [234, 488, 244, 497], "content": "B", "parent_index": 3, "subtype": "inline"}, {"bbox": [336, 487, 369, 500], "content": "f^{-1}(b)", "parent_index": 3, "subtype": "inline"}, {"bbox": [275, 517, 284, 525], "content": "\\Omega", "parent_index": 3, "subtype": "inline"}, {"bbox": [390, 531, 397, 542], "content": "f", "parent_index": 3, "subtype": "inline"}, {"bbox": [436, 143, 499, 153], "content": "\\pi:S\\to P^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [398, 157, 443, 167], "content": "C\\hookrightarrow P^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [451, 158, 459, 167], "content": "S", "parent_index": 4, "subtype": "inline"}, {"bbox": [336, 172, 350, 181], "content": "P^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [144, 187, 153, 196], "content": "C", "parent_index": 4, "subtype": "inline"}, {"bbox": [404, 186, 465, 196], "content": "\\pi:S\\to P^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [110, 215, 170, 227], "content": "f:\\mathcal{X}\\rightarrow P^{2}", "parent_index": 4, "subtype": "inline"}, {"bbox": [174, 244, 188, 254], "content": "P^{2}", "parent_index": 4, "subtype": "inline"}]
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[]
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Finally, we believe that our characterization of special Lagrangian submanifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions.
# References
[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and nonperturbative string theory, hep-th/9507158.
[2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782.
[3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004.
[4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157.
[5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alggeom/9705025.
[6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187.
[7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469.
[8] Matsushita D., On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), 79-83.
[9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045.
[10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099.
[11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259.
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<p data-bbox="110 256 500 341">Finally, we believe that our characterization of special Lagrangian submanifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions. </p>
<h1 data-bbox="109 363 202 382">References </h1>
<p data-bbox="115 393 502 657">[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and nonperturbative string theory, hep-th/9507158.
[2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782.
[3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004.
[4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157.
[5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alggeom/9705025.
[6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187.
[7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469.
[8] Matsushita D., On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), 79-83.
[9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045.
[10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099.
[11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259. </p>
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[{"type": "text", "text": "", "page_idx": 7}, {"type": "text", "text": "Finally, we believe that our characterization of special Lagrangian submanifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions. ", "page_idx": 7}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 7}, {"type": "text", "text": "[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and nonperturbative string theory, hep-th/9507158. \n[2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782. \n[3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of Calabi-Yau threefolds, alg-geom/9608004. \n[4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148 (1982), 47-157. \n[5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alggeom/9705025. \n[6] Markushevich D., Completely integrable projective symplectic 4- dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187. \n[7] Markushevich D., Integrable symplectic structures on compact complex manifolds, Math. USSR Sb. 59 (1988), 459-469. \n[8] Matsushita D., On fibre space structures of a projective irreducible symplectic manifold, Topology 38 (1999), 79-83. \n[9] Matsushita D., Addendum to: On fibre space structures of a projective irreducible symplectic manifold, math.ag/9903045. \n[10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and II, alg-geom/9708009, math.ag/9805099. \n[11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality, Nucl. Phys. B479, (1996), 243-259. ", "page_idx": 7}]
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Then the Altmann-Kleiman compactification of the relative Ja-", "type": "text"}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "cobian of the family turns out to be a simplectic projective irreducible 4-folds,", "type": "text"}], "index": 7}, {"bbox": [109, 243, 472, 257], "spans": [{"bbox": [109, 243, 173, 257], "score": 1.0, "content": "fibered over ", "type": "text"}, {"bbox": [174, 244, 188, 254], "score": 0.91, "content": "P^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [188, 243, 472, 257], "score": 1.0, "content": ", and in fact all fibres are Lagrangian Abelian varieties.", "type": "text"}], "index": 8}], "index": 4}, {"type": "text", "bbox": [110, 256, 500, 341], "lines": [{"bbox": [127, 258, 498, 271], "spans": [{"bbox": [127, 258, 498, 271], "score": 1.0, "content": "Finally, we believe that our characterization of special Lagrangian sub-", "type": "text"}], "index": 9}, {"bbox": [110, 273, 499, 285], "spans": [{"bbox": [110, 273, 499, 285], "score": 1.0, "content": "manifolds of irreducible symplectic 4-folds can be extended also to higher", "type": "text"}], "index": 10}, {"bbox": [110, 288, 500, 299], "spans": [{"bbox": [110, 288, 500, 299], "score": 1.0, "content": "dimensional irreducible symplectic manifolds: to this aim notice that the", "type": "text"}], "index": 11}, {"bbox": [109, 302, 500, 315], "spans": [{"bbox": [109, 302, 500, 315], "score": 1.0, "content": "proof we have given becomes longer and longer, since one has to deal with", "type": "text"}], "index": 12}, {"bbox": [109, 316, 502, 329], "spans": [{"bbox": [109, 316, 502, 329], "score": 1.0, "content": "new cases and subcases. 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[{"type": "text", "bbox": [109, 124, 501, 255], "content": "", "index": 0}, {"type": "text", "bbox": [110, 256, 500, 341], "content": "Finally, we believe that our characterization of special Lagrangian sub- manifolds of irreducible symplectic 4-folds can be extended also to higher dimensional irreducible symplectic manifolds: to this aim notice that the proof we have given becomes longer and longer, since one has to deal with new cases and subcases. It would be nice, instead, to find out a sort of inductive argument, which works for all dimensions.", "index": 1}, {"type": "title", "bbox": [109, 363, 202, 382], "content": "References", "index": 2}, {"type": "list", "bbox": [115, 393, 502, 657], "content": "", "index": 3}]
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[{"bbox": [127, 258, 498, 271], "content": "Finally, we believe that our characterization of special Lagrangian sub-", "parent_index": 1, "line_index": 0}, {"bbox": [110, 273, 499, 285], "content": "manifolds of irreducible symplectic 4-folds can be extended also to higher", "parent_index": 1, "line_index": 1}, {"bbox": [110, 288, 500, 299], "content": "dimensional irreducible symplectic manifolds: to this aim notice that the", "parent_index": 1, "line_index": 2}, {"bbox": [109, 302, 500, 315], "content": "proof we have given becomes longer and longer, since one has to deal with", "parent_index": 1, "line_index": 3}, {"bbox": [109, 316, 502, 329], "content": "new cases and subcases. It would be nice, instead, to find out a sort of", "parent_index": 1, "line_index": 4}, {"bbox": [109, 330, 378, 343], "content": "inductive argument, which works for all dimensions.", "parent_index": 1, "line_index": 5}, {"bbox": [110, 366, 203, 383], "content": "References", "parent_index": 2, "line_index": 0}, {"bbox": [116, 395, 500, 412], "content": "[1] Becker K., Becker M., Strominger A., Fivebranes, membranes and non-", "parent_index": 3, "line_index": 0}, {"bbox": [133, 411, 354, 425], "content": "perturbative string theory, hep-th/9507158.", "parent_index": 3, "line_index": 1}, {"bbox": [116, 434, 501, 450], "content": "[2] Beauville A., Varits Kahleriennes dont la premire classe de Chern est", "parent_index": 3, "line_index": 2}, {"bbox": [134, 450, 343, 463], "content": "nulle, J. Diff. Geom. 18 (1983), 755-782.", "parent_index": 3, "line_index": 3}, {"bbox": [115, 473, 502, 489], "content": "[3] Gross M., Wilson P.M.H., Mirror Symmetry via 3-tori for a class of", "parent_index": 3, "line_index": 4}, {"bbox": [135, 488, 347, 504], "content": "Calabi-Yau threefolds, alg-geom/9608004.", "parent_index": 3, "line_index": 5}, {"bbox": [115, 513, 500, 527], "content": "[4] Harvey R., Lawson Jr., H.B., Calibrated Geometries, Acta Math., 148", "parent_index": 3, "line_index": 6}, {"bbox": [134, 528, 212, 542], "content": "(1982), 47-157.", "parent_index": 3, "line_index": 7}, {"bbox": [115, 551, 498, 568], "content": "[5] Huybrechts D., Compact Hyperkahler manifolds: basic results, alg-", "parent_index": 3, "line_index": 8}, {"bbox": [132, 567, 212, 580], "content": "geom/9705025.", "parent_index": 3, "line_index": 9}, {"bbox": [115, 590, 500, 606], "content": "[6] Markushevich D., Completely integrable projective symplectic 4-", "parent_index": 3, "line_index": 10}, {"bbox": [135, 605, 471, 619], "content": "dimensional varieties, Izvestiya: Mathematics 59 (1995), 159-187.", "parent_index": 3, "line_index": 11}, {"bbox": [116, 629, 501, 644], "content": "[7] Markushevich D., Integrable symplectic structures on compact complex", "parent_index": 3, "line_index": 12}, {"bbox": [134, 644, 378, 659], "content": "manifolds, Math. USSR Sb. 59 (1988), 459-469.", "parent_index": 3, "line_index": 13}, {"bbox": [116, 127, 500, 143], "content": "[8] Matsushita D., On fibre space structures of a projective irreducible sym-", "parent_index": 3, "line_index": 14}, {"bbox": [134, 143, 360, 157], "content": "plectic manifold, Topology 38 (1999), 79-83.", "parent_index": 3, "line_index": 15}, {"bbox": [115, 165, 501, 182], "content": "[9] Matsushita D., Addendum to: On fibre space structures of a projective", "parent_index": 3, "line_index": 16}, {"bbox": [133, 181, 391, 196], "content": "irreducible symplectic manifold, math.ag/9903045.", "parent_index": 3, "line_index": 17}, {"bbox": [110, 205, 501, 221], "content": "[10] O’Grady K.G., Desingularized moduli spaces of sheaves on a K3, I and", "parent_index": 3, "line_index": 18}, {"bbox": [134, 220, 343, 235], "content": "II, alg-geom/9708009, math.ag/9805099.", "parent_index": 3, "line_index": 19}, {"bbox": [110, 244, 500, 259], "content": "[11] Strominger A., Yau S.-T., Zaslow E., Mirror Symmetry is T-duality,", "parent_index": 3, "line_index": 20}, {"bbox": [134, 259, 316, 273], "content": "Nucl. Phys. B479, (1996), 243-259.", "parent_index": 3, "line_index": 21}]
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# IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO
INNA SYSOEVA
Abstract. This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible representations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex representations $\rho$ of Artin braid group $B_{n}$ with the condition $r a n k(\rho(\sigma_{i})-1)=2$ where $\sigma_{i}$ are the standard generators. For $n\,\geq\,7$ they all belong to some one-parameter family of $n$ -dimensional representations.
# 1. Introduction.
In his paper [3] Edward Formanek classified all irreducible complex representations of Artin braid groups $B_{n}$ of dimension at most $n-1$ . This paper is the first in a series of papers aimed at extending this classification to irreducible representations of higher dimensions.
To describe our results, we need the following definition.
Definition 1.1. The corank of the representation $\rho:B_{n}\to G L_{r}(\mathbb{C})$ $i s\ r a n k(\rho(\sigma_{i})-1)$ where the $\sigma_{i}$ are the standard generators of the group $B_{n}$
Remark 1.1. Because the $\sigma_{i}$ are conjugate to each other ([2], p.655), the number $r a n k(\rho(\sigma_{i})-1)$ does not depend on $i$ , which justifies the above definition.
The corank of specializations of the reduced Burau representation ([1], p.121; [4], p.338) and of the standard one-dimensional representation is 1.
By the results of Formanek ([3], Theorem 23) almost all of the irreducible complex representations $B_{n}$ of degree at most $n-1$ of are the tensor product of a one-dimensional representation and a representation of corank 1. He also classified all the irreducible representations of corank 1 (see [3], Theorem 10). For $n$ large enough they are one of the following.
1. A one-dimensional representation $\chi(y):B_{n}\to\mathbb{C}^{*}$ , $\chi(y)(\sigma_{i})=y$
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<html><body>
<h1 data-bbox="149 141 462 170">IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO </h1>
<p data-bbox="267 187 344 199">INNA SYSOEVA </p>
<p data-bbox="161 216 450 301">Abstract. This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible representations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex representations $\rho$ of Artin braid group $B_{n}$ with the condition $r a n k(\rho(\sigma_{i})-1)=2$ where $\sigma_{i}$ are the standard generators. For $n\,\geq\,7$ they all belong to some one-parameter family of $n$ -dimensional representations. </p>
<h1 data-bbox="256 335 355 349">1. Introduction. </h1>
<p data-bbox="125 356 486 412">In his paper [3] Edward Formanek classified all irreducible complex representations of Artin braid groups $B_{n}$ of dimension at most $n-1$ . This paper is the first in a series of papers aimed at extending this classification to irreducible representations of higher dimensions. </p>
<p data-bbox="136 413 428 426">To describe our results, we need the following definition. </p>
<p data-bbox="126 433 486 475">Definition 1.1. The corank of the representation $\rho:B_{n}\to G L_{r}(\mathbb{C})$ $i s\ r a n k(\rho(\sigma_{i})-1)$ where the $\sigma_{i}$ are the standard generators of the group $B_{n}$ </p>
<p data-bbox="125 489 486 532">Remark 1.1. Because the $\sigma_{i}$ are conjugate to each other ([2], p.655), the number $r a n k(\rho(\sigma_{i})-1)$ does not depend on $i$ , which justifies the above definition. </p>
<p data-bbox="125 538 486 580">The corank of specializations of the reduced Burau representation ([1], p.121; [4], p.338) and of the standard one-dimensional representation is 1. </p>
<p data-bbox="125 581 486 664">By the results of Formanek ([3], Theorem 23) almost all of the irreducible complex representations $B_{n}$ of degree at most $n-1$ of are the tensor product of a one-dimensional representation and a representation of corank 1. He also classified all the irreducible representations of corank 1 (see [3], Theorem 10). For $n$ large enough they are one of the following. </p>
<p data-bbox="135 667 483 682">1. A one-dimensional representation $\chi(y):B_{n}\to\mathbb{C}^{*}$ , $\chi(y)(\sigma_{i})=y$ </p>
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0003047v1
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This paper is the first part of a series of papers aimed", "type": "text"}], "index": 3}, {"bbox": [162, 231, 450, 243], "spans": [{"bbox": [162, 231, 450, 243], "score": 1.0, "content": "at improving the classification by Formanek of the irreducible rep-", "type": "text"}], "index": 4}, {"bbox": [160, 243, 450, 255], "spans": [{"bbox": [160, 243, 450, 255], "score": 1.0, "content": "resentations of Artin braid groups of small dimension. In this paper", "type": "text"}], "index": 5}, {"bbox": [162, 255, 450, 266], "spans": [{"bbox": [162, 255, 403, 266], "score": 1.0, "content": "we classify all the irreducible complex representations ", "type": "text"}, {"bbox": [404, 259, 410, 266], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [410, 255, 450, 266], "score": 1.0, "content": " of Artin", "type": "text"}], "index": 6}, {"bbox": [161, 266, 448, 279], "spans": [{"bbox": [161, 266, 217, 279], "score": 1.0, "content": "braid group ", "type": "text"}, {"bbox": [217, 268, 230, 277], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [230, 266, 318, 279], "score": 1.0, "content": " with the condition ", "type": "text"}, {"bbox": [318, 268, 407, 278], "score": 0.94, "content": "r a n k(\\rho(\\sigma_{i})-1)=2", "type": "inline_equation", "height": 10, "width": 89}, {"bbox": [407, 266, 439, 279], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [440, 271, 448, 277], "score": 0.89, "content": "\\sigma_{i}", "type": "inline_equation", "height": 6, "width": 8}], "index": 7}, {"bbox": [161, 279, 450, 291], "spans": [{"bbox": [161, 279, 312, 291], "score": 1.0, "content": "are the standard generators. For ", "type": "text"}, {"bbox": [312, 281, 340, 289], "score": 0.9, "content": "n\\,\\geq\\,7", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [340, 279, 450, 291], "score": 1.0, "content": " they all belong to some", "type": "text"}], "index": 8}, {"bbox": [162, 291, 403, 303], "spans": [{"bbox": [162, 291, 269, 303], "score": 1.0, "content": "one-parameter family of ", "type": "text"}, {"bbox": [269, 295, 276, 300], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [276, 291, 403, 303], "score": 1.0, "content": "-dimensional representations.", "type": "text"}], "index": 9}], "index": 6}, {"type": "title", "bbox": [256, 335, 355, 349], "lines": [{"bbox": [255, 337, 356, 350], "spans": [{"bbox": [255, 337, 356, 350], "score": 1.0, "content": "1. Introduction.", "type": "text"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [125, 356, 486, 412], "lines": [{"bbox": [137, 358, 486, 373], "spans": [{"bbox": [137, 358, 486, 373], "score": 1.0, "content": "In his paper [3] Edward Formanek classified all irreducible complex", "type": "text"}], "index": 11}, {"bbox": [126, 372, 487, 387], "spans": [{"bbox": [126, 372, 335, 387], "score": 1.0, "content": "representations of Artin braid groups", "type": "text"}, {"bbox": [336, 374, 350, 385], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [351, 372, 487, 387], "score": 1.0, "content": "of dimension at most", "type": "text"}], "index": 12}, {"bbox": [126, 387, 485, 400], "spans": [{"bbox": [126, 389, 154, 398], "score": 0.91, "content": "n-1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [154, 387, 485, 400], "score": 1.0, "content": ". This paper is the first in a series of papers aimed at extending", "type": "text"}], "index": 13}, {"bbox": [126, 401, 479, 414], "spans": [{"bbox": [126, 401, 479, 414], "score": 1.0, "content": "this classification to irreducible representations of higher dimensions.", "type": "text"}], "index": 14}], "index": 12.5}, {"type": "text", "bbox": [136, 413, 428, 426], "lines": [{"bbox": [138, 414, 427, 428], "spans": [{"bbox": [138, 414, 427, 428], "score": 1.0, "content": "To describe our results, we need the following definition.", "type": "text"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [126, 433, 486, 475], "lines": [{"bbox": [125, 435, 485, 449], "spans": [{"bbox": [125, 435, 393, 449], "score": 1.0, "content": "Definition 1.1. The corank of the representation ", "type": "text"}, {"bbox": [394, 437, 485, 449], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 91}], "index": 16}, {"bbox": [126, 449, 486, 464], "spans": [{"bbox": [126, 450, 215, 463], "score": 0.71, "content": "i s\\ r a n k(\\rho(\\sigma_{i})-1)", "type": "inline_equation", "height": 13, "width": 89}, {"bbox": [216, 449, 269, 464], "score": 1.0, "content": " where the ", "type": "text"}, {"bbox": [270, 454, 280, 462], "score": 0.73, "content": "\\sigma_{i}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [280, 449, 486, 464], "score": 1.0, "content": " are the standard generators of the group", "type": "text"}], "index": 17}, {"bbox": [126, 465, 141, 476], "spans": [{"bbox": [126, 465, 141, 476], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}], "index": 18}], "index": 17}, {"type": "text", "bbox": [125, 489, 486, 532], "lines": [{"bbox": [124, 491, 485, 507], "spans": [{"bbox": [124, 491, 268, 507], "score": 1.0, "content": "Remark 1.1. Because the ", "type": "text"}, {"bbox": [268, 497, 279, 504], "score": 0.86, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [279, 491, 485, 507], "score": 1.0, "content": " are conjugate to each other ([2], p.655),", "type": "text"}], "index": 19}, {"bbox": [125, 505, 486, 520], "spans": [{"bbox": [125, 505, 189, 520], "score": 1.0, "content": "the number ", "type": "text"}, {"bbox": [189, 507, 270, 519], "score": 0.93, "content": "r a n k(\\rho(\\sigma_{i})-1)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [271, 505, 379, 520], "score": 1.0, "content": " does not depend on ", "type": "text"}, {"bbox": [380, 508, 384, 516], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [384, 505, 486, 520], "score": 1.0, "content": ", which justifies the", "type": "text"}], "index": 20}, {"bbox": [125, 519, 211, 534], "spans": [{"bbox": [125, 519, 211, 534], "score": 1.0, "content": "above definition.", "type": "text"}], "index": 21}], "index": 20}, {"type": "text", "bbox": [125, 538, 486, 580], "lines": [{"bbox": [137, 540, 486, 555], "spans": [{"bbox": [137, 540, 486, 555], "score": 1.0, "content": "The corank of specializations of the reduced Burau representation", "type": "text"}], "index": 22}, {"bbox": [127, 555, 484, 568], "spans": [{"bbox": [127, 555, 484, 568], "score": 1.0, "content": "([1], p.121; [4], p.338) and of the standard one-dimensional representa-", "type": "text"}], "index": 23}, {"bbox": [125, 570, 172, 581], "spans": [{"bbox": [125, 570, 172, 581], "score": 1.0, "content": "tion is 1.", "type": "text"}], "index": 24}], "index": 23}, {"type": "text", "bbox": [125, 581, 486, 664], "lines": [{"bbox": [137, 582, 485, 596], "spans": [{"bbox": [137, 582, 485, 596], "score": 1.0, "content": "By the results of Formanek ([3], Theorem 23) almost all of the irre-", "type": "text"}], "index": 25}, {"bbox": [126, 596, 486, 610], "spans": [{"bbox": [126, 596, 293, 610], "score": 1.0, "content": "ducible complex representations ", "type": "text"}, {"bbox": [293, 598, 308, 609], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [308, 596, 404, 610], "score": 1.0, "content": " of degree at most ", "type": "text"}, {"bbox": [404, 599, 432, 608], "score": 0.92, "content": "n-1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [432, 596, 486, 610], "score": 1.0, "content": " of are the", "type": "text"}], "index": 26}, {"bbox": [125, 611, 485, 624], "spans": [{"bbox": [125, 611, 485, 624], "score": 1.0, "content": "tensor product of a one-dimensional representation and a representa-", "type": "text"}], "index": 27}, {"bbox": [125, 624, 487, 639], "spans": [{"bbox": [125, 624, 487, 639], "score": 1.0, "content": "tion of corank 1. 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For ", "type": "text"}, {"bbox": [312, 281, 340, 289], "score": 0.9, "content": "n\\,\\geq\\,7", "type": "inline_equation", "height": 8, "width": 28}, {"bbox": [340, 279, 450, 291], "score": 1.0, "content": " they all belong to some", "type": "text"}], "index": 8}, {"bbox": [162, 291, 403, 303], "spans": [{"bbox": [162, 291, 269, 303], "score": 1.0, "content": "one-parameter family of ", "type": "text"}, {"bbox": [269, 295, 276, 300], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [276, 291, 403, 303], "score": 1.0, "content": "-dimensional representations.", "type": "text"}], "index": 9}], "index": 6, "bbox_fs": [160, 219, 451, 303]}, {"type": "title", "bbox": [256, 335, 355, 349], "lines": [{"bbox": [255, 337, 356, 350], "spans": [{"bbox": [255, 337, 356, 350], "score": 1.0, "content": "1. Introduction.", "type": "text"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [125, 356, 486, 412], "lines": [{"bbox": [137, 358, 486, 373], "spans": [{"bbox": [137, 358, 486, 373], "score": 1.0, "content": "In his paper [3] Edward Formanek classified all irreducible complex", "type": "text"}], "index": 11}, {"bbox": [126, 372, 487, 387], "spans": [{"bbox": [126, 372, 335, 387], "score": 1.0, "content": "representations of Artin braid groups", "type": "text"}, {"bbox": [336, 374, 350, 385], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [351, 372, 487, 387], "score": 1.0, "content": "of dimension at most", "type": "text"}], "index": 12}, {"bbox": [126, 387, 485, 400], "spans": [{"bbox": [126, 389, 154, 398], "score": 0.91, "content": "n-1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [154, 387, 485, 400], "score": 1.0, "content": ". This paper is the first in a series of papers aimed at extending", "type": "text"}], "index": 13}, {"bbox": [126, 401, 479, 414], "spans": [{"bbox": [126, 401, 479, 414], "score": 1.0, "content": "this classification to irreducible representations of higher dimensions.", "type": "text"}], "index": 14}], "index": 12.5, "bbox_fs": [126, 358, 487, 414]}, {"type": "text", "bbox": [136, 413, 428, 426], "lines": [{"bbox": [138, 414, 427, 428], "spans": [{"bbox": [138, 414, 427, 428], "score": 1.0, "content": "To describe our results, we need the following definition.", "type": "text"}], "index": 15}], "index": 15, "bbox_fs": [138, 414, 427, 428]}, {"type": "text", "bbox": [126, 433, 486, 475], "lines": [{"bbox": [125, 435, 485, 449], "spans": [{"bbox": [125, 435, 393, 449], "score": 1.0, "content": "Definition 1.1. The corank of the representation ", "type": "text"}, {"bbox": [394, 437, 485, 449], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 91}], "index": 16}, {"bbox": [126, 449, 486, 464], "spans": [{"bbox": [126, 450, 215, 463], "score": 0.71, "content": "i s\\ r a n k(\\rho(\\sigma_{i})-1)", "type": "inline_equation", "height": 13, "width": 89}, {"bbox": [216, 449, 269, 464], "score": 1.0, "content": " where the ", "type": "text"}, {"bbox": [270, 454, 280, 462], "score": 0.73, "content": "\\sigma_{i}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [280, 449, 486, 464], "score": 1.0, "content": " are the standard generators of the group", "type": "text"}], "index": 17}, {"bbox": [126, 465, 141, 476], "spans": [{"bbox": [126, 465, 141, 476], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}], "index": 18}], "index": 17, "bbox_fs": [125, 435, 486, 476]}, {"type": "text", "bbox": [125, 489, 486, 532], "lines": [{"bbox": [124, 491, 485, 507], "spans": [{"bbox": [124, 491, 268, 507], "score": 1.0, "content": "Remark 1.1. Because the ", "type": "text"}, {"bbox": [268, 497, 279, 504], "score": 0.86, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [279, 491, 485, 507], "score": 1.0, "content": " are conjugate to each other ([2], p.655),", "type": "text"}], "index": 19}, {"bbox": [125, 505, 486, 520], "spans": [{"bbox": [125, 505, 189, 520], "score": 1.0, "content": "the number ", "type": "text"}, {"bbox": [189, 507, 270, 519], "score": 0.93, "content": "r a n k(\\rho(\\sigma_{i})-1)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [271, 505, 379, 520], "score": 1.0, "content": " does not depend on ", "type": "text"}, {"bbox": [380, 508, 384, 516], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [384, 505, 486, 520], "score": 1.0, "content": ", which justifies the", "type": "text"}], "index": 20}, {"bbox": [125, 519, 211, 534], "spans": [{"bbox": [125, 519, 211, 534], "score": 1.0, "content": "above definition.", "type": "text"}], "index": 21}], "index": 20, "bbox_fs": [124, 491, 486, 534]}, {"type": "text", "bbox": [125, 538, 486, 580], "lines": [{"bbox": [137, 540, 486, 555], "spans": [{"bbox": [137, 540, 486, 555], "score": 1.0, "content": "The corank of specializations of the reduced Burau representation", "type": "text"}], "index": 22}, {"bbox": [127, 555, 484, 568], "spans": [{"bbox": [127, 555, 484, 568], "score": 1.0, "content": "([1], p.121; [4], p.338) and of the standard one-dimensional representa-", "type": "text"}], "index": 23}, {"bbox": [125, 570, 172, 581], "spans": [{"bbox": [125, 570, 172, 581], "score": 1.0, "content": "tion is 1.", "type": "text"}], "index": 24}], "index": 23, "bbox_fs": [125, 540, 486, 581]}, {"type": "text", "bbox": [125, 581, 486, 664], "lines": [{"bbox": [137, 582, 485, 596], "spans": [{"bbox": [137, 582, 485, 596], "score": 1.0, "content": "By the results of Formanek ([3], Theorem 23) almost all of the irre-", "type": "text"}], "index": 25}, {"bbox": [126, 596, 486, 610], "spans": [{"bbox": [126, 596, 293, 610], "score": 1.0, "content": "ducible complex representations ", "type": "text"}, {"bbox": [293, 598, 308, 609], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [308, 596, 404, 610], "score": 1.0, "content": " of degree at most ", "type": "text"}, {"bbox": [404, 599, 432, 608], "score": 0.92, "content": "n-1", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [432, 596, 486, 610], "score": 1.0, "content": " of are the", "type": "text"}], "index": 26}, {"bbox": [125, 611, 485, 624], "spans": [{"bbox": [125, 611, 485, 624], "score": 1.0, "content": "tensor product of a one-dimensional representation and a representa-", "type": "text"}], "index": 27}, {"bbox": [125, 624, 487, 639], "spans": [{"bbox": [125, 624, 487, 639], "score": 1.0, "content": "tion of corank 1. He also classified all the irreducible representations of", "type": "text"}], "index": 28}, {"bbox": [126, 639, 486, 653], "spans": [{"bbox": [126, 639, 309, 653], "score": 1.0, "content": "corank 1 (see [3], Theorem 10). For ", "type": "text"}, {"bbox": [310, 643, 317, 649], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [317, 639, 486, 653], "score": 1.0, "content": " large enough they are one of the", "type": "text"}], "index": 29}, {"bbox": [125, 652, 176, 667], "spans": [{"bbox": [125, 652, 176, 667], "score": 1.0, "content": "following.", "type": "text"}], "index": 30}], "index": 27.5, "bbox_fs": [125, 582, 487, 667]}, {"type": "text", "bbox": [135, 667, 483, 682], "lines": [{"bbox": [137, 668, 481, 684], "spans": [{"bbox": [137, 668, 326, 684], "score": 1.0, "content": "1. A one-dimensional representation ", "type": "text"}, {"bbox": [327, 670, 406, 682], "score": 0.78, "content": "\\chi(y):B_{n}\\to\\mathbb{C}^{*}", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [406, 668, 416, 684], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [416, 670, 481, 682], "score": 0.58, "content": "\\chi(y)(\\sigma_{i})=y", "type": "inline_equation", "height": 12, "width": 65}], "index": 31}], "index": 31, "bbox_fs": [137, 668, 481, 684]}]}
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[{"type": "title", "bbox": [149, 141, 462, 170], "content": "IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO", "index": 0}, {"type": "text", "bbox": [267, 187, 344, 199], "content": "INNA SYSOEVA", "index": 1}, {"type": "text", "bbox": [161, 216, 450, 301], "content": "Abstract. This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible rep- resentations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex representations of Artin braid group with the condition where are the standard generators. For they all belong to some one-parameter family of -dimensional representations.", "index": 2}, {"type": "title", "bbox": [256, 335, 355, 349], "content": "1. Introduction.", "index": 3}, {"type": "text", "bbox": [125, 356, 486, 412], "content": "In his paper [3] Edward Formanek classified all irreducible complex representations of Artin braid groups of dimension at most . This paper is the first in a series of papers aimed at extending this classification to irreducible representations of higher dimensions.", "index": 4}, {"type": "text", "bbox": [136, 413, 428, 426], "content": "To describe our results, we need the following definition.", "index": 5}, {"type": "text", "bbox": [126, 433, 486, 475], "content": "Definition 1.1. The corank of the representation where the are the standard generators of the group", "index": 6}, {"type": "text", "bbox": [125, 489, 486, 532], "content": "Remark 1.1. Because the are conjugate to each other ([2], p.655), the number does not depend on , which justifies the above definition.", "index": 7}, {"type": "text", "bbox": [125, 538, 486, 580], "content": "The corank of specializations of the reduced Burau representation ([1], p.121; [4], p.338) and of the standard one-dimensional representa- tion is 1.", "index": 8}, {"type": "text", "bbox": [125, 581, 486, 664], "content": "By the results of Formanek ([3], Theorem 23) almost all of the irre- ducible complex representations of degree at most of are the tensor product of a one-dimensional representation and a representa- tion of corank 1. He also classified all the irreducible representations of corank 1 (see [3], Theorem 10). For large enough they are one of the following.", "index": 9}, {"type": "text", "bbox": [135, 667, 483, 682], "content": "1. A one-dimensional representation ,", "index": 10}]
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[{"bbox": [150, 146, 460, 156], "content": "IRREDUCIBLE REPRESENTATIONS OF BRAID", "parent_index": 0, "line_index": 0}, {"bbox": [213, 160, 399, 171], "content": "GROUPS OF CORANK TWO", "parent_index": 0, "line_index": 1}, {"bbox": [266, 190, 344, 200], "content": "INNA SYSOEVA", "parent_index": 1, "line_index": 0}, {"bbox": [162, 219, 451, 231], "content": "Abstract. This paper is the first part of a series of papers aimed", "parent_index": 2, "line_index": 0}, {"bbox": [162, 231, 450, 243], "content": "at improving the classification by Formanek of the irreducible rep-", "parent_index": 2, "line_index": 1}, {"bbox": [160, 243, 450, 255], "content": "resentations of Artin braid groups of small dimension. In this paper", "parent_index": 2, "line_index": 2}, {"bbox": [162, 255, 450, 266], "content": "we classify all the irreducible complex representations of Artin", "parent_index": 2, "line_index": 3}, {"bbox": [161, 266, 448, 279], "content": "braid group with the condition where", "parent_index": 2, "line_index": 4}, {"bbox": [161, 279, 450, 291], "content": "are the standard generators. For they all belong to some", "parent_index": 2, "line_index": 5}, {"bbox": [162, 291, 403, 303], "content": "one-parameter family of -dimensional representations.", "parent_index": 2, "line_index": 6}, {"bbox": [255, 337, 356, 350], "content": "1. Introduction.", "parent_index": 3, "line_index": 0}, {"bbox": [137, 358, 486, 373], "content": "In his paper [3] Edward Formanek classified all irreducible complex", "parent_index": 4, "line_index": 0}, {"bbox": [126, 372, 487, 387], "content": "representations of Artin braid groups of dimension at most", "parent_index": 4, "line_index": 1}, {"bbox": [126, 387, 485, 400], "content": ". This paper is the first in a series of papers aimed at extending", "parent_index": 4, "line_index": 2}, {"bbox": [126, 401, 479, 414], "content": "this classification to irreducible representations of higher dimensions.", "parent_index": 4, "line_index": 3}, {"bbox": [138, 414, 427, 428], "content": "To describe our results, we need the following definition.", "parent_index": 5, "line_index": 0}, {"bbox": [125, 435, 485, 449], "content": "Definition 1.1. The corank of the representation", "parent_index": 6, "line_index": 0}, {"bbox": [126, 449, 486, 464], "content": "where the are the standard generators of the group", "parent_index": 6, "line_index": 1}, {"bbox": [126, 465, 141, 476], "content": "", "parent_index": 6, "line_index": 2}, {"bbox": [124, 491, 485, 507], "content": "Remark 1.1. Because the are conjugate to each other ([2], p.655),", "parent_index": 7, "line_index": 0}, {"bbox": [125, 505, 486, 520], "content": "the number does not depend on , which justifies the", "parent_index": 7, "line_index": 1}, {"bbox": [125, 519, 211, 534], "content": "above definition.", "parent_index": 7, "line_index": 2}, {"bbox": [137, 540, 486, 555], "content": "The corank of specializations of the reduced Burau representation", "parent_index": 8, "line_index": 0}, {"bbox": [127, 555, 484, 568], "content": "([1], p.121; [4], p.338) and of the standard one-dimensional representa-", "parent_index": 8, "line_index": 1}, {"bbox": [125, 570, 172, 581], "content": "tion is 1.", "parent_index": 8, "line_index": 2}, {"bbox": [137, 582, 485, 596], "content": "By the results of Formanek ([3], Theorem 23) almost all of the irre-", "parent_index": 9, "line_index": 0}, {"bbox": [126, 596, 486, 610], "content": "ducible complex representations of degree at most of are the", "parent_index": 9, "line_index": 1}, {"bbox": [125, 611, 485, 624], "content": "tensor product of a one-dimensional representation and a representa-", "parent_index": 9, "line_index": 2}, {"bbox": [125, 624, 487, 639], "content": "tion of corank 1. He also classified all the irreducible representations of", "parent_index": 9, "line_index": 3}, {"bbox": [126, 639, 486, 653], "content": "corank 1 (see [3], Theorem 10). For large enough they are one of the", "parent_index": 9, "line_index": 4}, {"bbox": [125, 652, 176, 667], "content": "following.", "parent_index": 9, "line_index": 5}, {"bbox": [137, 668, 481, 684], "content": "1. A one-dimensional representation ,", "parent_index": 10, "line_index": 0}]
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[]
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[{"bbox": [404, 259, 410, 266], "content": "\\rho", "parent_index": 2, "subtype": "inline"}, {"bbox": [217, 268, 230, 277], "content": "B_{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [318, 268, 407, 278], "content": "r a n k(\\rho(\\sigma_{i})-1)=2", "parent_index": 2, "subtype": "inline"}, {"bbox": [440, 271, 448, 277], "content": "\\sigma_{i}", "parent_index": 2, "subtype": "inline"}, {"bbox": [312, 281, 340, 289], "content": "n\\,\\geq\\,7", "parent_index": 2, "subtype": "inline"}, {"bbox": [269, 295, 276, 300], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [336, 374, 350, 385], "content": "B_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 389, 154, 398], "content": "n-1", "parent_index": 4, "subtype": "inline"}, {"bbox": [394, 437, 485, 449], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 6, "subtype": "inline"}, {"bbox": [126, 450, 215, 463], "content": "i s\\ r a n k(\\rho(\\sigma_{i})-1)", "parent_index": 6, "subtype": "inline"}, {"bbox": [270, 454, 280, 462], "content": "\\sigma_{i}", "parent_index": 6, "subtype": "inline"}, {"bbox": [126, 465, 141, 476], "content": "B_{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [268, 497, 279, 504], "content": "\\sigma_{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [189, 507, 270, 519], "content": "r a n k(\\rho(\\sigma_{i})-1)", "parent_index": 7, "subtype": "inline"}, {"bbox": [380, 508, 384, 516], "content": "i", "parent_index": 7, "subtype": "inline"}, {"bbox": [293, 598, 308, 609], "content": "B_{n}", "parent_index": 9, "subtype": "inline"}, {"bbox": [404, 599, 432, 608], "content": "n-1", "parent_index": 9, "subtype": "inline"}, {"bbox": [310, 643, 317, 649], "content": "n", "parent_index": 9, "subtype": "inline"}, {"bbox": [327, 670, 406, 682], "content": "\\chi(y):B_{n}\\to\\mathbb{C}^{*}", "parent_index": 10, "subtype": "inline"}, {"bbox": [416, 670, 481, 682], "content": "\\chi(y)(\\sigma_{i})=y", "parent_index": 10, "subtype": "inline"}]
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[]
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2. An irreducible $(n-1)-$ dimensional specialization of the reduced Burau representation 3. An irreducible $(n-2)-$ dimensional specialization of the composition factor of the reduced Burau representation
The main goal of this paper is to classify all the irreducible complex representations of corank 2. Apart from a number of exceptions for $n\leq6$ , they all are equivalent to specializations for $u\ne1$ , $u\in\mathbb{C}^{*}$ of the following representation $\rho:B_{n}\to G L_{n}(\mathbb{C}[u^{\pm1}])$ , first discovered by Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:
$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$
for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix.
The main tool we use is the friendship graph of a representation. Namely the (full) friendship graph of a representation $\rho$ of a braid group $B_{n}$ is a graph whose vertices are the set of generators $\left(\sigma_{0},\right)$ $\sigma_{1},\ldots,\sigma_{n-1}$ of $B_{n}$ . Two vertices $\sigma_{i}$ and $\sigma_{j}$ are joined by an edge if and only if $I m(\rho(\sigma_{i})-1)\cap I m(\rho(\sigma_{j})-1)\neq\{0\}$ .
Using the braid relations, we investigate the structure of the friendship graph. It turns out that every irreducible representation of $B_{n}$ of dimension at least $n$ and corank 2 the friendship graph is a chain, provided that $n\geq6$ . This means that $\sigma_{i}$ and $\sigma_{j}$ are joined by an edge if and only if $\vert i-j\vert=1$ .
For a given friendship graph it is relatively easy to classify all irreducible complex representations of $B_{n}$ for which it is the associated friendship graph.” When the graph is a chain, we get specializations of the representation discovered by Tong, Yang and Ma.
Now we are going to explain the place of this paper in the coming series. According to [3], Theorem 23, for $n$ large enough every irreducible complex representation of $B_{n}$ of dimension at most $n-1$ is a tensor product of a one-dimensional representation and a representation of corank 1. Using similar ideas one can show that for $n$ large enough every irreducible complex representation of $B_{n}$ of dimension at most $n$ is a tensor product of a one-dimensional representation and a representation of corank 2. Therefore one can use the results of this paper to extend the classification theorem of Formanek to the representations of $B_{n}$ of dimension $n$ . The proof of this result will appear elsewhere.
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<p data-bbox="136 110 486 167">2. An irreducible $(n-1)-$ dimensional specialization of the reduced Burau representation 3. An irreducible $(n-2)-$ dimensional specialization of the composition factor of the reduced Burau representation </p>
<p data-bbox="124 174 487 246">The main goal of this paper is to classify all the irreducible complex representations of corank 2. Apart from a number of exceptions for $n\leq6$ , they all are equivalent to specializations for $u\ne1$ , $u\in\mathbb{C}^{*}$ of the following representation $\rho:B_{n}\to G L_{n}(\mathbb{C}[u^{\pm1}])$ , first discovered by Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]: </p>
<div class="equation" data-bbox="221 265 388 322">$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$</div>
<p data-bbox="124 350 434 365">for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix. </p>
<p data-bbox="124 366 486 435">The main tool we use is the friendship graph of a representation. Namely the (full) friendship graph of a representation $\rho$ of a braid group $B_{n}$ is a graph whose vertices are the set of generators $\left(\sigma_{0},\right)$ $\sigma_{1},\ldots,\sigma_{n-1}$ of $B_{n}$ . Two vertices $\sigma_{i}$ and $\sigma_{j}$ are joined by an edge if and only if $I m(\rho(\sigma_{i})-1)\cap I m(\rho(\sigma_{j})-1)\neq\{0\}$ . </p>
<p data-bbox="124 435 486 505">Using the braid relations, we investigate the structure of the friendship graph. It turns out that every irreducible representation of $B_{n}$ of dimension at least $n$ and corank 2 the friendship graph is a chain, provided that $n\geq6$ . This means that $\sigma_{i}$ and $\sigma_{j}$ are joined by an edge if and only if $\vert i-j\vert=1$ . </p>
<p data-bbox="125 505 487 560">For a given friendship graph it is relatively easy to classify all irreducible complex representations of $B_{n}$ for which it is the associated friendship graph.” When the graph is a chain, we get specializations of the representation discovered by Tong, Yang and Ma. </p>
<p data-bbox="125 561 487 700">Now we are going to explain the place of this paper in the coming series. According to [3], Theorem 23, for $n$ large enough every irreducible complex representation of $B_{n}$ of dimension at most $n-1$ is a tensor product of a one-dimensional representation and a representation of corank 1. Using similar ideas one can show that for $n$ large enough every irreducible complex representation of $B_{n}$ of dimension at most $n$ is a tensor product of a one-dimensional representation and a representation of corank 2. Therefore one can use the results of this paper to extend the classification theorem of Formanek to the representations of $B_{n}$ of dimension $n$ . The proof of this result will appear elsewhere. </p>
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0003047v1
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[{"type": "text", "text": "2. An irreducible $(n-1)-$ dimensional specialization of the reduced Burau representation 3. An irreducible $(n-2)-$ dimensional specialization of the composition factor of the reduced Burau representation ", "page_idx": 1}, {"type": "text", "text": "The main goal of this paper is to classify all the irreducible complex representations of corank 2. Apart from a number of exceptions for $n\\leq6$ , they all are equivalent to specializations for $u\\ne1$ , $u\\in\\mathbb{C}^{*}$ of the following representation $\\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}])$ , first discovered by Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]: ", "page_idx": 1}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "for $i=1,2,\\dots,n-1$ , where $I_{k}$ is the $k\\times k$ identity matrix. ", "page_idx": 1}, {"type": "text", "text": "The main tool we use is the friendship graph of a representation. Namely the (full) friendship graph of a representation $\\rho$ of a braid group $B_{n}$ is a graph whose vertices are the set of generators $\\left(\\sigma_{0},\\right)$ $\\sigma_{1},\\ldots,\\sigma_{n-1}$ of $B_{n}$ . Two vertices $\\sigma_{i}$ and $\\sigma_{j}$ are joined by an edge if and only if $I m(\\rho(\\sigma_{i})-1)\\cap I m(\\rho(\\sigma_{j})-1)\\neq\\{0\\}$ . ", "page_idx": 1}, {"type": "text", "text": "Using the braid relations, we investigate the structure of the friendship graph. It turns out that every irreducible representation of $B_{n}$ of dimension at least $n$ and corank 2 the friendship graph is a chain, provided that $n\\geq6$ . This means that $\\sigma_{i}$ and $\\sigma_{j}$ are joined by an edge if and only if $\\vert i-j\\vert=1$ . ", "page_idx": 1}, {"type": "text", "text": "For a given friendship graph it is relatively easy to classify all irreducible complex representations of $B_{n}$ for which it is the associated friendship graph.” When the graph is a chain, we get specializations of the representation discovered by Tong, Yang and Ma. ", "page_idx": 1}, {"type": "text", "text": "Now we are going to explain the place of this paper in the coming series. According to [3], Theorem 23, for $n$ large enough every irreducible complex representation of $B_{n}$ of dimension at most $n-1$ is a tensor product of a one-dimensional representation and a representation of corank 1. Using similar ideas one can show that for $n$ large enough every irreducible complex representation of $B_{n}$ of dimension at most $n$ is a tensor product of a one-dimensional representation and a representation of corank 2. Therefore one can use the results of this paper to extend the classification theorem of Formanek to the representations of $B_{n}$ of dimension $n$ . The proof of this result will appear elsewhere. ", "page_idx": 1}]
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Apart from a number of exceptions for", "type": "text"}], "index": 5}, {"bbox": [126, 206, 487, 220], "spans": [{"bbox": [126, 208, 157, 218], "score": 0.92, "content": "n\\leq6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 206, 396, 220], "score": 1.0, "content": ", they all are equivalent to specializations for ", "type": "text"}, {"bbox": [396, 207, 426, 218], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [427, 206, 434, 220], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [434, 207, 471, 217], "score": 0.92, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [471, 206, 487, 220], "score": 1.0, "content": "of", "type": "text"}], "index": 6}, {"bbox": [125, 219, 486, 235], "spans": [{"bbox": [125, 219, 271, 235], "score": 1.0, "content": "the following representation ", "type": "text"}, {"bbox": [271, 220, 386, 233], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}])", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [386, 219, 486, 235], "score": 1.0, "content": ", first discovered by", "type": "text"}], "index": 7}, {"bbox": [125, 232, 421, 249], "spans": [{"bbox": [125, 232, 421, 249], "score": 1.0, "content": "Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:", "type": "text"}], "index": 8}], "index": 6}, {"type": "interline_equation", "bbox": [221, 265, 388, 322], "lines": [{"bbox": [221, 265, 388, 322], "spans": [{"bbox": [221, 265, 388, 322], "score": 0.92, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [124, 350, 434, 365], "lines": [{"bbox": [126, 353, 434, 367], "spans": [{"bbox": [126, 353, 144, 367], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 355, 235, 366], "score": 0.93, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [235, 353, 275, 367], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 355, 286, 365], "score": 0.91, "content": "I_{k}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [286, 353, 321, 367], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [321, 355, 349, 364], "score": 0.93, "content": "k\\times k", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [349, 353, 434, 367], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [124, 366, 486, 435], "lines": [{"bbox": [137, 366, 484, 381], "spans": [{"bbox": [137, 366, 484, 381], "score": 1.0, "content": "The main tool we use is the friendship graph of a representation.", "type": "text"}], "index": 11}, {"bbox": [126, 381, 486, 395], "spans": [{"bbox": [126, 381, 419, 395], "score": 1.0, "content": "Namely the (full) friendship graph of a representation ", "type": "text"}, {"bbox": [419, 385, 426, 394], "score": 0.83, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 381, 486, 395], "score": 1.0, "content": " of a braid", "type": "text"}], "index": 12}, {"bbox": [125, 395, 485, 409], "spans": [{"bbox": [125, 395, 160, 409], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [161, 397, 176, 407], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [176, 395, 459, 409], "score": 1.0, "content": " is a graph whose vertices are the set of generators ", "type": "text"}, {"bbox": [459, 396, 485, 408], "score": 0.91, "content": "\\left(\\sigma_{0},\\right)", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [126, 410, 486, 423], "spans": [{"bbox": [126, 414, 186, 422], "score": 0.86, "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 8, "width": 60}, {"bbox": [187, 410, 203, 423], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [203, 411, 218, 421], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [218, 410, 293, 423], "score": 1.0, "content": ". Two vertices ", "type": "text"}, {"bbox": [294, 414, 304, 421], "score": 0.9, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [304, 410, 329, 423], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [330, 414, 340, 423], "score": 0.88, "content": "\\sigma_{j}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [341, 410, 486, 423], "score": 1.0, "content": " are joined by an edge if and", "type": "text"}], "index": 14}, {"bbox": [126, 421, 357, 438], "spans": [{"bbox": [126, 421, 162, 438], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [162, 424, 353, 436], "score": 0.93, "content": "I m(\\rho(\\sigma_{i})-1)\\cap I m(\\rho(\\sigma_{j})-1)\\neq\\{0\\}", "type": "inline_equation", "height": 12, "width": 191}, {"bbox": [353, 421, 357, 438], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13}, {"type": "text", "bbox": [124, 435, 486, 505], "lines": [{"bbox": [137, 436, 485, 450], "spans": [{"bbox": [137, 436, 485, 450], "score": 1.0, "content": "Using the braid relations, we investigate the structure of the friend-", "type": "text"}], "index": 16}, {"bbox": [125, 450, 484, 465], "spans": [{"bbox": [125, 450, 469, 465], "score": 1.0, "content": "ship graph. It turns out that every irreducible representation of ", "type": "text"}, {"bbox": [470, 452, 484, 463], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}], "index": 17}, {"bbox": [125, 465, 484, 478], "spans": [{"bbox": [125, 465, 238, 478], "score": 1.0, "content": "of dimension at least ", "type": "text"}, {"bbox": [238, 470, 246, 475], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [246, 465, 484, 478], "score": 1.0, "content": " and corank 2 the friendship graph is a chain,", "type": "text"}], "index": 18}, {"bbox": [124, 479, 485, 493], "spans": [{"bbox": [124, 479, 199, 493], "score": 1.0, "content": "provided that ", "type": "text"}, {"bbox": [199, 480, 228, 490], "score": 0.92, "content": "n\\geq6", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [229, 479, 323, 493], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [324, 483, 333, 491], "score": 0.9, "content": "\\sigma_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [334, 479, 360, 493], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 483, 371, 492], "score": 0.91, "content": "\\sigma_{j}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [372, 479, 485, 493], "score": 1.0, "content": " are joined by an edge", "type": "text"}], "index": 19}, {"bbox": [125, 492, 252, 507], "spans": [{"bbox": [125, 492, 196, 507], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [196, 493, 249, 506], "score": 0.94, "content": "\\vert i-j\\vert=1", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [249, 492, 252, 507], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 18}, {"type": "text", "bbox": [125, 505, 487, 560], "lines": [{"bbox": [137, 506, 485, 521], "spans": [{"bbox": [137, 506, 485, 521], "score": 1.0, "content": "For a given friendship graph it is relatively easy to classify all ir-", "type": "text"}], "index": 21}, {"bbox": [125, 520, 486, 534], "spans": [{"bbox": [125, 520, 317, 534], "score": 1.0, "content": "reducible complex representations of ", "type": "text"}, {"bbox": [317, 522, 332, 533], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [333, 520, 486, 534], "score": 1.0, "content": " for which it is the associated", "type": "text"}], "index": 22}, {"bbox": [125, 533, 488, 549], "spans": [{"bbox": [125, 533, 488, 549], "score": 1.0, "content": "friendship graph.” When the graph is a chain, we get specializations of", "type": "text"}], "index": 23}, {"bbox": [126, 549, 401, 562], "spans": [{"bbox": [126, 549, 401, 562], "score": 1.0, "content": "the representation discovered by Tong, Yang and Ma.", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [125, 561, 487, 700], "lines": [{"bbox": [137, 562, 484, 576], "spans": [{"bbox": [137, 562, 484, 576], "score": 1.0, "content": "Now we are going to explain the place of this paper in the coming se-", "type": "text"}], "index": 25}, {"bbox": [124, 576, 486, 590], "spans": [{"bbox": [124, 576, 322, 590], "score": 1.0, "content": "ries. According to [3], Theorem 23, for", "type": "text"}, {"bbox": [323, 581, 330, 587], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [330, 576, 486, 590], "score": 1.0, "content": " large enough every irreducible", "type": "text"}], "index": 26}, {"bbox": [126, 591, 487, 603], "spans": [{"bbox": [126, 591, 263, 603], "score": 1.0, "content": "complex representation of ", "type": "text"}, {"bbox": [264, 592, 278, 602], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 591, 397, 603], "score": 1.0, "content": " of dimension at most ", "type": "text"}, {"bbox": [397, 592, 426, 601], "score": 0.93, "content": "n-1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [426, 591, 487, 603], "score": 1.0, "content": " is a tensor", "type": "text"}], "index": 27}, {"bbox": [124, 604, 488, 618], "spans": [{"bbox": [124, 604, 488, 618], "score": 1.0, "content": "product of a one-dimensional representation and a representation of", "type": "text"}], "index": 28}, {"bbox": [125, 618, 486, 632], "spans": [{"bbox": [125, 618, 406, 632], "score": 1.0, "content": "corank 1. Using similar ideas one can show that for ", "type": "text"}, {"bbox": [407, 623, 414, 628], "score": 0.9, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [414, 618, 486, 632], "score": 1.0, "content": " large enough", "type": "text"}], "index": 29}, {"bbox": [126, 633, 484, 646], "spans": [{"bbox": [126, 633, 349, 646], "score": 1.0, "content": "every irreducible complex representation of ", "type": "text"}, {"bbox": [349, 634, 363, 644], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [364, 633, 477, 646], "score": 1.0, "content": " of dimension at most ", "type": "text"}, {"bbox": [477, 637, 484, 642], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}], "index": 30}, {"bbox": [125, 646, 486, 659], "spans": [{"bbox": [125, 646, 486, 659], "score": 1.0, "content": "is a tensor product of a one-dimensional representation and a represen-", "type": "text"}], "index": 31}, {"bbox": [125, 660, 486, 673], "spans": [{"bbox": [125, 660, 486, 673], "score": 1.0, "content": "tation of corank 2. Therefore one can use the results of this paper to", "type": "text"}], "index": 32}, {"bbox": [126, 674, 486, 687], "spans": [{"bbox": [126, 674, 486, 687], "score": 1.0, "content": "extend the classification theorem of Formanek to the representations", "type": "text"}], "index": 33}, {"bbox": [125, 688, 476, 702], "spans": [{"bbox": [125, 688, 139, 702], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 690, 154, 700], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [154, 688, 226, 702], "score": 1.0, "content": " of dimension ", "type": "text"}, {"bbox": [226, 693, 234, 698], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [234, 688, 476, 702], "score": 1.0, "content": ". The proof of this result will appear elsewhere.", "type": "text"}], "index": 34}], "index": 29.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [221, 265, 388, 322], "lines": [{"bbox": [221, 265, 388, 322], "spans": [{"bbox": [221, 265, 388, 322], "score": 0.92, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [124, 91, 132, 100], "lines": [{"bbox": [126, 93, 132, 102], "spans": [{"bbox": [126, 93, 132, 102], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [136, 110, 486, 167], "lines": [{"bbox": [136, 113, 487, 128], "spans": [{"bbox": [136, 113, 229, 128], "score": 1.0, "content": "2. An irreducible ", "type": "text"}, {"bbox": [229, 114, 275, 127], "score": 0.93, "content": "(n-1)-", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [275, 113, 487, 128], "score": 1.0, "content": "dimensional specialization of the reduced", "type": "text"}], "index": 0}, {"bbox": [151, 127, 262, 141], "spans": [{"bbox": [151, 127, 262, 141], "score": 1.0, "content": "Burau representation", "type": "text"}], "index": 1}, {"bbox": [136, 140, 485, 155], "spans": [{"bbox": [136, 140, 229, 155], "score": 1.0, "content": "3. An irreducible ", "type": "text"}, {"bbox": [230, 142, 276, 154], "score": 0.92, "content": "(n-2)-", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [277, 140, 485, 155], "score": 1.0, "content": "dimensional specialization of the compo-", "type": "text"}], "index": 2}, {"bbox": [151, 155, 404, 169], "spans": [{"bbox": [151, 155, 404, 169], "score": 1.0, "content": "sition factor of the reduced Burau representation", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [136, 113, 487, 169]}, {"type": "text", "bbox": [124, 174, 487, 246], "lines": [{"bbox": [138, 178, 485, 191], "spans": [{"bbox": [138, 178, 485, 191], "score": 1.0, "content": "The main goal of this paper is to classify all the irreducible complex", "type": "text"}], "index": 4}, {"bbox": [126, 192, 485, 206], "spans": [{"bbox": [126, 192, 485, 206], "score": 1.0, "content": "representations of corank 2. Apart from a number of exceptions for", "type": "text"}], "index": 5}, {"bbox": [126, 206, 487, 220], "spans": [{"bbox": [126, 208, 157, 218], "score": 0.92, "content": "n\\leq6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [157, 206, 396, 220], "score": 1.0, "content": ", they all are equivalent to specializations for ", "type": "text"}, {"bbox": [396, 207, 426, 218], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [427, 206, 434, 220], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [434, 207, 471, 217], "score": 0.92, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [471, 206, 487, 220], "score": 1.0, "content": "of", "type": "text"}], "index": 6}, {"bbox": [125, 219, 486, 235], "spans": [{"bbox": [125, 219, 271, 235], "score": 1.0, "content": "the following representation ", "type": "text"}, {"bbox": [271, 220, 386, 233], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}])", "type": "inline_equation", "height": 13, "width": 115}, {"bbox": [386, 219, 486, 235], "score": 1.0, "content": ", first discovered by", "type": "text"}], "index": 7}, {"bbox": [125, 232, 421, 249], "spans": [{"bbox": [125, 232, 421, 249], "score": 1.0, "content": "Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:", "type": "text"}], "index": 8}], "index": 6, "bbox_fs": [125, 178, 487, 249]}, {"type": "interline_equation", "bbox": [221, 265, 388, 322], "lines": [{"bbox": [221, 265, 388, 322], "spans": [{"bbox": [221, 265, 388, 322], "score": 0.92, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [124, 350, 434, 365], "lines": [{"bbox": [126, 353, 434, 367], "spans": [{"bbox": [126, 353, 144, 367], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 355, 235, 366], "score": 0.93, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [235, 353, 275, 367], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 355, 286, 365], "score": 0.91, "content": "I_{k}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [286, 353, 321, 367], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [321, 355, 349, 364], "score": 0.93, "content": "k\\times k", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [349, 353, 434, 367], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 10}], "index": 10, "bbox_fs": [126, 353, 434, 367]}, {"type": "text", "bbox": [124, 366, 486, 435], "lines": [{"bbox": [137, 366, 484, 381], "spans": [{"bbox": [137, 366, 484, 381], "score": 1.0, "content": "The main tool we use is the friendship graph of a representation.", "type": "text"}], "index": 11}, {"bbox": [126, 381, 486, 395], "spans": [{"bbox": [126, 381, 419, 395], "score": 1.0, "content": "Namely the (full) friendship graph of a representation ", "type": "text"}, {"bbox": [419, 385, 426, 394], "score": 0.83, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 381, 486, 395], "score": 1.0, "content": " of a braid", "type": "text"}], "index": 12}, {"bbox": [125, 395, 485, 409], "spans": [{"bbox": [125, 395, 160, 409], "score": 1.0, "content": "group ", "type": "text"}, {"bbox": [161, 397, 176, 407], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [176, 395, 459, 409], "score": 1.0, "content": " is a graph whose vertices are the set of generators ", "type": "text"}, {"bbox": [459, 396, 485, 408], "score": 0.91, "content": "\\left(\\sigma_{0},\\right)", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [126, 410, 486, 423], "spans": [{"bbox": [126, 414, 186, 422], "score": 0.86, "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 8, "width": 60}, {"bbox": [187, 410, 203, 423], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [203, 411, 218, 421], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [218, 410, 293, 423], "score": 1.0, "content": ". Two vertices ", "type": "text"}, {"bbox": [294, 414, 304, 421], "score": 0.9, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [304, 410, 329, 423], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [330, 414, 340, 423], "score": 0.88, "content": "\\sigma_{j}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [341, 410, 486, 423], "score": 1.0, "content": " are joined by an edge if and", "type": "text"}], "index": 14}, {"bbox": [126, 421, 357, 438], "spans": [{"bbox": [126, 421, 162, 438], "score": 1.0, "content": "only if ", "type": "text"}, {"bbox": [162, 424, 353, 436], "score": 0.93, "content": "I m(\\rho(\\sigma_{i})-1)\\cap I m(\\rho(\\sigma_{j})-1)\\neq\\{0\\}", "type": "inline_equation", "height": 12, "width": 191}, {"bbox": [353, 421, 357, 438], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 13, "bbox_fs": [125, 366, 486, 438]}, {"type": "text", "bbox": [124, 435, 486, 505], "lines": [{"bbox": [137, 436, 485, 450], "spans": [{"bbox": [137, 436, 485, 450], "score": 1.0, "content": "Using the braid relations, we investigate the structure of the friend-", "type": "text"}], "index": 16}, {"bbox": [125, 450, 484, 465], "spans": [{"bbox": [125, 450, 469, 465], "score": 1.0, "content": "ship graph. It turns out that every irreducible representation of ", "type": "text"}, {"bbox": [470, 452, 484, 463], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}], "index": 17}, {"bbox": [125, 465, 484, 478], "spans": [{"bbox": [125, 465, 238, 478], "score": 1.0, "content": "of dimension at least ", "type": "text"}, {"bbox": [238, 470, 246, 475], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [246, 465, 484, 478], "score": 1.0, "content": " and corank 2 the friendship graph is a chain,", "type": "text"}], "index": 18}, {"bbox": [124, 479, 485, 493], "spans": [{"bbox": [124, 479, 199, 493], "score": 1.0, "content": "provided that ", "type": "text"}, {"bbox": [199, 480, 228, 490], "score": 0.92, "content": "n\\geq6", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [229, 479, 323, 493], "score": 1.0, "content": ". This means that ", "type": "text"}, {"bbox": [324, 483, 333, 491], "score": 0.9, "content": "\\sigma_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [334, 479, 360, 493], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 483, 371, 492], "score": 0.91, "content": "\\sigma_{j}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [372, 479, 485, 493], "score": 1.0, "content": " are joined by an edge", "type": "text"}], "index": 19}, {"bbox": [125, 492, 252, 507], "spans": [{"bbox": [125, 492, 196, 507], "score": 1.0, "content": "if and only if ", "type": "text"}, {"bbox": [196, 493, 249, 506], "score": 0.94, "content": "\\vert i-j\\vert=1", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [249, 492, 252, 507], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 18, "bbox_fs": [124, 436, 485, 507]}, {"type": "text", "bbox": [125, 505, 487, 560], "lines": [{"bbox": [137, 506, 485, 521], "spans": [{"bbox": [137, 506, 485, 521], "score": 1.0, "content": "For a given friendship graph it is relatively easy to classify all ir-", "type": "text"}], "index": 21}, {"bbox": [125, 520, 486, 534], "spans": [{"bbox": [125, 520, 317, 534], "score": 1.0, "content": "reducible complex representations of ", "type": "text"}, {"bbox": [317, 522, 332, 533], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [333, 520, 486, 534], "score": 1.0, "content": " for which it is the associated", "type": "text"}], "index": 22}, {"bbox": [125, 533, 488, 549], "spans": [{"bbox": [125, 533, 488, 549], "score": 1.0, "content": "friendship graph.” When the graph is a chain, we get specializations of", "type": "text"}], "index": 23}, {"bbox": [126, 549, 401, 562], "spans": [{"bbox": [126, 549, 401, 562], "score": 1.0, "content": "the representation discovered by Tong, Yang and Ma.", "type": "text"}], "index": 24}], "index": 22.5, "bbox_fs": [125, 506, 488, 562]}, {"type": "text", "bbox": [125, 561, 487, 700], "lines": [{"bbox": [137, 562, 484, 576], "spans": [{"bbox": [137, 562, 484, 576], "score": 1.0, "content": "Now we are going to explain the place of this paper in the coming se-", "type": "text"}], "index": 25}, {"bbox": [124, 576, 486, 590], "spans": [{"bbox": [124, 576, 322, 590], "score": 1.0, "content": "ries. According to [3], Theorem 23, for", "type": "text"}, {"bbox": [323, 581, 330, 587], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [330, 576, 486, 590], "score": 1.0, "content": " large enough every irreducible", "type": "text"}], "index": 26}, {"bbox": [126, 591, 487, 603], "spans": [{"bbox": [126, 591, 263, 603], "score": 1.0, "content": "complex representation of ", "type": "text"}, {"bbox": [264, 592, 278, 602], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 591, 397, 603], "score": 1.0, "content": " of dimension at most ", "type": "text"}, {"bbox": [397, 592, 426, 601], "score": 0.93, "content": "n-1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [426, 591, 487, 603], "score": 1.0, "content": " is a tensor", "type": "text"}], "index": 27}, {"bbox": [124, 604, 488, 618], "spans": [{"bbox": [124, 604, 488, 618], "score": 1.0, "content": "product of a one-dimensional representation and a representation of", "type": "text"}], "index": 28}, {"bbox": [125, 618, 486, 632], "spans": [{"bbox": [125, 618, 406, 632], "score": 1.0, "content": "corank 1. Using similar ideas one can show that for ", "type": "text"}, {"bbox": [407, 623, 414, 628], "score": 0.9, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [414, 618, 486, 632], "score": 1.0, "content": " large enough", "type": "text"}], "index": 29}, {"bbox": [126, 633, 484, 646], "spans": [{"bbox": [126, 633, 349, 646], "score": 1.0, "content": "every irreducible complex representation of ", "type": "text"}, {"bbox": [349, 634, 363, 644], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [364, 633, 477, 646], "score": 1.0, "content": " of dimension at most ", "type": "text"}, {"bbox": [477, 637, 484, 642], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}], "index": 30}, {"bbox": [125, 646, 486, 659], "spans": [{"bbox": [125, 646, 486, 659], "score": 1.0, "content": "is a tensor product of a one-dimensional representation and a represen-", "type": "text"}], "index": 31}, {"bbox": [125, 660, 486, 673], "spans": [{"bbox": [125, 660, 486, 673], "score": 1.0, "content": "tation of corank 2. Therefore one can use the results of this paper to", "type": "text"}], "index": 32}, {"bbox": [126, 674, 486, 687], "spans": [{"bbox": [126, 674, 486, 687], "score": 1.0, "content": "extend the classification theorem of Formanek to the representations", "type": "text"}], "index": 33}, {"bbox": [125, 688, 476, 702], "spans": [{"bbox": [125, 688, 139, 702], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 690, 154, 700], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [154, 688, 226, 702], "score": 1.0, "content": " of dimension ", "type": "text"}, {"bbox": [226, 693, 234, 698], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [234, 688, 476, 702], "score": 1.0, "content": ". The proof of this result will appear elsewhere.", "type": "text"}], "index": 34}], "index": 29.5, "bbox_fs": [124, 562, 488, 702]}]}
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[{"type": "text", "bbox": [136, 110, 486, 167], "content": "2. An irreducible dimensional specialization of the reduced Burau representation 3. An irreducible dimensional specialization of the compo- sition factor of the reduced Burau representation", "index": 0}, {"type": "text", "bbox": [124, 174, 487, 246], "content": "The main goal of this paper is to classify all the irreducible complex representations of corank 2. Apart from a number of exceptions for , they all are equivalent to specializations for , of the following representation , first discovered by Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:", "index": 1}, {"type": "interline_equation", "bbox": [221, 265, 388, 322], "content": "", "index": 2}, {"type": "text", "bbox": [124, 350, 434, 365], "content": "for , where is the identity matrix.", "index": 3}, {"type": "text", "bbox": [124, 366, 486, 435], "content": "The main tool we use is the friendship graph of a representation. Namely the (full) friendship graph of a representation of a braid group is a graph whose vertices are the set of generators of . Two vertices and are joined by an edge if and only if .", "index": 4}, {"type": "text", "bbox": [124, 435, 486, 505], "content": "Using the braid relations, we investigate the structure of the friend- ship graph. It turns out that every irreducible representation of of dimension at least and corank 2 the friendship graph is a chain, provided that . This means that and are joined by an edge if and only if .", "index": 5}, {"type": "text", "bbox": [125, 505, 487, 560], "content": "For a given friendship graph it is relatively easy to classify all ir- reducible complex representations of for which it is the associated friendship graph.” When the graph is a chain, we get specializations of the representation discovered by Tong, Yang and Ma.", "index": 6}, {"type": "text", "bbox": [125, 561, 487, 700], "content": "Now we are going to explain the place of this paper in the coming se- ries. According to [3], Theorem 23, for large enough every irreducible complex representation of of dimension at most is a tensor product of a one-dimensional representation and a representation of corank 1. Using similar ideas one can show that for large enough every irreducible complex representation of of dimension at most is a tensor product of a one-dimensional representation and a represen- tation of corank 2. Therefore one can use the results of this paper to extend the classification theorem of Formanek to the representations of of dimension . The proof of this result will appear elsewhere.", "index": 7}]
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[{"bbox": [136, 113, 487, 128], "content": "2. An irreducible dimensional specialization of the reduced", "parent_index": 0, "line_index": 0}, {"bbox": [151, 127, 262, 141], "content": "Burau representation", "parent_index": 0, "line_index": 1}, {"bbox": [136, 140, 485, 155], "content": "3. An irreducible dimensional specialization of the compo-", "parent_index": 0, "line_index": 2}, {"bbox": [151, 155, 404, 169], "content": "sition factor of the reduced Burau representation", "parent_index": 0, "line_index": 3}, {"bbox": [138, 178, 485, 191], "content": "The main goal of this paper is to classify all the irreducible complex", "parent_index": 1, "line_index": 0}, {"bbox": [126, 192, 485, 206], "content": "representations of corank 2. Apart from a number of exceptions for", "parent_index": 1, "line_index": 1}, {"bbox": [126, 206, 487, 220], "content": ", they all are equivalent to specializations for , of", "parent_index": 1, "line_index": 2}, {"bbox": [125, 219, 486, 235], "content": "the following representation , first discovered by", "parent_index": 1, "line_index": 3}, {"bbox": [125, 232, 421, 249], "content": "Dian-Ming Tong, Shan-De Yang and Zhong-Qi Ma in [6]:", "parent_index": 1, "line_index": 4}, {"bbox": [126, 353, 434, 367], "content": "for , where is the identity matrix.", "parent_index": 3, "line_index": 0}, {"bbox": [137, 366, 484, 381], "content": "The main tool we use is the friendship graph of a representation.", "parent_index": 4, "line_index": 0}, {"bbox": [126, 381, 486, 395], "content": "Namely the (full) friendship graph of a representation of a braid", "parent_index": 4, "line_index": 1}, {"bbox": [125, 395, 485, 409], "content": "group is a graph whose vertices are the set of generators", "parent_index": 4, "line_index": 2}, {"bbox": [126, 410, 486, 423], "content": "of . Two vertices and are joined by an edge if and", "parent_index": 4, "line_index": 3}, {"bbox": [126, 421, 357, 438], "content": "only if .", "parent_index": 4, "line_index": 4}, {"bbox": [137, 436, 485, 450], "content": "Using the braid relations, we investigate the structure of the friend-", "parent_index": 5, "line_index": 0}, {"bbox": [125, 450, 484, 465], "content": "ship graph. It turns out that every irreducible representation of", "parent_index": 5, "line_index": 1}, {"bbox": [125, 465, 484, 478], "content": "of dimension at least and corank 2 the friendship graph is a chain,", "parent_index": 5, "line_index": 2}, {"bbox": [124, 479, 485, 493], "content": "provided that . This means that and are joined by an edge", "parent_index": 5, "line_index": 3}, {"bbox": [125, 492, 252, 507], "content": "if and only if .", "parent_index": 5, "line_index": 4}, {"bbox": [137, 506, 485, 521], "content": "For a given friendship graph it is relatively easy to classify all ir-", "parent_index": 6, "line_index": 0}, {"bbox": [125, 520, 486, 534], "content": "reducible complex representations of for which it is the associated", "parent_index": 6, "line_index": 1}, {"bbox": [125, 533, 488, 549], "content": "friendship graph.” When the graph is a chain, we get specializations of", "parent_index": 6, "line_index": 2}, {"bbox": [126, 549, 401, 562], "content": "the representation discovered by Tong, Yang and Ma.", "parent_index": 6, "line_index": 3}, {"bbox": [137, 562, 484, 576], "content": "Now we are going to explain the place of this paper in the coming se-", "parent_index": 7, "line_index": 0}, {"bbox": [124, 576, 486, 590], "content": "ries. According to [3], Theorem 23, for large enough every irreducible", "parent_index": 7, "line_index": 1}, {"bbox": [126, 591, 487, 603], "content": "complex representation of of dimension at most is a tensor", "parent_index": 7, "line_index": 2}, {"bbox": [124, 604, 488, 618], "content": "product of a one-dimensional representation and a representation of", "parent_index": 7, "line_index": 3}, {"bbox": [125, 618, 486, 632], "content": "corank 1. Using similar ideas one can show that for large enough", "parent_index": 7, "line_index": 4}, {"bbox": [126, 633, 484, 646], "content": "every irreducible complex representation of of dimension at most", "parent_index": 7, "line_index": 5}, {"bbox": [125, 646, 486, 659], "content": "is a tensor product of a one-dimensional representation and a represen-", "parent_index": 7, "line_index": 6}, {"bbox": [125, 660, 486, 673], "content": "tation of corank 2. Therefore one can use the results of this paper to", "parent_index": 7, "line_index": 7}, {"bbox": [126, 674, 486, 687], "content": "extend the classification theorem of Formanek to the representations", "parent_index": 7, "line_index": 8}, {"bbox": [125, 688, 476, 702], "content": "of of dimension . The proof of this result will appear elsewhere.", "parent_index": 7, "line_index": 9}]
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[]
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[]
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Another result, which will appear elsewhere is that for $n$ large enough there are no irreducible complex representations of $B_{n}$ of corank 3 and no irreducible complex representations of $B_{n}$ of dimension $n+1$ .
Based on the above result we would like to make the following two conjectures.
Conjecture 1. For every $k\geq3$ for $n$ large enough there are no irreducible complex representations of $B_{n}$ of corank $k$ .
Conjecture 2. For every $k\geq1$ for $n$ large enough there are no irreducible complex representations of $B_{n}$ of dimension $n+k$ .
We should also note that for the purpose of brevity we did not include in this paper some of the details of the classification of representations of $B_{n}$ for small $n$ . The full proof can be found in our thesis [5], Chapters 6 and 7.
The paper is organized as follows. In section 2 we introduce some convenient notation that will be used throughout the rest of the paper. In section 3 we define the friendship graph of the representation and study its structure. We also study the case when the friendship graph is totally disconnected. In section 4 we prove that for $n\ge6$ for any irreducible complex representation of $B_{n}$ of corank 2 and dimension at least $n$ the associated friendship graph is a chain. In section 5 we determine all irreducible representations of corank 2 whose friendship graph is a chain.
Acknowledgments: The author would like to express her deep gratitude to professor Formanek for the numerous helpful discussions and comments on the preliminary versions of this paper, and for generous financial support of this research.
# 2. Notation and preliminary results
Let $B_{n}$ be the braid group on $n$ strings. It has a presentation
$$
B_{n}=<\sigma_{1},\ldots,\sigma_{n-1}|\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},1\leq i\leq n-2;\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},|i-j|\geq2
$$
Lemma 2.1. For the braid group $B_{n}$ set
$$
\tau=\sigma_{1}\sigma_{2}\dots\sigma_{n-1}\ a n d\ \sigma_{0}=\tau\sigma_{n-1}\tau^{-1}
$$
$$
\sigma_{i+1}=\tau\sigma_{i}\tau^{-1},
$$
$$
\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},
$$
|
<html><body>
<p data-bbox="125 111 486 152">Another result, which will appear elsewhere is that for $n$ large enough there are no irreducible complex representations of $B_{n}$ of corank 3 and no irreducible complex representations of $B_{n}$ of dimension $n+1$ . </p>
<p data-bbox="124 153 486 180">Based on the above result we would like to make the following two conjectures. </p>
<p data-bbox="125 187 485 215">Conjecture 1. For every $k\geq3$ for $n$ large enough there are no irreducible complex representations of $B_{n}$ of corank $k$ . </p>
<p data-bbox="125 228 485 256">Conjecture 2. For every $k\geq1$ for $n$ large enough there are no irreducible complex representations of $B_{n}$ of dimension $n+k$ . </p>
<p data-bbox="124 263 486 319">We should also note that for the purpose of brevity we did not include in this paper some of the details of the classification of representations of $B_{n}$ for small $n$ . The full proof can be found in our thesis [5], Chapters 6 and 7. </p>
<p data-bbox="125 320 486 444">The paper is organized as follows. In section 2 we introduce some convenient notation that will be used throughout the rest of the paper. In section 3 we define the friendship graph of the representation and study its structure. We also study the case when the friendship graph is totally disconnected. In section 4 we prove that for $n\ge6$ for any irreducible complex representation of $B_{n}$ of corank 2 and dimension at least $n$ the associated friendship graph is a chain. In section 5 we determine all irreducible representations of corank 2 whose friendship graph is a chain. </p>
<p data-bbox="125 445 486 501">Acknowledgments: The author would like to express her deep gratitude to professor Formanek for the numerous helpful discussions and comments on the preliminary versions of this paper, and for generous financial support of this research. </p>
<h1 data-bbox="193 511 418 524">2. Notation and preliminary results </h1>
<p data-bbox="135 531 453 545">Let $B_{n}$ be the braid group on $n$ strings. It has a presentation </p>
<div class="equation" data-bbox="125 551 527 566">$$
B_{n}=<\sigma_{1},\ldots,\sigma_{n-1}|\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},1\leq i\leq n-2;\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},|i-j|\geq2
$$</div>
<p data-bbox="124 575 339 590">Lemma 2.1. For the braid group $B_{n}$ set </p>
<div class="equation" data-bbox="209 595 398 610">$$
\tau=\sigma_{1}\sigma_{2}\dots\sigma_{n-1}\ a n d\ \sigma_{0}=\tau\sigma_{n-1}\tau^{-1}
$$</div>
<div class="equation" data-bbox="267 641 343 657">$$
\sigma_{i+1}=\tau\sigma_{i}\tau^{-1},
$$</div>
<div class="equation" data-bbox="248 690 361 701">$$
\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},
$$</div>
</body></html>
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0003047v1
| 2 | 612 | 792 | 1,275 | 1,650 |
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The full proof can be found in our thesis [5], Chapters", "type": "text"}], "index": 11}, {"bbox": [124, 307, 169, 321], "spans": [{"bbox": [124, 307, 169, 321], "score": 1.0, "content": "6 and 7.", "type": "text"}], "index": 12}], "index": 10.5}, {"type": "text", "bbox": [125, 320, 486, 444], "lines": [{"bbox": [137, 321, 487, 336], "spans": [{"bbox": [137, 321, 487, 336], "score": 1.0, "content": "The paper is organized as follows. In section 2 we introduce some", "type": "text"}], "index": 13}, {"bbox": [125, 335, 486, 349], "spans": [{"bbox": [125, 335, 486, 349], "score": 1.0, "content": "convenient notation that will be used throughout the rest of the paper.", "type": "text"}], "index": 14}, {"bbox": [124, 349, 487, 363], "spans": [{"bbox": [124, 349, 487, 363], "score": 1.0, "content": "In section 3 we define the friendship graph of the representation and", "type": "text"}], "index": 15}, {"bbox": [125, 364, 486, 377], "spans": [{"bbox": [125, 364, 486, 377], "score": 1.0, "content": "study its structure. We also study the case when the friendship graph", "type": "text"}], "index": 16}, {"bbox": [124, 376, 486, 392], "spans": [{"bbox": [124, 376, 412, 392], "score": 1.0, "content": "is totally disconnected. 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In section 5 we", "type": "text"}], "index": 19}, {"bbox": [126, 420, 485, 433], "spans": [{"bbox": [126, 420, 485, 433], "score": 1.0, "content": "determine all irreducible representations of corank 2 whose friendship", "type": "text"}], "index": 20}, {"bbox": [126, 434, 211, 446], "spans": [{"bbox": [126, 434, 211, 446], "score": 1.0, "content": "graph is a chain.", "type": "text"}], "index": 21}], "index": 17}, {"type": "text", "bbox": [125, 445, 486, 501], "lines": [{"bbox": [137, 446, 486, 461], "spans": [{"bbox": [137, 446, 486, 461], "score": 1.0, "content": "Acknowledgments: The author would like to express her deep", "type": "text"}], "index": 22}, {"bbox": [125, 461, 485, 474], "spans": [{"bbox": [125, 461, 485, 474], "score": 1.0, "content": "gratitude to professor Formanek for the numerous helpful discussions", "type": "text"}], "index": 23}, {"bbox": [126, 475, 485, 488], "spans": [{"bbox": [126, 475, 485, 488], "score": 1.0, "content": "and comments on the preliminary versions of this paper, and for gen-", "type": "text"}], "index": 24}, {"bbox": [126, 490, 327, 501], "spans": [{"bbox": [126, 490, 327, 501], "score": 1.0, "content": "erous financial support of this research.", "type": "text"}], "index": 25}], "index": 23.5}, {"type": "title", "bbox": [193, 511, 418, 524], "lines": [{"bbox": [193, 513, 418, 526], "spans": [{"bbox": [193, 513, 418, 526], "score": 1.0, "content": "2. 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It has a presentation", "type": "text"}], "index": 27}], "index": 27}, {"type": "interline_equation", "bbox": [125, 551, 527, 566], "lines": [{"bbox": [125, 551, 527, 566], "spans": [{"bbox": [125, 551, 527, 566], "score": 0.84, "content": "B_{n}=<\\sigma_{1},\\ldots,\\sigma_{n-1}|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},1\\leq i\\leq n-2;\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},|i-j|\\geq2", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [124, 575, 339, 590], "lines": [{"bbox": [125, 577, 339, 591], "spans": [{"bbox": [125, 577, 302, 591], "score": 1.0, "content": "Lemma 2.1. For the braid group ", "type": "text"}, {"bbox": [303, 578, 318, 590], "score": 0.89, "content": "B_{n}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [318, 577, 339, 591], "score": 1.0, "content": " set", "type": "text"}], "index": 29}], "index": 29}, {"type": "interline_equation", "bbox": [209, 595, 398, 610], "lines": [{"bbox": [209, 595, 398, 610], "spans": [{"bbox": [209, 595, 398, 610], "score": 0.84, "content": "\\tau=\\sigma_{1}\\sigma_{2}\\dots\\sigma_{n-1}\\ a n d\\ \\sigma_{0}=\\tau\\sigma_{n-1}\\tau^{-1}", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [267, 641, 343, 657], "lines": [{"bbox": [267, 641, 343, 657], "spans": [{"bbox": [267, 641, 343, 657], "score": 0.9, "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [248, 690, 361, 701], "lines": [{"bbox": [248, 690, 361, 701], "spans": [{"bbox": 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For every ", "type": "text"}, {"bbox": [264, 191, 293, 201], "score": 0.93, "content": "k\\geq3", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [294, 189, 315, 204], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [315, 194, 323, 200], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [323, 189, 486, 204], "score": 1.0, "content": " large enough there are no irre-", "type": "text"}], "index": 5}, {"bbox": [126, 203, 387, 217], "spans": [{"bbox": [126, 203, 306, 217], "score": 1.0, "content": "ducible complex representations of ", "type": "text"}, {"bbox": [306, 205, 321, 216], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [321, 203, 376, 217], "score": 1.0, "content": " of corank ", "type": "text"}, {"bbox": [376, 205, 383, 214], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [383, 203, 387, 217], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [126, 189, 486, 217]}, {"type": "text", "bbox": [125, 228, 485, 256], "lines": [{"bbox": [127, 231, 486, 244], "spans": [{"bbox": [127, 231, 263, 244], "score": 1.0, "content": "Conjecture 2. For every ", "type": "text"}, {"bbox": [264, 232, 293, 243], "score": 0.93, "content": "k\\geq1", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [294, 231, 315, 244], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [315, 235, 323, 241], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [323, 231, 486, 244], "score": 1.0, "content": " large enough there are no irre-", "type": "text"}], "index": 7}, {"bbox": [126, 245, 426, 258], "spans": [{"bbox": [126, 245, 306, 258], "score": 1.0, "content": "ducible complex representations of ", "type": "text"}, {"bbox": [307, 247, 321, 257], "score": 0.93, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [321, 245, 393, 258], "score": 1.0, "content": " of dimension ", "type": "text"}, {"bbox": [393, 246, 422, 256], "score": 0.94, "content": "n+k", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [422, 245, 426, 258], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [126, 231, 486, 258]}, {"type": "text", "bbox": [124, 263, 486, 319], "lines": [{"bbox": [138, 266, 485, 279], "spans": [{"bbox": [138, 266, 485, 279], "score": 1.0, "content": "We should also note that for the purpose of brevity we did not include", "type": "text"}], "index": 9}, {"bbox": [125, 280, 485, 293], "spans": [{"bbox": [125, 280, 485, 293], "score": 1.0, "content": "in this paper some of the details of the classification of representations", "type": "text"}], "index": 10}, {"bbox": [125, 293, 484, 307], "spans": [{"bbox": [125, 293, 138, 307], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [138, 295, 153, 306], "score": 0.95, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [153, 293, 202, 307], "score": 1.0, "content": " for small ", "type": "text"}, {"bbox": [203, 298, 209, 304], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [210, 293, 484, 307], "score": 1.0, "content": ". The full proof can be found in our thesis [5], Chapters", "type": "text"}], "index": 11}, {"bbox": [124, 307, 169, 321], "spans": [{"bbox": [124, 307, 169, 321], "score": 1.0, "content": "6 and 7.", "type": "text"}], "index": 12}], "index": 10.5, "bbox_fs": [124, 266, 485, 321]}, {"type": "text", "bbox": [125, 320, 486, 444], "lines": [{"bbox": [137, 321, 487, 336], "spans": [{"bbox": [137, 321, 487, 336], "score": 1.0, "content": "The paper is organized as follows. In section 2 we introduce some", "type": "text"}], "index": 13}, {"bbox": [125, 335, 486, 349], "spans": [{"bbox": [125, 335, 486, 349], "score": 1.0, "content": "convenient notation that will be used throughout the rest of the paper.", "type": "text"}], "index": 14}, {"bbox": [124, 349, 487, 363], "spans": [{"bbox": [124, 349, 487, 363], "score": 1.0, "content": "In section 3 we define the friendship graph of the representation and", "type": "text"}], "index": 15}, {"bbox": [125, 364, 486, 377], "spans": [{"bbox": [125, 364, 486, 377], "score": 1.0, "content": "study its structure. We also study the case when the friendship graph", "type": "text"}], "index": 16}, {"bbox": [124, 376, 486, 392], "spans": [{"bbox": [124, 376, 412, 392], "score": 1.0, "content": "is totally disconnected. In section 4 we prove that for ", "type": "text"}, {"bbox": [412, 379, 443, 389], "score": 0.92, "content": "n\\ge6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [444, 376, 486, 392], "score": 1.0, "content": " for any", "type": "text"}], "index": 17}, {"bbox": [125, 391, 486, 405], "spans": [{"bbox": [125, 391, 324, 405], "score": 1.0, "content": "irreducible complex representation of ", "type": "text"}, {"bbox": [325, 393, 339, 403], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [339, 391, 486, 405], "score": 1.0, "content": " of corank 2 and dimension", "type": "text"}], "index": 18}, {"bbox": [126, 405, 487, 419], "spans": [{"bbox": [126, 405, 168, 419], "score": 1.0, "content": "at least ", "type": "text"}, {"bbox": [169, 410, 176, 415], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [176, 405, 487, 419], "score": 1.0, "content": " the associated friendship graph is a chain. In section 5 we", "type": "text"}], "index": 19}, {"bbox": [126, 420, 485, 433], "spans": [{"bbox": [126, 420, 485, 433], "score": 1.0, "content": "determine all irreducible representations of corank 2 whose friendship", "type": "text"}], "index": 20}, {"bbox": [126, 434, 211, 446], "spans": [{"bbox": [126, 434, 211, 446], "score": 1.0, "content": "graph is a chain.", "type": "text"}], "index": 21}], "index": 17, "bbox_fs": [124, 321, 487, 446]}, {"type": "text", "bbox": [125, 445, 486, 501], "lines": [{"bbox": [137, 446, 486, 461], "spans": [{"bbox": [137, 446, 486, 461], "score": 1.0, "content": "Acknowledgments: The author would like to express her deep", "type": "text"}], "index": 22}, {"bbox": [125, 461, 485, 474], "spans": [{"bbox": [125, 461, 485, 474], "score": 1.0, "content": "gratitude to professor Formanek for the numerous helpful discussions", "type": "text"}], "index": 23}, {"bbox": [126, 475, 485, 488], "spans": [{"bbox": [126, 475, 485, 488], "score": 1.0, "content": "and comments on the preliminary versions of this paper, and for gen-", "type": "text"}], "index": 24}, {"bbox": [126, 490, 327, 501], "spans": [{"bbox": [126, 490, 327, 501], "score": 1.0, "content": "erous financial support of this research.", "type": "text"}], "index": 25}], "index": 23.5, "bbox_fs": [125, 446, 486, 501]}, {"type": "title", "bbox": [193, 511, 418, 524], "lines": [{"bbox": [193, 513, 418, 526], "spans": [{"bbox": [193, 513, 418, 526], "score": 1.0, "content": "2. Notation and preliminary results", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [135, 531, 453, 545], "lines": [{"bbox": [137, 533, 454, 548], "spans": [{"bbox": [137, 533, 159, 548], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [159, 535, 173, 546], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [174, 533, 293, 548], "score": 1.0, "content": " be the braid group on ", "type": "text"}, {"bbox": [293, 537, 301, 544], "score": 0.66, "content": "n", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [301, 533, 454, 548], "score": 1.0, "content": " strings. It has a presentation", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [137, 533, 454, 548]}, {"type": "interline_equation", "bbox": [125, 551, 527, 566], "lines": [{"bbox": [125, 551, 527, 566], "spans": [{"bbox": [125, 551, 527, 566], "score": 0.84, "content": "B_{n}=<\\sigma_{1},\\ldots,\\sigma_{n-1}|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},1\\leq i\\leq n-2;\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},|i-j|\\geq2", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [124, 575, 339, 590], "lines": [{"bbox": [125, 577, 339, 591], "spans": [{"bbox": [125, 577, 302, 591], "score": 1.0, "content": "Lemma 2.1. For the braid group ", "type": "text"}, {"bbox": [303, 578, 318, 590], "score": 0.89, "content": "B_{n}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [318, 577, 339, 591], "score": 1.0, "content": " set", "type": "text"}], "index": 29}], "index": 29, "bbox_fs": [125, 577, 339, 591]}, {"type": "interline_equation", "bbox": [209, 595, 398, 610], "lines": [{"bbox": [209, 595, 398, 610], "spans": [{"bbox": [209, 595, 398, 610], "score": 0.84, "content": "\\tau=\\sigma_{1}\\sigma_{2}\\dots\\sigma_{n-1}\\ a n d\\ \\sigma_{0}=\\tau\\sigma_{n-1}\\tau^{-1}", "type": "interline_equation"}], "index": 30}], "index": 30}, {"type": "interline_equation", "bbox": [267, 641, 343, 657], "lines": [{"bbox": [267, 641, 343, 657], "spans": [{"bbox": [267, 641, 343, 657], "score": 0.9, "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "type": "interline_equation"}], "index": 31}], "index": 31}, {"type": "interline_equation", "bbox": [248, 690, 361, 701], "lines": [{"bbox": [248, 690, 361, 701], "spans": [{"bbox": [248, 690, 361, 701], "score": 0.89, "content": "\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},", "type": "interline_equation"}], "index": 32}], "index": 32}]}
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[{"type": "text", "bbox": [125, 111, 486, 152], "content": "Another result, which will appear elsewhere is that for large enough there are no irreducible complex representations of of corank 3 and no irreducible complex representations of of dimension .", "index": 0}, {"type": "text", "bbox": [124, 153, 486, 180], "content": "Based on the above result we would like to make the following two conjectures.", "index": 1}, {"type": "text", "bbox": [125, 187, 485, 215], "content": "Conjecture 1. For every for large enough there are no irre- ducible complex representations of of corank .", "index": 2}, {"type": "text", "bbox": [125, 228, 485, 256], "content": "Conjecture 2. For every for large enough there are no irre- ducible complex representations of of dimension .", "index": 3}, {"type": "text", "bbox": [124, 263, 486, 319], "content": "We should also note that for the purpose of brevity we did not include in this paper some of the details of the classification of representations of for small . The full proof can be found in our thesis [5], Chapters 6 and 7.", "index": 4}, {"type": "text", "bbox": [125, 320, 486, 444], "content": "The paper is organized as follows. In section 2 we introduce some convenient notation that will be used throughout the rest of the paper. In section 3 we define the friendship graph of the representation and study its structure. We also study the case when the friendship graph is totally disconnected. In section 4 we prove that for for any irreducible complex representation of of corank 2 and dimension at least the associated friendship graph is a chain. In section 5 we determine all irreducible representations of corank 2 whose friendship graph is a chain.", "index": 5}, {"type": "text", "bbox": [125, 445, 486, 501], "content": "Acknowledgments: The author would like to express her deep gratitude to professor Formanek for the numerous helpful discussions and comments on the preliminary versions of this paper, and for gen- erous financial support of this research.", "index": 6}, {"type": "title", "bbox": [193, 511, 418, 524], "content": "2. Notation and preliminary results", "index": 7}, {"type": "text", "bbox": [135, 531, 453, 545], "content": "Let be the braid group on strings. It has a presentation", "index": 8}, {"type": "interline_equation", "bbox": [125, 551, 527, 566], "content": "", "index": 9}, {"type": "text", "bbox": [124, 575, 339, 590], "content": "Lemma 2.1. For the braid group set", "index": 10}, {"type": "interline_equation", "bbox": [209, 595, 398, 610], "content": "", "index": 11}, {"type": "interline_equation", "bbox": [267, 641, 343, 657], "content": "", "index": 12}, {"type": "interline_equation", "bbox": [248, 690, 361, 701], "content": "", "index": 13}]
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[{"bbox": [137, 113, 485, 127], "content": "Another result, which will appear elsewhere is that for large enough", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 486, 141], "content": "there are no irreducible complex representations of of corank 3 and", "parent_index": 0, "line_index": 1}, {"bbox": [126, 141, 460, 154], "content": "no irreducible complex representations of of dimension .", "parent_index": 0, "line_index": 2}, {"bbox": [137, 155, 486, 168], "content": "Based on the above result we would like to make the following two", "parent_index": 1, "line_index": 0}, {"bbox": [125, 169, 187, 182], "content": "conjectures.", "parent_index": 1, "line_index": 1}, {"bbox": [126, 189, 486, 204], "content": "Conjecture 1. For every for large enough there are no irre-", "parent_index": 2, "line_index": 0}, {"bbox": [126, 203, 387, 217], "content": "ducible complex representations of of corank .", "parent_index": 2, "line_index": 1}, {"bbox": [127, 231, 486, 244], "content": "Conjecture 2. For every for large enough there are no irre-", "parent_index": 3, "line_index": 0}, {"bbox": [126, 245, 426, 258], "content": "ducible complex representations of of dimension .", "parent_index": 3, "line_index": 1}, {"bbox": [138, 266, 485, 279], "content": "We should also note that for the purpose of brevity we did not include", "parent_index": 4, "line_index": 0}, {"bbox": [125, 280, 485, 293], "content": "in this paper some of the details of the classification of representations", "parent_index": 4, "line_index": 1}, {"bbox": [125, 293, 484, 307], "content": "of for small . The full proof can be found in our thesis [5], Chapters", "parent_index": 4, "line_index": 2}, {"bbox": [124, 307, 169, 321], "content": "6 and 7.", "parent_index": 4, "line_index": 3}, {"bbox": [137, 321, 487, 336], "content": "The paper is organized as follows. In section 2 we introduce some", "parent_index": 5, "line_index": 0}, {"bbox": [125, 335, 486, 349], "content": "convenient notation that will be used throughout the rest of the paper.", "parent_index": 5, "line_index": 1}, {"bbox": [124, 349, 487, 363], "content": "In section 3 we define the friendship graph of the representation and", "parent_index": 5, "line_index": 2}, {"bbox": [125, 364, 486, 377], "content": "study its structure. We also study the case when the friendship graph", "parent_index": 5, "line_index": 3}, {"bbox": [124, 376, 486, 392], "content": "is totally disconnected. In section 4 we prove that for for any", "parent_index": 5, "line_index": 4}, {"bbox": [125, 391, 486, 405], "content": "irreducible complex representation of of corank 2 and dimension", "parent_index": 5, "line_index": 5}, {"bbox": [126, 405, 487, 419], "content": "at least the associated friendship graph is a chain. In section 5 we", "parent_index": 5, "line_index": 6}, {"bbox": [126, 420, 485, 433], "content": "determine all irreducible representations of corank 2 whose friendship", "parent_index": 5, "line_index": 7}, {"bbox": [126, 434, 211, 446], "content": "graph is a chain.", "parent_index": 5, "line_index": 8}, {"bbox": [137, 446, 486, 461], "content": "Acknowledgments: The author would like to express her deep", "parent_index": 6, "line_index": 0}, {"bbox": [125, 461, 485, 474], "content": "gratitude to professor Formanek for the numerous helpful discussions", "parent_index": 6, "line_index": 1}, {"bbox": [126, 475, 485, 488], "content": "and comments on the preliminary versions of this paper, and for gen-", "parent_index": 6, "line_index": 2}, {"bbox": [126, 490, 327, 501], "content": "erous financial support of this research.", "parent_index": 6, "line_index": 3}, {"bbox": [193, 513, 418, 526], "content": "2. Notation and preliminary results", "parent_index": 7, "line_index": 0}, {"bbox": [137, 533, 454, 548], "content": "Let be the braid group on strings. It has a presentation", "parent_index": 8, "line_index": 0}, {"bbox": [125, 577, 339, 591], "content": "Lemma 2.1. For the braid group set", "parent_index": 10, "line_index": 0}]
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[]
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[{"bbox": [411, 118, 418, 123], "content": "n", "parent_index": 0, "subtype": "inline"}, {"bbox": [388, 128, 402, 139], "content": "B_{n}", "parent_index": 0, "subtype": "inline"}, {"bbox": [340, 142, 355, 153], "content": "B_{n}", "parent_index": 0, "subtype": "inline"}, {"bbox": [427, 143, 455, 152], "content": "n+1", "parent_index": 0, "subtype": "inline"}, {"bbox": [264, 191, 293, 201], "content": "k\\geq3", "parent_index": 2, "subtype": "inline"}, {"bbox": [315, 194, 323, 200], "content": "n", "parent_index": 2, "subtype": "inline"}, {"bbox": [306, 205, 321, 216], "content": "B_{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [376, 205, 383, 214], "content": "k", "parent_index": 2, "subtype": "inline"}, {"bbox": [264, 232, 293, 243], "content": "k\\geq1", "parent_index": 3, "subtype": "inline"}, {"bbox": [315, 235, 323, 241], "content": "n", "parent_index": 3, "subtype": "inline"}, {"bbox": [307, 247, 321, 257], "content": "B_{n}", "parent_index": 3, "subtype": "inline"}, {"bbox": [393, 246, 422, 256], "content": "n+k", "parent_index": 3, "subtype": "inline"}, {"bbox": [138, 295, 153, 306], "content": "B_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [203, 298, 209, 304], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [412, 379, 443, 389], "content": "n\\ge6", "parent_index": 5, "subtype": "inline"}, {"bbox": [325, 393, 339, 403], "content": "B_{n}", "parent_index": 5, "subtype": "inline"}, {"bbox": [169, 410, 176, 415], "content": "n", "parent_index": 5, "subtype": "inline"}, {"bbox": [159, 535, 173, 546], "content": "B_{n}", "parent_index": 8, "subtype": "inline"}, {"bbox": [293, 537, 301, 544], "content": "n", "parent_index": 8, "subtype": "inline"}, {"bbox": [125, 551, 527, 566], "content": "B_{n}=<\\sigma_{1},\\ldots,\\sigma_{n-1}|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},1\\leq i\\leq n-2;\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},|i-j|\\geq2", "parent_index": 9, "subtype": "interline"}, {"bbox": [303, 578, 318, 590], "content": "B_{n}", "parent_index": 10, "subtype": "inline"}, {"bbox": [209, 595, 398, 610], "content": "\\tau=\\sigma_{1}\\sigma_{2}\\dots\\sigma_{n-1}\\ a n d\\ \\sigma_{0}=\\tau\\sigma_{n-1}\\tau^{-1}", "parent_index": 11, "subtype": "interline"}, {"bbox": [267, 641, 343, 657], "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "parent_index": 12, "subtype": "interline"}, {"bbox": [248, 690, 361, 701], "content": "\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1},", "parent_index": 13, "subtype": "interline"}]
|
[]
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$$
\sigma_{i+1}=\tau\sigma_{i}\tau^{-1},
$$
and
$$
\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},|i-j|\geq2
$$
for all $i,j$ where indices are taken modulo $n$ .
Remark 2.2. Taking into account the above lemma, we also have the following presentation of $B_{n}$ :
$B_{n}=<\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}|\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1};\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},|i-j|\geq2;\sigma_{0}=\tau\sigma_{n-1}$ 1τ 1 >
for all $i,j$ where indices are taken modulo $n$ and $\tau$ is defined as above.
Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be a matrix representation of $B_{n}$ with
$$
\rho(\sigma_{i})=1+A_{i},
$$
and
$$
\rho(\tau)=T\in G L_{r}(\mathbb{C}).
$$
Then for any $i$ (indices are modulo $n$ ), the relation
$$
\tau\sigma_{i}\tau^{-1}=\sigma_{i+1}
$$
implies that
$$
T A_{i}T^{-1}=A_{i+1}.
$$
Hence all the $A_{i}$ are conjugate to each other, so they have the same rank, spectrum and Jordan normal form.
Lemma 2.3. For a representation $\rho$ of $B_{n}$ with
$$
\rho(\sigma_{i})=1+A_{i},
$$
we have:
1) $A_{i}A_{j}=A_{j}A_{i}$ , for $|i-j|\geq2$ ; $\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\end{array}$ for all $i=0,1,\dotsc,n-1$ , where indices are taken modulo $n$ .
Proof. This follows easily from the relations on the generators of $B_{n}$ .
# 3. The friendship graph.
In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of $B_{n}$ . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself.
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<html><body>
<div class="equation" data-bbox="268 112 342 126">$$
\sigma_{i+1}=\tau\sigma_{i}\tau^{-1},
$$</div>
<p data-bbox="125 128 148 142">and </p>
<div class="equation" data-bbox="247 146 365 160">$$
\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},|i-j|\geq2
$$</div>
<p data-bbox="124 162 354 176">for all $i,j$ where indices are taken modulo $n$ . </p>
<p data-bbox="124 190 487 219">Remark 2.2. Taking into account the above lemma, we also have the following presentation of $B_{n}$ : </p>
<p data-bbox="125 224 563 241">$B_{n}=<\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}|\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1};\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},|i-j|\geq2;\sigma_{0}=\tau\sigma_{n-1}$ 1τ 1 > </p>
<p data-bbox="125 245 486 259">for all $i,j$ where indices are taken modulo $n$ and $\tau$ is defined as above. </p>
<p data-bbox="135 280 448 295">Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be a matrix representation of $B_{n}$ with </p>
<div class="equation" data-bbox="267 304 343 317">$$
\rho(\sigma_{i})=1+A_{i},
$$</div>
<p data-bbox="125 321 147 335">and </p>
<div class="equation" data-bbox="254 339 357 353">$$
\rho(\tau)=T\in G L_{r}(\mathbb{C}).
$$</div>
<p data-bbox="125 354 389 369">Then for any $i$ (indices are modulo $n$ ), the relation </p>
<div class="equation" data-bbox="269 376 341 390">$$
\tau\sigma_{i}\tau^{-1}=\sigma_{i+1}
$$</div>
<p data-bbox="124 395 189 409">implies that </p>
<div class="equation" data-bbox="264 412 347 425">$$
T A_{i}T^{-1}=A_{i+1}.
$$</div>
<p data-bbox="125 428 487 456">Hence all the $A_{i}$ are conjugate to each other, so they have the same rank, spectrum and Jordan normal form. </p>
<p data-bbox="125 463 375 478">Lemma 2.3. For a representation $\rho$ of $B_{n}$ with </p>
<div class="equation" data-bbox="266 485 343 500">$$
\rho(\sigma_{i})=1+A_{i},
$$</div>
<p data-bbox="126 505 172 517">we have: </p>
<p data-bbox="136 518 447 560">1) $A_{i}A_{j}=A_{j}A_{i}$ , for $|i-j|\geq2$ ; $\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\end{array}$ for all $i=0,1,\dotsc,n-1$ , where indices are taken modulo $n$ . </p>
<p data-bbox="124 567 487 596">Proof. This follows easily from the relations on the generators of $B_{n}$ . </p>
<h1 data-bbox="229 609 381 623">3. The friendship graph. </h1>
<p data-bbox="124 630 487 699">In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of $B_{n}$ . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself. </p>
</body></html>
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0003047v1
| 3 | 612 | 792 | 1,275 | 1,650 |
[{"type": "equation", "text": "$$\n\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "and ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},|i-j|\\geq2\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "for all $i,j$ where indices are taken modulo $n$ . ", "page_idx": 3}, {"type": "text", "text": "Remark 2.2. Taking into account the above lemma, we also have the following presentation of $B_{n}$ : ", "page_idx": 3}, {"type": "text", "text": "$B_{n}=<\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1};\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},|i-j|\\geq2;\\sigma_{0}=\\tau\\sigma_{n-1}$ 1τ 1 > ", "page_idx": 3}, {"type": "text", "text": "for all $i,j$ where indices are taken modulo $n$ and $\\tau$ is defined as above. ", "page_idx": 3}, {"type": "text", "text": "Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be a matrix representation of $B_{n}$ with ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=1+A_{i},\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "and ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\rho(\\tau)=T\\in G L_{r}(\\mathbb{C}).\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "Then for any $i$ (indices are modulo $n$ ), the relation ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\tau\\sigma_{i}\\tau^{-1}=\\sigma_{i+1}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "implies that ", "page_idx": 3}, {"type": "equation", "text": "$$\nT A_{i}T^{-1}=A_{i+1}.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "Hence all the $A_{i}$ are conjugate to each other, so they have the same rank, spectrum and Jordan normal form. ", "page_idx": 3}, {"type": "text", "text": "Lemma 2.3. For a representation $\\rho$ of $B_{n}$ with ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=1+A_{i},\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "we have: ", "page_idx": 3}, {"type": "text", "text": "1) $A_{i}A_{j}=A_{j}A_{i}$ , for $|i-j|\\geq2$ ; $\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}$ for all $i=0,1,\\dotsc,n-1$ , where indices are taken modulo $n$ . ", "page_idx": 3}, {"type": "text", "text": "Proof. This follows easily from the relations on the generators of $B_{n}$ . ", "page_idx": 3}, {"type": "text", "text": "3. The friendship graph. ", "text_level": 1, "page_idx": 3}, {"type": "text", "text": "In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of $B_{n}$ . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself. ", "page_idx": 3}]
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For a representation ", "type": "text"}, {"bbox": [308, 469, 315, 479], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [316, 465, 332, 480], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [333, 468, 347, 478], "score": 0.89, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [347, 465, 373, 480], "score": 1.0, "content": " with", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [266, 485, 343, 500], "lines": [{"bbox": [266, 485, 343, 500], "spans": [{"bbox": [266, 485, 343, 500], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i},", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [126, 505, 172, 517], "lines": [{"bbox": [126, 506, 172, 519], "spans": [{"bbox": [126, 506, 172, 519], "score": 1.0, "content": "we have:", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [136, 518, 447, 560], "lines": [{"bbox": [139, 520, 302, 534], "spans": [{"bbox": [139, 521, 152, 533], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [153, 520, 219, 534], "score": 0.9, "content": "A_{i}A_{j}=A_{j}A_{i}", "type": "inline_equation", "height": 14, "width": 66}, {"bbox": [220, 521, 245, 533], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [245, 520, 299, 533], "score": 0.9, "content": "|i-j|\\geq2", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [299, 521, 302, 533], "score": 1.0, "content": ";", "type": "text"}], "index": 21}, {"bbox": [137, 534, 399, 548], "spans": [{"bbox": [137, 534, 399, 548], "score": 0.84, "content": "\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}", "type": "inline_equation", "height": 14, "width": 262}], "index": 22}, {"bbox": [137, 548, 445, 561], "spans": [{"bbox": [137, 549, 173, 561], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [173, 548, 264, 561], "score": 0.91, "content": "i=0,1,\\dotsc,n-1", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [264, 549, 434, 561], "score": 1.0, "content": ", where indices are taken modulo ", "type": "text"}, {"bbox": [434, 552, 442, 558], "score": 0.75, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [442, 549, 445, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22}, {"type": "text", "bbox": [124, 567, 487, 596], "lines": [{"bbox": [137, 568, 487, 583], "spans": [{"bbox": [137, 568, 487, 583], "score": 1.0, "content": "Proof. 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"score": 1.0, "content": "Hence all the ", "type": "text"}, {"bbox": [209, 432, 221, 442], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [222, 429, 486, 444], "score": 1.0, "content": " are conjugate to each other, so they have the same", "type": "text"}], "index": 16}, {"bbox": [126, 445, 337, 457], "spans": [{"bbox": [126, 445, 337, 457], "score": 1.0, "content": "rank, spectrum and Jordan normal form.", "type": "text"}], "index": 17}], "index": 16.5, "bbox_fs": [126, 429, 486, 457]}, {"type": "text", "bbox": [125, 463, 375, 478], "lines": [{"bbox": [125, 465, 373, 480], "spans": [{"bbox": [125, 465, 308, 480], "score": 1.0, "content": "Lemma 2.3. For a representation ", "type": "text"}, {"bbox": [308, 469, 315, 479], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [316, 465, 332, 480], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [333, 468, 347, 478], "score": 0.89, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [347, 465, 373, 480], "score": 1.0, "content": " with", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [125, 465, 373, 480]}, {"type": "interline_equation", "bbox": [266, 485, 343, 500], "lines": [{"bbox": [266, 485, 343, 500], "spans": [{"bbox": [266, 485, 343, 500], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i},", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [126, 505, 172, 517], "lines": [{"bbox": [126, 506, 172, 519], "spans": [{"bbox": [126, 506, 172, 519], "score": 1.0, "content": "we have:", "type": "text"}], "index": 20}], "index": 20, "bbox_fs": [126, 506, 172, 519]}, {"type": "text", "bbox": [136, 518, 447, 560], "lines": [{"bbox": [139, 520, 302, 534], "spans": [{"bbox": [139, 521, 152, 533], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [153, 520, 219, 534], "score": 0.9, "content": "A_{i}A_{j}=A_{j}A_{i}", "type": "inline_equation", "height": 14, "width": 66}, {"bbox": [220, 521, 245, 533], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [245, 520, 299, 533], "score": 0.9, "content": "|i-j|\\geq2", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [299, 521, 302, 533], "score": 1.0, "content": ";", "type": "text"}], "index": 21}, {"bbox": [137, 534, 399, 548], "spans": [{"bbox": [137, 534, 399, 548], "score": 0.84, "content": "\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}", "type": "inline_equation", "height": 14, "width": 262}], "index": 22}, {"bbox": [137, 548, 445, 561], "spans": [{"bbox": [137, 549, 173, 561], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [173, 548, 264, 561], "score": 0.91, "content": "i=0,1,\\dotsc,n-1", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [264, 549, 434, 561], "score": 1.0, "content": ", where indices are taken modulo ", "type": "text"}, {"bbox": [434, 552, 442, 558], "score": 0.75, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [442, 549, 445, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22, "bbox_fs": [137, 520, 445, 561]}, {"type": "text", "bbox": [124, 567, 487, 596], "lines": [{"bbox": [137, 568, 487, 583], "spans": [{"bbox": [137, 568, 487, 583], "score": 1.0, "content": "Proof. This follows easily from the relations on the generators of", "type": "text"}], "index": 24}, {"bbox": [126, 582, 146, 598], "spans": [{"bbox": [126, 585, 141, 596], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [141, 582, 146, 598], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [126, 568, 487, 598]}, {"type": "title", "bbox": [229, 609, 381, 623], "lines": [{"bbox": [230, 611, 380, 624], "spans": [{"bbox": [230, 611, 380, 624], "score": 1.0, "content": "3. The friendship graph.", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 630, 487, 699], "lines": [{"bbox": [137, 631, 486, 646], "spans": [{"bbox": [137, 631, 486, 646], "score": 1.0, "content": "In this section we define and prove some properties of the friendship", "type": "text"}], "index": 27}, {"bbox": [126, 646, 486, 660], "spans": [{"bbox": [126, 646, 466, 660], "score": 1.0, "content": "graph which is a finite graph associated with a representation of ", "type": "text"}, {"bbox": [467, 648, 482, 658], "score": 0.91, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [482, 646, 486, 660], "score": 1.0, "content": ".", "type": "text"}], "index": 28}, {"bbox": [126, 660, 487, 674], "spans": [{"bbox": [126, 660, 487, 674], "score": 1.0, "content": "Our graphs are simple-edged, which means that there is at most one", "type": "text"}], "index": 29}, {"bbox": [125, 675, 486, 688], "spans": [{"bbox": [125, 675, 486, 688], "score": 1.0, "content": "unoriented edge joining two vertices, and no edges joining a vertex to", "type": "text"}], "index": 30}, {"bbox": [124, 688, 155, 701], "spans": [{"bbox": [124, 688, 155, 701], "score": 1.0, "content": "itself.", "type": "text"}], "index": 31}], "index": 29, "bbox_fs": [124, 631, 487, 701]}]}
|
[{"type": "interline_equation", "bbox": [268, 112, 342, 126], "content": "", "index": 0}, {"type": "text", "bbox": [125, 128, 148, 142], "content": "and", "index": 1}, {"type": "interline_equation", "bbox": [247, 146, 365, 160], "content": "", "index": 2}, {"type": "text", "bbox": [124, 162, 354, 176], "content": "for all where indices are taken modulo .", "index": 3}, {"type": "text", "bbox": [124, 190, 487, 219], "content": "Remark 2.2. Taking into account the above lemma, we also have the following presentation of :", "index": 4}, {"type": "text", "bbox": [125, 224, 563, 241], "content": "1τ 1 >", "index": 5}, {"type": "text", "bbox": [125, 245, 486, 259], "content": "for all where indices are taken modulo and is defined as above.", "index": 6}, {"type": "text", "bbox": [135, 280, 448, 295], "content": "Let be a matrix representation of with", "index": 7}, {"type": "interline_equation", "bbox": [267, 304, 343, 317], "content": "", "index": 8}, {"type": "text", "bbox": [125, 321, 147, 335], "content": "and", "index": 9}, {"type": "interline_equation", "bbox": [254, 339, 357, 353], "content": "", "index": 10}, {"type": "text", "bbox": [125, 354, 389, 369], "content": "Then for any (indices are modulo ), the relation", "index": 11}, {"type": "interline_equation", "bbox": [269, 376, 341, 390], "content": "", "index": 12}, {"type": "text", "bbox": [124, 395, 189, 409], "content": "implies that", "index": 13}, {"type": "interline_equation", "bbox": [264, 412, 347, 425], "content": "", "index": 14}, {"type": "text", "bbox": [125, 428, 487, 456], "content": "Hence all the are conjugate to each other, so they have the same rank, spectrum and Jordan normal form.", "index": 15}, {"type": "text", "bbox": [125, 463, 375, 478], "content": "Lemma 2.3. For a representation of with", "index": 16}, {"type": "interline_equation", "bbox": [266, 485, 343, 500], "content": "", "index": 17}, {"type": "text", "bbox": [126, 505, 172, 517], "content": "we have:", "index": 18}, {"type": "text", "bbox": [136, 518, 447, 560], "content": "1) , for ; for all , where indices are taken modulo .", "index": 19}, {"type": "text", "bbox": [124, 567, 487, 596], "content": "Proof. This follows easily from the relations on the generators of .", "index": 20}, {"type": "title", "bbox": [229, 609, 381, 623], "content": "3. The friendship graph.", "index": 21}, {"type": "text", "bbox": [124, 630, 487, 699], "content": "In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself.", "index": 22}]
|
[{"bbox": [124, 132, 148, 142], "content": "and", "parent_index": 1, "line_index": 0}, {"bbox": [125, 164, 355, 177], "content": "for all where indices are taken modulo .", "parent_index": 3, "line_index": 0}, {"bbox": [125, 192, 486, 207], "content": "Remark 2.2. Taking into account the above lemma, we also have the", "parent_index": 4, "line_index": 0}, {"bbox": [125, 207, 278, 221], "content": "following presentation of :", "parent_index": 4, "line_index": 1}, {"bbox": [125, 226, 561, 244], "content": "1τ 1 >", "parent_index": 5, "line_index": 0}, {"bbox": [126, 248, 485, 260], "content": "for all where indices are taken modulo and is defined as above.", "parent_index": 6, "line_index": 0}, {"bbox": [137, 283, 447, 298], "content": "Let be a matrix representation of with", "parent_index": 7, "line_index": 0}, {"bbox": [124, 325, 146, 335], "content": "and", "parent_index": 9, "line_index": 0}, {"bbox": [127, 357, 387, 370], "content": "Then for any (indices are modulo ), the relation", "parent_index": 11, "line_index": 0}, {"bbox": [125, 396, 189, 411], "content": "implies that", "parent_index": 13, "line_index": 0}, {"bbox": [137, 429, 486, 444], "content": "Hence all the are conjugate to each other, so they have the same", "parent_index": 15, "line_index": 0}, {"bbox": [126, 445, 337, 457], "content": "rank, spectrum and Jordan normal form.", "parent_index": 15, "line_index": 1}, {"bbox": [125, 465, 373, 480], "content": "Lemma 2.3. For a representation of with", "parent_index": 16, "line_index": 0}, {"bbox": [126, 506, 172, 519], "content": "we have:", "parent_index": 18, "line_index": 0}, {"bbox": [139, 520, 302, 534], "content": "1) , for ;", "parent_index": 19, "line_index": 0}, {"bbox": [137, 534, 399, 548], "content": "", "parent_index": 19, "line_index": 1}, {"bbox": [137, 548, 445, 561], "content": "for all , where indices are taken modulo .", "parent_index": 19, "line_index": 2}, {"bbox": [137, 568, 487, 583], "content": "Proof. This follows easily from the relations on the generators of", "parent_index": 20, "line_index": 0}, {"bbox": [126, 582, 146, 598], "content": ".", "parent_index": 20, "line_index": 1}, {"bbox": [230, 611, 380, 624], "content": "3. The friendship graph.", "parent_index": 21, "line_index": 0}, {"bbox": [137, 631, 486, 646], "content": "In this section we define and prove some properties of the friendship", "parent_index": 22, "line_index": 0}, {"bbox": [126, 646, 486, 660], "content": "graph which is a finite graph associated with a representation of .", "parent_index": 22, "line_index": 1}, {"bbox": [126, 660, 487, 674], "content": "Our graphs are simple-edged, which means that there is at most one", "parent_index": 22, "line_index": 2}, {"bbox": [125, 675, 486, 688], "content": "unoriented edge joining two vertices, and no edges joining a vertex to", "parent_index": 22, "line_index": 3}, {"bbox": [124, 688, 155, 701], "content": "itself.", "parent_index": 22, "line_index": 4}]
|
[]
|
[{"bbox": [268, 112, 342, 126], "content": "\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},", "parent_index": 0, "subtype": "interline"}, {"bbox": [247, 146, 365, 160], "content": "\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},|i-j|\\geq2", "parent_index": 2, "subtype": "interline"}, {"bbox": [162, 166, 176, 177], "content": "i,j", "parent_index": 3, "subtype": "inline"}, {"bbox": [344, 169, 351, 174], "content": "n", "parent_index": 3, "subtype": "inline"}, {"bbox": [255, 209, 270, 219], "content": "B_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [125, 227, 527, 241], "content": "B_{n}=<\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}|\\sigma_{i}\\sigma_{i+1}\\sigma_{i}=\\sigma_{i+1}\\sigma_{i}\\sigma_{i+1};\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},|i-j|\\geq2;\\sigma_{0}=\\tau\\sigma_{n-1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [160, 249, 174, 260], "content": "i,j", "parent_index": 6, "subtype": "inline"}, {"bbox": [343, 252, 350, 258], "content": "n", "parent_index": 6, "subtype": "inline"}, {"bbox": [376, 252, 383, 258], "content": "\\tau", "parent_index": 6, "subtype": "inline"}, {"bbox": [159, 285, 247, 297], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 7, "subtype": "inline"}, {"bbox": [405, 285, 419, 295], "content": "B_{n}", "parent_index": 7, "subtype": "inline"}, {"bbox": [267, 304, 343, 317], "content": "\\rho(\\sigma_{i})=1+A_{i},", "parent_index": 8, "subtype": "interline"}, {"bbox": [254, 339, 357, 353], "content": "\\rho(\\tau)=T\\in G L_{r}(\\mathbb{C}).", "parent_index": 10, "subtype": "interline"}, {"bbox": [196, 359, 200, 367], "content": "i", "parent_index": 11, "subtype": "inline"}, {"bbox": [309, 362, 316, 367], "content": "n", "parent_index": 11, "subtype": "inline"}, {"bbox": [269, 376, 341, 390], "content": "\\tau\\sigma_{i}\\tau^{-1}=\\sigma_{i+1}", "parent_index": 12, "subtype": "interline"}, {"bbox": [264, 412, 347, 425], "content": "T A_{i}T^{-1}=A_{i+1}.", "parent_index": 14, "subtype": "interline"}, {"bbox": [209, 432, 221, 442], "content": "A_{i}", "parent_index": 15, "subtype": "inline"}, {"bbox": [308, 469, 315, 479], "content": "\\rho", "parent_index": 16, "subtype": "inline"}, {"bbox": [333, 468, 347, 478], "content": "B_{n}", "parent_index": 16, "subtype": "inline"}, {"bbox": [266, 485, 343, 500], "content": "\\rho(\\sigma_{i})=1+A_{i},", "parent_index": 17, "subtype": "interline"}, {"bbox": [153, 520, 219, 534], "content": "A_{i}A_{j}=A_{j}A_{i}", "parent_index": 19, "subtype": "inline"}, {"bbox": [245, 520, 299, 533], "content": "|i-j|\\geq2", "parent_index": 19, "subtype": "inline"}, {"bbox": [137, 534, 399, 548], "content": "\\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\\end{array}", "parent_index": 19, "subtype": "inline"}, {"bbox": [173, 548, 264, 561], "content": "i=0,1,\\dotsc,n-1", "parent_index": 19, "subtype": "inline"}, {"bbox": [434, 552, 442, 558], "content": "n", "parent_index": 19, "subtype": "inline"}, {"bbox": [126, 585, 141, 596], "content": "B_{n}", "parent_index": 20, "subtype": "inline"}, {"bbox": [467, 648, 482, 658], "content": "B_{n}", "parent_index": 22, "subtype": "inline"}]
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[]
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We assume throughout this section that we have a representation
$$
\rho:B_{n}\to G L_{r}(\mathbb{C}),
$$
with
$$
\rho(\sigma_{i})=1+A_{i},\;\;(i=0,1,\ldots,n-1).
$$
Definition 3.1. 1) $A_{i}$ , $A_{i+1}$ are neighbors (indices modulo $n$ ). 2) $A_{i}$ , $A_{j}$ are friends if
$$
I m(A_{i})\cap I m(A_{j})\neq\{0\}.
$$
3) $A_{i}$ , $A_{j}$ are true friends if either
$(a)\ A_{i}$ and $A_{j}$ are not neighbors, and
$$
A_{i}A_{j}=A_{j}A_{i}\neq0;
$$
or
(b) $A_{i}$ and $A_{j}$ are neighbors, and
$$
A_{i}+A_{i}^{2}+A_{i}A_{j}A_{i}=A_{j}+A_{j}^{2}+A_{j}A_{i}A_{j}\neq0.
$$
Lemma 3.1. If $A,B$ are true friends, then they are friends.
Proof. 1) If $A$ and $B$ are not neighbors, then $A B=B A\neq0$ , so,
$$
I m(A)\cap I m(B)\supseteq I m(A B)\cap I m(B A)=I m(A B)\neq\{0\}.
$$
2) If $A$ and $B$ are neighbors, then
$A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\neq0,$ and again
$$
I m(A)\cap I m(B)\supseteq I m(A+A^{2}+A B A)\neq\{0\}.
$$
Definition 3.2. The full friendship graph (associated with the representation $\rho:B_{n}\to G L_{n}(\mathbb{C})$ ) is the simple-edged graph with n vertices $A_{0},A_{1},\ldots,A_{n-1}$ and an edge joining $A_{i}$ and $A_{j}$ $\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}j,$ ) if and only if $A_{i}$ and $A_{j}$ are friends.
The friendship graph is the subgraph with vertices $A_{1},\dotsc,A_{n-1}$ obtained from the full friendship graph by deleting $A_{0}$ and all edges incident to it.
Our main interest is the friendship graph, but it is convenient to introduce the full friendship graph as a tool, because of the following lemma.
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<html><body>
<p data-bbox="137 110 475 125">We assume throughout this section that we have a representation </p>
<div class="equation" data-bbox="259 134 350 147">$$
\rho:B_{n}\to G L_{r}(\mathbb{C}),
$$</div>
<p data-bbox="125 153 150 165">with </p>
<div class="equation" data-bbox="210 171 400 185">$$
\rho(\sigma_{i})=1+A_{i},\;\;(i=0,1,\ldots,n-1).
$$</div>
<p data-bbox="126 195 461 224">Definition 3.1. 1) $A_{i}$ , $A_{i+1}$ are neighbors (indices modulo $n$ ). 2) $A_{i}$ , $A_{j}$ are friends if </p>
<div class="equation" data-bbox="241 231 369 246">$$
I m(A_{i})\cap I m(A_{j})\neq\{0\}.
$$</div>
<p data-bbox="137 251 329 265">3) $A_{i}$ , $A_{j}$ are true friends if either </p>
<p data-bbox="137 266 333 280">$(a)\ A_{i}$ and $A_{j}$ are not neighbors, and </p>
<div class="equation" data-bbox="259 287 351 302">$$
A_{i}A_{j}=A_{j}A_{i}\neq0;
$$</div>
<p data-bbox="126 310 139 320">or </p>
<p data-bbox="136 321 311 336">(b) $A_{i}$ and $A_{j}$ are neighbors, and </p>
<div class="equation" data-bbox="191 351 419 369">$$
A_{i}+A_{i}^{2}+A_{i}A_{j}A_{i}=A_{j}+A_{j}^{2}+A_{j}A_{i}A_{j}\neq0.
$$</div>
<p data-bbox="124 381 437 396">Lemma 3.1. If $A,B$ are true friends, then they are friends. </p>
<p data-bbox="136 403 473 418">Proof. 1) If $A$ and $B$ are not neighbors, then $A B=B A\neq0$ , so, </p>
<div class="equation" data-bbox="157 425 452 441">$$
I m(A)\cap I m(B)\supseteq I m(A B)\cap I m(B A)=I m(A B)\neq\{0\}.
$$</div>
<p data-bbox="136 445 313 460">2) If $A$ and $B$ are neighbors, then </p>
<p data-bbox="124 465 486 502">$A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\neq0,$ and again </p>
<div class="equation" data-bbox="185 509 425 525">$$
I m(A)\cap I m(B)\supseteq I m(A+A^{2}+A B A)\neq\{0\}.
$$</div>
<p data-bbox="124 551 488 608">Definition 3.2. The full friendship graph (associated with the representation $\rho:B_{n}\to G L_{n}(\mathbb{C})$ ) is the simple-edged graph with n vertices $A_{0},A_{1},\ldots,A_{n-1}$ and an edge joining $A_{i}$ and $A_{j}$ $\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}j,$ ) if and only if $A_{i}$ and $A_{j}$ are friends. </p>
<p data-bbox="125 608 487 650">The friendship graph is the subgraph with vertices $A_{1},\dotsc,A_{n-1}$ obtained from the full friendship graph by deleting $A_{0}$ and all edges incident to it. </p>
<p data-bbox="124 657 487 700">Our main interest is the friendship graph, but it is convenient to introduce the full friendship graph as a tool, because of the following lemma. </p>
</body></html>
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0003047v1
| 4 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "We assume throughout this section that we have a representation ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "with ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=1+A_{i},\\;\\;(i=0,1,\\ldots,n-1).\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Definition 3.1. 1) $A_{i}$ , $A_{i+1}$ are neighbors (indices modulo $n$ ). 2) $A_{i}$ , $A_{j}$ are friends if ", "page_idx": 4}, {"type": "equation", "text": "$$\nI m(A_{i})\\cap I m(A_{j})\\neq\\{0\\}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "3) $A_{i}$ , $A_{j}$ are true friends if either ", "page_idx": 4}, {"type": "text", "text": "$(a)\\ A_{i}$ and $A_{j}$ are not neighbors, and ", "page_idx": 4}, {"type": "equation", "text": "$$\nA_{i}A_{j}=A_{j}A_{i}\\neq0;\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "or ", "page_idx": 4}, {"type": "text", "text": "(b) $A_{i}$ and $A_{j}$ are neighbors, and ", "page_idx": 4}, {"type": "equation", "text": "$$\nA_{i}+A_{i}^{2}+A_{i}A_{j}A_{i}=A_{j}+A_{j}^{2}+A_{j}A_{i}A_{j}\\neq0.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Lemma 3.1. If $A,B$ are true friends, then they are friends. ", "page_idx": 4}, {"type": "text", "text": "Proof. 1) If $A$ and $B$ are not neighbors, then $A B=B A\\neq0$ , so, ", "page_idx": 4}, {"type": "equation", "text": "$$\nI m(A)\\cap I m(B)\\supseteq I m(A B)\\cap I m(B A)=I m(A B)\\neq\\{0\\}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "2) If $A$ and $B$ are neighbors, then ", "page_idx": 4}, {"type": "text", "text": "$A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\\neq0,$ and again ", "page_idx": 4}, {"type": "equation", "text": "$$\nI m(A)\\cap I m(B)\\supseteq I m(A+A^{2}+A B A)\\neq\\{0\\}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Definition 3.2. The full friendship graph (associated with the representation $\\rho:B_{n}\\to G L_{n}(\\mathbb{C})$ ) is the simple-edged graph with n vertices $A_{0},A_{1},\\ldots,A_{n-1}$ and an edge joining $A_{i}$ and $A_{j}$ $\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}j,$ ) if and only if $A_{i}$ and $A_{j}$ are friends. ", "page_idx": 4}, {"type": "text", "text": "The friendship graph is the subgraph with vertices $A_{1},\\dotsc,A_{n-1}$ obtained from the full friendship graph by deleting $A_{0}$ and all edges incident to it. ", "page_idx": 4}, {"type": "text", "text": "Our main interest is the friendship graph, but it is convenient to introduce the full friendship graph as a tool, because of the following lemma. ", "page_idx": 4}]
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[]
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[{"bbox": [259, 134, 350, 147], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "parent_index": 1, "subtype": "interline"}, {"bbox": [210, 171, 400, 185], "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;(i=0,1,\\ldots,n-1).", "parent_index": 3, "subtype": "interline"}, {"bbox": [230, 198, 243, 210], "content": "A_{i}", "parent_index": 4, "subtype": "inline"}, {"bbox": [251, 198, 275, 210], "content": "A_{i+1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [444, 203, 451, 208], "content": "n", "parent_index": 4, "subtype": "inline"}, {"bbox": [153, 211, 165, 224], "content": "A_{i}", "parent_index": 4, "subtype": "inline"}, {"bbox": [173, 211, 188, 225], "content": "A_{j}", "parent_index": 4, "subtype": "inline"}, {"bbox": [241, 231, 369, 246], "content": "I m(A_{i})\\cap I m(A_{j})\\neq\\{0\\}.", "parent_index": 5, "subtype": "interline"}, {"bbox": [152, 253, 165, 266], "content": "A_{i}", "parent_index": 6, "subtype": "inline"}, {"bbox": [173, 253, 188, 267], "content": "A_{j}", "parent_index": 6, "subtype": "inline"}, {"bbox": [138, 267, 171, 281], "content": "(a)\\ A_{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [196, 267, 210, 281], "content": "A_{j}", "parent_index": 7, "subtype": "inline"}, {"bbox": [259, 287, 351, 302], "content": "A_{i}A_{j}=A_{j}A_{i}\\neq0;", "parent_index": 8, "subtype": "interline"}, {"bbox": [156, 323, 170, 336], "content": "A_{i}", "parent_index": 10, "subtype": "inline"}, {"bbox": [195, 323, 210, 337], "content": "A_{j}", "parent_index": 10, "subtype": "inline"}, {"bbox": [191, 351, 419, 369], "content": "A_{i}+A_{i}^{2}+A_{i}A_{j}A_{i}=A_{j}+A_{j}^{2}+A_{j}A_{i}A_{j}\\neq0.", "parent_index": 11, "subtype": "interline"}, {"bbox": [212, 383, 237, 396], "content": "A,B", "parent_index": 12, "subtype": "inline"}, {"bbox": [206, 406, 216, 416], "content": "A", "parent_index": 13, "subtype": "inline"}, {"bbox": [241, 406, 252, 416], "content": "B", "parent_index": 13, "subtype": "inline"}, {"bbox": [378, 406, 452, 419], "content": "A B=B A\\neq0", "parent_index": 13, "subtype": "inline"}, {"bbox": [157, 425, 452, 441], "content": "I m(A)\\cap I m(B)\\supseteq I m(A B)\\cap I m(B A)=I m(A B)\\neq\\{0\\}.", "parent_index": 14, "subtype": "interline"}, {"bbox": [164, 450, 173, 459], "content": "A", "parent_index": 15, "subtype": "inline"}, {"bbox": [199, 450, 209, 459], "content": "B", "parent_index": 15, "subtype": "inline"}, {"bbox": [124, 466, 484, 483], "content": "A(1+A+B A)=A+A^{2}+A B A=B+B^{2}+B A B=B(1+B+A B)\\neq0,", "parent_index": 16, "subtype": "inline"}, {"bbox": [185, 509, 425, 525], "content": "I m(A)\\cap I m(B)\\supseteq I m(A+A^{2}+A B A)\\neq\\{0\\}.", "parent_index": 17, "subtype": "interline"}, {"bbox": [185, 568, 276, 581], "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "parent_index": 18, "subtype": "inline"}, {"bbox": [126, 582, 209, 595], "content": "A_{0},A_{1},\\ldots,A_{n-1}", "parent_index": 18, "subtype": "inline"}, {"bbox": [319, 582, 333, 594], "content": "A_{i}", "parent_index": 18, "subtype": "inline"}, {"bbox": [359, 582, 373, 596], "content": "A_{j}", "parent_index": 18, "subtype": "inline"}, {"bbox": [381, 581, 411, 595], "content": "\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}j,", "parent_index": 18, "subtype": "inline"}, {"bbox": [126, 597, 138, 608], "content": "A_{i}", "parent_index": 18, "subtype": "inline"}, {"bbox": [165, 597, 178, 610], "content": "A_{j}", "parent_index": 18, "subtype": "inline"}, {"bbox": [419, 611, 484, 623], "content": "A_{1},\\dotsc,A_{n-1}", "parent_index": 19, "subtype": "inline"}, {"bbox": [396, 625, 411, 636], "content": "A_{0}", "parent_index": 19, "subtype": "inline"}]
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[]
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Lemma 3.2. There is an edge between $A_{i}$ and $A_{j}$ in the full friendship graph if and only if there is an edge between $A_{i+k}$ and $A_{j+k}$ where indices are taken modulo $n$ . In other words, $\mathbb{Z}_{n}$ acts on the full friendship graph by permuting the vertices cyclically.
Proof. This follows immediately from the fact that conjugation by $T=\rho(\tau)=\rho(\sigma_{1}\dots\sigma_{n-1})$ permutes $\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}$ cyclically (Lemma 2.1).
Lemma 3.3 (Lemma about friends). Let $A$ and $B$ be neighbors which are not friends. If $C$ is not a neighbor of $A$ and $C$ is a friend of $B$ then $C$ is a true friend of $A$ .
Proof. By lemma 3.1, $A$ and $B$ are true not friends, because they are not friends, that is
$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$
Consider $y\in V$ such that $C y\;\in\;I m(B),C y\;=\;B z\;\neq\;0$ ( $y$ exists because $C$ and $B$ are friends). Then
$$
B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\neq0
$$
because $B z\neq0$ and $(1+B)$ is invertible.
So, $A C=C A\neq0$ ; that is, $A$ and $C$ are true friends.
Theorem 3.4. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be a representation. Then one of the following holds.
(a) The full friendship graph is totally disconnected (no friends at all).
(b) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ .
(c) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors.
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<html><body>
<p data-bbox="124 110 487 167">Lemma 3.2. There is an edge between $A_{i}$ and $A_{j}$ in the full friendship graph if and only if there is an edge between $A_{i+k}$ and $A_{j+k}$ where indices are taken modulo $n$ . In other words, $\mathbb{Z}_{n}$ acts on the full friendship graph by permuting the vertices cyclically. </p>
<p data-bbox="124 174 486 217">Proof. This follows immediately from the fact that conjugation by $T=\rho(\tau)=\rho(\sigma_{1}\dots\sigma_{n-1})$ permutes $\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}$ cyclically (Lemma 2.1). </p>
<p data-bbox="124 225 487 268">Lemma 3.3 (Lemma about friends). Let $A$ and $B$ be neighbors which are not friends. If $C$ is not a neighbor of $A$ and $C$ is a friend of $B$ then $C$ is a true friend of $A$ . </p>
<div class="image" data-bbox="125 307 300 371"><img data-bbox="125 307 300 371"/></div>
<p data-bbox="124 422 486 450">Proof. By lemma 3.1, $A$ and $B$ are true not friends, because they are not friends, that is </p>
<div class="equation" data-bbox="205 460 405 472">$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$</div>
<p data-bbox="124 479 486 508">Consider $y\in V$ such that $C y\;\in\;I m(B),C y\;=\;B z\;\neq\;0$ ( $y$ exists because $C$ and $B$ are friends). Then </p>
<div class="equation" data-bbox="174 517 437 531">$$
B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\neq0
$$</div>
<p data-bbox="124 537 339 551">because $B z\neq0$ and $(1+B)$ is invertible. </p>
<p data-bbox="136 552 408 566">So, $A C=C A\neq0$ ; that is, $A$ and $C$ are true friends. </p>
<p data-bbox="124 587 487 615">Theorem 3.4. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be a representation. Then one of the following holds. </p>
<p data-bbox="124 617 487 643">(a) The full friendship graph is totally disconnected (no friends at all). </p>
<p data-bbox="125 644 488 671">(b) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ . </p>
<p data-bbox="126 672 487 700">(c) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors. </p>
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0003047v1
| 5 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Lemma 3.2. There is an edge between $A_{i}$ and $A_{j}$ in the full friendship graph if and only if there is an edge between $A_{i+k}$ and $A_{j+k}$ where indices are taken modulo $n$ . In other words, $\\mathbb{Z}_{n}$ acts on the full friendship graph by permuting the vertices cyclically. ", "page_idx": 5}, {"type": "text", "text": "Proof. This follows immediately from the fact that conjugation by $T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})$ permutes $\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}$ cyclically (Lemma 2.1). ", "page_idx": 5}, {"type": "text", "text": "Lemma 3.3 (Lemma about friends). Let $A$ and $B$ be neighbors which are not friends. If $C$ is not a neighbor of $A$ and $C$ is a friend of $B$ then $C$ is a true friend of $A$ . ", "page_idx": 5}, {"type": "image", "img_path": "images/7782d0d127eba9149d6c475dbb5cc3b32b3a3f69a44ce9d8ed81904f6b0cbd1a.jpg", "img_caption": [], "img_footnote": [], "page_idx": 5}, {"type": "text", "text": "Proof. By lemma 3.1, $A$ and $B$ are true not friends, because they are not friends, that is ", "page_idx": 5}, {"type": "equation", "text": "$$\nA+A^{2}+A B A=B+B^{2}+B A B=0.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Consider $y\\in V$ such that $C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0$ ( $y$ exists because $C$ and $B$ are friends). Then ", "page_idx": 5}, {"type": "equation", "text": "$$\nB A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "because $B z\\neq0$ and $(1+B)$ is invertible. ", "page_idx": 5}, {"type": "text", "text": "So, $A C=C A\\neq0$ ; that is, $A$ and $C$ are true friends. ", "page_idx": 5}, {"type": "text", "text": "Theorem 3.4. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be a representation. Then one of the following holds. ", "page_idx": 5}, {"type": "text", "text": "(a) The full friendship graph is totally disconnected (no friends at all). ", "page_idx": 5}, {"type": "text", "text": "(b) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ . ", "page_idx": 5}, {"type": "text", "text": "(c) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors. ", "page_idx": 5}]
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There is an edge between ", "type": "text"}, {"bbox": [328, 115, 340, 125], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [340, 113, 365, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [366, 115, 379, 127], "score": 0.9, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [379, 113, 486, 127], "score": 1.0, "content": " in the full friendship", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 141], "spans": [{"bbox": [125, 127, 357, 141], "score": 1.0, "content": "graph if and only if there is an edge between ", "type": "text"}, {"bbox": [357, 128, 381, 140], "score": 0.92, "content": "A_{i+k}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [382, 127, 408, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [408, 128, 433, 140], "score": 0.91, "content": "A_{j+k}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [433, 127, 486, 141], "score": 1.0, "content": " where in-", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 243, 155], "score": 1.0, "content": "dices are taken modulo ", "type": "text"}, {"bbox": [243, 146, 250, 151], "score": 0.75, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [251, 141, 339, 155], "score": 1.0, "content": ". In other words, ", "type": "text"}, {"bbox": [339, 142, 353, 153], "score": 0.9, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [353, 141, 486, 155], "score": 1.0, "content": " acts on the full friendship", "type": "text"}], "index": 2}, {"bbox": [126, 155, 339, 169], "spans": [{"bbox": [126, 155, 339, 169], "score": 1.0, "content": "graph by permuting the vertices cyclically.", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "text", "bbox": [124, 174, 486, 217], "lines": [{"bbox": [137, 177, 484, 191], "spans": [{"bbox": [137, 177, 484, 191], "score": 1.0, "content": "Proof. This follows immediately from the fact that conjugation by", "type": "text"}], "index": 4}, {"bbox": [126, 189, 487, 208], "spans": [{"bbox": [126, 192, 255, 205], "score": 0.94, "content": "T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})", "type": "inline_equation", "height": 13, "width": 129}, {"bbox": [256, 189, 309, 208], "score": 1.0, "content": " permutes ", "type": "text"}, {"bbox": [310, 194, 388, 204], "score": 0.76, "content": "\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 10, "width": 78}, {"bbox": [388, 189, 487, 208], "score": 1.0, "content": " cyclically (Lemma", "type": "text"}], "index": 5}, {"bbox": [125, 204, 150, 220], "spans": [{"bbox": [125, 204, 150, 220], "score": 1.0, "content": "2.1).", "type": "text"}], "index": 6}], "index": 5}, {"type": "text", "bbox": [124, 225, 487, 268], "lines": [{"bbox": [125, 227, 486, 242], "spans": [{"bbox": [125, 227, 343, 242], "score": 1.0, "content": "Lemma 3.3 (Lemma about friends). Let ", "type": "text"}, {"bbox": [343, 228, 352, 239], "score": 0.54, "content": "A", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [353, 227, 377, 242], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [378, 229, 388, 239], "score": 0.6, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [388, 227, 486, 242], "score": 1.0, "content": " be neighbors which", "type": "text"}], "index": 7}, {"bbox": [126, 242, 485, 256], "spans": [{"bbox": [126, 242, 227, 256], "score": 1.0, "content": "are not friends. If ", "type": "text"}, {"bbox": [227, 244, 237, 253], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [237, 242, 349, 256], "score": 1.0, "content": " is not a neighbor of ", "type": "text"}, {"bbox": [349, 242, 359, 253], "score": 0.73, "content": "A", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [360, 242, 386, 256], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [387, 242, 397, 253], "score": 0.74, "content": "C", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [397, 242, 474, 256], "score": 1.0, "content": " is a friend of ", "type": "text"}, {"bbox": [475, 244, 485, 253], "score": 0.85, "content": "B", "type": "inline_equation", "height": 9, "width": 10}], "index": 8}, {"bbox": [126, 255, 272, 270], "spans": [{"bbox": [126, 255, 151, 270], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [152, 257, 161, 266], "score": 0.82, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [162, 255, 261, 270], "score": 1.0, "content": " is a true friend of ", "type": "text"}, {"bbox": [261, 257, 270, 266], "score": 0.84, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [270, 255, 272, 270], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8}, {"type": "image", "bbox": [125, 307, 300, 371], "blocks": [{"type": "image_body", "bbox": [125, 307, 300, 371], "group_id": 0, "lines": [{"bbox": [125, 307, 300, 371], "spans": [{"bbox": [125, 307, 300, 371], "score": 0.776, "type": "image", "image_path": "7782d0d127eba9149d6c475dbb5cc3b32b3a3f69a44ce9d8ed81904f6b0cbd1a.jpg"}]}], "index": 12, "virtual_lines": [{"bbox": [125, 307, 300, 321.0], "spans": [], "index": 10}, {"bbox": [125, 321.0, 300, 335.0], "spans": [], "index": 11}, {"bbox": [125, 335.0, 300, 349.0], "spans": [], "index": 12}, {"bbox": [125, 349.0, 300, 363.0], "spans": [], "index": 13}, {"bbox": [125, 363.0, 300, 377.0], "spans": [], "index": 14}]}], "index": 12}, {"type": "text", "bbox": [124, 422, 486, 450], "lines": [{"bbox": [137, 424, 485, 439], "spans": [{"bbox": [137, 424, 261, 439], "score": 1.0, "content": "Proof. By lemma 3.1, ", "type": "text"}, {"bbox": [261, 426, 270, 435], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [270, 424, 297, 439], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 426, 307, 435], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 424, 485, 439], "score": 1.0, "content": " are true not friends, because they", "type": "text"}], "index": 15}, {"bbox": [126, 438, 243, 451], "spans": [{"bbox": [126, 438, 243, 451], "score": 1.0, "content": "are not friends, that is", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "interline_equation", "bbox": [205, 460, 405, 472], "lines": [{"bbox": [205, 460, 405, 472], "spans": [{"bbox": [205, 460, 405, 472], "score": 0.86, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [124, 479, 486, 508], "lines": [{"bbox": [137, 481, 486, 497], "spans": [{"bbox": [137, 481, 188, 497], "score": 1.0, "content": "Consider ", "type": "text"}, {"bbox": [188, 484, 223, 495], "score": 0.94, "content": "y\\in V", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [223, 481, 282, 497], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [283, 483, 435, 495], "score": 0.93, "content": "C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0", "type": "inline_equation", "height": 12, "width": 152}, {"bbox": [435, 481, 444, 497], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [445, 487, 451, 495], "score": 0.84, "content": "y", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [452, 481, 486, 497], "score": 1.0, "content": " exists", "type": "text"}], "index": 18}, {"bbox": [125, 496, 312, 510], "spans": [{"bbox": [125, 496, 169, 510], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [169, 498, 179, 507], "score": 0.92, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [179, 496, 205, 510], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [205, 498, 215, 506], "score": 0.91, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [215, 496, 312, 510], "score": 1.0, "content": " are friends). Then", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "interline_equation", "bbox": [174, 517, 437, 531], "lines": [{"bbox": [174, 517, 437, 531], "spans": [{"bbox": [174, 517, 437, 531], "score": 0.89, "content": "B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [124, 537, 339, 551], "lines": [{"bbox": [126, 540, 338, 553], "spans": [{"bbox": [126, 540, 169, 552], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [169, 541, 207, 552], "score": 0.94, "content": "B z\\neq0", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [207, 540, 233, 552], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [233, 541, 272, 553], "score": 0.94, "content": "(1+B)", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [272, 540, 338, 552], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [136, 552, 408, 566], "lines": [{"bbox": [138, 554, 407, 567], "spans": [{"bbox": [138, 554, 157, 567], "score": 1.0, "content": "So, ", "type": "text"}, {"bbox": [158, 555, 231, 566], "score": 0.93, "content": "A C=C A\\neq0", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [231, 554, 278, 567], "score": 1.0, "content": "; that is, ", "type": "text"}, {"bbox": [278, 555, 287, 564], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [288, 554, 313, 567], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [314, 555, 323, 564], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [324, 554, 407, 567], "score": 1.0, "content": " are true friends.", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [124, 587, 487, 615], "lines": [{"bbox": [124, 588, 488, 605], "spans": [{"bbox": [124, 588, 230, 605], "score": 1.0, "content": "Theorem 3.4. Let ", "type": "text"}, {"bbox": [230, 591, 323, 604], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [323, 588, 488, 605], "score": 1.0, "content": " be a representation. Then one", "type": "text"}], "index": 23}, {"bbox": [127, 604, 238, 617], "spans": [{"bbox": [127, 604, 238, 617], "score": 1.0, "content": "of the following holds.", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [124, 617, 487, 643], "lines": [{"bbox": [139, 618, 487, 632], "spans": [{"bbox": [139, 618, 487, 632], "score": 1.0, "content": "(a) The full friendship graph is totally disconnected (no friends at", "type": "text"}], "index": 25}, {"bbox": [126, 631, 149, 645], "spans": [{"bbox": [126, 631, 149, 645], "score": 1.0, "content": "all).", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [125, 644, 488, 671], "lines": [{"bbox": [138, 645, 487, 660], "spans": [{"bbox": [138, 645, 389, 660], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [389, 648, 402, 658], "score": 0.9, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [402, 645, 427, 660], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [427, 648, 451, 659], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [451, 645, 487, 660], "score": 1.0, "content": " for all", "type": "text"}], "index": 27}, {"bbox": [126, 662, 133, 671], "spans": [{"bbox": [126, 662, 130, 671], "score": 0.65, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [131, 662, 133, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [126, 672, 487, 700], "lines": [{"bbox": [139, 673, 487, 689], "spans": [{"bbox": [139, 673, 384, 689], "score": 1.0, "content": "(c) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [385, 676, 397, 686], "score": 0.9, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [397, 673, 421, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 676, 435, 688], "score": 0.9, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [435, 673, 487, 689], "score": 1.0, "content": " whenever", "type": "text"}], "index": 29}, {"bbox": [126, 688, 275, 702], "spans": [{"bbox": [126, 688, 138, 700], "score": 0.85, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [139, 689, 164, 701], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 689, 178, 702], "score": 0.89, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [179, 689, 275, 701], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 30}], "index": 29.5}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [125, 307, 300, 371], "blocks": [{"type": "image_body", "bbox": [125, 307, 300, 371], "group_id": 0, "lines": [{"bbox": [125, 307, 300, 371], "spans": [{"bbox": [125, 307, 300, 371], "score": 0.776, "type": "image", "image_path": "7782d0d127eba9149d6c475dbb5cc3b32b3a3f69a44ce9d8ed81904f6b0cbd1a.jpg"}]}], "index": 12, "virtual_lines": [{"bbox": [125, 307, 300, 321.0], "spans": [], "index": 10}, {"bbox": [125, 321.0, 300, 335.0], "spans": [], "index": 11}, {"bbox": [125, 335.0, 300, 349.0], "spans": [], "index": 12}, {"bbox": [125, 349.0, 300, 363.0], "spans": [], "index": 13}, {"bbox": [125, 363.0, 300, 377.0], "spans": [], "index": 14}]}], "index": 12}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [205, 460, 405, 472], "lines": [{"bbox": [205, 460, 405, 472], "spans": [{"bbox": [205, 460, 405, 472], "score": 0.86, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [174, 517, 437, 531], "lines": [{"bbox": [174, 517, 437, 531], "spans": [{"bbox": [174, 517, 437, 531], "score": 0.89, "content": "B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0", "type": "interline_equation"}], "index": 20}], "index": 20}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [278, 92, 329, 102], "spans": [{"bbox": [278, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [124, 91, 132, 100], "lines": [{"bbox": [126, 93, 132, 102], "spans": [{"bbox": [126, 93, 132, 102], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 487, 167], "lines": [{"bbox": [125, 113, 486, 127], "spans": [{"bbox": [125, 113, 327, 127], "score": 1.0, "content": "Lemma 3.2. There is an edge between ", "type": "text"}, {"bbox": [328, 115, 340, 125], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [340, 113, 365, 127], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [366, 115, 379, 127], "score": 0.9, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [379, 113, 486, 127], "score": 1.0, "content": " in the full friendship", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 141], "spans": [{"bbox": [125, 127, 357, 141], "score": 1.0, "content": "graph if and only if there is an edge between ", "type": "text"}, {"bbox": [357, 128, 381, 140], "score": 0.92, "content": "A_{i+k}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [382, 127, 408, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [408, 128, 433, 140], "score": 0.91, "content": "A_{j+k}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [433, 127, 486, 141], "score": 1.0, "content": " where in-", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 243, 155], "score": 1.0, "content": "dices are taken modulo ", "type": "text"}, {"bbox": [243, 146, 250, 151], "score": 0.75, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [251, 141, 339, 155], "score": 1.0, "content": ". In other words, ", "type": "text"}, {"bbox": [339, 142, 353, 153], "score": 0.9, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [353, 141, 486, 155], "score": 1.0, "content": " acts on the full friendship", "type": "text"}], "index": 2}, {"bbox": [126, 155, 339, 169], "spans": [{"bbox": [126, 155, 339, 169], "score": 1.0, "content": "graph by permuting the vertices cyclically.", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [125, 113, 486, 169]}, {"type": "text", "bbox": [124, 174, 486, 217], "lines": [{"bbox": [137, 177, 484, 191], "spans": [{"bbox": [137, 177, 484, 191], "score": 1.0, "content": "Proof. This follows immediately from the fact that conjugation by", "type": "text"}], "index": 4}, {"bbox": [126, 189, 487, 208], "spans": [{"bbox": [126, 192, 255, 205], "score": 0.94, "content": "T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})", "type": "inline_equation", "height": 13, "width": 129}, {"bbox": [256, 189, 309, 208], "score": 1.0, "content": " permutes ", "type": "text"}, {"bbox": [310, 194, 388, 204], "score": 0.76, "content": "\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 10, "width": 78}, {"bbox": [388, 189, 487, 208], "score": 1.0, "content": " cyclically (Lemma", "type": "text"}], "index": 5}, {"bbox": [125, 204, 150, 220], "spans": [{"bbox": [125, 204, 150, 220], "score": 1.0, "content": "2.1).", "type": "text"}], "index": 6}], "index": 5, "bbox_fs": [125, 177, 487, 220]}, {"type": "text", "bbox": [124, 225, 487, 268], "lines": [{"bbox": [125, 227, 486, 242], "spans": [{"bbox": [125, 227, 343, 242], "score": 1.0, "content": "Lemma 3.3 (Lemma about friends). Let ", "type": "text"}, {"bbox": [343, 228, 352, 239], "score": 0.54, "content": "A", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [353, 227, 377, 242], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [378, 229, 388, 239], "score": 0.6, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [388, 227, 486, 242], "score": 1.0, "content": " be neighbors which", "type": "text"}], "index": 7}, {"bbox": [126, 242, 485, 256], "spans": [{"bbox": [126, 242, 227, 256], "score": 1.0, "content": "are not friends. If ", "type": "text"}, {"bbox": [227, 244, 237, 253], "score": 0.88, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [237, 242, 349, 256], "score": 1.0, "content": " is not a neighbor of ", "type": "text"}, {"bbox": [349, 242, 359, 253], "score": 0.73, "content": "A", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [360, 242, 386, 256], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [387, 242, 397, 253], "score": 0.74, "content": "C", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [397, 242, 474, 256], "score": 1.0, "content": " is a friend of ", "type": "text"}, {"bbox": [475, 244, 485, 253], "score": 0.85, "content": "B", "type": "inline_equation", "height": 9, "width": 10}], "index": 8}, {"bbox": [126, 255, 272, 270], "spans": [{"bbox": [126, 255, 151, 270], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [152, 257, 161, 266], "score": 0.82, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [162, 255, 261, 270], "score": 1.0, "content": " is a true friend of ", "type": "text"}, {"bbox": [261, 257, 270, 266], "score": 0.84, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [270, 255, 272, 270], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8, "bbox_fs": [125, 227, 486, 270]}, {"type": "image", "bbox": [125, 307, 300, 371], "blocks": [{"type": "image_body", "bbox": [125, 307, 300, 371], "group_id": 0, "lines": [{"bbox": [125, 307, 300, 371], "spans": [{"bbox": [125, 307, 300, 371], "score": 0.776, "type": "image", "image_path": "7782d0d127eba9149d6c475dbb5cc3b32b3a3f69a44ce9d8ed81904f6b0cbd1a.jpg"}]}], "index": 12, "virtual_lines": [{"bbox": [125, 307, 300, 321.0], "spans": [], "index": 10}, {"bbox": [125, 321.0, 300, 335.0], "spans": [], "index": 11}, {"bbox": [125, 335.0, 300, 349.0], "spans": [], "index": 12}, {"bbox": [125, 349.0, 300, 363.0], "spans": [], "index": 13}, {"bbox": [125, 363.0, 300, 377.0], "spans": [], "index": 14}]}], "index": 12}, {"type": "text", "bbox": [124, 422, 486, 450], "lines": [{"bbox": [137, 424, 485, 439], "spans": [{"bbox": [137, 424, 261, 439], "score": 1.0, "content": "Proof. By lemma 3.1, ", "type": "text"}, {"bbox": [261, 426, 270, 435], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [270, 424, 297, 439], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 426, 307, 435], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [307, 424, 485, 439], "score": 1.0, "content": " are true not friends, because they", "type": "text"}], "index": 15}, {"bbox": [126, 438, 243, 451], "spans": [{"bbox": [126, 438, 243, 451], "score": 1.0, "content": "are not friends, that is", "type": "text"}], "index": 16}], "index": 15.5, "bbox_fs": [126, 424, 485, 451]}, {"type": "interline_equation", "bbox": [205, 460, 405, 472], "lines": [{"bbox": [205, 460, 405, 472], "spans": [{"bbox": [205, 460, 405, 472], "score": 0.86, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [124, 479, 486, 508], "lines": [{"bbox": [137, 481, 486, 497], "spans": [{"bbox": [137, 481, 188, 497], "score": 1.0, "content": "Consider ", "type": "text"}, {"bbox": [188, 484, 223, 495], "score": 0.94, "content": "y\\in V", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [223, 481, 282, 497], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [283, 483, 435, 495], "score": 0.93, "content": "C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0", "type": "inline_equation", "height": 12, "width": 152}, {"bbox": [435, 481, 444, 497], "score": 1.0, "content": " (", "type": "text"}, {"bbox": [445, 487, 451, 495], "score": 0.84, "content": "y", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [452, 481, 486, 497], "score": 1.0, "content": " exists", "type": "text"}], "index": 18}, {"bbox": [125, 496, 312, 510], "spans": [{"bbox": [125, 496, 169, 510], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [169, 498, 179, 507], "score": 0.92, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [179, 496, 205, 510], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [205, 498, 215, 506], "score": 0.91, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [215, 496, 312, 510], "score": 1.0, "content": " are friends). Then", "type": "text"}], "index": 19}], "index": 18.5, "bbox_fs": [125, 481, 486, 510]}, {"type": "interline_equation", "bbox": [174, 517, 437, 531], "lines": [{"bbox": [174, 517, 437, 531], "spans": [{"bbox": [174, 517, 437, 531], "score": 0.89, "content": "B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [124, 537, 339, 551], "lines": [{"bbox": [126, 540, 338, 553], "spans": [{"bbox": [126, 540, 169, 552], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [169, 541, 207, 552], "score": 0.94, "content": "B z\\neq0", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [207, 540, 233, 552], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [233, 541, 272, 553], "score": 0.94, "content": "(1+B)", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [272, 540, 338, 552], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 21}], "index": 21, "bbox_fs": [126, 540, 338, 553]}, {"type": "text", "bbox": [136, 552, 408, 566], "lines": [{"bbox": [138, 554, 407, 567], "spans": [{"bbox": [138, 554, 157, 567], "score": 1.0, "content": "So, ", "type": "text"}, {"bbox": [158, 555, 231, 566], "score": 0.93, "content": "A C=C A\\neq0", "type": "inline_equation", "height": 11, "width": 73}, {"bbox": [231, 554, 278, 567], "score": 1.0, "content": "; that is, ", "type": "text"}, {"bbox": [278, 555, 287, 564], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [288, 554, 313, 567], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [314, 555, 323, 564], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [324, 554, 407, 567], "score": 1.0, "content": " are true friends.", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [138, 554, 407, 567]}, {"type": "text", "bbox": [124, 587, 487, 615], "lines": [{"bbox": [124, 588, 488, 605], "spans": [{"bbox": [124, 588, 230, 605], "score": 1.0, "content": "Theorem 3.4. Let ", "type": "text"}, {"bbox": [230, 591, 323, 604], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [323, 588, 488, 605], "score": 1.0, "content": " be a representation. Then one", "type": "text"}], "index": 23}, {"bbox": [127, 604, 238, 617], "spans": [{"bbox": [127, 604, 238, 617], "score": 1.0, "content": "of the following holds.", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [124, 588, 488, 617]}, {"type": "text", "bbox": [124, 617, 487, 643], "lines": [{"bbox": [139, 618, 487, 632], "spans": [{"bbox": [139, 618, 487, 632], "score": 1.0, "content": "(a) The full friendship graph is totally disconnected (no friends at", "type": "text"}], "index": 25}, {"bbox": [126, 631, 149, 645], "spans": [{"bbox": [126, 631, 149, 645], "score": 1.0, "content": "all).", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [126, 618, 487, 645]}, {"type": "text", "bbox": [125, 644, 488, 671], "lines": [{"bbox": [138, 645, 487, 660], "spans": [{"bbox": [138, 645, 389, 660], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [389, 648, 402, 658], "score": 0.9, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [402, 645, 427, 660], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [427, 648, 451, 659], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 11, "width": 24}, {"bbox": [451, 645, 487, 660], "score": 1.0, "content": " for all", "type": "text"}], "index": 27}, {"bbox": [126, 662, 133, 671], "spans": [{"bbox": [126, 662, 130, 671], "score": 0.65, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [131, 662, 133, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27.5, "bbox_fs": [126, 645, 487, 671]}, {"type": "text", "bbox": [126, 672, 487, 700], "lines": [{"bbox": [139, 673, 487, 689], "spans": [{"bbox": [139, 673, 384, 689], "score": 1.0, "content": "(c) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [385, 676, 397, 686], "score": 0.9, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [397, 673, 421, 689], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 676, 435, 688], "score": 0.9, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [435, 673, 487, 689], "score": 1.0, "content": " whenever", "type": "text"}], "index": 29}, {"bbox": [126, 688, 275, 702], "spans": [{"bbox": [126, 688, 138, 700], "score": 0.85, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [139, 689, 164, 701], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 689, 178, 702], "score": 0.89, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [179, 689, 275, 701], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 30}], "index": 29.5, "bbox_fs": [126, 673, 487, 702]}]}
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[{"type": "text", "bbox": [124, 110, 487, 167], "content": "Lemma 3.2. There is an edge between and in the full friendship graph if and only if there is an edge between and where in- dices are taken modulo . In other words, acts on the full friendship graph by permuting the vertices cyclically.", "index": 0}, {"type": "text", "bbox": [124, 174, 486, 217], "content": "Proof. This follows immediately from the fact that conjugation by permutes cyclically (Lemma 2.1).", "index": 1}, {"type": "text", "bbox": [124, 225, 487, 268], "content": "Lemma 3.3 (Lemma about friends). Let and be neighbors which are not friends. If is not a neighbor of and is a friend of then is a true friend of .", "index": 2}, {"type": "image", "bbox": [125, 307, 300, 371], "content": "", "index": 3}, {"type": "text", "bbox": [124, 422, 486, 450], "content": "Proof. By lemma 3.1, and are true not friends, because they are not friends, that is", "index": 4}, {"type": "interline_equation", "bbox": [205, 460, 405, 472], "content": "", "index": 5}, {"type": "text", "bbox": [124, 479, 486, 508], "content": "Consider such that ( exists because and are friends). Then", "index": 6}, {"type": "interline_equation", "bbox": [174, 517, 437, 531], "content": "", "index": 7}, {"type": "text", "bbox": [124, 537, 339, 551], "content": "because and is invertible.", "index": 8}, {"type": "text", "bbox": [136, 552, 408, 566], "content": "So, ; that is, and are true friends.", "index": 9}, {"type": "text", "bbox": [124, 587, 487, 615], "content": "Theorem 3.4. Let be a representation. Then one of the following holds.", "index": 10}, {"type": "text", "bbox": [124, 617, 487, 643], "content": "(a) The full friendship graph is totally disconnected (no friends at all).", "index": 11}, {"type": "text", "bbox": [125, 644, 488, 671], "content": "(b) The full friendship graph has an edge between and for all .", "index": 12}, {"type": "text", "bbox": [126, 672, 487, 700], "content": "(c) The full friendship graph has an edge between and whenever and are not neighbors.", "index": 13}]
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[{"bbox": [125, 113, 486, 127], "content": "Lemma 3.2. There is an edge between and in the full friendship", "parent_index": 0, "line_index": 0}, {"bbox": [125, 127, 486, 141], "content": "graph if and only if there is an edge between and where in-", "parent_index": 0, "line_index": 1}, {"bbox": [126, 141, 486, 155], "content": "dices are taken modulo . In other words, acts on the full friendship", "parent_index": 0, "line_index": 2}, {"bbox": [126, 155, 339, 169], "content": "graph by permuting the vertices cyclically.", "parent_index": 0, "line_index": 3}, {"bbox": [137, 177, 484, 191], "content": "Proof. This follows immediately from the fact that conjugation by", "parent_index": 1, "line_index": 0}, {"bbox": [126, 189, 487, 208], "content": "permutes cyclically (Lemma", "parent_index": 1, "line_index": 1}, {"bbox": [125, 204, 150, 220], "content": "2.1).", "parent_index": 1, "line_index": 2}, {"bbox": [125, 227, 486, 242], "content": "Lemma 3.3 (Lemma about friends). Let and be neighbors which", "parent_index": 2, "line_index": 0}, {"bbox": [126, 242, 485, 256], "content": "are not friends. If is not a neighbor of and is a friend of", "parent_index": 2, "line_index": 1}, {"bbox": [126, 255, 272, 270], "content": "then is a true friend of .", "parent_index": 2, "line_index": 2}, {"bbox": [137, 424, 485, 439], "content": "Proof. By lemma 3.1, and are true not friends, because they", "parent_index": 4, "line_index": 0}, {"bbox": [126, 438, 243, 451], "content": "are not friends, that is", "parent_index": 4, "line_index": 1}, {"bbox": [137, 481, 486, 497], "content": "Consider such that ( exists", "parent_index": 6, "line_index": 0}, {"bbox": [125, 496, 312, 510], "content": "because and are friends). Then", "parent_index": 6, "line_index": 1}, {"bbox": [126, 540, 338, 553], "content": "because and is invertible.", "parent_index": 8, "line_index": 0}, {"bbox": [138, 554, 407, 567], "content": "So, ; that is, and are true friends.", "parent_index": 9, "line_index": 0}, {"bbox": [124, 588, 488, 605], "content": "Theorem 3.4. Let be a representation. Then one", "parent_index": 10, "line_index": 0}, {"bbox": [127, 604, 238, 617], "content": "of the following holds.", "parent_index": 10, "line_index": 1}, {"bbox": [139, 618, 487, 632], "content": "(a) The full friendship graph is totally disconnected (no friends at", "parent_index": 11, "line_index": 0}, {"bbox": [126, 631, 149, 645], "content": "all).", "parent_index": 11, "line_index": 1}, {"bbox": [138, 645, 487, 660], "content": "(b) The full friendship graph has an edge between and for all", "parent_index": 12, "line_index": 0}, {"bbox": [126, 662, 133, 671], "content": ".", "parent_index": 12, "line_index": 1}, {"bbox": [139, 673, 487, 689], "content": "(c) The full friendship graph has an edge between and whenever", "parent_index": 13, "line_index": 0}, {"bbox": [126, 688, 275, 702], "content": "and are not neighbors.", "parent_index": 13, "line_index": 1}]
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[{"bbox": [125, 307, 300, 371], "content": "", "parent_index": 3, "subtype": "body"}]
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[{"bbox": [328, 115, 340, 125], "content": "A_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [366, 115, 379, 127], "content": "A_{j}", "parent_index": 0, "subtype": "inline"}, {"bbox": [357, 128, 381, 140], "content": "A_{i+k}", "parent_index": 0, "subtype": "inline"}, {"bbox": [408, 128, 433, 140], "content": "A_{j+k}", "parent_index": 0, "subtype": "inline"}, {"bbox": [243, 146, 250, 151], "content": "n", "parent_index": 0, "subtype": "inline"}, {"bbox": [339, 142, 353, 153], "content": "\\mathbb{Z}_{n}", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 192, 255, 205], "content": "T=\\rho(\\tau)=\\rho(\\sigma_{1}\\dots\\sigma_{n-1})", "parent_index": 1, "subtype": "inline"}, {"bbox": [310, 194, 388, 204], "content": "\\sigma_{0},\\sigma_{1},\\ldots,\\sigma_{n-1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [343, 228, 352, 239], "content": "A", "parent_index": 2, "subtype": "inline"}, {"bbox": [378, 229, 388, 239], "content": "B", "parent_index": 2, "subtype": "inline"}, {"bbox": [227, 244, 237, 253], "content": "C", "parent_index": 2, "subtype": "inline"}, {"bbox": [349, 242, 359, 253], "content": "A", "parent_index": 2, "subtype": "inline"}, {"bbox": [387, 242, 397, 253], "content": "C", "parent_index": 2, "subtype": "inline"}, {"bbox": [475, 244, 485, 253], "content": "B", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 257, 161, 266], "content": "C", "parent_index": 2, "subtype": "inline"}, {"bbox": [261, 257, 270, 266], "content": "A", "parent_index": 2, "subtype": "inline"}, {"bbox": [261, 426, 270, 435], "content": "A", "parent_index": 4, "subtype": "inline"}, {"bbox": [297, 426, 307, 435], "content": "B", "parent_index": 4, "subtype": "inline"}, {"bbox": [205, 460, 405, 472], "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "parent_index": 5, "subtype": "interline"}, {"bbox": [188, 484, 223, 495], "content": "y\\in V", "parent_index": 6, "subtype": "inline"}, {"bbox": [283, 483, 435, 495], "content": "C y\\;\\in\\;I m(B),C y\\;=\\;B z\\;\\neq\\;0", "parent_index": 6, "subtype": "inline"}, {"bbox": [445, 487, 451, 495], "content": "y", "parent_index": 6, "subtype": "inline"}, {"bbox": [169, 498, 179, 507], "content": "C", "parent_index": 6, "subtype": "inline"}, {"bbox": [205, 498, 215, 506], "content": "B", "parent_index": 6, "subtype": "inline"}, {"bbox": [174, 517, 437, 531], "content": "B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\\neq0", "parent_index": 7, "subtype": "interline"}, {"bbox": [169, 541, 207, 552], "content": "B z\\neq0", "parent_index": 8, "subtype": "inline"}, {"bbox": [233, 541, 272, 553], "content": "(1+B)", "parent_index": 8, "subtype": "inline"}, {"bbox": [158, 555, 231, 566], "content": "A C=C A\\neq0", "parent_index": 9, "subtype": "inline"}, {"bbox": [278, 555, 287, 564], "content": "A", "parent_index": 9, "subtype": "inline"}, {"bbox": [314, 555, 323, 564], "content": "C", "parent_index": 9, "subtype": "inline"}, {"bbox": [230, 591, 323, 604], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 10, "subtype": "inline"}, {"bbox": [389, 648, 402, 658], "content": "A_{i}", "parent_index": 12, "subtype": "inline"}, {"bbox": [427, 648, 451, 659], "content": "A_{i+1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 662, 130, 671], "content": "i", "parent_index": 12, "subtype": "inline"}, {"bbox": [385, 676, 397, 686], "content": "A_{i}", "parent_index": 13, "subtype": "inline"}, {"bbox": [421, 676, 435, 688], "content": "A_{j}", "parent_index": 13, "subtype": "inline"}, {"bbox": [126, 688, 138, 700], "content": "A_{i}", "parent_index": 13, "subtype": "inline"}, {"bbox": [165, 689, 178, 702], "content": "A_{j}", "parent_index": 13, "subtype": "inline"}]
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[]
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Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $B$ and $C$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $C$ and any neighbor of $B$ which is not a neighbor of $C$ . It follows inductively that there is an edge joining $C$ to every vertex which is not a neighbor of $C$ . Then (c) holds, because the full friendship graph is a $\mathbb{Z}_{n}$ -graph.
Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors.
Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph.
Corollary 3.5. For $n\neq4$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.
Remark 3.6. For $n=4$ there is a friendship graph which is neither totally disconnected nor connected:
By [5], Lemmas 6.2 and 6.3, every representation of $B_{4}$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible.
Now consider the case when the friendship graph is totally disconnected (that is, statement $(a)$ of theorem 3.4 holds).
Lemma 3.7. If $A$ and $B$ are neighbors and not friends then: (a) $A^{2}B=A B^{2}$ ; $B A^{2}=B^{2}A$ . $(b)$ If $x\in I m(A)\cap K e r(A-\lambda I)$ , then $B(B x)=\lambda(B x)$ and $A B x=$
$-(1+\lambda)x$ .
Proof. (a). By lemma 3.1, $A$ and $B$ are not true friends, so
$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$
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<p data-bbox="125 110 487 209">Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $B$ and $C$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $C$ and any neighbor of $B$ which is not a neighbor of $C$ . It follows inductively that there is an edge joining $C$ to every vertex which is not a neighbor of $C$ . Then (c) holds, because the full friendship graph is a $\mathbb{Z}_{n}$ -graph. </p>
<p data-bbox="124 232 486 261">Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors. </p>
<p data-bbox="136 270 417 299">Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph. </p>
<p data-bbox="125 321 486 351">Corollary 3.5. For $n\neq4$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected. </p>
<p data-bbox="124 369 486 398">Remark 3.6. For $n=4$ there is a friendship graph which is neither totally disconnected nor connected: </p>
<div class="image" data-bbox="126 424 209 477"><img data-bbox="126 424 209 477"/></div>
<p data-bbox="124 515 487 545">By [5], Lemmas 6.2 and 6.3, every representation of $B_{4}$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible. </p>
<p data-bbox="124 557 486 587">Now consider the case when the friendship graph is totally disconnected (that is, statement $(a)$ of theorem 3.4 holds). </p>
<p data-bbox="124 595 486 653">Lemma 3.7. If $A$ and $B$ are neighbors and not friends then: (a) $A^{2}B=A B^{2}$ ; $B A^{2}=B^{2}A$ . $(b)$ If $x\in I m(A)\cap K e r(A-\lambda I)$ , then $B(B x)=\lambda(B x)$ and $A B x=$
$-(1+\lambda)x$ . </p>
<p data-bbox="135 660 450 676">Proof. (a). By lemma 3.1, $A$ and $B$ are not true friends, so </p>
<div class="equation" data-bbox="205 686 405 700">$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$</div>
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0003047v1
| 6 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $B$ and $C$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $C$ and any neighbor of $B$ which is not a neighbor of $C$ . It follows inductively that there is an edge joining $C$ to every vertex which is not a neighbor of $C$ . Then (c) holds, because the full friendship graph is a $\\mathbb{Z}_{n}$ -graph. ", "page_idx": 6}, {"type": "text", "text": "Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors. ", "page_idx": 6}, {"type": "text", "text": "Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph. ", "page_idx": 6}, {"type": "text", "text": "Corollary 3.5. For $n\\neq4$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected. ", "page_idx": 6}, {"type": "text", "text": "Remark 3.6. For $n=4$ there is a friendship graph which is neither totally disconnected nor connected: ", "page_idx": 6}, {"type": "image", "img_path": "images/b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg", "img_caption": [], "img_footnote": [], "page_idx": 6}, {"type": "text", "text": "By [5], Lemmas 6.2 and 6.3, every representation of $B_{4}$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible. ", "page_idx": 6}, {"type": "text", "text": "Now consider the case when the friendship graph is totally disconnected (that is, statement $(a)$ of theorem 3.4 holds). ", "page_idx": 6}, {"type": "text", "text": "Lemma 3.7. If $A$ and $B$ are neighbors and not friends then: (a) $A^{2}B=A B^{2}$ ; $B A^{2}=B^{2}A$ . $(b)$ If $x\\in I m(A)\\cap K e r(A-\\lambda I)$ , then $B(B x)=\\lambda(B x)$ and $A B x=$ \n$-(1+\\lambda)x$ . ", "page_idx": 6}, {"type": "text", "text": "Proof. (a). By lemma 3.1, $A$ and $B$ are not true friends, so ", "page_idx": 6}, {"type": "equation", "text": "$$\nA+A^{2}+A B A=B+B^{2}+B A B=0.\n$$", "text_format": "latex", "page_idx": 6}]
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Since the graph is not", "type": "text"}], "index": 0}, {"bbox": [126, 127, 486, 140], "spans": [{"bbox": [126, 127, 435, 140], "score": 1.0, "content": "totally disconnected, there is an edge joining some vertices ", "type": "text"}, {"bbox": [435, 128, 445, 137], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 127, 473, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [473, 128, 482, 137], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [482, 127, 486, 140], "score": 1.0, "content": ".", "type": "text"}], "index": 1}, {"bbox": [125, 140, 487, 155], "spans": [{"bbox": [125, 140, 487, 155], "score": 1.0, "content": "Since (b) does not hold, no neighbors are joined by an edge. Lemma", "type": "text"}], "index": 2}, {"bbox": [126, 155, 484, 168], "spans": [{"bbox": [126, 155, 349, 168], "score": 1.0, "content": "3.3 implies that there is an edge between ", "type": "text"}, {"bbox": [349, 156, 359, 165], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 155, 474, 168], "score": 1.0, "content": " and any neighbor of ", "type": "text"}, {"bbox": [475, 156, 484, 165], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 9}], "index": 3}, {"bbox": [126, 169, 486, 182], "spans": [{"bbox": [126, 169, 270, 182], "score": 1.0, "content": "which is not a neighbor of ", "type": "text"}, {"bbox": [270, 171, 279, 180], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 169, 486, 182], "score": 1.0, "content": ". It follows inductively that there is an", "type": "text"}], "index": 4}, {"bbox": [126, 183, 484, 196], "spans": [{"bbox": [126, 183, 191, 196], "score": 1.0, "content": "edge joining ", "type": "text"}, {"bbox": [191, 184, 201, 193], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [201, 183, 423, 196], "score": 1.0, "content": " to every vertex which is not a neighbor of ", "type": "text"}, {"bbox": [424, 184, 433, 193], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [433, 183, 484, 196], "score": 1.0, "content": ". Then (c)", "type": "text"}], "index": 5}, {"bbox": [124, 195, 404, 212], "spans": [{"bbox": [124, 195, 352, 212], "score": 1.0, "content": "holds, because the full friendship graph is a ", "type": "text"}, {"bbox": [353, 198, 367, 209], "score": 0.89, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [367, 195, 404, 212], "score": 1.0, "content": "-graph.", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [124, 232, 486, 261], "lines": [{"bbox": [124, 234, 487, 250], "spans": [{"bbox": [124, 234, 487, 250], "score": 1.0, "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "type": "text"}], "index": 7}, {"bbox": [126, 250, 365, 262], "spans": [{"bbox": [126, 250, 365, 262], "score": 1.0, "content": "chain, if the only edges are between neighbors.", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [136, 270, 417, 299], "lines": [{"bbox": [138, 272, 391, 286], "spans": [{"bbox": [138, 272, 391, 286], "score": 1.0, "content": "Case (b) of the above theorem can be restated as", "type": "text"}], "index": 9}, {"bbox": [140, 288, 415, 299], "spans": [{"bbox": [140, 288, 415, 299], "score": 1.0, "content": "(b) The full friendship graph contains the chain graph.", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [125, 321, 486, 351], "lines": [{"bbox": [126, 324, 487, 339], "spans": [{"bbox": [126, 324, 234, 339], "score": 1.0, "content": "Corollary 3.5. For ", "type": "text"}, {"bbox": [234, 326, 263, 337], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [263, 324, 487, 339], "score": 1.0, "content": ", the friendship graph and the full friendship", "type": "text"}], "index": 11}, {"bbox": [126, 339, 439, 352], "spans": [{"bbox": [126, 339, 439, 352], "score": 1.0, "content": "graph are either totally disconnected (no edges) or connected.", "type": "text"}], "index": 12}], "index": 11.5}, {"type": "text", "bbox": [124, 369, 486, 398], "lines": [{"bbox": [125, 372, 485, 386], "spans": [{"bbox": [125, 372, 225, 386], "score": 1.0, "content": "Remark 3.6. For ", "type": "text"}, {"bbox": [225, 374, 256, 383], "score": 0.88, "content": "n=4", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [257, 372, 485, 386], "score": 1.0, "content": " there is a friendship graph which is neither", "type": "text"}], "index": 13}, {"bbox": [126, 386, 308, 399], "spans": [{"bbox": [126, 386, 308, 399], "score": 1.0, "content": "totally disconnected nor connected:", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5}, {"type": "text", "bbox": [124, 515, 487, 545], "lines": [{"bbox": [137, 518, 486, 532], "spans": [{"bbox": [137, 518, 409, 532], "score": 1.0, "content": "By [5], Lemmas 6.2 and 6.3, every representation of ", "type": "text"}, {"bbox": [409, 520, 423, 530], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [423, 518, 486, 532], "score": 1.0, "content": " of corank 2", "type": "text"}], "index": 17}, {"bbox": [126, 532, 485, 546], "spans": [{"bbox": [126, 532, 485, 546], "score": 1.0, "content": "and dimension at least 4, which has this friendship graph, is reducible.", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "text", "bbox": [124, 557, 486, 587], "lines": [{"bbox": [136, 559, 485, 575], "spans": [{"bbox": [136, 559, 485, 575], "score": 1.0, "content": "Now consider the case when the friendship graph is totally discon-", "type": "text"}], "index": 19}, {"bbox": [126, 574, 394, 588], "spans": [{"bbox": [126, 574, 262, 588], "score": 1.0, "content": "nected (that is, statement ", "type": "text"}, {"bbox": [262, 575, 278, 588], "score": 0.64, "content": "(a)", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [278, 574, 394, 588], "score": 1.0, "content": " of theorem 3.4 holds).", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "text", "bbox": [124, 595, 486, 653], "lines": [{"bbox": [125, 596, 443, 613], "spans": [{"bbox": [125, 596, 212, 613], "score": 1.0, "content": "Lemma 3.7. If ", "type": "text"}, {"bbox": [213, 600, 222, 608], "score": 0.78, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [222, 596, 248, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 600, 258, 608], "score": 0.85, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [258, 596, 443, 613], "score": 1.0, "content": " are neighbors and not friends then:", "type": "text"}], "index": 21}, {"bbox": [140, 612, 292, 624], "spans": [{"bbox": [140, 612, 157, 624], "score": 1.0, "content": "(a) ", "type": "text"}, {"bbox": [158, 613, 219, 623], "score": 0.9, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [220, 612, 226, 624], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [227, 613, 289, 622], "score": 0.92, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 9, "width": 62}, {"bbox": [289, 612, 292, 624], "score": 1.0, "content": ".", "type": "text"}], "index": 22}, {"bbox": [138, 626, 487, 640], "spans": [{"bbox": [138, 627, 154, 639], "score": 0.32, "content": "(b)", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [154, 626, 168, 640], "score": 1.0, "content": " If ", "type": "text"}, {"bbox": [169, 627, 305, 639], "score": 0.93, "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "type": "inline_equation", "height": 12, "width": 136}, {"bbox": [306, 626, 338, 640], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [338, 627, 421, 640], "score": 0.94, "content": "B(B x)=\\lambda(B x)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [421, 626, 447, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [447, 627, 487, 638], "score": 0.82, "content": "A B x=", "type": "inline_equation", "height": 11, "width": 40}], "index": 23}, {"bbox": [126, 639, 182, 654], "spans": [{"bbox": [126, 640, 178, 653], "score": 0.91, "content": "-(1+\\lambda)x", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [179, 639, 182, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [135, 660, 450, 676], "lines": [{"bbox": [138, 664, 448, 678], "spans": [{"bbox": [138, 664, 281, 678], "score": 1.0, "content": "Proof. (a). By lemma 3.1, ", "type": "text"}, {"bbox": [282, 665, 291, 675], "score": 0.82, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [291, 664, 317, 678], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 665, 327, 675], "score": 0.86, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [327, 664, 448, 678], "score": 1.0, "content": " are not true friends, so", "type": "text"}], "index": 25}], "index": 25}, {"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [222, 90, 389, 101], "lines": [{"bbox": [223, 92, 388, 102], "spans": [{"bbox": [223, 92, 388, 102], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [479, 91, 486, 100], "lines": [{"bbox": [480, 93, 486, 102], "spans": [{"bbox": [480, 93, 486, 102], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 110, 487, 209], "lines": [{"bbox": [137, 112, 486, 127], "spans": [{"bbox": [137, 112, 486, 127], "score": 1.0, "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not", "type": "text"}], "index": 0}, {"bbox": [126, 127, 486, 140], "spans": [{"bbox": [126, 127, 435, 140], "score": 1.0, "content": "totally disconnected, there is an edge joining some vertices ", "type": "text"}, {"bbox": [435, 128, 445, 137], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 127, 473, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [473, 128, 482, 137], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [482, 127, 486, 140], "score": 1.0, "content": ".", "type": "text"}], "index": 1}, {"bbox": [125, 140, 487, 155], "spans": [{"bbox": [125, 140, 487, 155], "score": 1.0, "content": "Since (b) does not hold, no neighbors are joined by an edge. Lemma", "type": "text"}], "index": 2}, {"bbox": [126, 155, 484, 168], "spans": [{"bbox": [126, 155, 349, 168], "score": 1.0, "content": "3.3 implies that there is an edge between ", "type": "text"}, {"bbox": [349, 156, 359, 165], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 155, 474, 168], "score": 1.0, "content": " and any neighbor of ", "type": "text"}, {"bbox": [475, 156, 484, 165], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 9}], "index": 3}, {"bbox": [126, 169, 486, 182], "spans": [{"bbox": [126, 169, 270, 182], "score": 1.0, "content": "which is not a neighbor of ", "type": "text"}, {"bbox": [270, 171, 279, 180], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [280, 169, 486, 182], "score": 1.0, "content": ". It follows inductively that there is an", "type": "text"}], "index": 4}, {"bbox": [126, 183, 484, 196], "spans": [{"bbox": [126, 183, 191, 196], "score": 1.0, "content": "edge joining ", "type": "text"}, {"bbox": [191, 184, 201, 193], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [201, 183, 423, 196], "score": 1.0, "content": " to every vertex which is not a neighbor of ", "type": "text"}, {"bbox": [424, 184, 433, 193], "score": 0.89, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [433, 183, 484, 196], "score": 1.0, "content": ". Then (c)", "type": "text"}], "index": 5}, {"bbox": [124, 195, 404, 212], "spans": [{"bbox": [124, 195, 352, 212], "score": 1.0, "content": "holds, because the full friendship graph is a ", "type": "text"}, {"bbox": [353, 198, 367, 209], "score": 0.89, "content": "\\mathbb{Z}_{n}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [367, 195, 404, 212], "score": 1.0, "content": "-graph.", "type": "text"}], "index": 6}], "index": 3, "bbox_fs": [124, 112, 487, 212]}, {"type": "text", "bbox": [124, 232, 486, 261], "lines": [{"bbox": [124, 234, 487, 250], "spans": [{"bbox": [124, 234, 487, 250], "score": 1.0, "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "type": "text"}], "index": 7}, {"bbox": [126, 250, 365, 262], "spans": [{"bbox": [126, 250, 365, 262], "score": 1.0, "content": "chain, if the only edges are between neighbors.", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [124, 234, 487, 262]}, {"type": "text", "bbox": [136, 270, 417, 299], "lines": [{"bbox": [138, 272, 391, 286], "spans": [{"bbox": [138, 272, 391, 286], "score": 1.0, "content": "Case (b) of the above theorem can be restated as", "type": "text"}], "index": 9}, {"bbox": [140, 288, 415, 299], "spans": [{"bbox": [140, 288, 415, 299], "score": 1.0, "content": "(b) The full friendship graph contains the chain graph.", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [138, 272, 415, 299]}, {"type": "text", "bbox": [125, 321, 486, 351], "lines": [{"bbox": [126, 324, 487, 339], "spans": [{"bbox": [126, 324, 234, 339], "score": 1.0, "content": "Corollary 3.5. For ", "type": "text"}, {"bbox": [234, 326, 263, 337], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [263, 324, 487, 339], "score": 1.0, "content": ", the friendship graph and the full friendship", "type": "text"}], "index": 11}, {"bbox": [126, 339, 439, 352], "spans": [{"bbox": [126, 339, 439, 352], "score": 1.0, "content": "graph are either totally disconnected (no edges) or connected.", "type": "text"}], "index": 12}], "index": 11.5, "bbox_fs": [126, 324, 487, 352]}, {"type": "text", "bbox": [124, 369, 486, 398], "lines": [{"bbox": [125, 372, 485, 386], "spans": [{"bbox": [125, 372, 225, 386], "score": 1.0, "content": "Remark 3.6. For ", "type": "text"}, {"bbox": [225, 374, 256, 383], "score": 0.88, "content": "n=4", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [257, 372, 485, 386], "score": 1.0, "content": " there is a friendship graph which is neither", "type": "text"}], "index": 13}, {"bbox": [126, 386, 308, 399], "spans": [{"bbox": [126, 386, 308, 399], "score": 1.0, "content": "totally disconnected nor connected:", "type": "text"}], "index": 14}], "index": 13.5, "bbox_fs": [125, 372, 485, 399]}, {"type": "image", "bbox": [126, 424, 209, 477], "blocks": [{"type": "image_body", "bbox": [126, 424, 209, 477], "group_id": 0, "lines": [{"bbox": [126, 424, 209, 477], "spans": [{"bbox": [126, 424, 209, 477], "score": 0.835, "type": "image", "image_path": "b1847cff0674dc1b5d2b85096e39aeb6f6360a3532b13d1f8d6025bc45278a72.jpg"}]}], "index": 15.5, "virtual_lines": [{"bbox": [126, 424, 209, 450.5], "spans": [], "index": 15}, {"bbox": [126, 450.5, 209, 477.0], "spans": [], "index": 16}]}], "index": 15.5}, {"type": "text", "bbox": [124, 515, 487, 545], "lines": [{"bbox": [137, 518, 486, 532], "spans": [{"bbox": [137, 518, 409, 532], "score": 1.0, "content": "By [5], Lemmas 6.2 and 6.3, every representation of ", "type": "text"}, {"bbox": [409, 520, 423, 530], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [423, 518, 486, 532], "score": 1.0, "content": " of corank 2", "type": "text"}], "index": 17}, {"bbox": [126, 532, 485, 546], "spans": [{"bbox": [126, 532, 485, 546], "score": 1.0, "content": "and dimension at least 4, which has this friendship graph, is reducible.", "type": "text"}], "index": 18}], "index": 17.5, "bbox_fs": [126, 518, 486, 546]}, {"type": "text", "bbox": [124, 557, 486, 587], "lines": [{"bbox": [136, 559, 485, 575], "spans": [{"bbox": [136, 559, 485, 575], "score": 1.0, "content": "Now consider the case when the friendship graph is totally discon-", "type": "text"}], "index": 19}, {"bbox": [126, 574, 394, 588], "spans": [{"bbox": [126, 574, 262, 588], "score": 1.0, "content": "nected (that is, statement ", "type": "text"}, {"bbox": [262, 575, 278, 588], "score": 0.64, "content": "(a)", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [278, 574, 394, 588], "score": 1.0, "content": " of theorem 3.4 holds).", "type": "text"}], "index": 20}], "index": 19.5, "bbox_fs": [126, 559, 485, 588]}, {"type": "list", "bbox": [124, 595, 486, 653], "lines": [{"bbox": [125, 596, 443, 613], "spans": [{"bbox": [125, 596, 212, 613], "score": 1.0, "content": "Lemma 3.7. If ", "type": "text"}, {"bbox": [213, 600, 222, 608], "score": 0.78, "content": "A", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [222, 596, 248, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [248, 600, 258, 608], "score": 0.85, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [258, 596, 443, 613], "score": 1.0, "content": " are neighbors and not friends then:", "type": "text"}], "index": 21, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [140, 612, 292, 624], "spans": [{"bbox": [140, 612, 157, 624], "score": 1.0, "content": "(a) ", "type": "text"}, {"bbox": [158, 613, 219, 623], "score": 0.9, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 61}, {"bbox": [220, 612, 226, 624], "score": 1.0, "content": "; ", "type": "text"}, {"bbox": [227, 613, 289, 622], "score": 0.92, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 9, "width": 62}, {"bbox": [289, 612, 292, 624], "score": 1.0, "content": ".", "type": "text"}], "index": 22, "is_list_end_line": true}, {"bbox": [138, 626, 487, 640], "spans": [{"bbox": [138, 627, 154, 639], "score": 0.32, "content": "(b)", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [154, 626, 168, 640], "score": 1.0, "content": " If ", "type": "text"}, {"bbox": [169, 627, 305, 639], "score": 0.93, "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "type": "inline_equation", "height": 12, "width": 136}, {"bbox": [306, 626, 338, 640], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [338, 627, 421, 640], "score": 0.94, "content": "B(B x)=\\lambda(B x)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [421, 626, 447, 640], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [447, 627, 487, 638], "score": 0.82, "content": "A B x=", "type": "inline_equation", "height": 11, "width": 40}], "index": 23}, {"bbox": [126, 639, 182, 654], "spans": [{"bbox": [126, 640, 178, 653], "score": 0.91, "content": "-(1+\\lambda)x", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [179, 639, 182, 654], "score": 1.0, "content": ".", "type": "text"}], "index": 24, "is_list_start_line": true, "is_list_end_line": true}], "index": 22.5, "bbox_fs": [125, 596, 487, 654]}, {"type": "text", "bbox": [135, 660, 450, 676], "lines": [{"bbox": [138, 664, 448, 678], "spans": [{"bbox": [138, 664, 281, 678], "score": 1.0, "content": "Proof. (a). By lemma 3.1, ", "type": "text"}, {"bbox": [282, 665, 291, 675], "score": 0.82, "content": "A", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [291, 664, 317, 678], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [317, 665, 327, 675], "score": 0.86, "content": "B", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [327, 664, 448, 678], "score": 1.0, "content": " are not true friends, so", "type": "text"}], "index": 25}], "index": 25, "bbox_fs": [138, 664, 448, 678]}, {"type": "interline_equation", "bbox": [205, 686, 405, 700], "lines": [{"bbox": [205, 686, 405, 700], "spans": [{"bbox": [205, 686, 405, 700], "score": 0.84, "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "type": "interline_equation"}], "index": 26}], "index": 26}]}
|
[{"type": "text", "bbox": [125, 110, 487, 209], "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices and . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between and any neighbor of which is not a neighbor of . It follows inductively that there is an edge joining to every vertex which is not a neighbor of . Then (c) holds, because the full friendship graph is a -graph.", "index": 0}, {"type": "text", "bbox": [124, 232, 486, 261], "content": "Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors.", "index": 1}, {"type": "text", "bbox": [136, 270, 417, 299], "content": "Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph.", "index": 2}, {"type": "text", "bbox": [125, 321, 486, 351], "content": "Corollary 3.5. For , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.", "index": 3}, {"type": "text", "bbox": [124, 369, 486, 398], "content": "Remark 3.6. For there is a friendship graph which is neither totally disconnected nor connected:", "index": 4}, {"type": "image", "bbox": [126, 424, 209, 477], "content": "", "index": 5}, {"type": "text", "bbox": [124, 515, 487, 545], "content": "By [5], Lemmas 6.2 and 6.3, every representation of of corank 2 and dimension at least 4, which has this friendship graph, is reducible.", "index": 6}, {"type": "text", "bbox": [124, 557, 486, 587], "content": "Now consider the case when the friendship graph is totally discon- nected (that is, statement of theorem 3.4 holds).", "index": 7}, {"type": "list", "bbox": [124, 595, 486, 653], "content": "", "index": 8}, {"type": "text", "bbox": [135, 660, 450, 676], "content": "Proof. (a). By lemma 3.1, and are not true friends, so", "index": 9}, {"type": "interline_equation", "bbox": [205, 686, 405, 700], "content": "", "index": 10}]
|
[{"bbox": [137, 112, 486, 127], "content": "Proof. Suppose neither (a) nor (b) holds. Since the graph is not", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 486, 140], "content": "totally disconnected, there is an edge joining some vertices and .", "parent_index": 0, "line_index": 1}, {"bbox": [125, 140, 487, 155], "content": "Since (b) does not hold, no neighbors are joined by an edge. Lemma", "parent_index": 0, "line_index": 2}, {"bbox": [126, 155, 484, 168], "content": "3.3 implies that there is an edge between and any neighbor of", "parent_index": 0, "line_index": 3}, {"bbox": [126, 169, 486, 182], "content": "which is not a neighbor of . It follows inductively that there is an", "parent_index": 0, "line_index": 4}, {"bbox": [126, 183, 484, 196], "content": "edge joining to every vertex which is not a neighbor of . Then (c)", "parent_index": 0, "line_index": 5}, {"bbox": [124, 195, 404, 212], "content": "holds, because the full friendship graph is a -graph.", "parent_index": 0, "line_index": 6}, {"bbox": [124, 234, 487, 250], "content": "Definition 3.3. The friendship graph (the full friendship graph) is a", "parent_index": 1, "line_index": 0}, {"bbox": [126, 250, 365, 262], "content": "chain, if the only edges are between neighbors.", "parent_index": 1, "line_index": 1}, {"bbox": [138, 272, 391, 286], "content": "Case (b) of the above theorem can be restated as", "parent_index": 2, "line_index": 0}, {"bbox": [140, 288, 415, 299], "content": "(b) The full friendship graph contains the chain graph.", "parent_index": 2, "line_index": 1}, {"bbox": [126, 324, 487, 339], "content": "Corollary 3.5. For , the friendship graph and the full friendship", "parent_index": 3, "line_index": 0}, {"bbox": [126, 339, 439, 352], "content": "graph are either totally disconnected (no edges) or connected.", "parent_index": 3, "line_index": 1}, {"bbox": [125, 372, 485, 386], "content": "Remark 3.6. For there is a friendship graph which is neither", "parent_index": 4, "line_index": 0}, {"bbox": [126, 386, 308, 399], "content": "totally disconnected nor connected:", "parent_index": 4, "line_index": 1}, {"bbox": [137, 518, 486, 532], "content": "By [5], Lemmas 6.2 and 6.3, every representation of of corank 2", "parent_index": 6, "line_index": 0}, {"bbox": [126, 532, 485, 546], "content": "and dimension at least 4, which has this friendship graph, is reducible.", "parent_index": 6, "line_index": 1}, {"bbox": [136, 559, 485, 575], "content": "Now consider the case when the friendship graph is totally discon-", "parent_index": 7, "line_index": 0}, {"bbox": [126, 574, 394, 588], "content": "nected (that is, statement of theorem 3.4 holds).", "parent_index": 7, "line_index": 1}, {"bbox": [125, 596, 443, 613], "content": "Lemma 3.7. If and are neighbors and not friends then:", "parent_index": 8, "line_index": 0}, {"bbox": [140, 612, 292, 624], "content": "(a) ; .", "parent_index": 8, "line_index": 1}, {"bbox": [138, 626, 487, 640], "content": "If , then and", "parent_index": 8, "line_index": 2}, {"bbox": [126, 639, 182, 654], "content": ".", "parent_index": 8, "line_index": 3}, {"bbox": [138, 664, 448, 678], "content": "Proof. (a). By lemma 3.1, and are not true friends, so", "parent_index": 9, "line_index": 0}]
|
[{"bbox": [126, 424, 209, 477], "content": "", "parent_index": 5, "subtype": "body"}]
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[{"bbox": [435, 128, 445, 137], "content": "B", "parent_index": 0, "subtype": "inline"}, {"bbox": [473, 128, 482, 137], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [349, 156, 359, 165], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [475, 156, 484, 165], "content": "B", "parent_index": 0, "subtype": "inline"}, {"bbox": [270, 171, 279, 180], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [191, 184, 201, 193], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [424, 184, 433, 193], "content": "C", "parent_index": 0, "subtype": "inline"}, {"bbox": [353, 198, 367, 209], "content": "\\mathbb{Z}_{n}", "parent_index": 0, "subtype": "inline"}, {"bbox": [234, 326, 263, 337], "content": "n\\neq4", "parent_index": 3, "subtype": "inline"}, {"bbox": [225, 374, 256, 383], "content": "n=4", "parent_index": 4, "subtype": "inline"}, {"bbox": [409, 520, 423, 530], "content": "B_{4}", "parent_index": 6, "subtype": "inline"}, {"bbox": [262, 575, 278, 588], "content": "(a)", "parent_index": 7, "subtype": "inline"}, {"bbox": [213, 600, 222, 608], "content": "A", "parent_index": 8, "subtype": "inline"}, {"bbox": [248, 600, 258, 608], "content": "B", "parent_index": 8, "subtype": "inline"}, {"bbox": [158, 613, 219, 623], "content": "A^{2}B=A B^{2}", "parent_index": 8, "subtype": "inline"}, {"bbox": [227, 613, 289, 622], "content": "B A^{2}=B^{2}A", "parent_index": 8, "subtype": "inline"}, {"bbox": [138, 627, 154, 639], "content": "(b)", "parent_index": 8, "subtype": "inline"}, {"bbox": [169, 627, 305, 639], "content": "x\\in I m(A)\\cap K e r(A-\\lambda I)", "parent_index": 8, "subtype": "inline"}, {"bbox": [338, 627, 421, 640], "content": "B(B x)=\\lambda(B x)", "parent_index": 8, "subtype": "inline"}, {"bbox": [447, 627, 487, 638], "content": "A B x=", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 640, 178, 653], "content": "-(1+\\lambda)x", "parent_index": 8, "subtype": "inline"}, {"bbox": [282, 665, 291, 675], "content": "A", "parent_index": 9, "subtype": "inline"}, {"bbox": [317, 665, 327, 675], "content": "B", "parent_index": 9, "subtype": "inline"}, {"bbox": [205, 686, 405, 700], "content": "A+A^{2}+A B A=B+B^{2}+B A B=0.", "parent_index": 10, "subtype": "interline"}]
|
[]
|
|
Multiplying the left hand side on the right by $B$ and the right hand side on the left by $A$ gives
$$
A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.
$$
Thus, $A^{2}B=A B^{2}$ ; by a symmetric argument $B A^{2}=B^{2}A$ .
$$
B(B x)=B^{2}A y=B A^{2}y=B A x=\lambda B x,
$$
and
$$
0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\lambda)x+A B x.
$$
Thus, $A B x=-(1+\lambda)x$ .
Theorem 3.8. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ , $(n\geq2,$ ) be an irreducible representation, whose associated friendship graph is totally disconnected. Then $r=d i m V\leq n-1$ .
Proof. If $A_{i}=0$ , $\rho$ is a trivial representation and $r=1$
If $A_{i}\neq0$ , choose an eigenvalue $\lambda$ for $A_{1}$ and a non-zero vector
$$
x_{1}\in I m(A_{1})\cap K e r(A_{1}-\lambda I).
$$
Set $x_{2}=A_{2}x_{1}$ $\begin{array}{r}{\mathrm{~}_{1},x_{3}=A_{3}x_{2},\,\cdot\,.\,.\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\{x_{1},x_{2},\,.\,.\,.\,,x_{n-1}\}}\end{array}$ By induction and lemma 3.7 (b) $x_{i}\in I m(A_{i})\cap K e r(A_{i}-\lambda I)$ .
Let $x_{i}\,=\,A_{i}y_{i}$ . Then by lemma 3.7 (b) and the fact that $A_{i}A_{j}\;=\;$ $A_{j}A_{i}=0$ , if $i$ and $j$ are not neighbors,
$$
A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\lambda)x_{i-1},\ \ i=2,\ldots,n-1,
$$
$$
A_{i}x_{i}=\lambda x_{i},\;\;\;i=1,\ldots,n-1,
$$
$$
A_{i+1}x_{i}=x_{i+1},\;\;\;i=1,\ldots,n-2,
$$
and
$$
A_{j}x_{i}=A_{j}A_{i}y_{i}=0\;\;j\neq i-1,i,i+1.
$$
Thus $U$ is invariant under $B_{n}$ . Hence $r=d i m U\le n-1$ , since $\rho$ is irreducible.
Corollary 3.9. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be irreducible, where $r=d i m V\ge$ $n$ , $n\neq4$ .
Then the associated friendship graph is connected.
Proof. By corollary 3.5 the friendship graph of $\rho$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected.
Corollary 3.10. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $:(A_{i})=k$ .
Then $r=d i m V\leq(n-1)(k-1)+1$ .
In particular, for $k=2$ , $r=d i m V=n$ , where $V=\mathbb{C}^{n}$ .
|
<html><body>
<p data-bbox="123 110 487 138">Multiplying the left hand side on the right by $B$ and the right hand side on the left by $A$ gives </p>
<div class="equation" data-bbox="177 144 432 158">$$
A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.
$$</div>
<p data-bbox="125 161 430 176">Thus, $A^{2}B=A B^{2}$ ; by a symmetric argument $B A^{2}=B^{2}A$ . </p>
<div class="equation" data-bbox="200 196 408 210">$$
B(B x)=B^{2}A y=B A^{2}y=B A x=\lambda B x,
$$</div>
<p data-bbox="125 215 147 227">and </p>
<div class="equation" data-bbox="149 233 461 249">$$
0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\lambda)x+A B x.
$$</div>
<p data-bbox="124 251 255 266">Thus, $A B x=-(1+\lambda)x$ . </p>
<p data-bbox="125 285 486 327">Theorem 3.8. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ , $(n\geq2,$ ) be an irreducible representation, whose associated friendship graph is totally disconnected. Then $r=d i m V\leq n-1$ . </p>
<p data-bbox="138 334 425 347">Proof. If $A_{i}=0$ , $\rho$ is a trivial representation and $r=1$ </p>
<p data-bbox="135 349 460 362">If $A_{i}\neq0$ , choose an eigenvalue $\lambda$ for $A_{1}$ and a non-zero vector </p>
<div class="equation" data-bbox="228 369 381 382">$$
x_{1}\in I m(A_{1})\cap K e r(A_{1}-\lambda I).
$$</div>
<p data-bbox="124 385 514 413">Set $x_{2}=A_{2}x_{1}$ $\begin{array}{r}{\mathrm{~}_{1},x_{3}=A_{3}x_{2},\,\cdot\,.\,.\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\{x_{1},x_{2},\,.\,.\,.\,,x_{n-1}\}}\end{array}$ By induction and lemma 3.7 (b) $x_{i}\in I m(A_{i})\cap K e r(A_{i}-\lambda I)$ . </p>
<p data-bbox="125 414 487 441">Let $x_{i}\,=\,A_{i}y_{i}$ . Then by lemma 3.7 (b) and the fact that $A_{i}A_{j}\;=\;$ $A_{j}A_{i}=0$ , if $i$ and $j$ are not neighbors, </p>
<div class="equation" data-bbox="163 448 446 462">$$
A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\lambda)x_{i-1},\ \ i=2,\ldots,n-1,
$$</div>
<div class="equation" data-bbox="229 468 379 480">$$
A_{i}x_{i}=\lambda x_{i},\;\;\;i=1,\ldots,n-1,
$$</div>
<div class="equation" data-bbox="222 484 388 497">$$
A_{i+1}x_{i}=x_{i+1},\;\;\;i=1,\ldots,n-2,
$$</div>
<p data-bbox="125 498 147 511">and </p>
<div class="equation" data-bbox="209 513 399 528">$$
A_{j}x_{i}=A_{j}A_{i}y_{i}=0\;\;j\neq i-1,i,i+1.
$$</div>
<p data-bbox="124 528 488 556">Thus $U$ is invariant under $B_{n}$ . Hence $r=d i m U\le n-1$ , since $\rho$ is irreducible. </p>
<p data-bbox="124 561 493 590">Corollary 3.9. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be irreducible, where $r=d i m V\ge$ $n$ , $n\neq4$ . </p>
<p data-bbox="140 590 395 604">Then the associated friendship graph is connected. </p>
<p data-bbox="124 609 486 638">Proof. By corollary 3.5 the friendship graph of $\rho$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected. </p>
<p data-bbox="124 643 486 672">Corollary 3.10. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $:(A_{i})=k$ . </p>
<p data-bbox="136 672 422 701">Then $r=d i m V\leq(n-1)(k-1)+1$ .
In particular, for $k=2$ , $r=d i m V=n$ , where $V=\mathbb{C}^{n}$ . </p>
</body></html>
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0003047v1
| 7 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Multiplying the left hand side on the right by $B$ and the right hand side on the left by $A$ gives ", "page_idx": 7}, {"type": "equation", "text": "$$\nA B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus, $A^{2}B=A B^{2}$ ; by a symmetric argument $B A^{2}=B^{2}A$ . ", "page_idx": 7}, {"type": "equation", "text": "$$\nB(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and ", "page_idx": 7}, {"type": "equation", "text": "$$\n0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus, $A B x=-(1+\\lambda)x$ . ", "page_idx": 7}, {"type": "text", "text": "Theorem 3.8. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ , $(n\\geq2,$ ) be an irreducible representation, whose associated friendship graph is totally disconnected. Then $r=d i m V\\leq n-1$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. If $A_{i}=0$ , $\\rho$ is a trivial representation and $r=1$ ", "page_idx": 7}, {"type": "text", "text": "If $A_{i}\\neq0$ , choose an eigenvalue $\\lambda$ for $A_{1}$ and a non-zero vector ", "page_idx": 7}, {"type": "equation", "text": "$$\nx_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Set $x_{2}=A_{2}x_{1}$ $\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}$ By induction and lemma 3.7 (b) $x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)$ . ", "page_idx": 7}, {"type": "text", "text": "Let $x_{i}\\,=\\,A_{i}y_{i}$ . Then by lemma 3.7 (b) and the fact that $A_{i}A_{j}\\;=\\;$ $A_{j}A_{i}=0$ , if $i$ and $j$ are not neighbors, ", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and ", "page_idx": 7}, {"type": "equation", "text": "$$\nA_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Thus $U$ is invariant under $B_{n}$ . Hence $r=d i m U\\le n-1$ , since $\\rho$ is irreducible. ", "page_idx": 7}, {"type": "text", "text": "Corollary 3.9. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be irreducible, where $r=d i m V\\ge$ $n$ , $n\\neq4$ . ", "page_idx": 7}, {"type": "text", "text": "Then the associated friendship graph is connected. ", "page_idx": 7}, {"type": "text", "text": "Proof. By corollary 3.5 the friendship graph of $\\rho$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected. ", "page_idx": 7}, {"type": "text", "text": "Corollary 3.10. Let $\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})$ be irreducible, where $r=$ $d i m V\\geq n$ , $n\\neq4$ . Suppose $\\rho(\\sigma_{i})=1+A_{i}$ , where rank $:(A_{i})=k$ . ", "page_idx": 7}, {"type": "text", "text": "Then $r=d i m V\\leq(n-1)(k-1)+1$ . \nIn particular, for $k=2$ , $r=d i m V=n$ , where $V=\\mathbb{C}^{n}$ . ", "page_idx": 7}]
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{"preproc_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 371, 127], "score": 1.0, "content": "Multiplying the left hand side on the right by ", "type": "text"}, {"bbox": [371, 115, 381, 124], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [381, 113, 486, 127], "score": 1.0, "content": " and the right hand", "type": "text"}], "index": 0}, {"bbox": [125, 126, 262, 141], "spans": [{"bbox": [125, 126, 222, 141], "score": 1.0, "content": "side on the left by ", "type": "text"}, {"bbox": [223, 128, 232, 137], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [232, 126, 262, 141], "score": 1.0, "content": " gives", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [125, 161, 430, 176], "lines": [{"bbox": [126, 163, 428, 178], "spans": [{"bbox": [126, 163, 159, 178], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 165, 221, 175], "score": 0.94, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [221, 163, 363, 178], "score": 1.0, "content": "; by a symmetric argument ", "type": "text"}, {"bbox": [363, 165, 425, 175], "score": 0.93, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [425, 163, 428, 178], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 215, 147, 227], "lines": [{"bbox": [125, 216, 147, 229], "spans": [{"bbox": [125, 216, 147, 229], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 251, 255, 266], "lines": [{"bbox": [126, 253, 254, 268], "spans": [{"bbox": [126, 253, 159, 268], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 255, 252, 267], "score": 0.91, "content": "A B x=-(1+\\lambda)x", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [252, 253, 254, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [125, 285, 486, 327], "lines": [{"bbox": [124, 287, 486, 303], "spans": [{"bbox": [124, 287, 230, 303], "score": 1.0, "content": "Theorem 3.8. Let ", "type": "text"}, {"bbox": [230, 288, 324, 301], "score": 0.87, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [324, 287, 333, 303], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [333, 288, 368, 301], "score": 0.56, "content": "(n\\geq2,", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [368, 287, 486, 303], "score": 1.0, "content": ") be an irreducible rep-", "type": "text"}], "index": 8}, {"bbox": [126, 303, 485, 316], "spans": [{"bbox": [126, 303, 485, 316], "score": 1.0, "content": "resentation, whose associated friendship graph is totally disconnected.", "type": "text"}], "index": 9}, {"bbox": [127, 316, 255, 329], "spans": [{"bbox": [127, 316, 156, 329], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [156, 318, 251, 329], "score": 0.9, "content": "r=d i m V\\leq n-1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [252, 316, 255, 329], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9}, {"type": "text", "bbox": [138, 334, 425, 347], "lines": [{"bbox": [137, 336, 426, 349], "spans": [{"bbox": [137, 336, 191, 349], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [192, 338, 226, 348], "score": 0.9, "content": "A_{i}=0", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [226, 336, 232, 349], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 341, 239, 349], "score": 0.85, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [240, 336, 398, 349], "score": 1.0, "content": " is a trivial representation and ", "type": "text"}, {"bbox": [398, 338, 426, 347], "score": 0.91, "content": "r=1", "type": "inline_equation", "height": 9, "width": 28}], "index": 11}], "index": 11}, {"type": "text", "bbox": [135, 349, 460, 362], "lines": [{"bbox": [137, 349, 458, 363], "spans": [{"bbox": [137, 349, 149, 363], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 352, 183, 363], "score": 0.93, "content": "A_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [184, 349, 299, 363], "score": 1.0, "content": ", choose an eigenvalue ", "type": "text"}, {"bbox": [300, 352, 307, 360], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 349, 328, 363], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [328, 352, 342, 362], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [343, 349, 458, 363], "score": 1.0, "content": " and a non-zero vector", "type": "text"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 385, 514, 413], "lines": [{"bbox": [125, 386, 513, 403], "spans": [{"bbox": [125, 386, 145, 403], "score": 1.0, "content": "Set ", "type": "text"}, {"bbox": [145, 389, 197, 400], "score": 0.83, "content": "x_{2}=A_{2}x_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [193, 389, 513, 401], "score": 0.82, "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "type": "inline_equation", "height": 12, "width": 320}], "index": 14}, {"bbox": [126, 401, 445, 415], "spans": [{"bbox": [126, 401, 294, 415], "score": 1.0, "content": "By induction and lemma 3.7 (b) ", "type": "text"}, {"bbox": [295, 402, 442, 415], "score": 0.92, "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "type": "inline_equation", "height": 13, "width": 147}, {"bbox": [442, 401, 445, 415], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "text", "bbox": [125, 414, 487, 441], "lines": [{"bbox": [136, 413, 486, 430], "spans": [{"bbox": [136, 413, 159, 430], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [160, 417, 212, 428], "score": 0.93, "content": "x_{i}\\,=\\,A_{i}y_{i}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [212, 413, 444, 430], "score": 1.0, "content": ". Then by lemma 3.7 (b) and the fact that ", "type": "text"}, {"bbox": [444, 417, 486, 429], "score": 0.87, "content": "A_{i}A_{j}\\;=\\;", "type": "inline_equation", "height": 12, "width": 42}], "index": 16}, {"bbox": [126, 429, 324, 444], "spans": [{"bbox": [126, 431, 173, 443], "score": 0.92, "content": "A_{j}A_{i}=0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [174, 429, 190, 444], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [191, 432, 195, 440], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 429, 221, 444], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [221, 432, 227, 442], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [227, 429, 324, 444], "score": 1.0, "content": " are not neighbors,", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [125, 498, 147, 511], "lines": [{"bbox": [125, 499, 147, 513], "spans": [{"bbox": [125, 499, 147, 513], "score": 1.0, "content": "and", "type": "text"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [124, 528, 488, 556], "lines": [{"bbox": [137, 529, 486, 544], "spans": [{"bbox": [137, 529, 168, 544], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [168, 532, 177, 541], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [177, 529, 276, 544], "score": 1.0, "content": " is invariant under ", "type": "text"}, {"bbox": [276, 531, 291, 542], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [291, 529, 333, 544], "score": 1.0, "content": ". Hence ", "type": "text"}, {"bbox": [333, 530, 430, 542], "score": 0.87, "content": "r=d i m U\\le n-1", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [431, 529, 465, 544], "score": 1.0, "content": ", since", "type": "text"}, {"bbox": [466, 533, 473, 543], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [474, 529, 486, 544], "score": 1.0, "content": " is", "type": "text"}], "index": 23}, {"bbox": [125, 544, 184, 557], "spans": [{"bbox": [125, 544, 184, 557], "score": 1.0, "content": "irreducible.", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "text", "bbox": [124, 561, 493, 590], "lines": [{"bbox": [126, 563, 493, 579], "spans": [{"bbox": [126, 563, 231, 579], "score": 1.0, "content": "Corollary 3.9. Let ", "type": "text"}, {"bbox": [231, 565, 320, 577], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [321, 563, 427, 579], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [427, 565, 493, 577], "score": 0.8, "content": "r=d i m V\\ge", "type": "inline_equation", "height": 12, "width": 66}], "index": 25}, {"bbox": [126, 578, 173, 592], "spans": [{"bbox": [126, 583, 133, 588], "score": 0.78, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [134, 578, 140, 592], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [140, 580, 169, 591], "score": 0.92, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [169, 578, 173, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [140, 590, 395, 604], "lines": [{"bbox": [139, 592, 393, 605], "spans": [{"bbox": [139, 592, 393, 605], "score": 1.0, "content": "Then the associated friendship graph is connected.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 609, 486, 638], "lines": [{"bbox": [137, 612, 484, 627], "spans": [{"bbox": [137, 612, 393, 627], "score": 1.0, "content": "Proof. By corollary 3.5 the friendship graph of ", "type": "text"}, {"bbox": [394, 614, 401, 625], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [401, 612, 484, 627], "score": 1.0, "content": " is either totally", "type": "text"}], "index": 28}, {"bbox": [126, 626, 464, 639], "spans": [{"bbox": [126, 626, 464, 639], "score": 1.0, "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "text", "bbox": [124, 643, 486, 672], "lines": [{"bbox": [126, 645, 486, 660], "spans": [{"bbox": [126, 645, 241, 660], "score": 1.0, "content": "Corollary 3.10. Let ", "type": "text"}, {"bbox": [241, 646, 344, 659], "score": 0.9, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [344, 645, 462, 660], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [463, 647, 486, 658], "score": 0.79, "content": "r=", "type": "inline_equation", "height": 11, "width": 23}], "index": 30}, {"bbox": [126, 660, 453, 674], "spans": [{"bbox": [126, 661, 178, 672], "score": 0.42, "content": "d i m V\\geq n", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [179, 660, 185, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 662, 215, 673], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 660, 266, 674], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 660, 340, 673], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [340, 660, 404, 674], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 660, 450, 673], "score": 0.54, "content": ":(A_{i})=k", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [450, 660, 453, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [136, 672, 422, 701], "lines": [{"bbox": [138, 673, 333, 689], "spans": [{"bbox": [138, 673, 168, 689], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 675, 329, 687], "score": 0.94, "content": "r=d i m V\\leq(n-1)(k-1)+1", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [329, 673, 333, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [137, 688, 420, 700], "spans": [{"bbox": [137, 688, 228, 700], "score": 1.0, "content": "In particular, for ", "type": "text"}, {"bbox": [229, 690, 257, 699], "score": 0.87, "content": "k=2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [258, 688, 264, 700], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [264, 689, 339, 699], "score": 0.85, "content": "r=d i m V=n", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [339, 688, 378, 700], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [378, 689, 418, 699], "score": 0.86, "content": "V=\\mathbb{C}^{n}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [419, 688, 420, 700], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32.5}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [124, 91, 132, 100], "lines": [{"bbox": [125, 93, 132, 103], "spans": [{"bbox": [125, 93, 132, 103], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 371, 127], "score": 1.0, "content": "Multiplying the left hand side on the right by ", "type": "text"}, {"bbox": [371, 115, 381, 124], "score": 0.91, "content": "B", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [381, 113, 486, 127], "score": 1.0, "content": " and the right hand", "type": "text"}], "index": 0}, {"bbox": [125, 126, 262, 141], "spans": [{"bbox": [125, 126, 222, 141], "score": 1.0, "content": "side on the left by ", "type": "text"}, {"bbox": [223, 128, 232, 137], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [232, 126, 262, 141], "score": 1.0, "content": " gives", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [125, 113, 486, 141]}, {"type": "interline_equation", "bbox": [177, 144, 432, 158], "lines": [{"bbox": [177, 144, 432, 158], "spans": [{"bbox": [177, 144, 432, 158], "score": 0.87, "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [125, 161, 430, 176], "lines": [{"bbox": [126, 163, 428, 178], "spans": [{"bbox": [126, 163, 159, 178], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 165, 221, 175], "score": 0.94, "content": "A^{2}B=A B^{2}", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [221, 163, 363, 178], "score": 1.0, "content": "; by a symmetric argument ", "type": "text"}, {"bbox": [363, 165, 425, 175], "score": 0.93, "content": "B A^{2}=B^{2}A", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [425, 163, 428, 178], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 3, "bbox_fs": [126, 163, 428, 178]}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "lines": [{"bbox": [200, 196, 408, 210], "spans": [{"bbox": [200, 196, 408, 210], "score": 0.86, "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 215, 147, 227], "lines": [{"bbox": [125, 216, 147, 229], "spans": [{"bbox": [125, 216, 147, 229], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [125, 216, 147, 229]}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "lines": [{"bbox": [149, 233, 461, 249], "spans": [{"bbox": [149, 233, 461, 249], "score": 0.89, "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 251, 255, 266], "lines": [{"bbox": [126, 253, 254, 268], "spans": [{"bbox": [126, 253, 159, 268], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [159, 255, 252, 267], "score": 0.91, "content": "A B x=-(1+\\lambda)x", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [252, 253, 254, 268], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [126, 253, 254, 268]}, {"type": "text", "bbox": [125, 285, 486, 327], "lines": [{"bbox": [124, 287, 486, 303], "spans": [{"bbox": [124, 287, 230, 303], "score": 1.0, "content": "Theorem 3.8. Let ", "type": "text"}, {"bbox": [230, 288, 324, 301], "score": 0.87, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 94}, {"bbox": [324, 287, 333, 303], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [333, 288, 368, 301], "score": 0.56, "content": "(n\\geq2,", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [368, 287, 486, 303], "score": 1.0, "content": ") be an irreducible rep-", "type": "text"}], "index": 8}, {"bbox": [126, 303, 485, 316], "spans": [{"bbox": [126, 303, 485, 316], "score": 1.0, "content": "resentation, whose associated friendship graph is totally disconnected.", "type": "text"}], "index": 9}, {"bbox": [127, 316, 255, 329], "spans": [{"bbox": [127, 316, 156, 329], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [156, 318, 251, 329], "score": 0.9, "content": "r=d i m V\\leq n-1", "type": "inline_equation", "height": 11, "width": 95}, {"bbox": [252, 316, 255, 329], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9, "bbox_fs": [124, 287, 486, 329]}, {"type": "text", "bbox": [138, 334, 425, 347], "lines": [{"bbox": [137, 336, 426, 349], "spans": [{"bbox": [137, 336, 191, 349], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [192, 338, 226, 348], "score": 0.9, "content": "A_{i}=0", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [226, 336, 232, 349], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [232, 341, 239, 349], "score": 0.85, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [240, 336, 398, 349], "score": 1.0, "content": " is a trivial representation and ", "type": "text"}, {"bbox": [398, 338, 426, 347], "score": 0.91, "content": "r=1", "type": "inline_equation", "height": 9, "width": 28}], "index": 11}], "index": 11, "bbox_fs": [137, 336, 426, 349]}, {"type": "text", "bbox": [135, 349, 460, 362], "lines": [{"bbox": [137, 349, 458, 363], "spans": [{"bbox": [137, 349, 149, 363], "score": 1.0, "content": "If ", "type": "text"}, {"bbox": [149, 352, 183, 363], "score": 0.93, "content": "A_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [184, 349, 299, 363], "score": 1.0, "content": ", choose an eigenvalue ", "type": "text"}, {"bbox": [300, 352, 307, 360], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [307, 349, 328, 363], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [328, 352, 342, 362], "score": 0.91, "content": "A_{1}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [343, 349, 458, 363], "score": 1.0, "content": " and a non-zero vector", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [137, 349, 458, 363]}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "lines": [{"bbox": [228, 369, 381, 382], "spans": [{"bbox": [228, 369, 381, 382], "score": 0.9, "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 385, 514, 413], "lines": [{"bbox": [125, 386, 513, 403], "spans": [{"bbox": [125, 386, 145, 403], "score": 1.0, "content": "Set ", "type": "text"}, {"bbox": [145, 389, 197, 400], "score": 0.83, "content": "x_{2}=A_{2}x_{1}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [193, 389, 513, 401], "score": 0.82, "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "type": "inline_equation", "height": 12, "width": 320}], "index": 14}, {"bbox": [126, 401, 445, 415], "spans": [{"bbox": [126, 401, 294, 415], "score": 1.0, "content": "By induction and lemma 3.7 (b) ", "type": "text"}, {"bbox": [295, 402, 442, 415], "score": 0.92, "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "type": "inline_equation", "height": 13, "width": 147}, {"bbox": [442, 401, 445, 415], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [125, 386, 513, 415]}, {"type": "text", "bbox": [125, 414, 487, 441], "lines": [{"bbox": [136, 413, 486, 430], "spans": [{"bbox": [136, 413, 159, 430], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [160, 417, 212, 428], "score": 0.93, "content": "x_{i}\\,=\\,A_{i}y_{i}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [212, 413, 444, 430], "score": 1.0, "content": ". Then by lemma 3.7 (b) and the fact that ", "type": "text"}, {"bbox": [444, 417, 486, 429], "score": 0.87, "content": "A_{i}A_{j}\\;=\\;", "type": "inline_equation", "height": 12, "width": 42}], "index": 16}, {"bbox": [126, 429, 324, 444], "spans": [{"bbox": [126, 431, 173, 443], "score": 0.92, "content": "A_{j}A_{i}=0", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [174, 429, 190, 444], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [191, 432, 195, 440], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 429, 221, 444], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [221, 432, 227, 442], "score": 0.88, "content": "j", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [227, 429, 324, 444], "score": 1.0, "content": " are not neighbors,", "type": "text"}], "index": 17}], "index": 16.5, "bbox_fs": [126, 413, 486, 444]}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "lines": [{"bbox": [163, 448, 446, 462], "spans": [{"bbox": [163, 448, 446, 462], "score": 0.62, "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "lines": [{"bbox": [229, 468, 379, 480], "spans": [{"bbox": [229, 468, 379, 480], "score": 0.72, "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "lines": [{"bbox": [222, 484, 388, 497], "spans": [{"bbox": [222, 484, 388, 497], "score": 0.67, "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [125, 498, 147, 511], "lines": [{"bbox": [125, 499, 147, 513], "spans": [{"bbox": [125, 499, 147, 513], "score": 1.0, "content": "and", "type": "text"}], "index": 21}], "index": 21, "bbox_fs": [125, 499, 147, 513]}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "lines": [{"bbox": [209, 513, 399, 528], "spans": [{"bbox": [209, 513, 399, 528], "score": 0.88, "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [124, 528, 488, 556], "lines": [{"bbox": [137, 529, 486, 544], "spans": [{"bbox": [137, 529, 168, 544], "score": 1.0, "content": "Thus ", "type": "text"}, {"bbox": [168, 532, 177, 541], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [177, 529, 276, 544], "score": 1.0, "content": " is invariant under ", "type": "text"}, {"bbox": [276, 531, 291, 542], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [291, 529, 333, 544], "score": 1.0, "content": ". Hence ", "type": "text"}, {"bbox": [333, 530, 430, 542], "score": 0.87, "content": "r=d i m U\\le n-1", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [431, 529, 465, 544], "score": 1.0, "content": ", since", "type": "text"}, {"bbox": [466, 533, 473, 543], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [474, 529, 486, 544], "score": 1.0, "content": " is", "type": "text"}], "index": 23}, {"bbox": [125, 544, 184, 557], "spans": [{"bbox": [125, 544, 184, 557], "score": 1.0, "content": "irreducible.", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [125, 529, 486, 557]}, {"type": "text", "bbox": [124, 561, 493, 590], "lines": [{"bbox": [126, 563, 493, 579], "spans": [{"bbox": [126, 563, 231, 579], "score": 1.0, "content": "Corollary 3.9. Let ", "type": "text"}, {"bbox": [231, 565, 320, 577], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [321, 563, 427, 579], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [427, 565, 493, 577], "score": 0.8, "content": "r=d i m V\\ge", "type": "inline_equation", "height": 12, "width": 66}], "index": 25}, {"bbox": [126, 578, 173, 592], "spans": [{"bbox": [126, 583, 133, 588], "score": 0.78, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [134, 578, 140, 592], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [140, 580, 169, 591], "score": 0.92, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [169, 578, 173, 592], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [126, 563, 493, 592]}, {"type": "text", "bbox": [140, 590, 395, 604], "lines": [{"bbox": [139, 592, 393, 605], "spans": [{"bbox": [139, 592, 393, 605], "score": 1.0, "content": "Then the associated friendship graph is connected.", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [139, 592, 393, 605]}, {"type": "text", "bbox": [124, 609, 486, 638], "lines": [{"bbox": [137, 612, 484, 627], "spans": [{"bbox": [137, 612, 393, 627], "score": 1.0, "content": "Proof. By corollary 3.5 the friendship graph of ", "type": "text"}, {"bbox": [394, 614, 401, 625], "score": 0.81, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [401, 612, 484, 627], "score": 1.0, "content": " is either totally", "type": "text"}], "index": 28}, {"bbox": [126, 626, 464, 639], "spans": [{"bbox": [126, 626, 464, 639], "score": 1.0, "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "type": "text"}], "index": 29}], "index": 28.5, "bbox_fs": [126, 612, 484, 639]}, {"type": "text", "bbox": [124, 643, 486, 672], "lines": [{"bbox": [126, 645, 486, 660], "spans": [{"bbox": [126, 645, 241, 660], "score": 1.0, "content": "Corollary 3.10. Let ", "type": "text"}, {"bbox": [241, 646, 344, 659], "score": 0.9, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [344, 645, 462, 660], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [463, 647, 486, 658], "score": 0.79, "content": "r=", "type": "inline_equation", "height": 11, "width": 23}], "index": 30}, {"bbox": [126, 660, 453, 674], "spans": [{"bbox": [126, 661, 178, 672], "score": 0.42, "content": "d i m V\\geq n", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [179, 660, 185, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 662, 215, 673], "score": 0.87, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 660, 266, 674], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 660, 340, 673], "score": 0.9, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [340, 660, 404, 674], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 660, 450, 673], "score": 0.54, "content": ":(A_{i})=k", "type": "inline_equation", "height": 13, "width": 45}, {"bbox": [450, 660, 453, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [126, 645, 486, 674]}, {"type": "list", "bbox": [136, 672, 422, 701], "lines": [{"bbox": [138, 673, 333, 689], "spans": [{"bbox": [138, 673, 168, 689], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 675, 329, 687], "score": 0.94, "content": "r=d i m V\\leq(n-1)(k-1)+1", "type": "inline_equation", "height": 12, "width": 161}, {"bbox": [329, 673, 333, 687], "score": 1.0, "content": ".", "type": "text"}], "index": 32, "is_list_end_line": true}, {"bbox": [137, 688, 420, 700], "spans": [{"bbox": [137, 688, 228, 700], "score": 1.0, "content": "In particular, for ", "type": "text"}, {"bbox": [229, 690, 257, 699], "score": 0.87, "content": "k=2", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [258, 688, 264, 700], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [264, 689, 339, 699], "score": 0.85, "content": "r=d i m V=n", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [339, 688, 378, 700], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [378, 689, 418, 699], "score": 0.86, "content": "V=\\mathbb{C}^{n}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [419, 688, 420, 700], "score": 1.0, "content": ".", "type": "text"}], "index": 33, "is_list_start_line": true, "is_list_end_line": true}], "index": 32.5, "bbox_fs": [137, 673, 420, 700]}]}
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[{"type": "text", "bbox": [123, 110, 487, 138], "content": "Multiplying the left hand side on the right by and the right hand side on the left by gives", "index": 0}, {"type": "interline_equation", "bbox": [177, 144, 432, 158], "content": "", "index": 1}, {"type": "text", "bbox": [125, 161, 430, 176], "content": "Thus, ; by a symmetric argument .", "index": 2}, {"type": "interline_equation", "bbox": [200, 196, 408, 210], "content": "", "index": 3}, {"type": "text", "bbox": [125, 215, 147, 227], "content": "and", "index": 4}, {"type": "interline_equation", "bbox": [149, 233, 461, 249], "content": "", "index": 5}, {"type": "text", "bbox": [124, 251, 255, 266], "content": "Thus, .", "index": 6}, {"type": "text", "bbox": [125, 285, 486, 327], "content": "Theorem 3.8. Let , ) be an irreducible rep- resentation, whose associated friendship graph is totally disconnected. Then .", "index": 7}, {"type": "text", "bbox": [138, 334, 425, 347], "content": "Proof. If , is a trivial representation and", "index": 8}, {"type": "text", "bbox": [135, 349, 460, 362], "content": "If , choose an eigenvalue for and a non-zero vector", "index": 9}, {"type": "interline_equation", "bbox": [228, 369, 381, 382], "content": "", "index": 10}, {"type": "text", "bbox": [124, 385, 514, 413], "content": "Set By induction and lemma 3.7 (b) .", "index": 11}, {"type": "text", "bbox": [125, 414, 487, 441], "content": "Let . Then by lemma 3.7 (b) and the fact that , if and are not neighbors,", "index": 12}, {"type": "interline_equation", "bbox": [163, 448, 446, 462], "content": "", "index": 13}, {"type": "interline_equation", "bbox": [229, 468, 379, 480], "content": "", "index": 14}, {"type": "interline_equation", "bbox": [222, 484, 388, 497], "content": "", "index": 15}, {"type": "text", "bbox": [125, 498, 147, 511], "content": "and", "index": 16}, {"type": "interline_equation", "bbox": [209, 513, 399, 528], "content": "", "index": 17}, {"type": "text", "bbox": [124, 528, 488, 556], "content": "Thus is invariant under . Hence , since is irreducible.", "index": 18}, {"type": "text", "bbox": [124, 561, 493, 590], "content": "Corollary 3.9. Let be irreducible, where , .", "index": 19}, {"type": "text", "bbox": [140, 590, 395, 604], "content": "Then the associated friendship graph is connected.", "index": 20}, {"type": "text", "bbox": [124, 609, 486, 638], "content": "Proof. By corollary 3.5 the friendship graph of is either totally disconnected or connected. By theorem 3.8 it is not disconnected.", "index": 21}, {"type": "text", "bbox": [124, 643, 486, 672], "content": "Corollary 3.10. Let be irreducible, where , . Suppose , where rank .", "index": 22}, {"type": "list", "bbox": [136, 672, 422, 701], "content": "", "index": 23}]
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[{"bbox": [126, 113, 486, 127], "content": "Multiplying the left hand side on the right by and the right hand", "parent_index": 0, "line_index": 0}, {"bbox": [125, 126, 262, 141], "content": "side on the left by gives", "parent_index": 0, "line_index": 1}, {"bbox": [126, 163, 428, 178], "content": "Thus, ; by a symmetric argument .", "parent_index": 2, "line_index": 0}, {"bbox": [125, 216, 147, 229], "content": "and", "parent_index": 4, "line_index": 0}, {"bbox": [126, 253, 254, 268], "content": "Thus, .", "parent_index": 6, "line_index": 0}, {"bbox": [124, 287, 486, 303], "content": "Theorem 3.8. Let , ) be an irreducible rep-", "parent_index": 7, "line_index": 0}, {"bbox": [126, 303, 485, 316], "content": "resentation, whose associated friendship graph is totally disconnected.", "parent_index": 7, "line_index": 1}, {"bbox": [127, 316, 255, 329], "content": "Then .", "parent_index": 7, "line_index": 2}, {"bbox": [137, 336, 426, 349], "content": "Proof. If , is a trivial representation and", "parent_index": 8, "line_index": 0}, {"bbox": [137, 349, 458, 363], "content": "If , choose an eigenvalue for and a non-zero vector", "parent_index": 9, "line_index": 0}, {"bbox": [125, 386, 513, 403], "content": "Set", "parent_index": 11, "line_index": 0}, {"bbox": [126, 401, 445, 415], "content": "By induction and lemma 3.7 (b) .", "parent_index": 11, "line_index": 1}, {"bbox": [136, 413, 486, 430], "content": "Let . Then by lemma 3.7 (b) and the fact that", "parent_index": 12, "line_index": 0}, {"bbox": [126, 429, 324, 444], "content": ", if and are not neighbors,", "parent_index": 12, "line_index": 1}, {"bbox": [125, 499, 147, 513], "content": "and", "parent_index": 16, "line_index": 0}, {"bbox": [137, 529, 486, 544], "content": "Thus is invariant under . Hence , since is", "parent_index": 18, "line_index": 0}, {"bbox": [125, 544, 184, 557], "content": "irreducible.", "parent_index": 18, "line_index": 1}, {"bbox": [126, 563, 493, 579], "content": "Corollary 3.9. Let be irreducible, where", "parent_index": 19, "line_index": 0}, {"bbox": [126, 578, 173, 592], "content": ", .", "parent_index": 19, "line_index": 1}, {"bbox": [139, 592, 393, 605], "content": "Then the associated friendship graph is connected.", "parent_index": 20, "line_index": 0}, {"bbox": [137, 612, 484, 627], "content": "Proof. By corollary 3.5 the friendship graph of is either totally", "parent_index": 21, "line_index": 0}, {"bbox": [126, 626, 464, 639], "content": "disconnected or connected. By theorem 3.8 it is not disconnected.", "parent_index": 21, "line_index": 1}, {"bbox": [126, 645, 486, 660], "content": "Corollary 3.10. Let be irreducible, where", "parent_index": 22, "line_index": 0}, {"bbox": [126, 660, 453, 674], "content": ", . Suppose , where rank .", "parent_index": 22, "line_index": 1}, {"bbox": [138, 673, 333, 689], "content": "Then .", "parent_index": 23, "line_index": 0}, {"bbox": [137, 688, 420, 700], "content": "In particular, for , , where .", "parent_index": 23, "line_index": 1}]
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[]
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[{"bbox": [371, 115, 381, 124], "content": "B", "parent_index": 0, "subtype": "inline"}, {"bbox": [223, 128, 232, 137], "content": "A", "parent_index": 0, "subtype": "inline"}, {"bbox": [177, 144, 432, 158], "content": "A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.", "parent_index": 1, "subtype": "interline"}, {"bbox": [159, 165, 221, 175], "content": "A^{2}B=A B^{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [363, 165, 425, 175], "content": "B A^{2}=B^{2}A", "parent_index": 2, "subtype": "inline"}, {"bbox": [200, 196, 408, 210], "content": "B(B x)=B^{2}A y=B A^{2}y=B A x=\\lambda B x,", "parent_index": 3, "subtype": "interline"}, {"bbox": [149, 233, 461, 249], "content": "0=(A+A^{2}+A B A)y=(1+A+A B)x=(1+\\lambda)x+A B x.", "parent_index": 5, "subtype": "interline"}, {"bbox": [159, 255, 252, 267], "content": "A B x=-(1+\\lambda)x", "parent_index": 6, "subtype": "inline"}, {"bbox": [230, 288, 324, 301], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 7, "subtype": "inline"}, {"bbox": [333, 288, 368, 301], "content": "(n\\geq2,", "parent_index": 7, "subtype": "inline"}, {"bbox": [156, 318, 251, 329], "content": "r=d i m V\\leq n-1", "parent_index": 7, "subtype": "inline"}, {"bbox": [192, 338, 226, 348], "content": "A_{i}=0", "parent_index": 8, "subtype": "inline"}, {"bbox": [232, 341, 239, 349], "content": "\\rho", "parent_index": 8, "subtype": "inline"}, {"bbox": [398, 338, 426, 347], "content": "r=1", "parent_index": 8, "subtype": "inline"}, {"bbox": [149, 352, 183, 363], "content": "A_{i}\\neq0", "parent_index": 9, "subtype": "inline"}, {"bbox": [300, 352, 307, 360], "content": "\\lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [328, 352, 342, 362], "content": "A_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [228, 369, 381, 382], "content": "x_{1}\\in I m(A_{1})\\cap K e r(A_{1}-\\lambda I).", "parent_index": 10, "subtype": "interline"}, {"bbox": [145, 389, 197, 400], "content": "x_{2}=A_{2}x_{1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [193, 389, 513, 401], "content": "\\begin{array}{r}{\\mathrm{~}_{1},x_{3}=A_{3}x_{2},\\,\\cdot\\,.\\,.\\,,x_{n-1}=A_{n-1}x_{n-2},U=s p a n\\{x_{1},x_{2},\\,.\\,.\\,.\\,,x_{n-1}\\}}\\end{array}", "parent_index": 11, "subtype": "inline"}, {"bbox": [295, 402, 442, 415], "content": "x_{i}\\in I m(A_{i})\\cap K e r(A_{i}-\\lambda I)", "parent_index": 11, "subtype": "inline"}, {"bbox": [160, 417, 212, 428], "content": "x_{i}\\,=\\,A_{i}y_{i}", "parent_index": 12, "subtype": "inline"}, {"bbox": [444, 417, 486, 429], "content": "A_{i}A_{j}\\;=\\;", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 431, 173, 443], "content": "A_{j}A_{i}=0", "parent_index": 12, "subtype": "inline"}, {"bbox": [191, 432, 195, 440], "content": "i", "parent_index": 12, "subtype": "inline"}, {"bbox": [221, 432, 227, 442], "content": "j", "parent_index": 12, "subtype": "inline"}, {"bbox": [163, 448, 446, 462], "content": "A_{i-1}x_{i}=A_{i-1}A_{i}x_{i-1}=-(1+\\lambda)x_{i-1},\\ \\ i=2,\\ldots,n-1,", "parent_index": 13, "subtype": "interline"}, {"bbox": [229, 468, 379, 480], "content": "A_{i}x_{i}=\\lambda x_{i},\\;\\;\\;i=1,\\ldots,n-1,", "parent_index": 14, "subtype": "interline"}, {"bbox": [222, 484, 388, 497], "content": "A_{i+1}x_{i}=x_{i+1},\\;\\;\\;i=1,\\ldots,n-2,", "parent_index": 15, "subtype": "interline"}, {"bbox": [209, 513, 399, 528], "content": "A_{j}x_{i}=A_{j}A_{i}y_{i}=0\\;\\;j\\neq i-1,i,i+1.", "parent_index": 17, "subtype": "interline"}, {"bbox": [168, 532, 177, 541], "content": "U", "parent_index": 18, "subtype": "inline"}, {"bbox": [276, 531, 291, 542], "content": "B_{n}", "parent_index": 18, "subtype": "inline"}, {"bbox": [333, 530, 430, 542], "content": "r=d i m U\\le n-1", "parent_index": 18, "subtype": "inline"}, {"bbox": [466, 533, 473, 543], "content": "\\rho", "parent_index": 18, "subtype": "inline"}, {"bbox": [231, 565, 320, 577], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 19, "subtype": "inline"}, {"bbox": [427, 565, 493, 577], "content": "r=d i m V\\ge", "parent_index": 19, "subtype": "inline"}, {"bbox": [126, 583, 133, 588], "content": "n", "parent_index": 19, "subtype": "inline"}, {"bbox": [140, 580, 169, 591], "content": "n\\neq4", "parent_index": 19, "subtype": "inline"}, {"bbox": [394, 614, 401, 625], "content": "\\rho", "parent_index": 21, "subtype": "inline"}, {"bbox": [241, 646, 344, 659], "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "parent_index": 22, "subtype": "inline"}, {"bbox": [463, 647, 486, 658], "content": "r=", "parent_index": 22, "subtype": "inline"}, {"bbox": [126, 661, 178, 672], "content": "d i m V\\geq n", "parent_index": 22, "subtype": "inline"}, {"bbox": [186, 662, 215, 673], "content": "n\\neq4", "parent_index": 22, "subtype": "inline"}, {"bbox": [266, 660, 340, 673], "content": "\\rho(\\sigma_{i})=1+A_{i}", "parent_index": 22, "subtype": "inline"}, {"bbox": [405, 660, 450, 673], "content": ":(A_{i})=k", "parent_index": 22, "subtype": "inline"}, {"bbox": [168, 675, 329, 687], "content": "r=d i m V\\leq(n-1)(k-1)+1", "parent_index": 23, "subtype": "inline"}, {"bbox": [229, 690, 257, 699], "content": "k=2", "parent_index": 23, "subtype": "inline"}, {"bbox": [264, 689, 339, 699], "content": "r=d i m V=n", "parent_index": 23, "subtype": "inline"}, {"bbox": [378, 689, 418, 699], "content": "V=\\mathbb{C}^{n}", "parent_index": 23, "subtype": "inline"}]
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[]
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Proof. By corollary 3.9, the friendship graph of the representation is connected. Arrange the vertices of the graph in a sequence $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{n-1}}$ such that each term $A_{i_{j}}$ , $2\leq j\leq n-1$ , is a friend of one the terms $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{j-1}}$ . Then
$$
\dim(I m(A_{i_{1}}))=k
$$
$$
\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\leq k+k-1=2k-1
$$
$$
\dim(I m(A_{i_{1}})+\cdot\cdot\cdot+I m(A_{i_{n-1}}))\leq k+(n-2)(k-1)=(n-1)(k-1)+1.
$$
Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following
Theorem 3.11. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $\cdot(A_{i})=2$ .
Then $r=n$ and one of the following holds.
(a) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ .
(b) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors.
# 4. For corank 2 the friendship graph is a chai
In this section, we assume throughout that we have an irreducible representation
$$
\rho:B_{n}\to G L_{r}(\mathbb{C}),
$$
where $r\geq n$ , and
$$
\rho(\sigma_{i})=1+A_{i},\;\;r a n k(A_{i})=2,\;\;1\leq i\leq n-1.
$$
Theorem 4.1. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation, where $r\geq n$ and $n\geq6$ . Let $r a n k(A_{1})=2$ .
Then $I m(A_{i})\cap I m(A_{i+1})\;\neq\;\{0\}$ for $1\,\leq\,i\,\leq\,n\,-\,2$ ; that is the friendship graph of $\rho$ contains the chain graph.
Proof. Suppose not. Then by Theorem $3.11\,\left(b\right),\,I m(A_{i})\cap I m(A_{j})\neq$ 0 whenever $A_{i}$ and $A_{j}$ are not neighbors. Consider
$$
U=I m(A_{1})+I m(A_{2})+I m(A_{3}).
$$
Since $I m(A_{1})\cap I m(A_{3})\neq0$ , $d i m U\leq5$ .
For $i=4,\dots,n-1$ , let $a_{i},\ b_{i}$ be, respectively, nonzero elements of $I m(A_{1})\cap I m(A_{i})$ and $I m(A_{2})\cap I m(A_{i})$ . Since $I m(A_{1})\cap I m(A_{2})=0$ ,
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<html><body>
<p data-bbox="123 110 486 168">Proof. By corollary 3.9, the friendship graph of the representation is connected. Arrange the vertices of the graph in a sequence $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{n-1}}$ such that each term $A_{i_{j}}$ , $2\leq j\leq n-1$ , is a friend of one the terms $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{j-1}}$ . Then </p>
<div class="equation" data-bbox="259 183 352 197">$$
\dim(I m(A_{i_{1}}))=k
$$</div>
<div class="equation" data-bbox="185 221 426 234">$$
\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\leq k+k-1=2k-1
$$</div>
<div class="equation" data-bbox="125 252 484 270">$$
\dim(I m(A_{i_{1}})+\cdot\cdot\cdot+I m(A_{i_{n-1}}))\leq k+(n-2)(k-1)=(n-1)(k-1)+1.
$$</div>
<p data-bbox="124 284 487 313">Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following </p>
<p data-bbox="124 319 486 348">Theorem 3.11. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $\cdot(A_{i})=2$ . </p>
<p data-bbox="137 348 358 361">Then $r=n$ and one of the following holds. </p>
<p data-bbox="125 362 487 389">(a) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ . </p>
<p data-bbox="126 390 487 418">(b) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors. </p>
<h1 data-bbox="157 429 444 443">4. For corank 2 the friendship graph is a chai </h1>
<p data-bbox="125 450 487 477">In this section, we assume throughout that we have an irreducible representation </p>
<div class="equation" data-bbox="259 480 351 494">$$
\rho:B_{n}\to G L_{r}(\mathbb{C}),
$$</div>
<p data-bbox="125 496 216 510">where $r\geq n$ , and </p>
<div class="equation" data-bbox="186 516 424 531">$$
\rho(\sigma_{i})=1+A_{i},\;\;r a n k(A_{i})=2,\;\;1\leq i\leq n-1.
$$</div>
<p data-bbox="124 541 486 569">Theorem 4.1. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation, where $r\geq n$ and $n\geq6$ . Let $r a n k(A_{1})=2$ . </p>
<p data-bbox="124 570 486 598">Then $I m(A_{i})\cap I m(A_{i+1})\;\neq\;\{0\}$ for $1\,\leq\,i\,\leq\,n\,-\,2$ ; that is the friendship graph of $\rho$ contains the chain graph. </p>
<p data-bbox="124 604 486 633">Proof. Suppose not. Then by Theorem $3.11\,\left(b\right),\,I m(A_{i})\cap I m(A_{j})\neq$ 0 whenever $A_{i}$ and $A_{j}$ are not neighbors. Consider </p>
<div class="equation" data-bbox="217 639 392 654">$$
U=I m(A_{1})+I m(A_{2})+I m(A_{3}).
$$</div>
<p data-bbox="124 657 332 672">Since $I m(A_{1})\cap I m(A_{3})\neq0$ , $d i m U\leq5$ . </p>
<p data-bbox="125 672 487 701">For $i=4,\dots,n-1$ , let $a_{i},\ b_{i}$ be, respectively, nonzero elements of $I m(A_{1})\cap I m(A_{i})$ and $I m(A_{2})\cap I m(A_{i})$ . Since $I m(A_{1})\cap I m(A_{2})=0$ , </p>
</body></html>
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0003047v1
| 8 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Proof. By corollary 3.9, the friendship graph of the representation is connected. Arrange the vertices of the graph in a sequence $A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}$ such that each term $A_{i_{j}}$ , $2\\leq j\\leq n-1$ , is a friend of one the terms $A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}$ . Then ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\dim(I m(A_{i_{1}}))=k\n$$", "text_format": "latex", "page_idx": 8}, {"type": "equation", "text": "$$\n\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1\n$$", "text_format": "latex", "page_idx": 8}, {"type": "equation", "text": "$$\n\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following ", "page_idx": 8}, {"type": "text", "text": "Theorem 3.11. Let $\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})$ be irreducible, where $r=$ $d i m V\\geq n$ , $n\\neq4$ . Suppose $\\rho(\\sigma_{i})=1+A_{i}$ , where rank $\\cdot(A_{i})=2$ . ", "page_idx": 8}, {"type": "text", "text": "Then $r=n$ and one of the following holds. ", "page_idx": 8}, {"type": "text", "text": "(a) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ . ", "page_idx": 8}, {"type": "text", "text": "(b) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors. ", "page_idx": 8}, {"type": "text", "text": "4. For corank 2 the friendship graph is a chai ", "text_level": 1, "page_idx": 8}, {"type": "text", "text": "In this section, we assume throughout that we have an irreducible representation ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "where $r\\geq n$ , and ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Theorem 4.1. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation, where $r\\geq n$ and $n\\geq6$ . Let $r a n k(A_{1})=2$ . ", "page_idx": 8}, {"type": "text", "text": "Then $I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}$ for $1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2$ ; that is the friendship graph of $\\rho$ contains the chain graph. ", "page_idx": 8}, {"type": "text", "text": "Proof. Suppose not. Then by Theorem $3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq$ 0 whenever $A_{i}$ and $A_{j}$ are not neighbors. Consider ", "page_idx": 8}, {"type": "equation", "text": "$$\nU=I m(A_{1})+I m(A_{2})+I m(A_{3}).\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Since $I m(A_{1})\\cap I m(A_{3})\\neq0$ , $d i m U\\leq5$ . ", "page_idx": 8}, {"type": "text", "text": "For $i=4,\\dots,n-1$ , let $a_{i},\\ b_{i}$ be, respectively, nonzero elements of $I m(A_{1})\\cap I m(A_{i})$ and $I m(A_{2})\\cap I m(A_{i})$ . Since $I m(A_{1})\\cap I m(A_{2})=0$ , ", "page_idx": 8}]
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Arrange the vertices of the graph in a sequence", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 157], "spans": [{"bbox": [126, 142, 216, 155], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 139, 327, 157], "score": 1.0, "content": " such that each term ", "type": "text"}, {"bbox": [327, 142, 343, 156], "score": 0.89, "content": "A_{i_{j}}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [344, 139, 353, 157], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [353, 142, 426, 154], "score": 0.85, "content": "2\\leq j\\leq n-1", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [427, 139, 487, 157], "score": 1.0, "content": ", is a friend", "type": "text"}], "index": 2}, {"bbox": [124, 154, 339, 171], "spans": [{"bbox": [124, 154, 213, 171], "score": 1.0, "content": "of one the terms ", "type": "text"}, {"bbox": [213, 156, 304, 169], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [304, 154, 339, 171], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 284, 487, 313], "lines": [{"bbox": [137, 286, 486, 300], "spans": [{"bbox": [137, 286, 486, 300], "score": 1.0, "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "type": "text"}], "index": 7}, {"bbox": [125, 299, 174, 316], "spans": [{"bbox": [125, 299, 174, 316], "score": 1.0, "content": "following", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [124, 319, 486, 348], "lines": [{"bbox": [125, 322, 486, 336], "spans": [{"bbox": [125, 322, 238, 336], "score": 1.0, "content": "Theorem 3.11. Let ", "type": "text"}, {"bbox": [238, 323, 342, 335], "score": 0.88, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [343, 322, 461, 336], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [462, 323, 486, 335], "score": 0.8, "content": "r=", "type": "inline_equation", "height": 12, "width": 24}], "index": 9}, {"bbox": [126, 336, 453, 350], "spans": [{"bbox": [126, 338, 179, 348], "score": 0.45, "content": "d i m V\\geq n", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [179, 336, 185, 350], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 338, 215, 349], "score": 0.88, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 336, 266, 350], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 337, 340, 350], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [341, 336, 405, 350], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 336, 449, 349], "score": 0.67, "content": "\\cdot(A_{i})=2", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [450, 336, 453, 350], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "text", "bbox": [137, 348, 358, 361], "lines": [{"bbox": [140, 350, 357, 363], "spans": [{"bbox": [140, 350, 168, 363], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 354, 197, 360], "score": 0.82, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [197, 350, 357, 363], "score": 1.0, "content": " and one of the following holds.", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 362, 487, 389], "lines": [{"bbox": [139, 363, 486, 378], "spans": [{"bbox": [139, 363, 401, 378], "score": 1.0, "content": "(a) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [401, 364, 414, 376], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [415, 363, 442, 378], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 364, 466, 377], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [466, 363, 486, 378], "score": 1.0, "content": " for", "type": "text"}], "index": 12}, {"bbox": [126, 377, 152, 391], "spans": [{"bbox": [126, 377, 142, 391], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [142, 380, 147, 388], "score": 0.67, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [147, 377, 152, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [126, 390, 487, 418], "lines": [{"bbox": [139, 392, 487, 406], "spans": [{"bbox": [139, 392, 384, 406], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [384, 392, 397, 404], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [398, 392, 421, 406], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 392, 435, 406], "score": 0.88, "content": "A_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [435, 392, 487, 406], "score": 1.0, "content": " whenever", "type": "text"}], "index": 14}, {"bbox": [126, 405, 275, 420], "spans": [{"bbox": [126, 407, 138, 418], "score": 0.89, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [138, 405, 164, 420], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 407, 178, 420], "score": 0.91, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [178, 405, 275, 420], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "title", "bbox": [157, 429, 444, 443], "lines": [{"bbox": [156, 431, 445, 444], "spans": [{"bbox": [156, 431, 445, 444], "score": 1.0, "content": "4. For corank 2 the friendship graph is a chai", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [125, 450, 487, 477], "lines": [{"bbox": [137, 452, 486, 465], "spans": [{"bbox": [137, 452, 486, 465], "score": 1.0, "content": "In this section, we assume throughout that we have an irreducible", "type": "text"}], "index": 17}, {"bbox": [125, 466, 200, 480], "spans": [{"bbox": [125, 466, 200, 480], "score": 1.0, "content": "representation", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 496, 216, 510], "lines": [{"bbox": [124, 496, 217, 512], "spans": [{"bbox": [124, 496, 159, 512], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 500, 189, 510], "score": 0.92, "content": "r\\geq n", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [189, 496, 217, 512], "score": 1.0, "content": ", and", "type": "text"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 541, 486, 569], "lines": [{"bbox": [126, 543, 485, 558], "spans": [{"bbox": [126, 543, 229, 558], "score": 1.0, "content": "Theorem 4.1. Let ", "type": "text"}, {"bbox": [229, 543, 318, 557], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 14, "width": 89}, {"bbox": [319, 543, 485, 558], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 22}, {"bbox": [127, 558, 344, 571], "spans": [{"bbox": [127, 558, 158, 571], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 559, 188, 570], "score": 0.87, "content": "r\\geq n", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [188, 558, 214, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 559, 244, 570], "score": 0.87, "content": "n\\geq6", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [244, 558, 270, 571], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [271, 558, 341, 571], "score": 0.78, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [341, 558, 344, 571], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [124, 570, 486, 598], "lines": [{"bbox": [137, 569, 487, 587], "spans": [{"bbox": [137, 569, 169, 587], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 571, 312, 585], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "type": "inline_equation", "height": 14, "width": 143}, {"bbox": [312, 569, 336, 587], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 572, 421, 584], "score": 0.89, "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [422, 569, 487, 587], "score": 1.0, "content": "; that is the", "type": "text"}], "index": 24}, {"bbox": [126, 585, 365, 600], "spans": [{"bbox": [126, 585, 225, 600], "score": 1.0, "content": "friendship graph of ", "type": "text"}, {"bbox": [226, 588, 233, 599], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [233, 585, 365, 600], "score": 1.0, "content": " contains the chain graph.", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "text", "bbox": [124, 604, 486, 633], "lines": [{"bbox": [136, 605, 487, 622], "spans": [{"bbox": [136, 605, 346, 622], "score": 1.0, "content": "Proof. Suppose not. Then by Theorem ", "type": "text"}, {"bbox": [347, 606, 487, 621], "score": 0.68, "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "type": "inline_equation", "height": 15, "width": 140}], "index": 26}, {"bbox": [125, 621, 386, 635], "spans": [{"bbox": [125, 621, 186, 635], "score": 1.0, "content": "0 whenever ", "type": "text"}, {"bbox": [187, 623, 199, 633], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [200, 621, 225, 635], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 622, 239, 635], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [240, 621, 386, 635], "score": 1.0, "content": " are not neighbors. Consider", "type": "text"}], "index": 27}], "index": 26.5}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [124, 657, 332, 672], "lines": [{"bbox": [126, 659, 331, 674], "spans": [{"bbox": [126, 659, 156, 674], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [156, 660, 270, 673], "score": 0.92, "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "type": "inline_equation", "height": 13, "width": 114}, {"bbox": [271, 659, 276, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [276, 660, 328, 672], "score": 0.8, "content": "d i m U\\leq5", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [329, 659, 331, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [125, 672, 487, 701], "lines": [{"bbox": [137, 673, 487, 688], "spans": [{"bbox": [137, 673, 159, 688], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [159, 676, 240, 687], "score": 0.9, "content": "i=4,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 81}, {"bbox": [240, 673, 263, 688], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [264, 674, 293, 686], "score": 0.87, "content": "a_{i},\\ b_{i}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [294, 673, 487, 688], "score": 1.0, "content": " be, respectively, nonzero elements of", "type": "text"}], "index": 30}, {"bbox": [126, 687, 484, 702], "spans": [{"bbox": [126, 689, 216, 702], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{i})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 687, 241, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [242, 688, 332, 702], "score": 0.91, "content": "I m(A_{2})\\cap I m(A_{i})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [332, 687, 368, 702], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [369, 689, 482, 702], "score": 0.94, "content": "I m(A_{1})\\cap I m(A_{2})=0", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [482, 687, 484, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 31}], "index": 30.5}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28}], "discarded_blocks": [{"type": "discarded", "bbox": [221, 89, 389, 101], "lines": [{"bbox": [223, 93, 388, 101], "spans": [{"bbox": [223, 93, 388, 101], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [478, 91, 486, 100], "lines": [{"bbox": [479, 93, 486, 102], "spans": [{"bbox": [479, 93, 486, 102], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 486, 168], "lines": [{"bbox": [137, 112, 485, 127], "spans": [{"bbox": [137, 112, 485, 127], "score": 1.0, "content": "Proof. By corollary 3.9, the friendship graph of the representa-", "type": "text"}], "index": 0}, {"bbox": [125, 127, 486, 141], "spans": [{"bbox": [125, 127, 486, 141], "score": 1.0, "content": "tion is connected. Arrange the vertices of the graph in a sequence", "type": "text"}], "index": 1}, {"bbox": [126, 139, 487, 157], "spans": [{"bbox": [126, 142, 216, 155], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 139, 327, 157], "score": 1.0, "content": " such that each term ", "type": "text"}, {"bbox": [327, 142, 343, 156], "score": 0.89, "content": "A_{i_{j}}", "type": "inline_equation", "height": 14, "width": 16}, {"bbox": [344, 139, 353, 157], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [353, 142, 426, 154], "score": 0.85, "content": "2\\leq j\\leq n-1", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [427, 139, 487, 157], "score": 1.0, "content": ", is a friend", "type": "text"}], "index": 2}, {"bbox": [124, 154, 339, 171], "spans": [{"bbox": [124, 154, 213, 171], "score": 1.0, "content": "of one the terms ", "type": "text"}, {"bbox": [213, 156, 304, 169], "score": 0.91, "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "type": "inline_equation", "height": 13, "width": 91}, {"bbox": [304, 154, 339, 171], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [124, 112, 487, 171]}, {"type": "interline_equation", "bbox": [259, 183, 352, 197], "lines": [{"bbox": [259, 183, 352, 197], "spans": [{"bbox": [259, 183, 352, 197], "score": 0.9, "content": "\\dim(I m(A_{i_{1}}))=k", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "lines": [{"bbox": [185, 221, 426, 234], "spans": [{"bbox": [185, 221, 426, 234], "score": 0.47, "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "lines": [{"bbox": [125, 254, 484, 270], "spans": [{"bbox": [125, 254, 484, 270], "score": 0.77, "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 284, 487, 313], "lines": [{"bbox": [137, 286, 486, 300], "spans": [{"bbox": [137, 286, 486, 300], "score": 1.0, "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "type": "text"}], "index": 7}, {"bbox": [125, 299, 174, 316], "spans": [{"bbox": [125, 299, 174, 316], "score": 1.0, "content": "following", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [125, 286, 486, 316]}, {"type": "text", "bbox": [124, 319, 486, 348], "lines": [{"bbox": [125, 322, 486, 336], "spans": [{"bbox": [125, 322, 238, 336], "score": 1.0, "content": "Theorem 3.11. Let ", "type": "text"}, {"bbox": [238, 323, 342, 335], "score": 0.88, "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 104}, {"bbox": [343, 322, 461, 336], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [462, 323, 486, 335], "score": 0.8, "content": "r=", "type": "inline_equation", "height": 12, "width": 24}], "index": 9}, {"bbox": [126, 336, 453, 350], "spans": [{"bbox": [126, 338, 179, 348], "score": 0.45, "content": "d i m V\\geq n", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [179, 336, 185, 350], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [186, 338, 215, 349], "score": 0.88, "content": "n\\neq4", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [216, 336, 266, 350], "score": 1.0, "content": ". Suppose ", "type": "text"}, {"bbox": [266, 337, 340, 350], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [341, 336, 405, 350], "score": 1.0, "content": ", where rank", "type": "text"}, {"bbox": [405, 336, 449, 349], "score": 0.67, "content": "\\cdot(A_{i})=2", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [450, 336, 453, 350], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [125, 322, 486, 350]}, {"type": "text", "bbox": [137, 348, 358, 361], "lines": [{"bbox": [140, 350, 357, 363], "spans": [{"bbox": [140, 350, 168, 363], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 354, 197, 360], "score": 0.82, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [197, 350, 357, 363], "score": 1.0, "content": " and one of the following holds.", "type": "text"}], "index": 11}], "index": 11, "bbox_fs": [140, 350, 357, 363]}, {"type": "text", "bbox": [125, 362, 487, 389], "lines": [{"bbox": [139, 363, 486, 378], "spans": [{"bbox": [139, 363, 401, 378], "score": 1.0, "content": "(a) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [401, 364, 414, 376], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [415, 363, 442, 378], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [442, 364, 466, 377], "score": 0.91, "content": "A_{i+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [466, 363, 486, 378], "score": 1.0, "content": " for", "type": "text"}], "index": 12}, {"bbox": [126, 377, 152, 391], "spans": [{"bbox": [126, 377, 142, 391], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [142, 380, 147, 388], "score": 0.67, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [147, 377, 152, 391], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 12.5, "bbox_fs": [126, 363, 486, 391]}, {"type": "text", "bbox": [126, 390, 487, 418], "lines": [{"bbox": [139, 392, 487, 406], "spans": [{"bbox": [139, 392, 384, 406], "score": 1.0, "content": "(b) The full friendship graph has an edge between ", "type": "text"}, {"bbox": [384, 392, 397, 404], "score": 0.88, "content": "A_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [398, 392, 421, 406], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [421, 392, 435, 406], "score": 0.88, "content": "A_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [435, 392, 487, 406], "score": 1.0, "content": " whenever", "type": "text"}], "index": 14}, {"bbox": [126, 405, 275, 420], "spans": [{"bbox": [126, 407, 138, 418], "score": 0.89, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [138, 405, 164, 420], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 407, 178, 420], "score": 0.91, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [178, 405, 275, 420], "score": 1.0, "content": " are not neighbors.", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [126, 392, 487, 420]}, {"type": "title", "bbox": [157, 429, 444, 443], "lines": [{"bbox": [156, 431, 445, 444], "spans": [{"bbox": [156, 431, 445, 444], "score": 1.0, "content": "4. For corank 2 the friendship graph is a chai", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [125, 450, 487, 477], "lines": [{"bbox": [137, 452, 486, 465], "spans": [{"bbox": [137, 452, 486, 465], "score": 1.0, "content": "In this section, we assume throughout that we have an irreducible", "type": "text"}], "index": 17}, {"bbox": [125, 466, 200, 480], "spans": [{"bbox": [125, 466, 200, 480], "score": 1.0, "content": "representation", "type": "text"}], "index": 18}], "index": 17.5, "bbox_fs": [125, 452, 486, 480]}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "lines": [{"bbox": [259, 480, 351, 494], "spans": [{"bbox": [259, 480, 351, 494], "score": 0.88, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [125, 496, 216, 510], "lines": [{"bbox": [124, 496, 217, 512], "spans": [{"bbox": [124, 496, 159, 512], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 500, 189, 510], "score": 0.92, "content": "r\\geq n", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [189, 496, 217, 512], "score": 1.0, "content": ", and", "type": "text"}], "index": 20}], "index": 20, "bbox_fs": [124, 496, 217, 512]}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "lines": [{"bbox": [186, 516, 424, 531], "spans": [{"bbox": [186, 516, 424, 531], "score": 0.88, "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [124, 541, 486, 569], "lines": [{"bbox": [126, 543, 485, 558], "spans": [{"bbox": [126, 543, 229, 558], "score": 1.0, "content": "Theorem 4.1. Let ", "type": "text"}, {"bbox": [229, 543, 318, 557], "score": 0.91, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 14, "width": 89}, {"bbox": [319, 543, 485, 558], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 22}, {"bbox": [127, 558, 344, 571], "spans": [{"bbox": [127, 558, 158, 571], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 559, 188, 570], "score": 0.87, "content": "r\\geq n", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [188, 558, 214, 571], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 559, 244, 570], "score": 0.87, "content": "n\\geq6", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [244, 558, 270, 571], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [271, 558, 341, 571], "score": 0.78, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [341, 558, 344, 571], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "bbox_fs": [126, 543, 485, 571]}, {"type": "text", "bbox": [124, 570, 486, 598], "lines": [{"bbox": [137, 569, 487, 587], "spans": [{"bbox": [137, 569, 169, 587], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 571, 312, 585], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "type": "inline_equation", "height": 14, "width": 143}, {"bbox": [312, 569, 336, 587], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 572, 421, 584], "score": 0.89, "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [422, 569, 487, 587], "score": 1.0, "content": "; that is the", "type": "text"}], "index": 24}, {"bbox": [126, 585, 365, 600], "spans": [{"bbox": [126, 585, 225, 600], "score": 1.0, "content": "friendship graph of ", "type": "text"}, {"bbox": [226, 588, 233, 599], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [233, 585, 365, 600], "score": 1.0, "content": " contains the chain graph.", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [126, 569, 487, 600]}, {"type": "text", "bbox": [124, 604, 486, 633], "lines": [{"bbox": [136, 605, 487, 622], "spans": [{"bbox": [136, 605, 346, 622], "score": 1.0, "content": "Proof. Suppose not. Then by Theorem ", "type": "text"}, {"bbox": [347, 606, 487, 621], "score": 0.68, "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "type": "inline_equation", "height": 15, "width": 140}], "index": 26}, {"bbox": [125, 621, 386, 635], "spans": [{"bbox": [125, 621, 186, 635], "score": 1.0, "content": "0 whenever ", "type": "text"}, {"bbox": [187, 623, 199, 633], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [200, 621, 225, 635], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [226, 622, 239, 635], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [240, 621, 386, 635], "score": 1.0, "content": " are not neighbors. Consider", "type": "text"}], "index": 27}], "index": 26.5, "bbox_fs": [125, 605, 487, 635]}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "lines": [{"bbox": [217, 639, 392, 654], "spans": [{"bbox": [217, 639, 392, 654], "score": 0.89, "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "type": "interline_equation"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [124, 657, 332, 672], "lines": [{"bbox": [126, 659, 331, 674], "spans": [{"bbox": [126, 659, 156, 674], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [156, 660, 270, 673], "score": 0.92, "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "type": "inline_equation", "height": 13, "width": 114}, {"bbox": [271, 659, 276, 674], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [276, 660, 328, 672], "score": 0.8, "content": "d i m U\\leq5", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [329, 659, 331, 674], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 29, "bbox_fs": [126, 659, 331, 674]}, {"type": "text", "bbox": [125, 672, 487, 701], "lines": [{"bbox": [137, 673, 487, 688], "spans": [{"bbox": [137, 673, 159, 688], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [159, 676, 240, 687], "score": 0.9, "content": "i=4,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 81}, {"bbox": [240, 673, 263, 688], "score": 1.0, "content": ", let ", "type": "text"}, {"bbox": [264, 674, 293, 686], "score": 0.87, "content": "a_{i},\\ b_{i}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [294, 673, 487, 688], "score": 1.0, "content": " be, respectively, nonzero elements of", "type": "text"}], "index": 30}, {"bbox": [126, 687, 484, 702], "spans": [{"bbox": [126, 689, 216, 702], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{i})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [217, 687, 241, 702], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [242, 688, 332, 702], "score": 0.91, "content": "I m(A_{2})\\cap I m(A_{i})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [332, 687, 368, 702], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [369, 689, 482, 702], "score": 0.94, "content": "I m(A_{1})\\cap I m(A_{2})=0", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [482, 687, 484, 702], "score": 1.0, "content": ",", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [126, 673, 487, 702]}]}
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[{"type": "text", "bbox": [123, 110, 486, 168], "content": "Proof. By corollary 3.9, the friendship graph of the representa- tion is connected. Arrange the vertices of the graph in a sequence such that each term , , is a friend of one the terms . Then", "index": 0}, {"type": "interline_equation", "bbox": [259, 183, 352, 197], "content": "", "index": 1}, {"type": "interline_equation", "bbox": [185, 221, 426, 234], "content": "", "index": 2}, {"type": "interline_equation", "bbox": [125, 252, 484, 270], "content": "", "index": 3}, {"type": "text", "bbox": [124, 284, 487, 313], "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following", "index": 4}, {"type": "text", "bbox": [124, 319, 486, 348], "content": "Theorem 3.11. Let be irreducible, where , . Suppose , where rank .", "index": 5}, {"type": "text", "bbox": [137, 348, 358, 361], "content": "Then and one of the following holds.", "index": 6}, {"type": "text", "bbox": [125, 362, 487, 389], "content": "(a) The full friendship graph has an edge between and for all .", "index": 7}, {"type": "text", "bbox": [126, 390, 487, 418], "content": "(b) The full friendship graph has an edge between and whenever and are not neighbors.", "index": 8}, {"type": "title", "bbox": [157, 429, 444, 443], "content": "4. For corank 2 the friendship graph is a chai", "index": 9}, {"type": "text", "bbox": [125, 450, 487, 477], "content": "In this section, we assume throughout that we have an irreducible representation", "index": 10}, {"type": "interline_equation", "bbox": [259, 480, 351, 494], "content": "", "index": 11}, {"type": "text", "bbox": [125, 496, 216, 510], "content": "where , and", "index": 12}, {"type": "interline_equation", "bbox": [186, 516, 424, 531], "content": "", "index": 13}, {"type": "text", "bbox": [124, 541, 486, 569], "content": "Theorem 4.1. Let be an irreducible representation, where and . Let .", "index": 14}, {"type": "text", "bbox": [124, 570, 486, 598], "content": "Then for ; that is the friendship graph of contains the chain graph.", "index": 15}, {"type": "text", "bbox": [124, 604, 486, 633], "content": "Proof. Suppose not. Then by Theorem 0 whenever and are not neighbors. Consider", "index": 16}, {"type": "interline_equation", "bbox": [217, 639, 392, 654], "content": "", "index": 17}, {"type": "text", "bbox": [124, 657, 332, 672], "content": "Since , .", "index": 18}, {"type": "text", "bbox": [125, 672, 487, 701], "content": "For , let be, respectively, nonzero elements of and . Since ,", "index": 19}]
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[{"bbox": [137, 112, 485, 127], "content": "Proof. By corollary 3.9, the friendship graph of the representa-", "parent_index": 0, "line_index": 0}, {"bbox": [125, 127, 486, 141], "content": "tion is connected. Arrange the vertices of the graph in a sequence", "parent_index": 0, "line_index": 1}, {"bbox": [126, 139, 487, 157], "content": "such that each term , , is a friend", "parent_index": 0, "line_index": 2}, {"bbox": [124, 154, 339, 171], "content": "of one the terms . Then", "parent_index": 0, "line_index": 3}, {"bbox": [137, 286, 486, 300], "content": "Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the", "parent_index": 4, "line_index": 0}, {"bbox": [125, 299, 174, 316], "content": "following", "parent_index": 4, "line_index": 1}, {"bbox": [125, 322, 486, 336], "content": "Theorem 3.11. Let be irreducible, where", "parent_index": 5, "line_index": 0}, {"bbox": [126, 336, 453, 350], "content": ", . Suppose , where rank .", "parent_index": 5, "line_index": 1}, {"bbox": [140, 350, 357, 363], "content": "Then and one of the following holds.", "parent_index": 6, "line_index": 0}, {"bbox": [139, 363, 486, 378], "content": "(a) The full friendship graph has an edge between and for", "parent_index": 7, "line_index": 0}, {"bbox": [126, 377, 152, 391], "content": "all .", "parent_index": 7, "line_index": 1}, {"bbox": [139, 392, 487, 406], "content": "(b) The full friendship graph has an edge between and whenever", "parent_index": 8, "line_index": 0}, {"bbox": [126, 405, 275, 420], "content": "and are not neighbors.", "parent_index": 8, "line_index": 1}, {"bbox": [156, 431, 445, 444], "content": "4. For corank 2 the friendship graph is a chai", "parent_index": 9, "line_index": 0}, {"bbox": [137, 452, 486, 465], "content": "In this section, we assume throughout that we have an irreducible", "parent_index": 10, "line_index": 0}, {"bbox": [125, 466, 200, 480], "content": "representation", "parent_index": 10, "line_index": 1}, {"bbox": [124, 496, 217, 512], "content": "where , and", "parent_index": 12, "line_index": 0}, {"bbox": [126, 543, 485, 558], "content": "Theorem 4.1. Let be an irreducible representation,", "parent_index": 14, "line_index": 0}, {"bbox": [127, 558, 344, 571], "content": "where and . Let .", "parent_index": 14, "line_index": 1}, {"bbox": [137, 569, 487, 587], "content": "Then for ; that is the", "parent_index": 15, "line_index": 0}, {"bbox": [126, 585, 365, 600], "content": "friendship graph of contains the chain graph.", "parent_index": 15, "line_index": 1}, {"bbox": [136, 605, 487, 622], "content": "Proof. Suppose not. Then by Theorem", "parent_index": 16, "line_index": 0}, {"bbox": [125, 621, 386, 635], "content": "0 whenever and are not neighbors. Consider", "parent_index": 16, "line_index": 1}, {"bbox": [126, 659, 331, 674], "content": "Since , .", "parent_index": 18, "line_index": 0}, {"bbox": [137, 673, 487, 688], "content": "For , let be, respectively, nonzero elements of", "parent_index": 19, "line_index": 0}, {"bbox": [126, 687, 484, 702], "content": "and . Since ,", "parent_index": 19, "line_index": 1}]
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[]
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[{"bbox": [126, 142, 216, 155], "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{n-1}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [327, 142, 343, 156], "content": "A_{i_{j}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [353, 142, 426, 154], "content": "2\\leq j\\leq n-1", "parent_index": 0, "subtype": "inline"}, {"bbox": [213, 156, 304, 169], "content": "A_{i_{1}},A_{i_{2}},\\ldots,A_{i_{j-1}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [259, 183, 352, 197], "content": "\\dim(I m(A_{i_{1}}))=k", "parent_index": 1, "subtype": "interline"}, {"bbox": [185, 221, 426, 234], "content": "\\mathrm{dim}(I m(A_{i_{1}})+I m(A_{i_{2}}))\\leq k+k-1=2k-1", "parent_index": 2, "subtype": "interline"}, {"bbox": [125, 252, 484, 270], "content": "\\dim(I m(A_{i_{1}})+\\cdot\\cdot\\cdot+I m(A_{i_{n-1}}))\\leq k+(n-2)(k-1)=(n-1)(k-1)+1.", "parent_index": 3, "subtype": "interline"}, {"bbox": [238, 323, 342, 335], "content": "\\rho~:~B_{n}~\\to~G L_{r}(\\mathbb{C})", "parent_index": 5, "subtype": "inline"}, {"bbox": [462, 323, 486, 335], "content": "r=", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 338, 179, 348], "content": "d i m V\\geq n", "parent_index": 5, "subtype": "inline"}, {"bbox": [186, 338, 215, 349], "content": "n\\neq4", "parent_index": 5, "subtype": "inline"}, {"bbox": [266, 337, 340, 350], "content": "\\rho(\\sigma_{i})=1+A_{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [405, 336, 449, 349], "content": "\\cdot(A_{i})=2", "parent_index": 5, "subtype": "inline"}, {"bbox": [168, 354, 197, 360], "content": "r=n", "parent_index": 6, "subtype": "inline"}, {"bbox": [401, 364, 414, 376], "content": "A_{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [442, 364, 466, 377], "content": "A_{i+1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [142, 380, 147, 388], "content": "i", "parent_index": 7, "subtype": "inline"}, {"bbox": [384, 392, 397, 404], "content": "A_{i}", "parent_index": 8, "subtype": "inline"}, {"bbox": [421, 392, 435, 406], "content": "A_{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 407, 138, 418], "content": "A_{i}", "parent_index": 8, "subtype": "inline"}, {"bbox": [165, 407, 178, 420], "content": "A_{j}", "parent_index": 8, "subtype": "inline"}, {"bbox": [259, 480, 351, 494], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C}),", "parent_index": 11, "subtype": "interline"}, {"bbox": [159, 500, 189, 510], "content": "r\\geq n", "parent_index": 12, "subtype": "inline"}, {"bbox": [186, 516, 424, 531], "content": "\\rho(\\sigma_{i})=1+A_{i},\\;\\;r a n k(A_{i})=2,\\;\\;1\\leq i\\leq n-1.", "parent_index": 13, "subtype": "interline"}, {"bbox": [229, 543, 318, 557], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 14, "subtype": "inline"}, {"bbox": [159, 559, 188, 570], "content": "r\\geq n", "parent_index": 14, "subtype": "inline"}, {"bbox": [214, 559, 244, 570], "content": "n\\geq6", "parent_index": 14, "subtype": "inline"}, {"bbox": [271, 558, 341, 571], "content": "r a n k(A_{1})=2", "parent_index": 14, "subtype": "inline"}, {"bbox": [169, 571, 312, 585], "content": "I m(A_{i})\\cap I m(A_{i+1})\\;\\neq\\;\\{0\\}", "parent_index": 15, "subtype": "inline"}, {"bbox": [336, 572, 421, 584], "content": "1\\,\\leq\\,i\\,\\leq\\,n\\,-\\,2", "parent_index": 15, "subtype": "inline"}, {"bbox": [226, 588, 233, 599], "content": "\\rho", "parent_index": 15, "subtype": "inline"}, {"bbox": [347, 606, 487, 621], "content": "3.11\\,\\left(b\\right),\\,I m(A_{i})\\cap I m(A_{j})\\neq", "parent_index": 16, "subtype": "inline"}, {"bbox": [187, 623, 199, 633], "content": "A_{i}", "parent_index": 16, "subtype": "inline"}, {"bbox": [226, 622, 239, 635], "content": "A_{j}", "parent_index": 16, "subtype": "inline"}, {"bbox": [217, 639, 392, 654], "content": "U=I m(A_{1})+I m(A_{2})+I m(A_{3}).", "parent_index": 17, "subtype": "interline"}, {"bbox": [156, 660, 270, 673], "content": "I m(A_{1})\\cap I m(A_{3})\\neq0", "parent_index": 18, "subtype": "inline"}, {"bbox": [276, 660, 328, 672], "content": "d i m U\\leq5", "parent_index": 18, "subtype": "inline"}, {"bbox": [159, 676, 240, 687], "content": "i=4,\\dots,n-1", "parent_index": 19, "subtype": "inline"}, {"bbox": [264, 674, 293, 686], "content": "a_{i},\\ b_{i}", "parent_index": 19, "subtype": "inline"}, {"bbox": [126, 689, 216, 702], "content": "I m(A_{1})\\cap I m(A_{i})", "parent_index": 19, "subtype": "inline"}, {"bbox": [242, 688, 332, 702], "content": "I m(A_{2})\\cap I m(A_{i})", "parent_index": 19, "subtype": "inline"}, {"bbox": [369, 689, 482, 702], "content": "I m(A_{1})\\cap I m(A_{2})=0", "parent_index": 19, "subtype": "inline"}]
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[]
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$a_{i}$ and $b_{i}$ are linearly independent, so they are a basis for $I m(A_{i})$ , and $I m(A_{i})\subseteq I m(A_{1})+I m(A_{2})$ . Thus
$$
U=I m(A_{1})+I m(A_{2})+\cdot\cdot\cdot+I m(A_{n-1}),
$$
which is invariant under $\rho(B_{n})$ . Thus $r\leq5$ , by the irreducibility of $\rho$ , a contradiction with $r\geq n\geq6$ .
Remark 4.2. For $n\,=\,5$ and $\rho$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case:
By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $B_{5}$ of the Jones’ representation (see [3], p. 296).
Lemma 4.3. Let $\rho:B_{n}\,\to\,G L_{r}(\mathbb{C})$ be an irreducible representation, where $r\,\geq\,n$ , $n\,\geq\,5$ , and $r a n k(A_{1})\,=\,2$ . Suppose that the associated friendship graph contains the chain.
Then $r=n$ and the associated friendship graph is the chain (that is, the only edges are between neighbors).
Proof. By corollary 3.10, $r=n$ . Consider the full friendship graph of $\rho$ . Then
$$
I m(A_{i})\cap I m(A_{i+1})\neq\{0\}
$$
for any $i$ where indices are taken modulo $n$ . If $I m(A_{i})\cap I m(A_{i+1})$ is two-dimensional, then $I m(A_{1})=I m(A_{2})=\ldots$ , and $I m(A_{1})$ is a twodimensional invariant subspace, contradicting the irreducibility of $\rho$ . Hence $I m(A_{i})\cap I m(A_{i+1})$ are one-dimensional.
For any $x\in I m(A_{i})$ , $x=A_{i}y$ , $x\neq0$ , we have that
$$
T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\in I m(A_{i+1})
$$
for $T=\rho(\tau)$ . Moreover, $T x\neq0$ because $T$ is invertible.
Choose $x_{1}~\neq~0$ to be a basis vector for $I m(A_{1})\cap I m(A_{2})$ . Define $x_{i+1}=T^{i}x_{1}$ for $1\leq i\leq n-1$ . Then $x_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ .
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<html><body>
<p data-bbox="124 110 486 139">$a_{i}$ and $b_{i}$ are linearly independent, so they are a basis for $I m(A_{i})$ , and $I m(A_{i})\subseteq I m(A_{1})+I m(A_{2})$ . Thus </p>
<div class="equation" data-bbox="198 147 411 161">$$
U=I m(A_{1})+I m(A_{2})+\cdot\cdot\cdot+I m(A_{n-1}),
$$</div>
<p data-bbox="124 165 486 194">which is invariant under $\rho(B_{n})$ . Thus $r\leq5$ , by the irreducibility of $\rho$ , a contradiction with $r\geq n\geq6$ . </p>
<p data-bbox="124 200 486 243">Remark 4.2. For $n\,=\,5$ and $\rho$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case: </p>
<div class="image" data-bbox="124 270 252 335"><img data-bbox="124 270 252 335"/></div>
<p data-bbox="124 365 486 408">By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $B_{5}$ of the Jones’ representation (see [3], p. 296). </p>
<p data-bbox="124 421 486 464">Lemma 4.3. Let $\rho:B_{n}\,\to\,G L_{r}(\mathbb{C})$ be an irreducible representation, where $r\,\geq\,n$ , $n\,\geq\,5$ , and $r a n k(A_{1})\,=\,2$ . Suppose that the associated friendship graph contains the chain. </p>
<p data-bbox="124 465 485 493">Then $r=n$ and the associated friendship graph is the chain (that is, the only edges are between neighbors). </p>
<p data-bbox="124 500 486 528">Proof. By corollary 3.10, $r=n$ . Consider the full friendship graph of $\rho$ . Then </p>
<div class="equation" data-bbox="239 532 372 545">$$
I m(A_{i})\cap I m(A_{i+1})\neq\{0\}
$$</div>
<p data-bbox="124 547 486 603">for any $i$ where indices are taken modulo $n$ . If $I m(A_{i})\cap I m(A_{i+1})$ is two-dimensional, then $I m(A_{1})=I m(A_{2})=\ldots$ , and $I m(A_{1})$ is a twodimensional invariant subspace, contradicting the irreducibility of $\rho$ . Hence $I m(A_{i})\cap I m(A_{i+1})$ are one-dimensional. </p>
<p data-bbox="138 604 398 618">For any $x\in I m(A_{i})$ , $x=A_{i}y$ , $x\neq0$ , we have that </p>
<div class="equation" data-bbox="173 625 437 639">$$
T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\in I m(A_{i+1})
$$</div>
<p data-bbox="124 643 412 658">for $T=\rho(\tau)$ . Moreover, $T x\neq0$ because $T$ is invertible. </p>
<p data-bbox="124 658 487 701">Choose $x_{1}~\neq~0$ to be a basis vector for $I m(A_{1})\cap I m(A_{2})$ . Define $x_{i+1}=T^{i}x_{1}$ for $1\leq i\leq n-1$ . Then $x_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ . </p>
</body></html>
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0003047v1
| 9 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "$a_{i}$ and $b_{i}$ are linearly independent, so they are a basis for $I m(A_{i})$ , and $I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})$ . Thus ", "page_idx": 9}, {"type": "equation", "text": "$$\nU=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "which is invariant under $\\rho(B_{n})$ . Thus $r\\leq5$ , by the irreducibility of $\\rho$ , a contradiction with $r\\geq n\\geq6$ . ", "page_idx": 9}, {"type": "text", "text": "Remark 4.2. For $n\\,=\\,5$ and $\\rho$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case: ", "page_idx": 9}, {"type": "image", "img_path": "images/37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg", "img_caption": [], "img_footnote": [], "page_idx": 9}, {"type": "text", "text": "By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $B_{5}$ of the Jones’ representation (see [3], p. 296). ", "page_idx": 9}, {"type": "text", "text": "Lemma 4.3. Let $\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})$ be an irreducible representation, where $r\\,\\geq\\,n$ , $n\\,\\geq\\,5$ , and $r a n k(A_{1})\\,=\\,2$ . Suppose that the associated friendship graph contains the chain. ", "page_idx": 9}, {"type": "text", "text": "Then $r=n$ and the associated friendship graph is the chain (that is, the only edges are between neighbors). ", "page_idx": 9}, {"type": "text", "text": "Proof. By corollary 3.10, $r=n$ . Consider the full friendship graph of $\\rho$ . Then ", "page_idx": 9}, {"type": "equation", "text": "$$\nI m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "for any $i$ where indices are taken modulo $n$ . If $I m(A_{i})\\cap I m(A_{i+1})$ is two-dimensional, then $I m(A_{1})=I m(A_{2})=\\ldots$ , and $I m(A_{1})$ is a twodimensional invariant subspace, contradicting the irreducibility of $\\rho$ . Hence $I m(A_{i})\\cap I m(A_{i+1})$ are one-dimensional. ", "page_idx": 9}, {"type": "text", "text": "For any $x\\in I m(A_{i})$ , $x=A_{i}y$ , $x\\neq0$ , we have that ", "page_idx": 9}, {"type": "equation", "text": "$$\nT x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "for $T=\\rho(\\tau)$ . Moreover, $T x\\neq0$ because $T$ is invertible. ", "page_idx": 9}, {"type": "text", "text": "Choose $x_{1}~\\neq~0$ to be a basis vector for $I m(A_{1})\\cap I m(A_{2})$ . Define $x_{i+1}=T^{i}x_{1}$ for $1\\leq i\\leq n-1$ . Then $x_{i}$ is a basis vector for $I m(A_{i})\\cap$ $I m(A_{i+1})$ . ", "page_idx": 9}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 110, 486, 139], "lines": [{"bbox": [126, 114, 485, 127], "spans": [{"bbox": [126, 118, 136, 125], "score": 0.9, "content": "a_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [136, 114, 162, 126], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 115, 171, 125], "score": 0.9, "content": "b_{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [171, 114, 421, 126], "score": 1.0, "content": " are linearly independent, so they are a basis for ", "type": "text"}, {"bbox": [421, 114, 459, 127], "score": 0.95, "content": "I m(A_{i})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [459, 114, 485, 126], "score": 1.0, "content": ", and", "type": "text"}], "index": 0}, {"bbox": [126, 126, 307, 141], "spans": [{"bbox": [126, 128, 272, 140], "score": 0.92, "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "type": "inline_equation", "height": 12, "width": 146}, {"bbox": [272, 126, 307, 141], "score": 1.0, "content": ". Thus", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [198, 147, 411, 161], "lines": [{"bbox": [198, 147, 411, 161], "spans": [{"bbox": [198, 147, 411, 161], "score": 0.87, "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [124, 165, 486, 194], "lines": [{"bbox": [125, 166, 485, 182], "spans": [{"bbox": [125, 166, 253, 182], "score": 1.0, "content": "which is invariant under ", "type": "text"}, {"bbox": [254, 168, 284, 181], "score": 0.94, "content": "\\rho(B_{n})", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [284, 166, 320, 182], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [320, 169, 349, 180], "score": 0.92, "content": "r\\leq5", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [349, 166, 475, 182], "score": 1.0, "content": ", by the irreducibility of ", "type": "text"}, {"bbox": [475, 172, 482, 180], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 166, 485, 182], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [126, 182, 288, 195], "spans": [{"bbox": [126, 182, 233, 195], "score": 1.0, "content": "a contradiction with ", "type": "text"}, {"bbox": [234, 184, 284, 194], "score": 0.92, "content": "r\\geq n\\geq6", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [285, 182, 288, 195], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "text", "bbox": [124, 200, 486, 243], "lines": [{"bbox": [125, 203, 486, 217], "spans": [{"bbox": [125, 203, 225, 217], "score": 1.0, "content": "Remark 4.2. For ", "type": "text"}, {"bbox": [225, 205, 257, 213], "score": 0.92, "content": "n\\,=\\,5", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [258, 203, 285, 217], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [285, 208, 292, 216], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [292, 203, 486, 217], "score": 1.0, "content": " satisfying the hypothesis of theorem", "type": "text"}], "index": 5}, {"bbox": [126, 217, 486, 231], "spans": [{"bbox": [126, 217, 486, 231], "score": 1.0, "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "type": "text"}], "index": 6}, {"bbox": [126, 231, 266, 244], "spans": [{"bbox": [126, 231, 266, 244], "score": 1.0, "content": "and 2) an exceptional case:", "type": "text"}], "index": 7}], "index": 6}, {"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5}, {"type": "text", "bbox": [124, 365, 486, 408], "lines": [{"bbox": [137, 368, 485, 381], "spans": [{"bbox": [137, 368, 485, 381], "score": 1.0, "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "type": "text"}], "index": 10}, {"bbox": [125, 381, 486, 396], "spans": [{"bbox": [125, 381, 437, 396], "score": 1.0, "content": "the above friendship graph is equivalent to the restriction to ", "type": "text"}, {"bbox": [438, 383, 451, 394], "score": 0.93, "content": "B_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [452, 381, 486, 396], "score": 1.0, "content": " of the", "type": "text"}], "index": 11}, {"bbox": [126, 396, 322, 409], "spans": [{"bbox": [126, 396, 322, 409], "score": 1.0, "content": "Jones’ representation (see [3], p. 296).", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [124, 421, 486, 464], "lines": [{"bbox": [125, 424, 485, 439], "spans": [{"bbox": [125, 424, 221, 439], "score": 1.0, "content": "Lemma 4.3. Let ", "type": "text"}, {"bbox": [221, 426, 314, 438], "score": 0.93, "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [315, 424, 485, 439], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 13}, {"bbox": [127, 439, 487, 453], "spans": [{"bbox": [127, 439, 159, 453], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [160, 441, 191, 451], "score": 0.88, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [192, 439, 199, 453], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 441, 231, 451], "score": 0.89, "content": "n\\,\\geq\\,5", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [232, 439, 265, 453], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [265, 440, 336, 452], "score": 0.74, "content": "r a n k(A_{1})\\,=\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [336, 439, 487, 453], "score": 1.0, "content": ". Suppose that the associated", "type": "text"}], "index": 14}, {"bbox": [127, 454, 310, 465], "spans": [{"bbox": [127, 454, 310, 465], "score": 1.0, "content": "friendship graph contains the chain.", "type": "text"}], "index": 15}], "index": 14}, {"type": "text", "bbox": [124, 465, 485, 493], "lines": [{"bbox": [138, 465, 485, 482], "spans": [{"bbox": [138, 465, 168, 482], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 471, 196, 477], "score": 0.83, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [197, 465, 485, 482], "score": 1.0, "content": " and the associated friendship graph is the chain (that is,", "type": "text"}], "index": 16}, {"bbox": [126, 480, 321, 495], "spans": [{"bbox": [126, 480, 321, 495], "score": 1.0, "content": "the only edges are between neighbors).", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "text", "bbox": [124, 500, 486, 528], "lines": [{"bbox": [137, 502, 485, 516], "spans": [{"bbox": [137, 502, 275, 516], "score": 1.0, "content": "Proof. By corollary 3.10, ", "type": "text"}, {"bbox": [275, 507, 304, 513], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [305, 502, 485, 516], "score": 1.0, "content": ". Consider the full friendship graph", "type": "text"}], "index": 18}, {"bbox": [126, 516, 181, 530], "spans": [{"bbox": [126, 516, 139, 530], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 522, 146, 529], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [146, 516, 181, 530], "score": 1.0, "content": ". Then", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [124, 547, 486, 603], "lines": [{"bbox": [126, 550, 486, 564], "spans": [{"bbox": [126, 550, 167, 564], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [167, 552, 171, 560], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [172, 550, 344, 564], "score": 1.0, "content": " where indices are taken modulo ", "type": "text"}, {"bbox": [345, 555, 352, 560], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [352, 550, 371, 564], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [372, 551, 473, 563], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [473, 550, 486, 564], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [125, 563, 484, 577], "spans": [{"bbox": [125, 563, 242, 577], "score": 1.0, "content": "two-dimensional, then ", "type": "text"}, {"bbox": [243, 564, 369, 577], "score": 0.92, "content": "I m(A_{1})=I m(A_{2})=\\ldots", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [370, 563, 397, 577], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [398, 564, 437, 577], "score": 0.94, "content": "I m(A_{1})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [437, 563, 484, 577], "score": 1.0, "content": " is a two-", "type": "text"}], "index": 22}, {"bbox": [126, 577, 485, 591], "spans": [{"bbox": [126, 577, 475, 591], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irreducibility of ", "type": "text"}, {"bbox": [475, 582, 482, 590], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 577, 485, 591], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [126, 591, 370, 605], "spans": [{"bbox": [126, 591, 160, 605], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [161, 592, 260, 605], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [261, 591, 370, 605], "score": 1.0, "content": " are one-dimensional.", "type": "text"}], "index": 24}], "index": 22.5}, {"type": "text", "bbox": [138, 604, 398, 618], "lines": [{"bbox": [137, 604, 398, 620], "spans": [{"bbox": [137, 604, 180, 620], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [181, 606, 240, 619], "score": 0.94, "content": "x\\in I m(A_{i})", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [241, 604, 246, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 607, 288, 618], "score": 0.91, "content": "x=A_{i}y", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [289, 604, 294, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [294, 607, 324, 618], "score": 0.92, "content": "x\\neq0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [324, 604, 398, 620], "score": 1.0, "content": ", we have that", "type": "text"}], "index": 25}], "index": 25}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 643, 412, 658], "lines": [{"bbox": [126, 645, 411, 660], "spans": [{"bbox": [126, 645, 143, 660], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 647, 190, 659], "score": 0.94, "content": "T=\\rho(\\tau)", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [190, 645, 251, 660], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [252, 648, 288, 659], "score": 0.93, "content": "T x\\neq0", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [289, 645, 335, 660], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [335, 648, 344, 656], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [344, 645, 411, 660], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 658, 487, 701], "lines": [{"bbox": [138, 659, 486, 674], "spans": [{"bbox": [138, 659, 179, 674], "score": 1.0, "content": "Choose ", "type": "text"}, {"bbox": [180, 662, 217, 673], "score": 0.94, "content": "x_{1}~\\neq~0", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [218, 659, 350, 674], "score": 1.0, "content": " to be a basis vector for ", "type": "text"}, {"bbox": [351, 661, 444, 673], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{2})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [445, 659, 486, 674], "score": 1.0, "content": ". Define", "type": "text"}], "index": 28}, {"bbox": [126, 674, 485, 688], "spans": [{"bbox": [126, 674, 187, 687], "score": 0.93, "content": "x_{i+1}=T^{i}x_{1}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [187, 674, 208, 688], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [208, 676, 280, 686], "score": 0.91, "content": "1\\leq i\\leq n-1", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [280, 674, 317, 688], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [318, 679, 328, 686], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [328, 674, 436, 688], "score": 1.0, "content": " is a basis vector for ", "type": "text"}, {"bbox": [436, 675, 485, 687], "score": 0.94, "content": "I m(A_{i})\\cap", "type": "inline_equation", "height": 12, "width": 49}], "index": 29}, {"bbox": [126, 686, 179, 703], "spans": [{"bbox": [126, 689, 174, 702], "score": 0.91, "content": "I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [175, 686, 179, 703], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [198, 147, 411, 161], "lines": [{"bbox": [198, 147, 411, 161], "spans": [{"bbox": [198, 147, 411, 161], "score": 0.87, "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [277, 92, 329, 102], "spans": [{"bbox": [277, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [126, 91, 137, 100], "lines": [{"bbox": [125, 93, 137, 103], "spans": [{"bbox": [125, 93, 137, 103], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 486, 139], "lines": [{"bbox": [126, 114, 485, 127], "spans": [{"bbox": [126, 118, 136, 125], "score": 0.9, "content": "a_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [136, 114, 162, 126], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [162, 115, 171, 125], "score": 0.9, "content": "b_{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [171, 114, 421, 126], "score": 1.0, "content": " are linearly independent, so they are a basis for ", "type": "text"}, {"bbox": [421, 114, 459, 127], "score": 0.95, "content": "I m(A_{i})", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [459, 114, 485, 126], "score": 1.0, "content": ", and", "type": "text"}], "index": 0}, {"bbox": [126, 126, 307, 141], "spans": [{"bbox": [126, 128, 272, 140], "score": 0.92, "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "type": "inline_equation", "height": 12, "width": 146}, {"bbox": [272, 126, 307, 141], "score": 1.0, "content": ". Thus", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [126, 114, 485, 141]}, {"type": "interline_equation", "bbox": [198, 147, 411, 161], "lines": [{"bbox": [198, 147, 411, 161], "spans": [{"bbox": [198, 147, 411, 161], "score": 0.87, "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [124, 165, 486, 194], "lines": [{"bbox": [125, 166, 485, 182], "spans": [{"bbox": [125, 166, 253, 182], "score": 1.0, "content": "which is invariant under ", "type": "text"}, {"bbox": [254, 168, 284, 181], "score": 0.94, "content": "\\rho(B_{n})", "type": "inline_equation", "height": 13, "width": 30}, {"bbox": [284, 166, 320, 182], "score": 1.0, "content": ". Thus ", "type": "text"}, {"bbox": [320, 169, 349, 180], "score": 0.92, "content": "r\\leq5", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [349, 166, 475, 182], "score": 1.0, "content": ", by the irreducibility of ", "type": "text"}, {"bbox": [475, 172, 482, 180], "score": 0.9, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 166, 485, 182], "score": 1.0, "content": ",", "type": "text"}], "index": 3}, {"bbox": [126, 182, 288, 195], "spans": [{"bbox": [126, 182, 233, 195], "score": 1.0, "content": "a contradiction with ", "type": "text"}, {"bbox": [234, 184, 284, 194], "score": 0.92, "content": "r\\geq n\\geq6", "type": "inline_equation", "height": 10, "width": 50}, {"bbox": [285, 182, 288, 195], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 3.5, "bbox_fs": [125, 166, 485, 195]}, {"type": "text", "bbox": [124, 200, 486, 243], "lines": [{"bbox": [125, 203, 486, 217], "spans": [{"bbox": [125, 203, 225, 217], "score": 1.0, "content": "Remark 4.2. For ", "type": "text"}, {"bbox": [225, 205, 257, 213], "score": 0.92, "content": "n\\,=\\,5", "type": "inline_equation", "height": 8, "width": 32}, {"bbox": [258, 203, 285, 217], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [285, 208, 292, 216], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [292, 203, 486, 217], "score": 1.0, "content": " satisfying the hypothesis of theorem", "type": "text"}], "index": 5}, {"bbox": [126, 217, 486, 231], "spans": [{"bbox": [126, 217, 486, 231], "score": 1.0, "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "type": "text"}], "index": 6}, {"bbox": [126, 231, 266, 244], "spans": [{"bbox": [126, 231, 266, 244], "score": 1.0, "content": "and 2) an exceptional case:", "type": "text"}], "index": 7}], "index": 6, "bbox_fs": [125, 203, 486, 244]}, {"type": "image", "bbox": [124, 270, 252, 335], "blocks": [{"type": "image_body", "bbox": [124, 270, 252, 335], "group_id": 0, "lines": [{"bbox": [124, 270, 252, 335], "spans": [{"bbox": [124, 270, 252, 335], "score": 0.918, "type": "image", "image_path": "37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg"}]}], "index": 8.5, "virtual_lines": [{"bbox": [124, 270, 252, 302.5], "spans": [], "index": 8}, {"bbox": [124, 302.5, 252, 335.0], "spans": [], "index": 9}]}], "index": 8.5}, {"type": "text", "bbox": [124, 365, 486, 408], "lines": [{"bbox": [137, 368, 485, 381], "spans": [{"bbox": [137, 368, 485, 381], "score": 1.0, "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "type": "text"}], "index": 10}, {"bbox": [125, 381, 486, 396], "spans": [{"bbox": [125, 381, 437, 396], "score": 1.0, "content": "the above friendship graph is equivalent to the restriction to ", "type": "text"}, {"bbox": [438, 383, 451, 394], "score": 0.93, "content": "B_{5}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [452, 381, 486, 396], "score": 1.0, "content": " of the", "type": "text"}], "index": 11}, {"bbox": [126, 396, 322, 409], "spans": [{"bbox": [126, 396, 322, 409], "score": 1.0, "content": "Jones’ representation (see [3], p. 296).", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [125, 368, 486, 409]}, {"type": "text", "bbox": [124, 421, 486, 464], "lines": [{"bbox": [125, 424, 485, 439], "spans": [{"bbox": [125, 424, 221, 439], "score": 1.0, "content": "Lemma 4.3. Let ", "type": "text"}, {"bbox": [221, 426, 314, 438], "score": 0.93, "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [315, 424, 485, 439], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 13}, {"bbox": [127, 439, 487, 453], "spans": [{"bbox": [127, 439, 159, 453], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [160, 441, 191, 451], "score": 0.88, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [192, 439, 199, 453], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [199, 441, 231, 451], "score": 0.89, "content": "n\\,\\geq\\,5", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [232, 439, 265, 453], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [265, 440, 336, 452], "score": 0.74, "content": "r a n k(A_{1})\\,=\\,2", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [336, 439, 487, 453], "score": 1.0, "content": ". Suppose that the associated", "type": "text"}], "index": 14}, {"bbox": [127, 454, 310, 465], "spans": [{"bbox": [127, 454, 310, 465], "score": 1.0, "content": "friendship graph contains the chain.", "type": "text"}], "index": 15}], "index": 14, "bbox_fs": [125, 424, 487, 465]}, {"type": "text", "bbox": [124, 465, 485, 493], "lines": [{"bbox": [138, 465, 485, 482], "spans": [{"bbox": [138, 465, 168, 482], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 471, 196, 477], "score": 0.83, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [197, 465, 485, 482], "score": 1.0, "content": " and the associated friendship graph is the chain (that is,", "type": "text"}], "index": 16}, {"bbox": [126, 480, 321, 495], "spans": [{"bbox": [126, 480, 321, 495], "score": 1.0, "content": "the only edges are between neighbors).", "type": "text"}], "index": 17}], "index": 16.5, "bbox_fs": [126, 465, 485, 495]}, {"type": "text", "bbox": [124, 500, 486, 528], "lines": [{"bbox": [137, 502, 485, 516], "spans": [{"bbox": [137, 502, 275, 516], "score": 1.0, "content": "Proof. By corollary 3.10, ", "type": "text"}, {"bbox": [275, 507, 304, 513], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 29}, {"bbox": [305, 502, 485, 516], "score": 1.0, "content": ". Consider the full friendship graph", "type": "text"}], "index": 18}, {"bbox": [126, 516, 181, 530], "spans": [{"bbox": [126, 516, 139, 530], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 522, 146, 529], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [146, 516, 181, 530], "score": 1.0, "content": ". Then", "type": "text"}], "index": 19}], "index": 18.5, "bbox_fs": [126, 502, 485, 530]}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "lines": [{"bbox": [239, 532, 372, 545], "spans": [{"bbox": [239, 532, 372, 545], "score": 0.91, "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [124, 547, 486, 603], "lines": [{"bbox": [126, 550, 486, 564], "spans": [{"bbox": [126, 550, 167, 564], "score": 1.0, "content": "for any ", "type": "text"}, {"bbox": [167, 552, 171, 560], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [172, 550, 344, 564], "score": 1.0, "content": " where indices are taken modulo ", "type": "text"}, {"bbox": [345, 555, 352, 560], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [352, 550, 371, 564], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [372, 551, 473, 563], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [473, 550, 486, 564], "score": 1.0, "content": " is", "type": "text"}], "index": 21}, {"bbox": [125, 563, 484, 577], "spans": [{"bbox": [125, 563, 242, 577], "score": 1.0, "content": "two-dimensional, then ", "type": "text"}, {"bbox": [243, 564, 369, 577], "score": 0.92, "content": "I m(A_{1})=I m(A_{2})=\\ldots", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [370, 563, 397, 577], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [398, 564, 437, 577], "score": 0.94, "content": "I m(A_{1})", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [437, 563, 484, 577], "score": 1.0, "content": " is a two-", "type": "text"}], "index": 22}, {"bbox": [126, 577, 485, 591], "spans": [{"bbox": [126, 577, 475, 591], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irreducibility of ", "type": "text"}, {"bbox": [475, 582, 482, 590], "score": 0.88, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [482, 577, 485, 591], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [126, 591, 370, 605], "spans": [{"bbox": [126, 591, 160, 605], "score": 1.0, "content": "Hence ", "type": "text"}, {"bbox": [161, 592, 260, 605], "score": 0.93, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [261, 591, 370, 605], "score": 1.0, "content": " are one-dimensional.", "type": "text"}], "index": 24}], "index": 22.5, "bbox_fs": [125, 550, 486, 605]}, {"type": "text", "bbox": [138, 604, 398, 618], "lines": [{"bbox": [137, 604, 398, 620], "spans": [{"bbox": [137, 604, 180, 620], "score": 1.0, "content": "For any ", "type": "text"}, {"bbox": [181, 606, 240, 619], "score": 0.94, "content": "x\\in I m(A_{i})", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [241, 604, 246, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 607, 288, 618], "score": 0.91, "content": "x=A_{i}y", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [289, 604, 294, 620], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [294, 607, 324, 618], "score": 0.92, "content": "x\\neq0", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [324, 604, 398, 620], "score": 1.0, "content": ", we have that", "type": "text"}], "index": 25}], "index": 25, "bbox_fs": [137, 604, 398, 620]}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "lines": [{"bbox": [173, 625, 437, 639], "spans": [{"bbox": [173, 625, 437, 639], "score": 0.89, "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "type": "interline_equation"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [124, 643, 412, 658], "lines": [{"bbox": [126, 645, 411, 660], "spans": [{"bbox": [126, 645, 143, 660], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 647, 190, 659], "score": 0.94, "content": "T=\\rho(\\tau)", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [190, 645, 251, 660], "score": 1.0, "content": ". Moreover, ", "type": "text"}, {"bbox": [252, 648, 288, 659], "score": 0.93, "content": "T x\\neq0", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [289, 645, 335, 660], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [335, 648, 344, 656], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [344, 645, 411, 660], "score": 1.0, "content": " is invertible.", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [126, 645, 411, 660]}, {"type": "text", "bbox": [124, 658, 487, 701], "lines": [{"bbox": [138, 659, 486, 674], "spans": [{"bbox": [138, 659, 179, 674], "score": 1.0, "content": "Choose ", "type": "text"}, {"bbox": [180, 662, 217, 673], "score": 0.94, "content": "x_{1}~\\neq~0", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [218, 659, 350, 674], "score": 1.0, "content": " to be a basis vector for ", "type": "text"}, {"bbox": [351, 661, 444, 673], "score": 0.93, "content": "I m(A_{1})\\cap I m(A_{2})", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [445, 659, 486, 674], "score": 1.0, "content": ". Define", "type": "text"}], "index": 28}, {"bbox": [126, 674, 485, 688], "spans": [{"bbox": [126, 674, 187, 687], "score": 0.93, "content": "x_{i+1}=T^{i}x_{1}", "type": "inline_equation", "height": 13, "width": 61}, {"bbox": [187, 674, 208, 688], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [208, 676, 280, 686], "score": 0.91, "content": "1\\leq i\\leq n-1", "type": "inline_equation", "height": 10, "width": 72}, {"bbox": [280, 674, 317, 688], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [318, 679, 328, 686], "score": 0.91, "content": "x_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [328, 674, 436, 688], "score": 1.0, "content": " is a basis vector for ", "type": "text"}, {"bbox": [436, 675, 485, 687], "score": 0.94, "content": "I m(A_{i})\\cap", "type": "inline_equation", "height": 12, "width": 49}], "index": 29}, {"bbox": [126, 686, 179, 703], "spans": [{"bbox": [126, 689, 174, 702], "score": 0.91, "content": "I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [175, 686, 179, 703], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29, "bbox_fs": [126, 659, 486, 703]}]}
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[{"type": "text", "bbox": [124, 110, 486, 139], "content": "and are linearly independent, so they are a basis for , and . Thus", "index": 0}, {"type": "interline_equation", "bbox": [198, 147, 411, 161], "content": "", "index": 1}, {"type": "text", "bbox": [124, 165, 486, 194], "content": "which is invariant under . Thus , by the irreducibility of , a contradiction with .", "index": 2}, {"type": "text", "bbox": [124, 200, 486, 243], "content": "Remark 4.2. For and satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case:", "index": 3}, {"type": "image", "bbox": [124, 270, 252, 335], "content": "", "index": 4}, {"type": "text", "bbox": [124, 365, 486, 408], "content": "By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to of the Jones’ representation (see [3], p. 296).", "index": 5}, {"type": "text", "bbox": [124, 421, 486, 464], "content": "Lemma 4.3. Let be an irreducible representation, where , , and . Suppose that the associated friendship graph contains the chain.", "index": 6}, {"type": "text", "bbox": [124, 465, 485, 493], "content": "Then and the associated friendship graph is the chain (that is, the only edges are between neighbors).", "index": 7}, {"type": "text", "bbox": [124, 500, 486, 528], "content": "Proof. By corollary 3.10, . Consider the full friendship graph of . Then", "index": 8}, {"type": "interline_equation", "bbox": [239, 532, 372, 545], "content": "", "index": 9}, {"type": "text", "bbox": [124, 547, 486, 603], "content": "for any where indices are taken modulo . If is two-dimensional, then , and is a two- dimensional invariant subspace, contradicting the irreducibility of . Hence are one-dimensional.", "index": 10}, {"type": "text", "bbox": [138, 604, 398, 618], "content": "For any , , , we have that", "index": 11}, {"type": "interline_equation", "bbox": [173, 625, 437, 639], "content": "", "index": 12}, {"type": "text", "bbox": [124, 643, 412, 658], "content": "for . Moreover, because is invertible.", "index": 13}, {"type": "text", "bbox": [124, 658, 487, 701], "content": "Choose to be a basis vector for . Define for . Then is a basis vector for .", "index": 14}]
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[{"bbox": [126, 114, 485, 127], "content": "and are linearly independent, so they are a basis for , and", "parent_index": 0, "line_index": 0}, {"bbox": [126, 126, 307, 141], "content": ". Thus", "parent_index": 0, "line_index": 1}, {"bbox": [125, 166, 485, 182], "content": "which is invariant under . Thus , by the irreducibility of ,", "parent_index": 2, "line_index": 0}, {"bbox": [126, 182, 288, 195], "content": "a contradiction with .", "parent_index": 2, "line_index": 1}, {"bbox": [125, 203, 486, 217], "content": "Remark 4.2. For and satisfying the hypothesis of theorem", "parent_index": 3, "line_index": 0}, {"bbox": [126, 217, 486, 231], "content": "4.1 there are two possible friendship graphs: 1) all neighbors are friends", "parent_index": 3, "line_index": 1}, {"bbox": [126, 231, 266, 244], "content": "and 2) an exceptional case:", "parent_index": 3, "line_index": 2}, {"bbox": [137, 368, 485, 381], "content": "By [5], Theorem 7.1, part 2, every irreducible representation with", "parent_index": 5, "line_index": 0}, {"bbox": [125, 381, 486, 396], "content": "the above friendship graph is equivalent to the restriction to of the", "parent_index": 5, "line_index": 1}, {"bbox": [126, 396, 322, 409], "content": "Jones’ representation (see [3], p. 296).", "parent_index": 5, "line_index": 2}, {"bbox": [125, 424, 485, 439], "content": "Lemma 4.3. Let be an irreducible representation,", "parent_index": 6, "line_index": 0}, {"bbox": [127, 439, 487, 453], "content": "where , , and . Suppose that the associated", "parent_index": 6, "line_index": 1}, {"bbox": [127, 454, 310, 465], "content": "friendship graph contains the chain.", "parent_index": 6, "line_index": 2}, {"bbox": [138, 465, 485, 482], "content": "Then and the associated friendship graph is the chain (that is,", "parent_index": 7, "line_index": 0}, {"bbox": [126, 480, 321, 495], "content": "the only edges are between neighbors).", "parent_index": 7, "line_index": 1}, {"bbox": [137, 502, 485, 516], "content": "Proof. By corollary 3.10, . Consider the full friendship graph", "parent_index": 8, "line_index": 0}, {"bbox": [126, 516, 181, 530], "content": "of . Then", "parent_index": 8, "line_index": 1}, {"bbox": [126, 550, 486, 564], "content": "for any where indices are taken modulo . If is", "parent_index": 10, "line_index": 0}, {"bbox": [125, 563, 484, 577], "content": "two-dimensional, then , and is a two-", "parent_index": 10, "line_index": 1}, {"bbox": [126, 577, 485, 591], "content": "dimensional invariant subspace, contradicting the irreducibility of .", "parent_index": 10, "line_index": 2}, {"bbox": [126, 591, 370, 605], "content": "Hence are one-dimensional.", "parent_index": 10, "line_index": 3}, {"bbox": [137, 604, 398, 620], "content": "For any , , , we have that", "parent_index": 11, "line_index": 0}, {"bbox": [126, 645, 411, 660], "content": "for . Moreover, because is invertible.", "parent_index": 13, "line_index": 0}, {"bbox": [138, 659, 486, 674], "content": "Choose to be a basis vector for . Define", "parent_index": 14, "line_index": 0}, {"bbox": [126, 674, 485, 688], "content": "for . Then is a basis vector for", "parent_index": 14, "line_index": 1}, {"bbox": [126, 686, 179, 703], "content": ".", "parent_index": 14, "line_index": 2}]
|
[{"bbox": [124, 270, 252, 335], "content": "", "parent_index": 4, "subtype": "body"}]
|
[{"bbox": [126, 118, 136, 125], "content": "a_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [162, 115, 171, 125], "content": "b_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [421, 114, 459, 127], "content": "I m(A_{i})", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 128, 272, 140], "content": "I m(A_{i})\\subseteq I m(A_{1})+I m(A_{2})", "parent_index": 0, "subtype": "inline"}, {"bbox": [198, 147, 411, 161], "content": "U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),", "parent_index": 1, "subtype": "interline"}, {"bbox": [254, 168, 284, 181], "content": "\\rho(B_{n})", "parent_index": 2, "subtype": "inline"}, {"bbox": [320, 169, 349, 180], "content": "r\\leq5", "parent_index": 2, "subtype": "inline"}, {"bbox": [475, 172, 482, 180], "content": "\\rho", "parent_index": 2, "subtype": "inline"}, {"bbox": [234, 184, 284, 194], "content": "r\\geq n\\geq6", "parent_index": 2, "subtype": "inline"}, {"bbox": [225, 205, 257, 213], "content": "n\\,=\\,5", "parent_index": 3, "subtype": "inline"}, {"bbox": [285, 208, 292, 216], "content": "\\rho", "parent_index": 3, "subtype": "inline"}, {"bbox": [438, 383, 451, 394], "content": "B_{5}", "parent_index": 5, "subtype": "inline"}, {"bbox": [221, 426, 314, 438], "content": "\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})", "parent_index": 6, "subtype": "inline"}, {"bbox": [160, 441, 191, 451], "content": "r\\,\\geq\\,n", "parent_index": 6, "subtype": "inline"}, {"bbox": [199, 441, 231, 451], "content": "n\\,\\geq\\,5", "parent_index": 6, "subtype": "inline"}, {"bbox": [265, 440, 336, 452], "content": "r a n k(A_{1})\\,=\\,2", "parent_index": 6, "subtype": "inline"}, {"bbox": [168, 471, 196, 477], "content": "r=n", "parent_index": 7, "subtype": "inline"}, {"bbox": [275, 507, 304, 513], "content": "r=n", "parent_index": 8, "subtype": "inline"}, {"bbox": [139, 522, 146, 529], "content": "\\rho", "parent_index": 8, "subtype": "inline"}, {"bbox": [239, 532, 372, 545], "content": "I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}", "parent_index": 9, "subtype": "interline"}, {"bbox": [167, 552, 171, 560], "content": "i", "parent_index": 10, "subtype": "inline"}, {"bbox": [345, 555, 352, 560], "content": "n", "parent_index": 10, "subtype": "inline"}, {"bbox": [372, 551, 473, 563], "content": "I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [243, 564, 369, 577], "content": "I m(A_{1})=I m(A_{2})=\\ldots", "parent_index": 10, "subtype": "inline"}, {"bbox": [398, 564, 437, 577], "content": "I m(A_{1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [475, 582, 482, 590], "content": "\\rho", "parent_index": 10, "subtype": "inline"}, {"bbox": [161, 592, 260, 605], "content": "I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 10, "subtype": "inline"}, {"bbox": [181, 606, 240, 619], "content": "x\\in I m(A_{i})", "parent_index": 11, "subtype": "inline"}, {"bbox": [247, 607, 288, 618], "content": "x=A_{i}y", "parent_index": 11, "subtype": "inline"}, {"bbox": [294, 607, 324, 618], "content": "x\\neq0", "parent_index": 11, "subtype": "inline"}, {"bbox": [173, 625, 437, 639], "content": "T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})", "parent_index": 12, "subtype": "interline"}, {"bbox": [144, 647, 190, 659], "content": "T=\\rho(\\tau)", "parent_index": 13, "subtype": "inline"}, {"bbox": [252, 648, 288, 659], "content": "T x\\neq0", "parent_index": 13, "subtype": "inline"}, {"bbox": [335, 648, 344, 656], "content": "T", "parent_index": 13, "subtype": "inline"}, {"bbox": [180, 662, 217, 673], "content": "x_{1}~\\neq~0", "parent_index": 14, "subtype": "inline"}, {"bbox": [351, 661, 444, 673], "content": "I m(A_{1})\\cap I m(A_{2})", "parent_index": 14, "subtype": "inline"}, {"bbox": [126, 674, 187, 687], "content": "x_{i+1}=T^{i}x_{1}", "parent_index": 14, "subtype": "inline"}, {"bbox": [208, 676, 280, 686], "content": "1\\leq i\\leq n-1", "parent_index": 14, "subtype": "inline"}, {"bbox": [318, 679, 328, 686], "content": "x_{i}", "parent_index": 14, "subtype": "inline"}, {"bbox": [436, 675, 485, 687], "content": "I m(A_{i})\\cap", "parent_index": 14, "subtype": "inline"}, {"bbox": [126, 689, 174, 702], "content": "I m(A_{i+1})", "parent_index": 14, "subtype": "inline"}]
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[]
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If for some $i$ , $x_{i}$ is proportional to $x_{i+1}$ then, because a full friendship graph is a $\mathbb{Z}_{n}$ -graph, all the $x_{j}$ are proportional to $x_{1}$ . Then, because we have 5 or more vertices in the full friendship graph, for any $A_{i}$ there exists $j$ such that both $A_{j}$ and $A_{j+1}$ are not neighbors of $A_{i}$ . Then
$$
A_{i}A_{j}=A_{j}A_{i}
$$
and
$$
A_{i}A_{j+1}=A_{j+1}A_{i}.
$$
So, if $x\in I m(A_{j})\cap I m(A_{j+1})$ then $A_{i}x\in I m(A_{j})\cap I m(A_{j+1})$ . But this means that $s p a n\{x_{1}\}$ is an invariant subspace and the representation is not irreducible.
So, if the representation is irreducible, then for any $i$ , $x_{i}\notin s p a n\{x_{i+1}\}$ . From this follows that for any $i$
$$
I m(A_{i})=s p a n\{x_{i-1},x_{i}\}
$$
and the $n$ vectors $x_{0},x_{1},\ldots,x_{n-1}$ form a basis of $V$ . Then for any two non-neighbors $A_{i}$ and $A_{j}$
$$
I m(A_{i})\cap I m(A_{j})=\{0\}.
$$
Now, we have the following
Theorem 4.4. Let $\rho\;:\;B_{n}\;\rightarrow\;G L_{r}(\mathbb{C})$ be irreducible, where $r\ \geq\ n$ .
Suppose that for any generator $\sigma_{i}$ , $\rho(\sigma_{i})=1+A_{i}$ , where $r a n k(A_{i})=2$ .
1) If $n\,\geq\,6$ , then $r\,=\,n$ and $\rho$ has a friendship graph which is a chain.
2) If $n=5$ , then $r=5$ and either $\rho$ has a friendship graph which is a chain or $\rho$ has the exceptional friendship graph (see Remark 4.2).
3) If $n=4$ , then either $r=4$ and $\rho$ has a friendship graph which is a chain; or $\rho$ has one of the following exceptional friendship graphs:
Proof. 1) If $n\ge6$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $r=n$ .
2) If $n=5$ , then by corollaries 3.9 and 3.10 the friendship graph of $\rho$ is connected and $r=n$ . If it contains a chain graph, then, by lemma
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<html><body>
<p data-bbox="124 110 487 167">If for some $i$ , $x_{i}$ is proportional to $x_{i+1}$ then, because a full friendship graph is a $\mathbb{Z}_{n}$ -graph, all the $x_{j}$ are proportional to $x_{1}$ . Then, because we have 5 or more vertices in the full friendship graph, for any $A_{i}$ there exists $j$ such that both $A_{j}$ and $A_{j+1}$ are not neighbors of $A_{i}$ . Then </p>
<div class="equation" data-bbox="271 177 339 190">$$
A_{i}A_{j}=A_{j}A_{i}
$$</div>
<p data-bbox="125 194 147 208">and </p>
<div class="equation" data-bbox="259 214 351 227">$$
A_{i}A_{j+1}=A_{j+1}A_{i}.
$$</div>
<p data-bbox="124 229 487 270">So, if $x\in I m(A_{j})\cap I m(A_{j+1})$ then $A_{i}x\in I m(A_{j})\cap I m(A_{j+1})$ . But this means that $s p a n\{x_{1}\}$ is an invariant subspace and the representation is not irreducible. </p>
<p data-bbox="124 271 488 299">So, if the representation is irreducible, then for any $i$ , $x_{i}\notin s p a n\{x_{i+1}\}$ . From this follows that for any $i$ </p>
<div class="equation" data-bbox="242 308 369 322">$$
I m(A_{i})=s p a n\{x_{i-1},x_{i}\}
$$</div>
<p data-bbox="124 327 486 356">and the $n$ vectors $x_{0},x_{1},\ldots,x_{n-1}$ form a basis of $V$ . Then for any two non-neighbors $A_{i}$ and $A_{j}$ </p>
<div class="equation" data-bbox="242 365 368 378">$$
I m(A_{i})\cap I m(A_{j})=\{0\}.
$$</div>
<p data-bbox="136 383 280 397">Now, we have the following </p>
<p data-bbox="124 405 486 433">Theorem 4.4. Let $\rho\;:\;B_{n}\;\rightarrow\;G L_{r}(\mathbb{C})$ be irreducible, where $r\ \geq\ n$ .
Suppose that for any generator $\sigma_{i}$ , $\rho(\sigma_{i})=1+A_{i}$ , where $r a n k(A_{i})=2$ . </p>
<p data-bbox="125 434 486 460">1) If $n\,\geq\,6$ , then $r\,=\,n$ and $\rho$ has a friendship graph which is a chain. </p>
<p data-bbox="126 461 487 489">2) If $n=5$ , then $r=5$ and either $\rho$ has a friendship graph which is a chain or $\rho$ has the exceptional friendship graph (see Remark 4.2). </p>
<p data-bbox="126 489 487 518">3) If $n=4$ , then either $r=4$ and $\rho$ has a friendship graph which is a chain; or $\rho$ has one of the following exceptional friendship graphs: </p>
<div class="image" data-bbox="122 541 464 602"><img data-bbox="122 541 464 602"/></div>
<p data-bbox="123 630 486 671">Proof. 1) If $n\ge6$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $r=n$ . </p>
<p data-bbox="124 672 487 700">2) If $n=5$ , then by corollaries 3.9 and 3.10 the friendship graph of $\rho$ is connected and $r=n$ . If it contains a chain graph, then, by lemma </p>
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0003047v1
| 10 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "If for some $i$ , $x_{i}$ is proportional to $x_{i+1}$ then, because a full friendship graph is a $\\mathbb{Z}_{n}$ -graph, all the $x_{j}$ are proportional to $x_{1}$ . Then, because we have 5 or more vertices in the full friendship graph, for any $A_{i}$ there exists $j$ such that both $A_{j}$ and $A_{j+1}$ are not neighbors of $A_{i}$ . Then ", "page_idx": 10}, {"type": "equation", "text": "$$\nA_{i}A_{j}=A_{j}A_{i}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "and ", "page_idx": 10}, {"type": "equation", "text": "$$\nA_{i}A_{j+1}=A_{j+1}A_{i}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "So, if $x\\in I m(A_{j})\\cap I m(A_{j+1})$ then $A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})$ . But this means that $s p a n\\{x_{1}\\}$ is an invariant subspace and the representation is not irreducible. ", "page_idx": 10}, {"type": "text", "text": "So, if the representation is irreducible, then for any $i$ , $x_{i}\\notin s p a n\\{x_{i+1}\\}$ . From this follows that for any $i$ ", "page_idx": 10}, {"type": "equation", "text": "$$\nI m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "and the $n$ vectors $x_{0},x_{1},\\ldots,x_{n-1}$ form a basis of $V$ . Then for any two non-neighbors $A_{i}$ and $A_{j}$ ", "page_idx": 10}, {"type": "equation", "text": "$$\nI m(A_{i})\\cap I m(A_{j})=\\{0\\}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "Now, we have the following ", "page_idx": 10}, {"type": "text", "text": "Theorem 4.4. Let $\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})$ be irreducible, where $r\\ \\geq\\ n$ . \nSuppose that for any generator $\\sigma_{i}$ , $\\rho(\\sigma_{i})=1+A_{i}$ , where $r a n k(A_{i})=2$ . ", "page_idx": 10}, {"type": "text", "text": "1) If $n\\,\\geq\\,6$ , then $r\\,=\\,n$ and $\\rho$ has a friendship graph which is a chain. ", "page_idx": 10}, {"type": "text", "text": "2) If $n=5$ , then $r=5$ and either $\\rho$ has a friendship graph which is a chain or $\\rho$ has the exceptional friendship graph (see Remark 4.2). ", "page_idx": 10}, {"type": "text", "text": "3) If $n=4$ , then either $r=4$ and $\\rho$ has a friendship graph which is a chain; or $\\rho$ has one of the following exceptional friendship graphs: ", "page_idx": 10}, {"type": "image", "img_path": "images/27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg", "img_caption": [], "img_footnote": [], "page_idx": 10}, {"type": "text", "text": "Proof. 1) If $n\\ge6$ , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and $r=n$ . ", "page_idx": 10}, {"type": "text", "text": "2) If $n=5$ , then by corollaries 3.9 and 3.10 the friendship graph of $\\rho$ is connected and $r=n$ . If it contains a chain graph, then, by lemma ", "page_idx": 10}]
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Then, because", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 443, 155], "score": 1.0, "content": "we have 5 or more vertices in the full friendship graph, for any ", "type": "text"}, {"bbox": [443, 142, 455, 153], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [456, 141, 486, 155], "score": 1.0, "content": " there", "type": "text"}], "index": 2}, {"bbox": [126, 155, 469, 169], "spans": [{"bbox": [126, 155, 158, 169], "score": 1.0, "content": "exists ", "type": "text"}, {"bbox": [158, 157, 164, 168], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [164, 155, 246, 169], "score": 1.0, "content": " such that both ", "type": "text"}, {"bbox": [247, 156, 261, 169], "score": 0.93, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [261, 155, 286, 169], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [287, 156, 311, 169], "score": 0.93, "content": "A_{j+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [311, 155, 421, 169], "score": 1.0, "content": " are not neighbors of ", "type": "text"}, {"bbox": [421, 156, 434, 167], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [434, 155, 469, 169], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [271, 177, 339, 190], "lines": [{"bbox": [271, 177, 339, 190], "spans": [{"bbox": [271, 177, 339, 190], "score": 0.92, "content": "A_{i}A_{j}=A_{j}A_{i}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 194, 147, 208], "lines": [{"bbox": [125, 197, 147, 209], "spans": [{"bbox": [125, 197, 147, 209], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [259, 214, 351, 227], "lines": [{"bbox": [259, 214, 351, 227], "spans": [{"bbox": [259, 214, 351, 227], "score": 0.9, "content": "A_{i}A_{j+1}=A_{j+1}A_{i}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 229, 487, 270], "lines": [{"bbox": [125, 231, 487, 246], "spans": [{"bbox": [125, 231, 154, 246], "score": 1.0, "content": "So, if", "type": "text"}, {"bbox": [155, 232, 275, 245], "score": 0.95, "content": "x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 120}, {"bbox": [276, 231, 304, 246], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [304, 232, 438, 245], "score": 0.95, "content": "A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [438, 231, 487, 246], "score": 1.0, "content": ". But this", "type": "text"}], "index": 7}, {"bbox": [125, 246, 485, 259], "spans": [{"bbox": [125, 246, 187, 259], "score": 1.0, "content": "means that ", "type": "text"}, {"bbox": [188, 246, 236, 259], "score": 0.92, "content": "s p a n\\{x_{1}\\}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [236, 246, 485, 259], "score": 1.0, "content": " is an invariant subspace and the representation", "type": "text"}], "index": 8}, {"bbox": [125, 260, 216, 272], "spans": [{"bbox": [125, 260, 216, 272], "score": 1.0, "content": "is not irreducible.", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [124, 271, 488, 299], "lines": [{"bbox": [137, 272, 489, 288], "spans": [{"bbox": [137, 272, 393, 288], "score": 1.0, "content": "So, if the representation is irreducible, then for any ", "type": "text"}, {"bbox": [393, 276, 397, 284], "score": 0.78, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [397, 272, 402, 288], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [403, 274, 486, 286], "score": 0.91, "content": "x_{i}\\notin s p a n\\{x_{i+1}\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [486, 272, 489, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 287, 287, 300], "spans": [{"bbox": [125, 287, 283, 300], "score": 1.0, "content": "From this follows that for any ", "type": "text"}, {"bbox": [283, 289, 287, 298], "score": 0.84, "content": "i", "type": "inline_equation", "height": 9, "width": 4}], "index": 11}], "index": 10.5}, {"type": "interline_equation", "bbox": [242, 308, 369, 322], "lines": [{"bbox": [242, 308, 369, 322], "spans": [{"bbox": [242, 308, 369, 322], "score": 0.92, "content": "I m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 327, 486, 356], "lines": [{"bbox": [125, 329, 486, 344], "spans": [{"bbox": [125, 329, 169, 344], "score": 1.0, "content": "and the ", "type": "text"}, {"bbox": [169, 334, 176, 340], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [176, 329, 219, 344], "score": 1.0, "content": " vectors ", "type": "text"}, {"bbox": [219, 334, 297, 342], "score": 0.91, "content": "x_{0},x_{1},\\ldots,x_{n-1}", "type": "inline_equation", "height": 8, "width": 78}, {"bbox": [297, 329, 380, 344], "score": 1.0, "content": " form a basis of ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.85, "content": "V", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [389, 329, 486, 344], "score": 1.0, "content": ". Then for any two", "type": "text"}], "index": 13}, {"bbox": [125, 342, 253, 357], "spans": [{"bbox": [125, 342, 201, 357], "score": 1.0, "content": "non-neighbors ", "type": "text"}, {"bbox": [201, 345, 214, 355], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [214, 342, 240, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 345, 253, 357], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}], "index": 14}], "index": 13.5}, {"type": "interline_equation", "bbox": [242, 365, 368, 378], "lines": [{"bbox": [242, 365, 368, 378], "spans": [{"bbox": [242, 365, 368, 378], "score": 0.92, "content": "I m(A_{i})\\cap I m(A_{j})=\\{0\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [136, 383, 280, 397], "lines": [{"bbox": [137, 384, 279, 399], "spans": [{"bbox": [137, 384, 279, 399], "score": 1.0, "content": "Now, we have the following", "type": "text"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [124, 405, 486, 433], "lines": [{"bbox": [126, 407, 486, 421], "spans": [{"bbox": [126, 407, 231, 421], "score": 1.0, "content": "Theorem 4.4. Let ", "type": "text"}, {"bbox": [231, 408, 330, 421], "score": 0.92, "content": "\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [331, 407, 447, 421], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [447, 410, 482, 420], "score": 0.9, "content": "r\\ \\geq\\ n", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 407, 486, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [126, 421, 485, 435], "spans": [{"bbox": [126, 421, 285, 435], "score": 1.0, "content": "Suppose that for any generator ", "type": "text"}, {"bbox": [285, 426, 296, 433], "score": 0.81, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [296, 421, 302, 435], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [302, 422, 374, 435], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [374, 421, 413, 435], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [414, 422, 482, 434], "score": 0.69, "content": "r a n k(A_{i})=2", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [482, 421, 485, 435], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 17.5}, {"type": "text", "bbox": [125, 434, 486, 460], "lines": [{"bbox": [139, 436, 487, 449], "spans": [{"bbox": [139, 436, 167, 449], "score": 1.0, "content": "1) If ", "type": "text"}, {"bbox": [167, 437, 201, 447], "score": 0.81, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [201, 436, 235, 449], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [236, 437, 269, 446], "score": 0.77, "content": "r\\,=\\,n", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [269, 436, 297, 449], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 438, 304, 448], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [304, 436, 487, 449], "score": 1.0, "content": " has a friendship graph which is a", "type": "text"}], "index": 19}, {"bbox": [126, 448, 158, 463], "spans": [{"bbox": [126, 448, 158, 463], "score": 1.0, "content": "chain.", "type": "text"}], "index": 20}], "index": 19.5}, {"type": "text", "bbox": [126, 461, 487, 489], "lines": [{"bbox": [138, 464, 487, 477], "spans": [{"bbox": [138, 464, 165, 477], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [165, 465, 194, 474], "score": 0.84, "content": "n=5", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 464, 227, 477], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [227, 464, 255, 474], "score": 0.85, "content": "r=5", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 464, 315, 477], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [315, 465, 322, 476], "score": 0.63, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [322, 464, 487, 477], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 21}, {"bbox": [127, 477, 471, 491], "spans": [{"bbox": [127, 477, 182, 491], "score": 1.0, "content": "a chain or ", "type": "text"}, {"bbox": [183, 482, 189, 490], "score": 0.76, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [190, 477, 471, 491], "score": 1.0, "content": " has the exceptional friendship graph (see Remark 4.2).", "type": "text"}], "index": 22}], "index": 21.5}, {"type": "text", "bbox": [126, 489, 487, 518], "lines": [{"bbox": [138, 491, 486, 504], "spans": [{"bbox": [138, 491, 164, 504], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [165, 493, 194, 502], "score": 0.84, "content": "n=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 491, 261, 504], "score": 1.0, "content": ", then either ", "type": "text"}, {"bbox": [261, 492, 289, 502], "score": 0.85, "content": "r=4", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [289, 491, 315, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [315, 495, 322, 504], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 491, 486, 504], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 23}, {"bbox": [127, 505, 474, 519], "spans": [{"bbox": [127, 505, 186, 519], "score": 1.0, "content": "a chain; or ", "type": "text"}, {"bbox": [186, 510, 193, 518], "score": 0.74, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [193, 505, 474, 519], "score": 1.0, "content": " has one of the following exceptional friendship graphs:", "type": "text"}], "index": 24}], "index": 23.5}, {"type": "image", "bbox": [122, 541, 464, 602], "blocks": [{"type": "image_body", "bbox": [122, 541, 464, 602], "group_id": 0, "lines": [{"bbox": [122, 541, 464, 602], "spans": [{"bbox": [122, 541, 464, 602], "score": 0.656, "type": "image", "image_path": "27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [122, 541, 464, 561.3333333333334], "spans": [], "index": 25}, {"bbox": [122, 561.3333333333334, 464, 581.6666666666667], "spans": [], "index": 26}, {"bbox": [122, 581.6666666666667, 464, 602.0000000000001], "spans": [], "index": 27}]}], "index": 26}, {"type": "text", "bbox": [123, 630, 486, 671], "lines": [{"bbox": [137, 632, 485, 646], "spans": [{"bbox": [137, 632, 209, 646], "score": 1.0, "content": "Proof. 1) If ", "type": "text"}, {"bbox": [209, 634, 240, 644], "score": 0.92, "content": "n\\ge6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [241, 632, 485, 646], "score": 1.0, "content": ", then by theorem 4.1 the associated friendship", "type": "text"}], "index": 28}, {"bbox": [124, 646, 486, 659], "spans": [{"bbox": [124, 646, 486, 659], "score": 1.0, "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "type": "text"}], "index": 29}, {"bbox": [126, 662, 158, 673], "spans": [{"bbox": [126, 664, 154, 670], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [155, 662, 158, 673], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [124, 672, 487, 700], "lines": [{"bbox": [137, 674, 487, 688], "spans": [{"bbox": [137, 674, 164, 688], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [164, 676, 194, 684], "score": 0.91, "content": "n=5", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [194, 674, 487, 688], "score": 1.0, "content": ", then by corollaries 3.9 and 3.10 the friendship graph of", "type": "text"}], "index": 31}, {"bbox": [126, 688, 486, 702], "spans": [{"bbox": [126, 693, 132, 701], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [133, 688, 224, 702], "score": 1.0, "content": " is connected and ", "type": "text"}, {"bbox": [224, 693, 253, 698], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [253, 688, 486, 702], "score": 1.0, "content": ". 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Then, because", "type": "text"}], "index": 1}, {"bbox": [126, 141, 486, 155], "spans": [{"bbox": [126, 141, 443, 155], "score": 1.0, "content": "we have 5 or more vertices in the full friendship graph, for any ", "type": "text"}, {"bbox": [443, 142, 455, 153], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [456, 141, 486, 155], "score": 1.0, "content": " there", "type": "text"}], "index": 2}, {"bbox": [126, 155, 469, 169], "spans": [{"bbox": [126, 155, 158, 169], "score": 1.0, "content": "exists ", "type": "text"}, {"bbox": [158, 157, 164, 168], "score": 0.88, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [164, 155, 246, 169], "score": 1.0, "content": " such that both ", "type": "text"}, {"bbox": [247, 156, 261, 169], "score": 0.93, "content": "A_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [261, 155, 286, 169], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [287, 156, 311, 169], "score": 0.93, "content": "A_{j+1}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [311, 155, 421, 169], "score": 1.0, "content": " are not neighbors of ", "type": "text"}, {"bbox": [421, 156, 434, 167], "score": 0.91, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [434, 155, 469, 169], "score": 1.0, "content": ". Then", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [125, 111, 486, 169]}, {"type": "interline_equation", "bbox": [271, 177, 339, 190], "lines": [{"bbox": [271, 177, 339, 190], "spans": [{"bbox": [271, 177, 339, 190], "score": 0.92, "content": "A_{i}A_{j}=A_{j}A_{i}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [125, 194, 147, 208], "lines": [{"bbox": [125, 197, 147, 209], "spans": [{"bbox": [125, 197, 147, 209], "score": 1.0, "content": "and", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [125, 197, 147, 209]}, {"type": "interline_equation", "bbox": [259, 214, 351, 227], "lines": [{"bbox": [259, 214, 351, 227], "spans": [{"bbox": [259, 214, 351, 227], "score": 0.9, "content": "A_{i}A_{j+1}=A_{j+1}A_{i}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [124, 229, 487, 270], "lines": [{"bbox": [125, 231, 487, 246], "spans": [{"bbox": [125, 231, 154, 246], "score": 1.0, "content": "So, if", "type": "text"}, {"bbox": [155, 232, 275, 245], "score": 0.95, "content": "x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 120}, {"bbox": [276, 231, 304, 246], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [304, 232, 438, 245], "score": 0.95, "content": "A_{i}x\\in I m(A_{j})\\cap I m(A_{j+1})", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [438, 231, 487, 246], "score": 1.0, "content": ". But this", "type": "text"}], "index": 7}, {"bbox": [125, 246, 485, 259], "spans": [{"bbox": [125, 246, 187, 259], "score": 1.0, "content": "means that ", "type": "text"}, {"bbox": [188, 246, 236, 259], "score": 0.92, "content": "s p a n\\{x_{1}\\}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [236, 246, 485, 259], "score": 1.0, "content": " is an invariant subspace and the representation", "type": "text"}], "index": 8}, {"bbox": [125, 260, 216, 272], "spans": [{"bbox": [125, 260, 216, 272], "score": 1.0, "content": "is not irreducible.", "type": "text"}], "index": 9}], "index": 8, "bbox_fs": [125, 231, 487, 272]}, {"type": "text", "bbox": [124, 271, 488, 299], "lines": [{"bbox": [137, 272, 489, 288], "spans": [{"bbox": [137, 272, 393, 288], "score": 1.0, "content": "So, if the representation is irreducible, then for any ", "type": "text"}, {"bbox": [393, 276, 397, 284], "score": 0.78, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [397, 272, 402, 288], "score": 1.0, "content": ",", "type": "text"}, {"bbox": [403, 274, 486, 286], "score": 0.91, "content": "x_{i}\\notin s p a n\\{x_{i+1}\\}", "type": "inline_equation", "height": 12, "width": 83}, {"bbox": [486, 272, 489, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [125, 287, 287, 300], "spans": [{"bbox": [125, 287, 283, 300], "score": 1.0, "content": "From this follows that for any ", "type": "text"}, {"bbox": [283, 289, 287, 298], "score": 0.84, "content": "i", "type": "inline_equation", "height": 9, "width": 4}], "index": 11}], "index": 10.5, "bbox_fs": [125, 272, 489, 300]}, {"type": "interline_equation", "bbox": [242, 308, 369, 322], "lines": [{"bbox": [242, 308, 369, 322], "spans": [{"bbox": [242, 308, 369, 322], "score": 0.92, "content": "I m(A_{i})=s p a n\\{x_{i-1},x_{i}\\}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [124, 327, 486, 356], "lines": [{"bbox": [125, 329, 486, 344], "spans": [{"bbox": [125, 329, 169, 344], "score": 1.0, "content": "and the ", "type": "text"}, {"bbox": [169, 334, 176, 340], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [176, 329, 219, 344], "score": 1.0, "content": " vectors ", "type": "text"}, {"bbox": [219, 334, 297, 342], "score": 0.91, "content": "x_{0},x_{1},\\ldots,x_{n-1}", "type": "inline_equation", "height": 8, "width": 78}, {"bbox": [297, 329, 380, 344], "score": 1.0, "content": " form a basis of ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.85, "content": "V", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [389, 329, 486, 344], "score": 1.0, "content": ". Then for any two", "type": "text"}], "index": 13}, {"bbox": [125, 342, 253, 357], "spans": [{"bbox": [125, 342, 201, 357], "score": 1.0, "content": "non-neighbors ", "type": "text"}, {"bbox": [201, 345, 214, 355], "score": 0.92, "content": "A_{i}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [214, 342, 240, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [240, 345, 253, 357], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}], "index": 14}], "index": 13.5, "bbox_fs": [125, 329, 486, 357]}, {"type": "interline_equation", "bbox": [242, 365, 368, 378], "lines": [{"bbox": [242, 365, 368, 378], "spans": [{"bbox": [242, 365, 368, 378], "score": 0.92, "content": "I m(A_{i})\\cap I m(A_{j})=\\{0\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [136, 383, 280, 397], "lines": [{"bbox": [137, 384, 279, 399], "spans": [{"bbox": [137, 384, 279, 399], "score": 1.0, "content": "Now, we have the following", "type": "text"}], "index": 16}], "index": 16, "bbox_fs": [137, 384, 279, 399]}, {"type": "list", "bbox": [124, 405, 486, 433], "lines": [{"bbox": [126, 407, 486, 421], "spans": [{"bbox": [126, 407, 231, 421], "score": 1.0, "content": "Theorem 4.4. Let ", "type": "text"}, {"bbox": [231, 408, 330, 421], "score": 0.92, "content": "\\rho\\;:\\;B_{n}\\;\\rightarrow\\;G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [331, 407, 447, 421], "score": 1.0, "content": " be irreducible, where ", "type": "text"}, {"bbox": [447, 410, 482, 420], "score": 0.9, "content": "r\\ \\geq\\ n", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 407, 486, 421], "score": 1.0, "content": ".", "type": "text"}], "index": 17, "is_list_end_line": true}, {"bbox": [126, 421, 485, 435], "spans": [{"bbox": [126, 421, 285, 435], "score": 1.0, "content": "Suppose that for any generator ", "type": "text"}, {"bbox": [285, 426, 296, 433], "score": 0.81, "content": "\\sigma_{i}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [296, 421, 302, 435], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [302, 422, 374, 435], "score": 0.92, "content": "\\rho(\\sigma_{i})=1+A_{i}", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [374, 421, 413, 435], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [414, 422, 482, 434], "score": 0.69, "content": "r a n k(A_{i})=2", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [482, 421, 485, 435], "score": 1.0, "content": ".", "type": "text"}], "index": 18, "is_list_start_line": true, "is_list_end_line": true}], "index": 17.5, "bbox_fs": [126, 407, 486, 435]}, {"type": "text", "bbox": [125, 434, 486, 460], "lines": [{"bbox": [139, 436, 487, 449], "spans": [{"bbox": [139, 436, 167, 449], "score": 1.0, "content": "1) If ", "type": "text"}, {"bbox": [167, 437, 201, 447], "score": 0.81, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [201, 436, 235, 449], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [236, 437, 269, 446], "score": 0.77, "content": "r\\,=\\,n", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [269, 436, 297, 449], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [297, 438, 304, 448], "score": 0.72, "content": "\\rho", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [304, 436, 487, 449], "score": 1.0, "content": " has a friendship graph which is a", "type": "text"}], "index": 19}, {"bbox": [126, 448, 158, 463], "spans": [{"bbox": [126, 448, 158, 463], "score": 1.0, "content": "chain.", "type": "text"}], "index": 20}], "index": 19.5, "bbox_fs": [126, 436, 487, 463]}, {"type": "text", "bbox": [126, 461, 487, 489], "lines": [{"bbox": [138, 464, 487, 477], "spans": [{"bbox": [138, 464, 165, 477], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [165, 465, 194, 474], "score": 0.84, "content": "n=5", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 464, 227, 477], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [227, 464, 255, 474], "score": 0.85, "content": "r=5", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [256, 464, 315, 477], "score": 1.0, "content": " and either ", "type": "text"}, {"bbox": [315, 465, 322, 476], "score": 0.63, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [322, 464, 487, 477], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 21}, {"bbox": [127, 477, 471, 491], "spans": [{"bbox": [127, 477, 182, 491], "score": 1.0, "content": "a chain or ", "type": "text"}, {"bbox": [183, 482, 189, 490], "score": 0.76, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [190, 477, 471, 491], "score": 1.0, "content": " has the exceptional friendship graph (see Remark 4.2).", "type": "text"}], "index": 22}], "index": 21.5, "bbox_fs": [127, 464, 487, 491]}, {"type": "text", "bbox": [126, 489, 487, 518], "lines": [{"bbox": [138, 491, 486, 504], "spans": [{"bbox": [138, 491, 164, 504], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [165, 493, 194, 502], "score": 0.84, "content": "n=4", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [195, 491, 261, 504], "score": 1.0, "content": ", then either ", "type": "text"}, {"bbox": [261, 492, 289, 502], "score": 0.85, "content": "r=4", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [289, 491, 315, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [315, 495, 322, 504], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 491, 486, 504], "score": 1.0, "content": " has a friendship graph which is", "type": "text"}], "index": 23}, {"bbox": [127, 505, 474, 519], "spans": [{"bbox": [127, 505, 186, 519], "score": 1.0, "content": "a chain; or ", "type": "text"}, {"bbox": [186, 510, 193, 518], "score": 0.74, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [193, 505, 474, 519], "score": 1.0, "content": " has one of the following exceptional friendship graphs:", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [127, 491, 486, 519]}, {"type": "image", "bbox": [122, 541, 464, 602], "blocks": [{"type": "image_body", "bbox": [122, 541, 464, 602], "group_id": 0, "lines": [{"bbox": [122, 541, 464, 602], "spans": [{"bbox": [122, 541, 464, 602], "score": 0.656, "type": "image", "image_path": "27c10b510c8bbbc3d069cb065d61e0381c4205bc59650bb665c9f4d98253a9b3.jpg"}]}], "index": 26, "virtual_lines": [{"bbox": [122, 541, 464, 561.3333333333334], "spans": [], "index": 25}, {"bbox": [122, 561.3333333333334, 464, 581.6666666666667], "spans": [], "index": 26}, {"bbox": [122, 581.6666666666667, 464, 602.0000000000001], "spans": [], "index": 27}]}], "index": 26}, {"type": "text", "bbox": [123, 630, 486, 671], "lines": [{"bbox": [137, 632, 485, 646], "spans": [{"bbox": [137, 632, 209, 646], "score": 1.0, "content": "Proof. 1) If ", "type": "text"}, {"bbox": [209, 634, 240, 644], "score": 0.92, "content": "n\\ge6", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [241, 632, 485, 646], "score": 1.0, "content": ", then by theorem 4.1 the associated friendship", "type": "text"}], "index": 28}, {"bbox": [124, 646, 486, 659], "spans": [{"bbox": [124, 646, 486, 659], "score": 1.0, "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "type": "text"}], "index": 29}, {"bbox": [126, 662, 158, 673], "spans": [{"bbox": [126, 664, 154, 670], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 6, "width": 28}, {"bbox": [155, 662, 158, 673], "score": 1.0, "content": ".", "type": "text"}], "index": 30}], "index": 29, "bbox_fs": [124, 632, 486, 673]}, {"type": "text", "bbox": [124, 672, 487, 700], "lines": [{"bbox": [137, 674, 487, 688], "spans": [{"bbox": [137, 674, 164, 688], "score": 1.0, "content": "2) If ", "type": "text"}, {"bbox": [164, 676, 194, 684], "score": 0.91, "content": "n=5", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [194, 674, 487, 688], "score": 1.0, "content": ", then by corollaries 3.9 and 3.10 the friendship graph of", "type": "text"}], "index": 31}, {"bbox": [126, 688, 486, 702], "spans": [{"bbox": [126, 693, 132, 701], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [133, 688, 224, 702], "score": 1.0, "content": " is connected and ", "type": "text"}, {"bbox": [224, 693, 253, 698], "score": 0.88, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [253, 688, 486, 702], "score": 1.0, "content": ". If it contains a chain graph, then, by lemma", "type": "text"}], "index": 32}], "index": 31.5, "bbox_fs": [126, 674, 487, 702]}]}
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[{"type": "text", "bbox": [124, 110, 487, 167], "content": "If for some , is proportional to then, because a full friendship graph is a -graph, all the are proportional to . Then, because we have 5 or more vertices in the full friendship graph, for any there exists such that both and are not neighbors of . Then", "index": 0}, {"type": "interline_equation", "bbox": [271, 177, 339, 190], "content": "", "index": 1}, {"type": "text", "bbox": [125, 194, 147, 208], "content": "and", "index": 2}, {"type": "interline_equation", "bbox": [259, 214, 351, 227], "content": "", "index": 3}, {"type": "text", "bbox": [124, 229, 487, 270], "content": "So, if then . But this means that is an invariant subspace and the representation is not irreducible.", "index": 4}, {"type": "text", "bbox": [124, 271, 488, 299], "content": "So, if the representation is irreducible, then for any , . From this follows that for any", "index": 5}, {"type": "interline_equation", "bbox": [242, 308, 369, 322], "content": "", "index": 6}, {"type": "text", "bbox": [124, 327, 486, 356], "content": "and the vectors form a basis of . Then for any two non-neighbors and", "index": 7}, {"type": "interline_equation", "bbox": [242, 365, 368, 378], "content": "", "index": 8}, {"type": "text", "bbox": [136, 383, 280, 397], "content": "Now, we have the following", "index": 9}, {"type": "list", "bbox": [124, 405, 486, 433], "content": "", "index": 10}, {"type": "text", "bbox": [125, 434, 486, 460], "content": "1) If , then and has a friendship graph which is a chain.", "index": 11}, {"type": "text", "bbox": [126, 461, 487, 489], "content": "2) If , then and either has a friendship graph which is a chain or has the exceptional friendship graph (see Remark 4.2).", "index": 12}, {"type": "text", "bbox": [126, 489, 487, 518], "content": "3) If , then either and has a friendship graph which is a chain; or has one of the following exceptional friendship graphs:", "index": 13}, {"type": "image", "bbox": [122, 541, 464, 602], "content": "", "index": 14}, {"type": "text", "bbox": [123, 630, 486, 671], "content": "Proof. 1) If , then by theorem 4.1 the associated friendship graph contains a chain, and, by lemma 4.3 has no other edges and .", "index": 15}, {"type": "text", "bbox": [124, 672, 487, 700], "content": "2) If , then by corollaries 3.9 and 3.10 the friendship graph of is connected and . If it contains a chain graph, then, by lemma", "index": 16}]
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[{"bbox": [136, 111, 485, 128], "content": "If for some , is proportional to then, because a full friendship", "parent_index": 0, "line_index": 0}, {"bbox": [125, 127, 486, 141], "content": "graph is a -graph, all the are proportional to . Then, because", "parent_index": 0, "line_index": 1}, {"bbox": [126, 141, 486, 155], "content": "we have 5 or more vertices in the full friendship graph, for any there", "parent_index": 0, "line_index": 2}, {"bbox": [126, 155, 469, 169], "content": "exists such that both and are not neighbors of . Then", "parent_index": 0, "line_index": 3}, {"bbox": [125, 197, 147, 209], "content": "and", "parent_index": 2, "line_index": 0}, {"bbox": [125, 231, 487, 246], "content": "So, if then . But this", "parent_index": 4, "line_index": 0}, {"bbox": [125, 246, 485, 259], "content": "means that is an invariant subspace and the representation", "parent_index": 4, "line_index": 1}, {"bbox": [125, 260, 216, 272], "content": "is not irreducible.", "parent_index": 4, "line_index": 2}, {"bbox": [137, 272, 489, 288], "content": "So, if the representation is irreducible, then for any , .", "parent_index": 5, "line_index": 0}, {"bbox": [125, 287, 287, 300], "content": "From this follows that for any", "parent_index": 5, "line_index": 1}, {"bbox": [125, 329, 486, 344], "content": "and the vectors form a basis of . Then for any two", "parent_index": 7, "line_index": 0}, {"bbox": [125, 342, 253, 357], "content": "non-neighbors and", "parent_index": 7, "line_index": 1}, {"bbox": [137, 384, 279, 399], "content": "Now, we have the following", "parent_index": 9, "line_index": 0}, {"bbox": [126, 407, 486, 421], "content": "Theorem 4.4. Let be irreducible, where .", "parent_index": 10, "line_index": 0}, {"bbox": [126, 421, 485, 435], "content": "Suppose that for any generator , , where .", "parent_index": 10, "line_index": 1}, {"bbox": [139, 436, 487, 449], "content": "1) If , then and has a friendship graph which is a", "parent_index": 11, "line_index": 0}, {"bbox": [126, 448, 158, 463], "content": "chain.", "parent_index": 11, "line_index": 1}, {"bbox": [138, 464, 487, 477], "content": "2) If , then and either has a friendship graph which is", "parent_index": 12, "line_index": 0}, {"bbox": [127, 477, 471, 491], "content": "a chain or has the exceptional friendship graph (see Remark 4.2).", "parent_index": 12, "line_index": 1}, {"bbox": [138, 491, 486, 504], "content": "3) If , then either and has a friendship graph which is", "parent_index": 13, "line_index": 0}, {"bbox": [127, 505, 474, 519], "content": "a chain; or has one of the following exceptional friendship graphs:", "parent_index": 13, "line_index": 1}, {"bbox": [137, 632, 485, 646], "content": "Proof. 1) If , then by theorem 4.1 the associated friendship", "parent_index": 15, "line_index": 0}, {"bbox": [124, 646, 486, 659], "content": "graph contains a chain, and, by lemma 4.3 has no other edges and", "parent_index": 15, "line_index": 1}, {"bbox": [126, 662, 158, 673], "content": ".", "parent_index": 15, "line_index": 2}, {"bbox": [137, 674, 487, 688], "content": "2) If , then by corollaries 3.9 and 3.10 the friendship graph of", "parent_index": 16, "line_index": 0}, {"bbox": [126, 688, 486, 702], "content": "is connected and . If it contains a chain graph, then, by lemma", "parent_index": 16, "line_index": 1}]
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4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case.
3) If $n=4$ , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible $\mathbb{Z}_{4}-$ graphs on 4 vertices.
Remark 4.5. It is proven in [5], Chapter 6, that any representation of $B_{4}$ with either of the exeptional friendship graphs in 3) of the above theorem is reducible.
5. Representations whose friendship graph is a chain
Definition 5.1. The standard representation is the representation
$$
\tau_{n}:B_{n}\to G L_{n}(\mathbb{Z}[t^{\pm1}]
$$
defined by
$$
\tau_{n}(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&&&\\ &{0}&{t}&\\ &{1}&{0}&\\ &&&{I_{n-1-i}}\end{array}\right),
$$
for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix.
Theorem 5.1. Let $\rho:B_{n}\to G L_{n}(\mathbb{C})$ be an irreducible representation, where $n\geq4$ . Suppose that $\rho(\sigma_{1})=1+A_{1}$ , where $r a n k(A_{1})=2$ , and the associated friendship graph of $\rho$ is a chain.
Then $\rho$ is equivalent to a specialization $\tau_{n}(u)$ of the standard representation for some $u\in\mathbb{C}^{*}$ .
Before proving the theorem, we will need the following technical lemma:
Lemma 5.2. Let A be a friend and a neighbor of B, $B$ be a friend and a neighbor of $C$ and suppose that A is not a friend of $C$ :
|
<html><body>
<p data-bbox="123 110 487 138">4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case. </p>
<p data-bbox="124 139 486 181">3) If $n=4$ , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible $\mathbb{Z}_{4}-$ graphs on 4 vertices. </p>
<p data-bbox="124 187 487 230">Remark 4.5. It is proven in [5], Chapter 6, that any representation of $B_{4}$ with either of the exeptional friendship graphs in 3) of the above theorem is reducible. </p>
<p data-bbox="140 246 471 259">5. Representations whose friendship graph is a chain </p>
<p data-bbox="124 265 486 293">Definition 5.1. The standard representation is the representation </p>
<div class="equation" data-bbox="249 296 362 312">$$
\tau_{n}:B_{n}\to G L_{n}(\mathbb{Z}[t^{\pm1}]
$$</div>
<p data-bbox="125 314 180 329">defined by </p>
<div class="equation" data-bbox="219 369 391 428">$$
\tau_{n}(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&&&\\ &{0}&{t}&\\ &{1}&{0}&\\ &&&{I_{n-1-i}}\end{array}\right),
$$</div>
<p data-bbox="124 442 434 458">for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix. </p>
<p data-bbox="125 472 487 515">Theorem 5.1. Let $\rho:B_{n}\to G L_{n}(\mathbb{C})$ be an irreducible representation, where $n\geq4$ . Suppose that $\rho(\sigma_{1})=1+A_{1}$ , where $r a n k(A_{1})=2$ , and the associated friendship graph of $\rho$ is a chain. </p>
<p data-bbox="126 516 485 543">Then $\rho$ is equivalent to a specialization $\tau_{n}(u)$ of the standard representation for some $u\in\mathbb{C}^{*}$ . </p>
<div class="image" data-bbox="124 567 404 607"><img data-bbox="124 567 404 607"/></div>
<p data-bbox="123 635 487 664">Before proving the theorem, we will need the following technical lemma: </p>
<p data-bbox="124 671 487 700">Lemma 5.2. Let A be a friend and a neighbor of B, $B$ be a friend and a neighbor of $C$ and suppose that A is not a friend of $C$ : </p>
</body></html>
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0003047v1
| 11 | 612 | 792 | 1,275 | 1,650 |
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{"category_id": 15, "poly": [797.0, 1237.0, 894.0, 1237.0, 894.0, 1278.0, 797.0, 1278.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [974.0, 1237.0, 1161.0, 1237.0, 1161.0, 1278.0, 974.0, 1278.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 11, "height": 2200, "width": 1700}}
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{"preproc_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 486, 127], "score": 1.0, "content": "4.3, it has no other edges. If it does not contain a chain graph, we", "type": "text"}], "index": 0}, {"bbox": [126, 127, 269, 140], "spans": [{"bbox": [126, 127, 269, 140], "score": 1.0, "content": "obtain the exceptional case.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [124, 139, 486, 181], "lines": [{"bbox": [137, 140, 484, 155], "spans": [{"bbox": [137, 140, 164, 155], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [164, 143, 194, 151], "score": 0.91, "content": "n=4", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [195, 140, 484, 155], "score": 1.0, "content": ", then by theorem 3.8 the friendship graph is not totally", "type": "text"}], "index": 2}, {"bbox": [125, 154, 487, 169], "spans": [{"bbox": [125, 154, 398, 169], "score": 1.0, "content": "disconnected. Hence, we have only three possible ", "type": "text"}, {"bbox": [398, 156, 421, 167], "score": 0.46, "content": "\\mathbb{Z}_{4}-", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [421, 154, 487, 169], "score": 1.0, "content": "graphs on 4", "type": "text"}], "index": 3}, {"bbox": [126, 170, 168, 182], "spans": [{"bbox": [126, 170, 168, 182], "score": 1.0, "content": "vertices.", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [124, 187, 487, 230], "lines": [{"bbox": [124, 190, 486, 206], "spans": [{"bbox": [124, 190, 486, 206], "score": 1.0, "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "type": "text"}], "index": 5}, {"bbox": [125, 204, 486, 218], "spans": [{"bbox": [125, 204, 138, 218], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 206, 153, 217], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [153, 204, 486, 218], "score": 1.0, "content": " with either of the exeptional friendship graphs in 3) of the above", "type": "text"}], "index": 6}, {"bbox": [126, 219, 232, 231], "spans": [{"bbox": [126, 219, 232, 231], "score": 1.0, "content": "theorem is reducible.", "type": "text"}], "index": 7}], "index": 6}, {"type": "text", "bbox": [140, 246, 471, 259], "lines": [{"bbox": [141, 248, 470, 261], "spans": [{"bbox": [141, 248, 470, 261], "score": 1.0, "content": "5. Representations whose friendship graph is a chain", "type": "text"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [124, 265, 486, 293], "lines": [{"bbox": [125, 268, 486, 282], "spans": [{"bbox": [125, 268, 486, 282], "score": 1.0, "content": "Definition 5.1. The standard representation is the representa-", "type": "text"}], "index": 9}, {"bbox": [125, 282, 150, 297], "spans": [{"bbox": [125, 282, 150, 297], "score": 1.0, "content": "tion", "type": "text"}], "index": 10}], "index": 9.5}, {"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 314, 180, 329], "lines": [{"bbox": [126, 314, 179, 331], "spans": [{"bbox": [126, 314, 179, 331], "score": 1.0, "content": "defined by", "type": "text"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 442, 434, 458], "lines": [{"bbox": [125, 445, 434, 459], "spans": [{"bbox": [125, 445, 144, 459], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 447, 235, 458], "score": 0.9, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [236, 445, 275, 459], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 446, 286, 457], "score": 0.87, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [286, 445, 321, 459], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [322, 447, 350, 457], "score": 0.9, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [350, 445, 434, 459], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [125, 472, 487, 515], "lines": [{"bbox": [126, 474, 486, 489], "spans": [{"bbox": [126, 474, 228, 489], "score": 1.0, "content": "Theorem 5.1. Let", "type": "text"}, {"bbox": [229, 475, 319, 488], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [320, 474, 486, 489], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 15}, {"bbox": [126, 489, 488, 504], "spans": [{"bbox": [126, 489, 159, 504], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 489, 191, 501], "score": 0.87, "content": "n\\geq4", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [191, 489, 267, 504], "score": 1.0, "content": ". Suppose that ", "type": "text"}, {"bbox": [267, 489, 346, 502], "score": 0.91, "content": "\\rho(\\sigma_{1})=1+A_{1}", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [347, 489, 386, 504], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [387, 490, 459, 503], "score": 0.73, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [459, 489, 488, 504], "score": 1.0, "content": ", and", "type": "text"}], "index": 16}, {"bbox": [126, 503, 365, 516], "spans": [{"bbox": [126, 503, 299, 516], "score": 1.0, "content": "the associated friendship graph of ", "type": "text"}, {"bbox": [300, 505, 307, 516], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [307, 503, 365, 516], "score": 1.0, "content": " is a chain.", "type": "text"}], "index": 17}], "index": 16}, {"type": "text", "bbox": [126, 516, 485, 543], "lines": [{"bbox": [139, 516, 485, 532], "spans": [{"bbox": [139, 516, 168, 532], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 519, 176, 530], "score": 0.68, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [176, 516, 342, 532], "score": 1.0, "content": " is equivalent to a specialization ", "type": "text"}, {"bbox": [342, 518, 370, 531], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [370, 516, 485, 532], "score": 1.0, "content": " of the standard repre-", "type": "text"}], "index": 18}, {"bbox": [127, 532, 264, 544], "spans": [{"bbox": [127, 532, 225, 544], "score": 1.0, "content": "sentation for some ", "type": "text"}, {"bbox": [225, 532, 260, 542], "score": 0.88, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [261, 532, 264, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20}, {"type": "text", "bbox": [123, 635, 487, 664], "lines": [{"bbox": [138, 637, 486, 651], "spans": [{"bbox": [138, 637, 486, 651], "score": 1.0, "content": "Before proving the theorem, we will need the following technical", "type": "text"}], "index": 21}, {"bbox": [125, 651, 164, 665], "spans": [{"bbox": [125, 651, 164, 665], "score": 1.0, "content": "lemma:", "type": "text"}], "index": 22}], "index": 21.5}, {"type": "text", "bbox": [124, 671, 487, 700], "lines": [{"bbox": [125, 674, 487, 687], "spans": [{"bbox": [125, 674, 396, 687], "score": 1.0, "content": "Lemma 5.2. Let A be a friend and a neighbor of B, ", "type": "text"}, {"bbox": [396, 676, 406, 684], "score": 0.41, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [406, 674, 487, 687], "score": 1.0, "content": " be a friend and", "type": "text"}], "index": 23}, {"bbox": [126, 687, 423, 702], "spans": [{"bbox": [126, 687, 196, 702], "score": 1.0, "content": "a neighbor of ", "type": "text"}, {"bbox": [197, 690, 206, 699], "score": 0.81, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [207, 687, 403, 702], "score": 1.0, "content": " and suppose that A is not a friend of ", "type": "text"}, {"bbox": [403, 690, 414, 699], "score": 0.44, "content": "C", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [414, 687, 423, 702], "score": 1.0, "content": " :", "type": "text"}], "index": 24}], "index": 23.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [{"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20}], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}], "discarded_blocks": [{"type": "discarded", "bbox": [278, 90, 329, 101], "lines": [{"bbox": [278, 92, 329, 102], "spans": [{"bbox": [278, 92, 329, 102], "score": 1.0, "content": "I.SYSOEVA", "type": "text"}]}]}, {"type": "discarded", "bbox": [126, 90, 136, 100], "lines": [{"bbox": [125, 92, 137, 103], "spans": [{"bbox": [125, 92, 137, 103], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [123, 110, 487, 138], "lines": [{"bbox": [126, 113, 486, 127], "spans": [{"bbox": [126, 113, 486, 127], "score": 1.0, "content": "4.3, it has no other edges. If it does not contain a chain graph, we", "type": "text"}], "index": 0}, {"bbox": [126, 127, 269, 140], "spans": [{"bbox": [126, 127, 269, 140], "score": 1.0, "content": "obtain the exceptional case.", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [126, 113, 486, 140]}, {"type": "text", "bbox": [124, 139, 486, 181], "lines": [{"bbox": [137, 140, 484, 155], "spans": [{"bbox": [137, 140, 164, 155], "score": 1.0, "content": "3) If ", "type": "text"}, {"bbox": [164, 143, 194, 151], "score": 0.91, "content": "n=4", "type": "inline_equation", "height": 8, "width": 30}, {"bbox": [195, 140, 484, 155], "score": 1.0, "content": ", then by theorem 3.8 the friendship graph is not totally", "type": "text"}], "index": 2}, {"bbox": [125, 154, 487, 169], "spans": [{"bbox": [125, 154, 398, 169], "score": 1.0, "content": "disconnected. Hence, we have only three possible ", "type": "text"}, {"bbox": [398, 156, 421, 167], "score": 0.46, "content": "\\mathbb{Z}_{4}-", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [421, 154, 487, 169], "score": 1.0, "content": "graphs on 4", "type": "text"}], "index": 3}, {"bbox": [126, 170, 168, 182], "spans": [{"bbox": [126, 170, 168, 182], "score": 1.0, "content": "vertices.", "type": "text"}], "index": 4}], "index": 3, "bbox_fs": [125, 140, 487, 182]}, {"type": "text", "bbox": [124, 187, 487, 230], "lines": [{"bbox": [124, 190, 486, 206], "spans": [{"bbox": [124, 190, 486, 206], "score": 1.0, "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "type": "text"}], "index": 5}, {"bbox": [125, 204, 486, 218], "spans": [{"bbox": [125, 204, 138, 218], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 206, 153, 217], "score": 0.92, "content": "B_{4}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [153, 204, 486, 218], "score": 1.0, "content": " with either of the exeptional friendship graphs in 3) of the above", "type": "text"}], "index": 6}, {"bbox": [126, 219, 232, 231], "spans": [{"bbox": [126, 219, 232, 231], "score": 1.0, "content": "theorem is reducible.", "type": "text"}], "index": 7}], "index": 6, "bbox_fs": [124, 190, 486, 231]}, {"type": "text", "bbox": [140, 246, 471, 259], "lines": [{"bbox": [141, 248, 470, 261], "spans": [{"bbox": [141, 248, 470, 261], "score": 1.0, "content": "5. Representations whose friendship graph is a chain", "type": "text"}], "index": 8}], "index": 8, "bbox_fs": [141, 248, 470, 261]}, {"type": "text", "bbox": [124, 265, 486, 293], "lines": [{"bbox": [125, 268, 486, 282], "spans": [{"bbox": [125, 268, 486, 282], "score": 1.0, "content": "Definition 5.1. The standard representation is the representa-", "type": "text"}], "index": 9}, {"bbox": [125, 282, 150, 297], "spans": [{"bbox": [125, 282, 150, 297], "score": 1.0, "content": "tion", "type": "text"}], "index": 10}], "index": 9.5, "bbox_fs": [125, 268, 486, 297]}, {"type": "interline_equation", "bbox": [249, 296, 362, 312], "lines": [{"bbox": [249, 296, 362, 312], "spans": [{"bbox": [249, 296, 362, 312], "score": 0.92, "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [125, 314, 180, 329], "lines": [{"bbox": [126, 314, 179, 331], "spans": [{"bbox": [126, 314, 179, 331], "score": 1.0, "content": "defined by", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [126, 314, 179, 331]}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "lines": [{"bbox": [219, 369, 391, 428], "spans": [{"bbox": [219, 369, 391, 428], "score": 0.93, "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 442, 434, 458], "lines": [{"bbox": [125, 445, 434, 459], "spans": [{"bbox": [125, 445, 144, 459], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 447, 235, 458], "score": 0.9, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 11, "width": 91}, {"bbox": [236, 445, 275, 459], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [275, 446, 286, 457], "score": 0.87, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [286, 445, 321, 459], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [322, 447, 350, 457], "score": 0.9, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [350, 445, 434, 459], "score": 1.0, "content": " identity matrix.", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [125, 445, 434, 459]}, {"type": "text", "bbox": [125, 472, 487, 515], "lines": [{"bbox": [126, 474, 486, 489], "spans": [{"bbox": [126, 474, 228, 489], "score": 1.0, "content": "Theorem 5.1. Let", "type": "text"}, {"bbox": [229, 475, 319, 488], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [320, 474, 486, 489], "score": 1.0, "content": " be an irreducible representation,", "type": "text"}], "index": 15}, {"bbox": [126, 489, 488, 504], "spans": [{"bbox": [126, 489, 159, 504], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [159, 489, 191, 501], "score": 0.87, "content": "n\\geq4", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [191, 489, 267, 504], "score": 1.0, "content": ". Suppose that ", "type": "text"}, {"bbox": [267, 489, 346, 502], "score": 0.91, "content": "\\rho(\\sigma_{1})=1+A_{1}", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [347, 489, 386, 504], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [387, 490, 459, 503], "score": 0.73, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 72}, {"bbox": [459, 489, 488, 504], "score": 1.0, "content": ", and", "type": "text"}], "index": 16}, {"bbox": [126, 503, 365, 516], "spans": [{"bbox": [126, 503, 299, 516], "score": 1.0, "content": "the associated friendship graph of ", "type": "text"}, {"bbox": [300, 505, 307, 516], "score": 0.7, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [307, 503, 365, 516], "score": 1.0, "content": " is a chain.", "type": "text"}], "index": 17}], "index": 16, "bbox_fs": [126, 474, 488, 516]}, {"type": "text", "bbox": [126, 516, 485, 543], "lines": [{"bbox": [139, 516, 485, 532], "spans": [{"bbox": [139, 516, 168, 532], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 519, 176, 530], "score": 0.68, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [176, 516, 342, 532], "score": 1.0, "content": " is equivalent to a specialization ", "type": "text"}, {"bbox": [342, 518, 370, 531], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [370, 516, 485, 532], "score": 1.0, "content": " of the standard repre-", "type": "text"}], "index": 18}, {"bbox": [127, 532, 264, 544], "spans": [{"bbox": [127, 532, 225, 544], "score": 1.0, "content": "sentation for some ", "type": "text"}, {"bbox": [225, 532, 260, 542], "score": 0.88, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [261, 532, 264, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18.5, "bbox_fs": [127, 516, 485, 544]}, {"type": "image", "bbox": [124, 567, 404, 607], "blocks": [{"type": "image_body", "bbox": [124, 567, 404, 607], "group_id": 0, "lines": [{"bbox": [124, 567, 404, 607], "spans": [{"bbox": [124, 567, 404, 607], "score": 0.614, "type": "image", "image_path": "46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg"}]}], "index": 20, "virtual_lines": [{"bbox": [124, 567, 404, 607], "spans": [], "index": 20}]}], "index": 20}, {"type": "text", "bbox": [123, 635, 487, 664], "lines": [{"bbox": [138, 637, 486, 651], "spans": [{"bbox": [138, 637, 486, 651], "score": 1.0, "content": "Before proving the theorem, we will need the following technical", "type": "text"}], "index": 21}, {"bbox": [125, 651, 164, 665], "spans": [{"bbox": [125, 651, 164, 665], "score": 1.0, "content": "lemma:", "type": "text"}], "index": 22}], "index": 21.5, "bbox_fs": [125, 637, 486, 665]}, {"type": "text", "bbox": [124, 671, 487, 700], "lines": [{"bbox": [125, 674, 487, 687], "spans": [{"bbox": [125, 674, 396, 687], "score": 1.0, "content": "Lemma 5.2. Let A be a friend and a neighbor of B, ", "type": "text"}, {"bbox": [396, 676, 406, 684], "score": 0.41, "content": "B", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [406, 674, 487, 687], "score": 1.0, "content": " be a friend and", "type": "text"}], "index": 23}, {"bbox": [126, 687, 423, 702], "spans": [{"bbox": [126, 687, 196, 702], "score": 1.0, "content": "a neighbor of ", "type": "text"}, {"bbox": [197, 690, 206, 699], "score": 0.81, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [207, 687, 403, 702], "score": 1.0, "content": " and suppose that A is not a friend of ", "type": "text"}, {"bbox": [403, 690, 414, 699], "score": 0.44, "content": "C", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [414, 687, 423, 702], "score": 1.0, "content": " :", "type": "text"}], "index": 24}], "index": 23.5, "bbox_fs": [125, 674, 487, 702]}]}
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[{"type": "text", "bbox": [123, 110, 487, 138], "content": "4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case.", "index": 0}, {"type": "text", "bbox": [124, 139, 486, 181], "content": "3) If , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible graphs on 4 vertices.", "index": 1}, {"type": "text", "bbox": [124, 187, 487, 230], "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation of with either of the exeptional friendship graphs in 3) of the above theorem is reducible.", "index": 2}, {"type": "text", "bbox": [140, 246, 471, 259], "content": "5. Representations whose friendship graph is a chain", "index": 3}, {"type": "text", "bbox": [124, 265, 486, 293], "content": "Definition 5.1. The standard representation is the representa- tion", "index": 4}, {"type": "interline_equation", "bbox": [249, 296, 362, 312], "content": "", "index": 5}, {"type": "text", "bbox": [125, 314, 180, 329], "content": "defined by", "index": 6}, {"type": "interline_equation", "bbox": [219, 369, 391, 428], "content": "", "index": 7}, {"type": "text", "bbox": [124, 442, 434, 458], "content": "for , where is the identity matrix.", "index": 8}, {"type": "text", "bbox": [125, 472, 487, 515], "content": "Theorem 5.1. Let be an irreducible representation, where . Suppose that , where , and the associated friendship graph of is a chain.", "index": 9}, {"type": "text", "bbox": [126, 516, 485, 543], "content": "Then is equivalent to a specialization of the standard repre- sentation for some .", "index": 10}, {"type": "image", "bbox": [124, 567, 404, 607], "content": "", "index": 11}, {"type": "text", "bbox": [123, 635, 487, 664], "content": "Before proving the theorem, we will need the following technical lemma:", "index": 12}, {"type": "text", "bbox": [124, 671, 487, 700], "content": "Lemma 5.2. Let A be a friend and a neighbor of B, be a friend and a neighbor of and suppose that A is not a friend of :", "index": 13}]
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[{"bbox": [126, 113, 486, 127], "content": "4.3, it has no other edges. If it does not contain a chain graph, we", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 269, 140], "content": "obtain the exceptional case.", "parent_index": 0, "line_index": 1}, {"bbox": [137, 140, 484, 155], "content": "3) If , then by theorem 3.8 the friendship graph is not totally", "parent_index": 1, "line_index": 0}, {"bbox": [125, 154, 487, 169], "content": "disconnected. Hence, we have only three possible graphs on 4", "parent_index": 1, "line_index": 1}, {"bbox": [126, 170, 168, 182], "content": "vertices.", "parent_index": 1, "line_index": 2}, {"bbox": [124, 190, 486, 206], "content": "Remark 4.5. It is proven in [5], Chapter 6, that any representation", "parent_index": 2, "line_index": 0}, {"bbox": [125, 204, 486, 218], "content": "of with either of the exeptional friendship graphs in 3) of the above", "parent_index": 2, "line_index": 1}, {"bbox": [126, 219, 232, 231], "content": "theorem is reducible.", "parent_index": 2, "line_index": 2}, {"bbox": [141, 248, 470, 261], "content": "5. Representations whose friendship graph is a chain", "parent_index": 3, "line_index": 0}, {"bbox": [125, 268, 486, 282], "content": "Definition 5.1. The standard representation is the representa-", "parent_index": 4, "line_index": 0}, {"bbox": [125, 282, 150, 297], "content": "tion", "parent_index": 4, "line_index": 1}, {"bbox": [126, 314, 179, 331], "content": "defined by", "parent_index": 6, "line_index": 0}, {"bbox": [125, 445, 434, 459], "content": "for , where is the identity matrix.", "parent_index": 8, "line_index": 0}, {"bbox": [126, 474, 486, 489], "content": "Theorem 5.1. Let be an irreducible representation,", "parent_index": 9, "line_index": 0}, {"bbox": [126, 489, 488, 504], "content": "where . Suppose that , where , and", "parent_index": 9, "line_index": 1}, {"bbox": [126, 503, 365, 516], "content": "the associated friendship graph of is a chain.", "parent_index": 9, "line_index": 2}, {"bbox": [139, 516, 485, 532], "content": "Then is equivalent to a specialization of the standard repre-", "parent_index": 10, "line_index": 0}, {"bbox": [127, 532, 264, 544], "content": "sentation for some .", "parent_index": 10, "line_index": 1}, {"bbox": [138, 637, 486, 651], "content": "Before proving the theorem, we will need the following technical", "parent_index": 12, "line_index": 0}, {"bbox": [125, 651, 164, 665], "content": "lemma:", "parent_index": 12, "line_index": 1}, {"bbox": [125, 674, 487, 687], "content": "Lemma 5.2. Let A be a friend and a neighbor of B, be a friend and", "parent_index": 13, "line_index": 0}, {"bbox": [126, 687, 423, 702], "content": "a neighbor of and suppose that A is not a friend of :", "parent_index": 13, "line_index": 1}]
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[{"bbox": [124, 567, 404, 607], "content": "", "parent_index": 11, "subtype": "body"}]
|
[{"bbox": [164, 143, 194, 151], "content": "n=4", "parent_index": 1, "subtype": "inline"}, {"bbox": [398, 156, 421, 167], "content": "\\mathbb{Z}_{4}-", "parent_index": 1, "subtype": "inline"}, {"bbox": [139, 206, 153, 217], "content": "B_{4}", "parent_index": 2, "subtype": "inline"}, {"bbox": [249, 296, 362, 312], "content": "\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]", "parent_index": 5, "subtype": "interline"}, {"bbox": [219, 369, 391, 428], "content": "\\tau_{n}(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&&&\\\\ &{0}&{t}&\\\\ &{1}&{0}&\\\\ &&&{I_{n-1-i}}\\end{array}\\right),", "parent_index": 7, "subtype": "interline"}, {"bbox": [144, 447, 235, 458], "content": "i=1,2,\\dots,n-1", "parent_index": 8, "subtype": "inline"}, {"bbox": [275, 446, 286, 457], "content": "I_{k}", "parent_index": 8, "subtype": "inline"}, {"bbox": [322, 447, 350, 457], "content": "k\\times k", "parent_index": 8, "subtype": "inline"}, {"bbox": [229, 475, 319, 488], "content": "\\rho:B_{n}\\to G L_{n}(\\mathbb{C})", "parent_index": 9, "subtype": "inline"}, {"bbox": [159, 489, 191, 501], "content": "n\\geq4", "parent_index": 9, "subtype": "inline"}, {"bbox": [267, 489, 346, 502], "content": "\\rho(\\sigma_{1})=1+A_{1}", "parent_index": 9, "subtype": "inline"}, {"bbox": [387, 490, 459, 503], "content": "r a n k(A_{1})=2", "parent_index": 9, "subtype": "inline"}, {"bbox": [300, 505, 307, 516], "content": "\\rho", "parent_index": 9, "subtype": "inline"}, {"bbox": [168, 519, 176, 530], "content": "\\rho", "parent_index": 10, "subtype": "inline"}, {"bbox": [342, 518, 370, 531], "content": "\\tau_{n}(u)", "parent_index": 10, "subtype": "inline"}, {"bbox": [225, 532, 260, 542], "content": "u\\in\\mathbb{C}^{*}", "parent_index": 10, "subtype": "inline"}, {"bbox": [396, 676, 406, 684], "content": "B", "parent_index": 13, "subtype": "inline"}, {"bbox": [197, 690, 206, 699], "content": "C", "parent_index": 13, "subtype": "inline"}, {"bbox": [403, 690, 414, 699], "content": "C", "parent_index": 13, "subtype": "inline"}]
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[]
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Let $a~\ne~0$ be such that s $\L)a n\{a\}\;=\;I m(A)\cap I m(B)$ , and let $b\,=$ $(1+B)a$ . Then:
1) $s p a n\{b\}=I m(C)\cap I m(B)$ .
$\mathcal{Q}_{g}$ ) $(1+B)b\in s p a n\{a\}$ and $(1+B)b\neq0$ .
3) The vectors $a$ and $b$ are linearly independent.
Proof. First of all, notice that the vector $b$ is non-zero, because $1+B$ is invertible and $a\ne0$ .
1) $b=(1+B)a\in I m(B)$ , because $a\in I m(B)$ .
$A$ and $C$ are not friends, that is $C A=0$ , so $C a=0$ . Let $a=B a_{1}$ . Then
$$
(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=
$$
$$
=(C+C^{2}+C B C)a_{1}\in I m(C);
$$
that is, $b\;\in\;I m(C)\cap I m(B)$ , and because $I m(C)\cap I m(B)$ is onedimensional and $b\neq0$ ,
$$
s p a n\{b\}=I m(C)\cap I m(B).
$$
2) Clearly, $(1+B)b\in I m(B)$ .
Note, that $A b=0$ , as $b\in I m(C)$ by the above, and $A C=0$ . Let $b=B a^{'}$ . Then
$$
(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\in I m(A).
$$
3) $a\in I m(A)$ , $b\in I m(C)$ by part 1), and $I m(A)\cap I m(C)=\{0\}$ by the hypothesis of the lemma.
Proof of Theorem 5.1 We include the redundant generator $\sigma_{0}$ , and indices are modulo $n$ . Consider $I m(A_{i})\cap I m(A_{i+1})$ , which is $\boldsymbol{0}$ , $1$ , or $2-$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then
$$
I m(A_{0})=I m(A_{1})=\cdots=I m(A_{n-1})
$$
would be a $2-$ dimensional invariant subspace, contradicting the irreducibility of $\rho$ . Hence, $I m(A_{i})\cap I m(A_{i+1})$ is one-dimensional.
Let $a_{0}$ be a basis vector for $I m(A_{0})\cap I m(A_{1})$ . Let
$$
a_{1}=(1+A_{1})a_{0},\;\;a_{2}=(1+A_{2})a_{1},\;\;.\;.\;.\;,\;\;a_{n-1}=(1+A_{n-1})a_{n-2}.
$$
|
<html><body>
<p data-bbox="124 166 487 195">Let $a~\ne~0$ be such that s $\L)a n\{a\}\;=\;I m(A)\cap I m(B)$ , and let $b\,=$ $(1+B)a$ . Then: </p>
<p data-bbox="136 195 385 237">1) $s p a n\{b\}=I m(C)\cap I m(B)$ .
$\mathcal{Q}_{g}$ ) $(1+B)b\in s p a n\{a\}$ and $(1+B)b\neq0$ .
3) The vectors $a$ and $b$ are linearly independent. </p>
<p data-bbox="126 245 486 273">Proof. First of all, notice that the vector $b$ is non-zero, because $1+B$ is invertible and $a\ne0$ . </p>
<p data-bbox="136 273 378 288">1) $b=(1+B)a\in I m(B)$ , because $a\in I m(B)$ . </p>
<p data-bbox="124 288 485 315">$A$ and $C$ are not friends, that is $C A=0$ , so $C a=0$ . Let $a=B a_{1}$ . Then </p>
<div class="equation" data-bbox="126 324 488 339">$$
(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=
$$</div>
<div class="equation" data-bbox="221 349 387 363">$$
=(C+C^{2}+C B C)a_{1}\in I m(C);
$$</div>
<p data-bbox="124 366 485 394">that is, $b\;\in\;I m(C)\cap I m(B)$ , and because $I m(C)\cap I m(B)$ is onedimensional and $b\neq0$ , </p>
<div class="equation" data-bbox="233 403 377 417">$$
s p a n\{b\}=I m(C)\cap I m(B).
$$</div>
<p data-bbox="136 423 292 437">2) Clearly, $(1+B)b\in I m(B)$ . </p>
<p data-bbox="123 438 487 466">Note, that $A b=0$ , as $b\in I m(C)$ by the above, and $A C=0$ . Let $b=B a^{'}$ . Then </p>
<div class="equation" data-bbox="126 473 495 489">$$
(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\in I m(A).
$$</div>
<p data-bbox="124 494 487 523">3) $a\in I m(A)$ , $b\in I m(C)$ by part 1), and $I m(A)\cap I m(C)=\{0\}$ by the hypothesis of the lemma. </p>
<p data-bbox="124 551 487 607">Proof of Theorem 5.1 We include the redundant generator $\sigma_{0}$ , and indices are modulo $n$ . Consider $I m(A_{i})\cap I m(A_{i+1})$ , which is $\boldsymbol{0}$ , $1$ , or $2-$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then </p>
<div class="equation" data-bbox="210 617 401 630">$$
I m(A_{0})=I m(A_{1})=\cdots=I m(A_{n-1})
$$</div>
<p data-bbox="124 636 487 664">would be a $2-$ dimensional invariant subspace, contradicting the irreducibility of $\rho$ . Hence, $I m(A_{i})\cap I m(A_{i+1})$ is one-dimensional. </p>
<p data-bbox="135 665 399 679">Let $a_{0}$ be a basis vector for $I m(A_{0})\cap I m(A_{1})$ . Let </p>
<div class="equation" data-bbox="141 688 470 702">$$
a_{1}=(1+A_{1})a_{0},\;\;a_{2}=(1+A_{2})a_{1},\;\;.\;.\;.\;,\;\;a_{n-1}=(1+A_{n-1})a_{n-2}.
$$</div>
</body></html>
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0003047v1
| 12 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Let $a~\\ne~0$ be such that s $\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)$ , and let $b\\,=$ $(1+B)a$ . Then: ", "page_idx": 12}, {"type": "text", "text": "1) $s p a n\\{b\\}=I m(C)\\cap I m(B)$ . \n$\\mathcal{Q}_{g}$ ) $(1+B)b\\in s p a n\\{a\\}$ and $(1+B)b\\neq0$ . \n3) The vectors $a$ and $b$ are linearly independent. ", "page_idx": 12}, {"type": "text", "text": "Proof. First of all, notice that the vector $b$ is non-zero, because $1+B$ is invertible and $a\\ne0$ . ", "page_idx": 12}, {"type": "text", "text": "1) $b=(1+B)a\\in I m(B)$ , because $a\\in I m(B)$ . ", "page_idx": 12}, {"type": "text", "text": "$A$ and $C$ are not friends, that is $C A=0$ , so $C a=0$ . Let $a=B a_{1}$ . Then ", "page_idx": 12}, {"type": "equation", "text": "$$\n(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=\n$$", "text_format": "latex", "page_idx": 12}, {"type": "equation", "text": "$$\n=(C+C^{2}+C B C)a_{1}\\in I m(C);\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "that is, $b\\;\\in\\;I m(C)\\cap I m(B)$ , and because $I m(C)\\cap I m(B)$ is onedimensional and $b\\neq0$ , ", "page_idx": 12}, {"type": "equation", "text": "$$\ns p a n\\{b\\}=I m(C)\\cap I m(B).\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "2) Clearly, $(1+B)b\\in I m(B)$ . ", "page_idx": 12}, {"type": "text", "text": "Note, that $A b=0$ , as $b\\in I m(C)$ by the above, and $A C=0$ . Let $b=B a^{'}$ . Then ", "page_idx": 12}, {"type": "equation", "text": "$$\n(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\\in I m(A).\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "3) $a\\in I m(A)$ , $b\\in I m(C)$ by part 1), and $I m(A)\\cap I m(C)=\\{0\\}$ by the hypothesis of the lemma. ", "page_idx": 12}, {"type": "text", "text": "Proof of Theorem 5.1 We include the redundant generator $\\sigma_{0}$ , and indices are modulo $n$ . Consider $I m(A_{i})\\cap I m(A_{i+1})$ , which is $\\boldsymbol{0}$ , $1$ , or $2-$ dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then ", "page_idx": 12}, {"type": "equation", "text": "$$\nI m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "would be a $2-$ dimensional invariant subspace, contradicting the irreducibility of $\\rho$ . Hence, $I m(A_{i})\\cap I m(A_{i+1})$ is one-dimensional. ", "page_idx": 12}, {"type": "text", "text": "Let $a_{0}$ be a basis vector for $I m(A_{0})\\cap I m(A_{1})$ . Let ", "page_idx": 12}, {"type": "equation", "text": "$$\na_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.\n$$", "text_format": "latex", "page_idx": 12}]
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First of all, notice that the vector ", "type": "text"}, {"bbox": [369, 248, 375, 258], "score": 0.8, "content": "b", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [376, 247, 486, 261], "score": 1.0, "content": " is non-zero, because", "type": "text"}], "index": 5}, {"bbox": [126, 262, 277, 274], "spans": [{"bbox": [126, 263, 156, 273], "score": 0.91, "content": "1+B", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 262, 245, 273], "score": 1.0, "content": " is invertible and ", "type": "text"}, {"bbox": [246, 263, 274, 274], "score": 0.89, "content": "a\\ne0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [274, 262, 277, 273], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [136, 273, 378, 288], "lines": [{"bbox": [138, 275, 377, 290], "spans": [{"bbox": [138, 275, 152, 290], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [152, 276, 268, 289], "score": 0.92, "content": "b=(1+B)a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [268, 275, 318, 290], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [318, 276, 374, 289], "score": 0.93, "content": "a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [374, 275, 377, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [124, 288, 485, 315], "lines": [{"bbox": [138, 288, 486, 303], "spans": [{"bbox": [138, 291, 147, 300], "score": 0.86, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [147, 288, 174, 303], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 291, 183, 300], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 288, 308, 303], "score": 1.0, "content": " are not friends, that is ", "type": "text"}, {"bbox": [308, 291, 349, 300], "score": 0.9, "content": "C A=0", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [349, 288, 370, 303], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 290, 410, 300], "score": 0.88, "content": "C a=0", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [410, 288, 437, 303], "score": 1.0, "content": ". 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Consider ", "type": "text"}, {"bbox": [311, 568, 411, 581], "score": 0.94, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 100}, {"bbox": [411, 567, 462, 581], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [462, 570, 469, 578], "score": 0.45, "content": "\\boldsymbol{0}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [469, 567, 475, 581], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [476, 570, 482, 578], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [482, 567, 485, 581], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [126, 582, 486, 595], "spans": [{"bbox": [126, 582, 141, 595], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [141, 583, 156, 593], "score": 0.88, "content": "2-", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [157, 582, 486, 595], "score": 1.0, "content": "dimensional. It is nonzero, because of the hypothesis that the", "type": "text"}], "index": 23}, {"bbox": [126, 595, 435, 608], "spans": [{"bbox": [126, 595, 435, 608], "score": 1.0, "content": "friendship graph is a chain. 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First of all, notice that the vector ", "type": "text"}, {"bbox": [369, 248, 375, 258], "score": 0.8, "content": "b", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [376, 247, 486, 261], "score": 1.0, "content": " is non-zero, because", "type": "text"}], "index": 5}, {"bbox": [126, 262, 277, 274], "spans": [{"bbox": [126, 263, 156, 273], "score": 0.91, "content": "1+B", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [156, 262, 245, 273], "score": 1.0, "content": " is invertible and ", "type": "text"}, {"bbox": [246, 263, 274, 274], "score": 0.89, "content": "a\\ne0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [274, 262, 277, 273], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [126, 247, 486, 274]}, {"type": "text", "bbox": [136, 273, 378, 288], "lines": [{"bbox": [138, 275, 377, 290], "spans": [{"bbox": [138, 275, 152, 290], "score": 1.0, "content": "1) ", "type": "text"}, {"bbox": [152, 276, 268, 289], "score": 0.92, "content": "b=(1+B)a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [268, 275, 318, 290], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [318, 276, 374, 289], "score": 0.93, "content": "a\\in I m(B)", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [374, 275, 377, 290], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [138, 275, 377, 290]}, {"type": "text", "bbox": [124, 288, 485, 315], "lines": [{"bbox": [138, 288, 486, 303], "spans": [{"bbox": [138, 291, 147, 300], "score": 0.86, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [147, 288, 174, 303], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [174, 291, 183, 300], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 288, 308, 303], "score": 1.0, "content": " are not friends, that is ", "type": "text"}, {"bbox": [308, 291, 349, 300], "score": 0.9, "content": "C A=0", "type": "inline_equation", "height": 9, "width": 41}, {"bbox": [349, 288, 370, 303], "score": 1.0, "content": ", so ", "type": "text"}, {"bbox": [371, 290, 410, 300], "score": 0.88, "content": "C a=0", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [410, 288, 437, 303], "score": 1.0, "content": ". 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Consider ", "type": "text"}, {"bbox": [311, 568, 411, 581], "score": 0.94, "content": "I m(A_{i})\\cap I m(A_{i+1})", "type": "inline_equation", "height": 13, "width": 100}, {"bbox": [411, 567, 462, 581], "score": 1.0, "content": ", which is ", "type": "text"}, {"bbox": [462, 570, 469, 578], "score": 0.45, "content": "\\boldsymbol{0}", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [469, 567, 475, 581], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [476, 570, 482, 578], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [482, 567, 485, 581], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [126, 582, 486, 595], "spans": [{"bbox": [126, 582, 141, 595], "score": 1.0, "content": "or ", "type": "text"}, {"bbox": [141, 583, 156, 593], "score": 0.88, "content": "2-", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [157, 582, 486, 595], "score": 1.0, "content": "dimensional. It is nonzero, because of the hypothesis that the", "type": "text"}], "index": 23}, {"bbox": [126, 595, 435, 608], "spans": [{"bbox": [126, 595, 435, 608], "score": 1.0, "content": "friendship graph is a chain. It is not 2-dimensional, for then", "type": "text"}], "index": 24}], "index": 22.5, "bbox_fs": [125, 550, 487, 608]}, {"type": "interline_equation", "bbox": [210, 617, 401, 630], "lines": [{"bbox": [210, 617, 401, 630], "spans": [{"bbox": [210, 617, 401, 630], "score": 0.89, "content": "I m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [124, 636, 487, 664], "lines": [{"bbox": [126, 639, 484, 651], "spans": [{"bbox": [126, 639, 187, 651], "score": 1.0, "content": "would be a ", "type": "text"}, {"bbox": [188, 641, 203, 650], "score": 0.89, "content": "2-", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [203, 639, 484, 651], "score": 1.0, "content": "dimensional invariant subspace, contradicting the irre-", "type": "text"}], "index": 26}, {"bbox": [127, 653, 444, 666], "spans": [{"bbox": [127, 653, 191, 666], "score": 1.0, "content": "ducibility of ", "type": "text"}, {"bbox": [191, 657, 198, 665], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [198, 653, 242, 666], "score": 1.0, "content": ". 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Let", "type": "text"}], "index": 28}], "index": 28, "bbox_fs": [137, 666, 397, 680]}, {"type": "interline_equation", "bbox": [141, 688, 470, 702], "lines": [{"bbox": [141, 688, 470, 702], "spans": [{"bbox": [141, 688, 470, 702], "score": 0.89, "content": "a_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.", "type": "interline_equation"}], "index": 29}], "index": 29}]}
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[{"type": "text", "bbox": [124, 166, 487, 195], "content": "Let be such that s , and let . Then:", "index": 0}, {"type": "list", "bbox": [136, 195, 385, 237], "content": "", "index": 1}, {"type": "text", "bbox": [126, 245, 486, 273], "content": "Proof. First of all, notice that the vector is non-zero, because is invertible and .", "index": 2}, {"type": "text", "bbox": [136, 273, 378, 288], "content": "1) , because .", "index": 3}, {"type": "text", "bbox": [124, 288, 485, 315], "content": "and are not friends, that is , so . Let . Then", "index": 4}, {"type": "interline_equation", "bbox": [126, 324, 488, 339], "content": "", "index": 5}, {"type": "interline_equation", "bbox": [221, 349, 387, 363], "content": "", "index": 6}, {"type": "text", "bbox": [124, 366, 485, 394], "content": "that is, , and because is one- dimensional and ,", "index": 7}, {"type": "interline_equation", "bbox": [233, 403, 377, 417], "content": "", "index": 8}, {"type": "text", "bbox": [136, 423, 292, 437], "content": "2) Clearly, .", "index": 9}, {"type": "text", "bbox": [123, 438, 487, 466], "content": "Note, that , as by the above, and . Let . Then", "index": 10}, {"type": "interline_equation", "bbox": [126, 473, 495, 489], "content": "", "index": 11}, {"type": "text", "bbox": [124, 494, 487, 523], "content": "3) , by part 1), and by the hypothesis of the lemma.", "index": 12}, {"type": "text", "bbox": [124, 551, 487, 607], "content": "Proof of Theorem 5.1 We include the redundant generator , and indices are modulo . Consider , which is , , or dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then", "index": 13}, {"type": "interline_equation", "bbox": [210, 617, 401, 630], "content": "", "index": 14}, {"type": "text", "bbox": [124, 636, 487, 664], "content": "would be a dimensional invariant subspace, contradicting the irre- ducibility of . Hence, is one-dimensional.", "index": 15}, {"type": "text", "bbox": [135, 665, 399, 679], "content": "Let be a basis vector for . Let", "index": 16}, {"type": "interline_equation", "bbox": [141, 688, 470, 702], "content": "", "index": 17}]
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[{"bbox": [137, 168, 486, 183], "content": "Let be such that s , and let", "parent_index": 0, "line_index": 0}, {"bbox": [126, 182, 212, 197], "content": ". Then:", "parent_index": 0, "line_index": 1}, {"bbox": [139, 198, 297, 211], "content": "1) .", "parent_index": 1, "line_index": 0}, {"bbox": [138, 211, 349, 225], "content": ") and .", "parent_index": 1, "line_index": 1}, {"bbox": [139, 225, 383, 237], "content": "3) The vectors and are linearly independent.", "parent_index": 1, "line_index": 2}, {"bbox": [137, 247, 486, 261], "content": "Proof. First of all, notice that the vector is non-zero, because", "parent_index": 2, "line_index": 0}, {"bbox": [126, 262, 277, 274], "content": "is invertible and .", "parent_index": 2, "line_index": 1}, {"bbox": [138, 275, 377, 290], "content": "1) , because .", "parent_index": 3, "line_index": 0}, {"bbox": [138, 288, 486, 303], "content": "and are not friends, that is , so . Let .", "parent_index": 4, "line_index": 0}, {"bbox": [124, 302, 155, 317], "content": "Then", "parent_index": 4, "line_index": 1}, {"bbox": [126, 367, 485, 383], "content": "that is, , and because is one-", "parent_index": 7, "line_index": 0}, {"bbox": [126, 382, 244, 396], "content": "dimensional and ,", "parent_index": 7, "line_index": 1}, {"bbox": [137, 424, 292, 439], "content": "2) Clearly, .", "parent_index": 9, "line_index": 0}, {"bbox": [136, 437, 487, 454], "content": "Note, that , as by the above, and . Let", "parent_index": 10, "line_index": 0}, {"bbox": [126, 452, 201, 467], "content": ". Then", "parent_index": 10, "line_index": 1}, {"bbox": [137, 496, 484, 512], "content": "3) , by part 1), and by", "parent_index": 12, "line_index": 0}, {"bbox": [126, 511, 275, 524], "content": "the hypothesis of the lemma.", "parent_index": 12, "line_index": 1}, {"bbox": [135, 550, 487, 570], "content": "Proof of Theorem 5.1 We include the redundant generator ,", "parent_index": 13, "line_index": 0}, {"bbox": [125, 567, 485, 581], "content": "and indices are modulo . Consider , which is , ,", "parent_index": 13, "line_index": 1}, {"bbox": [126, 582, 486, 595], "content": "or dimensional. It is nonzero, because of the hypothesis that the", "parent_index": 13, "line_index": 2}, {"bbox": [126, 595, 435, 608], "content": "friendship graph is a chain. It is not 2-dimensional, for then", "parent_index": 13, "line_index": 3}, {"bbox": [126, 639, 484, 651], "content": "would be a dimensional invariant subspace, contradicting the irre-", "parent_index": 15, "line_index": 0}, {"bbox": [127, 653, 444, 666], "content": "ducibility of . Hence, is one-dimensional.", "parent_index": 15, "line_index": 1}, {"bbox": [137, 666, 397, 680], "content": "Let be a basis vector for . Let", "parent_index": 16, "line_index": 0}]
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[]
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[{"bbox": [159, 170, 192, 182], "content": "a~\\ne~0", "parent_index": 0, "subtype": "inline"}, {"bbox": [275, 169, 415, 183], "content": "\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)", "parent_index": 0, "subtype": "inline"}, {"bbox": [465, 169, 486, 182], "content": "b\\,=", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 184, 171, 197], "content": "(1+B)a", "parent_index": 0, "subtype": "inline"}, {"bbox": [153, 198, 294, 210], "content": "s p a n\\{b\\}=I m(C)\\cap I m(B)", "parent_index": 1, "subtype": "inline"}, {"bbox": [138, 212, 145, 223], "content": "\\mathcal{Q}_{g}", "parent_index": 1, "subtype": "inline"}, {"bbox": [153, 212, 254, 225], "content": "(1+B)b\\in s p a n\\{a\\}", "parent_index": 1, "subtype": "inline"}, {"bbox": [281, 211, 347, 225], "content": "(1+B)b\\neq0", "parent_index": 1, "subtype": "inline"}, {"bbox": [216, 228, 223, 236], "content": "a", "parent_index": 1, "subtype": "inline"}, {"bbox": [249, 226, 255, 235], "content": "b", "parent_index": 1, "subtype": "inline"}, {"bbox": [369, 248, 375, 258], "content": "b", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 263, 156, 273], "content": "1+B", "parent_index": 2, "subtype": "inline"}, {"bbox": [246, 263, 274, 274], "content": "a\\ne0", "parent_index": 2, "subtype": "inline"}, {"bbox": [152, 276, 268, 289], "content": "b=(1+B)a\\in I m(B)", "parent_index": 3, "subtype": "inline"}, {"bbox": [318, 276, 374, 289], "content": "a\\in I m(B)", "parent_index": 3, "subtype": "inline"}, {"bbox": [138, 291, 147, 300], "content": "A", "parent_index": 4, "subtype": "inline"}, {"bbox": [174, 291, 183, 300], "content": "C", "parent_index": 4, "subtype": "inline"}, {"bbox": [308, 291, 349, 300], "content": "C A=0", "parent_index": 4, "subtype": "inline"}, {"bbox": [371, 290, 410, 300], "content": "C a=0", "parent_index": 4, "subtype": "inline"}, {"bbox": [438, 290, 482, 302], "content": "a=B a_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 324, 488, 339], "content": "(1+B)a=(1+B+B C)a=(1+B+B C)B a_{1}=(B+B^{2}+B C B)a_{1}=", "parent_index": 5, "subtype": "interline"}, {"bbox": [221, 349, 387, 363], "content": "=(C+C^{2}+C B C)a_{1}\\in I m(C);", "parent_index": 6, "subtype": "interline"}, {"bbox": [170, 370, 281, 382], "content": "b\\;\\in\\;I m(C)\\cap I m(B)", "parent_index": 7, "subtype": "inline"}, {"bbox": [358, 367, 444, 382], "content": "I m(C)\\cap I m(B)", "parent_index": 7, "subtype": "inline"}, {"bbox": [213, 384, 240, 395], "content": "b\\neq0", "parent_index": 7, "subtype": "inline"}, {"bbox": [233, 403, 377, 417], "content": "s p a n\\{b\\}=I m(C)\\cap I m(B).", "parent_index": 8, "subtype": "interline"}, {"bbox": [195, 426, 289, 439], "content": "(1+B)b\\in I m(B)", "parent_index": 9, "subtype": "inline"}, {"bbox": [196, 441, 235, 450], "content": "A b=0", "parent_index": 10, "subtype": "inline"}, {"bbox": [257, 439, 315, 453], "content": "b\\in I m(C)", "parent_index": 10, "subtype": "inline"}, {"bbox": [417, 439, 460, 450], "content": "A C=0", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 453, 165, 464], "content": "b=B a^{'}", "parent_index": 10, "subtype": "inline"}, {"bbox": [126, 473, 495, 489], "content": "(1+B)b=(1+B+B A)b=(1+B+B A)B a^{'}=(A+A^{2}+A B A)a^{'}\\in I m(A).", "parent_index": 11, "subtype": "interline"}, {"bbox": [152, 498, 207, 510], "content": "a\\in I m(A)", "parent_index": 12, "subtype": "inline"}, {"bbox": [214, 498, 268, 511], "content": "b\\in I m(C)", "parent_index": 12, "subtype": "inline"}, {"bbox": [353, 498, 469, 511], "content": "I m(A)\\cap I m(C)=\\{0\\}", "parent_index": 12, "subtype": "inline"}, {"bbox": [470, 558, 482, 565], "content": "\\sigma_{0}", "parent_index": 13, "subtype": "inline"}, {"bbox": [248, 572, 255, 578], "content": "n", "parent_index": 13, "subtype": "inline"}, {"bbox": [311, 568, 411, 581], "content": "I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 13, "subtype": "inline"}, {"bbox": [462, 570, 469, 578], "content": "\\boldsymbol{0}", "parent_index": 13, "subtype": "inline"}, {"bbox": [476, 570, 482, 578], "content": "1", "parent_index": 13, "subtype": "inline"}, {"bbox": [141, 583, 156, 593], "content": "2-", "parent_index": 13, "subtype": "inline"}, {"bbox": [210, 617, 401, 630], "content": "I m(A_{0})=I m(A_{1})=\\cdots=I m(A_{n-1})", "parent_index": 14, "subtype": "interline"}, {"bbox": [188, 641, 203, 650], "content": "2-", "parent_index": 15, "subtype": "inline"}, {"bbox": [191, 657, 198, 665], "content": "\\rho", "parent_index": 15, "subtype": "inline"}, {"bbox": [242, 653, 342, 666], "content": "I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 15, "subtype": "inline"}, {"bbox": [159, 671, 170, 679], "content": "a_{0}", "parent_index": 16, "subtype": "inline"}, {"bbox": [281, 667, 373, 680], "content": "I m(A_{0})\\cap I m(A_{1})", "parent_index": 16, "subtype": "inline"}, {"bbox": [141, 688, 470, 702], "content": "a_{1}=(1+A_{1})a_{0},\\;\\;a_{2}=(1+A_{2})a_{1},\\;\\;.\\;.\\;.\\;,\\;\\;a_{n-1}=(1+A_{n-1})a_{n-2}.", "parent_index": 17, "subtype": "interline"}]
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[]
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By induction and lemma 5.2, part 1), $a_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ , for $0\leq\,i\leq n-1$ . By lemma 5.2, part 3), $a_{i}$ and $a_{i+1}$ are linearly independent. Thus $\{a_{i},a_{i+1}\}$ is a basis for $I m(A_{i})$ .
Since
$$
s p a n\{a_{0},\ldots a_{n-1}\}=I m(A_{1})+\cdots+I m(A_{n-1})
$$
is invariant under $B_{n}$ and $\rho$ is an $n-$ dimensional irreducible representation, $\left\{a_{0},\ldots.a_{n-1}\right\}$ is a basis for $\mathbb{C}^{n}$ .
We now wish to determine the action of $\rho(\sigma_{1}),\;\;\rho(\sigma_{2}),\ldots,\rho(\sigma_{n-1})$ on this basis.
Consider $a_{i}\in I m(A_{i})\cap I m(A_{i+1})$ . If $j\neq i,\ \ i+1$ , then $A_{j}$ is not a neighbor of one of $A_{i}$ , $A_{i+1}$ (since $n\geq4$ ), say $A_{k}$ , and then $A_{k}A_{j}=$ $A_{j}A_{k}=0$ , so $A_{j}a_{i}=0$ , and
$$
\rho(\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.
$$
By our construction
$$
\rho(\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}
$$
for $0\leq i\leq n-2$ .
By lemma 5.2, part 2),
$$
\rho(\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},
$$
for $1\leq i\leq n-1$ , where $u_{i}\in\mathbb{C}^{*}$ .
By the above calculations the matrices of $\rho(\sigma_{1}),\dotsc,\rho(\sigma_{n-1})$ with respect to the basis $a_{0},\;\;a_{1},\ldots,a_{n-1}$ are
$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u_{i}}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$
for $i\;=\;1,2,\ldots,n\,-\,1$ , where $I_{k}$ is the $k\,\times\,k$ identity matrix, and $u_{1},\dotsc,u_{n-1}\in\mathbb{C}^{*}$ . Since $\sigma_{1},\ldots,\sigma_{n-1}$ are conjugate in $B_{n}$ , the $u_{i}$ are all equal, and we have the standard representation.
Now let us consider when the standard representation is irreducible.
Lemma 5.3. If $u=1$ then $\tau_{n}(u)$ is reducible.
Proof. If $u=1$ then the vector $v=(1,1,1,\ldots,1)^{T}$ is a fixed vector.
Lemma 5.4. If $u\ne1$ then $\tau_{n}(u)$ is irreducible.
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<html><body>
<p data-bbox="124 110 487 153">By induction and lemma 5.2, part 1), $a_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ , for $0\leq\,i\leq n-1$ . By lemma 5.2, part 3), $a_{i}$ and $a_{i+1}$ are linearly independent. Thus $\{a_{i},a_{i+1}\}$ is a basis for $I m(A_{i})$ . </p>
<p data-bbox="137 153 167 165">Since </p>
<div class="equation" data-bbox="186 174 425 187">$$
s p a n\{a_{0},\ldots a_{n-1}\}=I m(A_{1})+\cdots+I m(A_{n-1})
$$</div>
<p data-bbox="123 191 485 218">is invariant under $B_{n}$ and $\rho$ is an $n-$ dimensional irreducible representation, $\left\{a_{0},\ldots.a_{n-1}\right\}$ is a basis for $\mathbb{C}^{n}$ . </p>
<p data-bbox="124 219 486 245">We now wish to determine the action of $\rho(\sigma_{1}),\;\;\rho(\sigma_{2}),\ldots,\rho(\sigma_{n-1})$ on this basis. </p>
<p data-bbox="124 246 487 288">Consider $a_{i}\in I m(A_{i})\cap I m(A_{i+1})$ . If $j\neq i,\ \ i+1$ , then $A_{j}$ is not a neighbor of one of $A_{i}$ , $A_{i+1}$ (since $n\geq4$ ), say $A_{k}$ , and then $A_{k}A_{j}=$ $A_{j}A_{k}=0$ , so $A_{j}a_{i}=0$ , and </p>
<div class="equation" data-bbox="239 295 371 309">$$
\rho(\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.
$$</div>
<p data-bbox="137 312 242 325">By our construction </p>
<div class="equation" data-bbox="225 333 385 346">$$
\rho(\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}
$$</div>
<p data-bbox="124 349 217 363">for $0\leq i\leq n-2$ . </p>
<p data-bbox="137 364 255 377">By lemma 5.2, part 2), </p>
<div class="equation" data-bbox="229 384 380 398">$$
\rho(\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},
$$</div>
<p data-bbox="124 401 297 415">for $1\leq i\leq n-1$ , where $u_{i}\in\mathbb{C}^{*}$ . </p>
<p data-bbox="124 416 486 444">By the above calculations the matrices of $\rho(\sigma_{1}),\dotsc,\rho(\sigma_{n-1})$ with respect to the basis $a_{0},\;\;a_{1},\ldots,a_{n-1}$ are </p>
<div class="equation" data-bbox="219 483 390 540">$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u_{i}}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$</div>
<p data-bbox="123 553 487 596">for $i\;=\;1,2,\ldots,n\,-\,1$ , where $I_{k}$ is the $k\,\times\,k$ identity matrix, and $u_{1},\dotsc,u_{n-1}\in\mathbb{C}^{*}$ . Since $\sigma_{1},\ldots,\sigma_{n-1}$ are conjugate in $B_{n}$ , the $u_{i}$ are all equal, and we have the standard representation. </p>
<p data-bbox="135 609 485 624">Now let us consider when the standard representation is irreducible. </p>
<p data-bbox="124 630 362 645">Lemma 5.3. If $u=1$ then $\tau_{n}(u)$ is reducible. </p>
<p data-bbox="136 650 486 666">Proof. If $u=1$ then the vector $v=(1,1,1,\ldots,1)^{T}$ is a fixed vector. </p>
<p data-bbox="124 685 372 701">Lemma 5.4. If $u\ne1$ then $\tau_{n}(u)$ is irreducible. </p>
</body></html>
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0003047v1
| 13 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "By induction and lemma 5.2, part 1), $a_{i}$ is a basis vector for $I m(A_{i})\\cap$ $I m(A_{i+1})$ , for $0\\leq\\,i\\leq n-1$ . By lemma 5.2, part 3), $a_{i}$ and $a_{i+1}$ are linearly independent. Thus $\\{a_{i},a_{i+1}\\}$ is a basis for $I m(A_{i})$ . ", "page_idx": 13}, {"type": "text", "text": "Since ", "page_idx": 13}, {"type": "equation", "text": "$$\ns p a n\\{a_{0},\\ldots a_{n-1}\\}=I m(A_{1})+\\cdots+I m(A_{n-1})\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "is invariant under $B_{n}$ and $\\rho$ is an $n-$ dimensional irreducible representation, $\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}$ is a basis for $\\mathbb{C}^{n}$ . ", "page_idx": 13}, {"type": "text", "text": "We now wish to determine the action of $\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})$ on this basis. ", "page_idx": 13}, {"type": "text", "text": "Consider $a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})$ . If $j\\neq i,\\ \\ i+1$ , then $A_{j}$ is not a neighbor of one of $A_{i}$ , $A_{i+1}$ (since $n\\geq4$ ), say $A_{k}$ , and then $A_{k}A_{j}=$ $A_{j}A_{k}=0$ , so $A_{j}a_{i}=0$ , and ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "By our construction ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $0\\leq i\\leq n-2$ . ", "page_idx": 13}, {"type": "text", "text": "By lemma 5.2, part 2), ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $1\\leq i\\leq n-1$ , where $u_{i}\\in\\mathbb{C}^{*}$ . ", "page_idx": 13}, {"type": "text", "text": "By the above calculations the matrices of $\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})$ with respect to the basis $a_{0},\\;\\;a_{1},\\ldots,a_{n-1}$ are ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "for $i\\;=\\;1,2,\\ldots,n\\,-\\,1$ , where $I_{k}$ is the $k\\,\\times\\,k$ identity matrix, and $u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{*}$ . Since $\\sigma_{1},\\ldots,\\sigma_{n-1}$ are conjugate in $B_{n}$ , the $u_{i}$ are all equal, and we have the standard representation. ", "page_idx": 13}, {"type": "text", "text": "Now let us consider when the standard representation is irreducible. ", "page_idx": 13}, {"type": "text", "text": "Lemma 5.3. If $u=1$ then $\\tau_{n}(u)$ is reducible. ", "page_idx": 13}, {"type": "text", "text": "Proof. If $u=1$ then the vector $v=(1,1,1,\\ldots,1)^{T}$ is a fixed vector. ", "page_idx": 13}, {"type": "text", "text": "Lemma 5.4. If $u\\ne1$ then $\\tau_{n}(u)$ is irreducible. ", "page_idx": 13}]
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If ", "type": "text"}, {"bbox": [212, 632, 242, 644], "score": 0.88, "content": "u=1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [243, 632, 270, 646], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [271, 632, 298, 646], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [299, 632, 362, 646], "score": 1.0, "content": " is reducible.", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [136, 650, 486, 666], "lines": [{"bbox": [136, 652, 485, 668], "spans": [{"bbox": [136, 653, 190, 668], "score": 1.0, "content": "Proof. If", "type": "text"}, {"bbox": [190, 653, 220, 664], "score": 0.89, "content": "u=1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [220, 653, 301, 668], "score": 1.0, "content": " then the vector ", "type": "text"}, {"bbox": [302, 652, 400, 667], "score": 0.92, "content": "v=(1,1,1,\\ldots,1)^{T}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [400, 653, 485, 668], "score": 1.0, "content": " is a fixed vector.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [124, 685, 372, 701], "lines": [{"bbox": [125, 687, 370, 702], "spans": [{"bbox": [125, 688, 212, 701], "score": 1.0, "content": "Lemma 5.4. 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If ", "type": "text"}, {"bbox": [331, 250, 394, 261], "score": 0.81, "content": "j\\neq i,\\ \\ i+1", "type": "inline_equation", "height": 11, "width": 63}, {"bbox": [394, 248, 428, 263], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [428, 250, 441, 262], "score": 0.92, "content": "A_{j}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [442, 248, 486, 263], "score": 1.0, "content": " is not a", "type": "text"}], "index": 9}, {"bbox": [126, 262, 487, 276], "spans": [{"bbox": [126, 262, 223, 276], "score": 1.0, "content": "neighbor of one of ", "type": "text"}, {"bbox": [224, 264, 236, 275], "score": 0.84, "content": "A_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [237, 262, 247, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [247, 264, 270, 275], "score": 0.89, "content": "A_{i+1}", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [271, 262, 308, 276], "score": 1.0, "content": " (since ", "type": 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", and", "type": "text"}], "index": 11}], "index": 10, "bbox_fs": [126, 248, 487, 290]}, {"type": "interline_equation", "bbox": [239, 295, 371, 309], "lines": [{"bbox": [239, 295, 371, 309], "spans": [{"bbox": [239, 295, 371, 309], "score": 0.9, "content": "\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [137, 312, 242, 325], "lines": [{"bbox": [138, 314, 241, 326], "spans": [{"bbox": [138, 314, 241, 326], "score": 1.0, "content": "By our construction", "type": "text"}], "index": 13}], "index": 13, "bbox_fs": [138, 314, 241, 326]}, {"type": "interline_equation", "bbox": [225, 333, 385, 346], "lines": [{"bbox": [225, 333, 385, 346], "spans": [{"bbox": [225, 333, 385, 346], "score": 0.91, "content": "\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [124, 349, 217, 363], "lines": [{"bbox": [124, 350, 217, 365], 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"spans": [{"bbox": [125, 402, 144, 417], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 405, 213, 415], "score": 0.92, "content": "1\\leq i\\leq n-1", "type": "inline_equation", "height": 10, "width": 69}, {"bbox": [213, 402, 254, 417], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [254, 405, 292, 415], "score": 0.93, "content": "u_{i}\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [293, 402, 296, 417], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [125, 402, 296, 417]}, {"type": "text", "bbox": [124, 416, 486, 444], "lines": [{"bbox": [137, 416, 486, 432], "spans": [{"bbox": [137, 416, 365, 432], "score": 1.0, "content": "By the above calculations the matrices of ", "type": "text"}, {"bbox": [365, 417, 457, 430], "score": 0.92, "content": "\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [457, 416, 486, 432], "score": 1.0, "content": " with", "type": "text"}], "index": 19}, {"bbox": [125, 431, 333, 446], "spans": [{"bbox": [125, 431, 228, 446], "score": 1.0, "content": "respect to the basis ", "type": "text"}, {"bbox": [229, 435, 311, 444], "score": 0.51, "content": "a_{0},\\;\\;a_{1},\\ldots,a_{n-1}", "type": "inline_equation", "height": 9, "width": 82}, {"bbox": [311, 431, 333, 446], "score": 1.0, "content": " are", "type": "text"}], "index": 20}], "index": 19.5, "bbox_fs": [125, 416, 486, 446]}, {"type": "interline_equation", "bbox": [219, 483, 390, 540], "lines": [{"bbox": [219, 483, 390, 540], "spans": [{"bbox": [219, 483, 390, 540], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [123, 553, 487, 596], "lines": [{"bbox": [124, 555, 486, 570], "spans": [{"bbox": [124, 555, 145, 570], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [145, 556, 245, 569], "score": 0.9, "content": "i\\;=\\;1,2,\\ldots,n\\,-\\,1", "type": "inline_equation", "height": 13, "width": 100}, {"bbox": [246, 555, 289, 570], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [289, 556, 300, 568], "score": 0.89, "content": "I_{k}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [300, 555, 340, 570], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [340, 557, 372, 567], "score": 0.9, "content": "k\\,\\times\\,k", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [372, 555, 486, 570], "score": 1.0, "content": " identity matrix, and", "type": "text"}], "index": 22}, {"bbox": [126, 569, 487, 585], "spans": [{"bbox": [126, 570, 218, 583], "score": 0.9, "content": "u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 13, "width": 92}, {"bbox": [218, 569, 256, 585], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [256, 572, 318, 583], "score": 0.87, "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [318, 569, 410, 585], "score": 1.0, "content": " are conjugate in ", "type": "text"}, {"bbox": [411, 572, 425, 582], "score": 0.92, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [426, 569, 454, 585], "score": 1.0, "content": ", the ", "type": "text"}, {"bbox": [454, 575, 464, 582], "score": 0.9, "content": "u_{i}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [465, 569, 487, 585], "score": 1.0, "content": " are", "type": "text"}], "index": 23}, {"bbox": [126, 584, 388, 598], "spans": [{"bbox": [126, 584, 388, 598], "score": 1.0, "content": "all equal, and we have the standard representation.", "type": "text"}], "index": 24}], "index": 23, "bbox_fs": [124, 555, 487, 598]}, {"type": "text", "bbox": [135, 609, 485, 624], "lines": [{"bbox": [137, 612, 485, 626], "spans": [{"bbox": [137, 612, 485, 626], "score": 1.0, "content": "Now let us consider when the standard representation is irreducible.", "type": "text"}], "index": 25}], "index": 25, "bbox_fs": [137, 612, 485, 626]}, {"type": "text", "bbox": [124, 630, 362, 645], "lines": [{"bbox": [125, 632, 362, 646], "spans": [{"bbox": [125, 632, 212, 646], "score": 1.0, "content": "Lemma 5.3. If ", "type": "text"}, {"bbox": [212, 632, 242, 644], "score": 0.88, "content": "u=1", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [243, 632, 270, 646], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [271, 632, 298, 646], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [299, 632, 362, 646], "score": 1.0, "content": " is reducible.", "type": "text"}], "index": 26}], "index": 26, "bbox_fs": [125, 632, 362, 646]}, {"type": "text", "bbox": [136, 650, 486, 666], "lines": [{"bbox": [136, 652, 485, 668], "spans": [{"bbox": [136, 653, 190, 668], "score": 1.0, "content": "Proof. If", "type": "text"}, {"bbox": [190, 653, 220, 664], "score": 0.89, "content": "u=1", "type": "inline_equation", "height": 11, "width": 30}, {"bbox": [220, 653, 301, 668], "score": 1.0, "content": " then the vector ", "type": "text"}, {"bbox": [302, 652, 400, 667], "score": 0.92, "content": "v=(1,1,1,\\ldots,1)^{T}", "type": "inline_equation", "height": 15, "width": 98}, {"bbox": [400, 653, 485, 668], "score": 1.0, "content": " is a fixed vector.", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [136, 652, 485, 668]}, {"type": "text", "bbox": [124, 685, 372, 701], "lines": [{"bbox": [125, 687, 370, 702], "spans": [{"bbox": [125, 688, 212, 701], "score": 1.0, "content": "Lemma 5.4. If ", "type": "text"}, {"bbox": [212, 687, 242, 701], "score": 0.9, "content": "u\\ne1", "type": "inline_equation", "height": 14, "width": 30}, {"bbox": [242, 688, 270, 701], "score": 1.0, "content": " then ", "type": "text"}, {"bbox": [270, 687, 298, 702], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 15, "width": 28}, {"bbox": [299, 688, 370, 701], "score": 1.0, "content": " is irreducible.", "type": "text"}], "index": 28}], "index": 28, "bbox_fs": [125, 687, 370, 702]}]}
|
[{"type": "text", "bbox": [124, 110, 487, 153], "content": "By induction and lemma 5.2, part 1), is a basis vector for , for . By lemma 5.2, part 3), and are linearly independent. Thus is a basis for .", "index": 0}, {"type": "text", "bbox": [137, 153, 167, 165], "content": "Since", "index": 1}, {"type": "interline_equation", "bbox": [186, 174, 425, 187], "content": "", "index": 2}, {"type": "text", "bbox": [123, 191, 485, 218], "content": "is invariant under and is an dimensional irreducible represen- tation, is a basis for .", "index": 3}, {"type": "text", "bbox": [124, 219, 486, 245], "content": "We now wish to determine the action of on this basis.", "index": 4}, {"type": "text", "bbox": [124, 246, 487, 288], "content": "Consider . If , then is not a neighbor of one of , (since ), say , and then , so , and", "index": 5}, {"type": "interline_equation", "bbox": [239, 295, 371, 309], "content": "", "index": 6}, {"type": "text", "bbox": [137, 312, 242, 325], "content": "By our construction", "index": 7}, {"type": "interline_equation", "bbox": [225, 333, 385, 346], "content": "", "index": 8}, {"type": "text", "bbox": [124, 349, 217, 363], "content": "for .", "index": 9}, {"type": "text", "bbox": [137, 364, 255, 377], "content": "By lemma 5.2, part 2),", "index": 10}, {"type": "interline_equation", "bbox": [229, 384, 380, 398], "content": "", "index": 11}, {"type": "text", "bbox": [124, 401, 297, 415], "content": "for , where .", "index": 12}, {"type": "text", "bbox": [124, 416, 486, 444], "content": "By the above calculations the matrices of with respect to the basis are", "index": 13}, {"type": "interline_equation", "bbox": [219, 483, 390, 540], "content": "", "index": 14}, {"type": "text", "bbox": [123, 553, 487, 596], "content": "for , where is the identity matrix, and . Since are conjugate in , the are all equal, and we have the standard representation.", "index": 15}, {"type": "text", "bbox": [135, 609, 485, 624], "content": "Now let us consider when the standard representation is irreducible.", "index": 16}, {"type": "text", "bbox": [124, 630, 362, 645], "content": "Lemma 5.3. If then is reducible.", "index": 17}, {"type": "text", "bbox": [136, 650, 486, 666], "content": "Proof. If then the vector is a fixed vector.", "index": 18}, {"type": "text", "bbox": [124, 685, 372, 701], "content": "Lemma 5.4. If then is irreducible.", "index": 19}]
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[{"bbox": [125, 113, 486, 127], "content": "By induction and lemma 5.2, part 1), is a basis vector for", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 486, 141], "content": ", for . By lemma 5.2, part 3), and are", "parent_index": 0, "line_index": 1}, {"bbox": [125, 141, 429, 154], "content": "linearly independent. Thus is a basis for .", "parent_index": 0, "line_index": 2}, {"bbox": [137, 154, 167, 168], "content": "Since", "parent_index": 1, "line_index": 0}, {"bbox": [124, 193, 484, 206], "content": "is invariant under and is an dimensional irreducible represen-", "parent_index": 3, "line_index": 0}, {"bbox": [125, 206, 320, 220], "content": "tation, is a basis for .", "parent_index": 3, "line_index": 1}, {"bbox": [137, 219, 484, 235], "content": "We now wish to determine the action of", "parent_index": 4, "line_index": 0}, {"bbox": [125, 234, 194, 247], "content": "on this basis.", "parent_index": 4, "line_index": 1}, {"bbox": [138, 248, 486, 263], "content": "Consider . If , then is not a", "parent_index": 5, "line_index": 0}, {"bbox": [126, 262, 487, 276], "content": "neighbor of one of , (since ), say , and then", "parent_index": 5, "line_index": 1}, {"bbox": [126, 277, 268, 290], "content": ", so , and", "parent_index": 5, "line_index": 2}, {"bbox": [138, 314, 241, 326], "content": "By our construction", "parent_index": 7, "line_index": 0}, {"bbox": [124, 350, 217, 365], "content": "for .", "parent_index": 9, "line_index": 0}, {"bbox": [138, 366, 255, 378], "content": "By lemma 5.2, part 2),", "parent_index": 10, "line_index": 0}, {"bbox": [125, 402, 296, 417], "content": "for , where .", "parent_index": 12, "line_index": 0}, {"bbox": [137, 416, 486, 432], "content": "By the above calculations the matrices of with", "parent_index": 13, "line_index": 0}, {"bbox": [125, 431, 333, 446], "content": "respect to the basis are", "parent_index": 13, "line_index": 1}, {"bbox": [124, 555, 486, 570], "content": "for , where is the identity matrix, and", "parent_index": 15, "line_index": 0}, {"bbox": [126, 569, 487, 585], "content": ". Since are conjugate in , the are", "parent_index": 15, "line_index": 1}, {"bbox": [126, 584, 388, 598], "content": "all equal, and we have the standard representation.", "parent_index": 15, "line_index": 2}, {"bbox": [137, 612, 485, 626], "content": "Now let us consider when the standard representation is irreducible.", "parent_index": 16, "line_index": 0}, {"bbox": [125, 632, 362, 646], "content": "Lemma 5.3. If then is reducible.", "parent_index": 17, "line_index": 0}, {"bbox": [136, 652, 485, 668], "content": "Proof. If then the vector is a fixed vector.", "parent_index": 18, "line_index": 0}, {"bbox": [125, 687, 370, 702], "content": "Lemma 5.4. If then is irreducible.", "parent_index": 19, "line_index": 0}]
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[]
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[{"bbox": [321, 118, 330, 125], "content": "a_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [437, 114, 486, 127], "content": "I m(A_{i})\\cap", "parent_index": 0, "subtype": "inline"}, {"bbox": [126, 128, 174, 140], "content": "I m(A_{i+1})", "parent_index": 0, "subtype": "inline"}, {"bbox": [201, 129, 275, 139], "content": "0\\leq\\,i\\leq n-1", "parent_index": 0, "subtype": "inline"}, {"bbox": [407, 132, 416, 139], "content": "a_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [444, 132, 464, 140], "content": "a_{i+1}", "parent_index": 0, "subtype": "inline"}, {"bbox": [268, 142, 316, 154], "content": "\\{a_{i},a_{i+1}\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [387, 142, 425, 154], "content": "I m(A_{i})", "parent_index": 0, "subtype": "inline"}, {"bbox": [186, 174, 425, 187], "content": "s p a n\\{a_{0},\\ldots a_{n-1}\\}=I m(A_{1})+\\cdots+I m(A_{n-1})", "parent_index": 2, "subtype": "interline"}, {"bbox": [220, 194, 235, 205], "content": "B_{n}", "parent_index": 3, "subtype": "inline"}, {"bbox": [262, 198, 268, 205], "content": "\\rho", "parent_index": 3, "subtype": "inline"}, {"bbox": [299, 196, 316, 204], "content": "n-", "parent_index": 3, "subtype": "inline"}, {"bbox": [164, 207, 230, 220], "content": "\\left\\{a_{0},\\ldots.a_{n-1}\\right\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [302, 208, 317, 217], "content": "\\mathbb{C}^{n}", "parent_index": 3, "subtype": "inline"}, {"bbox": [356, 221, 484, 234], "content": "\\rho(\\sigma_{1}),\\;\\;\\rho(\\sigma_{2}),\\ldots,\\rho(\\sigma_{n-1})", "parent_index": 4, "subtype": "inline"}, {"bbox": [187, 249, 312, 262], "content": "a_{i}\\in I m(A_{i})\\cap I m(A_{i+1})", "parent_index": 5, "subtype": "inline"}, {"bbox": [331, 250, 394, 261], "content": "j\\neq i,\\ \\ i+1", "parent_index": 5, "subtype": "inline"}, {"bbox": [428, 250, 441, 262], "content": "A_{j}", "parent_index": 5, "subtype": "inline"}, {"bbox": [224, 264, 236, 275], "content": "A_{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [247, 264, 270, 275], "content": "A_{i+1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [308, 264, 340, 274], "content": "n\\geq4", "parent_index": 5, "subtype": "inline"}, {"bbox": [372, 264, 387, 274], "content": "A_{k}", "parent_index": 5, "subtype": "inline"}, {"bbox": [444, 263, 487, 276], "content": "A_{k}A_{j}=", "parent_index": 5, "subtype": "inline"}, {"bbox": [126, 278, 175, 290], "content": "A_{j}A_{k}=0", "parent_index": 5, "subtype": "inline"}, {"bbox": [196, 278, 241, 290], "content": "A_{j}a_{i}=0", "parent_index": 5, "subtype": "inline"}, {"bbox": [239, 295, 371, 309], "content": "\\rho(\\sigma_{j})a_{i}=(1+A_{j})a_{i}=a_{i}.", "parent_index": 6, "subtype": "interline"}, {"bbox": [225, 333, 385, 346], "content": "\\rho(\\sigma_{i+1})a_{i}=(1+A_{i+1})a_{i}=a_{i+1}", "parent_index": 8, "subtype": "interline"}, {"bbox": [144, 353, 213, 363], "content": "0\\leq i\\leq n-2", "parent_index": 9, "subtype": "inline"}, {"bbox": [229, 384, 380, 398], "content": "\\rho(\\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},", "parent_index": 11, "subtype": "interline"}, {"bbox": [144, 405, 213, 415], "content": "1\\leq i\\leq n-1", "parent_index": 12, "subtype": "inline"}, {"bbox": [254, 405, 292, 415], "content": "u_{i}\\in\\mathbb{C}^{*}", "parent_index": 12, "subtype": "inline"}, {"bbox": [365, 417, 457, 430], "content": "\\rho(\\sigma_{1}),\\dotsc,\\rho(\\sigma_{n-1})", "parent_index": 13, "subtype": "inline"}, {"bbox": [229, 435, 311, 444], "content": "a_{0},\\;\\;a_{1},\\ldots,a_{n-1}", "parent_index": 13, "subtype": "inline"}, {"bbox": [219, 483, 390, 540], "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u_{i}}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "parent_index": 14, "subtype": "interline"}, {"bbox": [145, 556, 245, 569], "content": "i\\;=\\;1,2,\\ldots,n\\,-\\,1", "parent_index": 15, "subtype": "inline"}, {"bbox": [289, 556, 300, 568], "content": "I_{k}", "parent_index": 15, "subtype": "inline"}, {"bbox": [340, 557, 372, 567], "content": "k\\,\\times\\,k", "parent_index": 15, "subtype": "inline"}, {"bbox": [126, 570, 218, 583], "content": "u_{1},\\dotsc,u_{n-1}\\in\\mathbb{C}^{*}", "parent_index": 15, "subtype": "inline"}, {"bbox": [256, 572, 318, 583], "content": "\\sigma_{1},\\ldots,\\sigma_{n-1}", "parent_index": 15, "subtype": "inline"}, {"bbox": [411, 572, 425, 582], "content": "B_{n}", "parent_index": 15, "subtype": "inline"}, {"bbox": [454, 575, 464, 582], "content": "u_{i}", "parent_index": 15, "subtype": "inline"}, {"bbox": [212, 632, 242, 644], "content": "u=1", "parent_index": 17, "subtype": "inline"}, {"bbox": [271, 632, 298, 646], "content": "\\tau_{n}(u)", "parent_index": 17, "subtype": "inline"}, {"bbox": [190, 653, 220, 664], "content": "u=1", "parent_index": 18, "subtype": "inline"}, {"bbox": [302, 652, 400, 667], "content": "v=(1,1,1,\\ldots,1)^{T}", "parent_index": 18, "subtype": "inline"}, {"bbox": [212, 687, 242, 701], "content": "u\\ne1", "parent_index": 19, "subtype": "inline"}, {"bbox": [270, 687, 298, 702], "content": "\\tau_{n}(u)", "parent_index": 19, "subtype": "inline"}]
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[]
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Proof. We need to prove that starting from any non-zero vector $x=\textstyle\sum a_{i}e_{i}$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $e_{i}$ . To do this, take $i$ such that $a_{i}\neq0$ . Consider the operator
$$
H=A+A^{2}+A B A=B+B^{2}+B A B,
$$
where $A\,=\,\rho(\sigma_{i-1})$ and $B\;=\;\rho(\sigma_{i})$ . By a direct calculation $H x\;=$ $(u-1)a_{i}e_{i}$ . Because $u\ne1$ the vector $H x$ is a non-zero multiple of $e_{i}$ .
Now, we have the main result of this paper:
Theorem 5.5 (The Main Theorem). Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\,\geq\,6$ . Let $r\,\geq\,n$ , and let $\rho(\sigma_{1})\,=$ $1+A_{1}$ with $r a n k(A_{1})=2$ .
Then $r=n$ and $\rho$ is equivalent to the following representation :
$$
\tau:B_{n}\to G L_{n}(\mathbb{C}),
$$
$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$
for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix, and $u\in\mathbb{C}^{*}$ , $u\ne1$ . These representations are non-equivalent for different values of $u$ .
Proof. By Theorem 4.4 the friendship graph of $\rho$ is a chain. Then, by theorem 5.1, $\rho$ is equivalent to a standard representation $\tau(u)$ for some $u\in\mathbb{C}^{*}$ . By Lemmas 5.3 and 5.4 $u\ne1$ .
Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following
Corollary 5.6. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\ge7$ . Let $c o r a n k(\rho)=2$ .
Then $\rho$ is equivalent to a specialization of the standard representation $\tau_{n}(u)$ , for some $u\neq1,\ u\in\mathbb{C}^{*}$ .
# References
[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.
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<html><body>
<p data-bbox="124 110 487 166">Proof. We need to prove that starting from any non-zero vector $x=\textstyle\sum a_{i}e_{i}$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $e_{i}$ . To do this, take $i$ such that $a_{i}\neq0$ . Consider the operator </p>
<div class="equation" data-bbox="203 173 407 186">$$
H=A+A^{2}+A B A=B+B^{2}+B A B,
$$</div>
<p data-bbox="124 191 486 220">where $A\,=\,\rho(\sigma_{i-1})$ and $B\;=\;\rho(\sigma_{i})$ . By a direct calculation $H x\;=$ $(u-1)a_{i}e_{i}$ . Because $u\ne1$ the vector $H x$ is a non-zero multiple of $e_{i}$ . </p>
<p data-bbox="136 234 363 248">Now, we have the main result of this paper: </p>
<p data-bbox="125 254 487 297">Theorem 5.5 (The Main Theorem). Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\,\geq\,6$ . Let $r\,\geq\,n$ , and let $\rho(\sigma_{1})\,=$ $1+A_{1}$ with $r a n k(A_{1})=2$ . </p>
<p data-bbox="137 297 466 311">Then $r=n$ and $\rho$ is equivalent to the following representation : </p>
<div class="equation" data-bbox="258 317 352 332">$$
\tau:B_{n}\to G L_{n}(\mathbb{C}),
$$</div>
<div class="equation" data-bbox="221 377 389 435">$$
\rho(\sigma_{i})=\left(\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\ {}&{0}&{u}&{}\\ {}&{1}&{0}&{}\\ {}&{}&{}&{I_{n-1-i}}\end{array}\right),
$$</div>
<p data-bbox="124 448 488 490">for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix, and $u\in\mathbb{C}^{*}$ , $u\ne1$ . These representations are non-equivalent for different values of $u$ . </p>
<p data-bbox="124 497 486 540">Proof. By Theorem 4.4 the friendship graph of $\rho$ is a chain. Then, by theorem 5.1, $\rho$ is equivalent to a standard representation $\tau(u)$ for some $u\in\mathbb{C}^{*}$ . By Lemmas 5.3 and 5.4 $u\ne1$ . </p>
<p data-bbox="124 540 486 569">Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following </p>
<p data-bbox="124 574 486 603">Corollary 5.6. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\ge7$ . Let $c o r a n k(\rho)=2$ . </p>
<p data-bbox="124 604 486 631">Then $\rho$ is equivalent to a specialization of the standard representation $\tau_{n}(u)$ , for some $u\neq1,\ u\in\mathbb{C}^{*}$ . </p>
<h1 data-bbox="270 643 342 657">References </h1>
<p data-bbox="128 662 486 701">[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658. </p>
</body></html>
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0003047v1
| 14 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Proof. We need to prove that starting from any non-zero vector $x=\\textstyle\\sum a_{i}e_{i}$ , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors $e_{i}$ . To do this, take $i$ such that $a_{i}\\neq0$ . Consider the operator ", "page_idx": 14}, {"type": "equation", "text": "$$\nH=A+A^{2}+A B A=B+B^{2}+B A B,\n$$", "text_format": "latex", "page_idx": 14}, {"type": "text", "text": "where $A\\,=\\,\\rho(\\sigma_{i-1})$ and $B\\;=\\;\\rho(\\sigma_{i})$ . By a direct calculation $H x\\;=$ $(u-1)a_{i}e_{i}$ . Because $u\\ne1$ the vector $H x$ is a non-zero multiple of $e_{i}$ . ", "page_idx": 14}, {"type": "text", "text": "Now, we have the main result of this paper: ", "page_idx": 14}, {"type": "text", "text": "Theorem 5.5 (The Main Theorem). Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\\,\\geq\\,6$ . Let $r\\,\\geq\\,n$ , and let $\\rho(\\sigma_{1})\\,=$ $1+A_{1}$ with $r a n k(A_{1})=2$ . ", "page_idx": 14}, {"type": "text", "text": "Then $r=n$ and $\\rho$ is equivalent to the following representation : ", "page_idx": 14}, {"type": "equation", "text": "$$\n\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),\n$$", "text_format": "latex", "page_idx": 14}, {"type": "equation", "text": "$$\n\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),\n$$", "text_format": "latex", "page_idx": 14}, {"type": "text", "text": "for $i=1,2,\\dots,n-1$ , where $I_{k}$ is the $k\\times k$ identity matrix, and $u\\in\\mathbb{C}^{*}$ , $u\\ne1$ . These representations are non-equivalent for different values of $u$ . ", "page_idx": 14}, {"type": "text", "text": "Proof. By Theorem 4.4 the friendship graph of $\\rho$ is a chain. Then, by theorem 5.1, $\\rho$ is equivalent to a standard representation $\\tau(u)$ for some $u\\in\\mathbb{C}^{*}$ . By Lemmas 5.3 and 5.4 $u\\ne1$ . ", "page_idx": 14}, {"type": "text", "text": "Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following ", "page_idx": 14}, {"type": "text", "text": "Corollary 5.6. Let $\\rho:B_{n}\\to G L_{r}(\\mathbb{C})$ be an irreducible representation of $B_{n}$ for $n\\ge7$ . Let $c o r a n k(\\rho)=2$ . ", "page_idx": 14}, {"type": "text", "text": "Then $\\rho$ is equivalent to a specialization of the standard representation $\\tau_{n}(u)$ , for some $u\\neq1,\\ u\\in\\mathbb{C}^{*}$ . ", "page_idx": 14}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658. ", "page_idx": 14}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 110, 487, 166], "lines": [{"bbox": [137, 113, 485, 126], "spans": [{"bbox": [137, 113, 485, 126], "score": 1.0, "content": "Proof. We need to prove that starting from any non-zero vector", "type": "text"}], "index": 0}, {"bbox": [126, 127, 484, 140], "spans": [{"bbox": [126, 128, 181, 140], "score": 0.93, "content": "x=\\textstyle\\sum a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 127, 484, 140], "score": 1.0, "content": ", we can generate the whole space. Obviously, it is enough", "type": "text"}], "index": 1}, {"bbox": [125, 141, 486, 154], "spans": [{"bbox": [125, 141, 453, 154], "score": 1.0, "content": "to show that we can generate one of the standard basis vectors ", "type": "text"}, {"bbox": [454, 146, 462, 153], "score": 0.9, "content": "e_{i}", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [463, 141, 486, 154], "score": 1.0, "content": ". To", "type": "text"}], "index": 2}, {"bbox": [125, 155, 405, 168], "spans": [{"bbox": [125, 155, 193, 168], "score": 1.0, "content": "do this, take ", "type": "text"}, {"bbox": [194, 157, 198, 165], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [198, 155, 253, 168], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [253, 156, 285, 167], "score": 0.93, "content": "a_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 155, 405, 168], "score": 1.0, "content": ". Consider the operator", "type": "text"}], "index": 3}], "index": 1.5}, {"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 191, 486, 220], "lines": [{"bbox": [126, 194, 486, 208], "spans": [{"bbox": [126, 194, 161, 208], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [161, 195, 227, 207], "score": 0.94, "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [228, 194, 257, 208], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [257, 195, 313, 208], "score": 0.95, "content": "B\\;=\\;\\rho(\\sigma_{i})", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [313, 194, 452, 208], "score": 1.0, "content": ". By a direct calculation ", "type": "text"}, {"bbox": [452, 195, 486, 207], "score": 0.82, "content": "H x\\;=", "type": "inline_equation", "height": 12, "width": 34}], "index": 5}, {"bbox": [126, 207, 484, 222], "spans": [{"bbox": [126, 209, 181, 221], "score": 0.93, "content": "(u-1)a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 207, 233, 222], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [234, 209, 262, 221], "score": 0.92, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [262, 207, 321, 222], "score": 1.0, "content": " the vector ", "type": "text"}, {"bbox": [321, 209, 339, 218], "score": 0.85, "content": "H x", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [339, 207, 470, 222], "score": 1.0, "content": " is a non-zero multiple of ", "type": "text"}, {"bbox": [470, 212, 479, 220], "score": 0.86, "content": "e_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [480, 207, 484, 222], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5}, {"type": "text", "bbox": [136, 234, 363, 248], "lines": [{"bbox": [137, 235, 363, 250], "spans": [{"bbox": [137, 235, 363, 250], "score": 1.0, "content": "Now, we have the main result of this paper:", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [125, 254, 487, 297], "lines": [{"bbox": [125, 257, 486, 272], "spans": [{"bbox": [125, 257, 343, 272], "score": 1.0, "content": "Theorem 5.5 (The Main Theorem). Let ", "type": "text"}, {"bbox": [343, 258, 435, 270], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [436, 257, 486, 272], "score": 1.0, "content": " be an ir-", "type": "text"}], "index": 8}, {"bbox": [126, 271, 487, 285], "spans": [{"bbox": [126, 272, 266, 285], "score": 1.0, "content": "reducible representation of ", "type": "text"}, {"bbox": [266, 271, 281, 284], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [281, 272, 304, 285], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 271, 337, 284], "score": 0.89, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [337, 272, 363, 285], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [364, 272, 396, 284], "score": 0.86, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [397, 272, 444, 285], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [444, 272, 487, 285], "score": 0.9, "content": "\\rho(\\sigma_{1})\\,=", "type": "inline_equation", "height": 13, "width": 43}], "index": 9}, {"bbox": [126, 285, 263, 299], "spans": [{"bbox": [126, 287, 160, 297], "score": 0.91, "content": "1+A_{1}", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [160, 285, 189, 298], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [190, 286, 259, 299], "score": 0.89, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [259, 285, 263, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9}, {"type": "text", "bbox": [137, 297, 466, 311], "lines": [{"bbox": [140, 299, 465, 313], "spans": [{"bbox": [140, 299, 168, 313], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 304, 197, 309], "score": 0.81, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [198, 299, 223, 313], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [223, 303, 230, 312], "score": 0.78, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 299, 465, 313], "score": 1.0, "content": " is equivalent to the following representation :", "type": "text"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 448, 488, 490], "lines": [{"bbox": [125, 451, 484, 464], "spans": [{"bbox": [125, 451, 144, 464], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 452, 231, 464], "score": 0.91, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [232, 451, 270, 464], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 452, 281, 463], "score": 0.89, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [281, 451, 314, 464], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [315, 452, 339, 462], "score": 0.92, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [339, 451, 447, 464], "score": 1.0, "content": " identity matrix, and ", "type": "text"}, {"bbox": [447, 452, 482, 462], "score": 0.89, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 451, 484, 464], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 465, 487, 479], "spans": [{"bbox": [126, 466, 154, 478], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [155, 465, 487, 479], "score": 1.0, "content": ". These representations are non-equivalent for different values of", "type": "text"}], "index": 15}, {"bbox": [126, 481, 137, 492], "spans": [{"bbox": [126, 483, 133, 489], "score": 0.86, "content": "u", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [133, 481, 137, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 15}, {"type": "text", "bbox": [124, 497, 486, 540], "lines": [{"bbox": [137, 500, 485, 514], "spans": [{"bbox": [137, 500, 388, 514], "score": 1.0, "content": "Proof. By Theorem 4.4 the friendship graph of ", "type": "text"}, {"bbox": [388, 505, 394, 513], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [394, 500, 485, 514], "score": 1.0, "content": " is a chain. Then,", "type": "text"}], "index": 17}, {"bbox": [126, 514, 485, 528], "spans": [{"bbox": [126, 514, 212, 528], "score": 1.0, "content": "by theorem 5.1, ", "type": "text"}, {"bbox": [212, 519, 218, 527], "score": 0.84, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [219, 514, 443, 528], "score": 1.0, "content": " is equivalent to a standard representation ", "type": "text"}, {"bbox": [444, 515, 466, 528], "score": 0.94, "content": "\\tau(u)", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [467, 514, 485, 528], "score": 1.0, "content": " for", "type": "text"}], "index": 18}, {"bbox": [126, 528, 355, 542], "spans": [{"bbox": [126, 528, 155, 542], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [155, 530, 190, 539], "score": 0.91, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [191, 528, 322, 542], "score": 1.0, "content": ". By Lemmas 5.3 and 5.4 ", "type": "text"}, {"bbox": [322, 529, 351, 541], "score": 0.61, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [351, 528, 355, 542], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [124, 540, 486, 569], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "type": "text"}], "index": 20}, {"bbox": [126, 554, 346, 571], "spans": [{"bbox": [126, 554, 346, 571], "score": 1.0, "content": "(see [3], Theorem 23), we get the following", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "text", "bbox": [124, 574, 486, 603], "lines": [{"bbox": [126, 577, 487, 592], "spans": [{"bbox": [126, 577, 232, 592], "score": 1.0, "content": "Corollary 5.6. Let ", "type": "text"}, {"bbox": [232, 578, 321, 590], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [322, 577, 487, 592], "score": 1.0, "content": " be an irreducible representation", "type": "text"}], "index": 22}, {"bbox": [127, 591, 310, 605], "spans": [{"bbox": [127, 592, 139, 605], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 591, 155, 604], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [155, 592, 176, 605], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [177, 591, 207, 603], "score": 0.8, "content": "n\\ge7", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [207, 592, 232, 605], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [233, 592, 307, 605], "score": 0.81, "content": "c o r a n k(\\rho)=2", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 592, 310, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "text", "bbox": [124, 604, 486, 631], "lines": [{"bbox": [139, 605, 487, 619], "spans": [{"bbox": [139, 605, 167, 619], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [167, 607, 174, 618], "score": 0.41, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [174, 605, 487, 619], "score": 1.0, "content": " is equivalent to a specialization of the standard representation", "type": "text"}], "index": 24}, {"bbox": [126, 619, 287, 632], "spans": [{"bbox": [126, 619, 153, 632], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [153, 620, 208, 632], "score": 1.0, "content": ", for some ", "type": "text"}, {"bbox": [208, 619, 283, 632], "score": 0.87, "content": "u\\neq1,\\ u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [283, 620, 287, 632], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "title", "bbox": [270, 643, 342, 657], "lines": [{"bbox": [270, 645, 342, 658], "spans": [{"bbox": [270, 645, 342, 658], "score": 1.0, "content": "References", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [128, 662, 486, 701], "lines": [{"bbox": [131, 665, 486, 678], "spans": [{"bbox": [131, 665, 486, 678], "score": 1.0, "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,", "type": "text"}], "index": 27}, {"bbox": [146, 678, 332, 689], "spans": [{"bbox": [146, 678, 332, 689], "score": 1.0, "content": "Princeton Univ.Press,Princeton,N.J.,1974.", "type": "text"}], "index": 28}, {"bbox": [131, 690, 484, 701], "spans": [{"bbox": [131, 690, 484, 701], "score": 1.0, "content": "[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "type": "text"}], "index": 29}], "index": 28}], "layout_bboxes": [], "page_idx": 14, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}], "discarded_blocks": [{"type": "discarded", "bbox": [221, 90, 389, 101], "lines": [{"bbox": [223, 93, 389, 102], "spans": [{"bbox": [223, 93, 389, 102], "score": 1.0, "content": "BRAID GROUP REPRESENTATIONS", "type": "text"}]}]}, {"type": "discarded", "bbox": [475, 91, 486, 100], "lines": [{"bbox": [473, 93, 486, 102], "spans": [{"bbox": [473, 93, 486, 102], "score": 1.0, "content": "15", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 110, 487, 166], "lines": [{"bbox": [137, 113, 485, 126], "spans": [{"bbox": [137, 113, 485, 126], "score": 1.0, "content": "Proof. We need to prove that starting from any non-zero vector", "type": "text"}], "index": 0}, {"bbox": [126, 127, 484, 140], "spans": [{"bbox": [126, 128, 181, 140], "score": 0.93, "content": "x=\\textstyle\\sum a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 127, 484, 140], "score": 1.0, "content": ", we can generate the whole space. Obviously, it is enough", "type": "text"}], "index": 1}, {"bbox": [125, 141, 486, 154], "spans": [{"bbox": [125, 141, 453, 154], "score": 1.0, "content": "to show that we can generate one of the standard basis vectors ", "type": "text"}, {"bbox": [454, 146, 462, 153], "score": 0.9, "content": "e_{i}", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [463, 141, 486, 154], "score": 1.0, "content": ". To", "type": "text"}], "index": 2}, {"bbox": [125, 155, 405, 168], "spans": [{"bbox": [125, 155, 193, 168], "score": 1.0, "content": "do this, take ", "type": "text"}, {"bbox": [194, 157, 198, 165], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [198, 155, 253, 168], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [253, 156, 285, 167], "score": 0.93, "content": "a_{i}\\neq0", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [285, 155, 405, 168], "score": 1.0, "content": ". Consider the operator", "type": "text"}], "index": 3}], "index": 1.5, "bbox_fs": [125, 113, 486, 168]}, {"type": "interline_equation", "bbox": [203, 173, 407, 186], "lines": [{"bbox": [203, 173, 407, 186], "spans": [{"bbox": [203, 173, 407, 186], "score": 0.87, "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [124, 191, 486, 220], "lines": [{"bbox": [126, 194, 486, 208], "spans": [{"bbox": [126, 194, 161, 208], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [161, 195, 227, 207], "score": 0.94, "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [228, 194, 257, 208], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [257, 195, 313, 208], "score": 0.95, "content": "B\\;=\\;\\rho(\\sigma_{i})", "type": "inline_equation", "height": 13, "width": 56}, {"bbox": [313, 194, 452, 208], "score": 1.0, "content": ". By a direct calculation ", "type": "text"}, {"bbox": [452, 195, 486, 207], "score": 0.82, "content": "H x\\;=", "type": "inline_equation", "height": 12, "width": 34}], "index": 5}, {"bbox": [126, 207, 484, 222], "spans": [{"bbox": [126, 209, 181, 221], "score": 0.93, "content": "(u-1)a_{i}e_{i}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [181, 207, 233, 222], "score": 1.0, "content": ". Because ", "type": "text"}, {"bbox": [234, 209, 262, 221], "score": 0.92, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [262, 207, 321, 222], "score": 1.0, "content": " the vector ", "type": "text"}, {"bbox": [321, 209, 339, 218], "score": 0.85, "content": "H x", "type": "inline_equation", "height": 9, "width": 18}, {"bbox": [339, 207, 470, 222], "score": 1.0, "content": " is a non-zero multiple of ", "type": "text"}, {"bbox": [470, 212, 479, 220], "score": 0.86, "content": "e_{i}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [480, 207, 484, 222], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 5.5, "bbox_fs": [126, 194, 486, 222]}, {"type": "text", "bbox": [136, 234, 363, 248], "lines": [{"bbox": [137, 235, 363, 250], "spans": [{"bbox": [137, 235, 363, 250], "score": 1.0, "content": "Now, we have the main result of this paper:", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [137, 235, 363, 250]}, {"type": "text", "bbox": [125, 254, 487, 297], "lines": [{"bbox": [125, 257, 486, 272], "spans": [{"bbox": [125, 257, 343, 272], "score": 1.0, "content": "Theorem 5.5 (The Main Theorem). Let ", "type": "text"}, {"bbox": [343, 258, 435, 270], "score": 0.9, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [436, 257, 486, 272], "score": 1.0, "content": " be an ir-", "type": "text"}], "index": 8}, {"bbox": [126, 271, 487, 285], "spans": [{"bbox": [126, 272, 266, 285], "score": 1.0, "content": "reducible representation of ", "type": "text"}, {"bbox": [266, 271, 281, 284], "score": 0.86, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [281, 272, 304, 285], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 271, 337, 284], "score": 0.89, "content": "n\\,\\geq\\,6", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [337, 272, 363, 285], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [364, 272, 396, 284], "score": 0.86, "content": "r\\,\\geq\\,n", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [397, 272, 444, 285], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [444, 272, 487, 285], "score": 0.9, "content": "\\rho(\\sigma_{1})\\,=", "type": "inline_equation", "height": 13, "width": 43}], "index": 9}, {"bbox": [126, 285, 263, 299], "spans": [{"bbox": [126, 287, 160, 297], "score": 0.91, "content": "1+A_{1}", "type": "inline_equation", "height": 10, "width": 34}, {"bbox": [160, 285, 189, 298], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [190, 286, 259, 299], "score": 0.89, "content": "r a n k(A_{1})=2", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [259, 285, 263, 298], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 9, "bbox_fs": [125, 257, 487, 299]}, {"type": "text", "bbox": [137, 297, 466, 311], "lines": [{"bbox": [140, 299, 465, 313], "spans": [{"bbox": [140, 299, 168, 313], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [168, 304, 197, 309], "score": 0.81, "content": "r=n", "type": "inline_equation", "height": 5, "width": 29}, {"bbox": [198, 299, 223, 313], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [223, 303, 230, 312], "score": 0.78, "content": "\\rho", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 299, 465, 313], "score": 1.0, "content": " is equivalent to the following representation :", "type": "text"}], "index": 11}], "index": 11, "bbox_fs": [140, 299, 465, 313]}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "lines": [{"bbox": [258, 317, 352, 332], "spans": [{"bbox": [258, 317, 352, 332], "score": 0.9, "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "lines": [{"bbox": [221, 377, 389, 435], "spans": [{"bbox": [221, 377, 389, 435], "score": 0.93, "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [124, 448, 488, 490], "lines": [{"bbox": [125, 451, 484, 464], "spans": [{"bbox": [125, 451, 144, 464], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [144, 452, 231, 464], "score": 0.91, "content": "i=1,2,\\dots,n-1", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [232, 451, 270, 464], "score": 1.0, "content": ", where ", "type": "text"}, {"bbox": [270, 452, 281, 463], "score": 0.89, "content": "I_{k}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [281, 451, 314, 464], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [315, 452, 339, 462], "score": 0.92, "content": "k\\times k", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [339, 451, 447, 464], "score": 1.0, "content": " identity matrix, and ", "type": "text"}, {"bbox": [447, 452, 482, 462], "score": 0.89, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [482, 451, 484, 464], "score": 1.0, "content": ",", "type": "text"}], "index": 14}, {"bbox": [126, 465, 487, 479], "spans": [{"bbox": [126, 466, 154, 478], "score": 0.91, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [155, 465, 487, 479], "score": 1.0, "content": ". These representations are non-equivalent for different values of", "type": "text"}], "index": 15}, {"bbox": [126, 481, 137, 492], "spans": [{"bbox": [126, 483, 133, 489], "score": 0.86, "content": "u", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [133, 481, 137, 492], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 15, "bbox_fs": [125, 451, 487, 492]}, {"type": "text", "bbox": [124, 497, 486, 540], "lines": [{"bbox": [137, 500, 485, 514], "spans": [{"bbox": [137, 500, 388, 514], "score": 1.0, "content": "Proof. By Theorem 4.4 the friendship graph of ", "type": "text"}, {"bbox": [388, 505, 394, 513], "score": 0.89, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [394, 500, 485, 514], "score": 1.0, "content": " is a chain. Then,", "type": "text"}], "index": 17}, {"bbox": [126, 514, 485, 528], "spans": [{"bbox": [126, 514, 212, 528], "score": 1.0, "content": "by theorem 5.1, ", "type": "text"}, {"bbox": [212, 519, 218, 527], "score": 0.84, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [219, 514, 443, 528], "score": 1.0, "content": " is equivalent to a standard representation ", "type": "text"}, {"bbox": [444, 515, 466, 528], "score": 0.94, "content": "\\tau(u)", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [467, 514, 485, 528], "score": 1.0, "content": " for", "type": "text"}], "index": 18}, {"bbox": [126, 528, 355, 542], "spans": [{"bbox": [126, 528, 155, 542], "score": 1.0, "content": "some ", "type": "text"}, {"bbox": [155, 530, 190, 539], "score": 0.91, "content": "u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [191, 528, 322, 542], "score": 1.0, "content": ". By Lemmas 5.3 and 5.4 ", "type": "text"}, {"bbox": [322, 529, 351, 541], "score": 0.61, "content": "u\\ne1", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [351, 528, 355, 542], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [126, 500, 485, 542]}, {"type": "text", "bbox": [124, 540, 486, 569], "lines": [{"bbox": [137, 542, 486, 555], "spans": [{"bbox": [137, 542, 486, 555], "score": 1.0, "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "type": "text"}], "index": 20}, {"bbox": [126, 554, 346, 571], "spans": [{"bbox": [126, 554, 346, 571], "score": 1.0, "content": "(see [3], Theorem 23), we get the following", "type": "text"}], "index": 21}], "index": 20.5, "bbox_fs": [126, 542, 486, 571]}, {"type": "text", "bbox": [124, 574, 486, 603], "lines": [{"bbox": [126, 577, 487, 592], "spans": [{"bbox": [126, 577, 232, 592], "score": 1.0, "content": "Corollary 5.6. Let ", "type": "text"}, {"bbox": [232, 578, 321, 590], "score": 0.92, "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "type": "inline_equation", "height": 12, "width": 89}, {"bbox": [322, 577, 487, 592], "score": 1.0, "content": " be an irreducible representation", "type": "text"}], "index": 22}, {"bbox": [127, 591, 310, 605], "spans": [{"bbox": [127, 592, 139, 605], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [139, 591, 155, 604], "score": 0.88, "content": "B_{n}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [155, 592, 176, 605], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [177, 591, 207, 603], "score": 0.8, "content": "n\\ge7", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [207, 592, 232, 605], "score": 1.0, "content": ". Let", "type": "text"}, {"bbox": [233, 592, 307, 605], "score": 0.81, "content": "c o r a n k(\\rho)=2", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [307, 592, 310, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 22.5, "bbox_fs": [126, 577, 487, 605]}, {"type": "text", "bbox": [124, 604, 486, 631], "lines": [{"bbox": [139, 605, 487, 619], "spans": [{"bbox": [139, 605, 167, 619], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [167, 607, 174, 618], "score": 0.41, "content": "\\rho", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [174, 605, 487, 619], "score": 1.0, "content": " is equivalent to a specialization of the standard representation", "type": "text"}], "index": 24}, {"bbox": [126, 619, 287, 632], "spans": [{"bbox": [126, 619, 153, 632], "score": 0.92, "content": "\\tau_{n}(u)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [153, 620, 208, 632], "score": 1.0, "content": ", for some ", "type": "text"}, {"bbox": [208, 619, 283, 632], "score": 0.87, "content": "u\\neq1,\\ u\\in\\mathbb{C}^{*}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [283, 620, 287, 632], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [126, 605, 487, 632]}, {"type": "title", "bbox": [270, 643, 342, 657], "lines": [{"bbox": [270, 645, 342, 658], "spans": [{"bbox": [270, 645, 342, 658], "score": 1.0, "content": "References", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [128, 662, 486, 701], "lines": [{"bbox": [131, 665, 486, 678], "spans": [{"bbox": [131, 665, 486, 678], "score": 1.0, "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,", "type": "text"}], "index": 27}, {"bbox": [146, 678, 332, 689], "spans": [{"bbox": [146, 678, 332, 689], "score": 1.0, "content": "Princeton Univ.Press,Princeton,N.J.,1974.", "type": "text"}], "index": 28}, {"bbox": [131, 690, 484, 701], "spans": [{"bbox": [131, 690, 484, 701], "score": 1.0, "content": "[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "type": "text"}], "index": 29}], "index": 28, "bbox_fs": [131, 665, 486, 701]}]}
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[{"type": "text", "bbox": [124, 110, 487, 166], "content": "Proof. We need to prove that starting from any non-zero vector , we can generate the whole space. Obviously, it is enough to show that we can generate one of the standard basis vectors . To do this, take such that . Consider the operator", "index": 0}, {"type": "interline_equation", "bbox": [203, 173, 407, 186], "content": "", "index": 1}, {"type": "text", "bbox": [124, 191, 486, 220], "content": "where and . By a direct calculation . Because the vector is a non-zero multiple of .", "index": 2}, {"type": "text", "bbox": [136, 234, 363, 248], "content": "Now, we have the main result of this paper:", "index": 3}, {"type": "text", "bbox": [125, 254, 487, 297], "content": "Theorem 5.5 (The Main Theorem). Let be an ir- reducible representation of for . Let , and let with .", "index": 4}, {"type": "text", "bbox": [137, 297, 466, 311], "content": "Then and is equivalent to the following representation :", "index": 5}, {"type": "interline_equation", "bbox": [258, 317, 352, 332], "content": "", "index": 6}, {"type": "interline_equation", "bbox": [221, 377, 389, 435], "content": "", "index": 7}, {"type": "text", "bbox": [124, 448, 488, 490], "content": "for , where is the identity matrix, and , . These representations are non-equivalent for different values of .", "index": 8}, {"type": "text", "bbox": [124, 497, 486, 540], "content": "Proof. By Theorem 4.4 the friendship graph of is a chain. Then, by theorem 5.1, is equivalent to a standard representation for some . By Lemmas 5.3 and 5.4 .", "index": 9}, {"type": "text", "bbox": [124, 540, 486, 569], "content": "Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following", "index": 10}, {"type": "text", "bbox": [124, 574, 486, 603], "content": "Corollary 5.6. Let be an irreducible representation of for . Let .", "index": 11}, {"type": "text", "bbox": [124, 604, 486, 631], "content": "Then is equivalent to a specialization of the standard representation , for some .", "index": 12}, {"type": "title", "bbox": [270, 643, 342, 657], "content": "References", "index": 13}, {"type": "text", "bbox": [128, 662, 486, 701], "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82, Princeton Univ.Press,Princeton,N.J.,1974. [2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "index": 14}]
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[{"bbox": [137, 113, 485, 126], "content": "Proof. We need to prove that starting from any non-zero vector", "parent_index": 0, "line_index": 0}, {"bbox": [126, 127, 484, 140], "content": ", we can generate the whole space. Obviously, it is enough", "parent_index": 0, "line_index": 1}, {"bbox": [125, 141, 486, 154], "content": "to show that we can generate one of the standard basis vectors . To", "parent_index": 0, "line_index": 2}, {"bbox": [125, 155, 405, 168], "content": "do this, take such that . Consider the operator", "parent_index": 0, "line_index": 3}, {"bbox": [126, 194, 486, 208], "content": "where and . By a direct calculation", "parent_index": 2, "line_index": 0}, {"bbox": [126, 207, 484, 222], "content": ". Because the vector is a non-zero multiple of .", "parent_index": 2, "line_index": 1}, {"bbox": [137, 235, 363, 250], "content": "Now, we have the main result of this paper:", "parent_index": 3, "line_index": 0}, {"bbox": [125, 257, 486, 272], "content": "Theorem 5.5 (The Main Theorem). Let be an ir-", "parent_index": 4, "line_index": 0}, {"bbox": [126, 271, 487, 285], "content": "reducible representation of for . Let , and let", "parent_index": 4, "line_index": 1}, {"bbox": [126, 285, 263, 299], "content": "with .", "parent_index": 4, "line_index": 2}, {"bbox": [140, 299, 465, 313], "content": "Then and is equivalent to the following representation :", "parent_index": 5, "line_index": 0}, {"bbox": [125, 451, 484, 464], "content": "for , where is the identity matrix, and ,", "parent_index": 8, "line_index": 0}, {"bbox": [126, 465, 487, 479], "content": ". These representations are non-equivalent for different values of", "parent_index": 8, "line_index": 1}, {"bbox": [126, 481, 137, 492], "content": ".", "parent_index": 8, "line_index": 2}, {"bbox": [137, 500, 485, 514], "content": "Proof. By Theorem 4.4 the friendship graph of is a chain. Then,", "parent_index": 9, "line_index": 0}, {"bbox": [126, 514, 485, 528], "content": "by theorem 5.1, is equivalent to a standard representation for", "parent_index": 9, "line_index": 1}, {"bbox": [126, 528, 355, 542], "content": "some . By Lemmas 5.3 and 5.4 .", "parent_index": 9, "line_index": 2}, {"bbox": [137, 542, 486, 555], "content": "Combining Theorem 5.5 and the classification theorem of Formanek", "parent_index": 10, "line_index": 0}, {"bbox": [126, 554, 346, 571], "content": "(see [3], Theorem 23), we get the following", "parent_index": 10, "line_index": 1}, {"bbox": [126, 577, 487, 592], "content": "Corollary 5.6. Let be an irreducible representation", "parent_index": 11, "line_index": 0}, {"bbox": [127, 591, 310, 605], "content": "of for . Let .", "parent_index": 11, "line_index": 1}, {"bbox": [139, 605, 487, 619], "content": "Then is equivalent to a specialization of the standard representation", "parent_index": 12, "line_index": 0}, {"bbox": [126, 619, 287, 632], "content": ", for some .", "parent_index": 12, "line_index": 1}, {"bbox": [270, 645, 342, 658], "content": "References", "parent_index": 13, "line_index": 0}, {"bbox": [131, 665, 486, 678], "content": "[1] J.S. Birman, Braids, Links, and Mapping Class Groups, Ann.of Math.Stud.,82,", "parent_index": 14, "line_index": 0}, {"bbox": [146, 678, 332, 689], "content": "Princeton Univ.Press,Princeton,N.J.,1974.", "parent_index": 14, "line_index": 1}, {"bbox": [131, 690, 484, 701], "content": "[2] W.-L. Chow, On the algebraical braid group, Ann. of Math. 49 (1948), 654-658.", "parent_index": 14, "line_index": 2}]
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[{"bbox": [126, 128, 181, 140], "content": "x=\\textstyle\\sum a_{i}e_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [454, 146, 462, 153], "content": "e_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [194, 157, 198, 165], "content": "i", "parent_index": 0, "subtype": "inline"}, {"bbox": [253, 156, 285, 167], "content": "a_{i}\\neq0", "parent_index": 0, "subtype": "inline"}, {"bbox": [203, 173, 407, 186], "content": "H=A+A^{2}+A B A=B+B^{2}+B A B,", "parent_index": 1, "subtype": "interline"}, {"bbox": [161, 195, 227, 207], "content": "A\\,=\\,\\rho(\\sigma_{i-1})", "parent_index": 2, "subtype": "inline"}, {"bbox": [257, 195, 313, 208], "content": "B\\;=\\;\\rho(\\sigma_{i})", "parent_index": 2, "subtype": "inline"}, {"bbox": [452, 195, 486, 207], "content": "H x\\;=", "parent_index": 2, "subtype": "inline"}, {"bbox": [126, 209, 181, 221], "content": "(u-1)a_{i}e_{i}", "parent_index": 2, "subtype": "inline"}, {"bbox": [234, 209, 262, 221], "content": "u\\ne1", "parent_index": 2, "subtype": "inline"}, {"bbox": [321, 209, 339, 218], "content": "H x", "parent_index": 2, "subtype": "inline"}, {"bbox": [470, 212, 479, 220], "content": "e_{i}", "parent_index": 2, "subtype": "inline"}, {"bbox": [343, 258, 435, 270], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 4, "subtype": "inline"}, {"bbox": [266, 271, 281, 284], "content": "B_{n}", "parent_index": 4, "subtype": "inline"}, {"bbox": [304, 271, 337, 284], "content": "n\\,\\geq\\,6", "parent_index": 4, "subtype": "inline"}, {"bbox": [364, 272, 396, 284], "content": "r\\,\\geq\\,n", "parent_index": 4, "subtype": "inline"}, {"bbox": [444, 272, 487, 285], "content": "\\rho(\\sigma_{1})\\,=", "parent_index": 4, "subtype": "inline"}, {"bbox": [126, 287, 160, 297], "content": "1+A_{1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [190, 286, 259, 299], "content": "r a n k(A_{1})=2", "parent_index": 4, "subtype": "inline"}, {"bbox": [168, 304, 197, 309], "content": "r=n", "parent_index": 5, "subtype": "inline"}, {"bbox": [223, 303, 230, 312], "content": "\\rho", "parent_index": 5, "subtype": "inline"}, {"bbox": [258, 317, 352, 332], "content": "\\tau:B_{n}\\to G L_{n}(\\mathbb{C}),", "parent_index": 6, "subtype": "interline"}, {"bbox": [221, 377, 389, 435], "content": "\\rho(\\sigma_{i})=\\left(\\begin{array}{c c c c}{I_{i-1}}&{}&{}&{}\\\\ {}&{0}&{u}&{}\\\\ {}&{1}&{0}&{}\\\\ {}&{}&{}&{I_{n-1-i}}\\end{array}\\right),", "parent_index": 7, "subtype": "interline"}, {"bbox": [144, 452, 231, 464], "content": "i=1,2,\\dots,n-1", "parent_index": 8, "subtype": "inline"}, {"bbox": [270, 452, 281, 463], "content": "I_{k}", "parent_index": 8, "subtype": "inline"}, {"bbox": [315, 452, 339, 462], "content": "k\\times k", "parent_index": 8, "subtype": "inline"}, {"bbox": [447, 452, 482, 462], "content": "u\\in\\mathbb{C}^{*}", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 466, 154, 478], "content": "u\\ne1", "parent_index": 8, "subtype": "inline"}, {"bbox": [126, 483, 133, 489], "content": "u", "parent_index": 8, "subtype": "inline"}, {"bbox": [388, 505, 394, 513], "content": "\\rho", "parent_index": 9, "subtype": "inline"}, {"bbox": [212, 519, 218, 527], "content": "\\rho", "parent_index": 9, "subtype": "inline"}, {"bbox": [444, 515, 466, 528], "content": "\\tau(u)", "parent_index": 9, "subtype": "inline"}, {"bbox": [155, 530, 190, 539], "content": "u\\in\\mathbb{C}^{*}", "parent_index": 9, "subtype": "inline"}, {"bbox": [322, 529, 351, 541], "content": "u\\ne1", "parent_index": 9, "subtype": "inline"}, {"bbox": [232, 578, 321, 590], "content": "\\rho:B_{n}\\to G L_{r}(\\mathbb{C})", "parent_index": 11, "subtype": "inline"}, {"bbox": [139, 591, 155, 604], "content": "B_{n}", "parent_index": 11, "subtype": "inline"}, {"bbox": [177, 591, 207, 603], "content": "n\\ge7", "parent_index": 11, "subtype": "inline"}, {"bbox": [233, 592, 307, 605], "content": "c o r a n k(\\rho)=2", "parent_index": 11, "subtype": "inline"}, {"bbox": [167, 607, 174, 618], "content": "\\rho", "parent_index": 12, "subtype": "inline"}, {"bbox": [126, 619, 153, 632], "content": "\\tau_{n}(u)", "parent_index": 12, "subtype": "inline"}, {"bbox": [208, 619, 283, 632], "content": "u\\neq1,\\ u\\in\\mathbb{C}^{*}", "parent_index": 12, "subtype": "inline"}]
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[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London Math.Soc. 73 (1996), 279-322.
[4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math.126 (1987), 335-388.
[5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis, 1999.
[6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483–486.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected]
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<p data-bbox="130 112 487 208">[3] E. Formanek, Braid Group Representations of Low Degree, Proc.London Math.Soc. 73 (1996), 279-322.
[4] V.F.R.Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math.126 (1987), 335-388.
[5] I. Sysoeva, On the irreducible representations of braid groups, Ph.D. thesis, 1999.
[6] Dian-Min Tong, Shan-De Yang, Zhong-Qi Ma, A new class of representations of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483–486. </p>
<p data-bbox="125 218 484 255">Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected] </p>
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# Stratification of the Generalized Gauge Orbit Space
Christian Fleischhack∗
Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany
Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany
January 5, 2000
# Abstract
The action of Ashtekar’s generalized gauge group $\overline{{\mathcal{G}}}$ on the space $\overline{{\mathcal{A}}}$ of generalized connections is investigated for compact structure groups $\mathbf{G}$ .
First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on $\overline{{\mathcal{A}}}$ . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, $\overline{{\mathcal{A}}}$ is topologically regularly stratified by $\overline{{\mathcal{G}}}$ . These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of $\mathbf{G}$ . Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1.
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<h1 data-bbox="83 55 516 79">Stratification of the Generalized Gauge Orbit Space </h1>
<p data-bbox="232 97 371 113">Christian Fleischhack∗ </p>
<p data-bbox="133 129 486 189">Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany </p>
<p data-bbox="135 203 466 248">Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany </p>
<p data-bbox="250 259 350 276">January 5, 2000 </p>
<h1 data-bbox="275 318 324 332">Abstract </h1>
<p data-bbox="92 339 507 366">The action of Ashtekar’s generalized gauge group $\overline{{\mathcal{G}}}$ on the space $\overline{{\mathcal{A}}}$ of generalized connections is investigated for compact structure groups $\mathbf{G}$ . </p>
<p data-bbox="92 367 508 474">First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on $\overline{{\mathcal{A}}}$ . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, $\overline{{\mathcal{A}}}$ is topologically regularly stratified by $\overline{{\mathcal{G}}}$ . These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of $\mathbf{G}$ . Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1. </p>
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Hence, ", "type": "text"}, {"bbox": [296, 409, 305, 419], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [306, 409, 496, 423], "score": 1.0, "content": " is topologically regularly stratified by ", "type": "text"}, {"bbox": [496, 409, 504, 420], "score": 0.9, "content": "\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [504, 409, 507, 423], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [92, 423, 507, 435], "spans": [{"bbox": [92, 423, 507, 435], "score": 1.0, "content": "These results coincide with those of Kondracki and Rogulski for Sobolev connections.", "type": "text"}], "index": 17}, {"bbox": [91, 435, 509, 450], "spans": [{"bbox": [91, 435, 509, 450], "score": 1.0, "content": "As a by-product, we prove that the set of all gauge orbit types equals the set of all", "type": "text"}], "index": 18}, {"bbox": [92, 450, 508, 464], "spans": [{"bbox": [92, 450, 292, 464], "score": 1.0, "content": "(conjugacy classes of) Howe subgroups of ", "type": "text"}, {"bbox": [293, 452, 303, 460], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [303, 450, 508, 464], "score": 1.0, "content": ". Finally, we show that the set of all gauge", "type": "text"}], "index": 19}, {"bbox": [92, 464, 395, 476], "spans": [{"bbox": [92, 464, 395, 476], "score": 1.0, "content": "orbits with maximal type has the full induced Haar measure 1.", "type": "text"}], "index": 20}], "index": 16.5, "bbox_fs": [91, 368, 509, 476]}]}
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[{"type": "title", "bbox": [83, 55, 516, 79], "content": "Stratification of the Generalized Gauge Orbit Space", "index": 0}, {"type": "text", "bbox": [232, 97, 371, 113], "content": "Christian Fleischhack∗", "index": 1}, {"type": "text", "bbox": [133, 129, 486, 189], "content": "Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany", "index": 2}, {"type": "text", "bbox": [135, 203, 466, 248], "content": "Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany", "index": 3}, {"type": "text", "bbox": [250, 259, 350, 276], "content": "January 5, 2000", "index": 4}, {"type": "title", "bbox": [275, 318, 324, 332], "content": "Abstract", "index": 5}, {"type": "text", "bbox": [92, 339, 507, 366], "content": "The action of Ashtekar’s generalized gauge group on the space of generalized connections is investigated for compact structure groups .", "index": 6}, {"type": "text", "bbox": [92, 367, 508, 474], "content": "First a stratum is defined to be the set of all connections of one and the same gauge orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice theorem is proven on . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, is topologically regularly stratified by . These results coincide with those of Kondracki and Rogulski for Sobolev connections. As a by-product, we prove that the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of . Finally, we show that the set of all gauge orbits with maximal type has the full induced Haar measure 1.", "index": 7}]
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[{"bbox": [85, 59, 514, 80], "content": "Stratification of the Generalized Gauge Orbit Space", "parent_index": 0, "line_index": 0}, {"bbox": [234, 100, 370, 114], "content": "Christian Fleischhack∗", "parent_index": 1, "line_index": 0}, {"bbox": [135, 131, 486, 146], "content": "Mathematisches Institut Institut fir Theoretische Physik", "parent_index": 2, "line_index": 0}, {"bbox": [147, 145, 453, 162], "content": "Universitat Leipzig Universitat Leipzig", "parent_index": 2, "line_index": 1}, {"bbox": [144, 160, 457, 176], "content": "Augustusplatz 10/11 Augustusplatz 10/11", "parent_index": 2, "line_index": 2}, {"bbox": [134, 176, 467, 191], "content": "04109 Leipzig, Germany 04109 Leipzig, Germany", "parent_index": 2, "line_index": 3}, {"bbox": [135, 206, 465, 219], "content": "Max-Planck-Institut fir Mathematik in den Naturwissenschaften", "parent_index": 3, "line_index": 0}, {"bbox": [255, 220, 346, 235], "content": "Inselstraße 22-26", "parent_index": 3, "line_index": 1}, {"bbox": [238, 236, 363, 249], "content": "04103 Leipzig, Germany", "parent_index": 3, "line_index": 2}, {"bbox": [250, 262, 348, 276], "content": "January 5, 2000", "parent_index": 4, "line_index": 0}, {"bbox": [275, 320, 325, 333], "content": "Abstract", "parent_index": 5, "line_index": 0}, {"bbox": [108, 340, 507, 355], "content": "The action of Ashtekar’s generalized gauge group on the space of generalized", "parent_index": 6, "line_index": 0}, {"bbox": [92, 355, 379, 369], "content": "connections is investigated for compact structure groups .", "parent_index": 6, "line_index": 1}, {"bbox": [108, 368, 507, 381], "content": "First a stratum is defined to be the set of all connections of one and the same gauge", "parent_index": 7, "line_index": 0}, {"bbox": [92, 381, 507, 396], "content": "orbit type, i.e. the conjugacy class of the centralizer of the holonomy group. Then a slice", "parent_index": 7, "line_index": 1}, {"bbox": [92, 396, 508, 408], "content": "theorem is proven on . This yields the openness of the strata. Afterwards, a denseness", "parent_index": 7, "line_index": 2}, {"bbox": [92, 409, 507, 423], "content": "theorem is proven for the strata. Hence, is topologically regularly stratified by .", "parent_index": 7, "line_index": 3}, {"bbox": [92, 423, 507, 435], "content": "These results coincide with those of Kondracki and Rogulski for Sobolev connections.", "parent_index": 7, "line_index": 4}, {"bbox": [91, 435, 509, 450], "content": "As a by-product, we prove that the set of all gauge orbit types equals the set of all", "parent_index": 7, "line_index": 5}, {"bbox": [92, 450, 508, 464], "content": "(conjugacy classes of) Howe subgroups of . Finally, we show that the set of all gauge", "parent_index": 7, "line_index": 6}, {"bbox": [92, 464, 395, 476], "content": "orbits with maximal type has the full induced Haar measure 1.", "parent_index": 7, "line_index": 7}]
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[]
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# 1 Introduction
For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set $\mathcal{A}$ of smooth connections (”gauge fields”) in a principle fiber bundle, then the set $\mathcal{G}$ of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of $\mathcal{G}$ on $\mathcal{A}$ . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in $\scriptstyle A/\mathcal G$ , is the configuration space for the gauge theory. Unfortunately, in contrast to $\overline{{\mathcal{A}}}$ , which is an affine space, the space $\scriptstyle A/\mathcal G$ has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $\mathcal{A}/\mathcal{G}$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $\scriptstyle A/\mathcal G$ . However, up to now, results are quite rare.
About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $\mathcal{G}$ is a Hilbert-Lie group and acts smoothly on $\mathcal{A}$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $\mathcal{A}$ . This means, for every orbit $A\circ\mathcal{G}\subseteq A$ there is an equivariant retraction from a (so-called tubular) neighborhood of $A$ onto $A\circ{\mathcal{G}}$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $\mathcal{G}$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $\mathcal{A}$ is regularly stratified by the action of $\mathcal{G}$ . In particular, all the strata are smooth submanifolds of $\mathcal{A}$ .
Despite these results the mathematically rigorous construction of a measure on $\scriptstyle A/\mathcal{G}$ has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for $\mathcal{A}/\mathcal{G}$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $\mathbf{G}$ to the paths in the base manifold $M$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $\mathcal{P}$ of paths to the structure group $\mathbf{G}$ . Analogously, the set $\overline{{g}}$ of generalized gauge transforms collects all functions from $M$ to $\mathbf{G}$ . Now the action of $\mathcal{G}$ to $\overline{{\mathcal{A}}}$ is defined purely algebraically. Given $\overline{{\mathcal{A}}}$ and $\mathcal{G}$ the topologies induced by the topology of $\mathbf{G}$ , one sees that, for compact $\mathbf{G}$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $\overline{{\mathcal{A}}}$ and $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ , the new configuration space for the path integral quantization.
Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $\mathcal{A}$ and $\mathcal{G}$ are dense subsets in $\overline{{\mathcal{A}}}$ and $\overline{{g}}$ , respectively [17]. Furthermore, $\mathcal{A}$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the twodimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski.
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<html><body>
<h1 data-bbox="63 10 200 29">1 Introduction </h1>
<p data-bbox="63 41 538 228">For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set $\mathcal{A}$ of smooth connections (”gauge fields”) in a principle fiber bundle, then the set $\mathcal{G}$ of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of $\mathcal{G}$ on $\mathcal{A}$ . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in $\scriptstyle A/\mathcal G$ , is the configuration space for the gauge theory. Unfortunately, in contrast to $\overline{{\mathcal{A}}}$ , which is an affine space, the space $\scriptstyle A/\mathcal G$ has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $\mathcal{A}/\mathcal{G}$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $\scriptstyle A/\mathcal G$ . However, up to now, results are quite rare. </p>
<p data-bbox="63 229 538 388">About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $\mathcal{G}$ is a Hilbert-Lie group and acts smoothly on $\mathcal{A}$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $\mathcal{A}$ . This means, for every orbit $A\circ\mathcal{G}\subseteq A$ there is an equivariant retraction from a (so-called tubular) neighborhood of $A$ onto $A\circ{\mathcal{G}}$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $\mathcal{G}$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $\mathcal{A}$ is regularly stratified by the action of $\mathcal{G}$ . In particular, all the strata are smooth submanifolds of $\mathcal{A}$ . </p>
<p data-bbox="63 388 537 605">Despite these results the mathematically rigorous construction of a measure on $\scriptstyle A/\mathcal{G}$ has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for $\mathcal{A}/\mathcal{G}$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $\mathbf{G}$ to the paths in the base manifold $M$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $\mathcal{P}$ of paths to the structure group $\mathbf{G}$ . Analogously, the set $\overline{{g}}$ of generalized gauge transforms collects all functions from $M$ to $\mathbf{G}$ . Now the action of $\mathcal{G}$ to $\overline{{\mathcal{A}}}$ is defined purely algebraically. Given $\overline{{\mathcal{A}}}$ and $\mathcal{G}$ the topologies induced by the topology of $\mathbf{G}$ , one sees that, for compact $\mathbf{G}$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $\overline{{\mathcal{A}}}$ and $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ , the new configuration space for the path integral quantization. </p>
<p data-bbox="64 605 537 677">Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $\mathcal{A}$ and $\mathcal{G}$ are dense subsets in $\overline{{\mathcal{A}}}$ and $\overline{{g}}$ , respectively [17]. Furthermore, $\mathcal{A}$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the twodimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski. </p>
</body></html>
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0001008v1
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[{"type": "text", "text": "1 Introduction ", "text_level": 1, "page_idx": 1}, {"type": "text", "text": "For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set $\\mathcal{A}$ of smooth connections (”gauge fields”) in a principle fiber bundle, then the set $\\mathcal{G}$ of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of $\\mathcal{G}$ on $\\mathcal{A}$ . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in $\\scriptstyle A/\\mathcal G$ , is the configuration space for the gauge theory. Unfortunately, in contrast to $\\overline{{\\mathcal{A}}}$ , which is an affine space, the space $\\scriptstyle A/\\mathcal G$ has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on $\\mathcal{A}/\\mathcal{G}$ . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of $\\scriptstyle A/\\mathcal G$ . However, up to now, results are quite rare. ", "page_idx": 1}, {"type": "text", "text": "About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, $\\mathcal{G}$ is a Hilbert-Lie group and acts smoothly on $\\mathcal{A}$ . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on $\\mathcal{A}$ . This means, for every orbit $A\\circ\\mathcal{G}\\subseteq A$ there is an equivariant retraction from a (so-called tubular) neighborhood of $A$ onto $A\\circ{\\mathcal{G}}$ . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of $\\mathcal{G}$ . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space $\\mathcal{A}$ is regularly stratified by the action of $\\mathcal{G}$ . In particular, all the strata are smooth submanifolds of $\\mathcal{A}$ . ", "page_idx": 1}, {"type": "text", "text": "Despite these results the mathematically rigorous construction of a measure on $\\scriptstyle A/\\mathcal{G}$ has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for $\\mathcal{A}/\\mathcal{G}$ itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group $\\mathbf{G}$ to the paths in the base manifold $M$ such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid $\\mathcal{P}$ of paths to the structure group $\\mathbf{G}$ . Analogously, the set $\\overline{{g}}$ of generalized gauge transforms collects all functions from $M$ to $\\mathbf{G}$ . Now the action of $\\mathcal{G}$ to $\\overline{{\\mathcal{A}}}$ is defined purely algebraically. Given $\\overline{{\\mathcal{A}}}$ and $\\mathcal{G}$ the topologies induced by the topology of $\\mathbf{G}$ , one sees that, for compact $\\mathbf{G}$ , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on $\\overline{{\\mathcal{A}}}$ and $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ , the new configuration space for the path integral quantization. ", "page_idx": 1}, {"type": "text", "text": "Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that $\\mathcal{A}$ and $\\mathcal{G}$ are dense subsets in $\\overline{{\\mathcal{A}}}$ and $\\overline{{g}}$ , respectively [17]. Furthermore, $\\mathcal{A}$ is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the twodimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski. ", "page_idx": 1}]
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{"preproc_blocks": [{"type": "title", "bbox": [63, 10, 200, 29], "lines": [{"bbox": [63, 13, 199, 29], "spans": [{"bbox": [63, 15, 73, 28], "score": 1.0, "content": "1", "type": "text"}, {"bbox": [90, 13, 199, 29], "score": 1.0, "content": "Introduction", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [63, 41, 538, 228], "lines": [{"bbox": [62, 43, 537, 58], "spans": [{"bbox": [62, 43, 537, 58], "score": 1.0, "content": "For quite a long time the geometric structure of gauge theories has been investigated. A", "type": "text"}], "index": 1}, {"bbox": [62, 58, 538, 73], "spans": [{"bbox": [62, 58, 445, 73], "score": 1.0, "content": "classical (pure) gauge theory consists of three basic objects: First the set ", "type": "text"}, {"bbox": [445, 60, 455, 68], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [455, 58, 538, 73], "score": 1.0, "content": " of smooth con-", "type": "text"}], "index": 2}, {"bbox": [62, 71, 537, 87], "spans": [{"bbox": [62, 71, 416, 87], "score": 1.0, "content": "nections (”gauge fields”) in a principle fiber bundle, then the set ", "type": "text"}, {"bbox": [417, 74, 425, 84], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [425, 71, 537, 87], "score": 1.0, "content": " of all smooth gauge", "type": "text"}], "index": 3}, {"bbox": [62, 87, 536, 101], "spans": [{"bbox": [62, 87, 434, 101], "score": 1.0, "content": "transforms, i.e. automorphisms of this bundle, and finally the action of ", "type": "text"}, {"bbox": [434, 88, 442, 98], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 87, 462, 101], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [463, 88, 473, 97], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [473, 87, 536, 101], "score": 1.0, "content": ". Physically,", "type": "text"}], "index": 4}, {"bbox": [62, 101, 538, 116], "spans": [{"bbox": [62, 101, 538, 116], "score": 1.0, "content": "two gauge fields that are related by a gauge transform describe one and the same situation.", "type": "text"}], "index": 5}, {"bbox": [63, 116, 537, 129], "spans": [{"bbox": [63, 116, 334, 129], "score": 1.0, "content": "Thus, the space of all gauge orbits, i.e. elements in ", "type": "text"}, {"bbox": [334, 117, 358, 129], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [358, 116, 537, 129], "score": 1.0, "content": ", is the configuration space for the", "type": "text"}], "index": 6}, {"bbox": [61, 130, 538, 145], "spans": [{"bbox": [61, 130, 291, 145], "score": 1.0, "content": "gauge theory. Unfortunately, in contrast to ", "type": "text"}, {"bbox": [292, 130, 302, 141], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [302, 130, 491, 145], "score": 1.0, "content": ", which is an affine space, the space ", "type": "text"}, {"bbox": [491, 131, 515, 144], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [515, 130, 538, 145], "score": 1.0, "content": " has", "type": "text"}], "index": 7}, {"bbox": [62, 145, 538, 159], "spans": [{"bbox": [62, 145, 538, 159], "score": 1.0, "content": "a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is", "type": "text"}], "index": 8}, {"bbox": [61, 158, 538, 174], "spans": [{"bbox": [61, 158, 538, 174], "score": 1.0, "content": "not a manifold. This causes enormous problems, in particular, when one wants to quantize a", "type": "text"}], "index": 9}, {"bbox": [61, 173, 538, 188], "spans": [{"bbox": [61, 173, 538, 188], "score": 1.0, "content": "gauge theory. One possible quantization method is the path integral quantization. Here one", "type": "text"}], "index": 10}, {"bbox": [62, 188, 538, 203], "spans": [{"bbox": [62, 188, 538, 203], "score": 1.0, "content": "has to find an appropriate measure on the configuration space of the classical theory, hence", "type": "text"}], "index": 11}, {"bbox": [61, 202, 538, 217], "spans": [{"bbox": [61, 202, 136, 217], "score": 1.0, "content": "a measure on ", "type": "text"}, {"bbox": [136, 204, 159, 216], "score": 0.95, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [160, 202, 538, 217], "score": 1.0, "content": ". As just indicated, this is very hard to find. Thus, one has hoped for a", "type": "text"}], "index": 12}, {"bbox": [62, 216, 523, 231], "spans": [{"bbox": [62, 216, 271, 231], "score": 1.0, "content": "better understanding of the structure of ", "type": "text"}, {"bbox": [272, 218, 295, 230], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [295, 216, 523, 231], "score": 1.0, "content": ". However, up to now, results are quite rare.", "type": "text"}], "index": 13}], "index": 7}, {"type": "text", "bbox": [63, 229, 538, 388], "lines": [{"bbox": [63, 231, 539, 246], "spans": [{"bbox": [63, 231, 539, 246], "score": 1.0, "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of", "type": "text"}], "index": 14}, {"bbox": [61, 244, 538, 263], "spans": [{"bbox": [61, 244, 538, 263], "score": 1.0, "content": "connections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,", "type": "text"}], "index": 15}, {"bbox": [63, 261, 537, 275], "spans": [{"bbox": [63, 261, 123, 275], "score": 1.0, "content": "[16]). Now, ", "type": "text"}, {"bbox": [124, 262, 131, 272], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [132, 261, 360, 275], "score": 1.0, "content": " is a Hilbert-Lie group and acts smoothly on ", "type": "text"}, {"bbox": [360, 262, 370, 271], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [370, 261, 537, 275], "score": 1.0, "content": ". About 15 years ago, Kondracki", "type": "text"}], "index": 16}, {"bbox": [63, 275, 538, 289], "spans": [{"bbox": [63, 275, 538, 289], "score": 1.0, "content": "and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most", "type": "text"}], "index": 17}, {"bbox": [62, 289, 537, 303], "spans": [{"bbox": [62, 289, 371, 303], "score": 1.0, "content": "remarkable theorem they obtained was a slice theorem on ", "type": "text"}, {"bbox": [371, 291, 381, 299], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [382, 289, 537, 303], "score": 1.0, "content": ". This means, for every orbit", "type": "text"}], "index": 18}, {"bbox": [63, 303, 538, 319], "spans": [{"bbox": [63, 305, 113, 316], "score": 0.93, "content": "A\\circ\\mathcal{G}\\subseteq A", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [113, 303, 501, 319], "score": 1.0, "content": " there is an equivariant retraction from a (so-called tubular) neighborhood of ", "type": "text"}, {"bbox": [501, 305, 510, 314], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [511, 303, 538, 319], "score": 1.0, "content": " onto", "type": "text"}], "index": 19}, {"bbox": [63, 318, 537, 332], "spans": [{"bbox": [63, 320, 91, 330], "score": 0.92, "content": "A\\circ{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [91, 318, 537, 332], "score": 1.0, "content": ". Using this theorem they could clarify the structure of the so-called strata. A stratum", "type": "text"}], "index": 20}, {"bbox": [61, 332, 538, 348], "spans": [{"bbox": [61, 332, 538, 348], "score": 1.0, "content": "contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)", "type": "text"}], "index": 21}, {"bbox": [62, 347, 538, 361], "spans": [{"bbox": [62, 347, 218, 361], "score": 1.0, "content": "stabilizer under the action of ", "type": "text"}, {"bbox": [219, 349, 227, 358], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [227, 347, 538, 361], "score": 1.0, "content": ". Using a denseness theorem for the strata, Kondracki and", "type": "text"}], "index": 22}, {"bbox": [61, 360, 538, 376], "spans": [{"bbox": [61, 360, 226, 376], "score": 1.0, "content": "Rogulski proved that the space ", "type": "text"}, {"bbox": [226, 363, 236, 372], "score": 0.91, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 360, 435, 376], "score": 1.0, "content": " is regularly stratified by the action of ", "type": "text"}, {"bbox": [435, 363, 443, 373], "score": 0.92, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 360, 538, 376], "score": 1.0, "content": ". In particular, all", "type": "text"}], "index": 23}, {"bbox": [63, 376, 276, 390], "spans": [{"bbox": [63, 376, 262, 390], "score": 1.0, "content": "the strata are smooth submanifolds of ", "type": "text"}, {"bbox": [262, 378, 272, 386], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [272, 376, 276, 390], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 19}, {"type": "text", "bbox": [63, 388, 537, 605], "lines": [{"bbox": [63, 390, 537, 405], "spans": [{"bbox": [63, 390, 471, 405], "score": 1.0, "content": "Despite these results the mathematically rigorous construction of a measure on ", "type": "text"}, {"bbox": [471, 391, 495, 403], "score": 0.94, "content": "\\scriptstyle A/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [495, 390, 537, 405], "score": 1.0, "content": " has not", "type": "text"}], "index": 25}, {"bbox": [63, 405, 536, 418], "spans": [{"bbox": [63, 405, 536, 418], "score": 1.0, "content": "been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2],", "type": "text"}], "index": 26}, {"bbox": [61, 419, 537, 433], "spans": [{"bbox": [61, 419, 173, 433], "score": 1.0, "content": "but, however, not for ", "type": "text"}, {"bbox": [173, 420, 196, 433], "score": 0.94, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [197, 419, 537, 433], "score": 1.0, "content": " itself. Their idea was to drop simply all smoothness conditions for", "type": "text"}], "index": 27}, {"bbox": [62, 434, 537, 448], "spans": [{"bbox": [62, 434, 537, 448], "score": 1.0, "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "type": "text"}], "index": 28}, {"bbox": [63, 449, 537, 462], "spans": [{"bbox": [63, 449, 537, 462], "score": 1.0, "content": "can always be reconstructed uniquely by its parallel transports. On the other hand, these", "type": "text"}], "index": 29}, {"bbox": [62, 462, 537, 477], "spans": [{"bbox": [62, 462, 537, 477], "score": 1.0, "content": "parallel transports can be identified with an assignment of elements of the structure group", "type": "text"}], "index": 30}, {"bbox": [63, 476, 538, 491], "spans": [{"bbox": [63, 479, 74, 487], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [74, 476, 252, 491], "score": 1.0, "content": " to the paths in the base manifold ", "type": "text"}, {"bbox": [253, 479, 265, 487], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [266, 476, 538, 491], "score": 1.0, "content": " such that the concatenation of paths corresponds to", "type": "text"}], "index": 31}, {"bbox": [63, 491, 538, 506], "spans": [{"bbox": [63, 491, 538, 506], "score": 1.0, "content": "the product of these group elements. It is intuitively clear that for smooth connections the", "type": "text"}], "index": 32}, {"bbox": [63, 507, 537, 520], "spans": [{"bbox": [63, 507, 537, 520], "score": 1.0, "content": "parallel transports additionally depend smoothly on the paths [14]. But now this restriction", "type": "text"}], "index": 33}, {"bbox": [62, 520, 536, 533], "spans": [{"bbox": [62, 520, 536, 533], "score": 1.0, "content": "is removed for the generalized connections. They are only homomorphisms from the groupoid", "type": "text"}], "index": 34}, {"bbox": [63, 534, 537, 549], "spans": [{"bbox": [63, 537, 72, 545], "score": 0.91, "content": "\\mathcal{P}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [73, 534, 237, 549], "score": 1.0, "content": " of paths to the structure group ", "type": "text"}, {"bbox": [238, 536, 249, 545], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [249, 534, 363, 549], "score": 1.0, "content": ". Analogously, the set ", "type": "text"}, {"bbox": [363, 534, 371, 546], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [372, 534, 537, 549], "score": 1.0, "content": " of generalized gauge transforms", "type": "text"}], "index": 35}, {"bbox": [62, 548, 537, 564], "spans": [{"bbox": [62, 548, 198, 564], "score": 1.0, "content": "collects all functions from ", "type": "text"}, {"bbox": [198, 551, 211, 559], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [211, 548, 228, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [228, 551, 239, 560], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [240, 548, 342, 564], "score": 1.0, "content": ". Now the action of ", "type": "text"}, {"bbox": [342, 549, 350, 561], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [351, 548, 368, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [368, 549, 378, 560], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [378, 548, 537, 564], "score": 1.0, "content": " is defined purely algebraically.", "type": "text"}], "index": 36}, {"bbox": [62, 563, 537, 578], "spans": [{"bbox": [62, 563, 97, 578], "score": 1.0, "content": "Given ", "type": "text"}, {"bbox": [97, 564, 107, 574], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [107, 563, 133, 578], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [134, 564, 142, 575], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [142, 563, 364, 578], "score": 1.0, "content": " the topologies induced by the topology of ", "type": "text"}, {"bbox": [364, 565, 375, 574], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [375, 563, 522, 578], "score": 1.0, "content": ", one sees that, for compact ", "type": "text"}, {"bbox": [522, 565, 533, 574], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [533, 563, 537, 578], "score": 1.0, "content": ",", "type": "text"}], "index": 37}, {"bbox": [62, 578, 537, 592], "spans": [{"bbox": [62, 578, 537, 592], "score": 1.0, "content": "these spaces are again compact. This guarantees the existence of a natural induced Haar", "type": "text"}], "index": 38}, {"bbox": [62, 591, 513, 608], "spans": [{"bbox": [62, 591, 124, 608], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [125, 592, 135, 603], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [135, 591, 161, 608], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [161, 592, 185, 606], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [185, 591, 513, 608], "score": 1.0, "content": ", the new configuration space for the path integral quantization.", "type": "text"}], "index": 39}], "index": 32}, {"type": "text", "bbox": [64, 605, 537, 677], "lines": [{"bbox": [62, 606, 537, 622], "spans": [{"bbox": [62, 606, 537, 622], "score": 1.0, "content": "Both from the mathematical and from the physical point of view it is very interesting how the", "type": "text"}], "index": 40}, {"bbox": [61, 619, 537, 637], "spans": [{"bbox": [61, 619, 537, 637], "score": 1.0, "content": "”classical” regular gauge theories are related to the generalized formulation in the Ashtekar", "type": "text"}], "index": 41}, {"bbox": [61, 634, 537, 650], "spans": [{"bbox": [61, 634, 324, 650], "score": 1.0, "content": "framework. First of all, it has been proven that ", "type": "text"}, {"bbox": [324, 637, 334, 646], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [334, 634, 362, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 637, 370, 647], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [371, 634, 486, 650], "score": 1.0, "content": " are dense subsets in ", "type": "text"}, {"bbox": [486, 636, 496, 646], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [496, 634, 524, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [524, 636, 533, 647], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [533, 634, 537, 650], "score": 1.0, "content": ",", "type": "text"}], "index": 42}, {"bbox": [63, 650, 536, 664], "spans": [{"bbox": [63, 650, 228, 664], "score": 1.0, "content": "respectively [17]. Furthermore, ", "type": "text"}, {"bbox": [229, 652, 239, 660], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [239, 650, 536, 664], "score": 1.0, "content": " is contained in a set of induced Haar measure zero [15].", "type": "text"}], "index": 43}, {"bbox": [63, 664, 538, 680], "spans": [{"bbox": [63, 664, 538, 680], "score": 1.0, "content": "These properties coincide exactly with the experiences known from the Wiener or Feynman", "type": "text"}], "index": 44}], "index": 42}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [295, 704, 303, 715], "lines": [{"bbox": [295, 705, 304, 718], "spans": [{"bbox": [295, 705, 304, 718], "score": 1.0, "content": "2", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [63, 10, 200, 29], "lines": [{"bbox": [63, 13, 199, 29], "spans": [{"bbox": [63, 15, 73, 28], "score": 1.0, "content": "1", "type": "text"}, {"bbox": [90, 13, 199, 29], "score": 1.0, "content": "Introduction", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [63, 41, 538, 228], "lines": [{"bbox": [62, 43, 537, 58], "spans": [{"bbox": [62, 43, 537, 58], "score": 1.0, "content": "For quite a long time the geometric structure of gauge theories has been investigated. A", "type": "text"}], "index": 1}, {"bbox": [62, 58, 538, 73], "spans": [{"bbox": [62, 58, 445, 73], "score": 1.0, "content": "classical (pure) gauge theory consists of three basic objects: First the set ", "type": "text"}, {"bbox": [445, 60, 455, 68], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [455, 58, 538, 73], "score": 1.0, "content": " of smooth con-", "type": "text"}], "index": 2}, {"bbox": [62, 71, 537, 87], "spans": [{"bbox": [62, 71, 416, 87], "score": 1.0, "content": "nections (”gauge fields”) in a principle fiber bundle, then the set ", "type": "text"}, {"bbox": [417, 74, 425, 84], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [425, 71, 537, 87], "score": 1.0, "content": " of all smooth gauge", "type": "text"}], "index": 3}, {"bbox": [62, 87, 536, 101], "spans": [{"bbox": [62, 87, 434, 101], "score": 1.0, "content": "transforms, i.e. automorphisms of this bundle, and finally the action of ", "type": "text"}, {"bbox": [434, 88, 442, 98], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 87, 462, 101], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [463, 88, 473, 97], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [473, 87, 536, 101], "score": 1.0, "content": ". Physically,", "type": "text"}], "index": 4}, {"bbox": [62, 101, 538, 116], "spans": [{"bbox": [62, 101, 538, 116], "score": 1.0, "content": "two gauge fields that are related by a gauge transform describe one and the same situation.", "type": "text"}], "index": 5}, {"bbox": [63, 116, 537, 129], "spans": [{"bbox": [63, 116, 334, 129], "score": 1.0, "content": "Thus, the space of all gauge orbits, i.e. elements in ", "type": "text"}, {"bbox": [334, 117, 358, 129], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [358, 116, 537, 129], "score": 1.0, "content": ", is the configuration space for the", "type": "text"}], "index": 6}, {"bbox": [61, 130, 538, 145], "spans": [{"bbox": [61, 130, 291, 145], "score": 1.0, "content": "gauge theory. Unfortunately, in contrast to ", "type": "text"}, {"bbox": [292, 130, 302, 141], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [302, 130, 491, 145], "score": 1.0, "content": ", which is an affine space, the space ", "type": "text"}, {"bbox": [491, 131, 515, 144], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [515, 130, 538, 145], "score": 1.0, "content": " has", "type": "text"}], "index": 7}, {"bbox": [62, 145, 538, 159], "spans": [{"bbox": [62, 145, 538, 159], "score": 1.0, "content": "a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is", "type": "text"}], "index": 8}, {"bbox": [61, 158, 538, 174], "spans": [{"bbox": [61, 158, 538, 174], "score": 1.0, "content": "not a manifold. This causes enormous problems, in particular, when one wants to quantize a", "type": "text"}], "index": 9}, {"bbox": [61, 173, 538, 188], "spans": [{"bbox": [61, 173, 538, 188], "score": 1.0, "content": "gauge theory. One possible quantization method is the path integral quantization. Here one", "type": "text"}], "index": 10}, {"bbox": [62, 188, 538, 203], "spans": [{"bbox": [62, 188, 538, 203], "score": 1.0, "content": "has to find an appropriate measure on the configuration space of the classical theory, hence", "type": "text"}], "index": 11}, {"bbox": [61, 202, 538, 217], "spans": [{"bbox": [61, 202, 136, 217], "score": 1.0, "content": "a measure on ", "type": "text"}, {"bbox": [136, 204, 159, 216], "score": 0.95, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [160, 202, 538, 217], "score": 1.0, "content": ". As just indicated, this is very hard to find. Thus, one has hoped for a", "type": "text"}], "index": 12}, {"bbox": [62, 216, 523, 231], "spans": [{"bbox": [62, 216, 271, 231], "score": 1.0, "content": "better understanding of the structure of ", "type": "text"}, {"bbox": [272, 218, 295, 230], "score": 0.94, "content": "\\scriptstyle A/\\mathcal G", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [295, 216, 523, 231], "score": 1.0, "content": ". However, up to now, results are quite rare.", "type": "text"}], "index": 13}], "index": 7, "bbox_fs": [61, 43, 538, 231]}, {"type": "text", "bbox": [63, 229, 538, 388], "lines": [{"bbox": [63, 231, 539, 246], "spans": [{"bbox": [63, 231, 539, 246], "score": 1.0, "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of", "type": "text"}], "index": 14}, {"bbox": [61, 244, 538, 263], "spans": [{"bbox": [61, 244, 538, 263], "score": 1.0, "content": "connections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,", "type": "text"}], "index": 15}, {"bbox": [63, 261, 537, 275], "spans": [{"bbox": [63, 261, 123, 275], "score": 1.0, "content": "[16]). Now, ", "type": "text"}, {"bbox": [124, 262, 131, 272], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 7}, {"bbox": [132, 261, 360, 275], "score": 1.0, "content": " is a Hilbert-Lie group and acts smoothly on ", "type": "text"}, {"bbox": [360, 262, 370, 271], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [370, 261, 537, 275], "score": 1.0, "content": ". About 15 years ago, Kondracki", "type": "text"}], "index": 16}, {"bbox": [63, 275, 538, 289], "spans": [{"bbox": [63, 275, 538, 289], "score": 1.0, "content": "and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most", "type": "text"}], "index": 17}, {"bbox": [62, 289, 537, 303], "spans": [{"bbox": [62, 289, 371, 303], "score": 1.0, "content": "remarkable theorem they obtained was a slice theorem on ", "type": "text"}, {"bbox": [371, 291, 381, 299], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [382, 289, 537, 303], "score": 1.0, "content": ". This means, for every orbit", "type": "text"}], "index": 18}, {"bbox": [63, 303, 538, 319], "spans": [{"bbox": [63, 305, 113, 316], "score": 0.93, "content": "A\\circ\\mathcal{G}\\subseteq A", "type": "inline_equation", "height": 11, "width": 50}, {"bbox": [113, 303, 501, 319], "score": 1.0, "content": " there is an equivariant retraction from a (so-called tubular) neighborhood of ", "type": "text"}, {"bbox": [501, 305, 510, 314], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [511, 303, 538, 319], "score": 1.0, "content": " onto", "type": "text"}], "index": 19}, {"bbox": [63, 318, 537, 332], "spans": [{"bbox": [63, 320, 91, 330], "score": 0.92, "content": "A\\circ{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 28}, {"bbox": [91, 318, 537, 332], "score": 1.0, "content": ". Using this theorem they could clarify the structure of the so-called strata. A stratum", "type": "text"}], "index": 20}, {"bbox": [61, 332, 538, 348], "spans": [{"bbox": [61, 332, 538, 348], "score": 1.0, "content": "contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)", "type": "text"}], "index": 21}, {"bbox": [62, 347, 538, 361], "spans": [{"bbox": [62, 347, 218, 361], "score": 1.0, "content": "stabilizer under the action of ", "type": "text"}, {"bbox": [219, 349, 227, 358], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [227, 347, 538, 361], "score": 1.0, "content": ". Using a denseness theorem for the strata, Kondracki and", "type": "text"}], "index": 22}, {"bbox": [61, 360, 538, 376], "spans": [{"bbox": [61, 360, 226, 376], "score": 1.0, "content": "Rogulski proved that the space ", "type": "text"}, {"bbox": [226, 363, 236, 372], "score": 0.91, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 360, 435, 376], "score": 1.0, "content": " is regularly stratified by the action of ", "type": "text"}, {"bbox": [435, 363, 443, 373], "score": 0.92, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [443, 360, 538, 376], "score": 1.0, "content": ". In particular, all", "type": "text"}], "index": 23}, {"bbox": [63, 376, 276, 390], "spans": [{"bbox": [63, 376, 262, 390], "score": 1.0, "content": "the strata are smooth submanifolds of ", "type": "text"}, {"bbox": [262, 378, 272, 386], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [272, 376, 276, 390], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 19, "bbox_fs": [61, 231, 539, 390]}, {"type": "text", "bbox": [63, 388, 537, 605], "lines": [{"bbox": [63, 390, 537, 405], "spans": [{"bbox": [63, 390, 471, 405], "score": 1.0, "content": "Despite these results the mathematically rigorous construction of a measure on ", "type": "text"}, {"bbox": [471, 391, 495, 403], "score": 0.94, "content": "\\scriptstyle A/\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [495, 390, 537, 405], "score": 1.0, "content": " has not", "type": "text"}], "index": 25}, {"bbox": [63, 405, 536, 418], "spans": [{"bbox": [63, 405, 536, 418], "score": 1.0, "content": "been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2],", "type": "text"}], "index": 26}, {"bbox": [61, 419, 537, 433], "spans": [{"bbox": [61, 419, 173, 433], "score": 1.0, "content": "but, however, not for ", "type": "text"}, {"bbox": [173, 420, 196, 433], "score": 0.94, "content": "\\mathcal{A}/\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [197, 419, 537, 433], "score": 1.0, "content": " itself. Their idea was to drop simply all smoothness conditions for", "type": "text"}], "index": 27}, {"bbox": [62, 434, 537, 448], "spans": [{"bbox": [62, 434, 537, 448], "score": 1.0, "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "type": "text"}], "index": 28}, {"bbox": [63, 449, 537, 462], "spans": [{"bbox": [63, 449, 537, 462], "score": 1.0, "content": "can always be reconstructed uniquely by its parallel transports. On the other hand, these", "type": "text"}], "index": 29}, {"bbox": [62, 462, 537, 477], "spans": [{"bbox": [62, 462, 537, 477], "score": 1.0, "content": "parallel transports can be identified with an assignment of elements of the structure group", "type": "text"}], "index": 30}, {"bbox": [63, 476, 538, 491], "spans": [{"bbox": [63, 479, 74, 487], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [74, 476, 252, 491], "score": 1.0, "content": " to the paths in the base manifold ", "type": "text"}, {"bbox": [253, 479, 265, 487], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [266, 476, 538, 491], "score": 1.0, "content": " such that the concatenation of paths corresponds to", "type": "text"}], "index": 31}, {"bbox": [63, 491, 538, 506], "spans": [{"bbox": [63, 491, 538, 506], "score": 1.0, "content": "the product of these group elements. It is intuitively clear that for smooth connections the", "type": "text"}], "index": 32}, {"bbox": [63, 507, 537, 520], "spans": [{"bbox": [63, 507, 537, 520], "score": 1.0, "content": "parallel transports additionally depend smoothly on the paths [14]. But now this restriction", "type": "text"}], "index": 33}, {"bbox": [62, 520, 536, 533], "spans": [{"bbox": [62, 520, 536, 533], "score": 1.0, "content": "is removed for the generalized connections. They are only homomorphisms from the groupoid", "type": "text"}], "index": 34}, {"bbox": [63, 534, 537, 549], "spans": [{"bbox": [63, 537, 72, 545], "score": 0.91, "content": "\\mathcal{P}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [73, 534, 237, 549], "score": 1.0, "content": " of paths to the structure group ", "type": "text"}, {"bbox": [238, 536, 249, 545], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [249, 534, 363, 549], "score": 1.0, "content": ". Analogously, the set ", "type": "text"}, {"bbox": [363, 534, 371, 546], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [372, 534, 537, 549], "score": 1.0, "content": " of generalized gauge transforms", "type": "text"}], "index": 35}, {"bbox": [62, 548, 537, 564], "spans": [{"bbox": [62, 548, 198, 564], "score": 1.0, "content": "collects all functions from ", "type": "text"}, {"bbox": [198, 551, 211, 559], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [211, 548, 228, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [228, 551, 239, 560], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [240, 548, 342, 564], "score": 1.0, "content": ". Now the action of ", "type": "text"}, {"bbox": [342, 549, 350, 561], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [351, 548, 368, 564], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [368, 549, 378, 560], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [378, 548, 537, 564], "score": 1.0, "content": " is defined purely algebraically.", "type": "text"}], "index": 36}, {"bbox": [62, 563, 537, 578], "spans": [{"bbox": [62, 563, 97, 578], "score": 1.0, "content": "Given ", "type": "text"}, {"bbox": [97, 564, 107, 574], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [107, 563, 133, 578], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [134, 564, 142, 575], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [142, 563, 364, 578], "score": 1.0, "content": " the topologies induced by the topology of ", "type": "text"}, {"bbox": [364, 565, 375, 574], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [375, 563, 522, 578], "score": 1.0, "content": ", one sees that, for compact ", "type": "text"}, {"bbox": [522, 565, 533, 574], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [533, 563, 537, 578], "score": 1.0, "content": ",", "type": "text"}], "index": 37}, {"bbox": [62, 578, 537, 592], "spans": [{"bbox": [62, 578, 537, 592], "score": 1.0, "content": "these spaces are again compact. This guarantees the existence of a natural induced Haar", "type": "text"}], "index": 38}, {"bbox": [62, 591, 513, 608], "spans": [{"bbox": [62, 591, 124, 608], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [125, 592, 135, 603], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [135, 591, 161, 608], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [161, 592, 185, 606], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [185, 591, 513, 608], "score": 1.0, "content": ", the new configuration space for the path integral quantization.", "type": "text"}], "index": 39}], "index": 32, "bbox_fs": [61, 390, 538, 608]}, {"type": "text", "bbox": [64, 605, 537, 677], "lines": [{"bbox": [62, 606, 537, 622], "spans": [{"bbox": [62, 606, 537, 622], "score": 1.0, "content": "Both from the mathematical and from the physical point of view it is very interesting how the", "type": "text"}], "index": 40}, {"bbox": [61, 619, 537, 637], "spans": [{"bbox": [61, 619, 537, 637], "score": 1.0, "content": "”classical” regular gauge theories are related to the generalized formulation in the Ashtekar", "type": "text"}], "index": 41}, {"bbox": [61, 634, 537, 650], "spans": [{"bbox": [61, 634, 324, 650], "score": 1.0, "content": "framework. First of all, it has been proven that ", "type": "text"}, {"bbox": [324, 637, 334, 646], "score": 0.9, "content": "\\mathcal{A}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [334, 634, 362, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [362, 637, 370, 647], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [371, 634, 486, 650], "score": 1.0, "content": " are dense subsets in ", "type": "text"}, {"bbox": [486, 636, 496, 646], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [496, 634, 524, 650], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [524, 636, 533, 647], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [533, 634, 537, 650], "score": 1.0, "content": ",", "type": "text"}], "index": 42}, {"bbox": [63, 650, 536, 664], "spans": [{"bbox": [63, 650, 228, 664], "score": 1.0, "content": "respectively [17]. Furthermore, ", "type": "text"}, {"bbox": [229, 652, 239, 660], "score": 0.89, "content": "\\mathcal{A}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [239, 650, 536, 664], "score": 1.0, "content": " is contained in a set of induced Haar measure zero [15].", "type": "text"}], "index": 43}, {"bbox": [63, 664, 538, 680], "spans": [{"bbox": [63, 664, 538, 680], "score": 1.0, "content": "These properties coincide exactly with the experiences known from the Wiener or Feynman", "type": "text"}], "index": 44}, {"bbox": [63, 18, 536, 31], "spans": [{"bbox": [63, 18, 536, 31], "score": 1.0, "content": "path integral. Then the Wilson loop expectation values have been determined for the two-", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [63, 32, 537, 45], "spans": [{"bbox": [63, 32, 537, 45], "score": 1.0, "content": "dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [61, 45, 537, 60], "spans": [{"bbox": [61, 45, 537, 60], "score": 1.0, "content": "standard framework. In the present paper we continue the investigations on how the results", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [62, 59, 537, 75], "spans": [{"bbox": [62, 59, 537, 75], "score": 1.0, "content": "of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [63, 75, 537, 90], "spans": [{"bbox": [63, 75, 537, 90], "score": 1.0, "content": "[9] we have already shown that the gauge orbit type is determined by the centralizer of the", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [61, 88, 538, 106], "spans": [{"bbox": [61, 88, 538, 106], "score": 1.0, "content": "holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [61, 104, 537, 119], "spans": [{"bbox": [61, 104, 537, 119], "score": 1.0, "content": "the present paper we are going to prove that there is a slice theorem and a denseness theorem", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [63, 119, 536, 132], "spans": [{"bbox": [63, 119, 536, 132], "score": 1.0, "content": "for the space of connections in the Ashtekar framework as well. However, our methods are", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [64, 133, 352, 146], "spans": [{"bbox": [64, 133, 352, 146], "score": 1.0, "content": "completely different to those of Kondracki and Rogulski.", "type": "text", "cross_page": true}], "index": 8}], "index": 42, "bbox_fs": [61, 606, 538, 680]}]}
|
[{"type": "title", "bbox": [63, 10, 200, 29], "content": "1 Introduction", "index": 0}, {"type": "text", "bbox": [63, 41, 538, 228], "content": "For quite a long time the geometric structure of gauge theories has been investigated. A classical (pure) gauge theory consists of three basic objects: First the set of smooth con- nections (”gauge fields”) in a principle fiber bundle, then the set of all smooth gauge transforms, i.e. automorphisms of this bundle, and finally the action of on . Physically, two gauge fields that are related by a gauge transform describe one and the same situation. Thus, the space of all gauge orbits, i.e. elements in , is the configuration space for the gauge theory. Unfortunately, in contrast to , which is an affine space, the space has a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is not a manifold. This causes enormous problems, in particular, when one wants to quantize a gauge theory. One possible quantization method is the path integral quantization. Here one has to find an appropriate measure on the configuration space of the classical theory, hence a measure on . As just indicated, this is very hard to find. Thus, one has hoped for a better understanding of the structure of . However, up to now, results are quite rare.", "index": 1}, {"type": "text", "bbox": [63, 229, 538, 388], "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of connections and gauge transforms that are contained in a certain Sobolev class (see, e.g., [16]). Now, is a Hilbert-Lie group and acts smoothly on . About 15 years ago, Kondracki and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most remarkable theorem they obtained was a slice theorem on . This means, for every orbit there is an equivariant retraction from a (so-called tubular) neighborhood of onto . Using this theorem they could clarify the structure of the so-called strata. A stratum contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the) stabilizer under the action of . Using a denseness theorem for the strata, Kondracki and Rogulski proved that the space is regularly stratified by the action of . In particular, all the strata are smooth submanifolds of .", "index": 2}, {"type": "text", "bbox": [63, 388, 537, 605], "content": "Despite these results the mathematically rigorous construction of a measure on has not been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2], but, however, not for itself. Their idea was to drop simply all smoothness conditions for the connections and gauge transforms. In detail, they first used the fact that a connection can always be reconstructed uniquely by its parallel transports. On the other hand, these parallel transports can be identified with an assignment of elements of the structure group to the paths in the base manifold such that the concatenation of paths corresponds to the product of these group elements. It is intuitively clear that for smooth connections the parallel transports additionally depend smoothly on the paths [14]. But now this restriction is removed for the generalized connections. They are only homomorphisms from the groupoid of paths to the structure group . Analogously, the set of generalized gauge transforms collects all functions from to . Now the action of to is defined purely algebraically. Given and the topologies induced by the topology of , one sees that, for compact , these spaces are again compact. This guarantees the existence of a natural induced Haar measure on and , the new configuration space for the path integral quantization.", "index": 3}, {"type": "text", "bbox": [64, 605, 537, 677], "content": "Both from the mathematical and from the physical point of view it is very interesting how the ”classical” regular gauge theories are related to the generalized formulation in the Ashtekar framework. First of all, it has been proven that and are dense subsets in and , respectively [17]. Furthermore, is contained in a set of induced Haar measure zero [15]. These properties coincide exactly with the experiences known from the Wiener or Feynman path integral. Then the Wilson loop expectation values have been determined for the two- dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the standard framework. In the present paper we continue the investigations on how the results of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper [9] we have already shown that the gauge orbit type is determined by the centralizer of the holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In the present paper we are going to prove that there is a slice theorem and a denseness theorem for the space of connections in the Ashtekar framework as well. However, our methods are completely different to those of Kondracki and Rogulski.", "index": 4}]
|
[{"bbox": [63, 13, 199, 29], "content": "1 Introduction", "parent_index": 0, "line_index": 0}, {"bbox": [62, 43, 537, 58], "content": "For quite a long time the geometric structure of gauge theories has been investigated. A", "parent_index": 1, "line_index": 0}, {"bbox": [62, 58, 538, 73], "content": "classical (pure) gauge theory consists of three basic objects: First the set of smooth con-", "parent_index": 1, "line_index": 1}, {"bbox": [62, 71, 537, 87], "content": "nections (”gauge fields”) in a principle fiber bundle, then the set of all smooth gauge", "parent_index": 1, "line_index": 2}, {"bbox": [62, 87, 536, 101], "content": "transforms, i.e. automorphisms of this bundle, and finally the action of on . Physically,", "parent_index": 1, "line_index": 3}, {"bbox": [62, 101, 538, 116], "content": "two gauge fields that are related by a gauge transform describe one and the same situation.", "parent_index": 1, "line_index": 4}, {"bbox": [63, 116, 537, 129], "content": "Thus, the space of all gauge orbits, i.e. elements in , is the configuration space for the", "parent_index": 1, "line_index": 5}, {"bbox": [61, 130, 538, 145], "content": "gauge theory. Unfortunately, in contrast to , which is an affine space, the space has", "parent_index": 1, "line_index": 6}, {"bbox": [62, 145, 538, 159], "content": "a very complicated structure: It is non-affin, non-compact and infinite-dimensional and it is", "parent_index": 1, "line_index": 7}, {"bbox": [61, 158, 538, 174], "content": "not a manifold. This causes enormous problems, in particular, when one wants to quantize a", "parent_index": 1, "line_index": 8}, {"bbox": [61, 173, 538, 188], "content": "gauge theory. One possible quantization method is the path integral quantization. Here one", "parent_index": 1, "line_index": 9}, {"bbox": [62, 188, 538, 203], "content": "has to find an appropriate measure on the configuration space of the classical theory, hence", "parent_index": 1, "line_index": 10}, {"bbox": [61, 202, 538, 217], "content": "a measure on . As just indicated, this is very hard to find. Thus, one has hoped for a", "parent_index": 1, "line_index": 11}, {"bbox": [62, 216, 523, 231], "content": "better understanding of the structure of . However, up to now, results are quite rare.", "parent_index": 1, "line_index": 12}, {"bbox": [63, 231, 539, 246], "content": "About 20 years ago, the efforts were focussed on a related problem: The consideration of", "parent_index": 2, "line_index": 0}, {"bbox": [61, 244, 538, 263], "content": "connections and gauge transforms that are contained in a certain Sobolev class (see, e.g.,", "parent_index": 2, "line_index": 1}, {"bbox": [63, 261, 537, 275], "content": "[16]). Now, is a Hilbert-Lie group and acts smoothly on . About 15 years ago, Kondracki", "parent_index": 2, "line_index": 2}, {"bbox": [63, 275, 538, 289], "content": "and Rogulski [12] found lots of fundamental properties of this action. Perhaps, the most", "parent_index": 2, "line_index": 3}, {"bbox": [62, 289, 537, 303], "content": "remarkable theorem they obtained was a slice theorem on . This means, for every orbit", "parent_index": 2, "line_index": 4}, {"bbox": [63, 303, 538, 319], "content": "there is an equivariant retraction from a (so-called tubular) neighborhood of onto", "parent_index": 2, "line_index": 5}, {"bbox": [63, 318, 537, 332], "content": ". Using this theorem they could clarify the structure of the so-called strata. A stratum", "parent_index": 2, "line_index": 6}, {"bbox": [61, 332, 538, 348], "content": "contains all connections that have the same, fixed type, i.e. the same (conjugacy class of the)", "parent_index": 2, "line_index": 7}, {"bbox": [62, 347, 538, 361], "content": "stabilizer under the action of . Using a denseness theorem for the strata, Kondracki and", "parent_index": 2, "line_index": 8}, {"bbox": [61, 360, 538, 376], "content": "Rogulski proved that the space is regularly stratified by the action of . In particular, all", "parent_index": 2, "line_index": 9}, {"bbox": [63, 376, 276, 390], "content": "the strata are smooth submanifolds of .", "parent_index": 2, "line_index": 10}, {"bbox": [63, 390, 537, 405], "content": "Despite these results the mathematically rigorous construction of a measure on has not", "parent_index": 3, "line_index": 0}, {"bbox": [63, 405, 536, 418], "content": "been achieved. This problem was solved – at least preliminary – by Ashtekar et al. [1, 2],", "parent_index": 3, "line_index": 1}, {"bbox": [61, 419, 537, 433], "content": "but, however, not for itself. Their idea was to drop simply all smoothness conditions for", "parent_index": 3, "line_index": 2}, {"bbox": [62, 434, 537, 448], "content": "the connections and gauge transforms. In detail, they first used the fact that a connection", "parent_index": 3, "line_index": 3}, {"bbox": [63, 449, 537, 462], "content": "can always be reconstructed uniquely by its parallel transports. On the other hand, these", "parent_index": 3, "line_index": 4}, {"bbox": [62, 462, 537, 477], "content": "parallel transports can be identified with an assignment of elements of the structure group", "parent_index": 3, "line_index": 5}, {"bbox": [63, 476, 538, 491], "content": "to the paths in the base manifold such that the concatenation of paths corresponds to", "parent_index": 3, "line_index": 6}, {"bbox": [63, 491, 538, 506], "content": "the product of these group elements. It is intuitively clear that for smooth connections the", "parent_index": 3, "line_index": 7}, {"bbox": [63, 507, 537, 520], "content": "parallel transports additionally depend smoothly on the paths [14]. But now this restriction", "parent_index": 3, "line_index": 8}, {"bbox": [62, 520, 536, 533], "content": "is removed for the generalized connections. They are only homomorphisms from the groupoid", "parent_index": 3, "line_index": 9}, {"bbox": [63, 534, 537, 549], "content": "of paths to the structure group . Analogously, the set of generalized gauge transforms", "parent_index": 3, "line_index": 10}, {"bbox": [62, 548, 537, 564], "content": "collects all functions from to . Now the action of to is defined purely algebraically.", "parent_index": 3, "line_index": 11}, {"bbox": [62, 563, 537, 578], "content": "Given and the topologies induced by the topology of , one sees that, for compact ,", "parent_index": 3, "line_index": 12}, {"bbox": [62, 578, 537, 592], "content": "these spaces are again compact. This guarantees the existence of a natural induced Haar", "parent_index": 3, "line_index": 13}, {"bbox": [62, 591, 513, 608], "content": "measure on and , the new configuration space for the path integral quantization.", "parent_index": 3, "line_index": 14}, {"bbox": [62, 606, 537, 622], "content": "Both from the mathematical and from the physical point of view it is very interesting how the", "parent_index": 4, "line_index": 0}, {"bbox": [61, 619, 537, 637], "content": "”classical” regular gauge theories are related to the generalized formulation in the Ashtekar", "parent_index": 4, "line_index": 1}, {"bbox": [61, 634, 537, 650], "content": "framework. First of all, it has been proven that and are dense subsets in and ,", "parent_index": 4, "line_index": 2}, {"bbox": [63, 650, 536, 664], "content": "respectively [17]. Furthermore, is contained in a set of induced Haar measure zero [15].", "parent_index": 4, "line_index": 3}, {"bbox": [63, 664, 538, 680], "content": "These properties coincide exactly with the experiences known from the Wiener or Feynman", "parent_index": 4, "line_index": 4}, {"bbox": [63, 18, 536, 31], "content": "path integral. Then the Wilson loop expectation values have been determined for the two-", "parent_index": 4, "line_index": 5}, {"bbox": [63, 32, 537, 45], "content": "dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the", "parent_index": 4, "line_index": 6}, {"bbox": [61, 45, 537, 60], "content": "standard framework. In the present paper we continue the investigations on how the results", "parent_index": 4, "line_index": 7}, {"bbox": [62, 59, 537, 75], "content": "of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper", "parent_index": 4, "line_index": 8}, {"bbox": [63, 75, 537, 90], "content": "[9] we have already shown that the gauge orbit type is determined by the centralizer of the", "parent_index": 4, "line_index": 9}, {"bbox": [61, 88, 538, 106], "content": "holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In", "parent_index": 4, "line_index": 10}, {"bbox": [61, 104, 537, 119], "content": "the present paper we are going to prove that there is a slice theorem and a denseness theorem", "parent_index": 4, "line_index": 11}, {"bbox": [63, 119, 536, 132], "content": "for the space of connections in the Ashtekar framework as well. However, our methods are", "parent_index": 4, "line_index": 12}, {"bbox": [64, 133, 352, 146], "content": "completely different to those of Kondracki and Rogulski.", "parent_index": 4, "line_index": 13}]
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The outline of the paper is as follows:
After fixing the notations we prove a very crucial lemma in section 4: Every centralizer in a compact Lie group is finitely generated. This implies that every orbit type (being the centralizer of the holonomy group) is determined by a finite set of holonomies of the corresponding connection.
Using the projection onto these holonomies we can lift the slice theorem from an appropriate finite-dimensional $\mathbf{G}^{n}$ to the space $\overline{{\mathcal{A}}}$ . This is proven in section 5 and it implies the openness of the strata as shown in the following section.
Afterwards, we prove a denseness theorem for the strata. For this we need a construction for new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals the set of all conjugacy classes of Howe subgroups of $\mathbf{G}$ . A Howe subgroup is a subgroup that is the centralizer of some subset of $\mathbf{G}$ . This way we completely determine all possible gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our knowlegde – only for ${\mathbf{G}}=S U(n)$ and low-dimensional $M$ [18].
In Section 8 we show that the slice and the denseness theorem yield again a topologically regular stratification of $\overline{{\mathcal{A}}}$ as well as of $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ . But, in contrast to the Sobolev case, the strata are not proved to be manifolds.
Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal type) is not only dense in $\overline{{\mathcal{A}}}$ , but has also the total induced Haar measure 1. This shows that the Faddeev-Popov determinant for the projection $\overline{{A}}\longrightarrow\overline{{A}}/\overline{{\mathcal{G}}}$ is equal to 1.
# 2 Preliminaries
As we indicated in [9] the present paper is the final one in a small series of three papers. In the first one [9] we extended the definitions and propositions for $\overline{{\mathcal{A}}}$ , $\overline{{g}}$ and $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ made by Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the second paper [10] we investigated properties of $\overline{{\mathcal{A}}}$ and proved, in particular, the existence of an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important notations, definitions and facts used in the following. For detailed information we refer the reader to the preceding papers [9, 10].
• Let $\mathbf{G}$ be a compact Lie group. • A path (usually denoted by $\gamma$ or $\delta$ ) is a piecewise $C^{r}$ -map from $[0,1]$ into a connected $C^{r}$ -manifold $M$ , $\dim M\geq2$ , $r\in\mathbb{N}^{+}\cup\{\infty\}\cup\{\omega\}$ arbitrary, but fixed. Additionally, we fix now the decision whether we restrict the paths to be piecewise immersive or not. Paths can be multiplied as usual by concatenation. A graph is a finite union of paths, such that different paths intersect each other at most in their end points. Paths in a graph are called simple. A path is called finite iff it is up to the parametrization a finite product of simple paths. Two paths are equivalent iff the first one can be reconstructed from the second one by a sequence of reparametrizations or of insertions or deletions of retracings. We will only consider equivalence classes of finite paths and graphs. The set of (classes of) paths is denoted by $\mathcal{P}$ , that of paths from $x$ to $y$ by $\mathcal{P}_{x y}$ and that of loops (paths with a fixed initial and terminal point $m$ ) by $\mathcal{H G}$ , the so-called hoop group.
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<html><body>
<p data-bbox="63 155 257 169">The outline of the paper is as follows: </p>
<p data-bbox="63 171 537 227">After fixing the notations we prove a very crucial lemma in section 4: Every centralizer in a compact Lie group is finitely generated. This implies that every orbit type (being the centralizer of the holonomy group) is determined by a finite set of holonomies of the corresponding connection. </p>
<p data-bbox="63 228 537 270">Using the projection onto these holonomies we can lift the slice theorem from an appropriate finite-dimensional $\mathbf{G}^{n}$ to the space $\overline{{\mathcal{A}}}$ . This is proven in section 5 and it implies the openness of the strata as shown in the following section. </p>
<p data-bbox="63 271 537 357">Afterwards, we prove a denseness theorem for the strata. For this we need a construction for new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals the set of all conjugacy classes of Howe subgroups of $\mathbf{G}$ . A Howe subgroup is a subgroup that is the centralizer of some subset of $\mathbf{G}$ . This way we completely determine all possible gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our knowlegde – only for ${\mathbf{G}}=S U(n)$ and low-dimensional $M$ [18]. </p>
<p data-bbox="63 358 537 400">In Section 8 we show that the slice and the denseness theorem yield again a topologically regular stratification of $\overline{{\mathcal{A}}}$ as well as of $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ . But, in contrast to the Sobolev case, the strata are not proved to be manifolds. </p>
<p data-bbox="63 401 537 444">Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal type) is not only dense in $\overline{{\mathcal{A}}}$ , but has also the total induced Haar measure 1. This shows that the Faddeev-Popov determinant for the projection $\overline{{A}}\longrightarrow\overline{{A}}/\overline{{\mathcal{G}}}$ is equal to 1. </p>
<h1 data-bbox="62 465 206 485">2 Preliminaries </h1>
<p data-bbox="63 496 538 612">As we indicated in [9] the present paper is the final one in a small series of three papers. In the first one [9] we extended the definitions and propositions for $\overline{{\mathcal{A}}}$ , $\overline{{g}}$ and $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ made by Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the second paper [10] we investigated properties of $\overline{{\mathcal{A}}}$ and proved, in particular, the existence of an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important notations, definitions and facts used in the following. For detailed information we refer the reader to the preceding papers [9, 10]. </p>
<p data-bbox="63 612 538 684">• Let $\mathbf{G}$ be a compact Lie group. • A path (usually denoted by $\gamma$ or $\delta$ ) is a piecewise $C^{r}$ -map from $[0,1]$ into a connected $C^{r}$ -manifold $M$ , $\dim M\geq2$ , $r\in\mathbb{N}^{+}\cup\{\infty\}\cup\{\omega\}$ arbitrary, but fixed. Additionally, we fix now the decision whether we restrict the paths to be piecewise immersive or not. Paths can be multiplied as usual by concatenation. A graph is a finite union of paths, such that different paths intersect each other at most in their end points. Paths in a graph are called simple. A path is called finite iff it is up to the parametrization a finite product of simple paths. Two paths are equivalent iff the first one can be reconstructed from the second one by a sequence of reparametrizations or of insertions or deletions of retracings. We will only consider equivalence classes of finite paths and graphs. The set of (classes of) paths is denoted by $\mathcal{P}$ , that of paths from $x$ to $y$ by $\mathcal{P}_{x y}$ and that of loops (paths with a fixed initial and terminal point $m$ ) by $\mathcal{H G}$ , the so-called hoop group. </p>
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[{"type": "text", "text": "", "page_idx": 2}, {"type": "text", "text": "The outline of the paper is as follows: ", "page_idx": 2}, {"type": "text", "text": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer in a compact Lie group is finitely generated. This implies that every orbit type (being the centralizer of the holonomy group) is determined by a finite set of holonomies of the corresponding connection. ", "page_idx": 2}, {"type": "text", "text": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate finite-dimensional $\\mathbf{G}^{n}$ to the space $\\overline{{\\mathcal{A}}}$ . This is proven in section 5 and it implies the openness of the strata as shown in the following section. ", "page_idx": 2}, {"type": "text", "text": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals the set of all conjugacy classes of Howe subgroups of $\\mathbf{G}$ . A Howe subgroup is a subgroup that is the centralizer of some subset of $\\mathbf{G}$ . This way we completely determine all possible gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our knowlegde – only for ${\\mathbf{G}}=S U(n)$ and low-dimensional $M$ [18]. ", "page_idx": 2}, {"type": "text", "text": "In Section 8 we show that the slice and the denseness theorem yield again a topologically regular stratification of $\\overline{{\\mathcal{A}}}$ as well as of $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ . But, in contrast to the Sobolev case, the strata are not proved to be manifolds. ", "page_idx": 2}, {"type": "text", "text": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal type) is not only dense in $\\overline{{\\mathcal{A}}}$ , but has also the total induced Haar measure 1. This shows that the Faddeev-Popov determinant for the projection $\\overline{{A}}\\longrightarrow\\overline{{A}}/\\overline{{\\mathcal{G}}}$ is equal to 1. ", "page_idx": 2}, {"type": "text", "text": "2 Preliminaries ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "As we indicated in [9] the present paper is the final one in a small series of three papers. In the first one [9] we extended the definitions and propositions for $\\overline{{\\mathcal{A}}}$ , $\\overline{{g}}$ and $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ made by Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the second paper [10] we investigated properties of $\\overline{{\\mathcal{A}}}$ and proved, in particular, the existence of an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important notations, definitions and facts used in the following. For detailed information we refer the reader to the preceding papers [9, 10]. ", "page_idx": 2}, {"type": "text", "text": "• Let $\\mathbf{G}$ be a compact Lie group. • A path (usually denoted by $\\gamma$ or $\\delta$ ) is a piecewise $C^{r}$ -map from $[0,1]$ into a connected $C^{r}$ -manifold $M$ , $\\dim M\\geq2$ , $r\\in\\mathbb{N}^{+}\\cup\\{\\infty\\}\\cup\\{\\omega\\}$ arbitrary, but fixed. Additionally, we fix now the decision whether we restrict the paths to be piecewise immersive or not. Paths can be multiplied as usual by concatenation. A graph is a finite union of paths, such that different paths intersect each other at most in their end points. Paths in a graph are called simple. A path is called finite iff it is up to the parametrization a finite product of simple paths. Two paths are equivalent iff the first one can be reconstructed from the second one by a sequence of reparametrizations or of insertions or deletions of retracings. We will only consider equivalence classes of finite paths and graphs. The set of (classes of) paths is denoted by $\\mathcal{P}$ , that of paths from $x$ to $y$ by $\\mathcal{P}_{x y}$ and that of loops (paths with a fixed initial and terminal point $m$ ) by $\\mathcal{H G}$ , the so-called hoop group. ", "page_idx": 2}]
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{"preproc_blocks": [{"type": "text", "bbox": [63, 15, 538, 145], "lines": [{"bbox": [63, 18, 536, 31], "spans": [{"bbox": [63, 18, 536, 31], "score": 1.0, "content": "path integral. Then the Wilson loop expectation values have been determined for the two-", "type": "text"}], "index": 0}, {"bbox": [63, 32, 537, 45], "spans": [{"bbox": [63, 32, 537, 45], "score": 1.0, "content": "dimensional pure Yang-Mills theory [5, 11] – in coincidence with the known results in the", "type": "text"}], "index": 1}, {"bbox": [61, 45, 537, 60], "spans": [{"bbox": [61, 45, 537, 60], "score": 1.0, "content": "standard framework. In the present paper we continue the investigations on how the results", "type": "text"}], "index": 2}, {"bbox": [62, 59, 537, 75], "spans": [{"bbox": [62, 59, 537, 75], "score": 1.0, "content": "of Kondracki and Rogulski can be extended to the Ashtekar framework. In a previous paper", "type": "text"}], "index": 3}, {"bbox": [63, 75, 537, 90], "spans": [{"bbox": [63, 75, 537, 90], "score": 1.0, "content": "[9] we have already shown that the gauge orbit type is determined by the centralizer of the", "type": "text"}], "index": 4}, {"bbox": [61, 88, 538, 106], "spans": [{"bbox": [61, 88, 538, 106], "score": 1.0, "content": "holonomy group. This closely related to the observations of Kondracki and Sadowski [13]. In", "type": "text"}], "index": 5}, {"bbox": [61, 104, 537, 119], "spans": [{"bbox": [61, 104, 537, 119], "score": 1.0, "content": "the present paper we are going to prove that there is a slice theorem and a denseness theorem", "type": "text"}], "index": 6}, {"bbox": [63, 119, 536, 132], "spans": [{"bbox": [63, 119, 536, 132], "score": 1.0, "content": "for the space of connections in the Ashtekar framework as well. However, our methods are", "type": "text"}], "index": 7}, {"bbox": [64, 133, 352, 146], "spans": [{"bbox": [64, 133, 352, 146], "score": 1.0, "content": "completely different to those of Kondracki and Rogulski.", "type": "text"}], "index": 8}], "index": 4}, {"type": "text", "bbox": [63, 155, 257, 169], "lines": [{"bbox": [63, 157, 257, 171], "spans": [{"bbox": [63, 157, 257, 171], "score": 1.0, "content": "The outline of the paper is as follows:", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [63, 171, 537, 227], "lines": [{"bbox": [63, 172, 537, 188], "spans": [{"bbox": [63, 172, 537, 188], "score": 1.0, "content": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer", "type": "text"}], "index": 10}, {"bbox": [60, 186, 537, 201], "spans": [{"bbox": [60, 186, 537, 201], "score": 1.0, "content": "in a compact Lie group is finitely generated. This implies that every orbit type (being", "type": "text"}], "index": 11}, {"bbox": [61, 200, 538, 218], "spans": [{"bbox": [61, 200, 538, 218], "score": 1.0, "content": "the centralizer of the holonomy group) is determined by a finite set of holonomies of the", "type": "text"}], "index": 12}, {"bbox": [63, 217, 196, 229], "spans": [{"bbox": [63, 217, 196, 229], "score": 1.0, "content": "corresponding connection.", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "text", "bbox": [63, 228, 537, 270], "lines": [{"bbox": [63, 229, 537, 245], "spans": [{"bbox": [63, 229, 537, 245], "score": 1.0, "content": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate", "type": "text"}], "index": 14}, {"bbox": [62, 244, 536, 258], "spans": [{"bbox": [62, 244, 157, 258], "score": 1.0, "content": "finite-dimensional ", "type": "text"}, {"bbox": [157, 246, 173, 254], "score": 0.92, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [174, 244, 241, 258], "score": 1.0, "content": " to the space ", "type": "text"}, {"bbox": [242, 244, 252, 255], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [252, 244, 536, 258], "score": 1.0, "content": ". This is proven in section 5 and it implies the openness", "type": "text"}], "index": 15}, {"bbox": [62, 258, 303, 272], "spans": [{"bbox": [62, 258, 303, 272], "score": 1.0, "content": "of the strata as shown in the following section.", "type": "text"}], "index": 16}], "index": 15}, {"type": "text", "bbox": [63, 271, 537, 357], "lines": [{"bbox": [63, 274, 536, 287], "spans": [{"bbox": [63, 274, 536, 287], "score": 1.0, "content": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for", "type": "text"}], "index": 17}, {"bbox": [62, 287, 537, 302], "spans": [{"bbox": [62, 287, 537, 302], "score": 1.0, "content": "new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals", "type": "text"}], "index": 18}, {"bbox": [61, 300, 538, 318], "spans": [{"bbox": [61, 300, 345, 318], "score": 1.0, "content": "the set of all conjugacy classes of Howe subgroups of ", "type": "text"}, {"bbox": [346, 304, 357, 312], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [357, 300, 538, 318], "score": 1.0, "content": ". A Howe subgroup is a subgroup", "type": "text"}], "index": 19}, {"bbox": [63, 316, 537, 331], "spans": [{"bbox": [63, 316, 273, 331], "score": 1.0, "content": "that is the centralizer of some subset of ", "type": "text"}, {"bbox": [274, 318, 285, 327], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [285, 316, 537, 331], "score": 1.0, "content": ". This way we completely determine all possible", "type": "text"}], "index": 20}, {"bbox": [61, 331, 537, 345], "spans": [{"bbox": [61, 331, 537, 345], "score": 1.0, "content": "gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our", "type": "text"}], "index": 21}, {"bbox": [63, 345, 384, 359], "spans": [{"bbox": [63, 345, 172, 359], "score": 1.0, "content": "knowlegde – only for ", "type": "text"}, {"bbox": [173, 346, 232, 358], "score": 0.95, "content": "{\\mathbf{G}}=S U(n)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [232, 345, 344, 359], "score": 1.0, "content": " and low-dimensional ", "type": "text"}, {"bbox": [345, 347, 357, 356], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [358, 345, 384, 359], "score": 1.0, "content": " [18].", "type": "text"}], "index": 22}], "index": 19.5}, {"type": "text", "bbox": [63, 358, 537, 400], "lines": [{"bbox": [62, 359, 536, 374], "spans": [{"bbox": [62, 359, 536, 374], "score": 1.0, "content": "In Section 8 we show that the slice and the denseness theorem yield again a topologically", "type": "text"}], "index": 23}, {"bbox": [63, 373, 538, 388], "spans": [{"bbox": [63, 373, 184, 388], "score": 1.0, "content": "regular stratification of ", "type": "text"}, {"bbox": [185, 374, 194, 384], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [195, 373, 263, 388], "score": 1.0, "content": " as well as of ", "type": "text"}, {"bbox": [263, 374, 287, 388], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [287, 373, 538, 388], "score": 1.0, "content": ". But, in contrast to the Sobolev case, the strata", "type": "text"}], "index": 24}, {"bbox": [63, 389, 224, 402], "spans": [{"bbox": [63, 389, 224, 402], "score": 1.0, "content": "are not proved to be manifolds.", "type": "text"}], "index": 25}], "index": 24}, {"type": "text", "bbox": [63, 401, 537, 444], "lines": [{"bbox": [62, 403, 537, 418], "spans": [{"bbox": [62, 403, 537, 418], "score": 1.0, "content": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal", "type": "text"}], "index": 26}, {"bbox": [62, 417, 537, 431], "spans": [{"bbox": [62, 417, 195, 431], "score": 1.0, "content": "type) is not only dense in ", "type": "text"}, {"bbox": [195, 417, 205, 428], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [205, 417, 537, 431], "score": 1.0, "content": ", but has also the total induced Haar measure 1. This shows that", "type": "text"}], "index": 27}, {"bbox": [62, 432, 456, 446], "spans": [{"bbox": [62, 432, 325, 446], "score": 1.0, "content": "the Faddeev-Popov determinant for the projection ", "type": "text"}, {"bbox": [325, 432, 384, 446], "score": 0.94, "content": "\\overline{{A}}\\longrightarrow\\overline{{A}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [384, 432, 456, 446], "score": 1.0, "content": " is equal to 1.", "type": "text"}], "index": 28}], "index": 27}, {"type": "title", "bbox": [62, 465, 206, 485], "lines": [{"bbox": [63, 468, 205, 484], "spans": [{"bbox": [63, 470, 74, 483], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [90, 468, 205, 484], "score": 1.0, "content": "Preliminaries", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [63, 496, 538, 612], "lines": [{"bbox": [63, 498, 537, 513], "spans": [{"bbox": [63, 498, 537, 513], "score": 1.0, "content": "As we indicated in [9] the present paper is the final one in a small series of three papers.", "type": "text"}], "index": 30}, {"bbox": [61, 511, 536, 527], "spans": [{"bbox": [61, 511, 412, 527], "score": 1.0, "content": "In the first one [9] we extended the definitions and propositions for ", "type": "text"}, {"bbox": [412, 513, 423, 523], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [423, 511, 429, 527], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [429, 513, 438, 524], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [438, 511, 464, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [464, 513, 488, 526], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [488, 511, 536, 527], "score": 1.0, "content": " made by", "type": "text"}], "index": 31}, {"bbox": [62, 526, 538, 542], "spans": [{"bbox": [62, 526, 538, 542], "score": 1.0, "content": "Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth", "type": "text"}], "index": 32}, {"bbox": [61, 541, 538, 556], "spans": [{"bbox": [61, 541, 538, 556], "score": 1.0, "content": "paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the", "type": "text"}], "index": 33}, {"bbox": [62, 556, 538, 570], "spans": [{"bbox": [62, 556, 307, 570], "score": 1.0, "content": "second paper [10] we investigated properties of ", "type": "text"}, {"bbox": [307, 556, 317, 567], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [317, 556, 538, 570], "score": 1.0, "content": " and proved, in particular, the existence of", "type": "text"}], "index": 34}, {"bbox": [61, 569, 537, 585], "spans": [{"bbox": [61, 569, 537, 585], "score": 1.0, "content": "an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important", "type": "text"}], "index": 35}, {"bbox": [62, 585, 537, 600], "spans": [{"bbox": [62, 585, 537, 600], "score": 1.0, "content": "notations, definitions and facts used in the following. For detailed information we refer the", "type": "text"}], "index": 36}, {"bbox": [62, 599, 259, 615], "spans": [{"bbox": [62, 599, 259, 615], "score": 1.0, "content": "reader to the preceding papers [9, 10].", "type": "text"}], "index": 37}], "index": 33.5}, {"type": "text", "bbox": [63, 612, 538, 684], "lines": [{"bbox": [61, 612, 239, 630], "spans": [{"bbox": [61, 612, 100, 630], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [101, 615, 111, 624], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [111, 612, 239, 630], "score": 1.0, "content": " be a compact Lie group.", "type": "text"}], "index": 38}, {"bbox": [62, 628, 537, 642], "spans": [{"bbox": [62, 628, 230, 642], "score": 1.0, "content": "• A path (usually denoted by ", "type": "text"}, {"bbox": [230, 633, 237, 641], "score": 0.89, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [237, 628, 257, 642], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [257, 630, 263, 639], "score": 0.83, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [263, 628, 347, 642], "score": 1.0, "content": ") is a piecewise ", "type": "text"}, {"bbox": [347, 630, 361, 639], "score": 0.92, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [362, 628, 421, 642], "score": 1.0, "content": "-map from ", "type": "text"}, {"bbox": [421, 629, 444, 642], "score": 0.92, "content": "[0,1]", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [445, 628, 537, 642], "score": 1.0, "content": " into a connected", "type": "text"}], "index": 39}, {"bbox": [80, 643, 537, 657], "spans": [{"bbox": [80, 644, 93, 653], "score": 0.91, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [93, 643, 145, 657], "score": 1.0, "content": "-manifold ", "type": "text"}, {"bbox": [145, 644, 158, 653], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [159, 643, 164, 657], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [164, 644, 221, 654], "score": 0.89, "content": "\\dim M\\geq2", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [221, 643, 227, 657], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [227, 643, 329, 656], "score": 0.92, "content": "r\\in\\mathbb{N}^{+}\\cup\\{\\infty\\}\\cup\\{\\omega\\}", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [329, 643, 537, 657], "score": 1.0, "content": " arbitrary, but fixed. Additionally, we fix", "type": "text"}], "index": 40}, {"bbox": [78, 657, 537, 671], "spans": [{"bbox": [78, 657, 537, 671], "score": 1.0, "content": "now the decision whether we restrict the paths to be piecewise immersive or not. Paths", "type": "text"}], "index": 41}, {"bbox": [79, 672, 537, 685], "spans": [{"bbox": [79, 672, 537, 685], "score": 1.0, "content": "can be multiplied as usual by concatenation. A graph is a finite union of paths, such that", "type": "text"}], "index": 42}], "index": 40}], "layout_bboxes": [], "page_idx": 2, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [295, 705, 303, 714], "lines": [{"bbox": [295, 705, 304, 717], "spans": [{"bbox": [295, 705, 304, 717], "score": 1.0, "content": "3", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [63, 15, 538, 145], "lines": [], "index": 4, "bbox_fs": [61, 18, 538, 146], "lines_deleted": true}, {"type": "text", "bbox": [63, 155, 257, 169], "lines": [{"bbox": [63, 157, 257, 171], "spans": [{"bbox": [63, 157, 257, 171], "score": 1.0, "content": "The outline of the paper is as follows:", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [63, 157, 257, 171]}, {"type": "text", "bbox": [63, 171, 537, 227], "lines": [{"bbox": [63, 172, 537, 188], "spans": [{"bbox": [63, 172, 537, 188], "score": 1.0, "content": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer", "type": "text"}], "index": 10}, {"bbox": [60, 186, 537, 201], "spans": [{"bbox": [60, 186, 537, 201], "score": 1.0, "content": "in a compact Lie group is finitely generated. This implies that every orbit type (being", "type": "text"}], "index": 11}, {"bbox": [61, 200, 538, 218], "spans": [{"bbox": [61, 200, 538, 218], "score": 1.0, "content": "the centralizer of the holonomy group) is determined by a finite set of holonomies of the", "type": "text"}], "index": 12}, {"bbox": [63, 217, 196, 229], "spans": [{"bbox": [63, 217, 196, 229], "score": 1.0, "content": "corresponding connection.", "type": "text"}], "index": 13}], "index": 11.5, "bbox_fs": [60, 172, 538, 229]}, {"type": "text", "bbox": [63, 228, 537, 270], "lines": [{"bbox": [63, 229, 537, 245], "spans": [{"bbox": [63, 229, 537, 245], "score": 1.0, "content": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate", "type": "text"}], "index": 14}, {"bbox": [62, 244, 536, 258], "spans": [{"bbox": [62, 244, 157, 258], "score": 1.0, "content": "finite-dimensional ", "type": "text"}, {"bbox": [157, 246, 173, 254], "score": 0.92, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [174, 244, 241, 258], "score": 1.0, "content": " to the space ", "type": "text"}, {"bbox": [242, 244, 252, 255], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [252, 244, 536, 258], "score": 1.0, "content": ". This is proven in section 5 and it implies the openness", "type": "text"}], "index": 15}, {"bbox": [62, 258, 303, 272], "spans": [{"bbox": [62, 258, 303, 272], "score": 1.0, "content": "of the strata as shown in the following section.", "type": "text"}], "index": 16}], "index": 15, "bbox_fs": [62, 229, 537, 272]}, {"type": "text", "bbox": [63, 271, 537, 357], "lines": [{"bbox": [63, 274, 536, 287], "spans": [{"bbox": [63, 274, 536, 287], "score": 1.0, "content": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for", "type": "text"}], "index": 17}, {"bbox": [62, 287, 537, 302], "spans": [{"bbox": [62, 287, 537, 302], "score": 1.0, "content": "new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals", "type": "text"}], "index": 18}, {"bbox": [61, 300, 538, 318], "spans": [{"bbox": [61, 300, 345, 318], "score": 1.0, "content": "the set of all conjugacy classes of Howe subgroups of ", "type": "text"}, {"bbox": [346, 304, 357, 312], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [357, 300, 538, 318], "score": 1.0, "content": ". A Howe subgroup is a subgroup", "type": "text"}], "index": 19}, {"bbox": [63, 316, 537, 331], "spans": [{"bbox": [63, 316, 273, 331], "score": 1.0, "content": "that is the centralizer of some subset of ", "type": "text"}, {"bbox": [274, 318, 285, 327], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [285, 316, 537, 331], "score": 1.0, "content": ". This way we completely determine all possible", "type": "text"}], "index": 20}, {"bbox": [61, 331, 537, 345], "spans": [{"bbox": [61, 331, 537, 345], "score": 1.0, "content": "gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our", "type": "text"}], "index": 21}, {"bbox": [63, 345, 384, 359], "spans": [{"bbox": [63, 345, 172, 359], "score": 1.0, "content": "knowlegde – only for ", "type": "text"}, {"bbox": [173, 346, 232, 358], "score": 0.95, "content": "{\\mathbf{G}}=S U(n)", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [232, 345, 344, 359], "score": 1.0, "content": " and low-dimensional ", "type": "text"}, {"bbox": [345, 347, 357, 356], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [358, 345, 384, 359], "score": 1.0, "content": " [18].", "type": "text"}], "index": 22}], "index": 19.5, "bbox_fs": [61, 274, 538, 359]}, {"type": "text", "bbox": [63, 358, 537, 400], "lines": [{"bbox": [62, 359, 536, 374], "spans": [{"bbox": [62, 359, 536, 374], "score": 1.0, "content": "In Section 8 we show that the slice and the denseness theorem yield again a topologically", "type": "text"}], "index": 23}, {"bbox": [63, 373, 538, 388], "spans": [{"bbox": [63, 373, 184, 388], "score": 1.0, "content": "regular stratification of ", "type": "text"}, {"bbox": [185, 374, 194, 384], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [195, 373, 263, 388], "score": 1.0, "content": " as well as of ", "type": "text"}, {"bbox": [263, 374, 287, 388], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [287, 373, 538, 388], "score": 1.0, "content": ". But, in contrast to the Sobolev case, the strata", "type": "text"}], "index": 24}, {"bbox": [63, 389, 224, 402], "spans": [{"bbox": [63, 389, 224, 402], "score": 1.0, "content": "are not proved to be manifolds.", "type": "text"}], "index": 25}], "index": 24, "bbox_fs": [62, 359, 538, 402]}, {"type": "text", "bbox": [63, 401, 537, 444], "lines": [{"bbox": [62, 403, 537, 418], "spans": [{"bbox": [62, 403, 537, 418], "score": 1.0, "content": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal", "type": "text"}], "index": 26}, {"bbox": [62, 417, 537, 431], "spans": [{"bbox": [62, 417, 195, 431], "score": 1.0, "content": "type) is not only dense in ", "type": "text"}, {"bbox": [195, 417, 205, 428], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [205, 417, 537, 431], "score": 1.0, "content": ", but has also the total induced Haar measure 1. This shows that", "type": "text"}], "index": 27}, {"bbox": [62, 432, 456, 446], "spans": [{"bbox": [62, 432, 325, 446], "score": 1.0, "content": "the Faddeev-Popov determinant for the projection ", "type": "text"}, {"bbox": [325, 432, 384, 446], "score": 0.94, "content": "\\overline{{A}}\\longrightarrow\\overline{{A}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [384, 432, 456, 446], "score": 1.0, "content": " is equal to 1.", "type": "text"}], "index": 28}], "index": 27, "bbox_fs": [62, 403, 537, 446]}, {"type": "title", "bbox": [62, 465, 206, 485], "lines": [{"bbox": [63, 468, 205, 484], "spans": [{"bbox": [63, 470, 74, 483], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [90, 468, 205, 484], "score": 1.0, "content": "Preliminaries", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [63, 496, 538, 612], "lines": [{"bbox": [63, 498, 537, 513], "spans": [{"bbox": [63, 498, 537, 513], "score": 1.0, "content": "As we indicated in [9] the present paper is the final one in a small series of three papers.", "type": "text"}], "index": 30}, {"bbox": [61, 511, 536, 527], "spans": [{"bbox": [61, 511, 412, 527], "score": 1.0, "content": "In the first one [9] we extended the definitions and propositions for ", "type": "text"}, {"bbox": [412, 513, 423, 523], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [423, 511, 429, 527], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [429, 513, 438, 524], "score": 0.89, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [438, 511, 464, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [464, 513, 488, 526], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [488, 511, 536, 527], "score": 1.0, "content": " made by", "type": "text"}], "index": 31}, {"bbox": [62, 526, 538, 542], "spans": [{"bbox": [62, 526, 538, 542], "score": 1.0, "content": "Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth", "type": "text"}], "index": 32}, {"bbox": [61, 541, 538, 556], "spans": [{"bbox": [61, 541, 538, 556], "score": 1.0, "content": "paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the", "type": "text"}], "index": 33}, {"bbox": [62, 556, 538, 570], "spans": [{"bbox": [62, 556, 307, 570], "score": 1.0, "content": "second paper [10] we investigated properties of ", "type": "text"}, {"bbox": [307, 556, 317, 567], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [317, 556, 538, 570], "score": 1.0, "content": " and proved, in particular, the existence of", "type": "text"}], "index": 34}, {"bbox": [61, 569, 537, 585], "spans": [{"bbox": [61, 569, 537, 585], "score": 1.0, "content": "an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important", "type": "text"}], "index": 35}, {"bbox": [62, 585, 537, 600], "spans": [{"bbox": [62, 585, 537, 600], "score": 1.0, "content": "notations, definitions and facts used in the following. For detailed information we refer the", "type": "text"}], "index": 36}, {"bbox": [62, 599, 259, 615], "spans": [{"bbox": [62, 599, 259, 615], "score": 1.0, "content": "reader to the preceding papers [9, 10].", "type": "text"}], "index": 37}], "index": 33.5, "bbox_fs": [61, 498, 538, 615]}, {"type": "text", "bbox": [63, 612, 538, 684], "lines": [{"bbox": [61, 612, 239, 630], "spans": [{"bbox": [61, 612, 100, 630], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [101, 615, 111, 624], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [111, 612, 239, 630], "score": 1.0, "content": " be a compact Lie group.", "type": "text"}], "index": 38}, {"bbox": [62, 628, 537, 642], "spans": [{"bbox": [62, 628, 230, 642], "score": 1.0, "content": "• A path (usually denoted by ", "type": "text"}, {"bbox": [230, 633, 237, 641], "score": 0.89, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [237, 628, 257, 642], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [257, 630, 263, 639], "score": 0.83, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [263, 628, 347, 642], "score": 1.0, "content": ") is a piecewise ", "type": "text"}, {"bbox": [347, 630, 361, 639], "score": 0.92, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [362, 628, 421, 642], "score": 1.0, "content": "-map from ", "type": "text"}, {"bbox": [421, 629, 444, 642], "score": 0.92, "content": "[0,1]", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [445, 628, 537, 642], "score": 1.0, "content": " into a connected", "type": "text"}], "index": 39}, {"bbox": [80, 643, 537, 657], "spans": [{"bbox": [80, 644, 93, 653], "score": 0.91, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [93, 643, 145, 657], "score": 1.0, "content": "-manifold ", "type": "text"}, {"bbox": [145, 644, 158, 653], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [159, 643, 164, 657], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [164, 644, 221, 654], "score": 0.89, "content": "\\dim M\\geq2", "type": "inline_equation", "height": 10, "width": 57}, {"bbox": [221, 643, 227, 657], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [227, 643, 329, 656], "score": 0.92, "content": "r\\in\\mathbb{N}^{+}\\cup\\{\\infty\\}\\cup\\{\\omega\\}", "type": "inline_equation", "height": 13, "width": 102}, {"bbox": [329, 643, 537, 657], "score": 1.0, "content": " arbitrary, but fixed. Additionally, we fix", "type": "text"}], "index": 40}, {"bbox": [78, 657, 537, 671], "spans": [{"bbox": [78, 657, 537, 671], "score": 1.0, "content": "now the decision whether we restrict the paths to be piecewise immersive or not. Paths", "type": "text"}], "index": 41}, {"bbox": [79, 672, 537, 685], "spans": [{"bbox": [79, 672, 537, 685], "score": 1.0, "content": "can be multiplied as usual by concatenation. A graph is a finite union of paths, such that", "type": "text"}], "index": 42}, {"bbox": [79, 16, 538, 33], "spans": [{"bbox": [79, 16, 538, 33], "score": 1.0, "content": "different paths intersect each other at most in their end points. Paths in a graph are called", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [80, 32, 536, 46], "spans": [{"bbox": [80, 32, 536, 46], "score": 1.0, "content": "simple. A path is called finite iff it is up to the parametrization a finite product of simple", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [79, 46, 538, 61], "spans": [{"bbox": [79, 46, 538, 61], "score": 1.0, "content": "paths. Two paths are equivalent iff the first one can be reconstructed from the second", "type": "text", "cross_page": true}], "index": 2}, {"bbox": [79, 60, 538, 74], "spans": [{"bbox": [79, 60, 538, 74], "score": 1.0, "content": "one by a sequence of reparametrizations or of insertions or deletions of retracings. We will", "type": "text", "cross_page": true}], "index": 3}, {"bbox": [80, 75, 537, 89], "spans": [{"bbox": [80, 75, 537, 89], "score": 1.0, "content": "only consider equivalence classes of finite paths and graphs. The set of (classes of) paths", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [78, 89, 537, 104], "spans": [{"bbox": [78, 89, 153, 104], "score": 1.0, "content": "is denoted by ", "type": "text", "cross_page": true}, {"bbox": [153, 91, 162, 100], "score": 0.9, "content": "\\mathcal{P}", "type": "inline_equation", "height": 9, "width": 9, "cross_page": true}, {"bbox": [163, 89, 269, 104], "score": 1.0, "content": ", that of paths from ", "type": "text", "cross_page": true}, {"bbox": [270, 94, 276, 100], "score": 0.88, "content": "x", "type": "inline_equation", "height": 6, "width": 6, "cross_page": true}, {"bbox": [276, 89, 295, 104], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [295, 94, 301, 102], "score": 0.88, "content": "y", "type": "inline_equation", "height": 8, "width": 6, "cross_page": true}, {"bbox": [302, 89, 322, 104], "score": 1.0, "content": " by ", "type": "text", "cross_page": true}, {"bbox": [322, 91, 340, 103], "score": 0.93, "content": "\\mathcal{P}_{x y}", "type": "inline_equation", "height": 12, "width": 18, "cross_page": true}, {"bbox": [341, 89, 537, 104], "score": 1.0, "content": " and that of loops (paths with a fixed", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [78, 103, 404, 119], "spans": [{"bbox": [78, 103, 213, 119], "score": 1.0, "content": "initial and terminal point ", "type": "text", "cross_page": true}, {"bbox": [213, 109, 224, 114], "score": 0.86, "content": "m", "type": "inline_equation", "height": 5, "width": 11, "cross_page": true}, {"bbox": [225, 103, 248, 119], "score": 1.0, "content": ") by ", "type": "text", "cross_page": true}, {"bbox": [249, 106, 267, 116], "score": 0.92, "content": "\\mathcal{H G}", "type": "inline_equation", "height": 10, "width": 18, "cross_page": true}, {"bbox": [267, 103, 404, 119], "score": 1.0, "content": ", the so-called hoop group.", "type": "text", "cross_page": true}], "index": 6}], "index": 40, "bbox_fs": [61, 612, 537, 685]}]}
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[{"type": "text", "bbox": [63, 15, 538, 145], "content": "", "index": 0}, {"type": "text", "bbox": [63, 155, 257, 169], "content": "The outline of the paper is as follows:", "index": 1}, {"type": "text", "bbox": [63, 171, 537, 227], "content": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer in a compact Lie group is finitely generated. This implies that every orbit type (being the centralizer of the holonomy group) is determined by a finite set of holonomies of the corresponding connection.", "index": 2}, {"type": "text", "bbox": [63, 228, 537, 270], "content": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate finite-dimensional to the space . This is proven in section 5 and it implies the openness of the strata as shown in the following section.", "index": 3}, {"type": "text", "bbox": [63, 271, 537, 357], "content": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals the set of all conjugacy classes of Howe subgroups of . A Howe subgroup is a subgroup that is the centralizer of some subset of . This way we completely determine all possible gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our knowlegde – only for and low-dimensional [18].", "index": 4}, {"type": "text", "bbox": [63, 358, 537, 400], "content": "In Section 8 we show that the slice and the denseness theorem yield again a topologically regular stratification of as well as of . But, in contrast to the Sobolev case, the strata are not proved to be manifolds.", "index": 5}, {"type": "text", "bbox": [63, 401, 537, 444], "content": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal type) is not only dense in , but has also the total induced Haar measure 1. This shows that the Faddeev-Popov determinant for the projection is equal to 1.", "index": 6}, {"type": "title", "bbox": [62, 465, 206, 485], "content": "2 Preliminaries", "index": 7}, {"type": "text", "bbox": [63, 496, 538, 612], "content": "As we indicated in [9] the present paper is the final one in a small series of three papers. In the first one [9] we extended the definitions and propositions for , and made by Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the second paper [10] we investigated properties of and proved, in particular, the existence of an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important notations, definitions and facts used in the following. For detailed information we refer the reader to the preceding papers [9, 10].", "index": 8}, {"type": "text", "bbox": [63, 612, 538, 684], "content": "• Let be a compact Lie group. • A path (usually denoted by or ) is a piecewise -map from into a connected -manifold , , arbitrary, but fixed. Additionally, we fix now the decision whether we restrict the paths to be piecewise immersive or not. Paths can be multiplied as usual by concatenation. A graph is a finite union of paths, such that different paths intersect each other at most in their end points. Paths in a graph are called simple. A path is called finite iff it is up to the parametrization a finite product of simple paths. Two paths are equivalent iff the first one can be reconstructed from the second one by a sequence of reparametrizations or of insertions or deletions of retracings. We will only consider equivalence classes of finite paths and graphs. The set of (classes of) paths is denoted by , that of paths from to by and that of loops (paths with a fixed initial and terminal point ) by , the so-called hoop group.", "index": 9}]
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[{"bbox": [63, 157, 257, 171], "content": "The outline of the paper is as follows:", "parent_index": 1, "line_index": 0}, {"bbox": [63, 172, 537, 188], "content": "After fixing the notations we prove a very crucial lemma in section 4: Every centralizer", "parent_index": 2, "line_index": 0}, {"bbox": [60, 186, 537, 201], "content": "in a compact Lie group is finitely generated. This implies that every orbit type (being", "parent_index": 2, "line_index": 1}, {"bbox": [61, 200, 538, 218], "content": "the centralizer of the holonomy group) is determined by a finite set of holonomies of the", "parent_index": 2, "line_index": 2}, {"bbox": [63, 217, 196, 229], "content": "corresponding connection.", "parent_index": 2, "line_index": 3}, {"bbox": [63, 229, 537, 245], "content": "Using the projection onto these holonomies we can lift the slice theorem from an appropriate", "parent_index": 3, "line_index": 0}, {"bbox": [62, 244, 536, 258], "content": "finite-dimensional to the space . This is proven in section 5 and it implies the openness", "parent_index": 3, "line_index": 1}, {"bbox": [62, 258, 303, 272], "content": "of the strata as shown in the following section.", "parent_index": 3, "line_index": 2}, {"bbox": [63, 274, 536, 287], "content": "Afterwards, we prove a denseness theorem for the strata. For this we need a construction for", "parent_index": 4, "line_index": 0}, {"bbox": [62, 287, 537, 302], "content": "new connections from [10]. As a corollary we obtain that the set of all gauge orbit types equals", "parent_index": 4, "line_index": 1}, {"bbox": [61, 300, 538, 318], "content": "the set of all conjugacy classes of Howe subgroups of . A Howe subgroup is a subgroup", "parent_index": 4, "line_index": 2}, {"bbox": [63, 316, 537, 331], "content": "that is the centralizer of some subset of . This way we completely determine all possible", "parent_index": 4, "line_index": 3}, {"bbox": [61, 331, 537, 345], "content": "gauge orbit types. This has been succeeded for the Sobolev connections – to the best of our", "parent_index": 4, "line_index": 4}, {"bbox": [63, 345, 384, 359], "content": "knowlegde – only for and low-dimensional [18].", "parent_index": 4, "line_index": 5}, {"bbox": [62, 359, 536, 374], "content": "In Section 8 we show that the slice and the denseness theorem yield again a topologically", "parent_index": 5, "line_index": 0}, {"bbox": [63, 373, 538, 388], "content": "regular stratification of as well as of . But, in contrast to the Sobolev case, the strata", "parent_index": 5, "line_index": 1}, {"bbox": [63, 389, 224, 402], "content": "are not proved to be manifolds.", "parent_index": 5, "line_index": 2}, {"bbox": [62, 403, 537, 418], "content": "Finally, we show in Section 9 that the generic stratum (it collects the connections of maximal", "parent_index": 6, "line_index": 0}, {"bbox": [62, 417, 537, 431], "content": "type) is not only dense in , but has also the total induced Haar measure 1. This shows that", "parent_index": 6, "line_index": 1}, {"bbox": [62, 432, 456, 446], "content": "the Faddeev-Popov determinant for the projection is equal to 1.", "parent_index": 6, "line_index": 2}, {"bbox": [63, 468, 205, 484], "content": "2 Preliminaries", "parent_index": 7, "line_index": 0}, {"bbox": [63, 498, 537, 513], "content": "As we indicated in [9] the present paper is the final one in a small series of three papers.", "parent_index": 8, "line_index": 0}, {"bbox": [61, 511, 536, 527], "content": "In the first one [9] we extended the definitions and propositions for , and made by", "parent_index": 8, "line_index": 1}, {"bbox": [62, 526, 538, 542], "content": "Ashtekar et al. from the case of graphs [1, 2, 4, 3, 15] and of webs [6] to arbitrarily smooth", "parent_index": 8, "line_index": 2}, {"bbox": [61, 541, 538, 556], "content": "paths. Moreover, in that paper we determined the gauge orbit type of a connection. In the", "parent_index": 8, "line_index": 3}, {"bbox": [62, 556, 538, 570], "content": "second paper [10] we investigated properties of and proved, in particular, the existence of", "parent_index": 8, "line_index": 4}, {"bbox": [61, 569, 537, 585], "content": "an Ashtekar-Lewandowski measure in our context. Now, we summarize the most important", "parent_index": 8, "line_index": 5}, {"bbox": [62, 585, 537, 600], "content": "notations, definitions and facts used in the following. For detailed information we refer the", "parent_index": 8, "line_index": 6}, {"bbox": [62, 599, 259, 615], "content": "reader to the preceding papers [9, 10].", "parent_index": 8, "line_index": 7}, {"bbox": [61, 612, 239, 630], "content": "• Let be a compact Lie group.", "parent_index": 9, "line_index": 0}, {"bbox": [62, 628, 537, 642], "content": "• A path (usually denoted by or ) is a piecewise -map from into a connected", "parent_index": 9, "line_index": 1}, {"bbox": [80, 643, 537, 657], "content": "-manifold , , arbitrary, but fixed. Additionally, we fix", "parent_index": 9, "line_index": 2}, {"bbox": [78, 657, 537, 671], "content": "now the decision whether we restrict the paths to be piecewise immersive or not. Paths", "parent_index": 9, "line_index": 3}, {"bbox": [79, 672, 537, 685], "content": "can be multiplied as usual by concatenation. A graph is a finite union of paths, such that", "parent_index": 9, "line_index": 4}, {"bbox": [79, 16, 538, 33], "content": "different paths intersect each other at most in their end points. Paths in a graph are called", "parent_index": 9, "line_index": 5}, {"bbox": [80, 32, 536, 46], "content": "simple. A path is called finite iff it is up to the parametrization a finite product of simple", "parent_index": 9, "line_index": 6}, {"bbox": [79, 46, 538, 61], "content": "paths. Two paths are equivalent iff the first one can be reconstructed from the second", "parent_index": 9, "line_index": 7}, {"bbox": [79, 60, 538, 74], "content": "one by a sequence of reparametrizations or of insertions or deletions of retracings. We will", "parent_index": 9, "line_index": 8}, {"bbox": [80, 75, 537, 89], "content": "only consider equivalence classes of finite paths and graphs. The set of (classes of) paths", "parent_index": 9, "line_index": 9}, {"bbox": [78, 89, 537, 104], "content": "is denoted by , that of paths from to by and that of loops (paths with a fixed", "parent_index": 9, "line_index": 10}, {"bbox": [78, 103, 404, 119], "content": "initial and terminal point ) by , the so-called hoop group.", "parent_index": 9, "line_index": 11}]
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[]
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[{"bbox": [157, 246, 173, 254], "content": "\\mathbf{G}^{n}", "parent_index": 3, "subtype": "inline"}, {"bbox": [242, 244, 252, 255], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [346, 304, 357, 312], "content": "\\mathbf{G}", "parent_index": 4, "subtype": "inline"}, {"bbox": [274, 318, 285, 327], "content": "\\mathbf{G}", "parent_index": 4, "subtype": "inline"}, {"bbox": [173, 346, 232, 358], "content": "{\\mathbf{G}}=S U(n)", "parent_index": 4, "subtype": "inline"}, {"bbox": [345, 347, 357, 356], "content": "M", "parent_index": 4, "subtype": "inline"}, {"bbox": [185, 374, 194, 384], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [263, 374, 287, 388], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [195, 417, 205, 428], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [325, 432, 384, 446], "content": "\\overline{{A}}\\longrightarrow\\overline{{A}}/\\overline{{\\mathcal{G}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [412, 513, 423, 523], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [429, 513, 438, 524], "content": "\\overline{{g}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [464, 513, 488, 526], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [307, 556, 317, 567], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [101, 615, 111, 624], "content": "\\mathbf{G}", "parent_index": 9, "subtype": "inline"}, {"bbox": [230, 633, 237, 641], "content": "\\gamma", "parent_index": 9, "subtype": "inline"}, {"bbox": [257, 630, 263, 639], "content": "\\delta", "parent_index": 9, "subtype": "inline"}, {"bbox": [347, 630, 361, 639], "content": "C^{r}", "parent_index": 9, "subtype": "inline"}, {"bbox": [421, 629, 444, 642], "content": "[0,1]", "parent_index": 9, "subtype": "inline"}, {"bbox": [80, 644, 93, 653], "content": "C^{r}", "parent_index": 9, "subtype": "inline"}, {"bbox": [145, 644, 158, 653], "content": "M", "parent_index": 9, "subtype": "inline"}, {"bbox": [164, 644, 221, 654], "content": "\\dim M\\geq2", "parent_index": 9, "subtype": "inline"}, {"bbox": [227, 643, 329, 656], "content": "r\\in\\mathbb{N}^{+}\\cup\\{\\infty\\}\\cup\\{\\omega\\}", "parent_index": 9, "subtype": "inline"}, {"bbox": [153, 91, 162, 100], "content": "\\mathcal{P}", "parent_index": 9, "subtype": "inline"}, {"bbox": [270, 94, 276, 100], "content": "x", "parent_index": 9, "subtype": "inline"}, {"bbox": [295, 94, 301, 102], "content": "y", "parent_index": 9, "subtype": "inline"}, {"bbox": [322, 91, 340, 103], "content": "\\mathcal{P}_{x y}", "parent_index": 9, "subtype": "inline"}, {"bbox": [213, 109, 224, 114], "content": "m", "parent_index": 9, "subtype": "inline"}, {"bbox": [249, 106, 267, 116], "content": "\\mathcal{H G}", "parent_index": 9, "subtype": "inline"}]
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• A generalized connection ${\overline{{A}}}\in{\overline{{A}}}$ is a homomorphism1 $h_{\overline{{A}}}:{\mathcal{P}}\longrightarrow\mathbf{G}$ . (We usually write $h_{\overline{{A}}}$ synonymously for $\overline{{A}}$ .) A generalized gauge transform ${\overline{{g}}}\,\in{\overline{{\mathcal{G}}}}$ is a map $\overline{{g}}:M\longrightarrow\mathbf{G}$ . The value $\overline{{g}}(x)$ of the gauge transform in the point $x$ is usually denoted by $g_{x}$ . The action of $\overline{{g}}$ on $\overline{{\mathcal{A}}}$ is given by
$$
h_{\overline{{A}}\circ\overline{{g}}}(\gamma):=g_{\gamma(0)}^{-1}\;h_{\overline{{A}}}(\gamma)\;g_{\gamma(1)}\mathrm{~for~all~}\gamma\in\mathcal{P}.
$$
We have $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}\cong\mathrm{Hom}(\mathcal{H}\mathcal{G},\mathbf{G})/\mathrm{Ad}$ .
• Now, let $\Gamma$ be a graph with $\mathbf{E}(\Gamma)\,=\,\{e_{1},\dots,e_{E}\}$ being the set of edges and ${\mathbf V}(\Gamma)\mathbf{\Sigma}=$ $\{v_{1},\ldots,v_{V}\}$ the set of vertices. The projections onto the lattice gauge theories are defined by
$$
\begin{array}{r l}{\tau_{\Gamma}:\;\;\overline{{\mathcal{A}}}\;\;\longrightarrow\;\;\;\;\overline{{\mathcal{A}}}_{\Gamma}\equiv\mathbf{G}^{E}\qquad\qquad\mathrm{and}\qquad\pi_{\Gamma}:\;\;\overline{{\mathcal{G}}}\;\;\longrightarrow\;\;\;\;\overline{{\mathcal{G}}}_{\Gamma}\equiv\mathbf{G}^{V}.}\\ {\overline{{\mathcal{A}}}\;\;\longmapsto\;\;\left(h_{\overline{{A}}}(e_{1}),\ldots,h_{\overline{{A}}}(e_{E})\right)\qquad\qquad\qquad\quad\overline{{g}}\;\;\longmapsto\;\;\left(g_{v_{1}},\ldots,g_{v_{V}}\right)}\end{array}
$$
The topologies on $\overline{{\mathcal{A}}}$ and $\overline{{g}}$ are the topologies generated by these projections. Using these topologies the action $\Theta:\overline{{\mathcal{A}}}\times\overline{{\mathcal{G}}}\longrightarrow\overline{{\mathcal{A}}}$ defined by (1) is continuous. Since $\mathbf{G}$ is compact Lie, $\overline{{\mathcal{A}}}$ and $\mathcal{G}$ are compact Hausdorff spaces and consequently completely regular.
• The holonomy group $\mathbf{H}_{\overline{{A}}}$ of a connection $\overline{{A}}$ is defined by $\mathbf{H}_{\overline{{A}}}:=h_{\overline{{A}}}(\mathcal{H}\mathcal{G})\subseteq\mathbf{G}$ , its centralizer is denoted by $Z(\mathbf{H}_{\overline{{A}}})$ . The stabilizer of a connection $\overline{{A}}\in\overline{{A}}$ under the action of $\overline{{g}}$ is denoted by $\mathbf{B}(\overline{{A}})$ . We have ${\overline{{g}}}\,\in\,{\bf B}({\overline{{A}}})$ iff $g_{m}\,\in\,Z(\mathbf{H}_{\overline{{A}}})$ and for all $x\,\in\,M$ there is a path $\gamma\in\mathcal{P}_{m x}$ with $h_{\overline{{{A}}}}(\gamma)\,=\,g_{m}^{-1}h_{\overline{{{A}}}}(\gamma)g_{x}$ . In [9] we proved that $\mathbf{B}(\overline{{A}})$ and $Z(\mathbf{H}_{\overline{{A}}})$ are homeomorphic.
• The type of a gauge orbit $\mathbf{E}_{\overline{{A}}}:=\overline{{A}}\circ\overline{{\mathcal{G}}}$ is the centralizer of the holonomy group of $\overline{{A}}$ modulo conjugation in $\mathbf{G}$ . (An equivalent definition uses the stabilizer $\mathbf{B}(\overline{{A}})$ itself.)
# 3 Partial Ordering of Types
Definition 3.1 A subgroup $U$ of $\mathbf{G}$ is called Howe subgroup iff there is a set $V\subseteq\mathbf{G}$ with $U=Z(V)$ .
Analogously to the general theory we define a partial ordering for the gauge orbit types [8].
Definition 3.2 Let $\tau$ denote the set of all Howe subgroups of $\mathbf{G}$ . Let $t_{1},t_{2}\in\mathcal{T}$ . Then $t_{1}\leq t_{2}$ holds iff there are $\mathbf{G}_{1}\in t_{1}$ and $\mathbf{G}_{2}\in t_{2}$ with $\mathbf{G}_{1}\supseteq\mathbf{G}_{2}$ .
Obviously, we have
Lemma 3.1 The maximal element in $\tau$ is the class $t_{\mathrm{max}}$ of the center $Z(\mathbf{G})$ of $\mathbf{G}$ , the minimal is the class $t_{\mathrm{min}}$ of $\mathbf{G}$ itself.
1Homomorphism means $h_{\overline{{{A}}}}(\gamma_{1}\gamma_{2})=h_{\overline{{{A}}}}(\gamma_{1})h_{\overline{{{A}}}}(\gamma_{2})$ supposed $\gamma_{1}\gamma_{2}$ is defined.
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<html><body>
<p data-bbox="65 116 538 173">• A generalized connection ${\overline{{A}}}\in{\overline{{A}}}$ is a homomorphism1 $h_{\overline{{A}}}:{\mathcal{P}}\longrightarrow\mathbf{G}$ . (We usually write $h_{\overline{{A}}}$ synonymously for $\overline{{A}}$ .) A generalized gauge transform ${\overline{{g}}}\,\in{\overline{{\mathcal{G}}}}$ is a map $\overline{{g}}:M\longrightarrow\mathbf{G}$ . The value $\overline{{g}}(x)$ of the gauge transform in the point $x$ is usually denoted by $g_{x}$ . The action of $\overline{{g}}$ on $\overline{{\mathcal{A}}}$ is given by </p>
<div class="equation" data-bbox="203 177 412 195">$$
h_{\overline{{A}}\circ\overline{{g}}}(\gamma):=g_{\gamma(0)}^{-1}\;h_{\overline{{A}}}(\gamma)\;g_{\gamma(1)}\mathrm{~for~all~}\gamma\in\mathcal{P}.
$$</div>
<p data-bbox="77 195 259 210">We have $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}\cong\mathrm{Hom}(\mathcal{H}\mathcal{G},\mathbf{G})/\mathrm{Ad}$ . </p>
<p data-bbox="66 210 538 250">• Now, let $\Gamma$ be a graph with $\mathbf{E}(\Gamma)\,=\,\{e_{1},\dots,e_{E}\}$ being the set of edges and ${\mathbf V}(\Gamma)\mathbf{\Sigma}=$ $\{v_{1},\ldots,v_{V}\}$ the set of vertices. The projections onto the lattice gauge theories are defined by </p>
<div class="equation" data-bbox="98 253 500 289">$$
\begin{array}{r l}{\tau_{\Gamma}:\;\;\overline{{\mathcal{A}}}\;\;\longrightarrow\;\;\;\;\overline{{\mathcal{A}}}_{\Gamma}\equiv\mathbf{G}^{E}\qquad\qquad\mathrm{and}\qquad\pi_{\Gamma}:\;\;\overline{{\mathcal{G}}}\;\;\longrightarrow\;\;\;\;\overline{{\mathcal{G}}}_{\Gamma}\equiv\mathbf{G}^{V}.}\\ {\overline{{\mathcal{A}}}\;\;\longmapsto\;\;\left(h_{\overline{{A}}}(e_{1}),\ldots,h_{\overline{{A}}}(e_{E})\right)\qquad\qquad\qquad\quad\overline{{g}}\;\;\longmapsto\;\;\left(g_{v_{1}},\ldots,g_{v_{V}}\right)}\end{array}
$$</div>
<p data-bbox="77 287 538 330">The topologies on $\overline{{\mathcal{A}}}$ and $\overline{{g}}$ are the topologies generated by these projections. Using these topologies the action $\Theta:\overline{{\mathcal{A}}}\times\overline{{\mathcal{G}}}\longrightarrow\overline{{\mathcal{A}}}$ defined by (1) is continuous. Since $\mathbf{G}$ is compact Lie, $\overline{{\mathcal{A}}}$ and $\mathcal{G}$ are compact Hausdorff spaces and consequently completely regular. </p>
<p data-bbox="65 330 538 401">• The holonomy group $\mathbf{H}_{\overline{{A}}}$ of a connection $\overline{{A}}$ is defined by $\mathbf{H}_{\overline{{A}}}:=h_{\overline{{A}}}(\mathcal{H}\mathcal{G})\subseteq\mathbf{G}$ , its centralizer is denoted by $Z(\mathbf{H}_{\overline{{A}}})$ . The stabilizer of a connection $\overline{{A}}\in\overline{{A}}$ under the action of $\overline{{g}}$ is denoted by $\mathbf{B}(\overline{{A}})$ . We have ${\overline{{g}}}\,\in\,{\bf B}({\overline{{A}}})$ iff $g_{m}\,\in\,Z(\mathbf{H}_{\overline{{A}}})$ and for all $x\,\in\,M$ there is a path $\gamma\in\mathcal{P}_{m x}$ with $h_{\overline{{{A}}}}(\gamma)\,=\,g_{m}^{-1}h_{\overline{{{A}}}}(\gamma)g_{x}$ . In [9] we proved that $\mathbf{B}(\overline{{A}})$ and $Z(\mathbf{H}_{\overline{{A}}})$ are homeomorphic. </p>
<p data-bbox="65 402 537 432">• The type of a gauge orbit $\mathbf{E}_{\overline{{A}}}:=\overline{{A}}\circ\overline{{\mathcal{G}}}$ is the centralizer of the holonomy group of $\overline{{A}}$ modulo conjugation in $\mathbf{G}$ . (An equivalent definition uses the stabilizer $\mathbf{B}(\overline{{A}})$ itself.) </p>
<h1 data-bbox="63 452 313 472">3 Partial Ordering of Types </h1>
<p data-bbox="63 482 537 512">Definition 3.1 A subgroup $U$ of $\mathbf{G}$ is called Howe subgroup iff there is a set $V\subseteq\mathbf{G}$ with $U=Z(V)$ . </p>
<p data-bbox="63 524 534 540">Analogously to the general theory we define a partial ordering for the gauge orbit types [8]. </p>
<p data-bbox="63 548 536 593">Definition 3.2 Let $\tau$ denote the set of all Howe subgroups of $\mathbf{G}$ . Let $t_{1},t_{2}\in\mathcal{T}$ . Then $t_{1}\leq t_{2}$ holds iff there are $\mathbf{G}_{1}\in t_{1}$ and $\mathbf{G}_{2}\in t_{2}$ with $\mathbf{G}_{1}\supseteq\mathbf{G}_{2}$ . </p>
<p data-bbox="62 603 162 617">Obviously, we have </p>
<p data-bbox="64 626 538 657">Lemma 3.1 The maximal element in $\tau$ is the class $t_{\mathrm{max}}$ of the center $Z(\mathbf{G})$ of $\mathbf{G}$ , the minimal is the class $t_{\mathrm{min}}$ of $\mathbf{G}$ itself. </p>
<p data-bbox="75 663 410 678">1Homomorphism means $h_{\overline{{{A}}}}(\gamma_{1}\gamma_{2})=h_{\overline{{{A}}}}(\gamma_{1})h_{\overline{{{A}}}}(\gamma_{2})$ supposed $\gamma_{1}\gamma_{2}$ is defined. </p>
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[{"type": "text", "text": "", "page_idx": 3}, {"type": "text", "text": "• A generalized connection ${\\overline{{A}}}\\in{\\overline{{A}}}$ is a homomorphism1 $h_{\\overline{{A}}}:{\\mathcal{P}}\\longrightarrow\\mathbf{G}$ . (We usually write $h_{\\overline{{A}}}$ synonymously for $\\overline{{A}}$ .) A generalized gauge transform ${\\overline{{g}}}\\,\\in{\\overline{{\\mathcal{G}}}}$ is a map $\\overline{{g}}:M\\longrightarrow\\mathbf{G}$ . The value $\\overline{{g}}(x)$ of the gauge transform in the point $x$ is usually denoted by $g_{x}$ . The action of $\\overline{{g}}$ on $\\overline{{\\mathcal{A}}}$ is given by ", "page_idx": 3}, {"type": "equation", "text": "$$\nh_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "We have $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}\\cong\\mathrm{Hom}(\\mathcal{H}\\mathcal{G},\\mathbf{G})/\\mathrm{Ad}$ . ", "page_idx": 3}, {"type": "text", "text": "• Now, let $\\Gamma$ be a graph with $\\mathbf{E}(\\Gamma)\\,=\\,\\{e_{1},\\dots,e_{E}\\}$ being the set of edges and ${\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=$ $\\{v_{1},\\ldots,v_{V}\\}$ the set of vertices. The projections onto the lattice gauge theories are defined by ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "The topologies on $\\overline{{\\mathcal{A}}}$ and $\\overline{{g}}$ are the topologies generated by these projections. Using these topologies the action $\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}$ defined by (1) is continuous. Since $\\mathbf{G}$ is compact Lie, $\\overline{{\\mathcal{A}}}$ and $\\mathcal{G}$ are compact Hausdorff spaces and consequently completely regular. ", "page_idx": 3}, {"type": "text", "text": "• The holonomy group $\\mathbf{H}_{\\overline{{A}}}$ of a connection $\\overline{{A}}$ is defined by $\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}$ , its centralizer is denoted by $Z(\\mathbf{H}_{\\overline{{A}}})$ . The stabilizer of a connection $\\overline{{A}}\\in\\overline{{A}}$ under the action of $\\overline{{g}}$ is denoted by $\\mathbf{B}(\\overline{{A}})$ . We have ${\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})$ iff $g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})$ and for all $x\\,\\in\\,M$ there is a path $\\gamma\\in\\mathcal{P}_{m x}$ with $h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}$ . In [9] we proved that $\\mathbf{B}(\\overline{{A}})$ and $Z(\\mathbf{H}_{\\overline{{A}}})$ are homeomorphic. ", "page_idx": 3}, {"type": "text", "text": "• The type of a gauge orbit $\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ is the centralizer of the holonomy group of $\\overline{{A}}$ modulo conjugation in $\\mathbf{G}$ . (An equivalent definition uses the stabilizer $\\mathbf{B}(\\overline{{A}})$ itself.) ", "page_idx": 3}, {"type": "text", "text": "3 Partial Ordering of Types ", "text_level": 1, "page_idx": 3}, {"type": "text", "text": "Definition 3.1 A subgroup $U$ of $\\mathbf{G}$ is called Howe subgroup iff there is a set $V\\subseteq\\mathbf{G}$ with $U=Z(V)$ . ", "page_idx": 3}, {"type": "text", "text": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8]. ", "page_idx": 3}, {"type": "text", "text": "Definition 3.2 Let $\\tau$ denote the set of all Howe subgroups of $\\mathbf{G}$ . Let $t_{1},t_{2}\\in\\mathcal{T}$ . Then $t_{1}\\leq t_{2}$ holds iff there are $\\mathbf{G}_{1}\\in t_{1}$ and $\\mathbf{G}_{2}\\in t_{2}$ with $\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}$ . ", "page_idx": 3}, {"type": "text", "text": "Obviously, we have ", "page_idx": 3}, {"type": "text", "text": "Lemma 3.1 The maximal element in $\\tau$ is the class $t_{\\mathrm{max}}$ of the center $Z(\\mathbf{G})$ of $\\mathbf{G}$ , the minimal is the class $t_{\\mathrm{min}}$ of $\\mathbf{G}$ itself. ", "page_idx": 3}, {"type": "text", "text": "1Homomorphism means $h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})$ supposed $\\gamma_{1}\\gamma_{2}$ is defined. ", "page_idx": 3}]
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{"preproc_blocks": [{"type": "text", "bbox": [77, 14, 538, 115], "lines": [{"bbox": [79, 16, 538, 33], "spans": [{"bbox": [79, 16, 538, 33], "score": 1.0, "content": "different paths intersect each other at most in their end points. Paths in a graph are called", "type": "text"}], "index": 0}, {"bbox": [80, 32, 536, 46], "spans": [{"bbox": [80, 32, 536, 46], "score": 1.0, "content": "simple. A path is called finite iff it is up to the parametrization a finite product of simple", "type": "text"}], "index": 1}, {"bbox": [79, 46, 538, 61], "spans": [{"bbox": [79, 46, 538, 61], "score": 1.0, "content": "paths. Two paths are equivalent iff the first one can be reconstructed from the second", "type": "text"}], "index": 2}, {"bbox": [79, 60, 538, 74], "spans": [{"bbox": [79, 60, 538, 74], "score": 1.0, "content": "one by a sequence of reparametrizations or of insertions or deletions of retracings. We will", "type": "text"}], "index": 3}, {"bbox": [80, 75, 537, 89], "spans": [{"bbox": [80, 75, 537, 89], "score": 1.0, "content": "only consider equivalence classes of finite paths and graphs. The set of (classes of) paths", "type": "text"}], "index": 4}, {"bbox": [78, 89, 537, 104], "spans": [{"bbox": [78, 89, 153, 104], "score": 1.0, "content": "is denoted by ", "type": "text"}, {"bbox": [153, 91, 162, 100], "score": 0.9, "content": "\\mathcal{P}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [163, 89, 269, 104], "score": 1.0, "content": ", that of paths from ", "type": "text"}, {"bbox": [270, 94, 276, 100], "score": 0.88, "content": "x", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [276, 89, 295, 104], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [295, 94, 301, 102], "score": 0.88, "content": "y", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [302, 89, 322, 104], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [322, 91, 340, 103], "score": 0.93, "content": "\\mathcal{P}_{x y}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [341, 89, 537, 104], "score": 1.0, "content": " and that of loops (paths with a fixed", "type": "text"}], "index": 5}, {"bbox": [78, 103, 404, 119], "spans": [{"bbox": [78, 103, 213, 119], "score": 1.0, "content": "initial and terminal point ", "type": "text"}, {"bbox": [213, 109, 224, 114], "score": 0.86, "content": "m", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [225, 103, 248, 119], "score": 1.0, "content": ") by ", "type": "text"}, {"bbox": [249, 106, 267, 116], "score": 0.92, "content": "\\mathcal{H G}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [267, 103, 404, 119], "score": 1.0, "content": ", the so-called hoop group.", "type": "text"}], "index": 6}], "index": 3}, {"type": "text", "bbox": [65, 116, 538, 173], "lines": [{"bbox": [61, 117, 538, 134], "spans": [{"bbox": [61, 117, 212, 134], "score": 1.0, "content": "• A generalized connection ", "type": "text"}, {"bbox": [213, 118, 248, 129], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [248, 117, 361, 134], "score": 1.0, "content": " is a homomorphism1 ", "type": "text"}, {"bbox": [362, 120, 435, 132], "score": 0.9, "content": "h_{\\overline{{A}}}:{\\mathcal{P}}\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [435, 117, 538, 134], "score": 1.0, "content": ". (We usually write", "type": "text"}], "index": 7}, {"bbox": [79, 131, 538, 149], "spans": [{"bbox": [79, 135, 95, 146], "score": 0.92, "content": "h_{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [95, 131, 192, 149], "score": 1.0, "content": " synonymously for ", "type": "text"}, {"bbox": [192, 133, 201, 143], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [202, 131, 378, 149], "score": 1.0, "content": ".) A generalized gauge transform ", "type": "text"}, {"bbox": [378, 133, 410, 146], "score": 0.92, "content": "{\\overline{{g}}}\\,\\in{\\overline{{\\mathcal{G}}}}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [410, 131, 462, 149], "score": 1.0, "content": " is a map ", "type": "text"}, {"bbox": [463, 135, 533, 145], "score": 0.9, "content": "\\overline{{g}}:M\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [533, 131, 538, 149], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [77, 145, 538, 164], "spans": [{"bbox": [77, 145, 132, 164], "score": 1.0, "content": "The value", "type": "text"}, {"bbox": [133, 148, 155, 160], "score": 0.94, "content": "\\overline{{g}}(x)", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [155, 145, 340, 164], "score": 1.0, "content": " of the gauge transform in the point ", "type": "text"}, {"bbox": [340, 152, 347, 158], "score": 0.87, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 145, 461, 164], "score": 1.0, "content": " is usually denoted by ", "type": "text"}, {"bbox": [462, 152, 473, 160], "score": 0.91, "content": "g_{x}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [473, 145, 538, 164], "score": 1.0, "content": ". The action", "type": "text"}], "index": 9}, {"bbox": [79, 160, 190, 177], "spans": [{"bbox": [79, 160, 92, 177], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [93, 162, 101, 173], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [101, 160, 120, 177], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [121, 162, 131, 172], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [131, 160, 190, 177], "score": 1.0, "content": " is given by", "type": "text"}], "index": 10}], "index": 8.5}, {"type": "interline_equation", "bbox": [203, 177, 412, 195], "lines": [{"bbox": [203, 177, 412, 195], "spans": [{"bbox": [203, 177, 412, 195], "score": 0.91, "content": "h_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [77, 195, 259, 210], "lines": [{"bbox": [79, 196, 258, 212], "spans": [{"bbox": [79, 196, 127, 211], "score": 1.0, "content": "We have ", "type": "text"}, {"bbox": [127, 198, 255, 212], "score": 0.95, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}\\cong\\mathrm{Hom}(\\mathcal{H}\\mathcal{G},\\mathbf{G})/\\mathrm{Ad}", "type": "inline_equation", "height": 14, "width": 128}, {"bbox": [255, 196, 258, 211], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [66, 210, 538, 250], "lines": [{"bbox": [63, 210, 537, 230], "spans": [{"bbox": [63, 210, 129, 230], "score": 1.0, "content": "• Now, let ", "type": "text"}, {"bbox": [129, 214, 137, 223], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [138, 210, 233, 230], "score": 1.0, "content": " be a graph with ", "type": "text"}, {"bbox": [234, 213, 343, 226], "score": 0.92, "content": "\\mathbf{E}(\\Gamma)\\,=\\,\\{e_{1},\\dots,e_{E}\\}", "type": "inline_equation", "height": 13, "width": 109}, {"bbox": [344, 210, 494, 230], "score": 1.0, "content": " being the set of edges and ", "type": "text"}, {"bbox": [494, 213, 537, 226], "score": 0.92, "content": "{\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=", "type": "inline_equation", "height": 13, "width": 43}], "index": 13}, {"bbox": [80, 226, 537, 242], "spans": [{"bbox": [80, 228, 144, 240], "score": 0.93, "content": "\\{v_{1},\\ldots,v_{V}\\}", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [144, 226, 537, 242], "score": 1.0, "content": " the set of vertices. The projections onto the lattice gauge theories are defined", "type": "text"}], "index": 14}, {"bbox": [80, 242, 93, 254], "spans": [{"bbox": [80, 242, 93, 254], "score": 1.0, "content": "by", "type": "text"}], "index": 15}], "index": 14}, {"type": "interline_equation", "bbox": [98, 253, 500, 289], "lines": [{"bbox": [98, 253, 500, 289], "spans": [{"bbox": [98, 253, 500, 289], "score": 0.86, "content": "\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [77, 287, 538, 330], "lines": [{"bbox": [79, 288, 537, 304], "spans": [{"bbox": [79, 288, 173, 304], "score": 1.0, "content": "The topologies on ", "type": "text"}, {"bbox": [174, 289, 184, 299], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [184, 288, 209, 304], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [209, 289, 217, 301], "score": 0.87, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [218, 288, 537, 304], "score": 1.0, "content": " are the topologies generated by these projections. Using these", "type": "text"}], "index": 17}, {"bbox": [79, 302, 538, 318], "spans": [{"bbox": [79, 302, 190, 318], "score": 1.0, "content": "topologies the action ", "type": "text"}, {"bbox": [191, 303, 279, 315], "score": 0.92, "content": "\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [280, 302, 466, 318], "score": 1.0, "content": " defined by (1) is continuous. Since ", "type": "text"}, {"bbox": [466, 305, 477, 314], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [477, 302, 538, 318], "score": 1.0, "content": " is compact", "type": "text"}], "index": 18}, {"bbox": [79, 317, 496, 333], "spans": [{"bbox": [79, 317, 102, 333], "score": 1.0, "content": "Lie, ", "type": "text"}, {"bbox": [102, 318, 112, 328], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [113, 317, 138, 333], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [139, 318, 147, 330], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [147, 317, 496, 333], "score": 1.0, "content": " are compact Hausdorff spaces and consequently completely regular.", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [65, 330, 538, 401], "lines": [{"bbox": [65, 331, 537, 349], "spans": [{"bbox": [65, 331, 192, 349], "score": 1.0, "content": "• The holonomy group ", "type": "text"}, {"bbox": [192, 334, 211, 346], "score": 0.91, "content": "\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [211, 331, 297, 349], "score": 1.0, "content": " of a connection ", "type": "text"}, {"bbox": [298, 333, 307, 343], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [307, 331, 381, 349], "score": 1.0, "content": " is defined by ", "type": "text"}, {"bbox": [382, 333, 490, 346], "score": 0.92, "content": "\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 108}, {"bbox": [490, 331, 537, 349], "score": 1.0, "content": ", its cen-", "type": "text"}], "index": 20}, {"bbox": [79, 346, 539, 361], "spans": [{"bbox": [79, 346, 194, 361], "score": 1.0, "content": "tralizer is denoted by ", "type": "text"}, {"bbox": [194, 348, 230, 361], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [230, 346, 397, 361], "score": 1.0, "content": ". The stabilizer of a connection ", "type": "text"}, {"bbox": [397, 347, 432, 358], "score": 0.9, "content": "\\overline{{A}}\\in\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [433, 346, 539, 361], "score": 1.0, "content": " under the action of", "type": "text"}], "index": 21}, {"bbox": [79, 360, 538, 376], "spans": [{"bbox": [79, 361, 88, 373], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [88, 360, 167, 376], "score": 1.0, "content": " is denoted by ", "type": "text"}, {"bbox": [167, 361, 195, 375], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [195, 360, 254, 376], "score": 1.0, "content": ". We have ", "type": "text"}, {"bbox": [254, 361, 306, 375], "score": 0.94, "content": "{\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [306, 360, 325, 376], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [325, 362, 392, 375], "score": 0.93, "content": "g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [392, 360, 455, 376], "score": 1.0, "content": " and for all ", "type": "text"}, {"bbox": [455, 362, 493, 372], "score": 0.89, "content": "x\\,\\in\\,M", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [493, 360, 538, 376], "score": 1.0, "content": " there is", "type": "text"}], "index": 22}, {"bbox": [77, 375, 538, 392], "spans": [{"bbox": [77, 375, 118, 392], "score": 1.0, "content": "a path ", "type": "text"}, {"bbox": [118, 378, 163, 389], "score": 0.92, "content": "\\gamma\\in\\mathcal{P}_{m x}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [163, 375, 194, 392], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [194, 376, 299, 389], "score": 0.93, "content": "h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [300, 375, 424, 392], "score": 1.0, "content": ". In [9] we proved that ", "type": "text"}, {"bbox": [424, 375, 452, 389], "score": 0.92, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [453, 375, 480, 392], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [480, 376, 516, 389], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [516, 375, 538, 392], "score": 1.0, "content": " are", "type": "text"}], "index": 23}, {"bbox": [79, 390, 158, 404], "spans": [{"bbox": [79, 390, 158, 404], "score": 1.0, "content": "homeomorphic.", "type": "text"}], "index": 24}], "index": 22}, {"type": "text", "bbox": [65, 402, 537, 432], "lines": [{"bbox": [62, 403, 536, 419], "spans": [{"bbox": [62, 403, 222, 419], "score": 1.0, "content": "• The type of a gauge orbit ", "type": "text"}, {"bbox": [222, 405, 291, 418], "score": 0.94, "content": "\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [291, 403, 527, 419], "score": 1.0, "content": " is the centralizer of the holonomy group of ", "type": "text"}, {"bbox": [527, 405, 536, 415], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}], "index": 25}, {"bbox": [78, 418, 508, 433], "spans": [{"bbox": [78, 418, 198, 433], "score": 1.0, "content": "modulo conjugation in ", "type": "text"}, {"bbox": [199, 421, 209, 430], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [210, 418, 443, 433], "score": 1.0, "content": ". (An equivalent definition uses the stabilizer ", "type": "text"}, {"bbox": [444, 418, 472, 433], "score": 0.92, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 28}, {"bbox": [472, 418, 508, 433], "score": 1.0, "content": " itself.)", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "title", "bbox": [63, 452, 313, 472], "lines": [{"bbox": [63, 454, 311, 474], "spans": [{"bbox": [63, 457, 74, 470], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [90, 454, 311, 474], "score": 1.0, "content": "Partial Ordering of Types", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [63, 482, 537, 512], "lines": [{"bbox": [62, 485, 537, 501], "spans": [{"bbox": [62, 485, 216, 501], "score": 1.0, "content": "Definition 3.1 A subgroup ", "type": "text"}, {"bbox": [216, 487, 225, 496], "score": 0.88, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [226, 485, 241, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [241, 487, 252, 496], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [253, 485, 473, 501], "score": 1.0, "content": " is called Howe subgroup iff there is a set ", "type": "text"}, {"bbox": [474, 487, 510, 497], "score": 0.91, "content": "V\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [510, 485, 537, 501], "score": 1.0, "content": " with", "type": "text"}], "index": 28}, {"bbox": [154, 498, 211, 515], "spans": [{"bbox": [154, 501, 206, 513], "score": 0.94, "content": "U=Z(V)", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [207, 498, 211, 515], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "text", "bbox": [63, 524, 534, 540], "lines": [{"bbox": [62, 526, 533, 543], "spans": [{"bbox": [62, 526, 533, 543], "score": 1.0, "content": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8].", "type": "text"}], "index": 30}], "index": 30}, {"type": "text", "bbox": [63, 548, 536, 593], "lines": [{"bbox": [62, 551, 408, 566], "spans": [{"bbox": [62, 551, 174, 566], "score": 1.0, "content": "Definition 3.2 Let ", "type": "text"}, {"bbox": [174, 553, 185, 563], "score": 0.88, "content": "\\tau", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [185, 551, 394, 566], "score": 1.0, "content": " denote the set of all Howe subgroups of ", "type": "text"}, {"bbox": [394, 553, 405, 562], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [405, 551, 408, 566], "score": 1.0, "content": ".", "type": "text"}], "index": 31}, {"bbox": [153, 566, 537, 580], "spans": [{"bbox": [153, 566, 174, 580], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [175, 568, 224, 578], "score": 0.92, "content": "t_{1},t_{2}\\in\\mathcal{T}", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [225, 566, 264, 580], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [264, 568, 300, 578], "score": 0.93, "content": "t_{1}\\leq t_{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [300, 566, 400, 580], "score": 1.0, "content": " holds iff there are ", "type": "text"}, {"bbox": [400, 568, 441, 578], "score": 0.92, "content": "\\mathbf{G}_{1}\\in t_{1}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [441, 566, 468, 580], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [468, 568, 509, 578], "score": 0.92, "content": "\\mathbf{G}_{2}\\in t_{2}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [509, 566, 537, 580], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [154, 578, 206, 597], "spans": [{"bbox": [154, 582, 200, 593], "score": 0.91, "content": "\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 578, 206, 597], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32}, {"type": "text", "bbox": [62, 603, 162, 617], "lines": [{"bbox": [63, 604, 162, 618], "spans": [{"bbox": [63, 604, 162, 618], "score": 1.0, "content": "Obviously, we have", "type": "text"}], "index": 34}], "index": 34}, {"type": "text", "bbox": [64, 626, 538, 657], "lines": [{"bbox": [61, 628, 538, 645], "spans": [{"bbox": [61, 628, 273, 645], "score": 1.0, "content": "Lemma 3.1 The maximal element in ", "type": "text"}, {"bbox": [273, 631, 284, 640], "score": 0.91, "content": "\\tau", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [284, 628, 352, 645], "score": 1.0, "content": " is the class ", "type": "text"}, {"bbox": [352, 632, 373, 641], "score": 0.89, "content": "t_{\\mathrm{max}}", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [374, 628, 451, 645], "score": 1.0, "content": " of the center ", "type": "text"}, {"bbox": [451, 630, 480, 643], "score": 0.92, "content": "Z(\\mathbf{G})", "type": "inline_equation", "height": 13, 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"spans": [{"bbox": [75, 664, 183, 681], "score": 1.0, "content": "1Homomorphism means ", "type": "text"}, {"bbox": [184, 667, 295, 678], "score": 0.92, "content": "h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})", "type": "inline_equation", "height": 11, "width": 111}, {"bbox": [296, 664, 342, 681], "score": 1.0, "content": " supposed ", "type": "text"}, {"bbox": [342, 670, 361, 677], "score": 0.9, "content": "\\gamma_{1}\\gamma_{2}", "type": "inline_equation", "height": 7, "width": 19}, {"bbox": [361, 664, 409, 681], "score": 1.0, "content": " is defined.", "type": "text"}], "index": 37}], "index": 37}], "layout_bboxes": [], "page_idx": 3, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [203, 177, 412, 195], "lines": [{"bbox": [203, 177, 412, 195], "spans": [{"bbox": [203, 177, 412, 195], "score": 0.91, "content": "h_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [98, 253, 500, 289], "lines": [{"bbox": [98, 253, 500, 289], "spans": [{"bbox": [98, 253, 500, 289], "score": 0.86, "content": "\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}], "discarded_blocks": [{"type": "discarded", "bbox": [296, 705, 303, 715], "lines": [{"bbox": [296, 706, 304, 717], "spans": [{"bbox": [296, 706, 304, 717], "score": 1.0, "content": "4", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [77, 14, 538, 115], "lines": [], "index": 3, "bbox_fs": [78, 16, 538, 119], "lines_deleted": true}, {"type": "text", "bbox": [65, 116, 538, 173], "lines": [{"bbox": [61, 117, 538, 134], "spans": [{"bbox": [61, 117, 212, 134], "score": 1.0, "content": "• A generalized connection ", "type": "text"}, {"bbox": [213, 118, 248, 129], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [248, 117, 361, 134], "score": 1.0, "content": " is a homomorphism1 ", "type": "text"}, {"bbox": [362, 120, 435, 132], "score": 0.9, "content": "h_{\\overline{{A}}}:{\\mathcal{P}}\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 73}, {"bbox": [435, 117, 538, 134], "score": 1.0, "content": ". (We usually write", "type": "text"}], "index": 7}, {"bbox": [79, 131, 538, 149], "spans": [{"bbox": [79, 135, 95, 146], "score": 0.92, "content": "h_{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [95, 131, 192, 149], "score": 1.0, "content": " synonymously for ", "type": "text"}, {"bbox": [192, 133, 201, 143], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [202, 131, 378, 149], "score": 1.0, "content": ".) A generalized gauge transform ", "type": "text"}, {"bbox": [378, 133, 410, 146], "score": 0.92, "content": "{\\overline{{g}}}\\,\\in{\\overline{{\\mathcal{G}}}}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [410, 131, 462, 149], "score": 1.0, "content": " is a map ", "type": "text"}, {"bbox": [463, 135, 533, 145], "score": 0.9, "content": "\\overline{{g}}:M\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [533, 131, 538, 149], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [77, 145, 538, 164], "spans": [{"bbox": [77, 145, 132, 164], "score": 1.0, "content": "The value", "type": "text"}, {"bbox": [133, 148, 155, 160], "score": 0.94, "content": "\\overline{{g}}(x)", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [155, 145, 340, 164], "score": 1.0, "content": " of the gauge transform in the point ", "type": "text"}, {"bbox": [340, 152, 347, 158], "score": 0.87, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 145, 461, 164], "score": 1.0, "content": " is usually denoted by ", "type": "text"}, {"bbox": [462, 152, 473, 160], "score": 0.91, "content": "g_{x}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [473, 145, 538, 164], "score": 1.0, "content": ". The action", "type": "text"}], "index": 9}, {"bbox": [79, 160, 190, 177], "spans": [{"bbox": [79, 160, 92, 177], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [93, 162, 101, 173], "score": 0.91, "content": "\\overline{{g}}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [101, 160, 120, 177], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [121, 162, 131, 172], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [131, 160, 190, 177], "score": 1.0, "content": " is given by", "type": "text"}], "index": 10}], "index": 8.5, "bbox_fs": [61, 117, 538, 177]}, {"type": "interline_equation", "bbox": [203, 177, 412, 195], "lines": [{"bbox": [203, 177, 412, 195], "spans": [{"bbox": [203, 177, 412, 195], "score": 0.91, "content": "h_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [77, 195, 259, 210], "lines": [{"bbox": [79, 196, 258, 212], "spans": [{"bbox": [79, 196, 127, 211], "score": 1.0, "content": "We have ", "type": "text"}, {"bbox": [127, 198, 255, 212], "score": 0.95, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}\\cong\\mathrm{Hom}(\\mathcal{H}\\mathcal{G},\\mathbf{G})/\\mathrm{Ad}", "type": "inline_equation", "height": 14, "width": 128}, {"bbox": [255, 196, 258, 211], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [79, 196, 258, 212]}, {"type": "text", "bbox": [66, 210, 538, 250], "lines": [{"bbox": [63, 210, 537, 230], "spans": [{"bbox": [63, 210, 129, 230], "score": 1.0, "content": "• Now, let ", "type": "text"}, {"bbox": [129, 214, 137, 223], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [138, 210, 233, 230], "score": 1.0, "content": " be a graph with ", "type": "text"}, {"bbox": [234, 213, 343, 226], "score": 0.92, "content": "\\mathbf{E}(\\Gamma)\\,=\\,\\{e_{1},\\dots,e_{E}\\}", "type": "inline_equation", "height": 13, "width": 109}, {"bbox": [344, 210, 494, 230], "score": 1.0, "content": " being the set of edges and ", "type": "text"}, {"bbox": [494, 213, 537, 226], "score": 0.92, "content": "{\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=", "type": "inline_equation", "height": 13, "width": 43}], "index": 13}, {"bbox": [80, 226, 537, 242], "spans": [{"bbox": [80, 228, 144, 240], "score": 0.93, "content": "\\{v_{1},\\ldots,v_{V}\\}", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [144, 226, 537, 242], "score": 1.0, "content": " the set of vertices. The projections onto the lattice gauge theories are defined", "type": "text"}], "index": 14}, {"bbox": [80, 242, 93, 254], "spans": [{"bbox": [80, 242, 93, 254], "score": 1.0, "content": "by", "type": "text"}], "index": 15}], "index": 14, "bbox_fs": [63, 210, 537, 254]}, {"type": "interline_equation", "bbox": [98, 253, 500, 289], "lines": [{"bbox": [98, 253, 500, 289], "spans": [{"bbox": [98, 253, 500, 289], "score": 0.86, "content": "\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [77, 287, 538, 330], "lines": [{"bbox": [79, 288, 537, 304], "spans": [{"bbox": [79, 288, 173, 304], "score": 1.0, "content": "The topologies on ", "type": "text"}, {"bbox": [174, 289, 184, 299], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [184, 288, 209, 304], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [209, 289, 217, 301], "score": 0.87, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [218, 288, 537, 304], "score": 1.0, "content": " are the topologies generated by these projections. Using these", "type": "text"}], "index": 17}, {"bbox": [79, 302, 538, 318], "spans": [{"bbox": [79, 302, 190, 318], "score": 1.0, "content": "topologies the action ", "type": "text"}, {"bbox": [191, 303, 279, 315], "score": 0.92, "content": "\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [280, 302, 466, 318], "score": 1.0, "content": " defined by (1) is continuous. Since ", "type": "text"}, {"bbox": [466, 305, 477, 314], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [477, 302, 538, 318], "score": 1.0, "content": " is compact", "type": "text"}], "index": 18}, {"bbox": [79, 317, 496, 333], "spans": [{"bbox": [79, 317, 102, 333], "score": 1.0, "content": "Lie, ", "type": "text"}, {"bbox": [102, 318, 112, 328], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [113, 317, 138, 333], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [139, 318, 147, 330], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [147, 317, 496, 333], "score": 1.0, "content": " are compact Hausdorff spaces and consequently completely regular.", "type": "text"}], "index": 19}], "index": 18, "bbox_fs": [79, 288, 538, 333]}, {"type": "text", "bbox": [65, 330, 538, 401], "lines": [{"bbox": [65, 331, 537, 349], "spans": [{"bbox": [65, 331, 192, 349], "score": 1.0, "content": "• The holonomy group ", "type": "text"}, {"bbox": [192, 334, 211, 346], "score": 0.91, "content": "\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [211, 331, 297, 349], "score": 1.0, "content": " of a connection ", "type": "text"}, {"bbox": [298, 333, 307, 343], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [307, 331, 381, 349], "score": 1.0, "content": " is defined by ", "type": "text"}, {"bbox": [382, 333, 490, 346], "score": 0.92, "content": "\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 108}, {"bbox": [490, 331, 537, 349], "score": 1.0, "content": ", its cen-", "type": "text"}], "index": 20}, {"bbox": [79, 346, 539, 361], "spans": [{"bbox": [79, 346, 194, 361], "score": 1.0, "content": "tralizer is denoted by ", "type": "text"}, {"bbox": [194, 348, 230, 361], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [230, 346, 397, 361], "score": 1.0, "content": ". The stabilizer of a connection ", "type": "text"}, {"bbox": [397, 347, 432, 358], "score": 0.9, "content": "\\overline{{A}}\\in\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [433, 346, 539, 361], "score": 1.0, "content": " under the action of", "type": "text"}], "index": 21}, {"bbox": [79, 360, 538, 376], "spans": [{"bbox": [79, 361, 88, 373], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [88, 360, 167, 376], "score": 1.0, "content": " is denoted by ", "type": "text"}, {"bbox": [167, 361, 195, 375], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [195, 360, 254, 376], "score": 1.0, "content": ". We have ", "type": "text"}, {"bbox": [254, 361, 306, 375], "score": 0.94, "content": "{\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [306, 360, 325, 376], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [325, 362, 392, 375], "score": 0.93, "content": "g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [392, 360, 455, 376], "score": 1.0, "content": " and for all ", "type": "text"}, {"bbox": [455, 362, 493, 372], "score": 0.89, "content": "x\\,\\in\\,M", "type": "inline_equation", "height": 10, "width": 38}, {"bbox": [493, 360, 538, 376], "score": 1.0, "content": " there is", "type": "text"}], "index": 22}, {"bbox": [77, 375, 538, 392], "spans": [{"bbox": [77, 375, 118, 392], "score": 1.0, "content": "a path ", "type": "text"}, {"bbox": [118, 378, 163, 389], "score": 0.92, "content": "\\gamma\\in\\mathcal{P}_{m x}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [163, 375, 194, 392], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [194, 376, 299, 389], "score": 0.93, "content": "h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [300, 375, 424, 392], "score": 1.0, "content": ". In [9] we proved that ", "type": "text"}, {"bbox": [424, 375, 452, 389], "score": 0.92, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [453, 375, 480, 392], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [480, 376, 516, 389], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [516, 375, 538, 392], "score": 1.0, "content": " are", "type": "text"}], "index": 23}, {"bbox": [79, 390, 158, 404], "spans": [{"bbox": [79, 390, 158, 404], "score": 1.0, "content": "homeomorphic.", "type": "text"}], "index": 24}], "index": 22, "bbox_fs": [65, 331, 539, 404]}, {"type": "text", "bbox": [65, 402, 537, 432], "lines": [{"bbox": [62, 403, 536, 419], "spans": [{"bbox": [62, 403, 222, 419], "score": 1.0, "content": "• The type of a gauge orbit ", "type": "text"}, {"bbox": [222, 405, 291, 418], "score": 0.94, "content": "\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [291, 403, 527, 419], "score": 1.0, "content": " is the centralizer of the holonomy group of ", "type": "text"}, {"bbox": [527, 405, 536, 415], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}], "index": 25}, {"bbox": [78, 418, 508, 433], "spans": [{"bbox": [78, 418, 198, 433], "score": 1.0, "content": "modulo conjugation in ", "type": "text"}, {"bbox": [199, 421, 209, 430], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [210, 418, 443, 433], "score": 1.0, "content": ". (An equivalent definition uses the stabilizer ", "type": "text"}, {"bbox": [444, 418, 472, 433], "score": 0.92, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 28}, {"bbox": [472, 418, 508, 433], "score": 1.0, "content": " itself.)", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [62, 403, 536, 433]}, {"type": "title", "bbox": [63, 452, 313, 472], "lines": [{"bbox": [63, 454, 311, 474], "spans": [{"bbox": [63, 457, 74, 470], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [90, 454, 311, 474], "score": 1.0, "content": "Partial Ordering of Types", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [63, 482, 537, 512], "lines": [{"bbox": [62, 485, 537, 501], "spans": [{"bbox": [62, 485, 216, 501], "score": 1.0, "content": "Definition 3.1 A subgroup ", "type": "text"}, {"bbox": [216, 487, 225, 496], "score": 0.88, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [226, 485, 241, 501], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [241, 487, 252, 496], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [253, 485, 473, 501], "score": 1.0, "content": " is called Howe subgroup iff there is a set ", "type": "text"}, {"bbox": [474, 487, 510, 497], "score": 0.91, "content": "V\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [510, 485, 537, 501], "score": 1.0, "content": " with", "type": "text"}], "index": 28}, {"bbox": [154, 498, 211, 515], "spans": [{"bbox": [154, 501, 206, 513], "score": 0.94, "content": "U=Z(V)", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [207, 498, 211, 515], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 28.5, "bbox_fs": [62, 485, 537, 515]}, {"type": "text", "bbox": [63, 524, 534, 540], "lines": [{"bbox": [62, 526, 533, 543], "spans": [{"bbox": [62, 526, 533, 543], "score": 1.0, "content": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8].", "type": "text"}], "index": 30}], "index": 30, "bbox_fs": [62, 526, 533, 543]}, {"type": "text", "bbox": [63, 548, 536, 593], "lines": [{"bbox": [62, 551, 408, 566], "spans": [{"bbox": [62, 551, 174, 566], "score": 1.0, "content": "Definition 3.2 Let ", "type": "text"}, {"bbox": [174, 553, 185, 563], "score": 0.88, "content": "\\tau", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [185, 551, 394, 566], "score": 1.0, "content": " denote the set of all Howe subgroups of ", "type": "text"}, {"bbox": [394, 553, 405, 562], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [405, 551, 408, 566], "score": 1.0, "content": ".", "type": "text"}], "index": 31}, {"bbox": [153, 566, 537, 580], "spans": [{"bbox": [153, 566, 174, 580], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [175, 568, 224, 578], "score": 0.92, "content": "t_{1},t_{2}\\in\\mathcal{T}", "type": "inline_equation", "height": 10, "width": 49}, {"bbox": [225, 566, 264, 580], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [264, 568, 300, 578], "score": 0.93, "content": "t_{1}\\leq t_{2}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [300, 566, 400, 580], "score": 1.0, "content": " holds iff there are ", "type": "text"}, {"bbox": [400, 568, 441, 578], "score": 0.92, "content": "\\mathbf{G}_{1}\\in t_{1}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [441, 566, 468, 580], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [468, 568, 509, 578], "score": 0.92, "content": "\\mathbf{G}_{2}\\in t_{2}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [509, 566, 537, 580], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [154, 578, 206, 597], "spans": [{"bbox": [154, 582, 200, 593], "score": 0.91, "content": "\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}", "type": "inline_equation", "height": 11, "width": 46}, {"bbox": [201, 578, 206, 597], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 32, "bbox_fs": [62, 551, 537, 597]}, {"type": "text", "bbox": [62, 603, 162, 617], "lines": [{"bbox": [63, 604, 162, 618], "spans": [{"bbox": [63, 604, 162, 618], "score": 1.0, "content": "Obviously, we have", "type": "text"}], "index": 34}], "index": 34, "bbox_fs": [63, 604, 162, 618]}, {"type": "text", "bbox": [64, 626, 538, 657], "lines": [{"bbox": [61, 628, 538, 645], "spans": [{"bbox": [61, 628, 273, 645], "score": 1.0, "content": "Lemma 3.1 The maximal element in ", "type": "text"}, {"bbox": [273, 631, 284, 640], "score": 0.91, "content": "\\tau", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [284, 628, 352, 645], "score": 1.0, "content": " is the class ", "type": "text"}, {"bbox": [352, 632, 373, 641], "score": 0.89, "content": "t_{\\mathrm{max}}", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [374, 628, 451, 645], "score": 1.0, "content": " of the center ", "type": "text"}, {"bbox": [451, 630, 480, 643], "score": 0.92, "content": "Z(\\mathbf{G})", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [480, 628, 499, 645], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [500, 631, 511, 640], "score": 0.85, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [511, 628, 538, 645], "score": 1.0, "content": ", the", "type": "text"}], "index": 35}, {"bbox": [137, 644, 322, 658], "spans": [{"bbox": [137, 644, 243, 658], "score": 1.0, "content": "minimal is the class ", "type": "text"}, {"bbox": [244, 646, 262, 656], "score": 0.91, "content": "t_{\\mathrm{min}}", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [263, 644, 279, 658], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [280, 645, 290, 654], "score": 0.91, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [290, 644, 322, 658], "score": 1.0, "content": " itself.", "type": "text"}], "index": 36}], "index": 35.5, "bbox_fs": [61, 628, 538, 658]}, {"type": "text", "bbox": [75, 663, 410, 678], "lines": [{"bbox": [75, 664, 409, 681], "spans": [{"bbox": [75, 664, 183, 681], "score": 1.0, "content": "1Homomorphism means ", "type": "text"}, {"bbox": [184, 667, 295, 678], "score": 0.92, "content": "h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})", "type": "inline_equation", "height": 11, "width": 111}, {"bbox": [296, 664, 342, 681], "score": 1.0, "content": " supposed ", "type": "text"}, {"bbox": [342, 670, 361, 677], "score": 0.9, "content": "\\gamma_{1}\\gamma_{2}", "type": "inline_equation", "height": 7, "width": 19}, {"bbox": [361, 664, 409, 681], "score": 1.0, "content": " is defined.", "type": "text"}], "index": 37}], "index": 37, "bbox_fs": [75, 664, 409, 681]}]}
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[{"type": "text", "bbox": [77, 14, 538, 115], "content": "", "index": 0}, {"type": "text", "bbox": [65, 116, 538, 173], "content": "• A generalized connection is a homomorphism1 . (We usually write synonymously for .) A generalized gauge transform is a map . The value of the gauge transform in the point is usually denoted by . The action of on is given by", "index": 1}, {"type": "interline_equation", "bbox": [203, 177, 412, 195], "content": "", "index": 2}, {"type": "text", "bbox": [77, 195, 259, 210], "content": "We have .", "index": 3}, {"type": "text", "bbox": [66, 210, 538, 250], "content": "• Now, let be a graph with being the set of edges and the set of vertices. The projections onto the lattice gauge theories are defined by", "index": 4}, {"type": "interline_equation", "bbox": [98, 253, 500, 289], "content": "", "index": 5}, {"type": "text", "bbox": [77, 287, 538, 330], "content": "The topologies on and are the topologies generated by these projections. Using these topologies the action defined by (1) is continuous. Since is compact Lie, and are compact Hausdorff spaces and consequently completely regular.", "index": 6}, {"type": "text", "bbox": [65, 330, 538, 401], "content": "• The holonomy group of a connection is defined by , its cen- tralizer is denoted by . The stabilizer of a connection under the action of is denoted by . We have iff and for all there is a path with . In [9] we proved that and are homeomorphic.", "index": 7}, {"type": "text", "bbox": [65, 402, 537, 432], "content": "• The type of a gauge orbit is the centralizer of the holonomy group of modulo conjugation in . (An equivalent definition uses the stabilizer itself.)", "index": 8}, {"type": "title", "bbox": [63, 452, 313, 472], "content": "3 Partial Ordering of Types", "index": 9}, {"type": "text", "bbox": [63, 482, 537, 512], "content": "Definition 3.1 A subgroup of is called Howe subgroup iff there is a set with .", "index": 10}, {"type": "text", "bbox": [63, 524, 534, 540], "content": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8].", "index": 11}, {"type": "text", "bbox": [63, 548, 536, 593], "content": "Definition 3.2 Let denote the set of all Howe subgroups of . Let . Then holds iff there are and with .", "index": 12}, {"type": "text", "bbox": [62, 603, 162, 617], "content": "Obviously, we have", "index": 13}, {"type": "text", "bbox": [64, 626, 538, 657], "content": "Lemma 3.1 The maximal element in is the class of the center of , the minimal is the class of itself.", "index": 14}, {"type": "text", "bbox": [75, 663, 410, 678], "content": "1Homomorphism means supposed is defined.", "index": 15}]
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[{"bbox": [61, 117, 538, 134], "content": "• A generalized connection is a homomorphism1 . (We usually write", "parent_index": 1, "line_index": 0}, {"bbox": [79, 131, 538, 149], "content": "synonymously for .) A generalized gauge transform is a map .", "parent_index": 1, "line_index": 1}, {"bbox": [77, 145, 538, 164], "content": "The value of the gauge transform in the point is usually denoted by . The action", "parent_index": 1, "line_index": 2}, {"bbox": [79, 160, 190, 177], "content": "of on is given by", "parent_index": 1, "line_index": 3}, {"bbox": [79, 196, 258, 212], "content": "We have .", "parent_index": 3, "line_index": 0}, {"bbox": [63, 210, 537, 230], "content": "• Now, let be a graph with being the set of edges and", "parent_index": 4, "line_index": 0}, {"bbox": [80, 226, 537, 242], "content": "the set of vertices. The projections onto the lattice gauge theories are defined", "parent_index": 4, "line_index": 1}, {"bbox": [80, 242, 93, 254], "content": "by", "parent_index": 4, "line_index": 2}, {"bbox": [79, 288, 537, 304], "content": "The topologies on and are the topologies generated by these projections. Using these", "parent_index": 6, "line_index": 0}, {"bbox": [79, 302, 538, 318], "content": "topologies the action defined by (1) is continuous. Since is compact", "parent_index": 6, "line_index": 1}, {"bbox": [79, 317, 496, 333], "content": "Lie, and are compact Hausdorff spaces and consequently completely regular.", "parent_index": 6, "line_index": 2}, {"bbox": [65, 331, 537, 349], "content": "• The holonomy group of a connection is defined by , its cen-", "parent_index": 7, "line_index": 0}, {"bbox": [79, 346, 539, 361], "content": "tralizer is denoted by . The stabilizer of a connection under the action of", "parent_index": 7, "line_index": 1}, {"bbox": [79, 360, 538, 376], "content": "is denoted by . We have iff and for all there is", "parent_index": 7, "line_index": 2}, {"bbox": [77, 375, 538, 392], "content": "a path with . In [9] we proved that and are", "parent_index": 7, "line_index": 3}, {"bbox": [79, 390, 158, 404], "content": "homeomorphic.", "parent_index": 7, "line_index": 4}, {"bbox": [62, 403, 536, 419], "content": "• The type of a gauge orbit is the centralizer of the holonomy group of", "parent_index": 8, "line_index": 0}, {"bbox": [78, 418, 508, 433], "content": "modulo conjugation in . (An equivalent definition uses the stabilizer itself.)", "parent_index": 8, "line_index": 1}, {"bbox": [63, 454, 311, 474], "content": "3 Partial Ordering of Types", "parent_index": 9, "line_index": 0}, {"bbox": [62, 485, 537, 501], "content": "Definition 3.1 A subgroup of is called Howe subgroup iff there is a set with", "parent_index": 10, "line_index": 0}, {"bbox": [154, 498, 211, 515], "content": ".", "parent_index": 10, "line_index": 1}, {"bbox": [62, 526, 533, 543], "content": "Analogously to the general theory we define a partial ordering for the gauge orbit types [8].", "parent_index": 11, "line_index": 0}, {"bbox": [62, 551, 408, 566], "content": "Definition 3.2 Let denote the set of all Howe subgroups of .", "parent_index": 12, "line_index": 0}, {"bbox": [153, 566, 537, 580], "content": "Let . Then holds iff there are and with", "parent_index": 12, "line_index": 1}, {"bbox": [154, 578, 206, 597], "content": ".", "parent_index": 12, "line_index": 2}, {"bbox": [63, 604, 162, 618], "content": "Obviously, we have", "parent_index": 13, "line_index": 0}, {"bbox": [61, 628, 538, 645], "content": "Lemma 3.1 The maximal element in is the class of the center of , the", "parent_index": 14, "line_index": 0}, {"bbox": [137, 644, 322, 658], "content": "minimal is the class of itself.", "parent_index": 14, "line_index": 1}, {"bbox": [75, 664, 409, 681], "content": "1Homomorphism means supposed is defined.", "parent_index": 15, "line_index": 0}]
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[]
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[{"bbox": [213, 118, 248, 129], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [362, 120, 435, 132], "content": "h_{\\overline{{A}}}:{\\mathcal{P}}\\longrightarrow\\mathbf{G}", "parent_index": 1, "subtype": "inline"}, {"bbox": [79, 135, 95, 146], "content": "h_{\\overline{{A}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [192, 133, 201, 143], "content": "\\overline{{A}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [378, 133, 410, 146], "content": "{\\overline{{g}}}\\,\\in{\\overline{{\\mathcal{G}}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [463, 135, 533, 145], "content": "\\overline{{g}}:M\\longrightarrow\\mathbf{G}", "parent_index": 1, "subtype": "inline"}, {"bbox": [133, 148, 155, 160], "content": "\\overline{{g}}(x)", "parent_index": 1, "subtype": "inline"}, {"bbox": [340, 152, 347, 158], "content": "x", "parent_index": 1, "subtype": "inline"}, {"bbox": [462, 152, 473, 160], "content": "g_{x}", "parent_index": 1, "subtype": "inline"}, {"bbox": [93, 162, 101, 173], "content": "\\overline{{g}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [121, 162, 131, 172], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [203, 177, 412, 195], "content": "h_{\\overline{{A}}\\circ\\overline{{g}}}(\\gamma):=g_{\\gamma(0)}^{-1}\\;h_{\\overline{{A}}}(\\gamma)\\;g_{\\gamma(1)}\\mathrm{~for~all~}\\gamma\\in\\mathcal{P}.", "parent_index": 2, "subtype": "interline"}, {"bbox": [127, 198, 255, 212], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}\\cong\\mathrm{Hom}(\\mathcal{H}\\mathcal{G},\\mathbf{G})/\\mathrm{Ad}", "parent_index": 3, "subtype": "inline"}, {"bbox": [129, 214, 137, 223], "content": "\\Gamma", "parent_index": 4, "subtype": "inline"}, {"bbox": [234, 213, 343, 226], "content": "\\mathbf{E}(\\Gamma)\\,=\\,\\{e_{1},\\dots,e_{E}\\}", "parent_index": 4, "subtype": "inline"}, {"bbox": [494, 213, 537, 226], "content": "{\\mathbf V}(\\Gamma)\\mathbf{\\Sigma}=", "parent_index": 4, "subtype": "inline"}, {"bbox": [80, 228, 144, 240], "content": "\\{v_{1},\\ldots,v_{V}\\}", "parent_index": 4, "subtype": "inline"}, {"bbox": [98, 253, 500, 289], "content": "\\begin{array}{r l}{\\tau_{\\Gamma}:\\;\\;\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{A}}}_{\\Gamma}\\equiv\\mathbf{G}^{E}\\qquad\\qquad\\mathrm{and}\\qquad\\pi_{\\Gamma}:\\;\\;\\overline{{\\mathcal{G}}}\\;\\;\\longrightarrow\\;\\;\\;\\;\\overline{{\\mathcal{G}}}_{\\Gamma}\\equiv\\mathbf{G}^{V}.}\\\\ {\\overline{{\\mathcal{A}}}\\;\\;\\longmapsto\\;\\;\\left(h_{\\overline{{A}}}(e_{1}),\\ldots,h_{\\overline{{A}}}(e_{E})\\right)\\qquad\\qquad\\qquad\\quad\\overline{{g}}\\;\\;\\longmapsto\\;\\;\\left(g_{v_{1}},\\ldots,g_{v_{V}}\\right)}\\end{array}", "parent_index": 5, "subtype": "interline"}, {"bbox": [174, 289, 184, 299], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [209, 289, 217, 301], "content": "\\overline{{g}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [191, 303, 279, 315], "content": "\\Theta:\\overline{{\\mathcal{A}}}\\times\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{\\mathcal{A}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [466, 305, 477, 314], "content": "\\mathbf{G}", "parent_index": 6, "subtype": "inline"}, {"bbox": [102, 318, 112, 328], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [139, 318, 147, 330], "content": "\\mathcal{G}", "parent_index": 6, "subtype": "inline"}, {"bbox": [192, 334, 211, 346], "content": "\\mathbf{H}_{\\overline{{A}}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [298, 333, 307, 343], "content": "\\overline{{A}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [382, 333, 490, 346], "content": "\\mathbf{H}_{\\overline{{A}}}:=h_{\\overline{{A}}}(\\mathcal{H}\\mathcal{G})\\subseteq\\mathbf{G}", "parent_index": 7, "subtype": "inline"}, {"bbox": [194, 348, 230, 361], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 7, "subtype": "inline"}, {"bbox": [397, 347, 432, 358], "content": "\\overline{{A}}\\in\\overline{{A}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [79, 361, 88, 373], "content": "\\overline{{g}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [167, 361, 195, 375], "content": "\\mathbf{B}(\\overline{{A}})", "parent_index": 7, "subtype": "inline"}, {"bbox": [254, 361, 306, 375], "content": "{\\overline{{g}}}\\,\\in\\,{\\bf B}({\\overline{{A}}})", "parent_index": 7, "subtype": "inline"}, {"bbox": [325, 362, 392, 375], "content": "g_{m}\\,\\in\\,Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 7, "subtype": "inline"}, {"bbox": [455, 362, 493, 372], "content": "x\\,\\in\\,M", "parent_index": 7, "subtype": "inline"}, {"bbox": [118, 378, 163, 389], "content": "\\gamma\\in\\mathcal{P}_{m x}", "parent_index": 7, "subtype": "inline"}, {"bbox": [194, 376, 299, 389], "content": "h_{\\overline{{{A}}}}(\\gamma)\\,=\\,g_{m}^{-1}h_{\\overline{{{A}}}}(\\gamma)g_{x}", "parent_index": 7, "subtype": "inline"}, {"bbox": [424, 375, 452, 389], "content": "\\mathbf{B}(\\overline{{A}})", "parent_index": 7, "subtype": "inline"}, {"bbox": [480, 376, 516, 389], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 7, "subtype": "inline"}, {"bbox": [222, 405, 291, 418], "content": "\\mathbf{E}_{\\overline{{A}}}:=\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [527, 405, 536, 415], "content": "\\overline{{A}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [199, 421, 209, 430], "content": "\\mathbf{G}", "parent_index": 8, "subtype": "inline"}, {"bbox": [444, 418, 472, 433], "content": "\\mathbf{B}(\\overline{{A}})", "parent_index": 8, "subtype": "inline"}, {"bbox": [216, 487, 225, 496], "content": "U", "parent_index": 10, "subtype": "inline"}, {"bbox": [241, 487, 252, 496], "content": "\\mathbf{G}", "parent_index": 10, "subtype": "inline"}, {"bbox": [474, 487, 510, 497], "content": "V\\subseteq\\mathbf{G}", "parent_index": 10, "subtype": "inline"}, {"bbox": [154, 501, 206, 513], "content": "U=Z(V)", "parent_index": 10, "subtype": "inline"}, {"bbox": [174, 553, 185, 563], "content": "\\tau", "parent_index": 12, "subtype": "inline"}, {"bbox": [394, 553, 405, 562], "content": "\\mathbf{G}", "parent_index": 12, "subtype": "inline"}, {"bbox": [175, 568, 224, 578], "content": "t_{1},t_{2}\\in\\mathcal{T}", "parent_index": 12, "subtype": "inline"}, {"bbox": [264, 568, 300, 578], "content": "t_{1}\\leq t_{2}", "parent_index": 12, "subtype": "inline"}, {"bbox": [400, 568, 441, 578], "content": "\\mathbf{G}_{1}\\in t_{1}", "parent_index": 12, "subtype": "inline"}, {"bbox": [468, 568, 509, 578], "content": "\\mathbf{G}_{2}\\in t_{2}", "parent_index": 12, "subtype": "inline"}, {"bbox": [154, 582, 200, 593], "content": "\\mathbf{G}_{1}\\supseteq\\mathbf{G}_{2}", "parent_index": 12, "subtype": "inline"}, {"bbox": [273, 631, 284, 640], "content": "\\tau", "parent_index": 14, "subtype": "inline"}, {"bbox": [352, 632, 373, 641], "content": "t_{\\mathrm{max}}", "parent_index": 14, "subtype": "inline"}, {"bbox": [451, 630, 480, 643], "content": "Z(\\mathbf{G})", "parent_index": 14, "subtype": "inline"}, {"bbox": [500, 631, 511, 640], "content": "\\mathbf{G}", "parent_index": 14, "subtype": "inline"}, {"bbox": [244, 646, 262, 656], "content": "t_{\\mathrm{min}}", "parent_index": 14, "subtype": "inline"}, {"bbox": [280, 645, 290, 654], "content": "\\mathbf{G}", "parent_index": 14, "subtype": "inline"}, {"bbox": [184, 667, 295, 678], "content": "h_{\\overline{{{A}}}}(\\gamma_{1}\\gamma_{2})=h_{\\overline{{{A}}}}(\\gamma_{1})h_{\\overline{{{A}}}}(\\gamma_{2})", "parent_index": 15, "subtype": "inline"}, {"bbox": [342, 670, 361, 677], "content": "\\gamma_{1}\\gamma_{2}", "parent_index": 15, "subtype": "inline"}]
|
[]
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|
Definition 3.3 Let $t\in\mathcal T$ . We define the following expressions:
$$
\begin{array}{r l r}{\overline{{A}}_{\geq t}}&{:=}&{\{\overline{{A}}\in\overline{{A}}\mid\mathrm{Typ}(\overline{{A}})\geq t\}}\\ {\overline{{A}}_{=t}}&{:=}&{\{\overline{{A}}\in\overline{{A}}\mid\mathrm{Typ}(\overline{{A}})=t\}}\\ {\overline{{A}}_{\leq t}}&{:=}&{\{\overline{{A}}\in\overline{{A}}\mid\mathrm{Typ}(\overline{{A}})\leq t\}.}\end{array}
$$
All the $\overline{{A}}_{=t}$ are called strata.2
# 4 Reducing the Problem to Finite-Dimensional GSpaces
# 4.1 Finiteness Lemma for Centralizers
We start with the crucial
Lemma 4.1 Let $U$ be a subset of a compact Lie group $\mathbf{G}$ . Then there exist an $n\in\mathbb N$ and $u_{1},\ldots,u_{n}\in U$ , such that $Z(\{u_{1},\dots,u_{n}\})=Z(U)$ .
Proof • The case $Z(U)={\bf G}=Z(\emptyset)$ is trivial.
Let $Z(U)\neq\mathbf{G}$ . Then there is a $u_{1}~\in~U$ with $Z(\{u_{1}\})\neq\mathbf{G}$ . Choose now for $i\geq1$ successively $u_{i+1}\in U$ with $Z(\{u_{1},\dots\,,u_{i}\})\supset Z(\{u_{1},\dots\,,u_{i+1}\})$ as long as there is such a $u_{i+1}$ . This procedure stops after a finite number of steps, since each non-increasing sequence of compact subgroups in $\mathbf{G}$ stabilizes [8]. (Centralizers are always closed, thus compact.) Therefore there is an $n\,\in\,\mathbb{N}$ , such that $Z(\{u_{1},\dots\,,u_{n}\})\;=\;Z(\{u_{1},\dots\,,u_{n}\}\cup\{u\})$ for all $u~\in~U$ . Thus, we have $Z(\{u_{1},\dots,u_{n}\})=\bigcap_{u\in U}Z(\{u_{1},\dots,u_{n}\}\cup\{u\})=Z(\{u_{1},\dots,u_{n}\}\cup U)=Z$ (U).
Corollary 4.2 Let ${\overline{{A}}}\in{\overline{{A}}}$ .
Then there is a finite set $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ , such that $Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$ .3
Proof Due to $\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}$ and the just proven lemma there are an $n\in\mathbb N$ and $g_{1},\dots,g_{n}\in\mathbf{H}_{\overline{{A}}}$ with $Z(\left\{g_{1},\dots,g_{n}\right\})=Z(\mathbf{H}_{\overline{{A}}})$ . On the other hand, since $g_{1},\dots,g_{n}\in\mathbf{H}_{\overline{{A}}}$ , there are $\alpha_{1},\ldots,\alpha_{n}\in\mathcal{H}\mathcal{G}$ with $g_{i}=h_{\overline{{A}}}(\alpha_{i})$ for all $i=1,\dots,n$ . qed
# 4.2 Reduction Mapping
Definition 4.1 Let $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ . Then the map
$$
\varphi_{\alpha}:\;{\overline{{\mathbf{\mathcal{A}}}}}\;\;\longrightarrow\;\;{\mathbf{G}}^{\#\alpha}
$$
is called reduction mapping.
Lemma 4.3 Let $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ be arbitrary.
Then $\varphi_{\alpha}$ is continuous, and for all $\overline{{A}}\in\overline{{A}}$ and ${\overline{{g}}}\,\in\,{\overline{{g}}}$ we have $\varphi_{\pmb{\alpha}}(\overline{{\cal A}}\circ\overline{{\boldsymbol{g}}})\mathrm{~=~}$ $\varphi_{\pmb{\alpha}}(\overline{{\cal A}})\circ g_{m}$ . Here $\mathbf{G}$ acts on $\mathbf{G}^{\#\alpha}$ by the adjoint map.
|
<html><body>
<p data-bbox="62 14 397 29">Definition 3.3 Let $t\in\mathcal T$ . We define the following expressions: </p>
<div class="equation" data-bbox="258 33 429 81">$$
\begin{array}{r l r}{\overline{{A}}_{\geq t}}&{:=}&{\{\overline{{A}}\in\overline{{A}}\mid\mathrm{Typ}(\overline{{A}})\geq t\}}\\ {\overline{{A}}_{=t}}&{:=}&{\{\overline{{A}}\in\overline{{A}}\mid\mathrm{Typ}(\overline{{A}})=t\}}\\ {\overline{{A}}_{\leq t}}&{:=}&{\{\overline{{A}}\in\overline{{A}}\mid\mathrm{Typ}(\overline{{A}})\leq t\}.}\end{array}
$$</div>
<p data-bbox="153 83 313 99">All the $\overline{{A}}_{=t}$ are called strata.2 </p>
<h1 data-bbox="64 119 536 160">4 Reducing the Problem to Finite-Dimensional GSpaces </h1>
<h1 data-bbox="63 172 344 190">4.1 Finiteness Lemma for Centralizers </h1>
<p data-bbox="63 197 193 212">We start with the crucial </p>
<p data-bbox="63 218 538 249">Lemma 4.1 Let $U$ be a subset of a compact Lie group $\mathbf{G}$ . Then there exist an $n\in\mathbb N$ and $u_{1},\ldots,u_{n}\in U$ , such that $Z(\{u_{1},\dots,u_{n}\})=Z(U)$ . </p>
<p data-bbox="63 257 315 271">Proof • The case $Z(U)={\bf G}=Z(\emptyset)$ is trivial. </p>
<p data-bbox="107 272 537 378">Let $Z(U)\neq\mathbf{G}$ . Then there is a $u_{1}~\in~U$ with $Z(\{u_{1}\})\neq\mathbf{G}$ . Choose now for $i\geq1$ successively $u_{i+1}\in U$ with $Z(\{u_{1},\dots\,,u_{i}\})\supset Z(\{u_{1},\dots\,,u_{i+1}\})$ as long as there is such a $u_{i+1}$ . This procedure stops after a finite number of steps, since each non-increasing sequence of compact subgroups in $\mathbf{G}$ stabilizes [8]. (Centralizers are always closed, thus compact.) Therefore there is an $n\,\in\,\mathbb{N}$ , such that $Z(\{u_{1},\dots\,,u_{n}\})\;=\;Z(\{u_{1},\dots\,,u_{n}\}\cup\{u\})$ for all $u~\in~U$ . Thus, we have $Z(\{u_{1},\dots,u_{n}\})=\bigcap_{u\in U}Z(\{u_{1},\dots,u_{n}\}\cup\{u\})=Z(\{u_{1},\dots,u_{n}\}\cup U)=Z$ (U). </p>
<p data-bbox="62 396 209 411">Corollary 4.2 Let ${\overline{{A}}}\in{\overline{{A}}}$ . </p>
<p data-bbox="148 412 494 428">Then there is a finite set $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ , such that $Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$ .3 </p>
<p data-bbox="62 436 538 481">Proof Due to $\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}$ and the just proven lemma there are an $n\in\mathbb N$ and $g_{1},\dots,g_{n}\in\mathbf{H}_{\overline{{A}}}$ with $Z(\left\{g_{1},\dots,g_{n}\right\})=Z(\mathbf{H}_{\overline{{A}}})$ . On the other hand, since $g_{1},\dots,g_{n}\in\mathbf{H}_{\overline{{A}}}$ , there are $\alpha_{1},\ldots,\alpha_{n}\in\mathcal{H}\mathcal{G}$ with $g_{i}=h_{\overline{{A}}}(\alpha_{i})$ for all $i=1,\dots,n$ . qed </p>
<h1 data-bbox="63 497 242 514">4.2 Reduction Mapping </h1>
<p data-bbox="63 521 301 536">Definition 4.1 Let $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ . Then the map </p>
<div class="equation" data-bbox="279 537 392 567">$$
\varphi_{\alpha}:\;{\overline{{\mathbf{\mathcal{A}}}}}\;\;\longrightarrow\;\;{\mathbf{G}}^{\#\alpha}
$$</div>
<p data-bbox="152 565 313 579">is called reduction mapping. </p>
<p data-bbox="63 588 272 603">Lemma 4.3 Let $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ be arbitrary. </p>
<p data-bbox="137 603 537 632">Then $\varphi_{\alpha}$ is continuous, and for all $\overline{{A}}\in\overline{{A}}$ and ${\overline{{g}}}\,\in\,{\overline{{g}}}$ we have $\varphi_{\pmb{\alpha}}(\overline{{\cal A}}\circ\overline{{\boldsymbol{g}}})\mathrm{~=~}$ $\varphi_{\pmb{\alpha}}(\overline{{\cal A}})\circ g_{m}$ . Here $\mathbf{G}$ acts on $\mathbf{G}^{\#\alpha}$ by the adjoint map. </p>
</body></html>
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0001008v1
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[{"type": "text", "text": "Definition 3.3 Let $t\\in\\mathcal T$ . We define the following expressions: ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "All the $\\overline{{A}}_{=t}$ are called strata.2 ", "page_idx": 4}, {"type": "text", "text": "4 Reducing the Problem to Finite-Dimensional GSpaces ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "4.1 Finiteness Lemma for Centralizers ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "We start with the crucial ", "page_idx": 4}, {"type": "text", "text": "Lemma 4.1 Let $U$ be a subset of a compact Lie group $\\mathbf{G}$ . Then there exist an $n\\in\\mathbb N$ and $u_{1},\\ldots,u_{n}\\in U$ , such that $Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)$ . ", "page_idx": 4}, {"type": "text", "text": "Proof • The case $Z(U)={\\bf G}=Z(\\emptyset)$ is trivial. ", "page_idx": 4}, {"type": "text", "text": "Let $Z(U)\\neq\\mathbf{G}$ . Then there is a $u_{1}~\\in~U$ with $Z(\\{u_{1}\\})\\neq\\mathbf{G}$ . Choose now for $i\\geq1$ successively $u_{i+1}\\in U$ with $Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})$ as long as there is such a $u_{i+1}$ . This procedure stops after a finite number of steps, since each non-increasing sequence of compact subgroups in $\\mathbf{G}$ stabilizes [8]. (Centralizers are always closed, thus compact.) Therefore there is an $n\\,\\in\\,\\mathbb{N}$ , such that $Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})$ for all $u~\\in~U$ . Thus, we have $Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z$ (U). ", "page_idx": 4}, {"type": "text", "text": "Corollary 4.2 Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ . ", "page_idx": 4}, {"type": "text", "text": "Then there is a finite set $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ , such that $Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$ .3 ", "page_idx": 4}, {"type": "text", "text": "Proof Due to $\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}$ and the just proven lemma there are an $n\\in\\mathbb N$ and $g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}$ with $Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})$ . On the other hand, since $g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}$ , there are $\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}$ with $g_{i}=h_{\\overline{{A}}}(\\alpha_{i})$ for all $i=1,\\dots,n$ . qed ", "page_idx": 4}, {"type": "text", "text": "4.2 Reduction Mapping ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "Definition 4.1 Let $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ . Then the map ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "is called reduction mapping. ", "page_idx": 4}, {"type": "text", "text": "Lemma 4.3 Let $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ be arbitrary. ", "page_idx": 4}, {"type": "text", "text": "Then $\\varphi_{\\alpha}$ is continuous, and for all $\\overline{{A}}\\in\\overline{{A}}$ and ${\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}$ we have $\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}$ $\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}$ . Here $\\mathbf{G}$ acts on $\\mathbf{G}^{\\#\\alpha}$ by the adjoint map. ", "page_idx": 4}]
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{"preproc_blocks": [{"type": "text", "bbox": [62, 14, 397, 29], "lines": [{"bbox": [62, 17, 396, 31], "spans": [{"bbox": [62, 17, 174, 31], "score": 1.0, "content": "Definition 3.3 Let ", "type": "text"}, {"bbox": [174, 19, 203, 28], "score": 0.93, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [204, 17, 396, 31], "score": 1.0, "content": ". We define the following expressions:", "type": "text"}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [258, 33, 429, 81], "lines": [{"bbox": [258, 33, 429, 81], "spans": [{"bbox": [258, 33, 429, 81], "score": 0.9, "content": "\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [153, 83, 313, 99], "lines": [{"bbox": [154, 85, 313, 99], "spans": [{"bbox": [154, 85, 192, 99], "score": 1.0, "content": "All the ", "type": "text"}, {"bbox": [193, 86, 213, 99], "score": 0.92, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [213, 85, 313, 99], "score": 1.0, "content": " are called strata.2", "type": "text"}], "index": 2}], "index": 2}, {"type": "title", "bbox": [64, 119, 536, 160], "lines": [{"bbox": [61, 122, 537, 140], "spans": [{"bbox": [61, 122, 537, 140], "score": 1.0, "content": "4 Reducing the Problem to Finite-Dimensional G-", "type": "text"}], "index": 3}, {"bbox": [91, 144, 150, 163], "spans": [{"bbox": [91, 144, 150, 163], "score": 1.0, "content": "Spaces", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "title", "bbox": [63, 172, 344, 190], "lines": [{"bbox": [63, 176, 343, 189], "spans": [{"bbox": [63, 176, 91, 189], "score": 1.0, "content": "4.1", "type": "text"}, {"bbox": [96, 176, 343, 189], "score": 1.0, "content": "Finiteness Lemma for Centralizers", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [63, 197, 193, 212], "lines": [{"bbox": [63, 200, 193, 212], "spans": [{"bbox": [63, 200, 193, 212], "score": 1.0, "content": "We start with the crucial", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [63, 218, 538, 249], "lines": [{"bbox": [61, 220, 538, 236], "spans": [{"bbox": [61, 220, 159, 236], "score": 1.0, "content": "Lemma 4.1 Let ", "type": "text"}, {"bbox": [160, 223, 169, 232], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [169, 220, 358, 236], "score": 1.0, "content": " be a subset of a compact Lie group ", "type": "text"}, {"bbox": [358, 223, 369, 232], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [369, 220, 482, 236], "score": 1.0, "content": ". Then there exist an ", "type": "text"}, {"bbox": [482, 223, 514, 232], "score": 0.9, "content": "n\\in\\mathbb N", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [514, 220, 538, 236], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [138, 236, 401, 250], "spans": [{"bbox": [138, 238, 214, 249], "score": 0.9, "content": "u_{1},\\ldots,u_{n}\\in U", "type": "inline_equation", "height": 11, "width": 76}, {"bbox": [215, 236, 273, 250], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [273, 237, 398, 250], "score": 0.93, "content": "Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [398, 236, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [63, 257, 315, 271], "lines": [{"bbox": [63, 260, 315, 274], "spans": [{"bbox": [63, 260, 172, 272], "score": 1.0, "content": "Proof • The case ", "type": "text"}, {"bbox": [172, 261, 265, 274], "score": 0.93, "content": "Z(U)={\\bf G}=Z(\\emptyset)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [266, 260, 315, 272], "score": 1.0, "content": " is trivial.", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [107, 272, 537, 378], "lines": [{"bbox": [112, 272, 536, 289], "spans": [{"bbox": [112, 272, 145, 289], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [145, 276, 202, 288], "score": 0.94, "content": "Z(U)\\neq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [203, 272, 299, 289], "score": 1.0, "content": ". Then there is a ", "type": "text"}, {"bbox": [300, 276, 339, 287], "score": 0.9, "content": "u_{1}~\\in~U", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [339, 272, 371, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [372, 276, 443, 288], "score": 0.94, "content": "Z(\\{u_{1}\\})\\neq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [443, 272, 536, 289], "score": 1.0, "content": ". Choose now for", "type": "text"}], "index": 10}, {"bbox": [123, 288, 537, 305], "spans": [{"bbox": [123, 291, 150, 302], "score": 0.9, "content": "i\\geq1", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [150, 288, 217, 305], "score": 1.0, "content": " successively ", "type": "text"}, {"bbox": [217, 290, 263, 302], "score": 0.91, "content": "u_{i+1}\\in U", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 288, 293, 305], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [293, 290, 480, 303], "score": 0.92, "content": "Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})", "type": "inline_equation", "height": 13, "width": 187}, {"bbox": [481, 288, 537, 305], "score": 1.0, "content": " as long as", "type": "text"}], "index": 11}, {"bbox": [122, 303, 538, 319], "spans": [{"bbox": [122, 303, 204, 319], "score": 1.0, "content": "there is such a ", "type": "text"}, {"bbox": [204, 308, 225, 317], "score": 0.9, "content": "u_{i+1}", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [226, 303, 538, 319], "score": 1.0, "content": ". This procedure stops after a finite number of steps, since", "type": "text"}], "index": 12}, {"bbox": [122, 318, 536, 333], "spans": [{"bbox": [122, 318, 414, 333], "score": 1.0, "content": "each non-increasing sequence of compact subgroups in ", "type": "text"}, {"bbox": [414, 320, 425, 329], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [426, 318, 536, 333], "score": 1.0, "content": " stabilizes [8]. (Cen-", "type": "text"}], "index": 13}, {"bbox": [123, 333, 537, 346], "spans": [{"bbox": [123, 333, 470, 346], "score": 1.0, "content": "tralizers are always closed, thus compact.) Therefore there is an ", "type": "text"}, {"bbox": [470, 334, 505, 344], "score": 0.89, "content": "n\\,\\in\\,\\mathbb{N}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [505, 333, 537, 346], "score": 1.0, "content": ", such", "type": "text"}], "index": 14}, {"bbox": [122, 344, 539, 363], "spans": [{"bbox": [122, 344, 149, 363], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [150, 348, 369, 361], "score": 0.9, "content": "Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})", "type": "inline_equation", "height": 13, "width": 219}, {"bbox": [369, 344, 410, 363], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [411, 349, 446, 358], "score": 0.91, "content": "u~\\in~U", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [447, 344, 539, 363], "score": 1.0, "content": ". Thus, we have", "type": "text"}], "index": 15}, {"bbox": [118, 361, 538, 377], "spans": [{"bbox": [118, 362, 513, 375], "score": 0.75, "content": "Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z", "type": "inline_equation", "height": 13, "width": 395}, {"bbox": [513, 361, 538, 377], "score": 1.0, "content": "(U).", "type": "text"}], "index": 16}], "index": 13}, {"type": "text", "bbox": [62, 396, 209, 411], "lines": [{"bbox": [63, 399, 208, 412], "spans": [{"bbox": [63, 399, 171, 412], "score": 1.0, "content": "Corollary 4.2 Let ", "type": "text"}, {"bbox": [171, 400, 205, 411], "score": 0.92, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [205, 399, 208, 412], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [148, 412, 494, 428], "lines": [{"bbox": [150, 414, 492, 430], "spans": [{"bbox": [150, 414, 280, 430], "score": 1.0, "content": "Then there is a finite set ", "type": "text"}, {"bbox": [281, 416, 324, 427], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [325, 414, 382, 430], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [383, 416, 484, 428], "score": 0.91, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [484, 414, 492, 430], "score": 1.0, "content": ".3", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [62, 436, 538, 481], "lines": [{"bbox": [62, 438, 538, 454], "spans": [{"bbox": [62, 438, 144, 454], "score": 1.0, "content": "Proof Due to ", "type": "text"}, {"bbox": [145, 439, 189, 452], "score": 0.9, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [189, 438, 397, 454], "score": 1.0, "content": " and the just proven lemma there are an ", "type": "text"}, {"bbox": [398, 441, 428, 450], "score": 0.92, "content": "n\\in\\mathbb N", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [429, 438, 453, 454], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [454, 440, 538, 452], "score": 0.88, "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 84}], "index": 19}, {"bbox": [105, 451, 539, 470], "spans": [{"bbox": [105, 451, 132, 470], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [132, 454, 264, 467], "score": 0.93, "content": "Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 132}, {"bbox": [264, 451, 402, 470], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [402, 455, 484, 467], "score": 0.9, "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [484, 451, 539, 470], "score": 1.0, "content": ", there are", "type": "text"}], "index": 20}, {"bbox": [106, 467, 537, 483], "spans": [{"bbox": [106, 469, 192, 480], "score": 0.88, "content": "\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 86}, {"bbox": [193, 467, 223, 483], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [223, 469, 281, 481], "score": 0.94, "content": "g_{i}=h_{\\overline{{A}}}(\\alpha_{i})", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [282, 467, 319, 483], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [320, 470, 380, 480], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [381, 467, 384, 483], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 469, 537, 481], "score": 1.0, "content": "qed", "type": "text"}], "index": 21}], "index": 20}, {"type": "title", "bbox": [63, 497, 242, 514], "lines": [{"bbox": [63, 499, 241, 516], "spans": [{"bbox": [63, 501, 85, 513], "score": 1.0, "content": "4.2", "type": "text"}, {"bbox": [97, 499, 241, 516], "score": 1.0, "content": "Reduction Mapping", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [63, 521, 301, 536], "lines": [{"bbox": [62, 523, 300, 538], "spans": [{"bbox": [62, 523, 174, 538], "score": 1.0, "content": "Definition 4.1 Let ", "type": "text"}, {"bbox": [175, 526, 218, 536], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [218, 523, 300, 538], "score": 1.0, "content": ". Then the map", "type": "text"}], "index": 23}], "index": 23}, {"type": "interline_equation", "bbox": [279, 537, 392, 567], "lines": [{"bbox": [279, 537, 392, 567], "spans": [{"bbox": [279, 537, 392, 567], "score": 0.79, "content": "\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [152, 565, 313, 579], "lines": [{"bbox": [152, 565, 312, 582], "spans": [{"bbox": [152, 565, 312, 582], "score": 1.0, "content": "is called reduction mapping.", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [63, 588, 272, 603], "lines": [{"bbox": [62, 590, 270, 604], "spans": [{"bbox": [62, 590, 159, 604], "score": 1.0, "content": "Lemma 4.3 Let ", "type": "text"}, {"bbox": [160, 593, 203, 603], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [203, 590, 270, 604], "score": 1.0, "content": " be arbitrary.", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [137, 603, 537, 632], "lines": [{"bbox": [137, 603, 538, 621], "spans": [{"bbox": [137, 603, 169, 621], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [170, 610, 185, 618], "score": 0.9, "content": "\\varphi_{\\alpha}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [185, 603, 324, 621], "score": 1.0, "content": " is continuous, and for all ", "type": "text"}, {"bbox": [324, 605, 361, 616], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [361, 603, 388, 621], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [388, 605, 420, 618], "score": 0.94, "content": "{\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [421, 603, 470, 621], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [471, 605, 538, 619], "score": 0.91, "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}", "type": "inline_equation", "height": 14, "width": 67}], "index": 27}, {"bbox": [139, 617, 419, 637], "spans": [{"bbox": [139, 619, 196, 633], "score": 0.93, "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}", "type": "inline_equation", "height": 14, "width": 57}, {"bbox": [197, 617, 232, 637], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [232, 621, 243, 630], "score": 0.71, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 617, 286, 637], "score": 1.0, "content": " acts on ", "type": "text"}, {"bbox": [287, 620, 312, 630], "score": 0.92, "content": "\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [312, 617, 419, 637], "score": 1.0, "content": " by the adjoint map.", "type": "text"}], "index": 28}], "index": 27.5}], "layout_bboxes": [], "page_idx": 4, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [258, 33, 429, 81], "lines": [{"bbox": [258, 33, 429, 81], "spans": [{"bbox": [258, 33, 429, 81], "score": 0.9, "content": "\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [279, 537, 392, 567], "lines": [{"bbox": [279, 537, 392, 567], "spans": [{"bbox": [279, 537, 392, 567], "score": 0.79, "content": "\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}", "type": "interline_equation"}], "index": 24}], "index": 24}], "discarded_blocks": [{"type": "discarded", "bbox": [63, 637, 538, 687], "lines": [{"bbox": [75, 636, 348, 654], "spans": [{"bbox": [75, 636, 348, 654], "score": 1.0, "content": "2The justification for that notation can be found in section 8.", "type": "text"}]}, {"bbox": [76, 648, 539, 668], "spans": [{"bbox": [76, 651, 248, 664], "score": 0.9, "content": "{}^{3}h_{\\overline{{A}}}(\\alpha):=\\left\\{h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{n})\\right\\}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 172}, {"bbox": [248, 648, 280, 668], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [280, 653, 320, 662], "score": 0.93, "content": "n:=\\#\\alpha", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [320, 648, 539, 668], "score": 1.0, "content": ". To avoid cumbersome notations we denote also", "type": "text"}]}, {"bbox": [63, 663, 536, 680], "spans": [{"bbox": [63, 664, 186, 677], "score": 0.89, "content": "{\\big(}h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{n}){\\big)}\\in\\mathbf{G}^{n}", "type": "inline_equation", "height": 13, "width": 123}, {"bbox": [186, 663, 203, 680], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [204, 666, 231, 676], "score": 0.93, "content": "h_{\\overline{{A}}}(\\alpha)", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [232, 663, 528, 680], "score": 1.0, "content": ". It should be clear from the context what is meant. Furthermore, ", "type": "text"}, {"bbox": [528, 669, 536, 673], "score": 0.83, "content": "\\alpha", "type": "inline_equation", "height": 4, "width": 8}]}, {"bbox": [63, 677, 131, 689], "spans": [{"bbox": [63, 677, 131, 689], "score": 1.0, "content": "is always finite.", "type": "text"}]}]}, {"type": "discarded", "bbox": [295, 704, 303, 715], "lines": [{"bbox": [296, 705, 303, 717], "spans": [{"bbox": [296, 705, 303, 717], "score": 1.0, "content": "5", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [62, 14, 397, 29], "lines": [{"bbox": [62, 17, 396, 31], "spans": [{"bbox": [62, 17, 174, 31], "score": 1.0, "content": "Definition 3.3 Let ", "type": "text"}, {"bbox": [174, 19, 203, 28], "score": 0.93, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [204, 17, 396, 31], "score": 1.0, "content": ". We define the following expressions:", "type": "text"}], "index": 0}], "index": 0, "bbox_fs": [62, 17, 396, 31]}, {"type": "interline_equation", "bbox": [258, 33, 429, 81], "lines": [{"bbox": [258, 33, 429, 81], "spans": [{"bbox": [258, 33, 429, 81], "score": 0.9, "content": "\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [153, 83, 313, 99], "lines": [{"bbox": [154, 85, 313, 99], "spans": [{"bbox": [154, 85, 192, 99], "score": 1.0, "content": "All the ", "type": "text"}, {"bbox": [193, 86, 213, 99], "score": 0.92, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [213, 85, 313, 99], "score": 1.0, "content": " are called strata.2", "type": "text"}], "index": 2}], "index": 2, "bbox_fs": [154, 85, 313, 99]}, {"type": "title", "bbox": [64, 119, 536, 160], "lines": [{"bbox": [61, 122, 537, 140], "spans": [{"bbox": [61, 122, 537, 140], "score": 1.0, "content": "4 Reducing the Problem to Finite-Dimensional G-", "type": "text"}], "index": 3}, {"bbox": [91, 144, 150, 163], "spans": [{"bbox": [91, 144, 150, 163], "score": 1.0, "content": "Spaces", "type": "text"}], "index": 4}], "index": 3.5}, {"type": "title", "bbox": [63, 172, 344, 190], "lines": [{"bbox": [63, 176, 343, 189], "spans": [{"bbox": [63, 176, 91, 189], "score": 1.0, "content": "4.1", "type": "text"}, {"bbox": [96, 176, 343, 189], "score": 1.0, "content": "Finiteness Lemma for Centralizers", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [63, 197, 193, 212], "lines": [{"bbox": [63, 200, 193, 212], "spans": [{"bbox": [63, 200, 193, 212], "score": 1.0, "content": "We start with the crucial", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [63, 200, 193, 212]}, {"type": "text", "bbox": [63, 218, 538, 249], "lines": [{"bbox": [61, 220, 538, 236], "spans": [{"bbox": [61, 220, 159, 236], "score": 1.0, "content": "Lemma 4.1 Let ", "type": "text"}, {"bbox": [160, 223, 169, 232], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [169, 220, 358, 236], "score": 1.0, "content": " be a subset of a compact Lie group ", "type": "text"}, {"bbox": [358, 223, 369, 232], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [369, 220, 482, 236], "score": 1.0, "content": ". Then there exist an ", "type": "text"}, {"bbox": [482, 223, 514, 232], "score": 0.9, "content": "n\\in\\mathbb N", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [514, 220, 538, 236], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [138, 236, 401, 250], "spans": [{"bbox": [138, 238, 214, 249], "score": 0.9, "content": "u_{1},\\ldots,u_{n}\\in U", "type": "inline_equation", "height": 11, "width": 76}, {"bbox": [215, 236, 273, 250], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [273, 237, 398, 250], "score": 0.93, "content": "Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)", "type": "inline_equation", "height": 13, "width": 125}, {"bbox": [398, 236, 401, 250], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [61, 220, 538, 250]}, {"type": "text", "bbox": [63, 257, 315, 271], "lines": [{"bbox": [63, 260, 315, 274], "spans": [{"bbox": [63, 260, 172, 272], "score": 1.0, "content": "Proof • The case ", "type": "text"}, {"bbox": [172, 261, 265, 274], "score": 0.93, "content": "Z(U)={\\bf G}=Z(\\emptyset)", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [266, 260, 315, 272], "score": 1.0, "content": " is trivial.", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [63, 260, 315, 274]}, {"type": "text", "bbox": [107, 272, 537, 378], "lines": [{"bbox": [112, 272, 536, 289], "spans": [{"bbox": [112, 272, 145, 289], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [145, 276, 202, 288], "score": 0.94, "content": "Z(U)\\neq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [203, 272, 299, 289], "score": 1.0, "content": ". Then there is a ", "type": "text"}, {"bbox": [300, 276, 339, 287], "score": 0.9, "content": "u_{1}~\\in~U", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [339, 272, 371, 289], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [372, 276, 443, 288], "score": 0.94, "content": "Z(\\{u_{1}\\})\\neq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [443, 272, 536, 289], "score": 1.0, "content": ". Choose now for", "type": "text"}], "index": 10}, {"bbox": [123, 288, 537, 305], "spans": [{"bbox": [123, 291, 150, 302], "score": 0.9, "content": "i\\geq1", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [150, 288, 217, 305], "score": 1.0, "content": " successively ", "type": "text"}, {"bbox": [217, 290, 263, 302], "score": 0.91, "content": "u_{i+1}\\in U", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [264, 288, 293, 305], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [293, 290, 480, 303], "score": 0.92, "content": "Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})", "type": "inline_equation", "height": 13, "width": 187}, {"bbox": [481, 288, 537, 305], "score": 1.0, "content": " as long as", "type": "text"}], "index": 11}, {"bbox": [122, 303, 538, 319], "spans": [{"bbox": [122, 303, 204, 319], "score": 1.0, "content": "there is such a ", "type": "text"}, {"bbox": [204, 308, 225, 317], "score": 0.9, "content": "u_{i+1}", "type": "inline_equation", "height": 9, "width": 21}, {"bbox": [226, 303, 538, 319], "score": 1.0, "content": ". This procedure stops after a finite number of steps, since", "type": "text"}], "index": 12}, {"bbox": [122, 318, 536, 333], "spans": [{"bbox": [122, 318, 414, 333], "score": 1.0, "content": "each non-increasing sequence of compact subgroups in ", "type": "text"}, {"bbox": [414, 320, 425, 329], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [426, 318, 536, 333], "score": 1.0, "content": " stabilizes [8]. (Cen-", "type": "text"}], "index": 13}, {"bbox": [123, 333, 537, 346], "spans": [{"bbox": [123, 333, 470, 346], "score": 1.0, "content": "tralizers are always closed, thus compact.) Therefore there is an ", "type": "text"}, {"bbox": [470, 334, 505, 344], "score": 0.89, "content": "n\\,\\in\\,\\mathbb{N}", "type": "inline_equation", "height": 10, "width": 35}, {"bbox": [505, 333, 537, 346], "score": 1.0, "content": ", such", "type": "text"}], "index": 14}, {"bbox": [122, 344, 539, 363], "spans": [{"bbox": [122, 344, 149, 363], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [150, 348, 369, 361], "score": 0.9, "content": "Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})", "type": "inline_equation", "height": 13, "width": 219}, {"bbox": [369, 344, 410, 363], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [411, 349, 446, 358], "score": 0.91, "content": "u~\\in~U", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [447, 344, 539, 363], "score": 1.0, "content": ". Thus, we have", "type": "text"}], "index": 15}, {"bbox": [118, 361, 538, 377], "spans": [{"bbox": [118, 362, 513, 375], "score": 0.75, "content": "Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z", "type": "inline_equation", "height": 13, "width": 395}, {"bbox": [513, 361, 538, 377], "score": 1.0, "content": "(U).", "type": "text"}], "index": 16}], "index": 13, "bbox_fs": [112, 272, 539, 377]}, {"type": "text", "bbox": [62, 396, 209, 411], "lines": [{"bbox": [63, 399, 208, 412], "spans": [{"bbox": [63, 399, 171, 412], "score": 1.0, "content": "Corollary 4.2 Let ", "type": "text"}, {"bbox": [171, 400, 205, 411], "score": 0.92, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [205, 399, 208, 412], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 17, "bbox_fs": [63, 399, 208, 412]}, {"type": "text", "bbox": [148, 412, 494, 428], "lines": [{"bbox": [150, 414, 492, 430], "spans": [{"bbox": [150, 414, 280, 430], "score": 1.0, "content": "Then there is a finite set ", "type": "text"}, {"bbox": [281, 416, 324, 427], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [325, 414, 382, 430], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [383, 416, 484, 428], "score": 0.91, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [484, 414, 492, 430], "score": 1.0, "content": ".3", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [150, 414, 492, 430]}, {"type": "text", "bbox": [62, 436, 538, 481], "lines": [{"bbox": [62, 438, 538, 454], "spans": [{"bbox": [62, 438, 144, 454], "score": 1.0, "content": "Proof Due to ", "type": "text"}, {"bbox": [145, 439, 189, 452], "score": 0.9, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [189, 438, 397, 454], "score": 1.0, "content": " and the just proven lemma there are an ", "type": "text"}, {"bbox": [398, 441, 428, 450], "score": 0.92, "content": "n\\in\\mathbb N", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [429, 438, 453, 454], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [454, 440, 538, 452], "score": 0.88, "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 84}], "index": 19}, {"bbox": [105, 451, 539, 470], "spans": [{"bbox": [105, 451, 132, 470], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [132, 454, 264, 467], "score": 0.93, "content": "Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 132}, {"bbox": [264, 451, 402, 470], "score": 1.0, "content": ". On the other hand, since ", "type": "text"}, {"bbox": [402, 455, 484, 467], "score": 0.9, "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [484, 451, 539, 470], "score": 1.0, "content": ", there are", "type": "text"}], "index": 20}, {"bbox": [106, 467, 537, 483], "spans": [{"bbox": [106, 469, 192, 480], "score": 0.88, "content": "\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 86}, {"bbox": [193, 467, 223, 483], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [223, 469, 281, 481], "score": 0.94, "content": "g_{i}=h_{\\overline{{A}}}(\\alpha_{i})", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [282, 467, 319, 483], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [320, 470, 380, 480], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 10, "width": 60}, {"bbox": [381, 467, 384, 483], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 469, 537, 481], "score": 1.0, "content": "qed", "type": "text"}], "index": 21}], "index": 20, "bbox_fs": [62, 438, 539, 483]}, {"type": "title", "bbox": [63, 497, 242, 514], "lines": [{"bbox": [63, 499, 241, 516], "spans": [{"bbox": [63, 501, 85, 513], "score": 1.0, "content": "4.2", "type": "text"}, {"bbox": [97, 499, 241, 516], "score": 1.0, "content": "Reduction Mapping", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [63, 521, 301, 536], "lines": [{"bbox": [62, 523, 300, 538], "spans": [{"bbox": [62, 523, 174, 538], "score": 1.0, "content": "Definition 4.1 Let ", "type": "text"}, {"bbox": [175, 526, 218, 536], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [218, 523, 300, 538], "score": 1.0, "content": ". Then the map", "type": "text"}], "index": 23}], "index": 23, "bbox_fs": [62, 523, 300, 538]}, {"type": "interline_equation", "bbox": [279, 537, 392, 567], "lines": [{"bbox": [279, 537, 392, 567], "spans": [{"bbox": [279, 537, 392, 567], "score": 0.79, "content": "\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [152, 565, 313, 579], "lines": [{"bbox": [152, 565, 312, 582], "spans": [{"bbox": [152, 565, 312, 582], "score": 1.0, "content": "is called reduction mapping.", "type": "text"}], "index": 25}], "index": 25, "bbox_fs": [152, 565, 312, 582]}, {"type": "text", "bbox": [63, 588, 272, 603], "lines": [{"bbox": [62, 590, 270, 604], "spans": [{"bbox": [62, 590, 159, 604], "score": 1.0, "content": "Lemma 4.3 Let ", "type": "text"}, {"bbox": [160, 593, 203, 603], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [203, 590, 270, 604], "score": 1.0, "content": " be arbitrary.", "type": "text"}], "index": 26}], "index": 26, "bbox_fs": [62, 590, 270, 604]}, {"type": "text", "bbox": [137, 603, 537, 632], "lines": [{"bbox": [137, 603, 538, 621], "spans": [{"bbox": [137, 603, 169, 621], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [170, 610, 185, 618], "score": 0.9, "content": "\\varphi_{\\alpha}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [185, 603, 324, 621], "score": 1.0, "content": " is continuous, and for all ", "type": "text"}, {"bbox": [324, 605, 361, 616], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [361, 603, 388, 621], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [388, 605, 420, 618], "score": 0.94, "content": "{\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [421, 603, 470, 621], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [471, 605, 538, 619], "score": 0.91, "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}", "type": "inline_equation", "height": 14, "width": 67}], "index": 27}, {"bbox": [139, 617, 419, 637], "spans": [{"bbox": [139, 619, 196, 633], "score": 0.93, "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}", "type": "inline_equation", "height": 14, "width": 57}, {"bbox": [197, 617, 232, 637], "score": 1.0, "content": ". Here ", "type": "text"}, {"bbox": [232, 621, 243, 630], "score": 0.71, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 617, 286, 637], "score": 1.0, "content": " acts on ", "type": "text"}, {"bbox": [287, 620, 312, 630], "score": 0.92, "content": "\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 10, "width": 25}, {"bbox": [312, 617, 419, 637], "score": 1.0, "content": " by the adjoint map.", "type": "text"}], "index": 28}], "index": 27.5, "bbox_fs": [137, 603, 538, 637]}]}
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[{"type": "text", "bbox": [62, 14, 397, 29], "content": "Definition 3.3 Let . We define the following expressions:", "index": 0}, {"type": "interline_equation", "bbox": [258, 33, 429, 81], "content": "", "index": 1}, {"type": "text", "bbox": [153, 83, 313, 99], "content": "All the are called strata.2", "index": 2}, {"type": "title", "bbox": [64, 119, 536, 160], "content": "4 Reducing the Problem to Finite-Dimensional G- Spaces", "index": 3}, {"type": "title", "bbox": [63, 172, 344, 190], "content": "4.1 Finiteness Lemma for Centralizers", "index": 4}, {"type": "text", "bbox": [63, 197, 193, 212], "content": "We start with the crucial", "index": 5}, {"type": "text", "bbox": [63, 218, 538, 249], "content": "Lemma 4.1 Let be a subset of a compact Lie group . Then there exist an and , such that .", "index": 6}, {"type": "text", "bbox": [63, 257, 315, 271], "content": "Proof • The case is trivial.", "index": 7}, {"type": "text", "bbox": [107, 272, 537, 378], "content": "Let . Then there is a with . Choose now for successively with as long as there is such a . This procedure stops after a finite number of steps, since each non-increasing sequence of compact subgroups in stabilizes [8]. (Cen- tralizers are always closed, thus compact.) Therefore there is an , such that for all . Thus, we have (U).", "index": 8}, {"type": "text", "bbox": [62, 396, 209, 411], "content": "Corollary 4.2 Let .", "index": 9}, {"type": "text", "bbox": [148, 412, 494, 428], "content": "Then there is a finite set , such that .3", "index": 10}, {"type": "text", "bbox": [62, 436, 538, 481], "content": "Proof Due to and the just proven lemma there are an and with . On the other hand, since , there are with for all . qed", "index": 11}, {"type": "title", "bbox": [63, 497, 242, 514], "content": "4.2 Reduction Mapping", "index": 12}, {"type": "text", "bbox": [63, 521, 301, 536], "content": "Definition 4.1 Let . Then the map", "index": 13}, {"type": "interline_equation", "bbox": [279, 537, 392, 567], "content": "", "index": 14}, {"type": "text", "bbox": [152, 565, 313, 579], "content": "is called reduction mapping.", "index": 15}, {"type": "text", "bbox": [63, 588, 272, 603], "content": "Lemma 4.3 Let be arbitrary.", "index": 16}, {"type": "text", "bbox": [137, 603, 537, 632], "content": "Then is continuous, and for all and we have . Here acts on by the adjoint map.", "index": 17}]
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[{"bbox": [62, 17, 396, 31], "content": "Definition 3.3 Let . We define the following expressions:", "parent_index": 0, "line_index": 0}, {"bbox": [154, 85, 313, 99], "content": "All the are called strata.2", "parent_index": 2, "line_index": 0}, {"bbox": [61, 122, 537, 140], "content": "4 Reducing the Problem to Finite-Dimensional G-", "parent_index": 3, "line_index": 0}, {"bbox": [91, 144, 150, 163], "content": "Spaces", "parent_index": 3, "line_index": 1}, {"bbox": [63, 176, 343, 189], "content": "4.1 Finiteness Lemma for Centralizers", "parent_index": 4, "line_index": 0}, {"bbox": [63, 200, 193, 212], "content": "We start with the crucial", "parent_index": 5, "line_index": 0}, {"bbox": [61, 220, 538, 236], "content": "Lemma 4.1 Let be a subset of a compact Lie group . Then there exist an and", "parent_index": 6, "line_index": 0}, {"bbox": [138, 236, 401, 250], "content": ", such that .", "parent_index": 6, "line_index": 1}, {"bbox": [63, 260, 315, 274], "content": "Proof • The case is trivial.", "parent_index": 7, "line_index": 0}, {"bbox": [112, 272, 536, 289], "content": "Let . Then there is a with . Choose now for", "parent_index": 8, "line_index": 0}, {"bbox": [123, 288, 537, 305], "content": "successively with as long as", "parent_index": 8, "line_index": 1}, {"bbox": [122, 303, 538, 319], "content": "there is such a . This procedure stops after a finite number of steps, since", "parent_index": 8, "line_index": 2}, {"bbox": [122, 318, 536, 333], "content": "each non-increasing sequence of compact subgroups in stabilizes [8]. (Cen-", "parent_index": 8, "line_index": 3}, {"bbox": [123, 333, 537, 346], "content": "tralizers are always closed, thus compact.) Therefore there is an , such", "parent_index": 8, "line_index": 4}, {"bbox": [122, 344, 539, 363], "content": "that for all . Thus, we have", "parent_index": 8, "line_index": 5}, {"bbox": [118, 361, 538, 377], "content": "(U).", "parent_index": 8, "line_index": 6}, {"bbox": [63, 399, 208, 412], "content": "Corollary 4.2 Let .", "parent_index": 9, "line_index": 0}, {"bbox": [150, 414, 492, 430], "content": "Then there is a finite set , such that .3", "parent_index": 10, "line_index": 0}, {"bbox": [62, 438, 538, 454], "content": "Proof Due to and the just proven lemma there are an and", "parent_index": 11, "line_index": 0}, {"bbox": [105, 451, 539, 470], "content": "with . On the other hand, since , there are", "parent_index": 11, "line_index": 1}, {"bbox": [106, 467, 537, 483], "content": "with for all . qed", "parent_index": 11, "line_index": 2}, {"bbox": [63, 499, 241, 516], "content": "4.2 Reduction Mapping", "parent_index": 12, "line_index": 0}, {"bbox": [62, 523, 300, 538], "content": "Definition 4.1 Let . Then the map", "parent_index": 13, "line_index": 0}, {"bbox": [152, 565, 312, 582], "content": "is called reduction mapping.", "parent_index": 15, "line_index": 0}, {"bbox": [62, 590, 270, 604], "content": "Lemma 4.3 Let be arbitrary.", "parent_index": 16, "line_index": 0}, {"bbox": [137, 603, 538, 621], "content": "Then is continuous, and for all and we have", "parent_index": 17, "line_index": 0}, {"bbox": [139, 617, 419, 637], "content": ". Here acts on by the adjoint map.", "parent_index": 17, "line_index": 1}]
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[]
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[{"bbox": [174, 19, 203, 28], "content": "t\\in\\mathcal T", "parent_index": 0, "subtype": "inline"}, {"bbox": [258, 33, 429, 81], "content": "\\begin{array}{r l r}{\\overline{{A}}_{\\geq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\geq t\\}}\\\\ {\\overline{{A}}_{=t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})=t\\}}\\\\ {\\overline{{A}}_{\\leq t}}&{:=}&{\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\mathrm{Typ}(\\overline{{A}})\\leq t\\}.}\\end{array}", "parent_index": 1, "subtype": "interline"}, {"bbox": [193, 86, 213, 99], "content": "\\overline{{A}}_{=t}", "parent_index": 2, "subtype": "inline"}, {"bbox": [160, 223, 169, 232], "content": "U", "parent_index": 6, "subtype": "inline"}, {"bbox": [358, 223, 369, 232], "content": "\\mathbf{G}", "parent_index": 6, "subtype": "inline"}, {"bbox": [482, 223, 514, 232], "content": "n\\in\\mathbb N", "parent_index": 6, "subtype": "inline"}, {"bbox": [138, 238, 214, 249], "content": "u_{1},\\ldots,u_{n}\\in U", "parent_index": 6, "subtype": "inline"}, {"bbox": [273, 237, 398, 250], "content": "Z(\\{u_{1},\\dots,u_{n}\\})=Z(U)", "parent_index": 6, "subtype": "inline"}, {"bbox": [172, 261, 265, 274], "content": "Z(U)={\\bf G}=Z(\\emptyset)", "parent_index": 7, "subtype": "inline"}, {"bbox": [145, 276, 202, 288], "content": "Z(U)\\neq\\mathbf{G}", "parent_index": 8, "subtype": "inline"}, {"bbox": [300, 276, 339, 287], "content": "u_{1}~\\in~U", "parent_index": 8, "subtype": "inline"}, {"bbox": [372, 276, 443, 288], "content": "Z(\\{u_{1}\\})\\neq\\mathbf{G}", "parent_index": 8, "subtype": "inline"}, {"bbox": [123, 291, 150, 302], "content": "i\\geq1", "parent_index": 8, "subtype": "inline"}, {"bbox": [217, 290, 263, 302], "content": "u_{i+1}\\in U", "parent_index": 8, "subtype": "inline"}, {"bbox": [293, 290, 480, 303], "content": "Z(\\{u_{1},\\dots\\,,u_{i}\\})\\supset Z(\\{u_{1},\\dots\\,,u_{i+1}\\})", "parent_index": 8, "subtype": "inline"}, {"bbox": [204, 308, 225, 317], "content": "u_{i+1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [414, 320, 425, 329], "content": "\\mathbf{G}", "parent_index": 8, "subtype": "inline"}, {"bbox": [470, 334, 505, 344], "content": "n\\,\\in\\,\\mathbb{N}", "parent_index": 8, "subtype": "inline"}, {"bbox": [150, 348, 369, 361], "content": "Z(\\{u_{1},\\dots\\,,u_{n}\\})\\;=\\;Z(\\{u_{1},\\dots\\,,u_{n}\\}\\cup\\{u\\})", "parent_index": 8, "subtype": "inline"}, {"bbox": [411, 349, 446, 358], "content": "u~\\in~U", "parent_index": 8, "subtype": "inline"}, {"bbox": [118, 362, 513, 375], "content": "Z(\\{u_{1},\\dots,u_{n}\\})=\\bigcap_{u\\in U}Z(\\{u_{1},\\dots,u_{n}\\}\\cup\\{u\\})=Z(\\{u_{1},\\dots,u_{n}\\}\\cup U)=Z", "parent_index": 8, "subtype": "inline"}, {"bbox": [171, 400, 205, 411], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "parent_index": 9, "subtype": "inline"}, {"bbox": [281, 416, 324, 427], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [383, 416, 484, 428], "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "parent_index": 10, "subtype": "inline"}, {"bbox": [145, 439, 189, 452], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}", "parent_index": 11, "subtype": "inline"}, {"bbox": [398, 441, 428, 450], "content": "n\\in\\mathbb N", "parent_index": 11, "subtype": "inline"}, {"bbox": [454, 440, 538, 452], "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [132, 454, 264, 467], "content": "Z(\\left\\{g_{1},\\dots,g_{n}\\right\\})=Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 11, "subtype": "inline"}, {"bbox": [402, 455, 484, 467], "content": "g_{1},\\dots,g_{n}\\in\\mathbf{H}_{\\overline{{A}}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [106, 469, 192, 480], "content": "\\alpha_{1},\\ldots,\\alpha_{n}\\in\\mathcal{H}\\mathcal{G}", "parent_index": 11, "subtype": "inline"}, {"bbox": [223, 469, 281, 481], "content": "g_{i}=h_{\\overline{{A}}}(\\alpha_{i})", "parent_index": 11, "subtype": "inline"}, {"bbox": [320, 470, 380, 480], "content": "i=1,\\dots,n", "parent_index": 11, "subtype": "inline"}, {"bbox": [175, 526, 218, 536], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 13, "subtype": "inline"}, {"bbox": [279, 537, 392, 567], "content": "\\varphi_{\\alpha}:\\;{\\overline{{\\mathbf{\\mathcal{A}}}}}\\;\\;\\longrightarrow\\;\\;{\\mathbf{G}}^{\\#\\alpha}", "parent_index": 14, "subtype": "interline"}, {"bbox": [160, 593, 203, 603], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 16, "subtype": "inline"}, {"bbox": [170, 610, 185, 618], "content": "\\varphi_{\\alpha}", "parent_index": 17, "subtype": "inline"}, {"bbox": [324, 605, 361, 616], "content": "\\overline{{A}}\\in\\overline{{A}}", "parent_index": 17, "subtype": "inline"}, {"bbox": [388, 605, 420, 618], "content": "{\\overline{{g}}}\\,\\in\\,{\\overline{{g}}}", "parent_index": 17, "subtype": "inline"}, {"bbox": [471, 605, 538, 619], "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}}\\circ\\overline{{\\boldsymbol{g}}})\\mathrm{~=~}", "parent_index": 17, "subtype": "inline"}, {"bbox": [139, 619, 196, 633], "content": "\\varphi_{\\pmb{\\alpha}}(\\overline{{\\cal A}})\\circ g_{m}", "parent_index": 17, "subtype": "inline"}, {"bbox": [232, 621, 243, 630], "content": "\\mathbf{G}", "parent_index": 17, "subtype": "inline"}, {"bbox": [287, 620, 312, 630], "content": "\\mathbf{G}^{\\#\\alpha}", "parent_index": 17, "subtype": "inline"}]
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Proof • $\varphi_{\alpha}:{\overline{{\mathcal{A}}}}\longrightarrow\mathbf{G}^{\#\alpha}$ is as a map into a product space continuous iff $\pi_{i}\circ\varphi_{\alpha}\equiv\varphi_{\{\alpha_{i}\}}$ is continuous for all projections $\pi_{i}:\mathbf{G}^{\#\alpha}\longrightarrow\mathbf{G}$ onto the $i$ th factor. Thus, it is sufficient to prove the continuity of $\varphi\{\alpha\}$ for all $\alpha\in{\mathcal{H}}{\mathcal{G}}$ . Now decompose $\alpha$ into a product of finitely many edges $e_{j}$ , $j\,=\,1,\ldots,J$ (i.e., into paths that can be represented as an edge in a graph). Then the mapping $\overline{{\mathcal{A}}}\longrightarrow\mathbf{G}^{J}$ with ${\overline{{A}}}\longmapsto\left(\pi_{e_{1}}({\overline{{A}}}),\dots,\pi_{e_{J}}({\overline{{A}}})\right)$ is continuous per definitionem. Since the multiplication in $\mathbf{G}$ is continuous, $\varphi_{\{\alpha\}}$ is continuous, too. • The compatibility with the group action follows from $h_{\overline{{{A}}}\circ\overline{{{g}}}}(\pmb{\alpha})\b{=}\,g_{m}^{-1}\,h_{\overline{{{A}}}}(\pmb{\alpha})\mathrm{~}g_{m}$ . qed
# 4.3 Adjoint Action of $\mathbf{G}$ on $\mathbf{G}^{n}$
In this short subsection we will summarize the most important facts about the adjoint action of $\mathbf{G}$ on $\mathbf{G}^{n}$ that can be deduced from the general theory of transformation groups (see, e.g., [7]).
Consequently, we have for the type of the corresponding orbit
$\mathrm{Typ}(\vec{g})=[\mathbf{G}_{\vec{g}}]=[Z(\{g_{1},\dots,g_{n}\})]$
The slice theorem reads now as follows:
Proposition 4.4 Let $\vec{g}\in\mathbf{G}^{n}$ . Then there is an $S\subseteq\mathbf{G}^{n}$ with $\vec{g}\in S$ , such that: • $S\circ\mathbf{G}$ is an open neighboorhood of $\vec{g}\circ\mathbf{G}$ and • there is an equivariant retraction $f:S\circ\mathbf{G}\longrightarrow\vec{g}\circ\mathbf{G}$ with $f^{-1}(\{\vec{g}\})=$ $S$ .
Both on $\overline{{\mathcal{A}}}$ and on $\mathbf{G}^{n}$ the type is a Howe subgroup of $\mathbf{G}$ . The transformation behaviour of the types under a reduction mapping is stated in the next
Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ and all ${\overline{{A}}}\in{\overline{{A}}}$ we have
$$
\mathrm{Typ}\big(\varphi_{\alpha}(\overline{{A}})\big)\leq\mathrm{Typ}(\overline{{A}}).
$$
Proof We have $\mathrm{Typ}\big(\varphi_{\alpha}(\overline{{A}})\big)=[Z(\varphi_{\alpha}(\overline{{A}}))]\equiv[Z(h_{\overline{{A}}}(\alpha))]\leq[Z(\mathbf{H}_{\overline{{A}}})]=\mathrm{Typ}(\overline{{A}}).$
# 5 Slice Theorem for $\overline{{\mathcal{A}}}$
We state now the main theorem of the present paper.
Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all ${\overline{{A}}}\in{\overline{{A}}}$ there is an ${\overline{{S}}}\subseteq{\overline{{A}}}$ with ${\overline{{A}}}\in{\overline{{S}}}$ , such that:
• $\overline{{S}}\circ\overline{{\mathcal{G}}}$ is an open neighbourhood of $\overline{{A}}\circ\overline{{\mathcal{G}}}$ and there is an equivariant retraction $F:\overline{{S}}\circ\overline{{\mathcal{G}}}\longrightarrow\overline{{A}}\circ\overline{{\mathcal{G}}}$ with $F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$ .
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<html><body>
<p data-bbox="61 12 539 146">Proof • $\varphi_{\alpha}:{\overline{{\mathcal{A}}}}\longrightarrow\mathbf{G}^{\#\alpha}$ is as a map into a product space continuous iff $\pi_{i}\circ\varphi_{\alpha}\equiv\varphi_{\{\alpha_{i}\}}$ is continuous for all projections $\pi_{i}:\mathbf{G}^{\#\alpha}\longrightarrow\mathbf{G}$ onto the $i$ th factor. Thus, it is sufficient to prove the continuity of $\varphi\{\alpha\}$ for all $\alpha\in{\mathcal{H}}{\mathcal{G}}$ . Now decompose $\alpha$ into a product of finitely many edges $e_{j}$ , $j\,=\,1,\ldots,J$ (i.e., into paths that can be represented as an edge in a graph). Then the mapping $\overline{{\mathcal{A}}}\longrightarrow\mathbf{G}^{J}$ with ${\overline{{A}}}\longmapsto\left(\pi_{e_{1}}({\overline{{A}}}),\dots,\pi_{e_{J}}({\overline{{A}}})\right)$ is continuous per definitionem. Since the multiplication in $\mathbf{G}$ is continuous, $\varphi_{\{\alpha\}}$ is continuous, too. • The compatibility with the group action follows from $h_{\overline{{{A}}}\circ\overline{{{g}}}}(\pmb{\alpha})\b{=}\,g_{m}^{-1}\,h_{\overline{{{A}}}}(\pmb{\alpha})\mathrm{~}g_{m}$ . qed </p>
<h1 data-bbox="62 164 290 181">4.3 Adjoint Action of $\mathbf{G}$ on $\mathbf{G}^{n}$ </h1>
<p data-bbox="62 189 537 232">In this short subsection we will summarize the most important facts about the adjoint action of $\mathbf{G}$ on $\mathbf{G}^{n}$ that can be deduced from the general theory of transformation groups (see, e.g., [7]). </p>
<p data-bbox="62 262 385 275">Consequently, we have for the type of the corresponding orbit </p>
<div class="equation" data-bbox="210 279 386 291">$\mathrm{Typ}(\vec{g})=[\mathbf{G}_{\vec{g}}]=[Z(\{g_{1},\dots,g_{n}\})]$ </div>
<p data-bbox="62 290 267 304">The slice theorem reads now as follows: </p>
<p data-bbox="62 313 538 371">Proposition 4.4 Let $\vec{g}\in\mathbf{G}^{n}$ . Then there is an $S\subseteq\mathbf{G}^{n}$ with $\vec{g}\in S$ , such that: • $S\circ\mathbf{G}$ is an open neighboorhood of $\vec{g}\circ\mathbf{G}$ and • there is an equivariant retraction $f:S\circ\mathbf{G}\longrightarrow\vec{g}\circ\mathbf{G}$ with $f^{-1}(\{\vec{g}\})=$ $S$ . </p>
<p data-bbox="62 381 538 411">Both on $\overline{{\mathcal{A}}}$ and on $\mathbf{G}^{n}$ the type is a Howe subgroup of $\mathbf{G}$ . The transformation behaviour of the types under a reduction mapping is stated in the next </p>
<p data-bbox="63 420 537 449">Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ and all ${\overline{{A}}}\in{\overline{{A}}}$ we have </p>
<div class="equation" data-bbox="288 450 411 469">$$
\mathrm{Typ}\big(\varphi_{\alpha}(\overline{{A}})\big)\leq\mathrm{Typ}(\overline{{A}}).
$$</div>
<p data-bbox="62 479 483 499">Proof We have $\mathrm{Typ}\big(\varphi_{\alpha}(\overline{{A}})\big)=[Z(\varphi_{\alpha}(\overline{{A}}))]\equiv[Z(h_{\overline{{A}}}(\alpha))]\leq[Z(\mathbf{H}_{\overline{{A}}})]=\mathrm{Typ}(\overline{{A}}).$ </p>
<h1 data-bbox="62 516 266 536">5 Slice Theorem for $\overline{{\mathcal{A}}}$ </h1>
<p data-bbox="62 548 338 563">We state now the main theorem of the present paper. </p>
<p data-bbox="61 571 539 615">Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all ${\overline{{A}}}\in{\overline{{A}}}$ there is an ${\overline{{S}}}\subseteq{\overline{{A}}}$ with ${\overline{{A}}}\in{\overline{{S}}}$ , such that: </p>
<p data-bbox="146 615 537 646">• $\overline{{S}}\circ\overline{{\mathcal{G}}}$ is an open neighbourhood of $\overline{{A}}\circ\overline{{\mathcal{G}}}$ and there is an equivariant retraction $F:\overline{{S}}\circ\overline{{\mathcal{G}}}\longrightarrow\overline{{A}}\circ\overline{{\mathcal{G}}}$ with $F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$ . </p>
</body></html>
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0001008v1
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[{"type": "text", "text": "Proof • $\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}$ is as a map into a product space continuous iff $\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}$ is continuous for all projections $\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}$ onto the $i$ th factor. Thus, it is sufficient to prove the continuity of $\\varphi\\{\\alpha\\}$ for all $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ . Now decompose $\\alpha$ into a product of finitely many edges $e_{j}$ , $j\\,=\\,1,\\ldots,J$ (i.e., into paths that can be represented as an edge in a graph). Then the mapping $\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}$ with ${\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)$ is continuous per definitionem. Since the multiplication in $\\mathbf{G}$ is continuous, $\\varphi_{\\{\\alpha\\}}$ is continuous, too. • The compatibility with the group action follows from $h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}$ . qed ", "page_idx": 5}, {"type": "text", "text": "4.3 Adjoint Action of $\\mathbf{G}$ on $\\mathbf{G}^{n}$ ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "In this short subsection we will summarize the most important facts about the adjoint action of $\\mathbf{G}$ on $\\mathbf{G}^{n}$ that can be deduced from the general theory of transformation groups (see, e.g., [7]). ", "page_idx": 5}, {"type": "text", "text": "Consequently, we have for the type of the corresponding orbit ", "page_idx": 5}, {"type": "equation", "text": "$\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]$ ", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "The slice theorem reads now as follows: ", "page_idx": 5}, {"type": "text", "text": "Proposition 4.4 Let $\\vec{g}\\in\\mathbf{G}^{n}$ . Then there is an $S\\subseteq\\mathbf{G}^{n}$ with $\\vec{g}\\in S$ , such that: • $S\\circ\\mathbf{G}$ is an open neighboorhood of $\\vec{g}\\circ\\mathbf{G}$ and • there is an equivariant retraction $f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}$ with $f^{-1}(\\{\\vec{g}\\})=$ $S$ . ", "page_idx": 5}, {"type": "text", "text": "Both on $\\overline{{\\mathcal{A}}}$ and on $\\mathbf{G}^{n}$ the type is a Howe subgroup of $\\mathbf{G}$ . The transformation behaviour of the types under a reduction mapping is stated in the next ", "page_idx": 5}, {"type": "text", "text": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ and all ${\\overline{{A}}}\\in{\\overline{{A}}}$ we have ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Proof We have $\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).$ ", "page_idx": 5}, {"type": "text", "text": "5 Slice Theorem for $\\overline{{\\mathcal{A}}}$ ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "We state now the main theorem of the present paper. ", "page_idx": 5}, {"type": "text", "text": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all ${\\overline{{A}}}\\in{\\overline{{A}}}$ there is an ${\\overline{{S}}}\\subseteq{\\overline{{A}}}$ with ${\\overline{{A}}}\\in{\\overline{{S}}}$ , such that: ", "page_idx": 5}, {"type": "text", "text": "• $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ is an open neighbourhood of $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ and there is an equivariant retraction $F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ with $F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}$ . ", "page_idx": 5}]
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{"preproc_blocks": [{"type": "text", "bbox": [61, 12, 539, 146], "lines": [{"bbox": [61, 14, 536, 35], "spans": [{"bbox": [61, 14, 122, 35], "score": 1.0, "content": "Proof •", "type": "text"}, {"bbox": [123, 17, 209, 30], "score": 0.92, "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 13, "width": 86}, {"bbox": [209, 14, 457, 35], "score": 1.0, "content": " is as a map into a product space continuous iff", "type": "text"}, {"bbox": [458, 21, 536, 32], "score": 0.87, "content": "\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}", "type": "inline_equation", "height": 11, "width": 78}], "index": 0}, {"bbox": [120, 30, 538, 46], "spans": [{"bbox": [120, 30, 289, 46], "score": 1.0, "content": "is continuous for all projections ", "type": "text"}, {"bbox": [289, 32, 373, 44], "score": 0.93, "content": "\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [373, 30, 424, 46], "score": 1.0, "content": " onto the ", "type": "text"}, {"bbox": [424, 34, 428, 42], "score": 0.8, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [428, 30, 538, 46], "score": 1.0, "content": "th factor. Thus, it is", "type": "text"}], "index": 1}, {"bbox": [122, 46, 412, 61], "spans": [{"bbox": [122, 46, 307, 61], "score": 1.0, "content": "sufficient to prove the continuity of ", "type": "text"}, {"bbox": [307, 51, 329, 60], "score": 0.92, "content": "\\varphi\\{\\alpha\\}", "type": "inline_equation", "height": 9, "width": 22}, {"bbox": [329, 46, 367, 61], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [367, 48, 407, 57], "score": 0.92, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [408, 46, 412, 61], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [121, 58, 537, 76], "spans": [{"bbox": [121, 58, 210, 76], "score": 1.0, "content": "Now decompose ", "type": "text"}, {"bbox": [211, 65, 218, 71], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [219, 58, 423, 76], "score": 1.0, "content": " into a product of finitely many edges ", "type": "text"}, {"bbox": [424, 65, 433, 74], "score": 0.87, "content": "e_{j}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [434, 58, 441, 76], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [441, 62, 509, 73], "score": 0.91, "content": "j\\,=\\,1,\\ldots,J", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [509, 58, 537, 76], "score": 1.0, "content": " (i.e.,", "type": "text"}], "index": 3}, {"bbox": [122, 74, 537, 91], "spans": [{"bbox": [122, 74, 537, 91], "score": 1.0, "content": "into paths that can be represented as an edge in a graph). Then the mapping", "type": "text"}], "index": 4}, {"bbox": [123, 88, 539, 108], "spans": [{"bbox": [123, 91, 175, 101], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [175, 88, 204, 108], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [204, 89, 346, 107], "score": 0.94, "content": "{\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)", "type": "inline_equation", "height": 18, "width": 142}, {"bbox": [347, 88, 539, 108], "score": 1.0, "content": "is continuous per definitionem. Since", "type": "text"}], "index": 5}, {"bbox": [119, 103, 443, 123], "spans": [{"bbox": [119, 103, 232, 123], "score": 1.0, "content": "the multiplication in ", "type": "text"}, {"bbox": [232, 107, 243, 116], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 103, 320, 123], "score": 1.0, "content": " is continuous, ", "type": "text"}, {"bbox": [320, 110, 342, 120], "score": 0.91, "content": "\\varphi_{\\{\\alpha\\}}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [342, 103, 443, 123], "score": 1.0, "content": " is continuous, too.", "type": "text"}], "index": 6}, {"bbox": [106, 118, 537, 136], "spans": [{"bbox": [106, 118, 405, 136], "score": 1.0, "content": "• The compatibility with the group action follows from ", "type": "text"}, {"bbox": [405, 120, 533, 135], "score": 0.92, "content": "h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}", "type": "inline_equation", "height": 15, "width": 128}, {"bbox": [534, 120, 537, 136], "score": 1.0, "content": ".", "type": "text"}], "index": 7}, {"bbox": [513, 136, 537, 147], "spans": [{"bbox": [513, 136, 537, 147], "score": 1.0, "content": "qed", "type": "text"}], "index": 8}], "index": 4}, {"type": "title", "bbox": [62, 164, 290, 181], "lines": [{"bbox": [63, 167, 289, 181], "spans": [{"bbox": [63, 168, 85, 180], "score": 1.0, "content": "4.3", "type": "text"}, {"bbox": [99, 167, 229, 181], "score": 1.0, "content": "Adjoint Action of ", "type": "text"}, {"bbox": [230, 169, 243, 180], "score": 0.28, "content": "\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [243, 167, 270, 181], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [270, 169, 289, 179], "score": 0.75, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 10, "width": 19}], "index": 9}], "index": 9}, {"type": "text", "bbox": [62, 189, 537, 232], "lines": [{"bbox": [62, 190, 537, 205], "spans": [{"bbox": [62, 190, 537, 205], "score": 1.0, "content": "In this short subsection we will summarize the most important facts about the adjoint action", "type": "text"}], "index": 10}, {"bbox": [62, 205, 537, 221], "spans": [{"bbox": [62, 205, 76, 221], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [76, 207, 87, 216], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [87, 205, 106, 221], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [106, 207, 123, 216], "score": 0.92, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [123, 205, 537, 221], "score": 1.0, "content": " that can be deduced from the general theory of transformation groups (see, e.g.,", "type": "text"}], "index": 11}, {"bbox": [62, 219, 85, 234], "spans": [{"bbox": [62, 219, 85, 234], "score": 1.0, "content": "[7]).", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [62, 262, 385, 275], "lines": [{"bbox": [64, 263, 380, 277], "spans": [{"bbox": [64, 263, 380, 277], "score": 1.0, "content": "Consequently, we have for the type of the corresponding orbit", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [210, 279, 386, 291], "lines": [{"bbox": [210, 279, 386, 291], "spans": [{"bbox": [210, 279, 386, 291], "score": 0.92, "content": "\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]", "type": "inline_equation", "height": 12, "width": 176}], "index": 14}], "index": 14}, {"type": "text", "bbox": [62, 290, 267, 304], "lines": [{"bbox": [63, 293, 266, 304], "spans": [{"bbox": [63, 293, 266, 304], "score": 1.0, "content": "The slice theorem reads now as follows:", "type": "text"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [62, 313, 538, 371], "lines": [{"bbox": [63, 316, 477, 331], "spans": [{"bbox": [63, 316, 184, 331], "score": 1.0, "content": "Proposition 4.4 Let ", "type": "text"}, {"bbox": [185, 318, 222, 329], "score": 0.93, "content": "\\vec{g}\\in\\mathbf{G}^{n}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [222, 316, 318, 331], "score": 1.0, "content": ". Then there is an ", "type": "text"}, {"bbox": [318, 318, 358, 329], "score": 0.93, "content": "S\\subseteq\\mathbf{G}^{n}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [359, 316, 389, 331], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [389, 318, 418, 329], "score": 0.92, "content": "\\vec{g}\\in S", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [418, 316, 477, 331], "score": 1.0, "content": ", such that:", "type": "text"}], "index": 16}, {"bbox": [161, 331, 415, 345], "spans": [{"bbox": [161, 331, 180, 345], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [180, 333, 210, 342], "score": 0.89, "content": "S\\circ\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [210, 331, 363, 345], "score": 1.0, "content": " is an open neighboorhood of ", "type": "text"}, {"bbox": [363, 333, 391, 344], "score": 0.93, "content": "\\vec{g}\\circ\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [392, 331, 415, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [161, 345, 538, 361], "spans": [{"bbox": [161, 345, 351, 361], "score": 1.0, "content": "• there is an equivariant retraction ", "type": "text"}, {"bbox": [351, 347, 448, 358], "score": 0.92, "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 97}, {"bbox": [449, 345, 478, 360], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [478, 346, 538, 359], "score": 0.92, "content": "f^{-1}(\\{\\vec{g}\\})=", "type": "inline_equation", "height": 13, "width": 60}], "index": 18}, {"bbox": [180, 361, 193, 373], "spans": [{"bbox": [180, 362, 188, 371], "score": 0.86, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [189, 361, 193, 373], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 17.5}, {"type": "text", "bbox": [62, 381, 538, 411], "lines": [{"bbox": [62, 383, 540, 400], "spans": [{"bbox": [62, 383, 109, 400], "score": 1.0, "content": "Both on ", "type": "text"}, {"bbox": [109, 384, 119, 395], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [119, 383, 162, 400], "score": 1.0, "content": " and on ", "type": "text"}, {"bbox": [162, 386, 178, 395], "score": 0.91, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [179, 383, 347, 400], "score": 1.0, "content": " the type is a Howe subgroup of ", "type": "text"}, {"bbox": [348, 386, 358, 395], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 383, 540, 400], "score": 1.0, "content": ". The transformation behaviour of", "type": "text"}], "index": 20}, {"bbox": [62, 398, 362, 414], "spans": [{"bbox": [62, 398, 362, 414], "score": 1.0, "content": "the types under a reduction mapping is stated in the next", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "text", "bbox": [63, 420, 537, 449], "lines": [{"bbox": [61, 422, 538, 438], "spans": [{"bbox": [61, 422, 451, 438], "score": 1.0, "content": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all ", "type": "text"}, {"bbox": [451, 425, 496, 435], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [496, 422, 538, 438], "score": 1.0, "content": " and all", "type": "text"}], "index": 22}, {"bbox": [164, 436, 244, 452], "spans": [{"bbox": [164, 438, 196, 448], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [197, 436, 244, 452], "score": 1.0, "content": " we have", "type": "text"}], "index": 23}], "index": 22.5}, {"type": "interline_equation", "bbox": [288, 450, 411, 469], "lines": [{"bbox": [288, 450, 411, 469], "spans": [{"bbox": [288, 450, 411, 469], "score": 0.91, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [62, 479, 483, 499], "lines": [{"bbox": [62, 482, 478, 500], "spans": [{"bbox": [62, 483, 151, 500], "score": 1.0, "content": "Proof We have", "type": "text"}, {"bbox": [153, 482, 478, 500], "score": 0.8, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).", "type": "inline_equation", "height": 18, "width": 325}], "index": 25}], "index": 25}, {"type": "title", "bbox": [62, 516, 266, 536], "lines": [{"bbox": [64, 519, 264, 536], "spans": [{"bbox": [64, 522, 74, 534], "score": 1.0, "content": "5", "type": "text"}, {"bbox": [90, 519, 249, 536], "score": 1.0, "content": "Slice Theorem for ", "type": "text"}, {"bbox": [249, 520, 264, 535], "score": 0.74, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 15, "width": 15}], "index": 26}], "index": 26}, {"type": "text", "bbox": [62, 548, 338, 563], "lines": [{"bbox": [63, 549, 338, 564], "spans": [{"bbox": [63, 549, 338, 564], "score": 1.0, "content": "We state now the main theorem of the present paper.", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [61, 571, 539, 615], "lines": [{"bbox": [62, 573, 426, 590], "spans": [{"bbox": [62, 573, 426, 590], "score": 1.0, "content": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit.", "type": "text"}], "index": 28}, {"bbox": [147, 587, 537, 602], "spans": [{"bbox": [147, 589, 305, 602], "score": 1.0, "content": "Equivalently we have: For all ", "type": "text"}, {"bbox": [305, 589, 341, 600], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [341, 589, 404, 602], "score": 1.0, "content": " there is an ", "type": "text"}, {"bbox": [404, 587, 441, 601], "score": 0.91, "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [442, 589, 471, 602], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [471, 587, 506, 600], "score": 0.91, "content": "{\\overline{{A}}}\\in{\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [506, 589, 537, 602], "score": 1.0, "content": ", such", "type": "text"}], "index": 29}, {"bbox": [147, 603, 175, 618], "spans": [{"bbox": [147, 603, 175, 618], "score": 1.0, "content": "that:", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [146, 615, 537, 646], "lines": [{"bbox": [148, 617, 398, 632], "spans": [{"bbox": [148, 617, 164, 632], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [164, 618, 192, 630], "score": 0.93, "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [192, 617, 345, 632], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [345, 618, 373, 630], "score": 0.92, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [374, 617, 398, 632], "score": 1.0, "content": " and", "type": "text"}], "index": 31}, {"bbox": [162, 630, 537, 647], "spans": [{"bbox": [162, 630, 334, 647], "score": 1.0, "content": "there is an equivariant retraction ", "type": "text"}, {"bbox": [334, 631, 430, 644], "score": 0.85, "content": "F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [430, 630, 457, 647], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [458, 630, 533, 646], "score": 0.93, "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [533, 630, 537, 647], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 31.5}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [210, 279, 386, 291], "lines": [{"bbox": [210, 279, 386, 291], "spans": [{"bbox": [210, 279, 386, 291], "score": 0.92, "content": "\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]", "type": "inline_equation", "height": 12, "width": 176}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [288, 450, 411, 469], "lines": [{"bbox": [288, 450, 411, 469], "spans": [{"bbox": [288, 450, 411, 469], "score": 0.91, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).", "type": "interline_equation"}], "index": 24}], "index": 24}], "discarded_blocks": [{"type": "discarded", "bbox": [295, 704, 304, 715], "lines": [{"bbox": [296, 705, 304, 717], "spans": [{"bbox": [296, 705, 304, 717], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [61, 12, 539, 146], "lines": [{"bbox": [61, 14, 536, 35], "spans": [{"bbox": [61, 14, 122, 35], "score": 1.0, "content": "Proof •", "type": "text"}, {"bbox": [123, 17, 209, 30], "score": 0.92, "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 13, "width": 86}, {"bbox": [209, 14, 457, 35], "score": 1.0, "content": " is as a map into a product space continuous iff", "type": "text"}, {"bbox": [458, 21, 536, 32], "score": 0.87, "content": "\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}", "type": "inline_equation", "height": 11, "width": 78}], "index": 0}, {"bbox": [120, 30, 538, 46], "spans": [{"bbox": [120, 30, 289, 46], "score": 1.0, "content": "is continuous for all projections ", "type": "text"}, {"bbox": [289, 32, 373, 44], "score": 0.93, "content": "\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 84}, {"bbox": [373, 30, 424, 46], "score": 1.0, "content": " onto the ", "type": "text"}, {"bbox": [424, 34, 428, 42], "score": 0.8, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [428, 30, 538, 46], "score": 1.0, "content": "th factor. Thus, it is", "type": "text"}], "index": 1}, {"bbox": [122, 46, 412, 61], "spans": [{"bbox": [122, 46, 307, 61], "score": 1.0, "content": "sufficient to prove the continuity of ", "type": "text"}, {"bbox": [307, 51, 329, 60], "score": 0.92, "content": "\\varphi\\{\\alpha\\}", "type": "inline_equation", "height": 9, "width": 22}, {"bbox": [329, 46, 367, 61], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [367, 48, 407, 57], "score": 0.92, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 9, "width": 40}, {"bbox": [408, 46, 412, 61], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [121, 58, 537, 76], "spans": [{"bbox": [121, 58, 210, 76], "score": 1.0, "content": "Now decompose ", "type": "text"}, {"bbox": [211, 65, 218, 71], "score": 0.87, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [219, 58, 423, 76], "score": 1.0, "content": " into a product of finitely many edges ", "type": "text"}, {"bbox": [424, 65, 433, 74], "score": 0.87, "content": "e_{j}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [434, 58, 441, 76], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [441, 62, 509, 73], "score": 0.91, "content": "j\\,=\\,1,\\ldots,J", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [509, 58, 537, 76], "score": 1.0, "content": " (i.e.,", "type": "text"}], "index": 3}, {"bbox": [122, 74, 537, 91], "spans": [{"bbox": [122, 74, 537, 91], "score": 1.0, "content": "into paths that can be represented as an edge in a graph). Then the mapping", "type": "text"}], "index": 4}, {"bbox": [123, 88, 539, 108], "spans": [{"bbox": [123, 91, 175, 101], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [175, 88, 204, 108], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [204, 89, 346, 107], "score": 0.94, "content": "{\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)", "type": "inline_equation", "height": 18, "width": 142}, {"bbox": [347, 88, 539, 108], "score": 1.0, "content": "is continuous per definitionem. Since", "type": "text"}], "index": 5}, {"bbox": [119, 103, 443, 123], "spans": [{"bbox": [119, 103, 232, 123], "score": 1.0, "content": "the multiplication in ", "type": "text"}, {"bbox": [232, 107, 243, 116], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 103, 320, 123], "score": 1.0, "content": " is continuous, ", "type": "text"}, {"bbox": [320, 110, 342, 120], "score": 0.91, "content": "\\varphi_{\\{\\alpha\\}}", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [342, 103, 443, 123], "score": 1.0, "content": " is continuous, too.", "type": "text"}], "index": 6}, {"bbox": [106, 118, 537, 136], "spans": [{"bbox": [106, 118, 405, 136], "score": 1.0, "content": "• The compatibility with the group action follows from ", "type": "text"}, {"bbox": [405, 120, 533, 135], "score": 0.92, "content": "h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}", "type": "inline_equation", "height": 15, "width": 128}, {"bbox": [534, 120, 537, 136], "score": 1.0, "content": ".", "type": "text"}], "index": 7}, {"bbox": [513, 136, 537, 147], "spans": [{"bbox": [513, 136, 537, 147], "score": 1.0, "content": "qed", "type": "text"}], "index": 8}], "index": 4, "bbox_fs": [61, 14, 539, 147]}, {"type": "title", "bbox": [62, 164, 290, 181], "lines": [{"bbox": [63, 167, 289, 181], "spans": [{"bbox": [63, 168, 85, 180], "score": 1.0, "content": "4.3", "type": "text"}, {"bbox": [99, 167, 229, 181], "score": 1.0, "content": "Adjoint Action of ", "type": "text"}, {"bbox": [230, 169, 243, 180], "score": 0.28, "content": "\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [243, 167, 270, 181], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [270, 169, 289, 179], "score": 0.75, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 10, "width": 19}], "index": 9}], "index": 9}, {"type": "text", "bbox": [62, 189, 537, 232], "lines": [{"bbox": [62, 190, 537, 205], "spans": [{"bbox": [62, 190, 537, 205], "score": 1.0, "content": "In this short subsection we will summarize the most important facts about the adjoint action", "type": "text"}], "index": 10}, {"bbox": [62, 205, 537, 221], "spans": [{"bbox": [62, 205, 76, 221], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [76, 207, 87, 216], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [87, 205, 106, 221], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [106, 207, 123, 216], "score": 0.92, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [123, 205, 537, 221], "score": 1.0, "content": " that can be deduced from the general theory of transformation groups (see, e.g.,", "type": "text"}], "index": 11}, {"bbox": [62, 219, 85, 234], "spans": [{"bbox": [62, 219, 85, 234], "score": 1.0, "content": "[7]).", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [62, 190, 537, 234]}, {"type": "text", "bbox": [62, 262, 385, 275], "lines": [{"bbox": [64, 263, 380, 277], "spans": [{"bbox": [64, 263, 380, 277], "score": 1.0, "content": "Consequently, we have for the type of the corresponding orbit", "type": "text"}], "index": 13}], "index": 13, "bbox_fs": [64, 263, 380, 277]}, {"type": "interline_equation", "bbox": [210, 279, 386, 291], "lines": [{"bbox": [210, 279, 386, 291], "spans": [{"bbox": [210, 279, 386, 291], "score": 0.92, "content": "\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]", "type": "inline_equation", "height": 12, "width": 176}], "index": 14}], "index": 14}, {"type": "text", "bbox": [62, 290, 267, 304], "lines": [{"bbox": [63, 293, 266, 304], "spans": [{"bbox": [63, 293, 266, 304], "score": 1.0, "content": "The slice theorem reads now as follows:", "type": "text"}], "index": 15}], "index": 15, "bbox_fs": [63, 293, 266, 304]}, {"type": "text", "bbox": [62, 313, 538, 371], "lines": [{"bbox": [63, 316, 477, 331], "spans": [{"bbox": [63, 316, 184, 331], "score": 1.0, "content": "Proposition 4.4 Let ", "type": "text"}, {"bbox": [185, 318, 222, 329], "score": 0.93, "content": "\\vec{g}\\in\\mathbf{G}^{n}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [222, 316, 318, 331], "score": 1.0, "content": ". Then there is an ", "type": "text"}, {"bbox": [318, 318, 358, 329], "score": 0.93, "content": "S\\subseteq\\mathbf{G}^{n}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [359, 316, 389, 331], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [389, 318, 418, 329], "score": 0.92, "content": "\\vec{g}\\in S", "type": "inline_equation", "height": 11, "width": 29}, {"bbox": [418, 316, 477, 331], "score": 1.0, "content": ", such that:", "type": "text"}], "index": 16}, {"bbox": [161, 331, 415, 345], "spans": [{"bbox": [161, 331, 180, 345], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [180, 333, 210, 342], "score": 0.89, "content": "S\\circ\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [210, 331, 363, 345], "score": 1.0, "content": " is an open neighboorhood of ", "type": "text"}, {"bbox": [363, 333, 391, 344], "score": 0.93, "content": "\\vec{g}\\circ\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [392, 331, 415, 345], "score": 1.0, "content": " and", "type": "text"}], "index": 17}, {"bbox": [161, 345, 538, 361], "spans": [{"bbox": [161, 345, 351, 361], "score": 1.0, "content": "• there is an equivariant retraction ", "type": "text"}, {"bbox": [351, 347, 448, 358], "score": 0.92, "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 97}, {"bbox": [449, 345, 478, 360], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [478, 346, 538, 359], "score": 0.92, "content": "f^{-1}(\\{\\vec{g}\\})=", "type": "inline_equation", "height": 13, "width": 60}], "index": 18}, {"bbox": [180, 361, 193, 373], "spans": [{"bbox": [180, 362, 188, 371], "score": 0.86, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [189, 361, 193, 373], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 17.5, "bbox_fs": [63, 316, 538, 373]}, {"type": "text", "bbox": [62, 381, 538, 411], "lines": [{"bbox": [62, 383, 540, 400], "spans": [{"bbox": [62, 383, 109, 400], "score": 1.0, "content": "Both on ", "type": "text"}, {"bbox": [109, 384, 119, 395], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [119, 383, 162, 400], "score": 1.0, "content": " and on ", "type": "text"}, {"bbox": [162, 386, 178, 395], "score": 0.91, "content": "\\mathbf{G}^{n}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [179, 383, 347, 400], "score": 1.0, "content": " the type is a Howe subgroup of ", "type": "text"}, {"bbox": [348, 386, 358, 395], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [359, 383, 540, 400], "score": 1.0, "content": ". The transformation behaviour of", "type": "text"}], "index": 20}, {"bbox": [62, 398, 362, 414], "spans": [{"bbox": [62, 398, 362, 414], "score": 1.0, "content": "the types under a reduction mapping is stated in the next", "type": "text"}], "index": 21}], "index": 20.5, "bbox_fs": [62, 383, 540, 414]}, {"type": "text", "bbox": [63, 420, 537, 449], "lines": [{"bbox": [61, 422, 538, 438], "spans": [{"bbox": [61, 422, 451, 438], "score": 1.0, "content": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all ", "type": "text"}, {"bbox": [451, 425, 496, 435], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [496, 422, 538, 438], "score": 1.0, "content": " and all", "type": "text"}], "index": 22}, {"bbox": [164, 436, 244, 452], "spans": [{"bbox": [164, 438, 196, 448], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [197, 436, 244, 452], "score": 1.0, "content": " we have", "type": "text"}], "index": 23}], "index": 22.5, "bbox_fs": [61, 422, 538, 452]}, {"type": "interline_equation", "bbox": [288, 450, 411, 469], "lines": [{"bbox": [288, 450, 411, 469], "spans": [{"bbox": [288, 450, 411, 469], "score": 0.91, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).", "type": "interline_equation"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [62, 479, 483, 499], "lines": [{"bbox": [62, 482, 478, 500], "spans": [{"bbox": [62, 483, 151, 500], "score": 1.0, "content": "Proof We have", "type": "text"}, {"bbox": [153, 482, 478, 500], "score": 0.8, "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).", "type": "inline_equation", "height": 18, "width": 325}], "index": 25}], "index": 25, "bbox_fs": [62, 482, 478, 500]}, {"type": "title", "bbox": [62, 516, 266, 536], "lines": [{"bbox": [64, 519, 264, 536], "spans": [{"bbox": [64, 522, 74, 534], "score": 1.0, "content": "5", "type": "text"}, {"bbox": [90, 519, 249, 536], "score": 1.0, "content": "Slice Theorem for ", "type": "text"}, {"bbox": [249, 520, 264, 535], "score": 0.74, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 15, "width": 15}], "index": 26}], "index": 26}, {"type": "text", "bbox": [62, 548, 338, 563], "lines": [{"bbox": [63, 549, 338, 564], "spans": [{"bbox": [63, 549, 338, 564], "score": 1.0, "content": "We state now the main theorem of the present paper.", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [63, 549, 338, 564]}, {"type": "text", "bbox": [61, 571, 539, 615], "lines": [{"bbox": [62, 573, 426, 590], "spans": [{"bbox": [62, 573, 426, 590], "score": 1.0, "content": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit.", "type": "text"}], "index": 28}, {"bbox": [147, 587, 537, 602], "spans": [{"bbox": [147, 589, 305, 602], "score": 1.0, "content": "Equivalently we have: For all ", "type": "text"}, {"bbox": [305, 589, 341, 600], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [341, 589, 404, 602], "score": 1.0, "content": " there is an ", "type": "text"}, {"bbox": [404, 587, 441, 601], "score": 0.91, "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [442, 589, 471, 602], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [471, 587, 506, 600], "score": 0.91, "content": "{\\overline{{A}}}\\in{\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [506, 589, 537, 602], "score": 1.0, "content": ", such", "type": "text"}], "index": 29}, {"bbox": [147, 603, 175, 618], "spans": [{"bbox": [147, 603, 175, 618], "score": 1.0, "content": "that:", "type": "text"}], "index": 30}], "index": 29, "bbox_fs": [62, 573, 537, 618]}, {"type": "text", "bbox": [146, 615, 537, 646], "lines": [{"bbox": [148, 617, 398, 632], "spans": [{"bbox": [148, 617, 164, 632], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [164, 618, 192, 630], "score": 0.93, "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [192, 617, 345, 632], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [345, 618, 373, 630], "score": 0.92, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [374, 617, 398, 632], "score": 1.0, "content": " and", "type": "text"}], "index": 31}, {"bbox": [162, 630, 537, 647], "spans": [{"bbox": [162, 630, 334, 647], "score": 1.0, "content": "there is an equivariant retraction ", "type": "text"}, {"bbox": [334, 631, 430, 644], "score": 0.85, "content": "F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [430, 630, 457, 647], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [458, 630, 533, 646], "score": 0.93, "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "type": "inline_equation", "height": 16, "width": 75}, {"bbox": [533, 630, 537, 647], "score": 1.0, "content": ".", "type": "text"}], "index": 32}], "index": 31.5, "bbox_fs": [148, 617, 537, 647]}]}
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[{"type": "text", "bbox": [61, 12, 539, 146], "content": "Proof • is as a map into a product space continuous iff is continuous for all projections onto the th factor. Thus, it is sufficient to prove the continuity of for all . Now decompose into a product of finitely many edges , (i.e., into paths that can be represented as an edge in a graph). Then the mapping with is continuous per definitionem. Since the multiplication in is continuous, is continuous, too. • The compatibility with the group action follows from . qed", "index": 0}, {"type": "title", "bbox": [62, 164, 290, 181], "content": "4.3 Adjoint Action of on", "index": 1}, {"type": "text", "bbox": [62, 189, 537, 232], "content": "In this short subsection we will summarize the most important facts about the adjoint action of on that can be deduced from the general theory of transformation groups (see, e.g., [7]).", "index": 2}, {"type": "text", "bbox": [62, 262, 385, 275], "content": "Consequently, we have for the type of the corresponding orbit", "index": 3}, {"type": "interline_equation", "bbox": [210, 279, 386, 291], "content": "", "index": 4}, {"type": "text", "bbox": [62, 290, 267, 304], "content": "The slice theorem reads now as follows:", "index": 5}, {"type": "text", "bbox": [62, 313, 538, 371], "content": "Proposition 4.4 Let . Then there is an with , such that: • is an open neighboorhood of and • there is an equivariant retraction with .", "index": 6}, {"type": "text", "bbox": [62, 381, 538, 411], "content": "Both on and on the type is a Howe subgroup of . The transformation behaviour of the types under a reduction mapping is stated in the next", "index": 7}, {"type": "text", "bbox": [63, 420, 537, 449], "content": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all and all we have", "index": 8}, {"type": "interline_equation", "bbox": [288, 450, 411, 469], "content": "", "index": 9}, {"type": "text", "bbox": [62, 479, 483, 499], "content": "Proof We have", "index": 10}, {"type": "title", "bbox": [62, 516, 266, 536], "content": "5 Slice Theorem for", "index": 11}, {"type": "text", "bbox": [62, 548, 338, 563], "content": "We state now the main theorem of the present paper.", "index": 12}, {"type": "text", "bbox": [61, 571, 539, 615], "content": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all there is an with , such that:", "index": 13}, {"type": "text", "bbox": [146, 615, 537, 646], "content": "• is an open neighbourhood of and there is an equivariant retraction with .", "index": 14}]
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[{"bbox": [61, 14, 536, 35], "content": "Proof • is as a map into a product space continuous iff", "parent_index": 0, "line_index": 0}, {"bbox": [120, 30, 538, 46], "content": "is continuous for all projections onto the th factor. Thus, it is", "parent_index": 0, "line_index": 1}, {"bbox": [122, 46, 412, 61], "content": "sufficient to prove the continuity of for all .", "parent_index": 0, "line_index": 2}, {"bbox": [121, 58, 537, 76], "content": "Now decompose into a product of finitely many edges , (i.e.,", "parent_index": 0, "line_index": 3}, {"bbox": [122, 74, 537, 91], "content": "into paths that can be represented as an edge in a graph). Then the mapping", "parent_index": 0, "line_index": 4}, {"bbox": [123, 88, 539, 108], "content": "with is continuous per definitionem. Since", "parent_index": 0, "line_index": 5}, {"bbox": [119, 103, 443, 123], "content": "the multiplication in is continuous, is continuous, too.", "parent_index": 0, "line_index": 6}, {"bbox": [106, 118, 537, 136], "content": "• The compatibility with the group action follows from .", "parent_index": 0, "line_index": 7}, {"bbox": [513, 136, 537, 147], "content": "qed", "parent_index": 0, "line_index": 8}, {"bbox": [63, 167, 289, 181], "content": "4.3 Adjoint Action of on", "parent_index": 1, "line_index": 0}, {"bbox": [62, 190, 537, 205], "content": "In this short subsection we will summarize the most important facts about the adjoint action", "parent_index": 2, "line_index": 0}, {"bbox": [62, 205, 537, 221], "content": "of on that can be deduced from the general theory of transformation groups (see, e.g.,", "parent_index": 2, "line_index": 1}, {"bbox": [62, 219, 85, 234], "content": "[7]).", "parent_index": 2, "line_index": 2}, {"bbox": [64, 263, 380, 277], "content": "Consequently, we have for the type of the corresponding orbit", "parent_index": 3, "line_index": 0}, {"bbox": [63, 293, 266, 304], "content": "The slice theorem reads now as follows:", "parent_index": 5, "line_index": 0}, {"bbox": [63, 316, 477, 331], "content": "Proposition 4.4 Let . Then there is an with , such that:", "parent_index": 6, "line_index": 0}, {"bbox": [161, 331, 415, 345], "content": "• is an open neighboorhood of and", "parent_index": 6, "line_index": 1}, {"bbox": [161, 345, 538, 361], "content": "• there is an equivariant retraction with", "parent_index": 6, "line_index": 2}, {"bbox": [180, 361, 193, 373], "content": ".", "parent_index": 6, "line_index": 3}, {"bbox": [62, 383, 540, 400], "content": "Both on and on the type is a Howe subgroup of . The transformation behaviour of", "parent_index": 7, "line_index": 0}, {"bbox": [62, 398, 362, 414], "content": "the types under a reduction mapping is stated in the next", "parent_index": 7, "line_index": 1}, {"bbox": [61, 422, 538, 438], "content": "Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all and all", "parent_index": 8, "line_index": 0}, {"bbox": [164, 436, 244, 452], "content": "we have", "parent_index": 8, "line_index": 1}, {"bbox": [62, 482, 478, 500], "content": "Proof We have", "parent_index": 10, "line_index": 0}, {"bbox": [64, 519, 264, 536], "content": "5 Slice Theorem for", "parent_index": 11, "line_index": 0}, {"bbox": [63, 549, 338, 564], "content": "We state now the main theorem of the present paper.", "parent_index": 12, "line_index": 0}, {"bbox": [62, 573, 426, 590], "content": "Theorem 5.1 There is a tubular neighbourhood for any gauge orbit.", "parent_index": 13, "line_index": 0}, {"bbox": [147, 587, 537, 602], "content": "Equivalently we have: For all there is an with , such", "parent_index": 13, "line_index": 1}, {"bbox": [147, 603, 175, 618], "content": "that:", "parent_index": 13, "line_index": 2}, {"bbox": [148, 617, 398, 632], "content": "• is an open neighbourhood of and", "parent_index": 14, "line_index": 0}, {"bbox": [162, 630, 537, 647], "content": "there is an equivariant retraction with .", "parent_index": 14, "line_index": 1}]
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[]
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[{"bbox": [123, 17, 209, 30], "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "parent_index": 0, "subtype": "inline"}, {"bbox": [458, 21, 536, 32], "content": "\\pi_{i}\\circ\\varphi_{\\alpha}\\equiv\\varphi_{\\{\\alpha_{i}\\}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [289, 32, 373, 44], "content": "\\pi_{i}:\\mathbf{G}^{\\#\\alpha}\\longrightarrow\\mathbf{G}", "parent_index": 0, "subtype": "inline"}, {"bbox": [424, 34, 428, 42], "content": "i", "parent_index": 0, "subtype": "inline"}, {"bbox": [307, 51, 329, 60], "content": "\\varphi\\{\\alpha\\}", "parent_index": 0, "subtype": "inline"}, {"bbox": [367, 48, 407, 57], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [211, 65, 218, 71], "content": "\\alpha", "parent_index": 0, "subtype": "inline"}, {"bbox": [424, 65, 433, 74], "content": "e_{j}", "parent_index": 0, "subtype": "inline"}, {"bbox": [441, 62, 509, 73], "content": "j\\,=\\,1,\\ldots,J", "parent_index": 0, "subtype": "inline"}, {"bbox": [123, 91, 175, 101], "content": "\\overline{{\\mathcal{A}}}\\longrightarrow\\mathbf{G}^{J}", "parent_index": 0, "subtype": "inline"}, {"bbox": [204, 89, 346, 107], "content": "{\\overline{{A}}}\\longmapsto\\left(\\pi_{e_{1}}({\\overline{{A}}}),\\dots,\\pi_{e_{J}}({\\overline{{A}}})\\right)", "parent_index": 0, "subtype": "inline"}, {"bbox": [232, 107, 243, 116], "content": "\\mathbf{G}", "parent_index": 0, "subtype": "inline"}, {"bbox": [320, 110, 342, 120], "content": "\\varphi_{\\{\\alpha\\}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [405, 120, 533, 135], "content": "h_{\\overline{{{A}}}\\circ\\overline{{{g}}}}(\\pmb{\\alpha})\\b{=}\\,g_{m}^{-1}\\,h_{\\overline{{{A}}}}(\\pmb{\\alpha})\\mathrm{~}g_{m}", "parent_index": 0, "subtype": "inline"}, {"bbox": [230, 169, 243, 180], "content": "\\mathbf{G}", "parent_index": 1, "subtype": "inline"}, {"bbox": [270, 169, 289, 179], "content": "\\mathbf{G}^{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [76, 207, 87, 216], "content": "\\mathbf{G}", "parent_index": 2, "subtype": "inline"}, {"bbox": [106, 207, 123, 216], "content": "\\mathbf{G}^{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [210, 279, 386, 291], "content": "", "parent_index": 4, "subtype": "interline"}, {"bbox": [185, 318, 222, 329], "content": "\\vec{g}\\in\\mathbf{G}^{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [318, 318, 358, 329], "content": "S\\subseteq\\mathbf{G}^{n}", "parent_index": 6, "subtype": "inline"}, {"bbox": [389, 318, 418, 329], "content": "\\vec{g}\\in S", "parent_index": 6, "subtype": "inline"}, {"bbox": [180, 333, 210, 342], "content": "S\\circ\\mathbf{G}", "parent_index": 6, "subtype": "inline"}, {"bbox": [363, 333, 391, 344], "content": "\\vec{g}\\circ\\mathbf{G}", "parent_index": 6, "subtype": "inline"}, {"bbox": [351, 347, 448, 358], "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\vec{g}\\circ\\mathbf{G}", "parent_index": 6, "subtype": "inline"}, {"bbox": [478, 346, 538, 359], "content": "f^{-1}(\\{\\vec{g}\\})=", "parent_index": 6, "subtype": "inline"}, {"bbox": [180, 362, 188, 371], "content": "S", "parent_index": 6, "subtype": "inline"}, {"bbox": [109, 384, 119, 395], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [162, 386, 178, 395], "content": "\\mathbf{G}^{n}", "parent_index": 7, "subtype": "inline"}, {"bbox": [348, 386, 358, 395], "content": "\\mathbf{G}", "parent_index": 7, "subtype": "inline"}, {"bbox": [451, 425, 496, 435], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [164, 438, 196, 448], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [288, 450, 411, 469], "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}}).", "parent_index": 9, "subtype": "interline"}, {"bbox": [153, 482, 478, 500], "content": "\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]\\equiv[Z(h_{\\overline{{A}}}(\\alpha))]\\leq[Z(\\mathbf{H}_{\\overline{{A}}})]=\\mathrm{Typ}(\\overline{{A}}).", "parent_index": 10, "subtype": "inline"}, {"bbox": [249, 520, 264, 535], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [305, 589, 341, 600], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "parent_index": 13, "subtype": "inline"}, {"bbox": [404, 587, 441, 601], "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "parent_index": 13, "subtype": "inline"}, {"bbox": [471, 587, 506, 600], "content": "{\\overline{{A}}}\\in{\\overline{{S}}}", "parent_index": 13, "subtype": "inline"}, {"bbox": [164, 618, 192, 630], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 14, "subtype": "inline"}, {"bbox": [345, 618, 373, 630], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 14, "subtype": "inline"}, {"bbox": [334, 631, 430, 644], "content": "F:\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 14, "subtype": "inline"}, {"bbox": [458, 630, 533, 646], "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "parent_index": 14, "subtype": "inline"}]
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[]
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# 5.1 The Idea
Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7]) which is valid for the action of a finite-dimensional compact $L i e$ group $G$ on a Hausdorff space $X$ . Let us review the main idea of this proof. Given $x\in X$ . Let $H\subseteq G$ be the stabilizer of $x$ , i.e., $[H]$ is an orbit type on the $G$ -space $X$ . Now, this situation is simulated on an $\mathbb{R}^{n}$ , i.e., for an appropriate action of $G$ on $\mathbb{R}^{n}$ one chooses a point with stabilizer $H$ . So the orbits on $X$ and on $\mathbb{R}^{n}$ can be identified. For the case of $\mathbb{R}^{n}$ the proof of a slice theorem is not very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason extension theorem because this yields an equivariant extension $\psi:X\longrightarrow\mathbb{R}^{n}$ , mapping one orbit onto the other. Finally, by means of $\psi$ the slice theorem can be lifted from $\mathbb{R}^{n}$ to $X$ .
What can we learn for our problem? Obviously, $\mathcal{G}$ is not a finite-dimensional Lie group. But, we know that the stabilizer $\mathbf{B}(\overline{{A}})$ of a connection is homeomorphic to the centralizer $Z(\mathbf{H}_{\overline{{A}}})$ of the holonomy group that is a subgroup of $\mathbf{G}$ . Since every centralizer is finitely generated, $Z(\mathbf{H}_{\overline{{A}}})$ equals $Z(h_{\overline{{{A}}}}(\pmb{\alpha}))$ with an appropriate finite $\alpha\in{\mathcal{H}}{\mathcal{G}}$ . This is nothing but the stabilizer of the adjoint action of $\mathbf{G}$ on ${\bf G}^{n}$ . Thus, the reduction mapping $\varphi_{\alpha}$ is the desired equivalent for $\psi$ .
We are now looking for an appropriate ${\overline{{S}}}\subseteq{\overline{{A}}}$ , such tha
$$
\begin{array}{r}{F:\mathrm{~}\,{\overline{{S}}}\circ{\overline{{\mathcal{G}}}}\enspace\longrightarrow\enspace{\overline{{A}}}\circ{\overline{{\mathcal{G}}}}}\\ {\overline{{A}}^{\prime}\circ\overline{{g}}\enspace\longmapsto\enspace{\overline{{A}}}\circ{\overline{{g}}}}\end{array}
$$
is well-defined and has the desired properties.
In order to make $F$ well-defined, we need $\overline{{A}}^{\prime}\circ\overline{{g}}\,=\,\overline{{A}}^{\prime}\implies\overline{{A}}\circ\overline{{g}}\,=\,\overline{{A}}$ for all ${\overline{{A}}}^{\prime}\in{\overline{{S}}}$ and ${\overline{{g}}}\in{\overline{{g}}}$ , i.e. $\mathbf{B}(\overline{{A}}^{\prime})\subseteq\mathbf{B}(\overline{{A}})$ . Applying the projections $\pi_{x}$ on the stabilizers (see [9]) we get for $\gamma_{x}\in\mathcal{P}_{m x}$ (let $\gamma_{m}$ be the trivial path)
$$
h_{\overline{{{A}}}^{\prime}}(\gamma_{m})^{-1}Z(\mathbf{H}_{\overline{{{A}}}^{\prime}})h_{\overline{{{A}}}^{\prime}}(\gamma_{x})=\pi_{x}(\mathbf{B}(\overline{{{A}}}^{\prime}))\subseteq\pi_{x}(\mathbf{B}(\overline{{{A}}}))=h_{\overline{{{A}}}}(\gamma_{m})^{-1}Z(\mathbf{H}_{\overline{{{A}}}})h_{\overline{{{A}}}}(\gamma_{x}),
$$
$$
Z({\bf H}_{\overline{{{A}}}^{\prime}})\subseteq h_{\overline{{{A}}}^{\prime}}(\gamma_{m})h_{\overline{{{A}}}}(\gamma_{m})^{-1}\,Z({\bf H}_{\overline{{{A}}}})\,h_{\overline{{{A}}}}(\gamma_{x})h_{\overline{{{A}}}^{\prime}}^{-1}(\gamma_{x})
$$
all $x\in M$ . In particular, we have $Z(\mathbf{H}_{\overline{{A}}^{\prime}})\subseteq Z(\mathbf{H}_{\overline{{A}}})$ for
Now we choose an $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ with $Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$ and an $S\subseteq\mathbf{G}^{\#\alpha}$ and an equivariant retraction $f:S\circ\mathbf{G}\longrightarrow\varphi_{\alpha}(\overline{{A}})\circ\mathbf{G}$ . Since equivariant mappings magnify stabilizers (or at least do not reduce them), we have $Z(\vec{g}^{\prime})\subseteq Z(\varphi_{\alpha}(\overline{{A}}))$ for all $\vec{g}^{\prime}\in S$ .
Therefore, the condition of (2) would be, e.g., fulfilled if we had for all $\overline{{A}}^{\prime}\in\overline{{S}}$ because the first condition implies $Z(\mathbf{H}_{\overline{{A}}^{\prime}})\subseteq Z(h_{\overline{{A}}^{\prime}}(\alpha))\equiv Z(\varphi_{\alpha}(\overline{{A}}^{\prime}))\subseteq Z(\varphi_{\alpha}(\overline{{A}}))=Z(\mathbf{H}_{\overline{{A}}})$ . We could now choose $\overline{S}$ such that these two conditions are fulfilled. However, this would imply $F^{-1}(\{A\})\supset{\overline{{S}}}$ in general because for $\overline{{g}}\in{\bf B}(\overline{{A}})$ together with $\overline{{A}}^{\prime}$ the connection $\overline{{A}}^{\prime}\circ\overline{{g}}$ is contained in $F^{-1}(\{A\})$ as well,4 but $\overline{{A}}^{\prime}\circ\overline{{g}}$ needs no longer fulfill the two conditions above. Now it is quite obvious to define $\overline{S}$ as the set of all connections fulfilling these conditions multiplied with $\mathbf{B}(\overline{{A}})$ . And indeed, the well-definedness remains valid.
|
<html><body>
<h1 data-bbox="62 12 165 29">5.1 The Idea </h1>
<p data-bbox="62 36 538 166">Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7]) which is valid for the action of a finite-dimensional compact $L i e$ group $G$ on a Hausdorff space $X$ . Let us review the main idea of this proof. Given $x\in X$ . Let $H\subseteq G$ be the stabilizer of $x$ , i.e., $[H]$ is an orbit type on the $G$ -space $X$ . Now, this situation is simulated on an $\mathbb{R}^{n}$ , i.e., for an appropriate action of $G$ on $\mathbb{R}^{n}$ one chooses a point with stabilizer $H$ . So the orbits on $X$ and on $\mathbb{R}^{n}$ can be identified. For the case of $\mathbb{R}^{n}$ the proof of a slice theorem is not very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason extension theorem because this yields an equivariant extension $\psi:X\longrightarrow\mathbb{R}^{n}$ , mapping one orbit onto the other. Finally, by means of $\psi$ the slice theorem can be lifted from $\mathbb{R}^{n}$ to $X$ . </p>
<p data-bbox="62 166 537 253">What can we learn for our problem? Obviously, $\mathcal{G}$ is not a finite-dimensional Lie group. But, we know that the stabilizer $\mathbf{B}(\overline{{A}})$ of a connection is homeomorphic to the centralizer $Z(\mathbf{H}_{\overline{{A}}})$ of the holonomy group that is a subgroup of $\mathbf{G}$ . Since every centralizer is finitely generated, $Z(\mathbf{H}_{\overline{{A}}})$ equals $Z(h_{\overline{{{A}}}}(\pmb{\alpha}))$ with an appropriate finite $\alpha\in{\mathcal{H}}{\mathcal{G}}$ . This is nothing but the stabilizer of the adjoint action of $\mathbf{G}$ on ${\bf G}^{n}$ . Thus, the reduction mapping $\varphi_{\alpha}$ is the desired equivalent for $\psi$ . </p>
<p data-bbox="63 254 348 268">We are now looking for an appropriate ${\overline{{S}}}\subseteq{\overline{{A}}}$ , such tha </p>
<div class="equation" data-bbox="231 269 354 300">$$
\begin{array}{r}{F:\mathrm{~}\,{\overline{{S}}}\circ{\overline{{\mathcal{G}}}}\enspace\longrightarrow\enspace{\overline{{A}}}\circ{\overline{{\mathcal{G}}}}}\\ {\overline{{A}}^{\prime}\circ\overline{{g}}\enspace\longmapsto\enspace{\overline{{A}}}\circ{\overline{{g}}}}\end{array}
$$</div>
<p data-bbox="62 298 299 311">is well-defined and has the desired properties. </p>
<p data-bbox="61 311 538 354">In order to make $F$ well-defined, we need $\overline{{A}}^{\prime}\circ\overline{{g}}\,=\,\overline{{A}}^{\prime}\implies\overline{{A}}\circ\overline{{g}}\,=\,\overline{{A}}$ for all ${\overline{{A}}}^{\prime}\in{\overline{{S}}}$ and ${\overline{{g}}}\in{\overline{{g}}}$ , i.e. $\mathbf{B}(\overline{{A}}^{\prime})\subseteq\mathbf{B}(\overline{{A}})$ . Applying the projections $\pi_{x}$ on the stabilizers (see [9]) we get for $\gamma_{x}\in\mathcal{P}_{m x}$ (let $\gamma_{m}$ be the trivial path) </p>
<div class="equation" data-bbox="100 356 496 372">$$
h_{\overline{{{A}}}^{\prime}}(\gamma_{m})^{-1}Z(\mathbf{H}_{\overline{{{A}}}^{\prime}})h_{\overline{{{A}}}^{\prime}}(\gamma_{x})=\pi_{x}(\mathbf{B}(\overline{{{A}}}^{\prime}))\subseteq\pi_{x}(\mathbf{B}(\overline{{{A}}}))=h_{\overline{{{A}}}}(\gamma_{m})^{-1}Z(\mathbf{H}_{\overline{{{A}}}})h_{\overline{{{A}}}}(\gamma_{x}),
$$</div>
<div class="equation" data-bbox="172 387 428 405">$$
Z({\bf H}_{\overline{{{A}}}^{\prime}})\subseteq h_{\overline{{{A}}}^{\prime}}(\gamma_{m})h_{\overline{{{A}}}}(\gamma_{m})^{-1}\,Z({\bf H}_{\overline{{{A}}}})\,h_{\overline{{{A}}}}(\gamma_{x})h_{\overline{{{A}}}^{\prime}}^{-1}(\gamma_{x})
$$</div>
<p data-bbox="81 406 367 420">all $x\in M$ . In particular, we have $Z(\mathbf{H}_{\overline{{A}}^{\prime}})\subseteq Z(\mathbf{H}_{\overline{{A}}})$ for </p>
<p data-bbox="61 420 537 463">Now we choose an $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ with $Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$ and an $S\subseteq\mathbf{G}^{\#\alpha}$ and an equivariant retraction $f:S\circ\mathbf{G}\longrightarrow\varphi_{\alpha}(\overline{{A}})\circ\mathbf{G}$ . Since equivariant mappings magnify stabilizers (or at least do not reduce them), we have $Z(\vec{g}^{\prime})\subseteq Z(\varphi_{\alpha}(\overline{{A}}))$ for all $\vec{g}^{\prime}\in S$ . </p>
<p data-bbox="63 464 464 477">Therefore, the condition of (2) would be, e.g., fulfilled if we had for all $\overline{{A}}^{\prime}\in\overline{{S}}$ because the first condition implies $Z(\mathbf{H}_{\overline{{A}}^{\prime}})\subseteq Z(h_{\overline{{A}}^{\prime}}(\alpha))\equiv Z(\varphi_{\alpha}(\overline{{A}}^{\prime}))\subseteq Z(\varphi_{\alpha}(\overline{{A}}))=Z(\mathbf{H}_{\overline{{A}}})$ . We could now choose $\overline{S}$ such that these two conditions are fulfilled. However, this would imply $F^{-1}(\{A\})\supset{\overline{{S}}}$ in general because for $\overline{{g}}\in{\bf B}(\overline{{A}})$ together with $\overline{{A}}^{\prime}$ the connection $\overline{{A}}^{\prime}\circ\overline{{g}}$ is contained in $F^{-1}(\{A\})$ as well,4 but $\overline{{A}}^{\prime}\circ\overline{{g}}$ needs no longer fulfill the two conditions above. Now it is quite obvious to define $\overline{S}$ as the set of all connections fulfilling these conditions multiplied with $\mathbf{B}(\overline{{A}})$ . And indeed, the well-definedness remains valid. </p>
</body></html>
|
0001008v1
| 6 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "5.1 The Idea ", "text_level": 1, "page_idx": 6}, {"type": "text", "text": "Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7]) which is valid for the action of a finite-dimensional compact $L i e$ group $G$ on a Hausdorff space $X$ . Let us review the main idea of this proof. Given $x\\in X$ . Let $H\\subseteq G$ be the stabilizer of $x$ , i.e., $[H]$ is an orbit type on the $G$ -space $X$ . Now, this situation is simulated on an $\\mathbb{R}^{n}$ , i.e., for an appropriate action of $G$ on $\\mathbb{R}^{n}$ one chooses a point with stabilizer $H$ . So the orbits on $X$ and on $\\mathbb{R}^{n}$ can be identified. For the case of $\\mathbb{R}^{n}$ the proof of a slice theorem is not very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason extension theorem because this yields an equivariant extension $\\psi:X\\longrightarrow\\mathbb{R}^{n}$ , mapping one orbit onto the other. Finally, by means of $\\psi$ the slice theorem can be lifted from $\\mathbb{R}^{n}$ to $X$ . ", "page_idx": 6}, {"type": "text", "text": "What can we learn for our problem? Obviously, $\\mathcal{G}$ is not a finite-dimensional Lie group. But, we know that the stabilizer $\\mathbf{B}(\\overline{{A}})$ of a connection is homeomorphic to the centralizer $Z(\\mathbf{H}_{\\overline{{A}}})$ of the holonomy group that is a subgroup of $\\mathbf{G}$ . Since every centralizer is finitely generated, $Z(\\mathbf{H}_{\\overline{{A}}})$ equals $Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))$ with an appropriate finite $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ . This is nothing but the stabilizer of the adjoint action of $\\mathbf{G}$ on ${\\bf G}^{n}$ . Thus, the reduction mapping $\\varphi_{\\alpha}$ is the desired equivalent for $\\psi$ . ", "page_idx": 6}, {"type": "text", "text": "We are now looking for an appropriate ${\\overline{{S}}}\\subseteq{\\overline{{A}}}$ , such tha ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "is well-defined and has the desired properties. ", "page_idx": 6}, {"type": "text", "text": "In order to make $F$ well-defined, we need $\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}$ for all ${\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}$ and ${\\overline{{g}}}\\in{\\overline{{g}}}$ , i.e. $\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})$ . Applying the projections $\\pi_{x}$ on the stabilizers (see [9]) we get for $\\gamma_{x}\\in\\mathcal{P}_{m x}$ (let $\\gamma_{m}$ be the trivial path) ", "page_idx": 6}, {"type": "equation", "text": "$$\nh_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),\n$$", "text_format": "latex", "page_idx": 6}, {"type": "equation", "text": "$$\nZ({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "all $x\\in M$ . In particular, we have $Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})$ for ", "page_idx": 6}, {"type": "text", "text": "Now we choose an $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ with $Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$ and an $S\\subseteq\\mathbf{G}^{\\#\\alpha}$ and an equivariant retraction $f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}$ . Since equivariant mappings magnify stabilizers (or at least do not reduce them), we have $Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))$ for all $\\vec{g}^{\\prime}\\in S$ . ", "page_idx": 6}, {"type": "text", "text": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all $\\overline{{A}}^{\\prime}\\in\\overline{{S}}$ because the first condition implies $Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})$ . We could now choose $\\overline{S}$ such that these two conditions are fulfilled. However, this would imply $F^{-1}(\\{A\\})\\supset{\\overline{{S}}}$ in general because for $\\overline{{g}}\\in{\\bf B}(\\overline{{A}})$ together with $\\overline{{A}}^{\\prime}$ the connection $\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$ is contained in $F^{-1}(\\{A\\})$ as well,4 but $\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$ needs no longer fulfill the two conditions above. Now it is quite obvious to define $\\overline{S}$ as the set of all connections fulfilling these conditions multiplied with $\\mathbf{B}(\\overline{{A}})$ . And indeed, the well-definedness remains valid. ", "page_idx": 6}, {"type": "text", "text": "", "page_idx": 6}]
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Let us review the main idea of this proof. Given ", "type": "text"}, {"bbox": [343, 70, 378, 79], "score": 0.93, "content": "x\\in X", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 69, 409, 82], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [410, 70, 448, 81], "score": 0.94, "content": "H\\subseteq G", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [448, 69, 537, 82], "score": 1.0, "content": " be the stabilizer", "type": "text"}], "index": 3}, {"bbox": [62, 83, 537, 96], "spans": [{"bbox": [62, 83, 76, 96], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [76, 88, 83, 93], "score": 0.87, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [83, 83, 113, 96], "score": 1.0, "content": ", i.e., ", "type": "text"}, {"bbox": [113, 84, 130, 96], "score": 0.93, "content": "[H]", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [130, 83, 254, 96], "score": 1.0, "content": " is an orbit type on the ", "type": "text"}, {"bbox": [254, 84, 264, 93], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [264, 83, 298, 96], "score": 1.0, "content": "-space ", "type": "text"}, {"bbox": [298, 85, 309, 93], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [310, 83, 518, 96], "score": 1.0, "content": ". Now, this situation is simulated on an ", "type": "text"}, {"bbox": [518, 84, 533, 93], "score": 0.87, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [533, 83, 537, 96], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [61, 97, 538, 113], "spans": [{"bbox": [61, 97, 228, 113], "score": 1.0, "content": "i.e., for an appropriate action of ", "type": "text"}, {"bbox": [228, 99, 237, 108], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [238, 97, 256, 113], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [257, 99, 271, 108], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [271, 97, 452, 113], "score": 1.0, "content": " one chooses a point with stabilizer ", "type": "text"}, {"bbox": [452, 99, 463, 108], "score": 0.91, "content": "H", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [463, 97, 538, 113], "score": 1.0, "content": ". So the orbits", "type": "text"}], "index": 5}, {"bbox": [62, 111, 537, 125], "spans": [{"bbox": [62, 111, 79, 125], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [80, 113, 91, 122], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [91, 111, 136, 125], "score": 1.0, "content": " and on ", "type": "text"}, {"bbox": [137, 113, 151, 122], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [151, 111, 336, 125], "score": 1.0, "content": " can be identified. For the case of ", "type": "text"}, {"bbox": [336, 113, 351, 122], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [351, 111, 537, 125], "score": 1.0, "content": " the proof of a slice theorem is not", "type": "text"}], "index": 6}, {"bbox": [63, 127, 537, 140], "spans": [{"bbox": [63, 127, 537, 140], "score": 1.0, "content": "very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason", "type": "text"}], "index": 7}, {"bbox": [62, 140, 538, 155], "spans": [{"bbox": [62, 140, 390, 155], "score": 1.0, "content": "extension theorem because this yields an equivariant extension ", "type": "text"}, {"bbox": [391, 142, 462, 153], "score": 0.92, "content": "\\psi:X\\longrightarrow\\mathbb{R}^{n}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [462, 140, 538, 155], "score": 1.0, "content": ", mapping one", "type": "text"}], "index": 8}, {"bbox": [63, 155, 527, 168], "spans": [{"bbox": [63, 155, 280, 168], "score": 1.0, "content": "orbit onto the other. Finally, by means of ", "type": "text"}, {"bbox": [281, 156, 289, 168], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [289, 155, 479, 168], "score": 1.0, "content": " the slice theorem can be lifted from ", "type": "text"}, {"bbox": [479, 156, 493, 165], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [494, 155, 511, 168], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [511, 156, 522, 165], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [523, 155, 527, 168], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 5}, {"type": "text", "bbox": [62, 166, 537, 253], "lines": [{"bbox": [62, 168, 536, 184], "spans": [{"bbox": [62, 168, 309, 184], "score": 1.0, "content": "What can we learn for our problem? Obviously, ", "type": "text"}, {"bbox": [309, 169, 317, 181], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [318, 168, 536, 184], "score": 1.0, "content": " is not a finite-dimensional Lie group. But,", "type": "text"}], "index": 10}, {"bbox": [63, 184, 536, 198], "spans": [{"bbox": [63, 184, 205, 198], "score": 1.0, "content": "we know that the stabilizer ", "type": "text"}, {"bbox": [206, 184, 234, 198], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [234, 184, 500, 198], "score": 1.0, "content": " of a connection is homeomorphic to the centralizer ", "type": "text"}, {"bbox": [500, 185, 536, 198], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}], "index": 11}, {"bbox": [62, 198, 537, 214], "spans": [{"bbox": [62, 198, 294, 214], "score": 1.0, "content": "of the holonomy group that is a subgroup of ", "type": "text"}, {"bbox": [294, 200, 305, 209], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [306, 198, 537, 214], "score": 1.0, "content": ". Since every centralizer is finitely generated,", "type": "text"}], "index": 12}, {"bbox": [63, 212, 537, 229], "spans": [{"bbox": [63, 214, 98, 226], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [99, 212, 136, 229], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [136, 214, 186, 226], "score": 0.94, "content": "Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [186, 212, 321, 229], "score": 1.0, "content": " with an appropriate finite ", "type": "text"}, {"bbox": [321, 214, 363, 225], "score": 0.92, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [363, 212, 537, 229], "score": 1.0, "content": ". This is nothing but the stabilizer", "type": "text"}], "index": 13}, {"bbox": [62, 226, 536, 243], "spans": [{"bbox": [62, 226, 185, 243], "score": 1.0, "content": "of the adjoint action of ", "type": "text"}, {"bbox": [185, 229, 195, 238], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [196, 226, 215, 243], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [216, 229, 232, 238], "score": 0.91, "content": "{\\bf G}^{n}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [232, 226, 393, 243], "score": 1.0, "content": ". Thus, the reduction mapping ", "type": "text"}, {"bbox": [393, 232, 408, 240], "score": 0.91, "content": "\\varphi_{\\alpha}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [409, 226, 536, 243], "score": 1.0, "content": " is the desired equivalent", "type": "text"}], "index": 14}, {"bbox": [62, 240, 95, 256], "spans": [{"bbox": [62, 240, 81, 256], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [81, 243, 89, 254], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [90, 240, 95, 256], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5}, {"type": "text", "bbox": [63, 254, 348, 268], "lines": [{"bbox": [63, 255, 349, 270], "spans": [{"bbox": [63, 255, 264, 270], "score": 1.0, "content": "We are now looking for an appropriate", "type": "text"}, {"bbox": [265, 256, 299, 268], "score": 0.92, "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [299, 255, 349, 270], "score": 1.0, "content": ", such tha", "type": "text"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [231, 269, 354, 300], "lines": [{"bbox": [231, 269, 354, 300], "spans": [{"bbox": [231, 269, 354, 300], "score": 0.75, "content": "\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [62, 298, 299, 311], "lines": [{"bbox": [62, 300, 296, 312], "spans": [{"bbox": [62, 300, 296, 312], "score": 1.0, "content": "is well-defined and has the desired properties.", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [61, 311, 538, 354], "lines": [{"bbox": [61, 313, 537, 327], "spans": [{"bbox": [61, 313, 157, 327], "score": 1.0, "content": "In order to make ", "type": "text"}, {"bbox": [157, 316, 167, 325], "score": 0.88, "content": "F", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [167, 313, 286, 327], "score": 1.0, "content": " well-defined, we need ", "type": "text"}, {"bbox": [286, 313, 433, 327], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}", "type": "inline_equation", "height": 14, "width": 147}, {"bbox": [434, 313, 474, 327], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [474, 313, 512, 325], "score": 0.94, "content": "{\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [513, 313, 537, 327], "score": 1.0, "content": " and", "type": "text"}], "index": 19}, {"bbox": [63, 326, 537, 344], "spans": [{"bbox": [63, 329, 92, 342], "score": 0.92, "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [92, 326, 119, 344], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [119, 327, 193, 342], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 74}, {"bbox": [194, 326, 333, 344], "score": 1.0, "content": ". Applying the projections ", "type": "text"}, {"bbox": [333, 334, 345, 341], "score": 0.9, "content": "\\pi_{x}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [345, 326, 537, 344], "score": 1.0, "content": " on the stabilizers (see [9]) we get for", "type": "text"}], "index": 20}, {"bbox": [63, 344, 253, 357], "spans": [{"bbox": [63, 345, 110, 356], "score": 0.89, "content": "\\gamma_{x}\\in\\mathcal{P}_{m x}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [110, 344, 135, 357], "score": 1.0, "content": " (let ", "type": "text"}, {"bbox": [135, 348, 150, 356], "score": 0.85, "content": "\\gamma_{m}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [150, 344, 253, 357], "score": 1.0, "content": " be the trivial path)", "type": "text"}], "index": 21}], "index": 20}, {"type": "interline_equation", "bbox": [100, 356, 496, 372], "lines": [{"bbox": [100, 356, 496, 372], "spans": [{"bbox": [100, 356, 496, 372], "score": 0.89, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [172, 387, 428, 405], "lines": [{"bbox": [172, 387, 428, 405], "spans": [{"bbox": [172, 387, 428, 405], "score": 0.87, "content": "Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [81, 406, 367, 420], "lines": [{"bbox": [81, 407, 366, 422], "spans": [{"bbox": [81, 407, 97, 422], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [97, 409, 131, 418], "score": 0.92, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [132, 407, 255, 422], "score": 1.0, "content": ". In particular, we have ", "type": "text"}, {"bbox": [255, 408, 345, 422], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [345, 407, 366, 422], "score": 1.0, "content": " for ", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [61, 420, 537, 463], "lines": [{"bbox": [62, 421, 537, 437], "spans": [{"bbox": [62, 421, 162, 437], "score": 1.0, "content": "Now we choose an ", "type": "text"}, {"bbox": [162, 424, 207, 434], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [207, 421, 237, 437], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [238, 423, 341, 435], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 12, "width": 103}, {"bbox": [341, 421, 384, 437], "score": 1.0, "content": " and an ", "type": "text"}, {"bbox": [384, 422, 435, 434], "score": 0.93, "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [435, 421, 537, 437], "score": 1.0, "content": " and an equivariant", "type": "text"}], "index": 25}, {"bbox": [60, 435, 538, 452], "spans": [{"bbox": [60, 435, 117, 452], "score": 1.0, "content": "retraction ", "type": "text"}, {"bbox": [118, 436, 249, 450], "score": 0.93, "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 131}, {"bbox": [250, 435, 538, 452], "score": 1.0, "content": ". Since equivariant mappings magnify stabilizers (or at", "type": "text"}], "index": 26}, {"bbox": [62, 451, 414, 465], "spans": [{"bbox": [62, 451, 246, 465], "score": 1.0, "content": "least do not reduce them), we have ", "type": "text"}, {"bbox": [246, 451, 339, 464], "score": 0.92, "content": "Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [340, 451, 378, 465], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [378, 452, 409, 464], "score": 0.92, "content": "\\vec{g}^{\\prime}\\in S", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [410, 451, 414, 465], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [63, 464, 464, 477], "lines": [{"bbox": [62, 463, 463, 480], "spans": [{"bbox": [62, 463, 428, 480], "score": 1.0, "content": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all ", "type": "text"}, {"bbox": [428, 464, 463, 476], "score": 0.89, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 12, "width": 35}], "index": 28}], "index": 28}, {"type": "text", "bbox": [62, 507, 538, 595], "lines": [{"bbox": [61, 509, 537, 527], "spans": [{"bbox": [61, 509, 240, 527], "score": 1.0, "content": "because the first condition implies ", "type": "text"}, {"bbox": [240, 509, 533, 525], "score": 0.85, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 16, "width": 293}, {"bbox": [533, 509, 537, 527], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [62, 525, 537, 539], "spans": [{"bbox": [62, 525, 180, 539], "score": 1.0, "content": "We could now choose ", "type": "text"}, {"bbox": [180, 525, 189, 536], "score": 0.9, "content": "\\overline{S}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [189, 525, 537, 539], "score": 1.0, "content": " such that these two conditions are fulfilled. However, this would", "type": "text"}], "index": 30}, {"bbox": [62, 538, 536, 554], "spans": [{"bbox": [62, 538, 95, 554], "score": 1.0, "content": "imply ", "type": "text"}, {"bbox": [96, 540, 170, 553], "score": 0.94, "content": "F^{-1}(\\{A\\})\\supset{\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [171, 538, 288, 554], "score": 1.0, "content": " in general because for ", "type": "text"}, {"bbox": [288, 540, 337, 553], "score": 0.95, "content": "\\overline{{g}}\\in{\\bf B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [337, 538, 413, 554], "score": 1.0, "content": " together with ", "type": "text"}, {"bbox": [413, 538, 425, 550], "score": 0.91, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [425, 538, 507, 554], "score": 1.0, "content": " the connection ", "type": "text"}, {"bbox": [507, 538, 536, 552], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 14, "width": 29}], "index": 31}, {"bbox": [61, 552, 538, 569], "spans": [{"bbox": [61, 552, 141, 569], "score": 1.0, "content": "is contained in ", "type": "text"}, {"bbox": [141, 555, 191, 568], "score": 0.94, "content": "F^{-1}(\\{A\\})", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [191, 552, 261, 569], "score": 1.0, "content": " as well,4 but ", "type": "text"}, {"bbox": [261, 553, 290, 567], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [290, 552, 538, 569], "score": 1.0, "content": " needs no longer fulfill the two conditions above.", "type": "text"}], "index": 32}, {"bbox": [61, 567, 538, 583], "spans": [{"bbox": [61, 567, 240, 583], "score": 1.0, "content": "Now it is quite obvious to define ", "type": "text"}, {"bbox": [241, 569, 249, 579], "score": 0.9, "content": "\\overline{S}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [249, 567, 538, 583], "score": 1.0, "content": " as the set of all connections fulfilling these conditions", "type": "text"}], "index": 33}, {"bbox": [62, 582, 425, 597], "spans": [{"bbox": [62, 582, 145, 597], "score": 1.0, "content": "multiplied with ", "type": "text"}, {"bbox": [145, 583, 173, 596], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [173, 582, 425, 597], "score": 1.0, "content": ". And indeed, the well-definedness remains valid.", "type": "text"}], "index": 34}], "index": 31.5}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [231, 269, 354, 300], "lines": [{"bbox": [231, 269, 354, 300], "spans": [{"bbox": [231, 269, 354, 300], "score": 0.75, "content": "\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [100, 356, 496, 372], "lines": [{"bbox": [100, 356, 496, 372], "spans": [{"bbox": [100, 356, 496, 372], "score": 0.89, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [172, 387, 428, 405], "lines": [{"bbox": [172, 387, 428, 405], "spans": [{"bbox": [172, 387, 428, 405], "score": 0.87, "content": "Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})", "type": "interline_equation"}], "index": 23}], "index": 23}], "discarded_blocks": [{"type": "discarded", "bbox": [74, 620, 260, 636], "lines": [{"bbox": [76, 622, 257, 636], "spans": [{"bbox": [76, 622, 120, 636], "score": 1.0, "content": "4We have", "type": "text"}, {"bbox": [121, 623, 257, 636], "score": 0.9, "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}=\\overline{{A}}\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})", "type": "inline_equation", "height": 13, "width": 136}]}]}, {"type": "discarded", "bbox": [295, 704, 303, 715], "lines": [{"bbox": [295, 705, 304, 717], "spans": [{"bbox": [295, 705, 304, 717], "score": 1.0, "content": "7", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [62, 12, 165, 29], "lines": [{"bbox": [61, 15, 164, 30], "spans": [{"bbox": [61, 15, 86, 30], "score": 1.0, "content": "5.1", "type": "text"}, {"bbox": [98, 15, 164, 29], "score": 1.0, "content": "The Idea", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [62, 36, 538, 166], "lines": [{"bbox": [62, 38, 537, 54], "spans": [{"bbox": [62, 38, 537, 54], "score": 1.0, "content": "Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7])", "type": "text"}], "index": 1}, {"bbox": [62, 53, 537, 68], "spans": [{"bbox": [62, 53, 367, 68], "score": 1.0, "content": "which is valid for the action of a finite-dimensional compact ", "type": "text"}, {"bbox": [367, 55, 384, 64], "score": 0.48, "content": "L i e", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [384, 53, 419, 68], "score": 1.0, "content": " group ", "type": "text"}, {"bbox": [419, 55, 428, 64], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [429, 53, 537, 68], "score": 1.0, "content": " on a Hausdorff space", "type": "text"}], "index": 2}, {"bbox": [63, 69, 537, 82], "spans": [{"bbox": [63, 70, 74, 79], "score": 0.89, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [74, 69, 343, 82], "score": 1.0, "content": ". Let us review the main idea of this proof. Given ", "type": "text"}, {"bbox": [343, 70, 378, 79], "score": 0.93, "content": "x\\in X", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [378, 69, 409, 82], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [410, 70, 448, 81], "score": 0.94, "content": "H\\subseteq G", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [448, 69, 537, 82], "score": 1.0, "content": " be the stabilizer", "type": "text"}], "index": 3}, {"bbox": [62, 83, 537, 96], "spans": [{"bbox": [62, 83, 76, 96], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [76, 88, 83, 93], "score": 0.87, "content": "x", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [83, 83, 113, 96], "score": 1.0, "content": ", i.e., ", "type": "text"}, {"bbox": [113, 84, 130, 96], "score": 0.93, "content": "[H]", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [130, 83, 254, 96], "score": 1.0, "content": " is an orbit type on the ", "type": "text"}, {"bbox": [254, 84, 264, 93], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [264, 83, 298, 96], "score": 1.0, "content": "-space ", "type": "text"}, {"bbox": [298, 85, 309, 93], "score": 0.91, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [310, 83, 518, 96], "score": 1.0, "content": ". Now, this situation is simulated on an ", "type": "text"}, {"bbox": [518, 84, 533, 93], "score": 0.87, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [533, 83, 537, 96], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [61, 97, 538, 113], "spans": [{"bbox": [61, 97, 228, 113], "score": 1.0, "content": "i.e., for an appropriate action of ", "type": "text"}, {"bbox": [228, 99, 237, 108], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [238, 97, 256, 113], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [257, 99, 271, 108], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [271, 97, 452, 113], "score": 1.0, "content": " one chooses a point with stabilizer ", "type": "text"}, {"bbox": [452, 99, 463, 108], "score": 0.91, "content": "H", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [463, 97, 538, 113], "score": 1.0, "content": ". So the orbits", "type": "text"}], "index": 5}, {"bbox": [62, 111, 537, 125], "spans": [{"bbox": [62, 111, 79, 125], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [80, 113, 91, 122], "score": 0.9, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [91, 111, 136, 125], "score": 1.0, "content": " and on ", "type": "text"}, {"bbox": [137, 113, 151, 122], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [151, 111, 336, 125], "score": 1.0, "content": " can be identified. For the case of ", "type": "text"}, {"bbox": [336, 113, 351, 122], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [351, 111, 537, 125], "score": 1.0, "content": " the proof of a slice theorem is not", "type": "text"}], "index": 6}, {"bbox": [63, 127, 537, 140], "spans": [{"bbox": [63, 127, 537, 140], "score": 1.0, "content": "very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason", "type": "text"}], "index": 7}, {"bbox": [62, 140, 538, 155], "spans": [{"bbox": [62, 140, 390, 155], "score": 1.0, "content": "extension theorem because this yields an equivariant extension ", "type": "text"}, {"bbox": [391, 142, 462, 153], "score": 0.92, "content": "\\psi:X\\longrightarrow\\mathbb{R}^{n}", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [462, 140, 538, 155], "score": 1.0, "content": ", mapping one", "type": "text"}], "index": 8}, {"bbox": [63, 155, 527, 168], "spans": [{"bbox": [63, 155, 280, 168], "score": 1.0, "content": "orbit onto the other. Finally, by means of ", "type": "text"}, {"bbox": [281, 156, 289, 168], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [289, 155, 479, 168], "score": 1.0, "content": " the slice theorem can be lifted from ", "type": "text"}, {"bbox": [479, 156, 493, 165], "score": 0.92, "content": "\\mathbb{R}^{n}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [494, 155, 511, 168], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [511, 156, 522, 165], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [523, 155, 527, 168], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 5, "bbox_fs": [61, 38, 538, 168]}, {"type": "text", "bbox": [62, 166, 537, 253], "lines": [{"bbox": [62, 168, 536, 184], "spans": [{"bbox": [62, 168, 309, 184], "score": 1.0, "content": "What can we learn for our problem? Obviously, ", "type": "text"}, {"bbox": [309, 169, 317, 181], "score": 0.91, "content": "\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [318, 168, 536, 184], "score": 1.0, "content": " is not a finite-dimensional Lie group. But,", "type": "text"}], "index": 10}, {"bbox": [63, 184, 536, 198], "spans": [{"bbox": [63, 184, 205, 198], "score": 1.0, "content": "we know that the stabilizer ", "type": "text"}, {"bbox": [206, 184, 234, 198], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [234, 184, 500, 198], "score": 1.0, "content": " of a connection is homeomorphic to the centralizer ", "type": "text"}, {"bbox": [500, 185, 536, 198], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}], "index": 11}, {"bbox": [62, 198, 537, 214], "spans": [{"bbox": [62, 198, 294, 214], "score": 1.0, "content": "of the holonomy group that is a subgroup of ", "type": "text"}, {"bbox": [294, 200, 305, 209], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [306, 198, 537, 214], "score": 1.0, "content": ". Since every centralizer is finitely generated,", "type": "text"}], "index": 12}, {"bbox": [63, 212, 537, 229], "spans": [{"bbox": [63, 214, 98, 226], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [99, 212, 136, 229], "score": 1.0, "content": " equals ", "type": "text"}, {"bbox": [136, 214, 186, 226], "score": 0.94, "content": "Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 12, "width": 50}, {"bbox": [186, 212, 321, 229], "score": 1.0, "content": " with an appropriate finite ", "type": "text"}, {"bbox": [321, 214, 363, 225], "score": 0.92, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [363, 212, 537, 229], "score": 1.0, "content": ". This is nothing but the stabilizer", "type": "text"}], "index": 13}, {"bbox": [62, 226, 536, 243], "spans": [{"bbox": [62, 226, 185, 243], "score": 1.0, "content": "of the adjoint action of ", "type": "text"}, {"bbox": [185, 229, 195, 238], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [196, 226, 215, 243], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [216, 229, 232, 238], "score": 0.91, "content": "{\\bf G}^{n}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [232, 226, 393, 243], "score": 1.0, "content": ". Thus, the reduction mapping ", "type": "text"}, {"bbox": [393, 232, 408, 240], "score": 0.91, "content": "\\varphi_{\\alpha}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [409, 226, 536, 243], "score": 1.0, "content": " is the desired equivalent", "type": "text"}], "index": 14}, {"bbox": [62, 240, 95, 256], "spans": [{"bbox": [62, 240, 81, 256], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [81, 243, 89, 254], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [90, 240, 95, 256], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 12.5, "bbox_fs": [62, 168, 537, 256]}, {"type": "text", "bbox": [63, 254, 348, 268], "lines": [{"bbox": [63, 255, 349, 270], "spans": [{"bbox": [63, 255, 264, 270], "score": 1.0, "content": "We are now looking for an appropriate", "type": "text"}, {"bbox": [265, 256, 299, 268], "score": 0.92, "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [299, 255, 349, 270], "score": 1.0, "content": ", such tha", "type": "text"}], "index": 16}], "index": 16, "bbox_fs": [63, 255, 349, 270]}, {"type": "interline_equation", "bbox": [231, 269, 354, 300], "lines": [{"bbox": [231, 269, 354, 300], "spans": [{"bbox": [231, 269, 354, 300], "score": 0.75, "content": "\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [62, 298, 299, 311], "lines": [{"bbox": [62, 300, 296, 312], "spans": [{"bbox": [62, 300, 296, 312], "score": 1.0, "content": "is well-defined and has the desired properties.", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [62, 300, 296, 312]}, {"type": "text", "bbox": [61, 311, 538, 354], "lines": [{"bbox": [61, 313, 537, 327], "spans": [{"bbox": [61, 313, 157, 327], "score": 1.0, "content": "In order to make ", "type": "text"}, {"bbox": [157, 316, 167, 325], "score": 0.88, "content": "F", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [167, 313, 286, 327], "score": 1.0, "content": " well-defined, we need ", "type": "text"}, {"bbox": [286, 313, 433, 327], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}", "type": "inline_equation", "height": 14, "width": 147}, {"bbox": [434, 313, 474, 327], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [474, 313, 512, 325], "score": 0.94, "content": "{\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [513, 313, 537, 327], "score": 1.0, "content": " and", "type": "text"}], "index": 19}, {"bbox": [63, 326, 537, 344], "spans": [{"bbox": [63, 329, 92, 342], "score": 0.92, "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [92, 326, 119, 344], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [119, 327, 193, 342], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 74}, {"bbox": [194, 326, 333, 344], "score": 1.0, "content": ". Applying the projections ", "type": "text"}, {"bbox": [333, 334, 345, 341], "score": 0.9, "content": "\\pi_{x}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [345, 326, 537, 344], "score": 1.0, "content": " on the stabilizers (see [9]) we get for", "type": "text"}], "index": 20}, {"bbox": [63, 344, 253, 357], "spans": [{"bbox": [63, 345, 110, 356], "score": 0.89, "content": "\\gamma_{x}\\in\\mathcal{P}_{m x}", "type": "inline_equation", "height": 11, "width": 47}, {"bbox": [110, 344, 135, 357], "score": 1.0, "content": " (let ", "type": "text"}, {"bbox": [135, 348, 150, 356], "score": 0.85, "content": "\\gamma_{m}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [150, 344, 253, 357], "score": 1.0, "content": " be the trivial path)", "type": "text"}], "index": 21}], "index": 20, "bbox_fs": [61, 313, 537, 357]}, {"type": "interline_equation", "bbox": [100, 356, 496, 372], "lines": [{"bbox": [100, 356, 496, 372], "spans": [{"bbox": [100, 356, 496, 372], "score": 0.89, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [172, 387, 428, 405], "lines": [{"bbox": [172, 387, 428, 405], "spans": [{"bbox": [172, 387, 428, 405], "score": 0.87, "content": "Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [81, 406, 367, 420], "lines": [{"bbox": [81, 407, 366, 422], "spans": [{"bbox": [81, 407, 97, 422], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [97, 409, 131, 418], "score": 0.92, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [132, 407, 255, 422], "score": 1.0, "content": ". In particular, we have ", "type": "text"}, {"bbox": [255, 408, 345, 422], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 90}, {"bbox": [345, 407, 366, 422], "score": 1.0, "content": " for ", "type": "text"}], "index": 24}], "index": 24, "bbox_fs": [81, 407, 366, 422]}, {"type": "text", "bbox": [61, 420, 537, 463], "lines": [{"bbox": [62, 421, 537, 437], "spans": [{"bbox": [62, 421, 162, 437], "score": 1.0, "content": "Now we choose an ", "type": "text"}, {"bbox": [162, 424, 207, 434], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [207, 421, 237, 437], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [238, 423, 341, 435], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 12, "width": 103}, {"bbox": [341, 421, 384, 437], "score": 1.0, "content": " and an ", "type": "text"}, {"bbox": [384, 422, 435, 434], "score": 0.93, "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [435, 421, 537, 437], "score": 1.0, "content": " and an equivariant", "type": "text"}], "index": 25}, {"bbox": [60, 435, 538, 452], "spans": [{"bbox": [60, 435, 117, 452], "score": 1.0, "content": "retraction ", "type": "text"}, {"bbox": [118, 436, 249, 450], "score": 0.93, "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 131}, {"bbox": [250, 435, 538, 452], "score": 1.0, "content": ". Since equivariant mappings magnify stabilizers (or at", "type": "text"}], "index": 26}, {"bbox": [62, 451, 414, 465], "spans": [{"bbox": [62, 451, 246, 465], "score": 1.0, "content": "least do not reduce them), we have ", "type": "text"}, {"bbox": [246, 451, 339, 464], "score": 0.92, "content": "Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))", "type": "inline_equation", "height": 13, "width": 93}, {"bbox": [340, 451, 378, 465], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [378, 452, 409, 464], "score": 0.92, "content": "\\vec{g}^{\\prime}\\in S", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [410, 451, 414, 465], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 26, "bbox_fs": [60, 421, 538, 465]}, {"type": "text", "bbox": [63, 464, 464, 477], "lines": [{"bbox": [62, 463, 463, 480], "spans": [{"bbox": [62, 463, 428, 480], "score": 1.0, "content": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all ", "type": "text"}, {"bbox": [428, 464, 463, 476], "score": 0.89, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 12, "width": 35}], "index": 28}, {"bbox": [61, 509, 537, 527], "spans": [{"bbox": [61, 509, 240, 527], "score": 1.0, "content": "because the first condition implies ", "type": "text"}, {"bbox": [240, 509, 533, 525], "score": 0.85, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 16, "width": 293}, {"bbox": [533, 509, 537, 527], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [62, 525, 537, 539], "spans": [{"bbox": [62, 525, 180, 539], "score": 1.0, "content": "We could now choose ", "type": "text"}, {"bbox": [180, 525, 189, 536], "score": 0.9, "content": "\\overline{S}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [189, 525, 537, 539], "score": 1.0, "content": " such that these two conditions are fulfilled. However, this would", "type": "text"}], "index": 30}, {"bbox": [62, 538, 536, 554], "spans": [{"bbox": [62, 538, 95, 554], "score": 1.0, "content": "imply ", "type": "text"}, {"bbox": [96, 540, 170, 553], "score": 0.94, "content": "F^{-1}(\\{A\\})\\supset{\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [171, 538, 288, 554], "score": 1.0, "content": " in general because for ", "type": "text"}, {"bbox": [288, 540, 337, 553], "score": 0.95, "content": "\\overline{{g}}\\in{\\bf B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 49}, {"bbox": [337, 538, 413, 554], "score": 1.0, "content": " together with ", "type": "text"}, {"bbox": [413, 538, 425, 550], "score": 0.91, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [425, 538, 507, 554], "score": 1.0, "content": " the connection ", "type": "text"}, {"bbox": [507, 538, 536, 552], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 14, "width": 29}], "index": 31}, {"bbox": [61, 552, 538, 569], "spans": [{"bbox": [61, 552, 141, 569], "score": 1.0, "content": "is contained in ", "type": "text"}, {"bbox": [141, 555, 191, 568], "score": 0.94, "content": "F^{-1}(\\{A\\})", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [191, 552, 261, 569], "score": 1.0, "content": " as well,4 but ", "type": "text"}, {"bbox": [261, 553, 290, 567], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 14, "width": 29}, {"bbox": [290, 552, 538, 569], "score": 1.0, "content": " needs no longer fulfill the two conditions above.", "type": "text"}], "index": 32}, {"bbox": [61, 567, 538, 583], "spans": [{"bbox": [61, 567, 240, 583], "score": 1.0, "content": "Now it is quite obvious to define ", "type": "text"}, {"bbox": [241, 569, 249, 579], "score": 0.9, "content": "\\overline{S}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [249, 567, 538, 583], "score": 1.0, "content": " as the set of all connections fulfilling these conditions", "type": "text"}], "index": 33}, {"bbox": [62, 582, 425, 597], "spans": [{"bbox": [62, 582, 145, 597], "score": 1.0, "content": "multiplied with ", "type": "text"}, {"bbox": [145, 583, 173, 596], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [173, 582, 425, 597], "score": 1.0, "content": ". And indeed, the well-definedness remains valid.", "type": "text"}], "index": 34}], "index": 28, "bbox_fs": [62, 463, 463, 480]}, {"type": "text", "bbox": [62, 507, 538, 595], "lines": [], "index": 31.5, "bbox_fs": [61, 509, 538, 597], "lines_deleted": true}]}
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[{"type": "title", "bbox": [62, 12, 165, 29], "content": "5.1 The Idea", "index": 0}, {"type": "text", "bbox": [62, 36, 538, 166], "content": "Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7]) which is valid for the action of a finite-dimensional compact group on a Hausdorff space . Let us review the main idea of this proof. Given . Let be the stabilizer of , i.e., is an orbit type on the -space . Now, this situation is simulated on an , i.e., for an appropriate action of on one chooses a point with stabilizer . So the orbits on and on can be identified. For the case of the proof of a slice theorem is not very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason extension theorem because this yields an equivariant extension , mapping one orbit onto the other. Finally, by means of the slice theorem can be lifted from to .", "index": 1}, {"type": "text", "bbox": [62, 166, 537, 253], "content": "What can we learn for our problem? Obviously, is not a finite-dimensional Lie group. But, we know that the stabilizer of a connection is homeomorphic to the centralizer of the holonomy group that is a subgroup of . Since every centralizer is finitely generated, equals with an appropriate finite . This is nothing but the stabilizer of the adjoint action of on . Thus, the reduction mapping is the desired equivalent for .", "index": 2}, {"type": "text", "bbox": [63, 254, 348, 268], "content": "We are now looking for an appropriate , such tha", "index": 3}, {"type": "interline_equation", "bbox": [231, 269, 354, 300], "content": "", "index": 4}, {"type": "text", "bbox": [62, 298, 299, 311], "content": "is well-defined and has the desired properties.", "index": 5}, {"type": "text", "bbox": [61, 311, 538, 354], "content": "In order to make well-defined, we need for all and , i.e. . Applying the projections on the stabilizers (see [9]) we get for (let be the trivial path)", "index": 6}, {"type": "interline_equation", "bbox": [100, 356, 496, 372], "content": "", "index": 7}, {"type": "interline_equation", "bbox": [172, 387, 428, 405], "content": "", "index": 8}, {"type": "text", "bbox": [81, 406, 367, 420], "content": "all . In particular, we have for", "index": 9}, {"type": "text", "bbox": [61, 420, 537, 463], "content": "Now we choose an with and an and an equivariant retraction . Since equivariant mappings magnify stabilizers (or at least do not reduce them), we have for all .", "index": 10}, {"type": "text", "bbox": [63, 464, 464, 477], "content": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all because the first condition implies . We could now choose such that these two conditions are fulfilled. However, this would imply in general because for together with the connection is contained in as well,4 but needs no longer fulfill the two conditions above. Now it is quite obvious to define as the set of all connections fulfilling these conditions multiplied with . And indeed, the well-definedness remains valid.", "index": 11}, {"type": "text", "bbox": [62, 507, 538, 595], "content": "", "index": 12}]
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[{"bbox": [61, 15, 164, 30], "content": "5.1 The Idea", "parent_index": 0, "line_index": 0}, {"bbox": [62, 38, 537, 54], "content": "Our proof imitates in a certain sense the proof of the standard slice theorem (see, e.g., [7])", "parent_index": 1, "line_index": 0}, {"bbox": [62, 53, 537, 68], "content": "which is valid for the action of a finite-dimensional compact group on a Hausdorff space", "parent_index": 1, "line_index": 1}, {"bbox": [63, 69, 537, 82], "content": ". Let us review the main idea of this proof. Given . Let be the stabilizer", "parent_index": 1, "line_index": 2}, {"bbox": [62, 83, 537, 96], "content": "of , i.e., is an orbit type on the -space . Now, this situation is simulated on an ,", "parent_index": 1, "line_index": 3}, {"bbox": [61, 97, 538, 113], "content": "i.e., for an appropriate action of on one chooses a point with stabilizer . So the orbits", "parent_index": 1, "line_index": 4}, {"bbox": [62, 111, 537, 125], "content": "on and on can be identified. For the case of the proof of a slice theorem is not", "parent_index": 1, "line_index": 5}, {"bbox": [63, 127, 537, 140], "content": "very complicated. The crucial point of the general proof is the usage of the Tietze-Gleason", "parent_index": 1, "line_index": 6}, {"bbox": [62, 140, 538, 155], "content": "extension theorem because this yields an equivariant extension , mapping one", "parent_index": 1, "line_index": 7}, {"bbox": [63, 155, 527, 168], "content": "orbit onto the other. Finally, by means of the slice theorem can be lifted from to .", "parent_index": 1, "line_index": 8}, {"bbox": [62, 168, 536, 184], "content": "What can we learn for our problem? Obviously, is not a finite-dimensional Lie group. But,", "parent_index": 2, "line_index": 0}, {"bbox": [63, 184, 536, 198], "content": "we know that the stabilizer of a connection is homeomorphic to the centralizer", "parent_index": 2, "line_index": 1}, {"bbox": [62, 198, 537, 214], "content": "of the holonomy group that is a subgroup of . Since every centralizer is finitely generated,", "parent_index": 2, "line_index": 2}, {"bbox": [63, 212, 537, 229], "content": "equals with an appropriate finite . This is nothing but the stabilizer", "parent_index": 2, "line_index": 3}, {"bbox": [62, 226, 536, 243], "content": "of the adjoint action of on . Thus, the reduction mapping is the desired equivalent", "parent_index": 2, "line_index": 4}, {"bbox": [62, 240, 95, 256], "content": "for .", "parent_index": 2, "line_index": 5}, {"bbox": [63, 255, 349, 270], "content": "We are now looking for an appropriate , such tha", "parent_index": 3, "line_index": 0}, {"bbox": [62, 300, 296, 312], "content": "is well-defined and has the desired properties.", "parent_index": 5, "line_index": 0}, {"bbox": [61, 313, 537, 327], "content": "In order to make well-defined, we need for all and", "parent_index": 6, "line_index": 0}, {"bbox": [63, 326, 537, 344], "content": ", i.e. . Applying the projections on the stabilizers (see [9]) we get for", "parent_index": 6, "line_index": 1}, {"bbox": [63, 344, 253, 357], "content": "(let be the trivial path)", "parent_index": 6, "line_index": 2}, {"bbox": [81, 407, 366, 422], "content": "all . In particular, we have for", "parent_index": 9, "line_index": 0}, {"bbox": [62, 421, 537, 437], "content": "Now we choose an with and an and an equivariant", "parent_index": 10, "line_index": 0}, {"bbox": [60, 435, 538, 452], "content": "retraction . Since equivariant mappings magnify stabilizers (or at", "parent_index": 10, "line_index": 1}, {"bbox": [62, 451, 414, 465], "content": "least do not reduce them), we have for all .", "parent_index": 10, "line_index": 2}, {"bbox": [62, 463, 463, 480], "content": "Therefore, the condition of (2) would be, e.g., fulfilled if we had for all", "parent_index": 11, "line_index": 0}, {"bbox": [61, 509, 537, 527], "content": "because the first condition implies .", "parent_index": 11, "line_index": 1}, {"bbox": [62, 525, 537, 539], "content": "We could now choose such that these two conditions are fulfilled. However, this would", "parent_index": 11, "line_index": 2}, {"bbox": [62, 538, 536, 554], "content": "imply in general because for together with the connection", "parent_index": 11, "line_index": 3}, {"bbox": [61, 552, 538, 569], "content": "is contained in as well,4 but needs no longer fulfill the two conditions above.", "parent_index": 11, "line_index": 4}, {"bbox": [61, 567, 538, 583], "content": "Now it is quite obvious to define as the set of all connections fulfilling these conditions", "parent_index": 11, "line_index": 5}, {"bbox": [62, 582, 425, 597], "content": "multiplied with . And indeed, the well-definedness remains valid.", "parent_index": 11, "line_index": 6}]
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[]
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[{"bbox": [367, 55, 384, 64], "content": "L i e", "parent_index": 1, "subtype": "inline"}, {"bbox": [419, 55, 428, 64], "content": "G", "parent_index": 1, "subtype": "inline"}, {"bbox": [63, 70, 74, 79], "content": "X", "parent_index": 1, "subtype": "inline"}, {"bbox": [343, 70, 378, 79], "content": "x\\in X", "parent_index": 1, "subtype": "inline"}, {"bbox": [410, 70, 448, 81], "content": "H\\subseteq G", "parent_index": 1, "subtype": "inline"}, {"bbox": [76, 88, 83, 93], "content": "x", "parent_index": 1, "subtype": "inline"}, {"bbox": [113, 84, 130, 96], "content": "[H]", "parent_index": 1, "subtype": "inline"}, {"bbox": [254, 84, 264, 93], "content": "G", "parent_index": 1, "subtype": "inline"}, {"bbox": [298, 85, 309, 93], "content": "X", "parent_index": 1, "subtype": "inline"}, {"bbox": [518, 84, 533, 93], "content": "\\mathbb{R}^{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [228, 99, 237, 108], "content": "G", "parent_index": 1, "subtype": "inline"}, {"bbox": [257, 99, 271, 108], "content": "\\mathbb{R}^{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [452, 99, 463, 108], "content": "H", "parent_index": 1, "subtype": "inline"}, {"bbox": [80, 113, 91, 122], "content": "X", "parent_index": 1, "subtype": "inline"}, {"bbox": [137, 113, 151, 122], "content": "\\mathbb{R}^{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [336, 113, 351, 122], "content": "\\mathbb{R}^{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [391, 142, 462, 153], "content": "\\psi:X\\longrightarrow\\mathbb{R}^{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [281, 156, 289, 168], "content": "\\psi", "parent_index": 1, "subtype": "inline"}, {"bbox": [479, 156, 493, 165], "content": "\\mathbb{R}^{n}", "parent_index": 1, "subtype": "inline"}, {"bbox": [511, 156, 522, 165], "content": "X", "parent_index": 1, "subtype": "inline"}, {"bbox": [309, 169, 317, 181], "content": "\\mathcal{G}", "parent_index": 2, "subtype": "inline"}, {"bbox": [206, 184, 234, 198], "content": "\\mathbf{B}(\\overline{{A}})", "parent_index": 2, "subtype": "inline"}, {"bbox": [500, 185, 536, 198], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 2, "subtype": "inline"}, {"bbox": [294, 200, 305, 209], "content": "\\mathbf{G}", "parent_index": 2, "subtype": "inline"}, {"bbox": [63, 214, 98, 226], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 2, "subtype": "inline"}, {"bbox": [136, 214, 186, 226], "content": "Z(h_{\\overline{{{A}}}}(\\pmb{\\alpha}))", "parent_index": 2, "subtype": "inline"}, {"bbox": [321, 214, 363, 225], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [185, 229, 195, 238], "content": "\\mathbf{G}", "parent_index": 2, "subtype": "inline"}, {"bbox": [216, 229, 232, 238], "content": "{\\bf G}^{n}", "parent_index": 2, "subtype": "inline"}, {"bbox": [393, 232, 408, 240], "content": "\\varphi_{\\alpha}", "parent_index": 2, "subtype": "inline"}, {"bbox": [81, 243, 89, 254], "content": "\\psi", "parent_index": 2, "subtype": "inline"}, {"bbox": [265, 256, 299, 268], "content": "{\\overline{{S}}}\\subseteq{\\overline{{A}}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [231, 269, 354, 300], "content": "\\begin{array}{r}{F:\\mathrm{~}\\,{\\overline{{S}}}\\circ{\\overline{{\\mathcal{G}}}}\\enspace\\longrightarrow\\enspace{\\overline{{A}}}\\circ{\\overline{{\\mathcal{G}}}}}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace{\\overline{{A}}}\\circ{\\overline{{g}}}}\\end{array}", "parent_index": 4, "subtype": "interline"}, {"bbox": [157, 316, 167, 325], "content": "F", "parent_index": 6, "subtype": "inline"}, {"bbox": [286, 313, 433, 327], "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}^{\\prime}\\implies\\overline{{A}}\\circ\\overline{{g}}\\,=\\,\\overline{{A}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [474, 313, 512, 325], "content": "{\\overline{{A}}}^{\\prime}\\in{\\overline{{S}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [63, 329, 92, 342], "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [119, 327, 193, 342], "content": "\\mathbf{B}(\\overline{{A}}^{\\prime})\\subseteq\\mathbf{B}(\\overline{{A}})", "parent_index": 6, "subtype": "inline"}, {"bbox": [333, 334, 345, 341], "content": "\\pi_{x}", "parent_index": 6, "subtype": "inline"}, {"bbox": [63, 345, 110, 356], "content": "\\gamma_{x}\\in\\mathcal{P}_{m x}", "parent_index": 6, "subtype": "inline"}, {"bbox": [135, 348, 150, 356], "content": "\\gamma_{m}", "parent_index": 6, "subtype": "inline"}, {"bbox": [100, 356, 496, 372], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})=\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}^{\\prime}))\\subseteq\\pi_{x}(\\mathbf{B}(\\overline{{{A}}}))=h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}Z(\\mathbf{H}_{\\overline{{{A}}}})h_{\\overline{{{A}}}}(\\gamma_{x}),", "parent_index": 7, "subtype": "interline"}, {"bbox": [172, 387, 428, 405], "content": "Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})\\subseteq h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{m})h_{\\overline{{{A}}}}(\\gamma_{m})^{-1}\\,Z({\\bf H}_{\\overline{{{A}}}})\\,h_{\\overline{{{A}}}}(\\gamma_{x})h_{\\overline{{{A}}}^{\\prime}}^{-1}(\\gamma_{x})", "parent_index": 8, "subtype": "interline"}, {"bbox": [97, 409, 131, 418], "content": "x\\in M", "parent_index": 9, "subtype": "inline"}, {"bbox": [255, 408, 345, 422], "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 9, "subtype": "inline"}, {"bbox": [162, 424, 207, 434], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [238, 423, 341, 435], "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "parent_index": 10, "subtype": "inline"}, {"bbox": [384, 422, 435, 434], "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "parent_index": 10, "subtype": "inline"}, {"bbox": [118, 436, 249, 450], "content": "f:S\\circ\\mathbf{G}\\longrightarrow\\varphi_{\\alpha}(\\overline{{A}})\\circ\\mathbf{G}", "parent_index": 10, "subtype": "inline"}, {"bbox": [246, 451, 339, 464], "content": "Z(\\vec{g}^{\\prime})\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))", "parent_index": 10, "subtype": "inline"}, {"bbox": [378, 452, 409, 464], "content": "\\vec{g}^{\\prime}\\in S", "parent_index": 10, "subtype": "inline"}, {"bbox": [428, 464, 463, 476], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [240, 509, 533, 525], "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})\\subseteq Z(h_{\\overline{{A}}^{\\prime}}(\\alpha))\\equiv Z(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\subseteq Z(\\varphi_{\\alpha}(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 11, "subtype": "inline"}, {"bbox": [180, 525, 189, 536], "content": "\\overline{S}", "parent_index": 11, "subtype": "inline"}, {"bbox": [96, 540, 170, 553], "content": "F^{-1}(\\{A\\})\\supset{\\overline{{S}}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [288, 540, 337, 553], "content": "\\overline{{g}}\\in{\\bf B}(\\overline{{A}})", "parent_index": 11, "subtype": "inline"}, {"bbox": [413, 538, 425, 550], "content": "\\overline{{A}}^{\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [507, 538, 536, 552], "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [141, 555, 191, 568], "content": "F^{-1}(\\{A\\})", "parent_index": 11, "subtype": "inline"}, {"bbox": [261, 553, 290, 567], "content": "\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [241, 569, 249, 579], "content": "\\overline{S}", "parent_index": 11, "subtype": "inline"}, {"bbox": [145, 583, 173, 596], "content": "\\mathbf{B}(\\overline{{A}})", "parent_index": 11, "subtype": "inline"}]
|
[]
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# 5.2 The Proof
Proof 1. Let ${\overline{{A}}}\in{\overline{{A}}}$ . Choose for $\overline{{A}}$ an $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ with $Z(\mathbf{H}_{\overline{{A}}})\,=\,Z(h_{\overline{{A}}}(\alpha))$ according to Corollary 4.2 and denote the corresponding reduction mapping $\varphi_{\alpha}:{\overline{{\mathcal{A}}}}\longrightarrow\mathbf{G}^{\#\alpha}$ shortly by $\varphi$ .
2. Due to Proposition 4.4 there is an $S\subseteq\mathbf{G}^{\#\alpha}$ with $\varphi({\overline{{A}}})\in S$ , such that
$S\circ\mathbf{G}$ is an open neighbourhood of $\varphi(\overline{{A}})\circ\mathbf{G}$ and there exists an equivariant mapping $f$ with $\begin{array}{r l}{-}&{{}f:S\circ\mathbf{G}\longrightarrow\varphi(\overline{{A}})\circ\mathbf{G}}\end{array}$ and $f^{-1}(\{\varphi({\overline{{A}}})\})=S$ .
3. We define the mapping
$$
\psi:\ {\overline{{\mathcal{A}}}}\ \ \longrightarrow\ \ \overline{{\mathcal{G}}},}
$$
whereas for all $x\in M\setminus\{m\}$ the (arbitrary, but fixed) path $\gamma_{x}$ runs from $m$ to $x$ and $\gamma_{m}$ is the trivial path.
4. As we motivated above we set
$$
\begin{array}{r c l}{\overline{{S}}_{0}}&{:=}&{\varphi^{-1}(S)\,\cap\,\psi^{-1}(\psi(\overline{{A}})),}\\ {\overline{{S}}}&{:=}&{\big(\varphi^{-1}(S)\,\cap\,\psi^{-1}(\psi(\overline{{A}}))\big)\circ{\bf B}(\overline{{A}})}&{\equiv}&{\overline{{S}}_{0}\circ{\bf B}(\overline{{A}})}\end{array}
$$
and
$$
\begin{array}{r}{F:\;\;\overline{{S}}\circ\overline{{\mathcal{G}}}\enspace\longrightarrow\enspace\overline{{A}}\circ\overline{{\mathcal{G}}}.}\\ {\overline{{A}}^{\prime}\circ\overline{{g}}\enspace\longmapsto\enspace\overline{{A}}\circ\overline{{g}}}\end{array}
$$
5. $F$ is well-defined.
• Let $\overline{{{A}}}^{\prime}\circ\overline{{{g}}}^{\prime}\,=\,\overline{{{A}}}^{\prime\prime}\circ\overline{{{g}}}^{\prime\prime}$ with $\overline{{A}}^{\prime},\overline{{A}}^{\prime\prime}\,\in\,\overline{{S}}$ and $\overline{{g}}^{\prime},\overline{{g}}^{\prime\prime}\,\in\,\overline{{g}}$ . Then there exist $\overline{{z}}^{\prime},\overline{{z}}^{\prime\prime}\in\mathbf{B}(\overline{{A}})$ with $\bar{\vec{A}}^{\prime}=\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}$ and $\overline{{A}}^{\prime\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\overline{{z}}^{\prime\prime}$ as well as $\overline{{A}}_{0}^{\prime},\overline{{A}}_{0}^{\prime\prime}\in\overline{{S}}_{0}$ . Due to ${\overline{{S}}}_{0}\,\subseteq\,\psi^{-1}(\psi({\overline{{A}}}))$ we have $\psi(\overline{{{A}}}_{0}^{\prime})\,=\,\bar{\psi}(\overline{{{A}}})\,=\,\psi(\overline{{{A}}}_{0}^{\prime\prime})$ , i.e. $h_{\overline{{{A}}}_{0}^{\prime}}(\gamma_{x})\;=\;$ $h_{\overline{{{A}}}}(\gamma_{x})=h_{\overline{{{A}}}_{0}^{\prime\prime}}(\gamma_{x})$ for all $x$ .
Furthermore, we have
$$
\begin{array}{r l r}{f(\varphi(\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime}))}&{=}&{f(\varphi(\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}\circ\overline{{g}}^{\prime}))}\\ &{=}&{f(\varphi(\overline{{A}}_{0}^{\prime})\circ z_{m}^{\prime}\circ g_{m}^{\prime})\ \ \ \ (\varphi\ ^{,}\mathrm{equivariant}^{,*})}\\ &{=}&{f(\varphi(\overline{{A}}_{0}^{\prime}))\circ z_{m}^{\prime}\circ g_{m}^{\prime}\ \ \ \ \ (f\ \mathrm{equivariant})}\\ &{=}&{\varphi(\overline{{A}})\circ z_{m}^{\prime}\circ g_{m}^{\prime}\ \ \ \ \ \ \ \ \ (\varphi(\overline{{A}}_{0}^{\prime})\in S)}\\ &{=}&{\varphi(\overline{{A}}\circ\overline{{z}}^{\prime})\circ g_{m}^{\prime}\ \ \ \ \ \ \ \ \ \ \ \ (\varphi\ ^{,}\mathrm{equivariant}^{,*})}\\ &{=}&{\varphi(\overline{{A}})\circ g_{m}^{\prime}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\overline{{z}}^{\prime}\in{\bf B}(\overline{{A}}))}\end{array}
$$
and analogously $f(\varphi(\overline{{A}}^{\prime\prime}\circ\overline{{g}}^{\prime\prime}))=\varphi(\overline{{A}})\circ g_{m}^{\prime\prime}$ .
Therefore, we have $\varphi(\overline{{A}})\circ g_{m}^{\prime}=\varphi(\overline{{A}})\circ g_{m}^{\prime\prime}$ , i.e. $g_{m}^{\prime\prime}\left(g_{m}^{\prime}\right)^{-1}$ is an element of the stabilizer of $\varphi(\overline{{A}})$ , thus $g_{m}^{\prime\prime}\left(g_{m}^{\prime}\right)^{-1}\in Z(\varphi(\overline{{A}}))=Z(\mathbf{H}_{\overline{{A}}})$ .
• Since $\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}\circ\overline{{g}}^{\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\overline{{z}}^{\prime\prime}\circ\overline{{g}}^{\prime\prime}$ , we have $\overline{{A}}_{0}^{\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\left(\overline{{z}}^{\prime\prime}\,\overline{{g}}^{\prime\prime}\,(\overline{{g}}^{\prime})^{-1}\,(\overline{{z}}^{\prime})^{-1}\right)$ , and so for all $x\in M$
Moreover, since $\left(\overline{{{g}}}^{\prime\prime}\ (\overline{{{g}}}^{\prime})^{-1}\right)_{m}\ \in\ Z(\mathbf{H}_{\overline{{{A}}}})$ , we have $\left(\overline{{{z}}}^{\prime\prime}\ \overline{{{g}}}^{\prime\prime}\ (\overline{{{g}}}^{\prime})^{-1}\ (\overline{{{z}}}^{\prime})^{-1}\right)_{m}\ \in$ $Z(\mathbf{H}_{\overline{{A}}})$ . From $h_{\overline{{{A}}}_{0}^{\prime}}(\gamma_{x})=h_{\overline{{{A}}}}(\gamma_{x})=h_{\overline{{{A}}}_{0}^{\prime\prime}}(\gamma_{x})$ for all $x$ now $\overline{{z}}^{\prime\prime}\,\overline{{g}}^{\prime\prime}\,(\overline{{g}}^{\prime})^{-1}\,(\overline{{z}}^{\prime})^{\overset{...}{-1}}\in$ $\mathbf{B}(\overline{{A}})$ follows, and thus $\overline{{g}}^{\prime\prime}\left(\overline{{g}}^{\prime}\right)^{-1}\in\mathbf{B}(\overline{{A}})$ .
By this we have $\overline{{A}}\circ\overline{{g}}^{\prime}=\overline{{A}}\circ\overline{{g}}^{\prime\prime}$ , i.e. $F^{'}$ is well-defined.
|
<html><body>
<h1 data-bbox="61 12 176 29">5.2 The Proof </h1>
<p data-bbox="62 35 539 80">Proof 1. Let ${\overline{{A}}}\in{\overline{{A}}}$ . Choose for $\overline{{A}}$ an $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ with $Z(\mathbf{H}_{\overline{{A}}})\,=\,Z(h_{\overline{{A}}}(\alpha))$ according to Corollary 4.2 and denote the corresponding reduction mapping $\varphi_{\alpha}:{\overline{{\mathcal{A}}}}\longrightarrow\mathbf{G}^{\#\alpha}$ shortly by $\varphi$ . </p>
<p data-bbox="112 80 495 95">2. Due to Proposition 4.4 there is an $S\subseteq\mathbf{G}^{\#\alpha}$ with $\varphi({\overline{{A}}})\in S$ , such that </p>
<p data-bbox="129 95 401 153">$S\circ\mathbf{G}$ is an open neighbourhood of $\varphi(\overline{{A}})\circ\mathbf{G}$ and there exists an equivariant mapping $f$ with $\begin{array}{r l}{-}&{{}f:S\circ\mathbf{G}\longrightarrow\varphi(\overline{{A}})\circ\mathbf{G}}\end{array}$ and $f^{-1}(\{\varphi({\overline{{A}}})\})=S$ . </p>
<p data-bbox="111 154 254 166">3. We define the mapping </p>
<div class="equation" data-bbox="254 168 400 204">$$
\psi:\ {\overline{{\mathcal{A}}}}\ \ \longrightarrow\ \ \overline{{\mathcal{G}}},}
$$</div>
<p data-bbox="131 201 537 229">whereas for all $x\in M\setminus\{m\}$ the (arbitrary, but fixed) path $\gamma_{x}$ runs from $m$ to $x$ and $\gamma_{m}$ is the trivial path. </p>
<p data-bbox="111 230 289 243">4. As we motivated above we set </p>
<div class="equation" data-bbox="190 248 478 291">$$
\begin{array}{r c l}{\overline{{S}}_{0}}&{:=}&{\varphi^{-1}(S)\,\cap\,\psi^{-1}(\psi(\overline{{A}})),}\\ {\overline{{S}}}&{:=}&{\big(\varphi^{-1}(S)\,\cap\,\psi^{-1}(\psi(\overline{{A}}))\big)\circ{\bf B}(\overline{{A}})}&{\equiv}&{\overline{{S}}_{0}\circ{\bf B}(\overline{{A}})}\end{array}
$$</div>
<p data-bbox="131 291 153 304">and </p>
<div class="equation" data-bbox="262 304 390 339">$$
\begin{array}{r}{F:\;\;\overline{{S}}\circ\overline{{\mathcal{G}}}\enspace\longrightarrow\enspace\overline{{A}}\circ\overline{{\mathcal{G}}}.}\\ {\overline{{A}}^{\prime}\circ\overline{{g}}\enspace\longmapsto\enspace\overline{{A}}\circ\overline{{g}}}\end{array}
$$</div>
<p data-bbox="110 335 222 348">5. $F$ is well-defined. </p>
<p data-bbox="128 347 538 415">• Let $\overline{{{A}}}^{\prime}\circ\overline{{{g}}}^{\prime}\,=\,\overline{{{A}}}^{\prime\prime}\circ\overline{{{g}}}^{\prime\prime}$ with $\overline{{A}}^{\prime},\overline{{A}}^{\prime\prime}\,\in\,\overline{{S}}$ and $\overline{{g}}^{\prime},\overline{{g}}^{\prime\prime}\,\in\,\overline{{g}}$ . Then there exist $\overline{{z}}^{\prime},\overline{{z}}^{\prime\prime}\in\mathbf{B}(\overline{{A}})$ with $\bar{\vec{A}}^{\prime}=\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}$ and $\overline{{A}}^{\prime\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\overline{{z}}^{\prime\prime}$ as well as $\overline{{A}}_{0}^{\prime},\overline{{A}}_{0}^{\prime\prime}\in\overline{{S}}_{0}$ . Due to ${\overline{{S}}}_{0}\,\subseteq\,\psi^{-1}(\psi({\overline{{A}}}))$ we have $\psi(\overline{{{A}}}_{0}^{\prime})\,=\,\bar{\psi}(\overline{{{A}}})\,=\,\psi(\overline{{{A}}}_{0}^{\prime\prime})$ , i.e. $h_{\overline{{{A}}}_{0}^{\prime}}(\gamma_{x})\;=\;$ $h_{\overline{{{A}}}}(\gamma_{x})=h_{\overline{{{A}}}_{0}^{\prime\prime}}(\gamma_{x})$ for all $x$ . </p>
<p data-bbox="131 411 263 423">Furthermore, we have </p>
<div class="equation" data-bbox="195 427 491 522">$$
\begin{array}{r l r}{f(\varphi(\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime}))}&{=}&{f(\varphi(\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}\circ\overline{{g}}^{\prime}))}\\ &{=}&{f(\varphi(\overline{{A}}_{0}^{\prime})\circ z_{m}^{\prime}\circ g_{m}^{\prime})\ \ \ \ (\varphi\ ^{,}\mathrm{equivariant}^{,*})}\\ &{=}&{f(\varphi(\overline{{A}}_{0}^{\prime}))\circ z_{m}^{\prime}\circ g_{m}^{\prime}\ \ \ \ \ (f\ \mathrm{equivariant})}\\ &{=}&{\varphi(\overline{{A}})\circ z_{m}^{\prime}\circ g_{m}^{\prime}\ \ \ \ \ \ \ \ \ (\varphi(\overline{{A}}_{0}^{\prime})\in S)}\\ &{=}&{\varphi(\overline{{A}}\circ\overline{{z}}^{\prime})\circ g_{m}^{\prime}\ \ \ \ \ \ \ \ \ \ \ \ (\varphi\ ^{,}\mathrm{equivariant}^{,*})}\\ &{=}&{\varphi(\overline{{A}})\circ g_{m}^{\prime}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\overline{{z}}^{\prime}\in{\bf B}(\overline{{A}}))}\end{array}
$$</div>
<p data-bbox="148 525 374 542">and analogously $f(\varphi(\overline{{A}}^{\prime\prime}\circ\overline{{g}}^{\prime\prime}))=\varphi(\overline{{A}})\circ g_{m}^{\prime\prime}$ . </p>
<p data-bbox="147 542 538 571">Therefore, we have $\varphi(\overline{{A}})\circ g_{m}^{\prime}=\varphi(\overline{{A}})\circ g_{m}^{\prime\prime}$ , i.e. $g_{m}^{\prime\prime}\left(g_{m}^{\prime}\right)^{-1}$ is an element of the stabilizer of $\varphi(\overline{{A}})$ , thus $g_{m}^{\prime\prime}\left(g_{m}^{\prime}\right)^{-1}\in Z(\varphi(\overline{{A}}))=Z(\mathbf{H}_{\overline{{A}}})$ . </p>
<p data-bbox="132 572 538 599">• Since $\overline{{A}}_{0}^{\prime}\circ\overline{{z}}^{\prime}\circ\overline{{g}}^{\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\overline{{z}}^{\prime\prime}\circ\overline{{g}}^{\prime\prime}$ , we have $\overline{{A}}_{0}^{\prime}=\overline{{A}}_{0}^{\prime\prime}\circ\left(\overline{{z}}^{\prime\prime}\,\overline{{g}}^{\prime\prime}\,(\overline{{g}}^{\prime})^{-1}\,(\overline{{z}}^{\prime})^{-1}\right)$ , and so for all $x\in M$ </p>
<p data-bbox="147 619 537 669">Moreover, since $\left(\overline{{{g}}}^{\prime\prime}\ (\overline{{{g}}}^{\prime})^{-1}\right)_{m}\ \in\ Z(\mathbf{H}_{\overline{{{A}}}})$ , we have $\left(\overline{{{z}}}^{\prime\prime}\ \overline{{{g}}}^{\prime\prime}\ (\overline{{{g}}}^{\prime})^{-1}\ (\overline{{{z}}}^{\prime})^{-1}\right)_{m}\ \in$ $Z(\mathbf{H}_{\overline{{A}}})$ . From $h_{\overline{{{A}}}_{0}^{\prime}}(\gamma_{x})=h_{\overline{{{A}}}}(\gamma_{x})=h_{\overline{{{A}}}_{0}^{\prime\prime}}(\gamma_{x})$ for all $x$ now $\overline{{z}}^{\prime\prime}\,\overline{{g}}^{\prime\prime}\,(\overline{{g}}^{\prime})^{-1}\,(\overline{{z}}^{\prime})^{\overset{...}{-1}}\in$ $\mathbf{B}(\overline{{A}})$ follows, and thus $\overline{{g}}^{\prime\prime}\left(\overline{{g}}^{\prime}\right)^{-1}\in\mathbf{B}(\overline{{A}})$ . </p>
<p data-bbox="132 669 427 684">By this we have $\overline{{A}}\circ\overline{{g}}^{\prime}=\overline{{A}}\circ\overline{{g}}^{\prime\prime}$ , i.e. $F^{'}$ is well-defined. </p>
</body></html>
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0001008v1
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[{"type": "text", "text": "5.2 The Proof ", "text_level": 1, "page_idx": 7}, {"type": "text", "text": "Proof 1. Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ . Choose for $\\overline{{A}}$ an $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ with $Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))$ according to Corollary 4.2 and denote the corresponding reduction mapping $\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}$ shortly by $\\varphi$ . ", "page_idx": 7}, {"type": "text", "text": "2. Due to Proposition 4.4 there is an $S\\subseteq\\mathbf{G}^{\\#\\alpha}$ with $\\varphi({\\overline{{A}}})\\in S$ , such that ", "page_idx": 7}, {"type": "text", "text": "$S\\circ\\mathbf{G}$ is an open neighbourhood of $\\varphi(\\overline{{A}})\\circ\\mathbf{G}$ and there exists an equivariant mapping $f$ with $\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}$ and $f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S$ . ", "page_idx": 7}, {"type": "text", "text": "3. We define the mapping ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "whereas for all $x\\in M\\setminus\\{m\\}$ the (arbitrary, but fixed) path $\\gamma_{x}$ runs from $m$ to $x$ and $\\gamma_{m}$ is the trivial path. ", "page_idx": 7}, {"type": "text", "text": "4. As we motivated above we set ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "5. $F$ is well-defined. ", "page_idx": 7}, {"type": "text", "text": "• Let $\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}$ with $\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}$ and $\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}$ . Then there exist $\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})$ with $\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}$ and $\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}$ as well as $\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}$ . Due to ${\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))$ we have $\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})$ , i.e. $h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;$ $h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})$ for all $x$ . ", "page_idx": 7}, {"type": "text", "text": "Furthermore, we have ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "and analogously $f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}$ . ", "page_idx": 7}, {"type": "text", "text": "Therefore, we have $\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}$ , i.e. $g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}$ is an element of the stabilizer of $\\varphi(\\overline{{A}})$ , thus $g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})$ . ", "page_idx": 7}, {"type": "text", "text": "• Since $\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}$ , we have $\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)$ , and so for all $x\\in M$ ", "page_idx": 7}, {"type": "text", "text": "Moreover, since $\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})$ , we have $\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in$ $Z(\\mathbf{H}_{\\overline{{A}}})$ . From $h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})$ for all $x$ now $\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in$ $\\mathbf{B}(\\overline{{A}})$ follows, and thus $\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})$ . ", "page_idx": 7}, {"type": "text", "text": "By this we have $\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}$ , i.e. $F^{'}$ is well-defined. ", "page_idx": 7}]
|
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Due to Proposition 4.4 there is an ", "type": "text"}, {"bbox": [311, 83, 360, 95], "score": 0.94, "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [360, 83, 389, 96], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [389, 82, 438, 96], "score": 0.94, "content": "\\varphi({\\overline{{A}}})\\in S", "type": "inline_equation", "height": 14, "width": 49}, {"bbox": [438, 83, 494, 96], "score": 1.0, "content": ", such that", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [129, 95, 401, 153], "lines": [{"bbox": [149, 97, 402, 110], "spans": [{"bbox": [149, 99, 178, 108], "score": 0.9, "content": "S\\circ\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [179, 97, 331, 110], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [332, 97, 380, 110], "score": 0.94, "content": "\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [380, 97, 402, 110], "score": 1.0, "content": " and", "type": "text"}], "index": 5}, {"bbox": [146, 111, 371, 126], "spans": [{"bbox": [146, 111, 336, 126], "score": 1.0, "content": "there exists an equivariant mapping ", "type": "text"}, {"bbox": [336, 113, 343, 124], "score": 0.9, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [344, 111, 371, 126], "score": 1.0, "content": " with", "type": "text"}], "index": 6}, {"bbox": [151, 125, 314, 140], "spans": [{"bbox": [151, 126, 289, 140], "score": 0.88, "content": "\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}", "type": "inline_equation", "height": 14, "width": 138}, {"bbox": [289, 125, 314, 139], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [168, 140, 262, 154], "spans": [{"bbox": [168, 141, 258, 154], "score": 0.89, "content": "f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [258, 140, 262, 154], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5}, {"type": "text", "bbox": [111, 154, 254, 166], "lines": [{"bbox": [111, 153, 252, 168], "spans": [{"bbox": [111, 153, 252, 168], "score": 1.0, "content": "3. We define the mapping", "type": "text"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [254, 168, 400, 204], "lines": [{"bbox": [254, 168, 400, 204], "spans": [{"bbox": [254, 168, 400, 204], "score": 0.84, "content": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [131, 201, 537, 229], "lines": [{"bbox": [131, 202, 538, 218], "spans": [{"bbox": [131, 202, 210, 218], "score": 1.0, "content": "whereas for all ", "type": "text"}, {"bbox": [211, 204, 279, 216], "score": 0.93, "content": "x\\in M\\setminus\\{m\\}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [279, 202, 441, 218], "score": 1.0, "content": " the (arbitrary, but fixed) path ", "type": "text"}, {"bbox": [442, 208, 453, 216], "score": 0.91, "content": "\\gamma_{x}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [453, 202, 510, 218], "score": 1.0, "content": " runs from ", "type": "text"}, {"bbox": [511, 208, 522, 213], "score": 0.89, "content": "m", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [522, 202, 538, 218], "score": 1.0, "content": " to", "type": "text"}], "index": 11}, {"bbox": [132, 218, 276, 231], "spans": [{"bbox": [132, 222, 138, 228], "score": 0.87, "content": "x", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [139, 218, 165, 231], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 222, 180, 230], "score": 0.93, "content": "\\gamma_{m}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [180, 218, 276, 231], "score": 1.0, "content": " is the trivial path.", "type": "text"}], "index": 12}], "index": 11.5}, {"type": "text", "bbox": [111, 230, 289, 243], "lines": [{"bbox": [111, 231, 288, 244], "spans": [{"bbox": [111, 231, 288, 244], "score": 1.0, "content": "4. As we motivated above we set", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [190, 248, 478, 291], "lines": [{"bbox": [190, 248, 478, 291], "spans": [{"bbox": [190, 248, 478, 291], "score": 0.79, "content": "\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [131, 291, 153, 304], "lines": [{"bbox": [130, 293, 153, 305], "spans": [{"bbox": [130, 293, 153, 305], "score": 1.0, "content": "and", "type": "text"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [262, 304, 390, 339], "lines": [{"bbox": [262, 304, 390, 339], "spans": [{"bbox": [262, 304, 390, 339], "score": 0.87, "content": "\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [110, 335, 222, 348], "lines": [{"bbox": [111, 337, 220, 349], "spans": [{"bbox": [111, 337, 131, 349], "score": 1.0, "content": "5.", "type": "text"}, {"bbox": [132, 339, 141, 348], "score": 0.89, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 337, 220, 349], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [128, 347, 538, 415], "lines": [{"bbox": [132, 348, 538, 365], "spans": [{"bbox": [132, 348, 170, 365], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [171, 349, 264, 364], "score": 0.9, "content": "\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}", "type": "inline_equation", "height": 15, "width": 93}, {"bbox": [264, 348, 296, 365], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [296, 349, 355, 364], "score": 0.89, "content": "\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [355, 348, 383, 365], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [383, 349, 437, 364], "score": 0.9, "content": "\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}", "type": "inline_equation", "height": 15, "width": 54}, {"bbox": [437, 348, 538, 365], "score": 1.0, "content": ". Then there exist", "type": "text"}], "index": 18}, {"bbox": [148, 363, 521, 380], "spans": [{"bbox": [148, 365, 216, 379], "score": 0.88, "content": "\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [216, 363, 246, 380], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [246, 363, 308, 379], "score": 0.91, "content": "\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}", "type": "inline_equation", "height": 16, "width": 62}, {"bbox": [308, 363, 333, 380], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 364, 401, 379], "score": 0.91, "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [401, 363, 456, 380], "score": 1.0, "content": " as well as ", "type": "text"}, {"bbox": [456, 363, 516, 379], "score": 0.88, "content": "\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 16, "width": 60}, {"bbox": [517, 363, 521, 380], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [137, 376, 538, 397], "spans": [{"bbox": [137, 376, 189, 397], "score": 1.0, "content": "Due to ", "type": "text"}, {"bbox": [189, 379, 276, 394], "score": 0.91, "content": "{\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))", "type": "inline_equation", "height": 15, "width": 87}, {"bbox": [276, 376, 326, 397], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [327, 378, 453, 394], "score": 0.9, "content": "\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})", "type": "inline_equation", "height": 16, "width": 126}, {"bbox": [453, 376, 482, 397], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [483, 380, 538, 397], "score": 0.9, "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;", "type": "inline_equation", "height": 17, "width": 55}], "index": 20}, {"bbox": [148, 396, 287, 414], "spans": [{"bbox": [148, 396, 238, 413], "score": 0.91, "content": "h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "type": "inline_equation", "height": 17, "width": 90}, {"bbox": [238, 396, 275, 414], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [275, 401, 282, 407], "score": 0.83, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [282, 396, 287, 414], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19.5}, {"type": "text", "bbox": [131, 411, 263, 423], "lines": [{"bbox": [148, 412, 261, 424], "spans": [{"bbox": [148, 412, 261, 424], "score": 1.0, "content": "Furthermore, we have", "type": "text"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [195, 427, 491, 522], "lines": [{"bbox": [195, 427, 491, 522], "spans": [{"bbox": [195, 427, 491, 522], "score": 0.94, "content": "\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [148, 525, 374, 542], "lines": [{"bbox": [147, 527, 373, 544], "spans": [{"bbox": [147, 527, 235, 544], "score": 1.0, "content": "and analogously ", "type": "text"}, {"bbox": [235, 528, 371, 542], "score": 0.92, "content": "f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 136}, {"bbox": [371, 527, 373, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [147, 542, 538, 571], "lines": [{"bbox": [148, 542, 539, 559], "spans": [{"bbox": [148, 542, 250, 559], "score": 1.0, "content": "Therefore, we have ", "type": "text"}, {"bbox": [250, 543, 370, 557], "score": 0.9, "content": "\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 120}, {"bbox": [371, 542, 398, 559], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [398, 542, 450, 557], "score": 0.91, "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}", "type": "inline_equation", "height": 15, "width": 52}, {"bbox": [450, 542, 539, 559], "score": 1.0, "content": " is an element of", "type": "text"}], "index": 25}, {"bbox": [147, 556, 455, 573], "spans": [{"bbox": [147, 556, 232, 573], "score": 1.0, "content": "the stabilizer of ", "type": "text"}, {"bbox": [232, 558, 258, 571], "score": 0.93, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [258, 556, 290, 573], "score": 1.0, "content": ", thus ", "type": "text"}, {"bbox": [291, 557, 451, 571], "score": 0.91, "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 160}, {"bbox": [451, 556, 455, 573], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [132, 572, 538, 599], "lines": [{"bbox": [133, 570, 536, 591], "spans": [{"bbox": [133, 570, 179, 591], "score": 1.0, "content": "• Since ", "type": "text"}, {"bbox": [179, 573, 306, 587], "score": 0.94, "content": "\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 127}, {"bbox": [306, 570, 356, 591], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [356, 571, 510, 591], "score": 0.93, "content": "\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)", "type": "inline_equation", "height": 20, "width": 154}, {"bbox": [511, 570, 536, 591], "score": 1.0, "content": ", and", "type": "text"}], "index": 27}, {"bbox": [148, 588, 231, 603], "spans": [{"bbox": [148, 588, 197, 603], "score": 1.0, "content": "so for all ", "type": "text"}, {"bbox": [197, 590, 231, 599], "score": 0.89, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [147, 619, 537, 669], "lines": [{"bbox": [147, 620, 537, 639], "spans": [{"bbox": [147, 620, 235, 638], "score": 1.0, "content": "Moreover, since", "type": "text"}, {"bbox": [235, 621, 355, 639], "score": 0.94, "content": "\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})", "type": "inline_equation", "height": 18, "width": 120}, {"bbox": [356, 620, 411, 638], "score": 1.0, "content": ", we have", "type": "text"}, {"bbox": [411, 620, 537, 638], "score": 0.79, "content": "\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in", "type": "inline_equation", "height": 18, "width": 126}], "index": 29}, {"bbox": [149, 637, 537, 656], "spans": [{"bbox": [149, 640, 184, 653], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [185, 638, 222, 655], "score": 1.0, "content": ". From ", "type": "text"}, {"bbox": [223, 640, 366, 656], "score": 0.91, "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "type": "inline_equation", "height": 16, "width": 143}, {"bbox": [366, 638, 402, 655], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [402, 642, 410, 650], "score": 0.3, "content": "x", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [410, 638, 437, 655], "score": 1.0, "content": " now ", "type": "text"}, {"bbox": [437, 637, 537, 653], "score": 0.72, "content": "\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in", "type": "inline_equation", "height": 16, "width": 100}], "index": 30}, {"bbox": [149, 656, 358, 670], "spans": [{"bbox": [149, 657, 176, 670], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [177, 656, 269, 670], "score": 1.0, "content": " follows, and thus", "type": "text"}, {"bbox": [270, 656, 355, 670], "score": 0.89, "content": "\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [356, 656, 358, 670], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30}, {"type": "text", "bbox": [132, 669, 427, 684], "lines": [{"bbox": [147, 671, 425, 684], "spans": [{"bbox": [147, 671, 234, 684], "score": 1.0, "content": "By this we have ", "type": "text"}, {"bbox": [234, 671, 310, 684], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [311, 671, 337, 684], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [337, 673, 347, 682], "score": 0.89, "content": "F^{'}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 671, 425, 684], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 32}], "index": 32}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [254, 168, 400, 204], "lines": [{"bbox": [254, 168, 400, 204], "spans": [{"bbox": [254, 168, 400, 204], "score": 0.84, "content": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "interline_equation", "bbox": [190, 248, 478, 291], "lines": [{"bbox": [190, 248, 478, 291], "spans": [{"bbox": [190, 248, 478, 291], "score": 0.79, "content": "\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [262, 304, 390, 339], "lines": [{"bbox": [262, 304, 390, 339], "spans": [{"bbox": [262, 304, 390, 339], "score": 0.87, "content": "\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [195, 427, 491, 522], "lines": [{"bbox": [195, 427, 491, 522], "spans": [{"bbox": [195, 427, 491, 522], "score": 0.94, "content": "\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}], "discarded_blocks": [{"type": "discarded", "bbox": [295, 705, 303, 715], "lines": [{"bbox": [296, 705, 304, 717], "spans": [{"bbox": [296, 705, 304, 717], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [61, 12, 176, 29], "lines": [{"bbox": [62, 15, 174, 29], "spans": [{"bbox": [62, 16, 86, 29], "score": 1.0, "content": "5.2", "type": "text"}, {"bbox": [97, 15, 174, 29], "score": 1.0, "content": "The Proof", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [62, 35, 539, 80], "lines": [{"bbox": [61, 36, 538, 56], "spans": [{"bbox": [61, 36, 153, 56], "score": 1.0, "content": "Proof 1. Let ", "type": "text"}, {"bbox": [153, 39, 189, 50], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [189, 36, 258, 56], "score": 1.0, "content": ". Choose for ", "type": "text"}, {"bbox": [258, 39, 267, 50], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [268, 36, 288, 56], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [288, 41, 333, 51], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [334, 36, 365, 56], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [365, 40, 468, 53], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 13, "width": 103}, {"bbox": [468, 36, 538, 56], "score": 1.0, "content": " according to", "type": "text"}], "index": 1}, {"bbox": [129, 51, 536, 70], "spans": [{"bbox": [129, 51, 451, 70], "score": 1.0, "content": "Corollary 4.2 and denote the corresponding reduction mapping ", "type": "text"}, {"bbox": [451, 54, 536, 66], "score": 0.93, "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 85}], "index": 2}, {"bbox": [131, 66, 201, 84], "spans": [{"bbox": [131, 66, 187, 84], "score": 1.0, "content": "shortly by ", "type": "text"}, {"bbox": [188, 73, 195, 81], "score": 0.9, "content": "\\varphi", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [196, 66, 201, 84], "score": 1.0, "content": ".", "type": "text"}], "index": 3}], "index": 2, "bbox_fs": [61, 36, 538, 84]}, {"type": "text", "bbox": [112, 80, 495, 95], "lines": [{"bbox": [111, 82, 494, 96], "spans": [{"bbox": [111, 83, 310, 96], "score": 1.0, "content": "2. Due to Proposition 4.4 there is an ", "type": "text"}, {"bbox": [311, 83, 360, 95], "score": 0.94, "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 49}, {"bbox": [360, 83, 389, 96], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [389, 82, 438, 96], "score": 0.94, "content": "\\varphi({\\overline{{A}}})\\in S", "type": "inline_equation", "height": 14, "width": 49}, {"bbox": [438, 83, 494, 96], "score": 1.0, "content": ", such that", "type": "text"}], "index": 4}], "index": 4, "bbox_fs": [111, 82, 494, 96]}, {"type": "text", "bbox": [129, 95, 401, 153], "lines": [{"bbox": [149, 97, 402, 110], "spans": [{"bbox": [149, 99, 178, 108], "score": 0.9, "content": "S\\circ\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [179, 97, 331, 110], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [332, 97, 380, 110], "score": 0.94, "content": "\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [380, 97, 402, 110], "score": 1.0, "content": " and", "type": "text"}], "index": 5}, {"bbox": [146, 111, 371, 126], "spans": [{"bbox": [146, 111, 336, 126], "score": 1.0, "content": "there exists an equivariant mapping ", "type": "text"}, {"bbox": [336, 113, 343, 124], "score": 0.9, "content": "f", "type": "inline_equation", "height": 11, "width": 7}, {"bbox": [344, 111, 371, 126], "score": 1.0, "content": " with", "type": "text"}], "index": 6}, {"bbox": [151, 125, 314, 140], "spans": [{"bbox": [151, 126, 289, 140], "score": 0.88, "content": "\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}", "type": "inline_equation", "height": 14, "width": 138}, {"bbox": [289, 125, 314, 139], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [168, 140, 262, 154], "spans": [{"bbox": [168, 141, 258, 154], "score": 0.89, "content": "f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [258, 140, 262, 154], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 6.5, "bbox_fs": [146, 97, 402, 154]}, {"type": "text", "bbox": [111, 154, 254, 166], "lines": [{"bbox": [111, 153, 252, 168], "spans": [{"bbox": [111, 153, 252, 168], "score": 1.0, "content": "3. We define the mapping", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [111, 153, 252, 168]}, {"type": "interline_equation", "bbox": [254, 168, 400, 204], "lines": [{"bbox": [254, 168, 400, 204], "spans": [{"bbox": [254, 168, 400, 204], "score": 0.84, "content": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [131, 201, 537, 229], "lines": [{"bbox": [131, 202, 538, 218], "spans": [{"bbox": [131, 202, 210, 218], "score": 1.0, "content": "whereas for all ", "type": "text"}, {"bbox": [211, 204, 279, 216], "score": 0.93, "content": "x\\in M\\setminus\\{m\\}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [279, 202, 441, 218], "score": 1.0, "content": " the (arbitrary, but fixed) path ", "type": "text"}, {"bbox": [442, 208, 453, 216], "score": 0.91, "content": "\\gamma_{x}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [453, 202, 510, 218], "score": 1.0, "content": " runs from ", "type": "text"}, {"bbox": [511, 208, 522, 213], "score": 0.89, "content": "m", "type": "inline_equation", "height": 5, "width": 11}, {"bbox": [522, 202, 538, 218], "score": 1.0, "content": " to", "type": "text"}], "index": 11}, {"bbox": [132, 218, 276, 231], "spans": [{"bbox": [132, 222, 138, 228], "score": 0.87, "content": "x", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [139, 218, 165, 231], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [165, 222, 180, 230], "score": 0.93, "content": "\\gamma_{m}", "type": "inline_equation", "height": 8, "width": 15}, {"bbox": [180, 218, 276, 231], "score": 1.0, "content": " is the trivial path.", "type": "text"}], "index": 12}], "index": 11.5, "bbox_fs": [131, 202, 538, 231]}, {"type": "text", "bbox": [111, 230, 289, 243], "lines": [{"bbox": [111, 231, 288, 244], "spans": [{"bbox": [111, 231, 288, 244], "score": 1.0, "content": "4. As we motivated above we set", "type": "text"}], "index": 13}], "index": 13, "bbox_fs": [111, 231, 288, 244]}, {"type": "interline_equation", "bbox": [190, 248, 478, 291], "lines": [{"bbox": [190, 248, 478, 291], "spans": [{"bbox": [190, 248, 478, 291], "score": 0.79, "content": "\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [131, 291, 153, 304], "lines": [{"bbox": [130, 293, 153, 305], "spans": [{"bbox": [130, 293, 153, 305], "score": 1.0, "content": "and", "type": "text"}], "index": 15}], "index": 15, "bbox_fs": [130, 293, 153, 305]}, {"type": "interline_equation", "bbox": [262, 304, 390, 339], "lines": [{"bbox": [262, 304, 390, 339], "spans": [{"bbox": [262, 304, 390, 339], "score": 0.87, "content": "\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [110, 335, 222, 348], "lines": [{"bbox": [111, 337, 220, 349], "spans": [{"bbox": [111, 337, 131, 349], "score": 1.0, "content": "5.", "type": "text"}, {"bbox": [132, 339, 141, 348], "score": 0.89, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 337, 220, 349], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 17}], "index": 17, "bbox_fs": [111, 337, 220, 349]}, {"type": "text", "bbox": [128, 347, 538, 415], "lines": [{"bbox": [132, 348, 538, 365], "spans": [{"bbox": [132, 348, 170, 365], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [171, 349, 264, 364], "score": 0.9, "content": "\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}", "type": "inline_equation", "height": 15, "width": 93}, {"bbox": [264, 348, 296, 365], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [296, 349, 355, 364], "score": 0.89, "content": "\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [355, 348, 383, 365], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [383, 349, 437, 364], "score": 0.9, "content": "\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}", "type": "inline_equation", "height": 15, "width": 54}, {"bbox": [437, 348, 538, 365], "score": 1.0, "content": ". Then there exist", "type": "text"}], "index": 18}, {"bbox": [148, 363, 521, 380], "spans": [{"bbox": [148, 365, 216, 379], "score": 0.88, "content": "\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [216, 363, 246, 380], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [246, 363, 308, 379], "score": 0.91, "content": "\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}", "type": "inline_equation", "height": 16, "width": 62}, {"bbox": [308, 363, 333, 380], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [334, 364, 401, 379], "score": 0.91, "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [401, 363, 456, 380], "score": 1.0, "content": " as well as ", "type": "text"}, {"bbox": [456, 363, 516, 379], "score": 0.88, "content": "\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 16, "width": 60}, {"bbox": [517, 363, 521, 380], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [137, 376, 538, 397], "spans": [{"bbox": [137, 376, 189, 397], "score": 1.0, "content": "Due to ", "type": "text"}, {"bbox": [189, 379, 276, 394], "score": 0.91, "content": "{\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))", "type": "inline_equation", "height": 15, "width": 87}, {"bbox": [276, 376, 326, 397], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [327, 378, 453, 394], "score": 0.9, "content": "\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})", "type": "inline_equation", "height": 16, "width": 126}, {"bbox": [453, 376, 482, 397], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [483, 380, 538, 397], "score": 0.9, "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;", "type": "inline_equation", "height": 17, "width": 55}], "index": 20}, {"bbox": [148, 396, 287, 414], "spans": [{"bbox": [148, 396, 238, 413], "score": 0.91, "content": "h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "type": "inline_equation", "height": 17, "width": 90}, {"bbox": [238, 396, 275, 414], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [275, 401, 282, 407], "score": 0.83, "content": "x", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [282, 396, 287, 414], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19.5, "bbox_fs": [132, 348, 538, 414]}, {"type": "text", "bbox": [131, 411, 263, 423], "lines": [{"bbox": [148, 412, 261, 424], "spans": [{"bbox": [148, 412, 261, 424], "score": 1.0, "content": "Furthermore, we have", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [148, 412, 261, 424]}, {"type": "interline_equation", "bbox": [195, 427, 491, 522], "lines": [{"bbox": [195, 427, 491, 522], "spans": [{"bbox": [195, 427, 491, 522], "score": 0.94, "content": "\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [148, 525, 374, 542], "lines": [{"bbox": [147, 527, 373, 544], "spans": [{"bbox": [147, 527, 235, 544], "score": 1.0, "content": "and analogously ", "type": "text"}, {"bbox": [235, 528, 371, 542], "score": 0.92, "content": "f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 136}, {"bbox": [371, 527, 373, 544], "score": 1.0, "content": ".", "type": "text"}], "index": 24}], "index": 24, "bbox_fs": [147, 527, 373, 544]}, {"type": "text", "bbox": [147, 542, 538, 571], "lines": [{"bbox": [148, 542, 539, 559], "spans": [{"bbox": [148, 542, 250, 559], "score": 1.0, "content": "Therefore, we have ", "type": "text"}, {"bbox": [250, 543, 370, 557], "score": 0.9, "content": "\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 120}, {"bbox": [371, 542, 398, 559], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [398, 542, 450, 557], "score": 0.91, "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}", "type": "inline_equation", "height": 15, "width": 52}, {"bbox": [450, 542, 539, 559], "score": 1.0, "content": " is an element of", "type": "text"}], "index": 25}, {"bbox": [147, 556, 455, 573], "spans": [{"bbox": [147, 556, 232, 573], "score": 1.0, "content": "the stabilizer of ", "type": "text"}, {"bbox": [232, 558, 258, 571], "score": 0.93, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [258, 556, 290, 573], "score": 1.0, "content": ", thus ", "type": "text"}, {"bbox": [291, 557, 451, 571], "score": 0.91, "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 14, "width": 160}, {"bbox": [451, 556, 455, 573], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 25.5, "bbox_fs": [147, 542, 539, 573]}, {"type": "text", "bbox": [132, 572, 538, 599], "lines": [{"bbox": [133, 570, 536, 591], "spans": [{"bbox": [133, 570, 179, 591], "score": 1.0, "content": "• Since ", "type": "text"}, {"bbox": [179, 573, 306, 587], "score": 0.94, "content": "\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}", "type": "inline_equation", "height": 14, "width": 127}, {"bbox": [306, 570, 356, 591], "score": 1.0, "content": ", we have ", "type": "text"}, {"bbox": [356, 571, 510, 591], "score": 0.93, "content": "\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)", "type": "inline_equation", "height": 20, "width": 154}, {"bbox": [511, 570, 536, 591], "score": 1.0, "content": ", and", "type": "text"}], "index": 27}, {"bbox": [148, 588, 231, 603], "spans": [{"bbox": [148, 588, 197, 603], "score": 1.0, "content": "so for all ", "type": "text"}, {"bbox": [197, 590, 231, 599], "score": 0.89, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}], "index": 28}], "index": 27.5, "bbox_fs": [133, 570, 536, 603]}, {"type": "text", "bbox": [147, 619, 537, 669], "lines": [{"bbox": [147, 620, 537, 639], "spans": [{"bbox": [147, 620, 235, 638], "score": 1.0, "content": "Moreover, since", "type": "text"}, {"bbox": [235, 621, 355, 639], "score": 0.94, "content": "\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})", "type": "inline_equation", "height": 18, "width": 120}, {"bbox": [356, 620, 411, 638], "score": 1.0, "content": ", we have", "type": "text"}, {"bbox": [411, 620, 537, 638], "score": 0.79, "content": "\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in", "type": "inline_equation", "height": 18, "width": 126}], "index": 29}, {"bbox": [149, 637, 537, 656], "spans": [{"bbox": [149, 640, 184, 653], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [185, 638, 222, 655], "score": 1.0, "content": ". From ", "type": "text"}, {"bbox": [223, 640, 366, 656], "score": 0.91, "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "type": "inline_equation", "height": 16, "width": 143}, {"bbox": [366, 638, 402, 655], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [402, 642, 410, 650], "score": 0.3, "content": "x", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [410, 638, 437, 655], "score": 1.0, "content": " now ", "type": "text"}, {"bbox": [437, 637, 537, 653], "score": 0.72, "content": "\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in", "type": "inline_equation", "height": 16, "width": 100}], "index": 30}, {"bbox": [149, 656, 358, 670], "spans": [{"bbox": [149, 657, 176, 670], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [177, 656, 269, 670], "score": 1.0, "content": " follows, and thus", "type": "text"}, {"bbox": [270, 656, 355, 670], "score": 0.89, "content": "\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [356, 656, 358, 670], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 30, "bbox_fs": [147, 620, 537, 670]}, {"type": "text", "bbox": [132, 669, 427, 684], "lines": [{"bbox": [147, 671, 425, 684], "spans": [{"bbox": [147, 671, 234, 684], "score": 1.0, "content": "By this we have ", "type": "text"}, {"bbox": [234, 671, 310, 684], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [311, 671, 337, 684], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [337, 673, 347, 682], "score": 0.89, "content": "F^{'}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [347, 671, 425, 684], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 32}], "index": 32, "bbox_fs": [147, 671, 425, 684]}]}
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[{"type": "title", "bbox": [61, 12, 176, 29], "content": "5.2 The Proof", "index": 0}, {"type": "text", "bbox": [62, 35, 539, 80], "content": "Proof 1. Let . Choose for an with according to Corollary 4.2 and denote the corresponding reduction mapping shortly by .", "index": 1}, {"type": "text", "bbox": [112, 80, 495, 95], "content": "2. Due to Proposition 4.4 there is an with , such that", "index": 2}, {"type": "text", "bbox": [129, 95, 401, 153], "content": "is an open neighbourhood of and there exists an equivariant mapping with and .", "index": 3}, {"type": "text", "bbox": [111, 154, 254, 166], "content": "3. We define the mapping", "index": 4}, {"type": "interline_equation", "bbox": [254, 168, 400, 204], "content": "", "index": 5}, {"type": "text", "bbox": [131, 201, 537, 229], "content": "whereas for all the (arbitrary, but fixed) path runs from to and is the trivial path.", "index": 6}, {"type": "text", "bbox": [111, 230, 289, 243], "content": "4. As we motivated above we set", "index": 7}, {"type": "interline_equation", "bbox": [190, 248, 478, 291], "content": "", "index": 8}, {"type": "text", "bbox": [131, 291, 153, 304], "content": "and", "index": 9}, {"type": "interline_equation", "bbox": [262, 304, 390, 339], "content": "", "index": 10}, {"type": "text", "bbox": [110, 335, 222, 348], "content": "5. is well-defined.", "index": 11}, {"type": "text", "bbox": [128, 347, 538, 415], "content": "• Let with and . Then there exist with and as well as . Due to we have , i.e. for all .", "index": 12}, {"type": "text", "bbox": [131, 411, 263, 423], "content": "Furthermore, we have", "index": 13}, {"type": "interline_equation", "bbox": [195, 427, 491, 522], "content": "", "index": 14}, {"type": "text", "bbox": [148, 525, 374, 542], "content": "and analogously .", "index": 15}, {"type": "text", "bbox": [147, 542, 538, 571], "content": "Therefore, we have , i.e. is an element of the stabilizer of , thus .", "index": 16}, {"type": "text", "bbox": [132, 572, 538, 599], "content": "• Since , we have , and so for all", "index": 17}, {"type": "text", "bbox": [147, 619, 537, 669], "content": "Moreover, since , we have . From for all now follows, and thus .", "index": 18}, {"type": "text", "bbox": [132, 669, 427, 684], "content": "By this we have , i.e. is well-defined.", "index": 19}]
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[{"bbox": [62, 15, 174, 29], "content": "5.2 The Proof", "parent_index": 0, "line_index": 0}, {"bbox": [61, 36, 538, 56], "content": "Proof 1. Let . Choose for an with according to", "parent_index": 1, "line_index": 0}, {"bbox": [129, 51, 536, 70], "content": "Corollary 4.2 and denote the corresponding reduction mapping", "parent_index": 1, "line_index": 1}, {"bbox": [131, 66, 201, 84], "content": "shortly by .", "parent_index": 1, "line_index": 2}, {"bbox": [111, 82, 494, 96], "content": "2. Due to Proposition 4.4 there is an with , such that", "parent_index": 2, "line_index": 0}, {"bbox": [149, 97, 402, 110], "content": "is an open neighbourhood of and", "parent_index": 3, "line_index": 0}, {"bbox": [146, 111, 371, 126], "content": "there exists an equivariant mapping with", "parent_index": 3, "line_index": 1}, {"bbox": [151, 125, 314, 140], "content": "and", "parent_index": 3, "line_index": 2}, {"bbox": [168, 140, 262, 154], "content": ".", "parent_index": 3, "line_index": 3}, {"bbox": [111, 153, 252, 168], "content": "3. We define the mapping", "parent_index": 4, "line_index": 0}, {"bbox": [131, 202, 538, 218], "content": "whereas for all the (arbitrary, but fixed) path runs from to", "parent_index": 6, "line_index": 0}, {"bbox": [132, 218, 276, 231], "content": "and is the trivial path.", "parent_index": 6, "line_index": 1}, {"bbox": [111, 231, 288, 244], "content": "4. As we motivated above we set", "parent_index": 7, "line_index": 0}, {"bbox": [130, 293, 153, 305], "content": "and", "parent_index": 9, "line_index": 0}, {"bbox": [111, 337, 220, 349], "content": "5. is well-defined.", "parent_index": 11, "line_index": 0}, {"bbox": [132, 348, 538, 365], "content": "• Let with and . Then there exist", "parent_index": 12, "line_index": 0}, {"bbox": [148, 363, 521, 380], "content": "with and as well as .", "parent_index": 12, "line_index": 1}, {"bbox": [137, 376, 538, 397], "content": "Due to we have , i.e.", "parent_index": 12, "line_index": 2}, {"bbox": [148, 396, 287, 414], "content": "for all .", "parent_index": 12, "line_index": 3}, {"bbox": [148, 412, 261, 424], "content": "Furthermore, we have", "parent_index": 13, "line_index": 0}, {"bbox": [147, 527, 373, 544], "content": "and analogously .", "parent_index": 15, "line_index": 0}, {"bbox": [148, 542, 539, 559], "content": "Therefore, we have , i.e. is an element of", "parent_index": 16, "line_index": 0}, {"bbox": [147, 556, 455, 573], "content": "the stabilizer of , thus .", "parent_index": 16, "line_index": 1}, {"bbox": [133, 570, 536, 591], "content": "• Since , we have , and", "parent_index": 17, "line_index": 0}, {"bbox": [148, 588, 231, 603], "content": "so for all", "parent_index": 17, "line_index": 1}, {"bbox": [147, 620, 537, 639], "content": "Moreover, since , we have", "parent_index": 18, "line_index": 0}, {"bbox": [149, 637, 537, 656], "content": ". From for all now", "parent_index": 18, "line_index": 1}, {"bbox": [149, 656, 358, 670], "content": "follows, and thus .", "parent_index": 18, "line_index": 2}, {"bbox": [147, 671, 425, 684], "content": "By this we have , i.e. is well-defined.", "parent_index": 19, "line_index": 0}]
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[]
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[{"bbox": [153, 39, 189, 50], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [258, 39, 267, 50], "content": "\\overline{{A}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [288, 41, 333, 51], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [365, 40, 468, 53], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "parent_index": 1, "subtype": "inline"}, {"bbox": [451, 54, 536, 66], "content": "\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}", "parent_index": 1, "subtype": "inline"}, {"bbox": [188, 73, 195, 81], "content": "\\varphi", "parent_index": 1, "subtype": "inline"}, {"bbox": [311, 83, 360, 95], "content": "S\\subseteq\\mathbf{G}^{\\#\\alpha}", "parent_index": 2, "subtype": "inline"}, {"bbox": [389, 82, 438, 96], "content": "\\varphi({\\overline{{A}}})\\in S", "parent_index": 2, "subtype": "inline"}, {"bbox": [149, 99, 178, 108], "content": "S\\circ\\mathbf{G}", "parent_index": 3, "subtype": "inline"}, {"bbox": [332, 97, 380, 110], "content": "\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "parent_index": 3, "subtype": "inline"}, {"bbox": [336, 113, 343, 124], "content": "f", "parent_index": 3, "subtype": "inline"}, {"bbox": [151, 126, 289, 140], "content": "\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}", "parent_index": 3, "subtype": "inline"}, {"bbox": [168, 141, 258, 154], "content": "f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S", "parent_index": 3, "subtype": "inline"}, {"bbox": [254, 168, 400, 204], "content": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}", "parent_index": 5, "subtype": "interline"}, {"bbox": [211, 204, 279, 216], "content": "x\\in M\\setminus\\{m\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [442, 208, 453, 216], "content": "\\gamma_{x}", "parent_index": 6, "subtype": "inline"}, {"bbox": [511, 208, 522, 213], "content": "m", "parent_index": 6, "subtype": "inline"}, {"bbox": [132, 222, 138, 228], "content": "x", "parent_index": 6, "subtype": "inline"}, {"bbox": [165, 222, 180, 230], "content": "\\gamma_{m}", "parent_index": 6, "subtype": "inline"}, {"bbox": [190, 248, 478, 291], "content": "\\begin{array}{r c l}{\\overline{{S}}_{0}}&{:=}&{\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}})),}\\\\ {\\overline{{S}}}&{:=}&{\\big(\\varphi^{-1}(S)\\,\\cap\\,\\psi^{-1}(\\psi(\\overline{{A}}))\\big)\\circ{\\bf B}(\\overline{{A}})}&{\\equiv}&{\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})}\\end{array}", "parent_index": 8, "subtype": "interline"}, {"bbox": [262, 304, 390, 339], "content": "\\begin{array}{r}{F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}.}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\enspace\\longmapsto\\enspace\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "parent_index": 10, "subtype": "interline"}, {"bbox": [132, 339, 141, 348], "content": "F", "parent_index": 11, "subtype": "inline"}, {"bbox": [171, 349, 264, 364], "content": "\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\,=\\,\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}}^{\\prime\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [296, 349, 355, 364], "content": "\\overline{{A}}^{\\prime},\\overline{{A}}^{\\prime\\prime}\\,\\in\\,\\overline{{S}}", "parent_index": 12, "subtype": "inline"}, {"bbox": [383, 349, 437, 364], "content": "\\overline{{g}}^{\\prime},\\overline{{g}}^{\\prime\\prime}\\,\\in\\,\\overline{{g}}", "parent_index": 12, "subtype": "inline"}, {"bbox": [148, 365, 216, 379], "content": "\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})", "parent_index": 12, "subtype": "inline"}, {"bbox": [246, 363, 308, 379], "content": "\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [334, 364, 401, 379], "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [456, 363, 516, 379], "content": "\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "parent_index": 12, "subtype": "inline"}, {"bbox": [189, 379, 276, 394], "content": "{\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))", "parent_index": 12, "subtype": "inline"}, {"bbox": [327, 378, 453, 394], "content": "\\psi(\\overline{{{A}}}_{0}^{\\prime})\\,=\\,\\bar{\\psi}(\\overline{{{A}}})\\,=\\,\\psi(\\overline{{{A}}}_{0}^{\\prime\\prime})", "parent_index": 12, "subtype": "inline"}, {"bbox": [483, 380, 538, 397], "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})\\;=\\;", "parent_index": 12, "subtype": "inline"}, {"bbox": [148, 396, 238, 413], "content": "h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "parent_index": 12, "subtype": "inline"}, {"bbox": [275, 401, 282, 407], "content": "x", "parent_index": 12, "subtype": "inline"}, {"bbox": [195, 427, 491, 522], "content": "\\begin{array}{r l r}{f(\\varphi(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}&{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}))}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime})\\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{f(\\varphi(\\overline{{A}}_{0}^{\\prime}))\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ (f\\ \\mathrm{equivariant})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ z_{m}^{\\prime}\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi(\\overline{{A}}_{0}^{\\prime})\\in S)}\\\\ &{=}&{\\varphi(\\overline{{A}}\\circ\\overline{{z}}^{\\prime})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\varphi\\ ^{,}\\mathrm{equivariant}^{,*})}\\\\ &{=}&{\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\overline{{z}}^{\\prime}\\in{\\bf B}(\\overline{{A}}))}\\end{array}", "parent_index": 14, "subtype": "interline"}, {"bbox": [235, 528, 371, 542], "content": "f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "parent_index": 15, "subtype": "inline"}, {"bbox": [250, 543, 370, 557], "content": "\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime}=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}", "parent_index": 16, "subtype": "inline"}, {"bbox": [398, 542, 450, 557], "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}", "parent_index": 16, "subtype": "inline"}, {"bbox": [232, 558, 258, 571], "content": "\\varphi(\\overline{{A}})", "parent_index": 16, "subtype": "inline"}, {"bbox": [291, 557, 451, 571], "content": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 16, "subtype": "inline"}, {"bbox": [179, 573, 306, 587], "content": "\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}", "parent_index": 17, "subtype": "inline"}, {"bbox": [356, 571, 510, 591], "content": "\\overline{{A}}_{0}^{\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\left(\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{-1}\\right)", "parent_index": 17, "subtype": "inline"}, {"bbox": [197, 590, 231, 599], "content": "x\\in M", "parent_index": 17, "subtype": "inline"}, {"bbox": [235, 621, 355, 639], "content": "\\left(\\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\right)_{m}\\ \\in\\ Z(\\mathbf{H}_{\\overline{{{A}}}})", "parent_index": 18, "subtype": "inline"}, {"bbox": [411, 620, 537, 638], "content": "\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in", "parent_index": 18, "subtype": "inline"}, {"bbox": [149, 640, 184, 653], "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 18, "subtype": "inline"}, {"bbox": [223, 640, 366, 656], "content": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})", "parent_index": 18, "subtype": "inline"}, {"bbox": [402, 642, 410, 650], "content": "x", "parent_index": 18, "subtype": "inline"}, {"bbox": [437, 637, 537, 653], "content": "\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in", "parent_index": 18, "subtype": "inline"}, {"bbox": [149, 657, 176, 670], "content": "\\mathbf{B}(\\overline{{A}})", "parent_index": 18, "subtype": "inline"}, {"bbox": [270, 656, 355, 670], "content": "\\overline{{g}}^{\\prime\\prime}\\left(\\overline{{g}}^{\\prime}\\right)^{-1}\\in\\mathbf{B}(\\overline{{A}})", "parent_index": 18, "subtype": "inline"}, {"bbox": [234, 671, 310, 684], "content": "\\overline{{A}}\\circ\\overline{{g}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}^{\\prime\\prime}", "parent_index": 19, "subtype": "inline"}, {"bbox": [337, 673, 347, 682], "content": "F^{'}", "parent_index": 19, "subtype": "inline"}]
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[]
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6. $F$ is equivariant.
Let $\overline{{A}}^{\prime\prime}=\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime}\in\overline{{S}}\circ\overline{{\mathcal{G}}}$ . Then
$$
\begin{array}{l l l}{{F(\overline{{{A}}}^{\prime\prime}\circ\overline{{{g}}})}}&{{=}}&{{F(\overline{{{A}}}^{\prime}\circ(\overline{{{g}}}^{\prime}\circ\overline{{{g}}}))}}\\ {{}}&{{=}}&{{\overline{{{A}}}\circ(\overline{{{g}}}^{\prime}\circ\overline{{{g}}})}}\\ {{}}&{{=}}&{{(\overline{{{A}}}\circ\overline{{{g}}}^{\prime})\circ\overline{{{g}}}}}\\ {{}}&{{=}}&{{F(\overline{{{A}}}^{\prime}\circ\overline{{{g}}}^{\prime})\circ\overline{{{g}}}}}\\ {{}}&{{=}}&{{F(\overline{{{A}}}^{\prime\prime})\circ\overline{{{g}}}.}}\end{array}
$$
7. $F$ is retracting.
• Let $\overline{{A}}^{\prime}=\overline{{A}}\circ\overline{{g}}\in\overline{{A}}\circ\overline{{g}}$ . Then $F(\overline{{A}}^{\prime})=F(\overline{{A}}\circ\overline{{g}})=\overline{{A}}\circ\overline{{g}}=\overline{{A}}^{\prime}$ .
8. $\overline{{S}}\circ\overline{{\mathcal{G}}}$ is an open neighbourhood of $\overline{{A}}\circ\overline{{\mathcal{G}}}$ .
Obviously, $\overline{{A}}\circ\overline{{\mathcal{G}}}\subseteq\overline{{S}}\circ\overline{{\mathcal{G}}}$ .
Consequently, $\overline{{S}}\circ\overline{{\mathcal{G}}}=\varphi^{-1}(S\circ\mathbf{G})$ is as a preimage of an open set again open because of the continuity of $\varphi$ .
9. $F$ is continuous.
We consider the following diagram
$$
\begin{array}{r l}{\lefteqn{\overline{{S}}\circ\overline{{\mathcal{G}}}\xrightarrow{\ F}\quad}&{{}\overline{{A}}\circ\overline{{\mathcal{G}}}}\\ {\Bigg\downarrow\varphi\quad}&{{}\varphi\Bigg\downarrow}\\ {\quad S\circ\mathbf{G}\xrightarrow{\ f\quad}\varphi(\overline{{A}})\circ\mathbf{G}\xrightarrow{\ \tau_{\mathbf{G}}}\chi_{\mathbf{G}}}&{{}\big.}\end{array}
$$
$$
\begin{array}{r}{\overbrace{A^{\prime}\circ\overline{{g}}\longmapsto\overrightarrow{F}}^{\overline{{A}}^{\prime}\circ\overline{{g}}\longmapsto\overrightarrow{A}\circ\overline{{g}}}}\\ {\left[\overset{\qquad\qquad\qquad\qquad\varphi}{\qquad\qquad\qquad\varphi}\right]^{\overline{{\varphi}}}}\\ {\varphi(\overrightarrow{A}^{\prime})\circ g_{m}\vdash\overrightarrow{f}\rightarrow\varphi(\overrightarrow{A})\circ g_{m}\vdash\cdots\overrightarrow{[g_{m}]}_{\overline{{Z}}(\mathbf{H}_{\overline{{A}}})}}\end{array}
$$
It is commutative due to $\varphi(\overline{{S}}\circ\overline{{\mathcal{G}}})\subseteq S\circ\mathbf{G}$ , $\varphi(\overline{{A}}\circ\overline{{\mathcal{G}}})\subseteq\varphi(\overline{{A}})\circ\mathbf{G}$ and the definition of $F$ . $\tau_{\mathbf G}$ is the canonical homeomorphism between the orbit of $\varphi(\overline{{A}})$ and the quotient of the acting group $\mathbf{G}$ by the stabilizer of $\varphi(\overline{{A}})$ .
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<html><body>
<p data-bbox="110 15 218 28">6. $F$ is equivariant. </p>
<p data-bbox="132 28 309 43">Let $\overline{{A}}^{\prime\prime}=\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime}\in\overline{{S}}\circ\overline{{\mathcal{G}}}$ . Then </p>
<div class="equation" data-bbox="263 47 421 127">$$
\begin{array}{l l l}{{F(\overline{{{A}}}^{\prime\prime}\circ\overline{{{g}}})}}&{{=}}&{{F(\overline{{{A}}}^{\prime}\circ(\overline{{{g}}}^{\prime}\circ\overline{{{g}}}))}}\\ {{}}&{{=}}&{{\overline{{{A}}}\circ(\overline{{{g}}}^{\prime}\circ\overline{{{g}}})}}\\ {{}}&{{=}}&{{(\overline{{{A}}}\circ\overline{{{g}}}^{\prime})\circ\overline{{{g}}}}}\\ {{}}&{{=}}&{{F(\overline{{{A}}}^{\prime}\circ\overline{{{g}}}^{\prime})\circ\overline{{{g}}}}}\\ {{}}&{{=}}&{{F(\overline{{{A}}}^{\prime\prime})\circ\overline{{{g}}}.}}\end{array}
$$</div>
<p data-bbox="111 128 212 141">7. $F$ is retracting. </p>
<p data-bbox="129 141 468 158">• Let $\overline{{A}}^{\prime}=\overline{{A}}\circ\overline{{g}}\in\overline{{A}}\circ\overline{{g}}$ . Then $F(\overline{{A}}^{\prime})=F(\overline{{A}}\circ\overline{{g}})=\overline{{A}}\circ\overline{{g}}=\overline{{A}}^{\prime}$ . </p>
<p data-bbox="111 158 344 171">8. $\overline{{S}}\circ\overline{{\mathcal{G}}}$ is an open neighbourhood of $\overline{{A}}\circ\overline{{\mathcal{G}}}$ . </p>
<p data-bbox="132 172 280 185">Obviously, $\overline{{A}}\circ\overline{{\mathcal{G}}}\subseteq\overline{{S}}\circ\overline{{\mathcal{G}}}$ . </p>
<p data-bbox="132 397 537 426">Consequently, $\overline{{S}}\circ\overline{{\mathcal{G}}}=\varphi^{-1}(S\circ\mathbf{G})$ is as a preimage of an open set again open because of the continuity of $\varphi$ . </p>
<p data-bbox="110 427 216 439">9. $F$ is continuous. </p>
<p data-bbox="132 441 327 454">We consider the following diagram </p>
<div class="equation" data-bbox="241 459 466 540">$$
\begin{array}{r l}{\lefteqn{\overline{{S}}\circ\overline{{\mathcal{G}}}\xrightarrow{\ F}\quad}&{{}\overline{{A}}\circ\overline{{\mathcal{G}}}}\\ {\Bigg\downarrow\varphi\quad}&{{}\varphi\Bigg\downarrow}\\ {\quad S\circ\mathbf{G}\xrightarrow{\ f\quad}\varphi(\overline{{A}})\circ\mathbf{G}\xrightarrow{\ \tau_{\mathbf{G}}}\chi_{\mathbf{G}}}&{{}\big.}\end{array}
$$</div>
<div class="equation" data-bbox="222 556 462 634">$$
\begin{array}{r}{\overbrace{A^{\prime}\circ\overline{{g}}\longmapsto\overrightarrow{F}}^{\overline{{A}}^{\prime}\circ\overline{{g}}\longmapsto\overrightarrow{A}\circ\overline{{g}}}}\\ {\left[\overset{\qquad\qquad\qquad\qquad\varphi}{\qquad\qquad\qquad\varphi}\right]^{\overline{{\varphi}}}}\\ {\varphi(\overrightarrow{A}^{\prime})\circ g_{m}\vdash\overrightarrow{f}\rightarrow\varphi(\overrightarrow{A})\circ g_{m}\vdash\cdots\overrightarrow{[g_{m}]}_{\overline{{Z}}(\mathbf{H}_{\overline{{A}}})}}\end{array}
$$</div>
<p data-bbox="147 632 539 677">It is commutative due to $\varphi(\overline{{S}}\circ\overline{{\mathcal{G}}})\subseteq S\circ\mathbf{G}$ , $\varphi(\overline{{A}}\circ\overline{{\mathcal{G}}})\subseteq\varphi(\overline{{A}})\circ\mathbf{G}$ and the definition of $F$ . $\tau_{\mathbf G}$ is the canonical homeomorphism between the orbit of $\varphi(\overline{{A}})$ and the quotient of the acting group $\mathbf{G}$ by the stabilizer of $\varphi(\overline{{A}})$ . </p>
</body></html>
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0001008v1
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[{"type": "text", "text": "6. $F$ is equivariant. ", "page_idx": 8}, {"type": "text", "text": "Let $\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}\\in\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ . Then ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{F(\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}})}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}}))}}\\\\ {{}}&{{=}}&{{\\overline{{{A}}}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})}}\\\\ {{}}&{{=}}&{{(\\overline{{{A}}}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime\\prime})\\circ\\overline{{{g}}}.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "7. $F$ is retracting. ", "page_idx": 8}, {"type": "text", "text": "• Let $\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}$ . Then $F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}$ . ", "page_idx": 8}, {"type": "text", "text": "8. $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ is an open neighbourhood of $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ . ", "page_idx": 8}, {"type": "text", "text": "Obviously, $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ . ", "page_idx": 8}, {"type": "text", "text": "Consequently, $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})$ is as a preimage of an open set again open because of the continuity of $\\varphi$ . ", "page_idx": 8}, {"type": "text", "text": "9. $F$ is continuous. ", "page_idx": 8}, {"type": "text", "text": "We consider the following diagram ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\begin{array}{r l}{\\lefteqn{\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\xrightarrow{\\ F}\\quad}&{{}\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}}\\\\ {\\Bigg\\downarrow\\varphi\\quad}&{{}\\varphi\\Bigg\\downarrow}\\\\ {\\quad S\\circ\\mathbf{G}\\xrightarrow{\\ f\\quad}\\varphi(\\overline{{A}})\\circ\\mathbf{G}\\xrightarrow{\\ \\tau_{\\mathbf{G}}}\\chi_{\\mathbf{G}}}&{{}\\big.}\\end{array}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "equation", "text": "$$\n\\begin{array}{r}{\\overbrace{A^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{F}}^{\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{A}\\circ\\overline{{g}}}}\\\\ {\\left[\\overset{\\qquad\\qquad\\qquad\\qquad\\varphi}{\\qquad\\qquad\\qquad\\varphi}\\right]^{\\overline{{\\varphi}}}}\\\\ {\\varphi(\\overrightarrow{A}^{\\prime})\\circ g_{m}\\vdash\\overrightarrow{f}\\rightarrow\\varphi(\\overrightarrow{A})\\circ g_{m}\\vdash\\cdots\\overrightarrow{[g_{m}]}_{\\overline{{Z}}(\\mathbf{H}_{\\overline{{A}}})}}\\end{array}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "It is commutative due to $\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}$ , $\\varphi(\\overline{{A}}\\circ\\overline{{\\mathcal{G}}})\\subseteq\\varphi(\\overline{{A}})\\circ\\mathbf{G}$ and the definition of $F$ . $\\tau_{\\mathbf G}$ is the canonical homeomorphism between the orbit of $\\varphi(\\overline{{A}})$ and the quotient of the acting group $\\mathbf{G}$ by the stabilizer of $\\varphi(\\overline{{A}})$ . ", "page_idx": 8}]
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Then ", "type": "text"}, {"bbox": [304, 143, 465, 158], "score": 0.92, "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 15, "width": 161}, {"bbox": [465, 141, 468, 160], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 4, "bbox_fs": [135, 140, 468, 160]}, {"type": "text", "bbox": [111, 158, 344, 171], "lines": [{"bbox": [112, 158, 343, 173], "spans": [{"bbox": [112, 158, 131, 173], "score": 1.0, "content": "8.", "type": "text"}, {"bbox": [132, 159, 159, 171], "score": 0.91, "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [159, 158, 312, 173], "score": 1.0, "content": " is an open neighbourhood of ", "type": "text"}, {"bbox": [312, 159, 341, 171], "score": 0.91, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [341, 158, 343, 173], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [112, 158, 343, 173]}, {"type": "text", "bbox": [132, 172, 280, 185], "lines": [{"bbox": [148, 173, 279, 186], "spans": [{"bbox": [148, 173, 205, 186], "score": 1.0, "content": "Obviously, ", "type": "text"}, {"bbox": [206, 174, 277, 186], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [277, 173, 279, 186], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [148, 173, 279, 186]}, {"type": "text", "bbox": [132, 397, 537, 426], "lines": [{"bbox": [147, 399, 537, 414], "spans": [{"bbox": [147, 400, 222, 414], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [223, 399, 320, 412], "score": 0.94, "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [320, 400, 537, 414], "score": 1.0, "content": " is as a preimage of an open set again open", "type": "text"}], "index": 7}, {"bbox": [148, 413, 306, 429], "spans": [{"bbox": [148, 413, 294, 429], "score": 1.0, "content": "because of the continuity of ", "type": "text"}, {"bbox": [294, 419, 302, 426], "score": 0.89, "content": "\\varphi", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [302, 413, 306, 429], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [147, 399, 537, 429]}, {"type": "text", "bbox": [110, 427, 216, 439], "lines": [{"bbox": [111, 428, 215, 441], "spans": [{"bbox": [111, 428, 131, 441], "score": 1.0, "content": "9.", "type": "text"}, {"bbox": [132, 430, 141, 439], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 428, 215, 441], "score": 1.0, "content": " is continuous.", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [111, 428, 215, 441]}, {"type": "text", "bbox": [132, 441, 327, 454], "lines": [{"bbox": [148, 443, 326, 455], "spans": [{"bbox": [148, 443, 326, 455], "score": 1.0, "content": "We consider the following diagram", "type": "text"}], "index": 10}], "index": 10, "bbox_fs": [148, 443, 326, 455]}, {"type": "interline_equation", "bbox": [241, 459, 466, 540], "lines": [{"bbox": [241, 459, 466, 540], "spans": [{"bbox": [241, 459, 466, 540], "score": 0.56, "content": "\\begin{array}{r l}{\\lefteqn{\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\xrightarrow{\\ F}\\quad}&{{}\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}}\\\\ {\\Bigg\\downarrow\\varphi\\quad}&{{}\\varphi\\Bigg\\downarrow}\\\\ {\\quad S\\circ\\mathbf{G}\\xrightarrow{\\ f\\quad}\\varphi(\\overline{{A}})\\circ\\mathbf{G}\\xrightarrow{\\ \\tau_{\\mathbf{G}}}\\chi_{\\mathbf{G}}}&{{}\\big.}\\end{array}", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [222, 556, 462, 634], "lines": [{"bbox": [222, 556, 462, 634], "spans": [{"bbox": [222, 556, 462, 634], "score": 0.28, "content": "\\begin{array}{r}{\\overbrace{A^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{F}}^{\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{A}\\circ\\overline{{g}}}}\\\\ {\\left[\\overset{\\qquad\\qquad\\qquad\\qquad\\varphi}{\\qquad\\qquad\\qquad\\varphi}\\right]^{\\overline{{\\varphi}}}}\\\\ {\\varphi(\\overrightarrow{A}^{\\prime})\\circ g_{m}\\vdash\\overrightarrow{f}\\rightarrow\\varphi(\\overrightarrow{A})\\circ g_{m}\\vdash\\cdots\\overrightarrow{[g_{m}]}_{\\overline{{Z}}(\\mathbf{H}_{\\overline{{A}}})}}\\end{array}", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [147, 632, 539, 677], "lines": [{"bbox": [146, 635, 537, 649], "spans": [{"bbox": [146, 635, 282, 649], "score": 1.0, "content": "It is commutative due to ", "type": "text"}, {"bbox": [282, 635, 374, 649], "score": 0.92, "content": "\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 92}, {"bbox": [374, 635, 381, 649], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [381, 635, 492, 649], "score": 0.9, "content": "\\varphi(\\overline{{A}}\\circ\\overline{{\\mathcal{G}}})\\subseteq\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 111}, {"bbox": [493, 635, 537, 649], "score": 1.0, "content": " and the", "type": "text"}], "index": 13}, {"bbox": [148, 649, 538, 664], "spans": [{"bbox": [148, 649, 216, 664], "score": 1.0, "content": "definition of ", "type": "text"}, {"bbox": [216, 651, 225, 660], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [226, 649, 236, 664], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [236, 655, 249, 662], "score": 0.88, "content": "\\tau_{\\mathbf G}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [250, 649, 538, 664], "score": 1.0, "content": " is the canonical homeomorphism between the orbit of", "type": "text"}], "index": 14}, {"bbox": [149, 664, 513, 678], "spans": [{"bbox": [149, 664, 174, 677], "score": 0.94, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [175, 664, 369, 678], "score": 1.0, "content": " and the quotient of the acting group ", "type": "text"}, {"bbox": [369, 666, 380, 675], "score": 0.89, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [380, 664, 483, 678], "score": 1.0, "content": " by the stabilizer of ", "type": "text"}, {"bbox": [483, 664, 509, 678], "score": 0.94, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 26}, {"bbox": [510, 664, 513, 678], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 14, "bbox_fs": [146, 635, 538, 678]}]}
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[]
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[{"bbox": [132, 19, 141, 28], "content": "F", "parent_index": 0, "subtype": "inline"}, {"bbox": [170, 30, 273, 45], "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}\\in\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [263, 47, 421, 127], "content": "\\begin{array}{l l l}{{F(\\overline{{{A}}}^{\\prime\\prime}\\circ\\overline{{{g}}})}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}}))}}\\\\ {{}}&{{=}}&{{\\overline{{{A}}}\\circ(\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})}}\\\\ {{}}&{{=}}&{{(\\overline{{{A}}}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}}^{\\prime})\\circ\\overline{{{g}}}}}\\\\ {{}}&{{=}}&{{F(\\overline{{{A}}}^{\\prime\\prime})\\circ\\overline{{{g}}}.}}\\end{array}", "parent_index": 2, "subtype": "interline"}, {"bbox": [132, 132, 141, 141], "content": "F", "parent_index": 3, "subtype": "inline"}, {"bbox": [169, 144, 266, 157], "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [304, 143, 465, 158], "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}", "parent_index": 4, "subtype": "inline"}, {"bbox": [132, 159, 159, 171], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [312, 159, 341, 171], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [206, 174, 277, 186], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [223, 399, 320, 412], "content": "\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})", "parent_index": 7, "subtype": "inline"}, {"bbox": [294, 419, 302, 426], "content": "\\varphi", "parent_index": 7, "subtype": "inline"}, {"bbox": [132, 430, 141, 439], "content": "F", "parent_index": 8, "subtype": "inline"}, {"bbox": [241, 459, 466, 540], "content": "\\begin{array}{r l}{\\lefteqn{\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\xrightarrow{\\ F}\\quad}&{{}\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}}\\\\ {\\Bigg\\downarrow\\varphi\\quad}&{{}\\varphi\\Bigg\\downarrow}\\\\ {\\quad S\\circ\\mathbf{G}\\xrightarrow{\\ f\\quad}\\varphi(\\overline{{A}})\\circ\\mathbf{G}\\xrightarrow{\\ \\tau_{\\mathbf{G}}}\\chi_{\\mathbf{G}}}&{{}\\big.}\\end{array}", "parent_index": 10, "subtype": "interline"}, {"bbox": [222, 556, 462, 634], "content": "\\begin{array}{r}{\\overbrace{A^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{F}}^{\\overline{{A}}^{\\prime}\\circ\\overline{{g}}\\longmapsto\\overrightarrow{A}\\circ\\overline{{g}}}}\\\\ {\\left[\\overset{\\qquad\\qquad\\qquad\\qquad\\varphi}{\\qquad\\qquad\\qquad\\varphi}\\right]^{\\overline{{\\varphi}}}}\\\\ {\\varphi(\\overrightarrow{A}^{\\prime})\\circ g_{m}\\vdash\\overrightarrow{f}\\rightarrow\\varphi(\\overrightarrow{A})\\circ g_{m}\\vdash\\cdots\\overrightarrow{[g_{m}]}_{\\overline{{Z}}(\\mathbf{H}_{\\overline{{A}}})}}\\end{array}", "parent_index": 11, "subtype": "interline"}, {"bbox": [282, 635, 374, 649], "content": "\\varphi(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\subseteq S\\circ\\mathbf{G}", "parent_index": 12, "subtype": "inline"}, {"bbox": [381, 635, 492, 649], "content": "\\varphi(\\overline{{A}}\\circ\\overline{{\\mathcal{G}}})\\subseteq\\varphi(\\overline{{A}})\\circ\\mathbf{G}", "parent_index": 12, "subtype": "inline"}, {"bbox": [216, 651, 225, 660], "content": "F", "parent_index": 12, "subtype": "inline"}, {"bbox": [236, 655, 249, 662], "content": "\\tau_{\\mathbf G}", "parent_index": 12, "subtype": "inline"}, {"bbox": [149, 664, 174, 677], "content": "\\varphi(\\overline{{A}})", "parent_index": 12, "subtype": "inline"}, {"bbox": [369, 666, 380, 675], "content": "\\mathbf{G}", "parent_index": 12, "subtype": "inline"}, {"bbox": [483, 664, 509, 678], "content": "\\varphi(\\overline{{A}})", "parent_index": 12, "subtype": "inline"}]
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[]
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Since $\varphi$ , $f$ and $\tau_{\mathbf G}$ are continuous, the map
$$
F^{\prime}:=\tau_{\mathbf{G}}\circ\varphi\circ F:\;\;\overline{{S}}\circ\overline{{\mathcal{G}}}\enspace\longrightarrow\enspace Z(\mathbf{H}_{\overline{{A}}})\backslash\mathbf{G}
$$
is continuous.
Now, we consider the map
$$
\begin{array}{c c c c}{F^{\prime\prime}:}&{(\overline{{S}}\circ\overline{{\mathcal{G}}})\times\mathbf{G}}&{\longrightarrow}&{\overline{{\mathcal{G}}}.}\\ &{(\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime},g_{m})}&{\longmapsto}&{\bigl(h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime})\bigr)_{x\in M}}\end{array}
$$
$F^{\prime\prime}$ is continuous because
$$
\begin{array}{r l}{\pi_{x}\circ F^{\prime\prime}:}&{(\overline{{S}}\circ\overline{{\mathcal{G}}})\times\mathbf{G}\quad\longrightarrow\quad\mathbf{G}\times\mathbf{G}\quad\xrightarrow{\mathrm{mult}_{*}}\quad\xrightarrow{\mathbf{G}}}\\ &{\quad(\overline{{A}}^{\prime\prime},g_{m})\quad\longmapsto\quad(h_{\gamma_{x}}(\overline{{A}}^{\prime\prime}),g_{m})\quad\longmapsto\quad h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})}\end{array}
$$
is obviously continuous for all $x\in M$ .
$F^{\prime\prime}$ induces a map $F^{\prime\prime\prime}$ via the following commutative diagram
$$
\begin{array}{r l r}&{}&{(\overline{{S}}\circ\overline{{\mathcal{G}}})\times{\mathbf{G}}\xrightarrow{F^{\prime\prime}}\quad\xrightarrow{\longrightarrow}\overline{{\mathcal{G}}}}\\ &{}&{\quad\quad\quad\mathrm{id}\times\pi_{Z(\mathbf{H}_{\overline{{A}}})}\,,}\\ &{}&{\quad\quad\quad\quad(\overline{{S}}\circ\overline{{\mathcal{G}}})\times Z(\mathbf{H}_{\overline{{A}}})\setminus{\mathbf{G}}\xrightarrow{F^{\prime\prime\prime}}{\substack{\mathbf{B}}}(\overline{{A}})\setminus\overline{{\mathcal{G}}}}\\ &{}&{\quad\quad\quad\quad\left[g_{m}\big]_{Z(\mathbf{H}_{\overline{{A}}})})=\left[\left(h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})\right)_{x\in M}\right]_{\mathbf{B}(\overline{{A}})}.}\end{array}
$$
i.e.,
$-\mathrm{~\textit~{~F'~}~}^{\prime\prime\prime}$ is well-defined.
Let $g_{2,m}=z g_{1,m}$ with $z\in Z(\mathbf{H}_{\overline{{A}}})$ . Then
$$
\begin{array}{r c l}{F^{\prime\prime\prime}(\overline{{A}}^{\prime\prime},[g_{2,m}]_{Z({\bf H}_{\overline{{A}}})})}&{=}&{\left[\left(h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{2,m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})\right)_{x\in M}\right]_{{\bf B}(\overline{{A}})}}\\ &{=}&{\left[\left(h_{\gamma_{x}}(\overline{{A}})^{-1}\,z\,g_{1,m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})\right)_{x\in M}\right]_{{\bf B}(\overline{{A}})}}\\ &{=}&{\left[\left(z_{x}\,h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{1,m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})\right)_{x\in M}\right]_{{\bf B}(\overline{{A}})}}\\ &{=}&{F^{\prime\prime\prime}(\overline{{A}}^{\prime\prime},[g_{1,m}]_{Z({\bf H}_{\overline{{A}}})}),}\end{array}
$$
because $(z_{x})_{x\in{\cal M}}:=(h_{\gamma_{x}}(\overline{{{\cal A}}})^{-1}\,z\,h_{\gamma_{x}}(\overline{{{\cal A}}}))_{x\in{\cal M}}\in{\bf B}(\overline{{{\cal A}}})$ for $z\in Z(\mathbf{H}_{\overline{{A}}})$ .
$F^{\prime\prime\prime}$ is continuous, because $\mathrm{id}\times\pi_{Z(\mathbf{H}_{\overline{{A}}})}$ is open and surjective and $\pi_{\mathbf{B}(\overline{{A}})}$ and $F^{\prime\prime}$ are continuous.
For $\overline{{A}}^{\prime}\in\overline{{S}}$ there is an $\overline{{A}}_{0}^{\prime}\in\overline{{S}}_{0}$ and a $\overline{{g}}^{\prime}\in{\bf B}(\overline{{A}})$ with $\overline{{{A}}}^{\prime}=\overline{{{A}}}_{0}^{\prime}\circ\overline{{{g}}}^{\prime}$ . Thus, we have $h_{\gamma_{x}}(\overline{{A}}_{0}^{\prime})=h_{\gamma_{x}}(\overline{{A}})$ and
$$
\begin{array}{r c l}{{F^{\prime\prime\prime}(\overline{{{A}}}^{\prime}\circ\overline{{{g}}},[g_{m}])}}&{{=}}&{{\left[\left(h_{\gamma_{x}}(\overline{{{A}}})^{-1}\,g_{m}\;h_{\gamma_{x}}(\overline{{{A}}}_{0}^{\prime}\circ\overline{{{g}}}^{\prime}\circ\overline{{{g}}})\right)_{x\in M}\right]_{\mathbf{B}(\overline{{{A}}})}}}\\ {{}}&{{=}}&{{\left[\left(h_{\gamma_{x}}(\overline{{{A}}})^{-1}\,g_{m}\;g_{m}^{-1}(g_{m}^{\prime})^{-1}h_{\gamma_{x}}(\overline{{{A}}})g_{x}^{\prime}g_{x}\right)_{x\in M}\right]_{\mathbf{B}(\overline{{{A}}})}}}\\ {{}}&{{=}}&{{\left[\left(h_{\gamma_{x}}(\overline{{{A}}})^{-1}h_{\gamma_{x}}(\overline{{{A}}}\circ g^{\prime})\;g_{x}\right)_{x\in M}\right]_{\mathbf{B}(\overline{{{A}}})}}}\\ {{}}&{{=}}&{{\left[\left(g_{x}\right)_{x\in M}\right]_{\mathbf{B}(\overline{{{A}}})}}}\\ {{}}&{{=}}&{{[\overline{{g}}]_{\mathbf{B}(\overline{{{A}}})}}}\end{array}
$$
where we used $\overline{{g}}^{\prime}\in{\bf B}(\overline{{A}})$ .
Now, $F$ is the concatenation of the following continuous maps:
$$
\begin{array}{r l r}{F:\,\,\overline{{S}}\circ\overline{{\mathcal{G}}}}&{\xrightarrow{\mathrm{id}\times F^{\prime}}}&{\big(\overline{{S}}\circ\overline{{\mathcal{G}}}\big)\times Z(\mathbf{H}_{\overline{{A}}})\big\backslash\,\mathbf{G}}&{\xrightarrow{F^{\prime\prime\prime}}}&{\mathbf{B}(\overline{{A}})\setminus\overline{{\mathcal{G}}}\,\,\xrightarrow{\tau_{\overline{{\mathcal{G}}}}}}&{\overline{{A}}\circ\overline{{\mathcal{G}}},}\\ {\overline{{A}}^{\prime}\circ\overline{{g}}}&{\longmapsto}&{\ \ (\overline{{A}}^{\prime}\circ\overline{{g}},[g_{m}]_{Z(\mathbf{H}_{\overline{{A}}})})}&{\longmapsto}&{\ \ [\overline{{g}}]_{\mathbf{B}(\overline{{A}})}}&{\longmapsto}&{\overline{{A}}\circ\overline{{g}}}\end{array}
$$
where $\tau_{\overline{{{\mathcal G}}}}$ is the canonical homeomorphism between the orbit $\overline{{A}}\circ\overline{{\mathcal{G}}}$ and the
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<html><body>
<p data-bbox="147 15 371 29">Since $\varphi$ , $f$ and $\tau_{\mathbf G}$ are continuous, the map </p>
<div class="equation" data-bbox="199 29 427 66">$$
F^{\prime}:=\tau_{\mathbf{G}}\circ\varphi\circ F:\;\;\overline{{S}}\circ\overline{{\mathcal{G}}}\enspace\longrightarrow\enspace Z(\mathbf{H}_{\overline{{A}}})\backslash\mathbf{G}
$$</div>
<p data-bbox="147 61 219 74">is continuous. </p>
<p data-bbox="132 75 286 88">Now, we consider the map </p>
<div class="equation" data-bbox="190 90 477 127">$$
\begin{array}{c c c c}{F^{\prime\prime}:}&{(\overline{{S}}\circ\overline{{\mathcal{G}}})\times\mathbf{G}}&{\longrightarrow}&{\overline{{\mathcal{G}}}.}\\ &{(\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime},g_{m})}&{\longmapsto}&{\bigl(h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime}\circ\overline{{g}}^{\prime})\bigr)_{x\in M}}\end{array}
$$</div>
<p data-bbox="149 123 279 136">$F^{\prime\prime}$ is continuous because </p>
<div class="equation" data-bbox="153 136 527 170">$$
\begin{array}{r l}{\pi_{x}\circ F^{\prime\prime}:}&{(\overline{{S}}\circ\overline{{\mathcal{G}}})\times\mathbf{G}\quad\longrightarrow\quad\mathbf{G}\times\mathbf{G}\quad\xrightarrow{\mathrm{mult}_{*}}\quad\xrightarrow{\mathbf{G}}}\\ &{\quad(\overline{{A}}^{\prime\prime},g_{m})\quad\longmapsto\quad(h_{\gamma_{x}}(\overline{{A}}^{\prime\prime}),g_{m})\quad\longmapsto\quad h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})}\end{array}
$$</div>
<p data-bbox="147 168 343 180">is obviously continuous for all $x\in M$ . </p>
<p data-bbox="131 181 468 195">$F^{\prime\prime}$ induces a map $F^{\prime\prime\prime}$ via the following commutative diagram </p>
<div class="equation" data-bbox="213 199 447 298">$$
\begin{array}{r l r}&{}&{(\overline{{S}}\circ\overline{{\mathcal{G}}})\times{\mathbf{G}}\xrightarrow{F^{\prime\prime}}\quad\xrightarrow{\longrightarrow}\overline{{\mathcal{G}}}}\\ &{}&{\quad\quad\quad\mathrm{id}\times\pi_{Z(\mathbf{H}_{\overline{{A}}})}\,,}\\ &{}&{\quad\quad\quad\quad(\overline{{S}}\circ\overline{{\mathcal{G}}})\times Z(\mathbf{H}_{\overline{{A}}})\setminus{\mathbf{G}}\xrightarrow{F^{\prime\prime\prime}}{\substack{\mathbf{B}}}(\overline{{A}})\setminus\overline{{\mathcal{G}}}}\\ &{}&{\quad\quad\quad\quad\left[g_{m}\big]_{Z(\mathbf{H}_{\overline{{A}}})})=\left[\left(h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})\right)_{x\in M}\right]_{\mathbf{B}(\overline{{A}})}.}\end{array}
$$</div>
<p data-bbox="146 274 448 295">i.e., </p>
<p data-bbox="148 294 265 308">$-\mathrm{~\textit~{~F'~}~}^{\prime\prime\prime}$ is well-defined. </p>
<p data-bbox="167 309 376 324">Let $g_{2,m}=z g_{1,m}$ with $z\in Z(\mathbf{H}_{\overline{{A}}})$ . Then </p>
<div class="equation" data-bbox="195 329 509 410">$$
\begin{array}{r c l}{F^{\prime\prime\prime}(\overline{{A}}^{\prime\prime},[g_{2,m}]_{Z({\bf H}_{\overline{{A}}})})}&{=}&{\left[\left(h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{2,m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})\right)_{x\in M}\right]_{{\bf B}(\overline{{A}})}}\\ &{=}&{\left[\left(h_{\gamma_{x}}(\overline{{A}})^{-1}\,z\,g_{1,m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})\right)_{x\in M}\right]_{{\bf B}(\overline{{A}})}}\\ &{=}&{\left[\left(z_{x}\,h_{\gamma_{x}}(\overline{{A}})^{-1}\,g_{1,m}\,h_{\gamma_{x}}(\overline{{A}}^{\prime\prime})\right)_{x\in M}\right]_{{\bf B}(\overline{{A}})}}\\ &{=}&{F^{\prime\prime\prime}(\overline{{A}}^{\prime\prime},[g_{1,m}]_{Z({\bf H}_{\overline{{A}}})}),}\end{array}
$$</div>
<p data-bbox="167 411 514 428">because $(z_{x})_{x\in{\cal M}}:=(h_{\gamma_{x}}(\overline{{{\cal A}}})^{-1}\,z\,h_{\gamma_{x}}(\overline{{{\cal A}}}))_{x\in{\cal M}}\in{\bf B}(\overline{{{\cal A}}})$ for $z\in Z(\mathbf{H}_{\overline{{A}}})$ . </p>
<p data-bbox="149 429 538 455">$F^{\prime\prime\prime}$ is continuous, because $\mathrm{id}\times\pi_{Z(\mathbf{H}_{\overline{{A}}})}$ is open and surjective and $\pi_{\mathbf{B}(\overline{{A}})}$ and $F^{\prime\prime}$ are continuous. </p>
<p data-bbox="132 456 538 486">For $\overline{{A}}^{\prime}\in\overline{{S}}$ there is an $\overline{{A}}_{0}^{\prime}\in\overline{{S}}_{0}$ and a $\overline{{g}}^{\prime}\in{\bf B}(\overline{{A}})$ with $\overline{{{A}}}^{\prime}=\overline{{{A}}}_{0}^{\prime}\circ\overline{{{g}}}^{\prime}$ . Thus, we have $h_{\gamma_{x}}(\overline{{A}}_{0}^{\prime})=h_{\gamma_{x}}(\overline{{A}})$ and </p>
<div class="equation" data-bbox="169 489 516 591">$$
\begin{array}{r c l}{{F^{\prime\prime\prime}(\overline{{{A}}}^{\prime}\circ\overline{{{g}}},[g_{m}])}}&{{=}}&{{\left[\left(h_{\gamma_{x}}(\overline{{{A}}})^{-1}\,g_{m}\;h_{\gamma_{x}}(\overline{{{A}}}_{0}^{\prime}\circ\overline{{{g}}}^{\prime}\circ\overline{{{g}}})\right)_{x\in M}\right]_{\mathbf{B}(\overline{{{A}}})}}}\\ {{}}&{{=}}&{{\left[\left(h_{\gamma_{x}}(\overline{{{A}}})^{-1}\,g_{m}\;g_{m}^{-1}(g_{m}^{\prime})^{-1}h_{\gamma_{x}}(\overline{{{A}}})g_{x}^{\prime}g_{x}\right)_{x\in M}\right]_{\mathbf{B}(\overline{{{A}}})}}}\\ {{}}&{{=}}&{{\left[\left(h_{\gamma_{x}}(\overline{{{A}}})^{-1}h_{\gamma_{x}}(\overline{{{A}}}\circ g^{\prime})\;g_{x}\right)_{x\in M}\right]_{\mathbf{B}(\overline{{{A}}})}}}\\ {{}}&{{=}}&{{\left[\left(g_{x}\right)_{x\in M}\right]_{\mathbf{B}(\overline{{{A}}})}}}\\ {{}}&{{=}}&{{[\overline{{g}}]_{\mathbf{B}(\overline{{{A}}})}}}\end{array}
$$</div>
<p data-bbox="145 593 281 608">where we used $\overline{{g}}^{\prime}\in{\bf B}(\overline{{A}})$ . </p>
<p data-bbox="131 608 471 622">Now, $F$ is the concatenation of the following continuous maps: </p>
<div class="equation" data-bbox="153 623 524 661">$$
\begin{array}{r l r}{F:\,\,\overline{{S}}\circ\overline{{\mathcal{G}}}}&{\xrightarrow{\mathrm{id}\times F^{\prime}}}&{\big(\overline{{S}}\circ\overline{{\mathcal{G}}}\big)\times Z(\mathbf{H}_{\overline{{A}}})\big\backslash\,\mathbf{G}}&{\xrightarrow{F^{\prime\prime\prime}}}&{\mathbf{B}(\overline{{A}})\setminus\overline{{\mathcal{G}}}\,\,\xrightarrow{\tau_{\overline{{\mathcal{G}}}}}}&{\overline{{A}}\circ\overline{{\mathcal{G}}},}\\ {\overline{{A}}^{\prime}\circ\overline{{g}}}&{\longmapsto}&{\ \ (\overline{{A}}^{\prime}\circ\overline{{g}},[g_{m}]_{Z(\mathbf{H}_{\overline{{A}}})})}&{\longmapsto}&{\ \ [\overline{{g}}]_{\mathbf{B}(\overline{{A}})}}&{\longmapsto}&{\overline{{A}}\circ\overline{{g}}}\end{array}
$$</div>
<p data-bbox="148 660 539 674">where $\tau_{\overline{{{\mathcal G}}}}$ is the canonical homeomorphism between the orbit $\overline{{A}}\circ\overline{{\mathcal{G}}}$ and the </p>
</body></html>
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0001008v1
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[{"type": "text", "text": "Since $\\varphi$ , $f$ and $\\tau_{\\mathbf G}$ are continuous, the map ", "page_idx": 9}, {"type": "equation", "text": "$$\nF^{\\prime}:=\\tau_{\\mathbf{G}}\\circ\\varphi\\circ F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace Z(\\mathbf{H}_{\\overline{{A}}})\\backslash\\mathbf{G}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "is continuous. ", "page_idx": 9}, {"type": "text", "text": "Now, we consider the map ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{c c c c}{F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}}&{\\longrightarrow}&{\\overline{{\\mathcal{G}}}.}\\\\ &{(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime},g_{m})}&{\\longmapsto}&{\\bigl(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime})\\bigr)_{x\\in M}}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "$F^{\\prime\\prime}$ is continuous because ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r l}{\\pi_{x}\\circ F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}\\quad\\longrightarrow\\quad\\mathbf{G}\\times\\mathbf{G}\\quad\\xrightarrow{\\mathrm{mult}_{*}}\\quad\\xrightarrow{\\mathbf{G}}}\\\\ &{\\quad(\\overline{{A}}^{\\prime\\prime},g_{m})\\quad\\longmapsto\\quad(h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime}),g_{m})\\quad\\longmapsto\\quad h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "is obviously continuous for all $x\\in M$ . ", "page_idx": 9}, {"type": "text", "text": "$F^{\\prime\\prime}$ induces a map $F^{\\prime\\prime\\prime}$ via the following commutative diagram ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}&{}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime}}\\quad\\xrightarrow{\\longrightarrow}\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}\\,,}\\\\ &{}&{\\quad\\quad\\quad\\quad(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times Z(\\mathbf{H}_{\\overline{{A}}})\\setminus{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime\\prime}}{\\substack{\\mathbf{B}}}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\quad\\left[g_{m}\\big]_{Z(\\mathbf{H}_{\\overline{{A}}})})=\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{A}})}.}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "i.e., ", "page_idx": 9}, {"type": "text", "text": "$-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}$ is well-defined. ", "page_idx": 9}, {"type": "text", "text": "Let $g_{2,m}=z g_{1,m}$ with $z\\in Z(\\mathbf{H}_{\\overline{{A}}})$ . Then ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{2,m}]_{Z({\\bf H}_{\\overline{{A}}})})}&{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{2,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,z\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(z_{x}\\,h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{1,m}]_{Z({\\bf H}_{\\overline{{A}}})}),}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "because $(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})$ for $z\\in Z(\\mathbf{H}_{\\overline{{A}}})$ . ", "page_idx": 9}, {"type": "text", "text": "$F^{\\prime\\prime\\prime}$ is continuous, because $\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}$ is open and surjective and $\\pi_{\\mathbf{B}(\\overline{{A}})}$ and $F^{\\prime\\prime}$ are continuous. ", "page_idx": 9}, {"type": "text", "text": "For $\\overline{{A}}^{\\prime}\\in\\overline{{S}}$ there is an $\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}$ and a $\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})$ with $\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}$ . Thus, we have $h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})$ and ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{{F^{\\prime\\prime\\prime}(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}},[g_{m}])}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;h_{\\gamma_{x}}(\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;g_{m}^{-1}(g_{m}^{\\prime})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}})g_{x}^{\\prime}g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}}\\circ g^{\\prime})\\;g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{[\\overline{{g}}]_{\\mathbf{B}(\\overline{{{A}}})}}}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "where we used $\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})$ . ", "page_idx": 9}, {"type": "text", "text": "Now, $F$ is the concatenation of the following continuous maps: ", "page_idx": 9}, {"type": "equation", "text": "$$\n\\begin{array}{r l r}{F:\\,\\,\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}}&{\\xrightarrow{\\mathrm{id}\\times F^{\\prime}}}&{\\big(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\big)\\times Z(\\mathbf{H}_{\\overline{{A}}})\\big\\backslash\\,\\mathbf{G}}&{\\xrightarrow{F^{\\prime\\prime\\prime}}}&{\\mathbf{B}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}\\,\\,\\xrightarrow{\\tau_{\\overline{{\\mathcal{G}}}}}}&{\\overline{{A}}\\circ\\overline{{\\mathcal{G}}},}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}}&{\\longmapsto}&{\\ \\ (\\overline{{A}}^{\\prime}\\circ\\overline{{g}},[g_{m}]_{Z(\\mathbf{H}_{\\overline{{A}}})})}&{\\longmapsto}&{\\ \\ [\\overline{{g}}]_{\\mathbf{B}(\\overline{{A}})}}&{\\longmapsto}&{\\overline{{A}}\\circ\\overline{{g}}}\\end{array}\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "where $\\tau_{\\overline{{{\\mathcal G}}}}$ is the canonical homeomorphism between the orbit $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ and the ", "page_idx": 9}]
|
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M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{[\\overline{{g}}]_{\\mathbf{B}(\\overline{{{A}}})}}}\\end{array}"}, {"category_id": 13, "poly": [785, 869, 942, 869, 942, 904, 785, 904], "score": 0.94, "latex": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})"}, {"category_id": 13, "poly": [489, 1311, 737, 1311, 737, 1354, 489, 1354], "score": 0.94, "latex": "h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})"}, {"category_id": 13, "poly": [630, 1656, 773, 1656, 773, 1694, 630, 1694], "score": 0.94, "latex": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})"}, {"category_id": 13, "poly": [1263, 1157, 1420, 1157, 1420, 1192, 1263, 1192], "score": 0.94, "latex": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})"}, {"category_id": 13, "poly": [735, 1271, 849, 1271, 849, 1312, 735, 1312], "score": 0.94, "latex": 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obviously continuous for all ", "type": "text"}, {"bbox": [305, 171, 339, 180], "score": 0.86, "content": "x\\in M", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [339, 169, 342, 181], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 7, "bbox_fs": [147, 169, 342, 181]}, {"type": "text", "bbox": [131, 181, 468, 195], "lines": [{"bbox": [149, 182, 466, 198], "spans": [{"bbox": [149, 185, 163, 194], "score": 0.91, "content": "F^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [164, 182, 244, 198], "score": 1.0, "content": " induces a map ", "type": "text"}, {"bbox": [244, 185, 261, 194], "score": 0.91, "content": "F^{\\prime\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [262, 182, 466, 198], "score": 1.0, "content": " via the following commutative diagram", "type": "text"}], "index": 8}], "index": 8, "bbox_fs": [149, 182, 466, 198]}, {"type": "interline_equation", "bbox": [213, 199, 447, 298], "lines": [{"bbox": [213, 199, 447, 298], "spans": [{"bbox": [213, 199, 447, 298], "score": 0.54, "content": "\\begin{array}{r l r}&{}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime}}\\quad\\xrightarrow{\\longrightarrow}\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}\\,,}\\\\ &{}&{\\quad\\quad\\quad\\quad(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times Z(\\mathbf{H}_{\\overline{{A}}})\\setminus{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime\\prime}}{\\substack{\\mathbf{B}}}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\quad\\left[g_{m}\\big]_{Z(\\mathbf{H}_{\\overline{{A}}})})=\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{A}})}.}\\end{array}", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [146, 274, 448, 295], "lines": [{"bbox": [146, 275, 171, 297], "spans": [{"bbox": [146, 275, 171, 297], "score": 1.0, "content": "i.e., ", "type": "text"}], "index": 10}], "index": 10, "bbox_fs": [146, 275, 171, 297]}, {"type": "text", "bbox": [148, 294, 265, 308], "lines": [{"bbox": [147, 295, 264, 310], "spans": [{"bbox": [147, 296, 186, 309], "score": 0.53, "content": "-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 39}, {"bbox": [186, 295, 264, 310], "score": 1.0, "content": " is well-defined.", "type": "text"}], "index": 11}], "index": 11, "bbox_fs": [147, 295, 264, 310]}, {"type": "text", "bbox": [167, 309, 376, 324], "lines": [{"bbox": [167, 311, 374, 327], "spans": [{"bbox": [167, 311, 189, 327], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [190, 315, 252, 325], "score": 0.87, "content": "g_{2,m}=z g_{1,m}", "type": "inline_equation", "height": 10, "width": 62}, {"bbox": [252, 311, 282, 327], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [282, 312, 339, 325], "score": 0.94, "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [339, 311, 374, 327], "score": 1.0, "content": ". Then", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [167, 311, 374, 327]}, {"type": "interline_equation", "bbox": [195, 329, 509, 410], "lines": [{"bbox": [195, 329, 509, 410], "spans": [{"bbox": [195, 329, 509, 410], "score": 0.95, "content": "\\begin{array}{r c l}{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{2,m}]_{Z({\\bf H}_{\\overline{{A}}})})}&{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{2,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,z\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(z_{x}\\,h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{1,m}]_{Z({\\bf H}_{\\overline{{A}}})}),}\\end{array}", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [167, 411, 514, 428], "lines": [{"bbox": [168, 414, 513, 430], "spans": [{"bbox": [168, 414, 211, 430], "score": 1.0, "content": "because ", "type": "text"}, {"bbox": [212, 415, 433, 429], "score": 0.89, "content": "(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})", "type": "inline_equation", "height": 14, "width": 221}, {"bbox": [433, 414, 454, 430], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [454, 416, 511, 429], "score": 0.94, "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [511, 414, 513, 430], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [168, 414, 513, 430]}, {"type": "text", "bbox": [149, 429, 538, 455], "lines": [{"bbox": [169, 426, 536, 448], "spans": [{"bbox": [169, 431, 186, 440], "score": 0.85, "content": "F^{\\prime\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 17}, {"bbox": [186, 426, 308, 448], "score": 1.0, "content": " is continuous, because ", "type": "text"}, {"bbox": [308, 432, 367, 445], "score": 0.89, "content": "\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [367, 426, 509, 448], "score": 1.0, "content": " is open and surjective and ", "type": "text"}, {"bbox": [509, 434, 536, 445], "score": 0.92, "content": "\\pi_{\\mathbf{B}(\\overline{{A}})}", "type": "inline_equation", "height": 11, "width": 27}], "index": 15}, {"bbox": [168, 443, 288, 458], "spans": [{"bbox": [168, 443, 191, 458], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [191, 445, 206, 454], "score": 0.91, "content": "F^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [207, 443, 288, 458], "score": 1.0, "content": " are continuous.", "type": "text"}], "index": 16}], "index": 15.5, "bbox_fs": [168, 426, 536, 458]}, {"type": "text", "bbox": [132, 456, 538, 486], "lines": [{"bbox": [137, 455, 536, 473], "spans": [{"bbox": [137, 455, 169, 473], "score": 1.0, "content": "For ", "type": "text"}, {"bbox": [169, 457, 204, 469], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [204, 455, 264, 473], "score": 1.0, "content": " there is an ", "type": "text"}, {"bbox": [264, 457, 305, 472], "score": 0.94, "content": "\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 15, "width": 41}, {"bbox": [306, 455, 340, 473], "score": 1.0, "content": " and a ", "type": "text"}, {"bbox": [340, 458, 392, 472], "score": 0.93, "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 52}, {"bbox": [392, 455, 421, 473], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [421, 457, 482, 472], "score": 0.95, "content": "\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}", "type": "inline_equation", "height": 15, "width": 61}, {"bbox": [482, 455, 536, 473], "score": 1.0, "content": ". Thus, we", "type": "text"}], "index": 17}, {"bbox": [148, 471, 289, 488], "spans": [{"bbox": [148, 471, 175, 488], "score": 1.0, "content": "have ", "type": "text"}, {"bbox": [176, 471, 265, 487], "score": 0.94, "content": "h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})", "type": "inline_equation", "height": 16, "width": 89}, {"bbox": [265, 471, 289, 488], "score": 1.0, "content": " and", "type": "text"}], "index": 18}], "index": 17.5, "bbox_fs": [137, 455, 536, 488]}, {"type": "interline_equation", "bbox": [169, 489, 516, 591], "lines": [{"bbox": [169, 489, 516, 591], "spans": [{"bbox": [169, 489, 516, 591], "score": 0.95, "content": "\\begin{array}{r c l}{{F^{\\prime\\prime\\prime}(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}},[g_{m}])}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;h_{\\gamma_{x}}(\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;g_{m}^{-1}(g_{m}^{\\prime})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}})g_{x}^{\\prime}g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}}\\circ g^{\\prime})\\;g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{[\\overline{{g}}]_{\\mathbf{B}(\\overline{{{A}}})}}}\\end{array}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [145, 593, 281, 608], "lines": [{"bbox": [149, 595, 281, 609], "spans": [{"bbox": [149, 595, 226, 609], "score": 1.0, "content": "where we used ", "type": "text"}, {"bbox": [226, 596, 278, 609], "score": 0.94, "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 52}, {"bbox": [278, 595, 281, 609], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 20, "bbox_fs": [149, 595, 281, 609]}, {"type": "text", "bbox": [131, 608, 471, 622], "lines": [{"bbox": [138, 610, 470, 624], "spans": [{"bbox": [138, 610, 178, 624], "score": 1.0, "content": "Now, ", "type": "text"}, {"bbox": [179, 612, 188, 621], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [188, 610, 470, 624], "score": 1.0, "content": " is the concatenation of the following continuous maps:", "type": "text"}], "index": 21}], "index": 21, "bbox_fs": [138, 610, 470, 624]}, {"type": "interline_equation", "bbox": [153, 623, 524, 661], "lines": [{"bbox": [153, 623, 524, 661], "spans": [{"bbox": [153, 623, 524, 661], "score": 0.92, "content": "\\begin{array}{r l r}{F:\\,\\,\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}}&{\\xrightarrow{\\mathrm{id}\\times F^{\\prime}}}&{\\big(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\big)\\times Z(\\mathbf{H}_{\\overline{{A}}})\\big\\backslash\\,\\mathbf{G}}&{\\xrightarrow{F^{\\prime\\prime\\prime}}}&{\\mathbf{B}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}\\,\\,\\xrightarrow{\\tau_{\\overline{{\\mathcal{G}}}}}}&{\\overline{{A}}\\circ\\overline{{\\mathcal{G}}},}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}}&{\\longmapsto}&{\\ \\ (\\overline{{A}}^{\\prime}\\circ\\overline{{g}},[g_{m}]_{Z(\\mathbf{H}_{\\overline{{A}}})})}&{\\longmapsto}&{\\ \\ [\\overline{{g}}]_{\\mathbf{B}(\\overline{{A}})}}&{\\longmapsto}&{\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [148, 660, 539, 674], "lines": [{"bbox": [149, 661, 536, 676], "spans": [{"bbox": [149, 661, 182, 674], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [182, 666, 194, 676], "score": 0.91, "content": "\\tau_{\\overline{{{\\mathcal G}}}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [195, 661, 464, 674], "score": 1.0, "content": " is the canonical homeomorphism between the orbit ", "type": "text"}, {"bbox": [464, 661, 493, 673], "score": 0.92, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [493, 661, 536, 674], "score": 1.0, "content": " and the", "type": "text"}], "index": 23}], "index": 23, "bbox_fs": [149, 661, 536, 676]}]}
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[{"type": "text", "bbox": [147, 15, 371, 29], "content": "Since , and are continuous, the map", "index": 0}, {"type": "interline_equation", "bbox": [199, 29, 427, 66], "content": "", "index": 1}, {"type": "text", "bbox": [147, 61, 219, 74], "content": "is continuous.", "index": 2}, {"type": "text", "bbox": [132, 75, 286, 88], "content": "Now, we consider the map", "index": 3}, {"type": "interline_equation", "bbox": [190, 90, 477, 127], "content": "", "index": 4}, {"type": "text", "bbox": [149, 123, 279, 136], "content": "is continuous because", "index": 5}, {"type": "interline_equation", "bbox": [153, 136, 527, 170], "content": "", "index": 6}, {"type": "text", "bbox": [147, 168, 343, 180], "content": "is obviously continuous for all .", "index": 7}, {"type": "text", "bbox": [131, 181, 468, 195], "content": "induces a map via the following commutative diagram", "index": 8}, {"type": "interline_equation", "bbox": [213, 199, 447, 298], "content": "", "index": 9}, {"type": "text", "bbox": [146, 274, 448, 295], "content": "i.e.,", "index": 10}, {"type": "text", "bbox": [148, 294, 265, 308], "content": "is well-defined.", "index": 11}, {"type": "text", "bbox": [167, 309, 376, 324], "content": "Let with . Then", "index": 12}, {"type": "interline_equation", "bbox": [195, 329, 509, 410], "content": "", "index": 13}, {"type": "text", "bbox": [167, 411, 514, 428], "content": "because for .", "index": 14}, {"type": "text", "bbox": [149, 429, 538, 455], "content": "is continuous, because is open and surjective and and are continuous.", "index": 15}, {"type": "text", "bbox": [132, 456, 538, 486], "content": "For there is an and a with . Thus, we have and", "index": 16}, {"type": "interline_equation", "bbox": [169, 489, 516, 591], "content": "", "index": 17}, {"type": "text", "bbox": [145, 593, 281, 608], "content": "where we used .", "index": 18}, {"type": "text", "bbox": [131, 608, 471, 622], "content": "Now, is the concatenation of the following continuous maps:", "index": 19}, {"type": "interline_equation", "bbox": [153, 623, 524, 661], "content": "", "index": 20}, {"type": "text", "bbox": [148, 660, 539, 674], "content": "where is the canonical homeomorphism between the orbit and the", "index": 21}]
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[{"bbox": [149, 17, 369, 31], "content": "Since , and are continuous, the map", "parent_index": 0, "line_index": 0}, {"bbox": [147, 64, 218, 75], "content": "is continuous.", "parent_index": 2, "line_index": 0}, {"bbox": [148, 77, 285, 90], "content": "Now, we consider the map", "parent_index": 3, "line_index": 0}, {"bbox": [149, 124, 279, 138], "content": "is continuous because", "parent_index": 5, "line_index": 0}, {"bbox": [147, 169, 342, 181], "content": "is obviously continuous for all .", "parent_index": 7, "line_index": 0}, {"bbox": [149, 182, 466, 198], "content": "induces a map via the following commutative diagram", "parent_index": 8, "line_index": 0}, {"bbox": [146, 275, 171, 297], "content": "i.e.,", "parent_index": 10, "line_index": 0}, {"bbox": [147, 295, 264, 310], "content": "is well-defined.", "parent_index": 11, "line_index": 0}, {"bbox": [167, 311, 374, 327], "content": "Let with . Then", "parent_index": 12, "line_index": 0}, {"bbox": [168, 414, 513, 430], "content": "because for .", "parent_index": 14, "line_index": 0}, {"bbox": [169, 426, 536, 448], "content": "is continuous, because is open and surjective and", "parent_index": 15, "line_index": 0}, {"bbox": [168, 443, 288, 458], "content": "and are continuous.", "parent_index": 15, "line_index": 1}, {"bbox": [137, 455, 536, 473], "content": "For there is an and a with . Thus, we", "parent_index": 16, "line_index": 0}, {"bbox": [148, 471, 289, 488], "content": "have and", "parent_index": 16, "line_index": 1}, {"bbox": [149, 595, 281, 609], "content": "where we used .", "parent_index": 18, "line_index": 0}, {"bbox": [138, 610, 470, 624], "content": "Now, is the concatenation of the following continuous maps:", "parent_index": 19, "line_index": 0}, {"bbox": [149, 661, 536, 676], "content": "where is the canonical homeomorphism between the orbit and the", "parent_index": 21, "line_index": 0}]
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[]
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[{"bbox": [179, 22, 187, 30], "content": "\\varphi", "parent_index": 0, "subtype": "inline"}, {"bbox": [194, 19, 201, 30], "content": "f", "parent_index": 0, "subtype": "inline"}, {"bbox": [228, 22, 241, 29], "content": "\\tau_{\\mathbf G}", "parent_index": 0, "subtype": "inline"}, {"bbox": [199, 29, 427, 66], "content": "F^{\\prime}:=\\tau_{\\mathbf{G}}\\circ\\varphi\\circ F:\\;\\;\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\enspace\\longrightarrow\\enspace Z(\\mathbf{H}_{\\overline{{A}}})\\backslash\\mathbf{G}", "parent_index": 1, "subtype": "interline"}, {"bbox": [190, 90, 477, 127], "content": "\\begin{array}{c c c c}{F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}}&{\\longrightarrow}&{\\overline{{\\mathcal{G}}}.}\\\\ &{(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime},g_{m})}&{\\longmapsto}&{\\bigl(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime})\\bigr)_{x\\in M}}\\end{array}", "parent_index": 4, "subtype": "interline"}, {"bbox": [149, 126, 163, 135], "content": "F^{\\prime\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [153, 136, 527, 170], "content": "\\begin{array}{r l}{\\pi_{x}\\circ F^{\\prime\\prime}:}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times\\mathbf{G}\\quad\\longrightarrow\\quad\\mathbf{G}\\times\\mathbf{G}\\quad\\xrightarrow{\\mathrm{mult}_{*}}\\quad\\xrightarrow{\\mathbf{G}}}\\\\ &{\\quad(\\overline{{A}}^{\\prime\\prime},g_{m})\\quad\\longmapsto\\quad(h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime}),g_{m})\\quad\\longmapsto\\quad h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})}\\end{array}", "parent_index": 6, "subtype": "interline"}, {"bbox": [305, 171, 339, 180], "content": "x\\in M", "parent_index": 7, "subtype": "inline"}, {"bbox": [149, 185, 163, 194], "content": "F^{\\prime\\prime}", "parent_index": 8, "subtype": "inline"}, {"bbox": [244, 185, 261, 194], "content": "F^{\\prime\\prime\\prime}", "parent_index": 8, "subtype": "inline"}, {"bbox": [213, 199, 447, 298], "content": "\\begin{array}{r l r}&{}&{(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime}}\\quad\\xrightarrow{\\longrightarrow}\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}\\,,}\\\\ &{}&{\\quad\\quad\\quad\\quad(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}})\\times Z(\\mathbf{H}_{\\overline{{A}}})\\setminus{\\mathbf{G}}\\xrightarrow{F^{\\prime\\prime\\prime}}{\\substack{\\mathbf{B}}}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}}\\\\ &{}&{\\quad\\quad\\quad\\quad\\left[g_{m}\\big]_{Z(\\mathbf{H}_{\\overline{{A}}})})=\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{A}})}.}\\end{array}", "parent_index": 9, "subtype": "interline"}, {"bbox": [147, 296, 186, 309], "content": "-\\mathrm{~\\textit~{~F'~}~}^{\\prime\\prime\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [190, 315, 252, 325], "content": "g_{2,m}=z g_{1,m}", "parent_index": 12, "subtype": "inline"}, {"bbox": [282, 312, 339, 325], "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 12, "subtype": "inline"}, {"bbox": [195, 329, 509, 410], "content": "\\begin{array}{r c l}{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{2,m}]_{Z({\\bf H}_{\\overline{{A}}})})}&{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{2,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,z\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{\\left[\\left(z_{x}\\,h_{\\gamma_{x}}(\\overline{{A}})^{-1}\\,g_{1,m}\\,h_{\\gamma_{x}}(\\overline{{A}}^{\\prime\\prime})\\right)_{x\\in M}\\right]_{{\\bf B}(\\overline{{A}})}}\\\\ &{=}&{F^{\\prime\\prime\\prime}(\\overline{{A}}^{\\prime\\prime},[g_{1,m}]_{Z({\\bf H}_{\\overline{{A}}})}),}\\end{array}", "parent_index": 13, "subtype": "interline"}, {"bbox": [212, 415, 433, 429], "content": "(z_{x})_{x\\in{\\cal M}}:=(h_{\\gamma_{x}}(\\overline{{{\\cal A}}})^{-1}\\,z\\,h_{\\gamma_{x}}(\\overline{{{\\cal A}}}))_{x\\in{\\cal M}}\\in{\\bf B}(\\overline{{{\\cal A}}})", "parent_index": 14, "subtype": "inline"}, {"bbox": [454, 416, 511, 429], "content": "z\\in Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 14, "subtype": "inline"}, {"bbox": [169, 431, 186, 440], "content": "F^{\\prime\\prime\\prime}", "parent_index": 15, "subtype": "inline"}, {"bbox": [308, 432, 367, 445], "content": "\\mathrm{id}\\times\\pi_{Z(\\mathbf{H}_{\\overline{{A}}})}", "parent_index": 15, "subtype": "inline"}, {"bbox": [509, 434, 536, 445], "content": "\\pi_{\\mathbf{B}(\\overline{{A}})}", "parent_index": 15, "subtype": "inline"}, {"bbox": [191, 445, 206, 454], "content": "F^{\\prime\\prime}", "parent_index": 15, "subtype": "inline"}, {"bbox": [169, 457, 204, 469], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "parent_index": 16, "subtype": "inline"}, {"bbox": [264, 457, 305, 472], "content": "\\overline{{A}}_{0}^{\\prime}\\in\\overline{{S}}_{0}", "parent_index": 16, "subtype": "inline"}, {"bbox": [340, 458, 392, 472], "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "parent_index": 16, "subtype": "inline"}, {"bbox": [421, 457, 482, 472], "content": "\\overline{{{A}}}^{\\prime}=\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}", "parent_index": 16, "subtype": "inline"}, {"bbox": [176, 471, 265, 487], "content": "h_{\\gamma_{x}}(\\overline{{A}}_{0}^{\\prime})=h_{\\gamma_{x}}(\\overline{{A}})", "parent_index": 16, "subtype": "inline"}, {"bbox": [169, 489, 516, 591], "content": "\\begin{array}{r c l}{{F^{\\prime\\prime\\prime}(\\overline{{{A}}}^{\\prime}\\circ\\overline{{{g}}},[g_{m}])}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;h_{\\gamma_{x}}(\\overline{{{A}}}_{0}^{\\prime}\\circ\\overline{{{g}}}^{\\prime}\\circ\\overline{{{g}}})\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}\\,g_{m}\\;g_{m}^{-1}(g_{m}^{\\prime})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}})g_{x}^{\\prime}g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(h_{\\gamma_{x}}(\\overline{{{A}}})^{-1}h_{\\gamma_{x}}(\\overline{{{A}}}\\circ g^{\\prime})\\;g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{\\left[\\left(g_{x}\\right)_{x\\in M}\\right]_{\\mathbf{B}(\\overline{{{A}}})}}}\\\\ {{}}&{{=}}&{{[\\overline{{g}}]_{\\mathbf{B}(\\overline{{{A}}})}}}\\end{array}", "parent_index": 17, "subtype": "interline"}, {"bbox": [226, 596, 278, 609], "content": "\\overline{{g}}^{\\prime}\\in{\\bf B}(\\overline{{A}})", "parent_index": 18, "subtype": "inline"}, {"bbox": [179, 612, 188, 621], "content": "F", "parent_index": 19, "subtype": "inline"}, {"bbox": [153, 623, 524, 661], "content": "\\begin{array}{r l r}{F:\\,\\,\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}}&{\\xrightarrow{\\mathrm{id}\\times F^{\\prime}}}&{\\big(\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}\\big)\\times Z(\\mathbf{H}_{\\overline{{A}}})\\big\\backslash\\,\\mathbf{G}}&{\\xrightarrow{F^{\\prime\\prime\\prime}}}&{\\mathbf{B}(\\overline{{A}})\\setminus\\overline{{\\mathcal{G}}}\\,\\,\\xrightarrow{\\tau_{\\overline{{\\mathcal{G}}}}}}&{\\overline{{A}}\\circ\\overline{{\\mathcal{G}}},}\\\\ {\\overline{{A}}^{\\prime}\\circ\\overline{{g}}}&{\\longmapsto}&{\\ \\ (\\overline{{A}}^{\\prime}\\circ\\overline{{g}},[g_{m}]_{Z(\\mathbf{H}_{\\overline{{A}}})})}&{\\longmapsto}&{\\ \\ [\\overline{{g}}]_{\\mathbf{B}(\\overline{{A}})}}&{\\longmapsto}&{\\overline{{A}}\\circ\\overline{{g}}}\\end{array}", "parent_index": 20, "subtype": "interline"}, {"bbox": [182, 666, 194, 676], "content": "\\tau_{\\overline{{{\\mathcal G}}}}", "parent_index": 21, "subtype": "inline"}, {"bbox": [464, 661, 493, 673], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 21, "subtype": "inline"}]
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acting group $\overline{{g}}$ modulo the stabilizer $\mathbf{B}(\overline{{A}})$ of $\overline{{A}}$ .
Hence, $F$ is continuous.
We have $F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$ . ”⊆” Let $\overline{{A}}^{\prime}\in F^{-1}(\{\overline{{A}}\})$ , i.e. $F(\overline{{A}}^{\prime})=\overline{{A}}$ . By the commutativity of (3) we have $f(\varphi(\overline{{{A}}}^{\prime}))\;=\;\varphi(F(\overline{{{A}}}^{\prime}))\;=$ $\varphi(\overline{{A}})$ , hence $\overline{{A}}^{\prime}\in\varphi^{-1}(f^{-1}(\varphi(\overline{{A}})))=\varphi_{..}^{-1}(S)$ . Define $g_{x}\,:=\,h_{\overline{{{A}}}^{\prime}}(\gamma_{x})^{-1}\,\,h_{\overline{{{A}}}}(\gamma_{x})$ and $\overline{{A}}^{\prime\prime}:=\overline{{A}}^{\prime}\circ\overline{{g}}$ . Then we have $\varphi(\overline{{A}}^{\prime\prime})=\varphi(\overline{{A}}^{\prime})\in S$ , i.e. $\overline{{A}}^{\prime\prime}\in\varphi^{-1}(S)$ , and $h_{\overline{{{A}}}^{\prime\prime}}(\gamma_{x})=h_{\overline{{{A}}}}(\gamma_{x})$ for all $x$ , i.e. ${\overline{{A}}}^{\prime\prime}\in\psi^{-1}(\psi({\overline{{A}}}))$ . By this, $\overline{{A}}^{\prime\prime}\in\overline{{S}}_{0}$ . Consequently, $F(\overline{{A}}^{\prime\prime})\:=\:\overline{{A}}\,=\,F(\overline{{A}}^{\prime})$ and therefore also $\overline{{{A}}}\circ\overline{{{g}}}\ =$ $F(\overline{{A}}_{.}^{\prime})\circ\overline{{g}}=F(\overline{{A}}^{\prime}\circ\overline{{g}})=F(\overline{{A}}^{\prime\prime})=\overline{{A}}$ , i.e. ${\overline{{g}}}\in{\mathbf{B}}({\overline{{A}}})$ . Thus, $\overline{{A}}^{\prime}=\overline{{A}}^{\prime\prime}\circ\overline{{g}}^{-1}\in\overline{{S}}_{0}\circ{\bf B}(\overline{{A}})=\overline{{S}}$ . ”⊇” Let $\overline{{A}}^{\prime}\in\overline{{S}}$ . Then $F(\overline{{A}}^{\prime})=F(\overline{{A}}^{\prime}\circ{1})=\overline{{A}}\circ{1}=\overline{{A}}$ , i.e. $\overline{{A}}^{\prime}\in F^{-1}(\{\overline{{A}}\})$ .
# 6 Openness of the Strata
Proposition 6.1 $\overline{{\mathcal{A}}}_{\geq t}$ is open for all $t\in\mathcal T$ .
Corollary 6.2 $\scriptstyle A_{=t}$ is open in $\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$ for all $t\in\mathcal T$ .
Proof Since $\overline{{\mathcal{A}}}_{=t}=\overline{{\mathcal{A}}}_{\geq t}\cap\overline{{\mathcal{A}}}_{\leq t}$ , $\overline{{\mathcal{A}}}_{=t}$ is open w.r.t. to the relative topology on $\overline{{\mathcal{A}}}_{\leq t}$ . qed Corollary 6.3 $\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$ is compact for all $t\in\mathcal T$ .
Proof $\begin{array}{r}{\overline{{\mathcal{A}}}\backslash\overline{{\mathcal{A}}}_{\leq t}=\bigcup_{t^{\prime}\in\mathcal{T},t^{\prime}\not\leq\;t}\overline{{\mathcal{A}}}_{=t^{\prime}}=\bigcup_{t^{\prime}\in\mathcal{T},t^{\prime}\not\leq\;t}\overline{{\mathcal{A}}}_{\geq t^{\prime}}}\end{array}$ is open because $\overline{{\mathcal{A}}}_{\geq t^{\prime}}$ is open for all $t^{\prime}\in\mathcal T$ . Thus, $\overline{{\mathcal{A}}}_{\leq t}$ is closed and the refore compact. qed
The proposition on the openness of the strata can be proven in two ways: first as a simple corollary of the slice theorem on $\overline{{\mathcal{A}}}$ , but second directly using the reduction mapping. Thus, altogether the second variant needs less effort.
# Proof Proposition 6.1
We have to show that any $\overline{{A}}\in\overline{{A}}_{\geq t}$ has a neighbourhood that again is contained in $\overline{{\mathcal{A}}}_{\geq t}$ . So, let $\overline{{A}}\in\overline{{A}}_{\geq t}$ .
• Variant 1 Due to the slice theorem there is an open neighbourhood $U$ of $\overline{{A}}\circ\overline{{\mathcal{G}}}$ , and so of $\overline{{A}}$ , too, and an equivariant retraction $F:U\longrightarrow\overline{{A}}\circ\overline{{\mathcal{G}}}$ . Since every equivariant mapping reduces types, we have $\mathrm{Typ}(\overline{{A}}^{\prime})\,\geq\,\mathrm{Typ}(\overline{{A}})\,=\,t$ for all $\overline{{A}}^{\prime}\,\in\,U$ , thus $U\subseteq{\overline{{A}}}_{\geq t}$ .
• Variant 2 Choose again for $\overline{{A}}$ an $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ with $\mathrm{Typ}(\overline{{A}})=[Z(\mathbf{H}_{\overline{{A}}})]=[Z(h_{\overline{{A}}}(\alpha))]\equiv[Z(\varphi_{\alpha}(\overline{{A}}))]=\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}))$ . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood $U^{\prime}$ of $\varphi_{\alpha}(\overline{{A}})$ in $\mathbf{G}^{\#\alpha}$ and an equivariant retraction $f:$ $U^{\prime}\longrightarrow\varphi_{\pmb{\alpha}}(\overline{{{A}}})\circ\mathbf{G}$ . Since $\varphi_{\alpha}(\overline{{A}})$ and $f$ are type-reducing, we have $\mathrm{Typ}(\overline{{A}}^{\prime})\geq\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}^{\prime}))\geq\mathrm{Typ}\big(f(\varphi_{\alpha}(\overline{{A}}^{\prime}))\big)=\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}))=\mathrm{Typ}(\overline{{A}})$ for all $\overline{{A}}^{\prime}\in U:=\varphi_{\pmb{\alpha}}^{-1}(U^{\prime})$ , i.e. $U\subseteq{\overline{{A}}}_{\geq t}$ . Obviously, $U$ contains $\overline{{A}}$ and is open as a preimage of an open set. qed
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<p data-bbox="144 13 399 42">acting group $\overline{{g}}$ modulo the stabilizer $\mathbf{B}(\overline{{A}})$ of $\overline{{A}}$ .
Hence, $F$ is continuous. </p>
<p data-bbox="118 44 539 214">We have $F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$ . ”⊆” Let $\overline{{A}}^{\prime}\in F^{-1}(\{\overline{{A}}\})$ , i.e. $F(\overline{{A}}^{\prime})=\overline{{A}}$ . By the commutativity of (3) we have $f(\varphi(\overline{{{A}}}^{\prime}))\;=\;\varphi(F(\overline{{{A}}}^{\prime}))\;=$ $\varphi(\overline{{A}})$ , hence $\overline{{A}}^{\prime}\in\varphi^{-1}(f^{-1}(\varphi(\overline{{A}})))=\varphi_{..}^{-1}(S)$ . Define $g_{x}\,:=\,h_{\overline{{{A}}}^{\prime}}(\gamma_{x})^{-1}\,\,h_{\overline{{{A}}}}(\gamma_{x})$ and $\overline{{A}}^{\prime\prime}:=\overline{{A}}^{\prime}\circ\overline{{g}}$ . Then we have $\varphi(\overline{{A}}^{\prime\prime})=\varphi(\overline{{A}}^{\prime})\in S$ , i.e. $\overline{{A}}^{\prime\prime}\in\varphi^{-1}(S)$ , and $h_{\overline{{{A}}}^{\prime\prime}}(\gamma_{x})=h_{\overline{{{A}}}}(\gamma_{x})$ for all $x$ , i.e. ${\overline{{A}}}^{\prime\prime}\in\psi^{-1}(\psi({\overline{{A}}}))$ . By this, $\overline{{A}}^{\prime\prime}\in\overline{{S}}_{0}$ . Consequently, $F(\overline{{A}}^{\prime\prime})\:=\:\overline{{A}}\,=\,F(\overline{{A}}^{\prime})$ and therefore also $\overline{{{A}}}\circ\overline{{{g}}}\ =$ $F(\overline{{A}}_{.}^{\prime})\circ\overline{{g}}=F(\overline{{A}}^{\prime}\circ\overline{{g}})=F(\overline{{A}}^{\prime\prime})=\overline{{A}}$ , i.e. ${\overline{{g}}}\in{\mathbf{B}}({\overline{{A}}})$ . Thus, $\overline{{A}}^{\prime}=\overline{{A}}^{\prime\prime}\circ\overline{{g}}^{-1}\in\overline{{S}}_{0}\circ{\bf B}(\overline{{A}})=\overline{{S}}$ . ”⊇” Let $\overline{{A}}^{\prime}\in\overline{{S}}$ . Then $F(\overline{{A}}^{\prime})=F(\overline{{A}}^{\prime}\circ{1})=\overline{{A}}\circ{1}=\overline{{A}}$ , i.e. $\overline{{A}}^{\prime}\in F^{-1}(\{\overline{{A}}\})$ . </p>
<h1 data-bbox="62 241 289 261">6 Openness of the Strata </h1>
<p data-bbox="63 271 294 287">Proposition 6.1 $\overline{{\mathcal{A}}}_{\geq t}$ is open for all $t\in\mathcal T$ . </p>
<p data-bbox="63 291 319 307">Corollary 6.2 $\scriptstyle A_{=t}$ is open in $\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$ for all $t\in\mathcal T$ . </p>
<p data-bbox="63 310 539 346">Proof Since $\overline{{\mathcal{A}}}_{=t}=\overline{{\mathcal{A}}}_{\geq t}\cap\overline{{\mathcal{A}}}_{\leq t}$ , $\overline{{\mathcal{A}}}_{=t}$ is open w.r.t. to the relative topology on $\overline{{\mathcal{A}}}_{\leq t}$ . qed Corollary 6.3 $\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$ is compact for all $t\in\mathcal T$ . </p>
<p data-bbox="63 349 538 381">Proof $\begin{array}{r}{\overline{{\mathcal{A}}}\backslash\overline{{\mathcal{A}}}_{\leq t}=\bigcup_{t^{\prime}\in\mathcal{T},t^{\prime}\not\leq\;t}\overline{{\mathcal{A}}}_{=t^{\prime}}=\bigcup_{t^{\prime}\in\mathcal{T},t^{\prime}\not\leq\;t}\overline{{\mathcal{A}}}_{\geq t^{\prime}}}\end{array}$ is open because $\overline{{\mathcal{A}}}_{\geq t^{\prime}}$ is open for all $t^{\prime}\in\mathcal T$ . Thus, $\overline{{\mathcal{A}}}_{\leq t}$ is closed and the refore compact. qed </p>
<p data-bbox="63 386 537 429">The proposition on the openness of the strata can be proven in two ways: first as a simple corollary of the slice theorem on $\overline{{\mathcal{A}}}$ , but second directly using the reduction mapping. Thus, altogether the second variant needs less effort. </p>
<h1 data-bbox="62 434 197 447">Proof Proposition 6.1 </h1>
<p data-bbox="104 448 537 477">We have to show that any $\overline{{A}}\in\overline{{A}}_{\geq t}$ has a neighbourhood that again is contained in $\overline{{\mathcal{A}}}_{\geq t}$ . So, let $\overline{{A}}\in\overline{{A}}_{\geq t}$ . </p>
<p data-bbox="105 478 540 685">• Variant 1 Due to the slice theorem there is an open neighbourhood $U$ of $\overline{{A}}\circ\overline{{\mathcal{G}}}$ , and so of $\overline{{A}}$ , too, and an equivariant retraction $F:U\longrightarrow\overline{{A}}\circ\overline{{\mathcal{G}}}$ . Since every equivariant mapping reduces types, we have $\mathrm{Typ}(\overline{{A}}^{\prime})\,\geq\,\mathrm{Typ}(\overline{{A}})\,=\,t$ for all $\overline{{A}}^{\prime}\,\in\,U$ , thus $U\subseteq{\overline{{A}}}_{\geq t}$ .
• Variant 2 Choose again for $\overline{{A}}$ an $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ with $\mathrm{Typ}(\overline{{A}})=[Z(\mathbf{H}_{\overline{{A}}})]=[Z(h_{\overline{{A}}}(\alpha))]\equiv[Z(\varphi_{\alpha}(\overline{{A}}))]=\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}))$ . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood $U^{\prime}$ of $\varphi_{\alpha}(\overline{{A}})$ in $\mathbf{G}^{\#\alpha}$ and an equivariant retraction $f:$ $U^{\prime}\longrightarrow\varphi_{\pmb{\alpha}}(\overline{{{A}}})\circ\mathbf{G}$ . Since $\varphi_{\alpha}(\overline{{A}})$ and $f$ are type-reducing, we have $\mathrm{Typ}(\overline{{A}}^{\prime})\geq\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}^{\prime}))\geq\mathrm{Typ}\big(f(\varphi_{\alpha}(\overline{{A}}^{\prime}))\big)=\mathrm{Typ}(\varphi_{\alpha}(\overline{{A}}))=\mathrm{Typ}(\overline{{A}})$ for all $\overline{{A}}^{\prime}\in U:=\varphi_{\pmb{\alpha}}^{-1}(U^{\prime})$ , i.e. $U\subseteq{\overline{{A}}}_{\geq t}$ . Obviously, $U$ contains $\overline{{A}}$ and is open as a preimage of an open set. qed </p>
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0001008v1
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[{"type": "text", "text": "acting group $\\overline{{g}}$ modulo the stabilizer $\\mathbf{B}(\\overline{{A}})$ of $\\overline{{A}}$ . \nHence, $F$ is continuous. ", "page_idx": 10}, {"type": "text", "text": "We have $F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}$ . ”⊆” Let $\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})$ , i.e. $F(\\overline{{A}}^{\\prime})=\\overline{{A}}$ . By the commutativity of (3) we have $f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=$ $\\varphi(\\overline{{A}})$ , hence $\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)$ . Define $g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})$ and $\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}$ . Then we have $\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S$ , i.e. $\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)$ , and $h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})$ for all $x$ , i.e. ${\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))$ . By this, $\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}$ . Consequently, $F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})$ and therefore also $\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =$ $F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}$ , i.e. ${\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})$ . Thus, $\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}$ . ”⊇” Let $\\overline{{A}}^{\\prime}\\in\\overline{{S}}$ . Then $F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}$ , i.e. $\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})$ . ", "page_idx": 10}, {"type": "text", "text": "6 Openness of the Strata ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "Proposition 6.1 $\\overline{{\\mathcal{A}}}_{\\geq t}$ is open for all $t\\in\\mathcal T$ . ", "page_idx": 10}, {"type": "text", "text": "Corollary 6.2 $\\scriptstyle A_{=t}$ is open in $\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$ for all $t\\in\\mathcal T$ . ", "page_idx": 10}, {"type": "text", "text": "Proof Since $\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}$ , $\\overline{{\\mathcal{A}}}_{=t}$ is open w.r.t. to the relative topology on $\\overline{{\\mathcal{A}}}_{\\leq t}$ . qed Corollary 6.3 $\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$ is compact for all $t\\in\\mathcal T$ . ", "page_idx": 10}, {"type": "text", "text": "Proof $\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}$ is open because $\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}$ is open for all $t^{\\prime}\\in\\mathcal T$ . Thus, $\\overline{{\\mathcal{A}}}_{\\leq t}$ is closed and the refore compact. qed ", "page_idx": 10}, {"type": "text", "text": "The proposition on the openness of the strata can be proven in two ways: first as a simple corollary of the slice theorem on $\\overline{{\\mathcal{A}}}$ , but second directly using the reduction mapping. Thus, altogether the second variant needs less effort. ", "page_idx": 10}, {"type": "text", "text": "Proof Proposition 6.1 ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "We have to show that any $\\overline{{A}}\\in\\overline{{A}}_{\\geq t}$ has a neighbourhood that again is contained in $\\overline{{\\mathcal{A}}}_{\\geq t}$ . So, let $\\overline{{A}}\\in\\overline{{A}}_{\\geq t}$ . ", "page_idx": 10}, {"type": "text", "text": "• Variant 1 Due to the slice theorem there is an open neighbourhood $U$ of $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ , and so of $\\overline{{A}}$ , too, and an equivariant retraction $F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ . Since every equivariant mapping reduces types, we have $\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t$ for all $\\overline{{A}}^{\\prime}\\,\\in\\,U$ , thus $U\\subseteq{\\overline{{A}}}_{\\geq t}$ . \n• Variant 2 Choose again for $\\overline{{A}}$ an $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ with $\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))$ . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood $U^{\\prime}$ of $\\varphi_{\\alpha}(\\overline{{A}})$ in $\\mathbf{G}^{\\#\\alpha}$ and an equivariant retraction $f:$ $U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}$ . Since $\\varphi_{\\alpha}(\\overline{{A}})$ and $f$ are type-reducing, we have $\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})$ for all $\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})$ , i.e. $U\\subseteq{\\overline{{A}}}_{\\geq t}$ . Obviously, $U$ contains $\\overline{{A}}$ and is open as a preimage of an open set. qed ", "page_idx": 10}]
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|
{"preproc_blocks": [{"type": "text", "bbox": [144, 13, 399, 42], "lines": [{"bbox": [148, 16, 397, 31], "spans": [{"bbox": [148, 16, 216, 29], "score": 1.0, "content": "acting group ", "type": "text"}, {"bbox": [217, 17, 225, 29], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [225, 16, 340, 29], "score": 1.0, "content": " modulo the stabilizer ", "type": "text"}, {"bbox": [341, 17, 369, 31], "score": 0.95, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [369, 16, 385, 29], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [386, 17, 395, 28], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [395, 16, 397, 29], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [149, 32, 270, 44], "spans": [{"bbox": [149, 32, 186, 44], "score": 1.0, "content": "Hence, ", "type": "text"}, {"bbox": [187, 33, 196, 42], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [196, 32, 270, 44], "score": 1.0, "content": " is continuous.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [118, 44, 539, 214], "lines": [{"bbox": [128, 44, 257, 59], "spans": [{"bbox": [128, 44, 179, 59], "score": 1.0, "content": "We have ", "type": "text"}, {"bbox": [179, 46, 254, 59], "score": 0.92, "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [254, 44, 257, 59], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [138, 59, 363, 74], "spans": [{"bbox": [138, 59, 201, 73], "score": 1.0, "content": "”⊆” Let ", "type": "text"}, {"bbox": [201, 59, 278, 74], "score": 0.9, "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "type": "inline_equation", "height": 15, "width": 77}, {"bbox": [279, 59, 305, 73], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [305, 59, 360, 74], "score": 0.93, "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 55}, {"bbox": [360, 59, 363, 73], "score": 1.0, "content": ".", "type": "text"}], "index": 3}, {"bbox": [199, 73, 537, 88], "spans": [{"bbox": [199, 73, 407, 88], "score": 1.0, "content": "By the commutativity of (3) we have ", "type": "text"}, {"bbox": [407, 73, 537, 88], "score": 0.92, "content": "f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=", "type": "inline_equation", "height": 15, "width": 130}], "index": 4}, {"bbox": [200, 86, 432, 104], "spans": [{"bbox": [200, 88, 226, 102], "score": 0.84, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 26}, {"bbox": [227, 86, 265, 104], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [265, 87, 425, 102], "score": 0.88, "content": "\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)", "type": "inline_equation", "height": 15, "width": 160}, {"bbox": [426, 86, 432, 104], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [198, 102, 538, 118], "spans": [{"bbox": [198, 102, 237, 118], "score": 1.0, "content": "Define ", "type": "text"}, {"bbox": [237, 103, 358, 118], "score": 0.92, "content": "g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})", "type": "inline_equation", "height": 15, "width": 121}, {"bbox": [358, 102, 385, 118], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [385, 102, 452, 117], "score": 0.92, "content": "\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [453, 102, 538, 118], "score": 1.0, "content": ". Then we have", "type": "text"}], "index": 6}, {"bbox": [200, 117, 538, 136], "spans": [{"bbox": [200, 118, 302, 134], "score": 0.92, "content": "\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S", "type": "inline_equation", "height": 16, "width": 102}, {"bbox": [302, 117, 329, 136], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [330, 118, 397, 134], "score": 0.93, "content": "\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)", "type": "inline_equation", "height": 16, "width": 67}, {"bbox": [397, 117, 426, 136], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [426, 120, 518, 135], "score": 0.92, "content": "h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})", "type": "inline_equation", "height": 15, "width": 92}, {"bbox": [518, 117, 538, 136], "score": 1.0, "content": " for", "type": "text"}], "index": 7}, {"bbox": [199, 134, 432, 151], "spans": [{"bbox": [199, 135, 216, 151], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [216, 141, 224, 148], "score": 0.61, "content": "x", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [224, 135, 250, 151], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [250, 135, 334, 150], "score": 0.87, "content": "{\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))", "type": "inline_equation", "height": 15, "width": 84}, {"bbox": [334, 135, 385, 151], "score": 1.0, "content": ". By this,", "type": "text"}, {"bbox": [386, 134, 428, 150], "score": 0.93, "content": "\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 16, "width": 42}, {"bbox": [428, 135, 432, 151], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [199, 149, 538, 165], "spans": [{"bbox": [199, 149, 276, 165], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [277, 149, 389, 164], "score": 0.89, "content": "F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})", "type": "inline_equation", "height": 15, "width": 112}, {"bbox": [389, 149, 492, 165], "score": 1.0, "content": " and therefore also ", "type": "text"}, {"bbox": [493, 150, 538, 164], "score": 0.9, "content": "\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =", "type": "inline_equation", "height": 14, "width": 45}], "index": 9}, {"bbox": [200, 164, 462, 180], "spans": [{"bbox": [200, 164, 384, 179], "score": 0.87, "content": "F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 184}, {"bbox": [385, 164, 409, 180], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [410, 164, 459, 179], "score": 0.93, "content": "{\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})", "type": "inline_equation", "height": 15, "width": 49}, {"bbox": [459, 164, 462, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [179, 178, 378, 194], "spans": [{"bbox": [179, 178, 212, 194], "score": 1.0, "content": "Thus,", "type": "text"}, {"bbox": [213, 179, 374, 193], "score": 0.89, "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}", "type": "inline_equation", "height": 14, "width": 161}, {"bbox": [374, 178, 378, 194], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [147, 192, 536, 208], "spans": [{"bbox": [147, 192, 200, 208], "score": 1.0, "content": "”⊇” Let ", "type": "text"}, {"bbox": [201, 193, 235, 206], "score": 0.87, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [236, 192, 273, 208], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [273, 193, 430, 208], "score": 0.9, "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 157}, {"bbox": [430, 192, 455, 208], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [456, 192, 533, 208], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "type": "inline_equation", "height": 16, "width": 77}, {"bbox": [533, 192, 536, 208], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 7}, {"type": "title", "bbox": [62, 241, 289, 261], "lines": [{"bbox": [64, 245, 289, 262], "spans": [{"bbox": [64, 246, 74, 259], "score": 1.0, "content": "6", "type": "text"}, {"bbox": [91, 245, 289, 262], "score": 1.0, "content": "Openness of the Strata", "type": "text"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [63, 271, 294, 287], "lines": [{"bbox": [62, 274, 294, 289], "spans": [{"bbox": [62, 274, 163, 288], "score": 1.0, "content": "Proposition 6.1", "type": "text"}, {"bbox": [163, 275, 183, 289], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [184, 274, 261, 288], "score": 1.0, "content": " is open for all ", "type": "text"}, {"bbox": [261, 276, 290, 286], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [290, 274, 294, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [63, 291, 319, 307], "lines": [{"bbox": [64, 293, 317, 308], "spans": [{"bbox": [64, 293, 150, 308], "score": 1.0, "content": "Corollary 6.2", "type": "text"}, {"bbox": [150, 295, 171, 306], "score": 0.9, "content": "\\scriptstyle A_{=t}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [171, 293, 228, 308], "score": 1.0, "content": " is open in ", "type": "text"}, {"bbox": [228, 294, 248, 308], "score": 0.93, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [248, 293, 285, 308], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [285, 296, 314, 305], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [315, 293, 317, 308], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [63, 310, 539, 346], "lines": [{"bbox": [62, 313, 539, 330], "spans": [{"bbox": [62, 313, 136, 330], "score": 1.0, "content": "Proof Since ", "type": "text"}, {"bbox": [137, 314, 224, 327], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [224, 313, 230, 330], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [230, 314, 250, 326], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [250, 313, 463, 330], "score": 1.0, "content": " is open w.r.t. to the relative topology on ", "type": "text"}, {"bbox": [464, 314, 484, 327], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [484, 313, 508, 330], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [510, 314, 539, 330], "score": 1.0, "content": "qed", "type": "text"}], "index": 16}, {"bbox": [63, 332, 300, 349], "spans": [{"bbox": [63, 332, 150, 349], "score": 1.0, "content": "Corollary 6.3", "type": "text"}, {"bbox": [150, 334, 171, 347], "score": 0.92, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [171, 332, 267, 349], "score": 1.0, "content": " is compact for all ", "type": "text"}, {"bbox": [267, 335, 296, 345], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [296, 332, 300, 349], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16.5}, {"type": "text", "bbox": [63, 349, 538, 381], "lines": [{"bbox": [61, 350, 538, 371], "spans": [{"bbox": [61, 350, 105, 371], "score": 1.0, "content": "Proof", "type": "text"}, {"bbox": [105, 353, 318, 368], "score": 0.91, "content": "\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}", "type": "inline_equation", "height": 15, "width": 213}, {"bbox": [319, 350, 403, 371], "score": 1.0, "content": " is open because ", "type": "text"}, {"bbox": [403, 353, 426, 367], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [426, 350, 501, 371], "score": 1.0, "content": " is open for all ", "type": "text"}, {"bbox": [501, 354, 533, 364], "score": 0.93, "content": "t^{\\prime}\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [533, 350, 538, 371], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [106, 367, 540, 384], "spans": [{"bbox": [106, 367, 139, 384], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [139, 369, 159, 383], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [159, 367, 328, 384], "score": 1.0, "content": " is closed and the refore compact.", "type": "text"}, {"bbox": [513, 369, 540, 384], "score": 1.0, "content": "qed", "type": "text"}], "index": 19}], "index": 18.5}, {"type": "text", "bbox": [63, 386, 537, 429], "lines": [{"bbox": [62, 387, 537, 403], "spans": [{"bbox": [62, 387, 537, 403], "score": 1.0, "content": "The proposition on the openness of the strata can be proven in two ways: first as a simple", "type": "text"}], "index": 20}, {"bbox": [63, 403, 536, 417], "spans": [{"bbox": [63, 403, 232, 417], "score": 1.0, "content": "corollary of the slice theorem on ", "type": "text"}, {"bbox": [232, 403, 242, 413], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [243, 403, 536, 417], "score": 1.0, "content": ", but second directly using the reduction mapping. Thus,", "type": "text"}], "index": 21}, {"bbox": [63, 417, 301, 432], "spans": [{"bbox": [63, 417, 301, 432], "score": 1.0, "content": "altogether the second variant needs less effort.", "type": "text"}], "index": 22}], "index": 21}, {"type": "title", "bbox": [62, 434, 197, 447], "lines": [{"bbox": [63, 436, 196, 449], "spans": [{"bbox": [63, 436, 196, 449], "score": 1.0, "content": "Proof Proposition 6.1", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [104, 448, 537, 477], "lines": [{"bbox": [105, 449, 537, 466], "spans": [{"bbox": [105, 449, 245, 466], "score": 1.0, "content": "We have to show that any ", "type": "text"}, {"bbox": [245, 451, 289, 464], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [289, 449, 537, 466], "score": 1.0, "content": " has a neighbourhood that again is contained in", "type": "text"}], "index": 24}, {"bbox": [106, 463, 218, 479], "spans": [{"bbox": [106, 465, 126, 478], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [127, 463, 170, 479], "score": 1.0, "content": ". So, let ", "type": "text"}, {"bbox": [171, 465, 214, 478], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [214, 463, 218, 479], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5}, {"type": "text", "bbox": [105, 478, 540, 685], "lines": [{"bbox": [106, 480, 171, 492], "spans": [{"bbox": [106, 480, 171, 492], "score": 1.0, "content": "• Variant 1", "type": "text"}], "index": 26}, {"bbox": [121, 493, 539, 507], "spans": [{"bbox": [121, 493, 424, 507], "score": 1.0, "content": "Due to the slice theorem there is an open neighbourhood ", "type": "text"}, {"bbox": [424, 496, 434, 505], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [434, 493, 451, 507], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [452, 494, 481, 506], "score": 0.93, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [481, 493, 539, 507], "score": 1.0, "content": ", and so of", "type": "text"}], "index": 27}, {"bbox": [123, 506, 537, 523], "spans": [{"bbox": [123, 508, 132, 519], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [132, 506, 319, 523], "score": 1.0, "content": ", too, and an equivariant retraction ", "type": "text"}, {"bbox": [320, 508, 407, 520], "score": 0.92, "content": "F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [407, 506, 537, 523], "score": 1.0, "content": ". Since every equivariant", "type": "text"}], "index": 28}, {"bbox": [121, 521, 538, 537], "spans": [{"bbox": [121, 521, 298, 537], "score": 1.0, "content": "mapping reduces types, we have ", "type": "text"}, {"bbox": [298, 522, 424, 536], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t", "type": "inline_equation", "height": 14, "width": 126}, {"bbox": [424, 521, 465, 537], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [465, 521, 505, 534], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\,\\in\\,U", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [506, 521, 538, 537], "score": 1.0, "content": ", thus", "type": "text"}], "index": 29}, {"bbox": [123, 535, 172, 552], "spans": [{"bbox": [123, 537, 168, 551], "score": 0.94, "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 45}, {"bbox": [169, 535, 172, 552], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [109, 552, 172, 564], "spans": [{"bbox": [109, 552, 172, 564], "score": 1.0, "content": "• Variant 2", "type": "text"}], "index": 31}, {"bbox": [122, 565, 313, 580], "spans": [{"bbox": [122, 565, 213, 580], "score": 1.0, "content": "Choose again for ", "type": "text"}, {"bbox": [213, 566, 222, 576], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [222, 565, 241, 580], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [242, 568, 285, 578], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [285, 565, 313, 580], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [168, 579, 492, 595], "spans": [{"bbox": [168, 581, 489, 594], "score": 0.88, "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))", "type": "inline_equation", "height": 13, "width": 321}, {"bbox": [489, 579, 492, 595], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [122, 594, 537, 610], "spans": [{"bbox": [122, 594, 537, 610], "score": 1.0, "content": "Due to the slice theorem for general transformation groups there is an open,", "type": "text"}], "index": 34}, {"bbox": [122, 608, 537, 624], "spans": [{"bbox": [122, 608, 253, 624], "score": 1.0, "content": "invariant neighbourhood ", "type": "text"}, {"bbox": [253, 611, 266, 620], "score": 0.88, "content": "U^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [266, 608, 284, 624], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [284, 609, 317, 623], "score": 0.93, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [317, 608, 335, 624], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [335, 609, 361, 621], "score": 0.88, "content": "\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [361, 608, 520, 624], "score": 1.0, "content": " and an equivariant retraction ", "type": "text"}, {"bbox": [521, 609, 537, 622], "score": 0.64, "content": "f:", "type": "inline_equation", "height": 13, "width": 16}], "index": 35}, {"bbox": [123, 622, 464, 639], "spans": [{"bbox": [123, 623, 216, 637], "score": 0.92, "content": "U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 93}, {"bbox": [216, 622, 254, 639], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [254, 623, 287, 637], "score": 0.94, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [288, 622, 313, 639], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [314, 625, 321, 637], "score": 0.87, "content": "f", "type": "inline_equation", "height": 12, "width": 7}, {"bbox": [321, 622, 464, 639], "score": 1.0, "content": " are type-reducing, we have", "type": "text"}], "index": 36}, {"bbox": [151, 638, 508, 656], "spans": [{"bbox": [151, 638, 508, 656], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 18, "width": 357}], "index": 37}, {"bbox": [121, 654, 538, 675], "spans": [{"bbox": [121, 654, 157, 675], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [157, 657, 253, 672], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})", "type": "inline_equation", "height": 15, "width": 96}, {"bbox": [253, 655, 279, 674], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [279, 658, 325, 672], "score": 0.93, "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [325, 655, 390, 674], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [390, 660, 400, 669], "score": 0.89, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [400, 655, 449, 674], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [450, 658, 459, 668], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [459, 655, 538, 674], "score": 1.0, "content": " and is open as", "type": "text"}], "index": 38}, {"bbox": [122, 672, 539, 687], "spans": [{"bbox": [122, 672, 259, 687], "score": 1.0, "content": "a preimage of an open set.", "type": "text"}, {"bbox": [513, 673, 539, 687], "score": 1.0, "content": "qed", "type": "text"}], "index": 39}], "index": 32.5}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 305, 715], "lines": [{"bbox": [293, 705, 307, 717], "spans": [{"bbox": [293, 705, 307, 717], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "list", "bbox": [144, 13, 399, 42], "lines": [{"bbox": [148, 16, 397, 31], "spans": [{"bbox": [148, 16, 216, 29], "score": 1.0, "content": "acting group ", "type": "text"}, {"bbox": [217, 17, 225, 29], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [225, 16, 340, 29], "score": 1.0, "content": " modulo the stabilizer ", "type": "text"}, {"bbox": [341, 17, 369, 31], "score": 0.95, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 28}, {"bbox": [369, 16, 385, 29], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [386, 17, 395, 28], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [395, 16, 397, 29], "score": 1.0, "content": ".", "type": "text"}], "index": 0, "is_list_end_line": true}, {"bbox": [149, 32, 270, 44], "spans": [{"bbox": [149, 32, 186, 44], "score": 1.0, "content": "Hence, ", "type": "text"}, {"bbox": [187, 33, 196, 42], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [196, 32, 270, 44], "score": 1.0, "content": " is continuous.", "type": "text"}], "index": 1, "is_list_start_line": true, "is_list_end_line": true}], "index": 0.5, "bbox_fs": [148, 16, 397, 44]}, {"type": "text", "bbox": [118, 44, 539, 214], "lines": [{"bbox": [128, 44, 257, 59], "spans": [{"bbox": [128, 44, 179, 59], "score": 1.0, "content": "We have ", "type": "text"}, {"bbox": [179, 46, 254, 59], "score": 0.92, "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [254, 44, 257, 59], "score": 1.0, "content": ".", "type": "text"}], "index": 2}, {"bbox": [138, 59, 363, 74], "spans": [{"bbox": [138, 59, 201, 73], "score": 1.0, "content": "”⊆” Let ", "type": "text"}, {"bbox": [201, 59, 278, 74], "score": 0.9, "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "type": "inline_equation", "height": 15, "width": 77}, {"bbox": [279, 59, 305, 73], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [305, 59, 360, 74], "score": 0.93, "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 55}, {"bbox": [360, 59, 363, 73], "score": 1.0, "content": ".", "type": "text"}], "index": 3}, {"bbox": [199, 73, 537, 88], "spans": [{"bbox": [199, 73, 407, 88], "score": 1.0, "content": "By the commutativity of (3) we have ", "type": "text"}, {"bbox": [407, 73, 537, 88], "score": 0.92, "content": "f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=", "type": "inline_equation", "height": 15, "width": 130}], "index": 4}, {"bbox": [200, 86, 432, 104], "spans": [{"bbox": [200, 88, 226, 102], "score": 0.84, "content": "\\varphi(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 26}, {"bbox": [227, 86, 265, 104], "score": 1.0, "content": ", hence ", "type": "text"}, {"bbox": [265, 87, 425, 102], "score": 0.88, "content": "\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)", "type": "inline_equation", "height": 15, "width": 160}, {"bbox": [426, 86, 432, 104], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [198, 102, 538, 118], "spans": [{"bbox": [198, 102, 237, 118], "score": 1.0, "content": "Define ", "type": "text"}, {"bbox": [237, 103, 358, 118], "score": 0.92, "content": "g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})", "type": "inline_equation", "height": 15, "width": 121}, {"bbox": [358, 102, 385, 118], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [385, 102, 452, 117], "score": 0.92, "content": "\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "type": "inline_equation", "height": 15, "width": 67}, {"bbox": [453, 102, 538, 118], "score": 1.0, "content": ". Then we have", "type": "text"}], "index": 6}, {"bbox": [200, 117, 538, 136], "spans": [{"bbox": [200, 118, 302, 134], "score": 0.92, "content": "\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S", "type": "inline_equation", "height": 16, "width": 102}, {"bbox": [302, 117, 329, 136], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [330, 118, 397, 134], "score": 0.93, "content": "\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)", "type": "inline_equation", "height": 16, "width": 67}, {"bbox": [397, 117, 426, 136], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [426, 120, 518, 135], "score": 0.92, "content": "h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})", "type": "inline_equation", "height": 15, "width": 92}, {"bbox": [518, 117, 538, 136], "score": 1.0, "content": " for", "type": "text"}], "index": 7}, {"bbox": [199, 134, 432, 151], "spans": [{"bbox": [199, 135, 216, 151], "score": 1.0, "content": "all ", "type": "text"}, {"bbox": [216, 141, 224, 148], "score": 0.61, "content": "x", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [224, 135, 250, 151], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [250, 135, 334, 150], "score": 0.87, "content": "{\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))", "type": "inline_equation", "height": 15, "width": 84}, {"bbox": [334, 135, 385, 151], "score": 1.0, "content": ". By this,", "type": "text"}, {"bbox": [386, 134, 428, 150], "score": 0.93, "content": "\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "type": "inline_equation", "height": 16, "width": 42}, {"bbox": [428, 135, 432, 151], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [199, 149, 538, 165], "spans": [{"bbox": [199, 149, 276, 165], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [277, 149, 389, 164], "score": 0.89, "content": "F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})", "type": "inline_equation", "height": 15, "width": 112}, {"bbox": [389, 149, 492, 165], "score": 1.0, "content": " and therefore also ", "type": "text"}, {"bbox": [493, 150, 538, 164], "score": 0.9, "content": "\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =", "type": "inline_equation", "height": 14, "width": 45}], "index": 9}, {"bbox": [200, 164, 462, 180], "spans": [{"bbox": [200, 164, 384, 179], "score": 0.87, "content": "F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 184}, {"bbox": [385, 164, 409, 180], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [410, 164, 459, 179], "score": 0.93, "content": "{\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})", "type": "inline_equation", "height": 15, "width": 49}, {"bbox": [459, 164, 462, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [179, 178, 378, 194], "spans": [{"bbox": [179, 178, 212, 194], "score": 1.0, "content": "Thus,", "type": "text"}, {"bbox": [213, 179, 374, 193], "score": 0.89, "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}", "type": "inline_equation", "height": 14, "width": 161}, {"bbox": [374, 178, 378, 194], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [147, 192, 536, 208], "spans": [{"bbox": [147, 192, 200, 208], "score": 1.0, "content": "”⊇” Let ", "type": "text"}, {"bbox": [201, 193, 235, 206], "score": 0.87, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [236, 192, 273, 208], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [273, 193, 430, 208], "score": 0.9, "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}", "type": "inline_equation", "height": 15, "width": 157}, {"bbox": [430, 192, 455, 208], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [456, 192, 533, 208], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "type": "inline_equation", "height": 16, "width": 77}, {"bbox": [533, 192, 536, 208], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 7, "bbox_fs": [128, 44, 538, 208]}, {"type": "title", "bbox": [62, 241, 289, 261], "lines": [{"bbox": [64, 245, 289, 262], "spans": [{"bbox": [64, 246, 74, 259], "score": 1.0, "content": "6", "type": "text"}, {"bbox": [91, 245, 289, 262], "score": 1.0, "content": "Openness of the Strata", "type": "text"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [63, 271, 294, 287], "lines": [{"bbox": [62, 274, 294, 289], "spans": [{"bbox": [62, 274, 163, 288], "score": 1.0, "content": "Proposition 6.1", "type": "text"}, {"bbox": [163, 275, 183, 289], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [184, 274, 261, 288], "score": 1.0, "content": " is open for all ", "type": "text"}, {"bbox": [261, 276, 290, 286], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [290, 274, 294, 288], "score": 1.0, "content": ".", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [62, 274, 294, 289]}, {"type": "text", "bbox": [63, 291, 319, 307], "lines": [{"bbox": [64, 293, 317, 308], "spans": [{"bbox": [64, 293, 150, 308], "score": 1.0, "content": "Corollary 6.2", "type": "text"}, {"bbox": [150, 295, 171, 306], "score": 0.9, "content": "\\scriptstyle A_{=t}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [171, 293, 228, 308], "score": 1.0, "content": " is open in ", "type": "text"}, {"bbox": [228, 294, 248, 308], "score": 0.93, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [248, 293, 285, 308], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [285, 296, 314, 305], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [315, 293, 317, 308], "score": 1.0, "content": ".", "type": "text"}], "index": 15}], "index": 15, "bbox_fs": [64, 293, 317, 308]}, {"type": "text", "bbox": [63, 310, 539, 346], "lines": [{"bbox": [62, 313, 539, 330], "spans": [{"bbox": [62, 313, 136, 330], "score": 1.0, "content": "Proof Since ", "type": "text"}, {"bbox": [137, 314, 224, 327], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [224, 313, 230, 330], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [230, 314, 250, 326], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [250, 313, 463, 330], "score": 1.0, "content": " is open w.r.t. to the relative topology on ", "type": "text"}, {"bbox": [464, 314, 484, 327], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [484, 313, 508, 330], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [510, 314, 539, 330], "score": 1.0, "content": "qed", "type": "text"}], "index": 16}, {"bbox": [63, 332, 300, 349], "spans": [{"bbox": [63, 332, 150, 349], "score": 1.0, "content": "Corollary 6.3", "type": "text"}, {"bbox": [150, 334, 171, 347], "score": 0.92, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [171, 332, 267, 349], "score": 1.0, "content": " is compact for all ", "type": "text"}, {"bbox": [267, 335, 296, 345], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [296, 332, 300, 349], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 16.5, "bbox_fs": [62, 313, 539, 349]}, {"type": "text", "bbox": [63, 349, 538, 381], "lines": [{"bbox": [61, 350, 538, 371], "spans": [{"bbox": [61, 350, 105, 371], "score": 1.0, "content": "Proof", "type": "text"}, {"bbox": [105, 353, 318, 368], "score": 0.91, "content": "\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}", "type": "inline_equation", "height": 15, "width": 213}, {"bbox": [319, 350, 403, 371], "score": 1.0, "content": " is open because ", "type": "text"}, {"bbox": [403, 353, 426, 367], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [426, 350, 501, 371], "score": 1.0, "content": " is open for all ", "type": "text"}, {"bbox": [501, 354, 533, 364], "score": 0.93, "content": "t^{\\prime}\\in\\mathcal T", "type": "inline_equation", "height": 10, "width": 32}, {"bbox": [533, 350, 538, 371], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [106, 367, 540, 384], "spans": [{"bbox": [106, 367, 139, 384], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [139, 369, 159, 383], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [159, 367, 328, 384], "score": 1.0, "content": " is closed and the refore compact.", "type": "text"}, {"bbox": [513, 369, 540, 384], "score": 1.0, "content": "qed", "type": "text"}], "index": 19}], "index": 18.5, "bbox_fs": [61, 350, 540, 384]}, {"type": "text", "bbox": [63, 386, 537, 429], "lines": [{"bbox": [62, 387, 537, 403], "spans": [{"bbox": [62, 387, 537, 403], "score": 1.0, "content": "The proposition on the openness of the strata can be proven in two ways: first as a simple", "type": "text"}], "index": 20}, {"bbox": [63, 403, 536, 417], "spans": [{"bbox": [63, 403, 232, 417], "score": 1.0, "content": "corollary of the slice theorem on ", "type": "text"}, {"bbox": [232, 403, 242, 413], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [243, 403, 536, 417], "score": 1.0, "content": ", but second directly using the reduction mapping. Thus,", "type": "text"}], "index": 21}, {"bbox": [63, 417, 301, 432], "spans": [{"bbox": [63, 417, 301, 432], "score": 1.0, "content": "altogether the second variant needs less effort.", "type": "text"}], "index": 22}], "index": 21, "bbox_fs": [62, 387, 537, 432]}, {"type": "title", "bbox": [62, 434, 197, 447], "lines": [{"bbox": [63, 436, 196, 449], "spans": [{"bbox": [63, 436, 196, 449], "score": 1.0, "content": "Proof Proposition 6.1", "type": "text"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [104, 448, 537, 477], "lines": [{"bbox": [105, 449, 537, 466], "spans": [{"bbox": [105, 449, 245, 466], "score": 1.0, "content": "We have to show that any ", "type": "text"}, {"bbox": [245, 451, 289, 464], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [289, 449, 537, 466], "score": 1.0, "content": " has a neighbourhood that again is contained in", "type": "text"}], "index": 24}, {"bbox": [106, 463, 218, 479], "spans": [{"bbox": [106, 465, 126, 478], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [127, 463, 170, 479], "score": 1.0, "content": ". So, let ", "type": "text"}, {"bbox": [171, 465, 214, 478], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [214, 463, 218, 479], "score": 1.0, "content": ".", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [105, 449, 537, 479]}, {"type": "list", "bbox": [105, 478, 540, 685], "lines": [{"bbox": [106, 480, 171, 492], "spans": [{"bbox": [106, 480, 171, 492], "score": 1.0, "content": "• Variant 1", "type": "text"}], "index": 26, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [121, 493, 539, 507], "spans": [{"bbox": [121, 493, 424, 507], "score": 1.0, "content": "Due to the slice theorem there is an open neighbourhood ", "type": "text"}, {"bbox": [424, 496, 434, 505], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [434, 493, 451, 507], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [452, 494, 481, 506], "score": 0.93, "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [481, 493, 539, 507], "score": 1.0, "content": ", and so of", "type": "text"}], "index": 27}, {"bbox": [123, 506, 537, 523], "spans": [{"bbox": [123, 508, 132, 519], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [132, 506, 319, 523], "score": 1.0, "content": ", too, and an equivariant retraction ", "type": "text"}, {"bbox": [320, 508, 407, 520], "score": 0.92, "content": "F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [407, 506, 537, 523], "score": 1.0, "content": ". Since every equivariant", "type": "text"}], "index": 28}, {"bbox": [121, 521, 538, 537], "spans": [{"bbox": [121, 521, 298, 537], "score": 1.0, "content": "mapping reduces types, we have ", "type": "text"}, {"bbox": [298, 522, 424, 536], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t", "type": "inline_equation", "height": 14, "width": 126}, {"bbox": [424, 521, 465, 537], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [465, 521, 505, 534], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\,\\in\\,U", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [506, 521, 538, 537], "score": 1.0, "content": ", thus", "type": "text"}], "index": 29}, {"bbox": [123, 535, 172, 552], "spans": [{"bbox": [123, 537, 168, 551], "score": 0.94, "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 45}, {"bbox": [169, 535, 172, 552], "score": 1.0, "content": ".", "type": "text"}], "index": 30, "is_list_end_line": true}, {"bbox": [109, 552, 172, 564], "spans": [{"bbox": [109, 552, 172, 564], "score": 1.0, "content": "• Variant 2", "type": "text"}], "index": 31, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [122, 565, 313, 580], "spans": [{"bbox": [122, 565, 213, 580], "score": 1.0, "content": "Choose again for ", "type": "text"}, {"bbox": [213, 566, 222, 576], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [222, 565, 241, 580], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [242, 568, 285, 578], "score": 0.92, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [285, 565, 313, 580], "score": 1.0, "content": " with", "type": "text"}], "index": 32, "is_list_end_line": true}, {"bbox": [168, 579, 492, 595], "spans": [{"bbox": [168, 581, 489, 594], "score": 0.88, "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))", "type": "inline_equation", "height": 13, "width": 321}, {"bbox": [489, 579, 492, 595], "score": 1.0, "content": ".", "type": "text"}], "index": 33, "is_list_end_line": true}, {"bbox": [122, 594, 537, 610], "spans": [{"bbox": [122, 594, 537, 610], "score": 1.0, "content": "Due to the slice theorem for general transformation groups there is an open,", "type": "text"}], "index": 34}, {"bbox": [122, 608, 537, 624], "spans": [{"bbox": [122, 608, 253, 624], "score": 1.0, "content": "invariant neighbourhood ", "type": "text"}, {"bbox": [253, 611, 266, 620], "score": 0.88, "content": "U^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [266, 608, 284, 624], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [284, 609, 317, 623], "score": 0.93, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [317, 608, 335, 624], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [335, 609, 361, 621], "score": 0.88, "content": "\\mathbf{G}^{\\#\\alpha}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [361, 608, 520, 624], "score": 1.0, "content": " and an equivariant retraction ", "type": "text"}, {"bbox": [521, 609, 537, 622], "score": 0.64, "content": "f:", "type": "inline_equation", "height": 13, "width": 16}], "index": 35}, {"bbox": [123, 622, 464, 639], "spans": [{"bbox": [123, 623, 216, 637], "score": 0.92, "content": "U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 93}, {"bbox": [216, 622, 254, 639], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [254, 623, 287, 637], "score": 0.94, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [288, 622, 313, 639], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [314, 625, 321, 637], "score": 0.87, "content": "f", "type": "inline_equation", "height": 12, "width": 7}, {"bbox": [321, 622, 464, 639], "score": 1.0, "content": " are type-reducing, we have", "type": "text"}], "index": 36, "is_list_end_line": true}, {"bbox": [151, 638, 508, 656], "spans": [{"bbox": [151, 638, 508, 656], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 18, "width": 357}], "index": 37, "is_list_end_line": true}, {"bbox": [121, 654, 538, 675], "spans": [{"bbox": [121, 654, 157, 675], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [157, 657, 253, 672], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})", "type": "inline_equation", "height": 15, "width": 96}, {"bbox": [253, 655, 279, 674], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [279, 658, 325, 672], "score": 0.93, "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [325, 655, 390, 674], "score": 1.0, "content": ". Obviously, ", "type": "text"}, {"bbox": [390, 660, 400, 669], "score": 0.89, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [400, 655, 449, 674], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [450, 658, 459, 668], "score": 0.87, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [459, 655, 538, 674], "score": 1.0, "content": " and is open as", "type": "text"}], "index": 38}, {"bbox": [122, 672, 539, 687], "spans": [{"bbox": [122, 672, 259, 687], "score": 1.0, "content": "a preimage of an open set.", "type": "text"}, {"bbox": [513, 673, 539, 687], "score": 1.0, "content": "qed", "type": "text"}], "index": 39}], "index": 32.5, "bbox_fs": [106, 480, 539, 687]}]}
|
[{"type": "list", "bbox": [144, 13, 399, 42], "content": "", "index": 0}, {"type": "text", "bbox": [118, 44, 539, 214], "content": "We have . ”⊆” Let , i.e. . By the commutativity of (3) we have , hence . Define and . Then we have , i.e. , and for all , i.e. . By this, . Consequently, and therefore also , i.e. . Thus, . ”⊇” Let . Then , i.e. .", "index": 1}, {"type": "title", "bbox": [62, 241, 289, 261], "content": "6 Openness of the Strata", "index": 2}, {"type": "text", "bbox": [63, 271, 294, 287], "content": "Proposition 6.1 is open for all .", "index": 3}, {"type": "text", "bbox": [63, 291, 319, 307], "content": "Corollary 6.2 is open in for all .", "index": 4}, {"type": "text", "bbox": [63, 310, 539, 346], "content": "Proof Since , is open w.r.t. to the relative topology on . qed Corollary 6.3 is compact for all .", "index": 5}, {"type": "text", "bbox": [63, 349, 538, 381], "content": "Proof is open because is open for all . Thus, is closed and the refore compact. qed", "index": 6}, {"type": "text", "bbox": [63, 386, 537, 429], "content": "The proposition on the openness of the strata can be proven in two ways: first as a simple corollary of the slice theorem on , but second directly using the reduction mapping. Thus, altogether the second variant needs less effort.", "index": 7}, {"type": "title", "bbox": [62, 434, 197, 447], "content": "Proof Proposition 6.1", "index": 8}, {"type": "text", "bbox": [104, 448, 537, 477], "content": "We have to show that any has a neighbourhood that again is contained in . So, let .", "index": 9}, {"type": "list", "bbox": [105, 478, 540, 685], "content": "", "index": 10}]
|
[{"bbox": [148, 16, 397, 31], "content": "acting group modulo the stabilizer of .", "parent_index": 0, "line_index": 0}, {"bbox": [149, 32, 270, 44], "content": "Hence, is continuous.", "parent_index": 0, "line_index": 1}, {"bbox": [128, 44, 257, 59], "content": "We have .", "parent_index": 1, "line_index": 0}, {"bbox": [138, 59, 363, 74], "content": "”⊆” Let , i.e. .", "parent_index": 1, "line_index": 1}, {"bbox": [199, 73, 537, 88], "content": "By the commutativity of (3) we have", "parent_index": 1, "line_index": 2}, {"bbox": [200, 86, 432, 104], "content": ", hence .", "parent_index": 1, "line_index": 3}, {"bbox": [198, 102, 538, 118], "content": "Define and . Then we have", "parent_index": 1, "line_index": 4}, {"bbox": [200, 117, 538, 136], "content": ", i.e. , and for", "parent_index": 1, "line_index": 5}, {"bbox": [199, 134, 432, 151], "content": "all , i.e. . By this, .", "parent_index": 1, "line_index": 6}, {"bbox": [199, 149, 538, 165], "content": "Consequently, and therefore also", "parent_index": 1, "line_index": 7}, {"bbox": [200, 164, 462, 180], "content": ", i.e. .", "parent_index": 1, "line_index": 8}, {"bbox": [179, 178, 378, 194], "content": "Thus, .", "parent_index": 1, "line_index": 9}, {"bbox": [147, 192, 536, 208], "content": "”⊇” Let . Then , i.e. .", "parent_index": 1, "line_index": 10}, {"bbox": [64, 245, 289, 262], "content": "6 Openness of the Strata", "parent_index": 2, "line_index": 0}, {"bbox": [62, 274, 294, 289], "content": "Proposition 6.1 is open for all .", "parent_index": 3, "line_index": 0}, {"bbox": [64, 293, 317, 308], "content": "Corollary 6.2 is open in for all .", "parent_index": 4, "line_index": 0}, {"bbox": [62, 313, 539, 330], "content": "Proof Since , is open w.r.t. to the relative topology on . qed", "parent_index": 5, "line_index": 0}, {"bbox": [63, 332, 300, 349], "content": "Corollary 6.3 is compact for all .", "parent_index": 5, "line_index": 1}, {"bbox": [61, 350, 538, 371], "content": "Proof is open because is open for all .", "parent_index": 6, "line_index": 0}, {"bbox": [106, 367, 540, 384], "content": "Thus, is closed and the refore compact. qed", "parent_index": 6, "line_index": 1}, {"bbox": [62, 387, 537, 403], "content": "The proposition on the openness of the strata can be proven in two ways: first as a simple", "parent_index": 7, "line_index": 0}, {"bbox": [63, 403, 536, 417], "content": "corollary of the slice theorem on , but second directly using the reduction mapping. Thus,", "parent_index": 7, "line_index": 1}, {"bbox": [63, 417, 301, 432], "content": "altogether the second variant needs less effort.", "parent_index": 7, "line_index": 2}, {"bbox": [63, 436, 196, 449], "content": "Proof Proposition 6.1", "parent_index": 8, "line_index": 0}, {"bbox": [105, 449, 537, 466], "content": "We have to show that any has a neighbourhood that again is contained in", "parent_index": 9, "line_index": 0}, {"bbox": [106, 463, 218, 479], "content": ". So, let .", "parent_index": 9, "line_index": 1}, {"bbox": [106, 480, 171, 492], "content": "• Variant 1", "parent_index": 10, "line_index": 0}, {"bbox": [121, 493, 539, 507], "content": "Due to the slice theorem there is an open neighbourhood of , and so of", "parent_index": 10, "line_index": 1}, {"bbox": [123, 506, 537, 523], "content": ", too, and an equivariant retraction . Since every equivariant", "parent_index": 10, "line_index": 2}, {"bbox": [121, 521, 538, 537], "content": "mapping reduces types, we have for all , thus", "parent_index": 10, "line_index": 3}, {"bbox": [123, 535, 172, 552], "content": ".", "parent_index": 10, "line_index": 4}, {"bbox": [109, 552, 172, 564], "content": "• Variant 2", "parent_index": 10, "line_index": 5}, {"bbox": [122, 565, 313, 580], "content": "Choose again for an with", "parent_index": 10, "line_index": 6}, {"bbox": [168, 579, 492, 595], "content": ".", "parent_index": 10, "line_index": 7}, {"bbox": [122, 594, 537, 610], "content": "Due to the slice theorem for general transformation groups there is an open,", "parent_index": 10, "line_index": 8}, {"bbox": [122, 608, 537, 624], "content": "invariant neighbourhood of in and an equivariant retraction", "parent_index": 10, "line_index": 9}, {"bbox": [123, 622, 464, 639], "content": ". Since and are type-reducing, we have", "parent_index": 10, "line_index": 10}, {"bbox": [151, 638, 508, 656], "content": "", "parent_index": 10, "line_index": 11}, {"bbox": [121, 654, 538, 675], "content": "for all , i.e. . Obviously, contains and is open as", "parent_index": 10, "line_index": 12}, {"bbox": [122, 672, 539, 687], "content": "a preimage of an open set. qed", "parent_index": 10, "line_index": 13}]
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[]
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[{"bbox": [217, 17, 225, 29], "content": "\\overline{{g}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [341, 17, 369, 31], "content": "\\mathbf{B}(\\overline{{A}})", "parent_index": 0, "subtype": "inline"}, {"bbox": [386, 17, 395, 28], "content": "\\overline{{A}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [187, 33, 196, 42], "content": "F", "parent_index": 0, "subtype": "inline"}, {"bbox": [179, 46, 254, 59], "content": "F^{-1}(\\{{\\overline{{A}}}\\})={\\overline{{S}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [201, 59, 278, 74], "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "parent_index": 1, "subtype": "inline"}, {"bbox": [305, 59, 360, 74], "content": "F(\\overline{{A}}^{\\prime})=\\overline{{A}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [407, 73, 537, 88], "content": "f(\\varphi(\\overline{{{A}}}^{\\prime}))\\;=\\;\\varphi(F(\\overline{{{A}}}^{\\prime}))\\;=", "parent_index": 1, "subtype": "inline"}, {"bbox": [200, 88, 226, 102], "content": "\\varphi(\\overline{{A}})", "parent_index": 1, "subtype": "inline"}, {"bbox": [265, 87, 425, 102], "content": "\\overline{{A}}^{\\prime}\\in\\varphi^{-1}(f^{-1}(\\varphi(\\overline{{A}})))=\\varphi_{..}^{-1}(S)", "parent_index": 1, "subtype": "inline"}, {"bbox": [237, 103, 358, 118], "content": "g_{x}\\,:=\\,h_{\\overline{{{A}}}^{\\prime}}(\\gamma_{x})^{-1}\\,\\,h_{\\overline{{{A}}}}(\\gamma_{x})", "parent_index": 1, "subtype": "inline"}, {"bbox": [385, 102, 452, 117], "content": "\\overline{{A}}^{\\prime\\prime}:=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [200, 118, 302, 134], "content": "\\varphi(\\overline{{A}}^{\\prime\\prime})=\\varphi(\\overline{{A}}^{\\prime})\\in S", "parent_index": 1, "subtype": "inline"}, {"bbox": [330, 118, 397, 134], "content": "\\overline{{A}}^{\\prime\\prime}\\in\\varphi^{-1}(S)", "parent_index": 1, "subtype": "inline"}, {"bbox": [426, 120, 518, 135], "content": "h_{\\overline{{{A}}}^{\\prime\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})", "parent_index": 1, "subtype": "inline"}, {"bbox": [216, 141, 224, 148], "content": "x", "parent_index": 1, "subtype": "inline"}, {"bbox": [250, 135, 334, 150], "content": "{\\overline{{A}}}^{\\prime\\prime}\\in\\psi^{-1}(\\psi({\\overline{{A}}}))", "parent_index": 1, "subtype": "inline"}, {"bbox": [386, 134, 428, 150], "content": "\\overline{{A}}^{\\prime\\prime}\\in\\overline{{S}}_{0}", "parent_index": 1, "subtype": "inline"}, {"bbox": [277, 149, 389, 164], "content": "F(\\overline{{A}}^{\\prime\\prime})\\:=\\:\\overline{{A}}\\,=\\,F(\\overline{{A}}^{\\prime})", "parent_index": 1, "subtype": "inline"}, {"bbox": [493, 150, 538, 164], "content": "\\overline{{{A}}}\\circ\\overline{{{g}}}\\ =", "parent_index": 1, "subtype": "inline"}, {"bbox": [200, 164, 384, 179], "content": "F(\\overline{{A}}_{.}^{\\prime})\\circ\\overline{{g}}=F(\\overline{{A}}^{\\prime}\\circ\\overline{{g}})=F(\\overline{{A}}^{\\prime\\prime})=\\overline{{A}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [410, 164, 459, 179], "content": "{\\overline{{g}}}\\in{\\mathbf{B}}({\\overline{{A}}})", "parent_index": 1, "subtype": "inline"}, {"bbox": [213, 179, 374, 193], "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{-1}\\in\\overline{{S}}_{0}\\circ{\\bf B}(\\overline{{A}})=\\overline{{S}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [201, 193, 235, 206], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{S}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [273, 193, 430, 208], "content": "F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}^{\\prime}\\circ{1})=\\overline{{A}}\\circ{1}=\\overline{{A}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [456, 192, 533, 208], "content": "\\overline{{A}}^{\\prime}\\in F^{-1}(\\{\\overline{{A}}\\})", "parent_index": 1, "subtype": "inline"}, {"bbox": [163, 275, 183, 289], "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "parent_index": 3, "subtype": "inline"}, {"bbox": [261, 276, 290, 286], "content": "t\\in\\mathcal T", "parent_index": 3, "subtype": "inline"}, {"bbox": [150, 295, 171, 306], "content": "\\scriptstyle A_{=t}", "parent_index": 4, "subtype": "inline"}, {"bbox": [228, 294, 248, 308], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "parent_index": 4, "subtype": "inline"}, {"bbox": [285, 296, 314, 305], "content": "t\\in\\mathcal T", "parent_index": 4, "subtype": "inline"}, {"bbox": [137, 314, 224, 327], "content": "\\overline{{\\mathcal{A}}}_{=t}=\\overline{{\\mathcal{A}}}_{\\geq t}\\cap\\overline{{\\mathcal{A}}}_{\\leq t}", "parent_index": 5, "subtype": "inline"}, {"bbox": [230, 314, 250, 326], "content": "\\overline{{\\mathcal{A}}}_{=t}", "parent_index": 5, "subtype": "inline"}, {"bbox": [464, 314, 484, 327], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "parent_index": 5, "subtype": "inline"}, {"bbox": [150, 334, 171, 347], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "parent_index": 5, "subtype": "inline"}, {"bbox": [267, 335, 296, 345], "content": "t\\in\\mathcal T", "parent_index": 5, "subtype": "inline"}, {"bbox": [105, 353, 318, 368], "content": "\\begin{array}{r}{\\overline{{\\mathcal{A}}}\\backslash\\overline{{\\mathcal{A}}}_{\\leq t}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{=t^{\\prime}}=\\bigcup_{t^{\\prime}\\in\\mathcal{T},t^{\\prime}\\not\\leq\\;t}\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}}\\end{array}", "parent_index": 6, "subtype": "inline"}, {"bbox": [403, 353, 426, 367], "content": "\\overline{{\\mathcal{A}}}_{\\geq t^{\\prime}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [501, 354, 533, 364], "content": "t^{\\prime}\\in\\mathcal T", "parent_index": 6, "subtype": "inline"}, {"bbox": [139, 369, 159, 383], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "parent_index": 6, "subtype": "inline"}, {"bbox": [232, 403, 242, 413], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [245, 451, 289, 464], "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "parent_index": 9, "subtype": "inline"}, {"bbox": [106, 465, 126, 478], "content": "\\overline{{\\mathcal{A}}}_{\\geq t}", "parent_index": 9, "subtype": "inline"}, {"bbox": [171, 465, 214, 478], "content": "\\overline{{A}}\\in\\overline{{A}}_{\\geq t}", "parent_index": 9, "subtype": "inline"}, {"bbox": [424, 496, 434, 505], "content": "U", "parent_index": 10, "subtype": "inline"}, {"bbox": [452, 494, 481, 506], "content": "\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [123, 508, 132, 519], "content": "\\overline{{A}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [320, 508, 407, 520], "content": "F:U\\longrightarrow\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [298, 522, 424, 536], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})\\,=\\,t", "parent_index": 10, "subtype": "inline"}, {"bbox": [465, 521, 505, 534], "content": "\\overline{{A}}^{\\prime}\\,\\in\\,U", "parent_index": 10, "subtype": "inline"}, {"bbox": [123, 537, 168, 551], "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "parent_index": 10, "subtype": "inline"}, {"bbox": [213, 566, 222, 576], "content": "\\overline{{A}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [242, 568, 285, 578], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [168, 581, 489, 594], "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(h_{\\overline{{A}}}(\\alpha))]\\equiv[Z(\\varphi_{\\alpha}(\\overline{{A}}))]=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))", "parent_index": 10, "subtype": "inline"}, {"bbox": [253, 611, 266, 620], "content": "U^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [284, 609, 317, 623], "content": "\\varphi_{\\alpha}(\\overline{{A}})", "parent_index": 10, "subtype": "inline"}, {"bbox": [335, 609, 361, 621], "content": "\\mathbf{G}^{\\#\\alpha}", "parent_index": 10, "subtype": "inline"}, {"bbox": [521, 609, 537, 622], "content": "f:", "parent_index": 10, "subtype": "inline"}, {"bbox": [123, 623, 216, 637], "content": "U^{\\prime}\\longrightarrow\\varphi_{\\pmb{\\alpha}}(\\overline{{{A}}})\\circ\\mathbf{G}", "parent_index": 10, "subtype": "inline"}, {"bbox": [254, 623, 287, 637], "content": "\\varphi_{\\alpha}(\\overline{{A}})", "parent_index": 10, "subtype": "inline"}, {"bbox": [314, 625, 321, 637], "content": "f", "parent_index": 10, "subtype": "inline"}, {"bbox": [151, 638, 508, 656], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\geq\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\geq\\mathrm{Typ}\\big(f(\\varphi_{\\alpha}(\\overline{{A}}^{\\prime}))\\big)=\\mathrm{Typ}(\\varphi_{\\alpha}(\\overline{{A}}))=\\mathrm{Typ}(\\overline{{A}})", "parent_index": 10, "subtype": "inline"}, {"bbox": [157, 657, 253, 672], "content": "\\overline{{A}}^{\\prime}\\in U:=\\varphi_{\\pmb{\\alpha}}^{-1}(U^{\\prime})", "parent_index": 10, "subtype": "inline"}, {"bbox": [279, 658, 325, 672], "content": "U\\subseteq{\\overline{{A}}}_{\\geq t}", "parent_index": 10, "subtype": "inline"}, {"bbox": [390, 660, 400, 669], "content": "U", "parent_index": 10, "subtype": "inline"}, {"bbox": [450, 658, 459, 668], "content": "\\overline{{A}}", "parent_index": 10, "subtype": "inline"}]
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[]
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# 7 Denseness of the Strata
The next theorem we want to prove is that the set $\overline{{A}}_{=t}$ is not only open, but also dense in $\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$ . This assertion does – in contrast to the slice theorem and the openness of the strata – not follow from the general theory of transformation groups. We have to show this directly on the level of $\overline{{\mathcal{A}}}$ .
As we will see in a moment, the next proposition will be very helpful.
Proposition 7.1 Let ${\overline{{A}}}\in{\overline{{A}}}$ and $\Gamma_{i}$ be finitely many graphs. Then there is for any $t\,\geq\,\mathrm{Typ}(\overline{{A}})$ an $\overline{{A}}^{\prime}\,\in\,\overline{{A}}$ with $\mathrm{Typ}(\overline{{A}}^{\prime})\;=\;t$ and $\pi_{\Gamma_{i}}(\overline{{{A}}})=\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})$ for all $i$ .
Namely, we have
Corollary 7.2 $\overline{{A}}_{=t}$ is dense in $\overline{{\mathcal{A}}}_{\leq t}$ for all $t\in\mathcal T$ .
Proof Let $\overline{{A}}\in\overline{{A}}_{\leq t}\subseteq\overline{{A}}$ . We have to show that any neighbourhood $U$ of $\overline{{A}}$ contains an $\overline{{A}}^{\prime}$ having type $t$ . It is sufficient to prove this assertion for all graphs $\Gamma_{i}$ and all $\begin{array}{r}{U=\bigcap_{i}\pi_{\Gamma_{i}}^{-1}(W_{i})}\end{array}$ with open $W_{i}\subseteq\mathbf{G}^{\#\mathbf{E}(\Gamma_{i})}$ and $\pi_{\Gamma_{i}}(\overline{{A}})\in W_{i}$ for all $i\in I$ with finite $I$ , beca use any general open $U$ contains such a set. Now let $\Gamma_{i}$ and $U$ be chosen as just described. Due to Proposition 7.1 above there exists an $\overline{{A}}^{\prime}\in\overline{{A}}$ with $\mathrm{Typ}(\overline{{A}}^{\prime})=t\geq\mathrm{Typ}(\overline{{A}})$ and $\pi_{\Gamma_{i}}(\overline{{{A}}})=\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})$ for all $i$ , i.e. with $\overline{{A}}^{\prime}\in\overline{{A}}_{=t}$ and $\overline{{A}}^{\prime}\in\pi_{\Gamma_{i}}^{-1}\Big(\pi_{\Gamma_{i}}\big(\{\overline{{A}}\}\big)\Big)\subseteq\pi_{\Gamma_{i}}^{-1}(W_{i})$ for all $i$ , thus, $\overline{{A}}^{\prime}\in\cap_{i}\pi_{\Gamma_{i}}^{-1}(W_{i})=U$ .
Along with the proposition about the openness of the strata we get
Corollary 7.3 For all $t\in\mathcal T$ the closure of $\overline{{\mathcal{A}}}_{=t}$ w.r.t. $\overline{{\mathcal{A}}}$ is equal to $\overline{{\mathcal{A}}}_{\leq t}$ .
Proof Denote the closure of $F$ w.r.t. $E$ by $\operatorname{Cl}_{E}(F)$ . Due to the denseness of $\overline{{A}}_{=t}$ in $\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$ we have $\mathrm{Cl}_{\overline{{A}}_{\leq t}}(\overline{{A}}_{=t})=\overline{{A}}_{\leq t}$ . Since the closure is compatible with the relative topology, we have $\overline{{\mathcal{A}}}_{\leq t}=\mathrm{Cl}_{\overline{{\mathcal{A}}}_{\leq t}}(\overline{{\mathcal{A}}}_{=t})=\overline{{\mathcal{A}}}_{\leq t}\cap\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$ , i.e. $\overline{{\mathcal{A}}}_{\leq t}\,\subseteq\,\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$ . But, due to Corollary 6.3, $\overline{{\mathcal{A}}}_{\leq t}\supseteq\overline{{\mathcal{A}}}_{=t}$ itself is closed in $\overline{{\mathcal{A}}}$ . Hence, $\overline{{\mathcal{A}}}_{\leq t}\supseteq\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$ . qed
# 7.1 How to Prove Proposition 7.1?
Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we get help from the finiteness lemma for centralizers. Namely, let $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ be chosen such that $\mathrm{Typ}(\overline{{A}})=[Z(\mathbf{H}_{\overline{{A}}})]=[Z(\varphi_{\alpha}(\overline{{A}}))]$ . $t\geq\mathrm{Typ}(\overline{{A}})$ is finitely generated as well. Thus, we have to construct a connection whose type is determined by $\varphi_{\alpha}(\overline{{A}})$ and the generators of $t$ . For this we use the induction on the number of generators of $t$ . In conclusion, we have to construct inductively from $\overline{{A}}$ new connections ${\overline{{A}}}_{i}$ , such that $\overline{{A}}_{i-1}$ coincides with ${\overline{{A}}}_{i}$ at least along the paths that pass $\alpha$ or that lie in the graphs $\Gamma_{i}$ . But, at the same time, there has to exist a path ${e}$ , such that $h_{\overline{{A}}_{i}}(e)$ equals the $i$ th generator of $t$ .
Now, it should be obvious that we get help from the construction method for new connections introduced in [10]. Before we do this we recall an important notation used there.
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<html><body>
<h1 data-bbox="63 10 294 29">7 Denseness of the Strata </h1>
<p data-bbox="62 40 538 98">The next theorem we want to prove is that the set $\overline{{A}}_{=t}$ is not only open, but also dense in $\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$ . This assertion does – in contrast to the slice theorem and the openness of the strata – not follow from the general theory of transformation groups. We have to show this directly on the level of $\overline{{\mathcal{A}}}$ . </p>
<p data-bbox="63 98 421 114">As we will see in a moment, the next proposition will be very helpful. </p>
<p data-bbox="63 121 538 168">Proposition 7.1 Let ${\overline{{A}}}\in{\overline{{A}}}$ and $\Gamma_{i}$ be finitely many graphs. Then there is for any $t\,\geq\,\mathrm{Typ}(\overline{{A}})$ an $\overline{{A}}^{\prime}\,\in\,\overline{{A}}$ with $\mathrm{Typ}(\overline{{A}}^{\prime})\;=\;t$ and $\pi_{\Gamma_{i}}(\overline{{{A}}})=\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})$ for all $i$ . </p>
<p data-bbox="62 176 150 191">Namely, we have </p>
<p data-bbox="62 199 322 216">Corollary 7.2 $\overline{{A}}_{=t}$ is dense in $\overline{{\mathcal{A}}}_{\leq t}$ for all $t\in\mathcal T$ . </p>
<p data-bbox="62 226 538 336">Proof Let $\overline{{A}}\in\overline{{A}}_{\leq t}\subseteq\overline{{A}}$ . We have to show that any neighbourhood $U$ of $\overline{{A}}$ contains an $\overline{{A}}^{\prime}$ having type $t$ . It is sufficient to prove this assertion for all graphs $\Gamma_{i}$ and all $\begin{array}{r}{U=\bigcap_{i}\pi_{\Gamma_{i}}^{-1}(W_{i})}\end{array}$ with open $W_{i}\subseteq\mathbf{G}^{\#\mathbf{E}(\Gamma_{i})}$ and $\pi_{\Gamma_{i}}(\overline{{A}})\in W_{i}$ for all $i\in I$ with finite $I$ , beca use any general open $U$ contains such a set. Now let $\Gamma_{i}$ and $U$ be chosen as just described. Due to Proposition 7.1 above there exists an $\overline{{A}}^{\prime}\in\overline{{A}}$ with $\mathrm{Typ}(\overline{{A}}^{\prime})=t\geq\mathrm{Typ}(\overline{{A}})$ and $\pi_{\Gamma_{i}}(\overline{{{A}}})=\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})$ for all $i$ , i.e. with $\overline{{A}}^{\prime}\in\overline{{A}}_{=t}$ and $\overline{{A}}^{\prime}\in\pi_{\Gamma_{i}}^{-1}\Big(\pi_{\Gamma_{i}}\big(\{\overline{{A}}\}\big)\Big)\subseteq\pi_{\Gamma_{i}}^{-1}(W_{i})$ for all $i$ , thus, $\overline{{A}}^{\prime}\in\cap_{i}\pi_{\Gamma_{i}}^{-1}(W_{i})=U$ . </p>
<p data-bbox="63 359 411 374">Along with the proposition about the openness of the strata we get </p>
<p data-bbox="62 382 444 399">Corollary 7.3 For all $t\in\mathcal T$ the closure of $\overline{{\mathcal{A}}}_{=t}$ w.r.t. $\overline{{\mathcal{A}}}$ is equal to $\overline{{\mathcal{A}}}_{\leq t}$ . </p>
<p data-bbox="63 410 538 491">Proof Denote the closure of $F$ w.r.t. $E$ by $\operatorname{Cl}_{E}(F)$ . Due to the denseness of $\overline{{A}}_{=t}$ in $\overline{{\mathbf{\mathcal{A}}}}_{\leq t}$ we have $\mathrm{Cl}_{\overline{{A}}_{\leq t}}(\overline{{A}}_{=t})=\overline{{A}}_{\leq t}$ . Since the closure is compatible with the relative topology, we have $\overline{{\mathcal{A}}}_{\leq t}=\mathrm{Cl}_{\overline{{\mathcal{A}}}_{\leq t}}(\overline{{\mathcal{A}}}_{=t})=\overline{{\mathcal{A}}}_{\leq t}\cap\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$ , i.e. $\overline{{\mathcal{A}}}_{\leq t}\,\subseteq\,\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$ . But, due to Corollary 6.3, $\overline{{\mathcal{A}}}_{\leq t}\supseteq\overline{{\mathcal{A}}}_{=t}$ itself is closed in $\overline{{\mathcal{A}}}$ . Hence, $\overline{{\mathcal{A}}}_{\leq t}\supseteq\mathrm{Cl}_{\overline{{\mathcal{A}}}}(\overline{{\mathcal{A}}}_{=t})$ . qed </p>
<h1 data-bbox="63 507 320 524">7.1 How to Prove Proposition 7.1? </h1>
<p data-bbox="62 531 538 647">Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we get help from the finiteness lemma for centralizers. Namely, let $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ be chosen such that $\mathrm{Typ}(\overline{{A}})=[Z(\mathbf{H}_{\overline{{A}}})]=[Z(\varphi_{\alpha}(\overline{{A}}))]$ . $t\geq\mathrm{Typ}(\overline{{A}})$ is finitely generated as well. Thus, we have to construct a connection whose type is determined by $\varphi_{\alpha}(\overline{{A}})$ and the generators of $t$ . For this we use the induction on the number of generators of $t$ . In conclusion, we have to construct inductively from $\overline{{A}}$ new connections ${\overline{{A}}}_{i}$ , such that $\overline{{A}}_{i-1}$ coincides with ${\overline{{A}}}_{i}$ at least along the paths that pass $\alpha$ or that lie in the graphs $\Gamma_{i}$ . But, at the same time, there has to exist a path ${e}$ , such that $h_{\overline{{A}}_{i}}(e)$ equals the $i$ th generator of $t$ . </p>
<p data-bbox="63 648 537 676">Now, it should be obvious that we get help from the construction method for new connections introduced in [10]. Before we do this we recall an important notation used there. </p>
</body></html>
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0001008v1
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[{"type": "text", "text": "7 Denseness of the Strata ", "text_level": 1, "page_idx": 11}, {"type": "text", "text": "The next theorem we want to prove is that the set $\\overline{{A}}_{=t}$ is not only open, but also dense in $\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$ . This assertion does – in contrast to the slice theorem and the openness of the strata – not follow from the general theory of transformation groups. We have to show this directly on the level of $\\overline{{\\mathcal{A}}}$ . ", "page_idx": 11}, {"type": "text", "text": "As we will see in a moment, the next proposition will be very helpful. ", "page_idx": 11}, {"type": "text", "text": "Proposition 7.1 Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ and $\\Gamma_{i}$ be finitely many graphs. Then there is for any $t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})$ an $\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}$ with $\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t$ and $\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})$ for all $i$ . ", "page_idx": 11}, {"type": "text", "text": "Namely, we have ", "page_idx": 11}, {"type": "text", "text": "Corollary 7.2 $\\overline{{A}}_{=t}$ is dense in $\\overline{{\\mathcal{A}}}_{\\leq t}$ for all $t\\in\\mathcal T$ . ", "page_idx": 11}, {"type": "text", "text": "Proof Let $\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}$ . We have to show that any neighbourhood $U$ of $\\overline{{A}}$ contains an $\\overline{{A}}^{\\prime}$ having type $t$ . It is sufficient to prove this assertion for all graphs $\\Gamma_{i}$ and all $\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}$ with open $W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}$ and $\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}$ for all $i\\in I$ with finite $I$ , beca use any general open $U$ contains such a set. Now let $\\Gamma_{i}$ and $U$ be chosen as just described. Due to Proposition 7.1 above there exists an $\\overline{{A}}^{\\prime}\\in\\overline{{A}}$ with $\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})$ and $\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})$ for all $i$ , i.e. with $\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}$ and $\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})$ for all $i$ , thus, $\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U$ . ", "page_idx": 11}, {"type": "text", "text": "Along with the proposition about the openness of the strata we get ", "page_idx": 11}, {"type": "text", "text": "Corollary 7.3 For all $t\\in\\mathcal T$ the closure of $\\overline{{\\mathcal{A}}}_{=t}$ w.r.t. $\\overline{{\\mathcal{A}}}$ is equal to $\\overline{{\\mathcal{A}}}_{\\leq t}$ . ", "page_idx": 11}, {"type": "text", "text": "Proof Denote the closure of $F$ w.r.t. $E$ by $\\operatorname{Cl}_{E}(F)$ . Due to the denseness of $\\overline{{A}}_{=t}$ in $\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}$ we have $\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}$ . Since the closure is compatible with the relative topology, we have $\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$ , i.e. $\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$ . But, due to Corollary 6.3, $\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}$ itself is closed in $\\overline{{\\mathcal{A}}}$ . Hence, $\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})$ . qed ", "page_idx": 11}, {"type": "text", "text": "7.1 How to Prove Proposition 7.1? ", "text_level": 1, "page_idx": 11}, {"type": "text", "text": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we get help from the finiteness lemma for centralizers. Namely, let $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ be chosen such that $\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]$ . $t\\geq\\mathrm{Typ}(\\overline{{A}})$ is finitely generated as well. Thus, we have to construct a connection whose type is determined by $\\varphi_{\\alpha}(\\overline{{A}})$ and the generators of $t$ . For this we use the induction on the number of generators of $t$ . In conclusion, we have to construct inductively from $\\overline{{A}}$ new connections ${\\overline{{A}}}_{i}$ , such that $\\overline{{A}}_{i-1}$ coincides with ${\\overline{{A}}}_{i}$ at least along the paths that pass $\\alpha$ or that lie in the graphs $\\Gamma_{i}$ . But, at the same time, there has to exist a path ${e}$ , such that $h_{\\overline{{A}}_{i}}(e)$ equals the $i$ th generator of $t$ . ", "page_idx": 11}, {"type": "text", "text": "Now, it should be obvious that we get help from the construction method for new connections introduced in [10]. Before we do this we recall an important notation used there. ", "page_idx": 11}]
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{"preproc_blocks": [{"type": "title", "bbox": [63, 10, 294, 29], "lines": [{"bbox": [65, 14, 293, 29], "spans": [{"bbox": [65, 16, 75, 27], "score": 1.0, "content": "7", "type": "text"}, {"bbox": [87, 14, 293, 29], "score": 1.0, "content": "Denseness of the Strata", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [62, 40, 538, 98], "lines": [{"bbox": [62, 43, 537, 57], "spans": [{"bbox": [62, 43, 331, 57], "score": 1.0, "content": "The next theorem we want to prove is that the set ", "type": "text"}, {"bbox": [332, 43, 351, 56], "score": 0.93, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [352, 43, 537, 57], "score": 1.0, "content": " is not only open, but also dense in", "type": "text"}], "index": 1}, {"bbox": [63, 56, 537, 73], "spans": [{"bbox": [63, 58, 83, 72], "score": 0.93, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [83, 56, 537, 73], "score": 1.0, "content": ". This assertion does – in contrast to the slice theorem and the openness of the strata –", "type": "text"}], "index": 2}, {"bbox": [62, 72, 537, 87], "spans": [{"bbox": [62, 72, 537, 87], "score": 1.0, "content": "not follow from the general theory of transformation groups. We have to show this directly", "type": "text"}], "index": 3}, {"bbox": [62, 86, 154, 100], "spans": [{"bbox": [62, 86, 139, 100], "score": 1.0, "content": "on the level of ", "type": "text"}, {"bbox": [140, 87, 149, 97], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [150, 86, 154, 100], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 2.5}, {"type": "text", "bbox": [63, 98, 421, 114], "lines": [{"bbox": [63, 101, 420, 115], "spans": [{"bbox": [63, 101, 420, 115], "score": 1.0, "content": "As we will see in a moment, the next proposition will be very helpful.", "type": "text"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [63, 121, 538, 168], "lines": [{"bbox": [62, 124, 383, 141], "spans": [{"bbox": [62, 124, 184, 141], "score": 1.0, "content": "Proposition 7.1 Let ", "type": "text"}, {"bbox": [184, 126, 218, 137], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [218, 124, 244, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 128, 255, 138], "score": 0.9, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [255, 124, 383, 141], "score": 1.0, "content": " be finitely many graphs.", "type": "text"}], "index": 6}, {"bbox": [162, 139, 537, 154], "spans": [{"bbox": [162, 139, 282, 154], "score": 1.0, "content": "Then there is for any ", "type": "text"}, {"bbox": [283, 140, 348, 154], "score": 0.89, "content": "t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 65}, {"bbox": [348, 139, 370, 154], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [370, 139, 412, 151], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [412, 139, 444, 154], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 139, 512, 154], "score": 0.84, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t", "type": "inline_equation", "height": 15, "width": 68}, {"bbox": [513, 139, 537, 154], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [164, 153, 296, 168], "spans": [{"bbox": [164, 154, 249, 168], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [249, 153, 286, 168], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [287, 156, 291, 165], "score": 0.87, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [292, 153, 296, 168], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7}, {"type": "text", "bbox": [62, 176, 150, 191], "lines": [{"bbox": [62, 178, 150, 193], "spans": [{"bbox": [62, 178, 150, 193], "score": 1.0, "content": "Namely, we have", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [62, 199, 322, 216], "lines": [{"bbox": [63, 202, 322, 217], "spans": [{"bbox": [63, 202, 150, 216], "score": 1.0, "content": "Corollary 7.2", "type": "text"}, {"bbox": [150, 203, 171, 216], "score": 0.9, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [171, 202, 231, 216], "score": 1.0, "content": " is dense in ", "type": "text"}, {"bbox": [232, 203, 252, 217], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [252, 202, 289, 216], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [289, 205, 318, 214], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [318, 202, 322, 216], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [62, 226, 538, 336], "lines": [{"bbox": [61, 228, 538, 245], "spans": [{"bbox": [61, 228, 128, 245], "score": 1.0, "content": "Proof Let ", "type": "text"}, {"bbox": [128, 231, 202, 244], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [202, 228, 435, 245], "score": 1.0, "content": ". We have to show that any neighbourhood ", "type": "text"}, {"bbox": [435, 232, 445, 241], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [445, 228, 463, 245], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [463, 231, 473, 241], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [473, 228, 538, 245], "score": 1.0, "content": " contains an", "type": "text"}], "index": 11}, {"bbox": [106, 243, 539, 262], "spans": [{"bbox": [106, 245, 118, 257], "score": 0.87, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [119, 243, 189, 262], "score": 1.0, "content": " having type ", "type": "text"}, {"bbox": [190, 249, 194, 257], "score": 0.84, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 243, 483, 262], "score": 1.0, "content": ". It is sufficient to prove this assertion for all graphs ", "type": "text"}, {"bbox": [483, 248, 494, 258], "score": 0.91, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [495, 243, 539, 262], "score": 1.0, "content": " and all", "type": "text"}], "index": 12}, {"bbox": [106, 257, 539, 277], "spans": [{"bbox": [106, 261, 187, 275], "score": 0.93, "content": "\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [187, 257, 245, 277], "score": 1.0, "content": " with open ", "type": "text"}, {"bbox": [245, 260, 315, 273], "score": 0.94, "content": "W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [316, 257, 341, 277], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [342, 260, 404, 274], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [405, 257, 441, 277], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [442, 263, 466, 271], "score": 0.92, "content": "i\\in I", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [467, 257, 526, 277], "score": 1.0, "content": " with finite ", "type": "text"}, {"bbox": [527, 263, 533, 271], "score": 0.86, "content": "I", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [533, 257, 539, 277], "score": 1.0, "content": ",", "type": "text"}], "index": 13}, {"bbox": [104, 274, 355, 292], "spans": [{"bbox": [104, 274, 240, 292], "score": 1.0, "content": "beca use any general open ", "type": "text"}, {"bbox": [240, 277, 250, 286], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [250, 274, 355, 292], "score": 1.0, "content": " contains such a set.", "type": "text"}], "index": 14}, {"bbox": [105, 289, 538, 305], "spans": [{"bbox": [105, 289, 150, 305], "score": 1.0, "content": "Now let ", "type": "text"}, {"bbox": [151, 290, 162, 302], "score": 0.85, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [162, 289, 189, 305], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [189, 291, 199, 300], "score": 0.86, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [199, 289, 538, 305], "score": 1.0, "content": " be chosen as just described. Due to Proposition 7.1 above there", "type": "text"}], "index": 15}, {"bbox": [105, 300, 538, 319], "spans": [{"bbox": [105, 300, 153, 319], "score": 1.0, "content": "exists an ", "type": "text"}, {"bbox": [154, 303, 190, 315], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [190, 300, 218, 319], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [218, 303, 335, 317], "score": 0.93, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 117}, {"bbox": [335, 300, 360, 319], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 303, 446, 317], "score": 0.93, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [447, 300, 482, 319], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [482, 307, 487, 315], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [487, 300, 538, 319], "score": 1.0, "content": ", i.e. with", "type": "text"}], "index": 16}, {"bbox": [106, 316, 538, 336], "spans": [{"bbox": [106, 319, 154, 332], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [154, 316, 180, 336], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [181, 318, 344, 336], "score": 0.94, "content": "\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})", "type": "inline_equation", "height": 18, "width": 163}, {"bbox": [344, 316, 381, 336], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [382, 321, 387, 331], "score": 0.74, "content": "i", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [387, 316, 423, 336], "score": 1.0, "content": ", thus, ", "type": "text"}, {"bbox": [423, 319, 533, 335], "score": 0.94, "content": "\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U", "type": "inline_equation", "height": 16, "width": 110}, {"bbox": [533, 316, 538, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 14}, {"type": "text", "bbox": [63, 359, 411, 374], "lines": [{"bbox": [63, 361, 410, 376], "spans": [{"bbox": [63, 361, 410, 376], "score": 1.0, "content": "Along with the proposition about the openness of the strata we get", "type": "text"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [62, 382, 444, 399], "lines": [{"bbox": [62, 384, 443, 401], "spans": [{"bbox": [62, 384, 187, 401], "score": 1.0, "content": "Corollary 7.3 For all ", "type": "text"}, {"bbox": [188, 388, 217, 397], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [217, 384, 292, 401], "score": 1.0, "content": " the closure of ", "type": "text"}, {"bbox": [293, 385, 313, 398], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [313, 384, 348, 401], "score": 1.0, "content": " w.r.t. ", "type": "text"}, {"bbox": [348, 385, 359, 397], "score": 0.76, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 384, 419, 401], "score": 1.0, "content": " is equal to ", "type": "text"}, {"bbox": [419, 385, 439, 400], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 15, "width": 20}, {"bbox": [440, 384, 443, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [63, 410, 538, 491], "lines": [{"bbox": [62, 413, 334, 428], "spans": [{"bbox": [62, 413, 218, 428], "score": 1.0, "content": "Proof Denote the closure of ", "type": "text"}, {"bbox": [219, 415, 228, 424], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [228, 413, 264, 428], "score": 1.0, "content": " w.r.t. ", "type": "text"}, {"bbox": [265, 415, 274, 424], "score": 0.88, "content": "E", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [275, 413, 294, 428], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [294, 414, 332, 427], "score": 0.92, "content": "\\operatorname{Cl}_{E}(F)", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [332, 413, 334, 428], "score": 1.0, "content": ".", "type": "text"}], "index": 20}, {"bbox": [104, 425, 540, 447], "spans": [{"bbox": [104, 425, 231, 447], "score": 1.0, "content": "Due to the denseness of ", "type": "text"}, {"bbox": [232, 428, 252, 440], "score": 0.92, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [252, 425, 268, 447], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [269, 428, 289, 442], "score": 0.92, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [289, 425, 336, 447], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [337, 427, 430, 444], "score": 0.91, "content": "\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}", "type": "inline_equation", "height": 17, "width": 93}, {"bbox": [430, 425, 540, 447], "score": 1.0, "content": ". Since the closure is", "type": "text"}], "index": 21}, {"bbox": [105, 443, 538, 463], "spans": [{"bbox": [105, 443, 346, 463], "score": 1.0, "content": "compatible with the relative topology, we have ", "type": "text"}, {"bbox": [346, 444, 533, 461], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 17, "width": 187}, {"bbox": [534, 443, 538, 463], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [105, 461, 538, 478], "spans": [{"bbox": [105, 461, 128, 478], "score": 1.0, "content": "i.e. ", "type": "text"}, {"bbox": [129, 462, 216, 476], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 14, "width": 87}, {"bbox": [216, 461, 367, 478], "score": 1.0, "content": ". But, due to Corollary 6.3, ", "type": "text"}, {"bbox": [368, 462, 427, 477], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [427, 461, 523, 478], "score": 1.0, "content": " itself is closed in ", "type": "text"}, {"bbox": [523, 462, 533, 473], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [533, 461, 538, 478], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [106, 477, 537, 492], "spans": [{"bbox": [106, 477, 144, 492], "score": 1.0, "content": "Hence, ", "type": "text"}, {"bbox": [144, 477, 228, 491], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [228, 477, 232, 492], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 478, 537, 491], "score": 1.0, "content": "qed", "type": "text"}], "index": 24}], "index": 22}, {"type": "title", "bbox": [63, 507, 320, 524], "lines": [{"bbox": [63, 510, 318, 524], "spans": [{"bbox": [63, 510, 87, 523], "score": 1.0, "content": "7.1", "type": "text"}, {"bbox": [98, 510, 318, 524], "score": 1.0, "content": "How to Prove Proposition 7.1?", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [62, 531, 538, 647], "lines": [{"bbox": [62, 533, 537, 548], "spans": [{"bbox": [62, 533, 537, 548], "score": 1.0, "content": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we", "type": "text"}], "index": 26}, {"bbox": [62, 548, 537, 561], "spans": [{"bbox": [62, 548, 388, 561], "score": 1.0, "content": "get help from the finiteness lemma for centralizers. Namely, let ", "type": "text"}, {"bbox": [389, 550, 431, 560], "score": 0.94, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [432, 548, 537, 561], "score": 1.0, "content": " be chosen such that", "type": "text"}], "index": 27}, {"bbox": [63, 562, 538, 577], "spans": [{"bbox": [63, 562, 232, 576], "score": 0.91, "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]", "type": "inline_equation", "height": 14, "width": 169}, {"bbox": [232, 562, 240, 577], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [240, 562, 299, 576], "score": 0.9, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [300, 562, 538, 577], "score": 1.0, "content": " is finitely generated as well. Thus, we have to", "type": "text"}], "index": 28}, {"bbox": [61, 577, 538, 591], "spans": [{"bbox": [61, 577, 333, 591], "score": 1.0, "content": "construct a connection whose type is determined by ", "type": "text"}, {"bbox": [333, 577, 366, 591], "score": 0.94, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [366, 577, 483, 591], "score": 1.0, "content": " and the generators of ", "type": "text"}, {"bbox": [483, 580, 487, 588], "score": 0.89, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [488, 577, 538, 591], "score": 1.0, "content": ". For this", "type": "text"}], "index": 29}, {"bbox": [63, 592, 537, 605], "spans": [{"bbox": [63, 592, 339, 605], "score": 1.0, "content": "we use the induction on the number of generators of ", "type": "text"}, {"bbox": [339, 594, 343, 602], "score": 0.88, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [344, 592, 537, 605], "score": 1.0, "content": ". In conclusion, we have to construct", "type": "text"}], "index": 30}, {"bbox": [62, 605, 537, 619], "spans": [{"bbox": [62, 605, 151, 619], "score": 1.0, "content": "inductively from ", "type": "text"}, {"bbox": [151, 606, 160, 616], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [161, 605, 252, 619], "score": 1.0, "content": " new connections ", "type": "text"}, {"bbox": [252, 606, 265, 618], "score": 0.92, "content": "{\\overline{{A}}}_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [265, 605, 325, 619], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [325, 606, 348, 619], "score": 0.94, "content": "\\overline{{A}}_{i-1}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [348, 605, 429, 619], "score": 1.0, "content": " coincides with ", "type": "text"}, {"bbox": [429, 606, 441, 618], "score": 0.93, "content": "{\\overline{{A}}}_{i}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [442, 605, 537, 619], "score": 1.0, "content": " at least along the", "type": "text"}], "index": 31}, {"bbox": [62, 620, 538, 635], "spans": [{"bbox": [62, 620, 147, 635], "score": 1.0, "content": "paths that pass ", "type": "text"}, {"bbox": [147, 625, 156, 631], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [157, 620, 291, 635], "score": 1.0, "content": " or that lie in the graphs ", "type": "text"}, {"bbox": [291, 622, 302, 632], "score": 0.91, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [302, 620, 538, 635], "score": 1.0, "content": ". But, at the same time, there has to exist a", "type": "text"}], "index": 32}, {"bbox": [62, 635, 339, 650], "spans": [{"bbox": [62, 635, 90, 650], "score": 1.0, "content": "path ", "type": "text"}, {"bbox": [90, 640, 96, 645], "score": 0.86, "content": "{e}", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [96, 635, 154, 650], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [154, 636, 186, 650], "score": 0.95, "content": "h_{\\overline{{A}}_{i}}(e)", "type": "inline_equation", "height": 14, "width": 32}, {"bbox": [186, 635, 245, 650], "score": 1.0, "content": " equals the ", "type": "text"}, {"bbox": [245, 637, 249, 645], "score": 0.84, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [250, 635, 329, 650], "score": 1.0, "content": "th generator of ", "type": "text"}, {"bbox": [330, 637, 334, 645], "score": 0.89, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [335, 635, 339, 650], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 29.5}, {"type": "text", "bbox": [63, 648, 537, 676], "lines": [{"bbox": [63, 649, 536, 663], "spans": [{"bbox": [63, 649, 536, 663], "score": 1.0, "content": "Now, it should be obvious that we get help from the construction method for new connections", "type": "text"}], "index": 34}, {"bbox": [63, 664, 478, 678], "spans": [{"bbox": [63, 664, 478, 678], "score": 1.0, "content": "introduced in [10]. Before we do this we recall an important notation used there.", "type": "text"}], "index": 35}], "index": 34.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [63, 10, 294, 29], "lines": [{"bbox": [65, 14, 293, 29], "spans": [{"bbox": [65, 16, 75, 27], "score": 1.0, "content": "7", "type": "text"}, {"bbox": [87, 14, 293, 29], "score": 1.0, "content": "Denseness of the Strata", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [62, 40, 538, 98], "lines": [{"bbox": [62, 43, 537, 57], "spans": [{"bbox": [62, 43, 331, 57], "score": 1.0, "content": "The next theorem we want to prove is that the set ", "type": "text"}, {"bbox": [332, 43, 351, 56], "score": 0.93, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [352, 43, 537, 57], "score": 1.0, "content": " is not only open, but also dense in", "type": "text"}], "index": 1}, {"bbox": [63, 56, 537, 73], "spans": [{"bbox": [63, 58, 83, 72], "score": 0.93, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [83, 56, 537, 73], "score": 1.0, "content": ". This assertion does – in contrast to the slice theorem and the openness of the strata –", "type": "text"}], "index": 2}, {"bbox": [62, 72, 537, 87], "spans": [{"bbox": [62, 72, 537, 87], "score": 1.0, "content": "not follow from the general theory of transformation groups. We have to show this directly", "type": "text"}], "index": 3}, {"bbox": [62, 86, 154, 100], "spans": [{"bbox": [62, 86, 139, 100], "score": 1.0, "content": "on the level of ", "type": "text"}, {"bbox": [140, 87, 149, 97], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [150, 86, 154, 100], "score": 1.0, "content": ".", "type": "text"}], "index": 4}], "index": 2.5, "bbox_fs": [62, 43, 537, 100]}, {"type": "text", "bbox": [63, 98, 421, 114], "lines": [{"bbox": [63, 101, 420, 115], "spans": [{"bbox": [63, 101, 420, 115], "score": 1.0, "content": "As we will see in a moment, the next proposition will be very helpful.", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [63, 101, 420, 115]}, {"type": "text", "bbox": [63, 121, 538, 168], "lines": [{"bbox": [62, 124, 383, 141], "spans": [{"bbox": [62, 124, 184, 141], "score": 1.0, "content": "Proposition 7.1 Let ", "type": "text"}, {"bbox": [184, 126, 218, 137], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [218, 124, 244, 141], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [244, 128, 255, 138], "score": 0.9, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [255, 124, 383, 141], "score": 1.0, "content": " be finitely many graphs.", "type": "text"}], "index": 6}, {"bbox": [162, 139, 537, 154], "spans": [{"bbox": [162, 139, 282, 154], "score": 1.0, "content": "Then there is for any ", "type": "text"}, {"bbox": [283, 140, 348, 154], "score": 0.89, "content": "t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 65}, {"bbox": [348, 139, 370, 154], "score": 1.0, "content": " an ", "type": "text"}, {"bbox": [370, 139, 412, 151], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [412, 139, 444, 154], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [444, 139, 512, 154], "score": 0.84, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t", "type": "inline_equation", "height": 15, "width": 68}, {"bbox": [513, 139, 537, 154], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [164, 153, 296, 168], "spans": [{"bbox": [164, 154, 249, 168], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [249, 153, 286, 168], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [287, 156, 291, 165], "score": 0.87, "content": "i", "type": "inline_equation", "height": 9, "width": 4}, {"bbox": [292, 153, 296, 168], "score": 1.0, "content": ".", "type": "text"}], "index": 8}], "index": 7, "bbox_fs": [62, 124, 537, 168]}, {"type": "text", "bbox": [62, 176, 150, 191], "lines": [{"bbox": [62, 178, 150, 193], "spans": [{"bbox": [62, 178, 150, 193], "score": 1.0, "content": "Namely, we have", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [62, 178, 150, 193]}, {"type": "text", "bbox": [62, 199, 322, 216], "lines": [{"bbox": [63, 202, 322, 217], "spans": [{"bbox": [63, 202, 150, 216], "score": 1.0, "content": "Corollary 7.2", "type": "text"}, {"bbox": [150, 203, 171, 216], "score": 0.9, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [171, 202, 231, 216], "score": 1.0, "content": " is dense in ", "type": "text"}, {"bbox": [232, 203, 252, 217], "score": 0.92, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [252, 202, 289, 216], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [289, 205, 318, 214], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [318, 202, 322, 216], "score": 1.0, "content": ".", "type": "text"}], "index": 10}], "index": 10, "bbox_fs": [63, 202, 322, 217]}, {"type": "text", "bbox": [62, 226, 538, 336], "lines": [{"bbox": [61, 228, 538, 245], "spans": [{"bbox": [61, 228, 128, 245], "score": 1.0, "content": "Proof Let ", "type": "text"}, {"bbox": [128, 231, 202, 244], "score": 0.94, "content": "\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [202, 228, 435, 245], "score": 1.0, "content": ". We have to show that any neighbourhood ", "type": "text"}, {"bbox": [435, 232, 445, 241], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [445, 228, 463, 245], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [463, 231, 473, 241], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [473, 228, 538, 245], "score": 1.0, "content": " contains an", "type": "text"}], "index": 11}, {"bbox": [106, 243, 539, 262], "spans": [{"bbox": [106, 245, 118, 257], "score": 0.87, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [119, 243, 189, 262], "score": 1.0, "content": " having type ", "type": "text"}, {"bbox": [190, 249, 194, 257], "score": 0.84, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [195, 243, 483, 262], "score": 1.0, "content": ". It is sufficient to prove this assertion for all graphs ", "type": "text"}, {"bbox": [483, 248, 494, 258], "score": 0.91, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [495, 243, 539, 262], "score": 1.0, "content": " and all", "type": "text"}], "index": 12}, {"bbox": [106, 257, 539, 277], "spans": [{"bbox": [106, 261, 187, 275], "score": 0.93, "content": "\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [187, 257, 245, 277], "score": 1.0, "content": " with open ", "type": "text"}, {"bbox": [245, 260, 315, 273], "score": 0.94, "content": "W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [316, 257, 341, 277], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [342, 260, 404, 274], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [405, 257, 441, 277], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [442, 263, 466, 271], "score": 0.92, "content": "i\\in I", "type": "inline_equation", "height": 8, "width": 24}, {"bbox": [467, 257, 526, 277], "score": 1.0, "content": " with finite ", "type": "text"}, {"bbox": [527, 263, 533, 271], "score": 0.86, "content": "I", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [533, 257, 539, 277], "score": 1.0, "content": ",", "type": "text"}], "index": 13}, {"bbox": [104, 274, 355, 292], "spans": [{"bbox": [104, 274, 240, 292], "score": 1.0, "content": "beca use any general open ", "type": "text"}, {"bbox": [240, 277, 250, 286], "score": 0.91, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [250, 274, 355, 292], "score": 1.0, "content": " contains such a set.", "type": "text"}], "index": 14}, {"bbox": [105, 289, 538, 305], "spans": [{"bbox": [105, 289, 150, 305], "score": 1.0, "content": "Now let ", "type": "text"}, {"bbox": [151, 290, 162, 302], "score": 0.85, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [162, 289, 189, 305], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [189, 291, 199, 300], "score": 0.86, "content": "U", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [199, 289, 538, 305], "score": 1.0, "content": " be chosen as just described. Due to Proposition 7.1 above there", "type": "text"}], "index": 15}, {"bbox": [105, 300, 538, 319], "spans": [{"bbox": [105, 300, 153, 319], "score": 1.0, "content": "exists an ", "type": "text"}, {"bbox": [154, 303, 190, 315], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [190, 300, 218, 319], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [218, 303, 335, 317], "score": 0.93, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 117}, {"bbox": [335, 300, 360, 319], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [360, 303, 446, 317], "score": 0.93, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [447, 300, 482, 319], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [482, 307, 487, 315], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 5}, {"bbox": [487, 300, 538, 319], "score": 1.0, "content": ", i.e. with", "type": "text"}], "index": 16}, {"bbox": [106, 316, 538, 336], "spans": [{"bbox": [106, 319, 154, 332], "score": 0.92, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 48}, {"bbox": [154, 316, 180, 336], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [181, 318, 344, 336], "score": 0.94, "content": "\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})", "type": "inline_equation", "height": 18, "width": 163}, {"bbox": [344, 316, 381, 336], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [382, 321, 387, 331], "score": 0.74, "content": "i", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [387, 316, 423, 336], "score": 1.0, "content": ", thus, ", "type": "text"}, {"bbox": [423, 319, 533, 335], "score": 0.94, "content": "\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U", "type": "inline_equation", "height": 16, "width": 110}, {"bbox": [533, 316, 538, 336], "score": 1.0, "content": ".", "type": "text"}], "index": 17}], "index": 14, "bbox_fs": [61, 228, 539, 336]}, {"type": "text", "bbox": [63, 359, 411, 374], "lines": [{"bbox": [63, 361, 410, 376], "spans": [{"bbox": [63, 361, 410, 376], "score": 1.0, "content": "Along with the proposition about the openness of the strata we get", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [63, 361, 410, 376]}, {"type": "text", "bbox": [62, 382, 444, 399], "lines": [{"bbox": [62, 384, 443, 401], "spans": [{"bbox": [62, 384, 187, 401], "score": 1.0, "content": "Corollary 7.3 For all ", "type": "text"}, {"bbox": [188, 388, 217, 397], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [217, 384, 292, 401], "score": 1.0, "content": " the closure of ", "type": "text"}, {"bbox": [293, 385, 313, 398], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 13, "width": 20}, {"bbox": [313, 384, 348, 401], "score": 1.0, "content": " w.r.t. ", "type": "text"}, {"bbox": [348, 385, 359, 397], "score": 0.76, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [359, 384, 419, 401], "score": 1.0, "content": " is equal to ", "type": "text"}, {"bbox": [419, 385, 439, 400], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "type": "inline_equation", "height": 15, "width": 20}, {"bbox": [440, 384, 443, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 19, "bbox_fs": [62, 384, 443, 401]}, {"type": "text", "bbox": [63, 410, 538, 491], "lines": [{"bbox": [62, 413, 334, 428], "spans": [{"bbox": [62, 413, 218, 428], "score": 1.0, "content": "Proof Denote the closure of ", "type": "text"}, {"bbox": [219, 415, 228, 424], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [228, 413, 264, 428], "score": 1.0, "content": " w.r.t. ", "type": "text"}, {"bbox": [265, 415, 274, 424], "score": 0.88, "content": "E", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [275, 413, 294, 428], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [294, 414, 332, 427], "score": 0.92, "content": "\\operatorname{Cl}_{E}(F)", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [332, 413, 334, 428], "score": 1.0, "content": ".", "type": "text"}], "index": 20}, {"bbox": [104, 425, 540, 447], "spans": [{"bbox": [104, 425, 231, 447], "score": 1.0, "content": "Due to the denseness of ", "type": "text"}, {"bbox": [232, 428, 252, 440], "score": 0.92, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [252, 425, 268, 447], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [269, 428, 289, 442], "score": 0.92, "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "type": "inline_equation", "height": 14, "width": 20}, {"bbox": [289, 425, 336, 447], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [337, 427, 430, 444], "score": 0.91, "content": "\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}", "type": "inline_equation", "height": 17, "width": 93}, {"bbox": [430, 425, 540, 447], "score": 1.0, "content": ". Since the closure is", "type": "text"}], "index": 21}, {"bbox": [105, 443, 538, 463], "spans": [{"bbox": [105, 443, 346, 463], "score": 1.0, "content": "compatible with the relative topology, we have ", "type": "text"}, {"bbox": [346, 444, 533, 461], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 17, "width": 187}, {"bbox": [534, 443, 538, 463], "score": 1.0, "content": ",", "type": "text"}], "index": 22}, {"bbox": [105, 461, 538, 478], "spans": [{"bbox": [105, 461, 128, 478], "score": 1.0, "content": "i.e. ", "type": "text"}, {"bbox": [129, 462, 216, 476], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 14, "width": 87}, {"bbox": [216, 461, 367, 478], "score": 1.0, "content": ". But, due to Corollary 6.3, ", "type": "text"}, {"bbox": [368, 462, 427, 477], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}", "type": "inline_equation", "height": 15, "width": 59}, {"bbox": [427, 461, 523, 478], "score": 1.0, "content": " itself is closed in ", "type": "text"}, {"bbox": [523, 462, 533, 473], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [533, 461, 538, 478], "score": 1.0, "content": ".", "type": "text"}], "index": 23}, {"bbox": [106, 477, 537, 492], "spans": [{"bbox": [106, 477, 144, 492], "score": 1.0, "content": "Hence, ", "type": "text"}, {"bbox": [144, 477, 228, 491], "score": 0.93, "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [228, 477, 232, 492], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 478, 537, 491], "score": 1.0, "content": "qed", "type": "text"}], "index": 24}], "index": 22, "bbox_fs": [62, 413, 540, 492]}, {"type": "title", "bbox": [63, 507, 320, 524], "lines": [{"bbox": [63, 510, 318, 524], "spans": [{"bbox": [63, 510, 87, 523], "score": 1.0, "content": "7.1", "type": "text"}, {"bbox": [98, 510, 318, 524], "score": 1.0, "content": "How to Prove Proposition 7.1?", "type": "text"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [62, 531, 538, 647], "lines": [{"bbox": [62, 533, 537, 548], "spans": [{"bbox": [62, 533, 537, 548], "score": 1.0, "content": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we", "type": "text"}], "index": 26}, {"bbox": [62, 548, 537, 561], "spans": [{"bbox": [62, 548, 388, 561], "score": 1.0, "content": "get help from the finiteness lemma for centralizers. Namely, let ", "type": "text"}, {"bbox": [389, 550, 431, 560], "score": 0.94, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [432, 548, 537, 561], "score": 1.0, "content": " be chosen such that", "type": "text"}], "index": 27}, {"bbox": [63, 562, 538, 577], "spans": [{"bbox": [63, 562, 232, 576], "score": 0.91, "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]", "type": "inline_equation", "height": 14, "width": 169}, {"bbox": [232, 562, 240, 577], "score": 1.0, "content": ". ", "type": "text"}, {"bbox": [240, 562, 299, 576], "score": 0.9, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [300, 562, 538, 577], "score": 1.0, "content": " is finitely generated as well. Thus, we have to", "type": "text"}], "index": 28}, {"bbox": [61, 577, 538, 591], "spans": [{"bbox": [61, 577, 333, 591], "score": 1.0, "content": "construct a connection whose type is determined by ", "type": "text"}, {"bbox": [333, 577, 366, 591], "score": 0.94, "content": "\\varphi_{\\alpha}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [366, 577, 483, 591], "score": 1.0, "content": " and the generators of ", "type": "text"}, {"bbox": [483, 580, 487, 588], "score": 0.89, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [488, 577, 538, 591], "score": 1.0, "content": ". For this", "type": "text"}], "index": 29}, {"bbox": [63, 592, 537, 605], "spans": [{"bbox": [63, 592, 339, 605], "score": 1.0, "content": "we use the induction on the number of generators of ", "type": "text"}, {"bbox": [339, 594, 343, 602], "score": 0.88, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [344, 592, 537, 605], "score": 1.0, "content": ". In conclusion, we have to construct", "type": "text"}], "index": 30}, {"bbox": [62, 605, 537, 619], "spans": [{"bbox": [62, 605, 151, 619], "score": 1.0, "content": "inductively from ", "type": "text"}, {"bbox": [151, 606, 160, 616], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [161, 605, 252, 619], "score": 1.0, "content": " new connections ", "type": "text"}, {"bbox": [252, 606, 265, 618], "score": 0.92, "content": "{\\overline{{A}}}_{i}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [265, 605, 325, 619], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [325, 606, 348, 619], "score": 0.94, "content": "\\overline{{A}}_{i-1}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [348, 605, 429, 619], "score": 1.0, "content": " coincides with ", "type": "text"}, {"bbox": [429, 606, 441, 618], "score": 0.93, "content": "{\\overline{{A}}}_{i}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [442, 605, 537, 619], "score": 1.0, "content": " at least along the", "type": "text"}], "index": 31}, {"bbox": [62, 620, 538, 635], "spans": [{"bbox": [62, 620, 147, 635], "score": 1.0, "content": "paths that pass ", "type": "text"}, {"bbox": [147, 625, 156, 631], "score": 0.88, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [157, 620, 291, 635], "score": 1.0, "content": " or that lie in the graphs ", "type": "text"}, {"bbox": [291, 622, 302, 632], "score": 0.91, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [302, 620, 538, 635], "score": 1.0, "content": ". But, at the same time, there has to exist a", "type": "text"}], "index": 32}, {"bbox": [62, 635, 339, 650], "spans": [{"bbox": [62, 635, 90, 650], "score": 1.0, "content": "path ", "type": "text"}, {"bbox": [90, 640, 96, 645], "score": 0.86, "content": "{e}", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [96, 635, 154, 650], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [154, 636, 186, 650], "score": 0.95, "content": "h_{\\overline{{A}}_{i}}(e)", "type": "inline_equation", "height": 14, "width": 32}, {"bbox": [186, 635, 245, 650], "score": 1.0, "content": " equals the ", "type": "text"}, {"bbox": [245, 637, 249, 645], "score": 0.84, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [250, 635, 329, 650], "score": 1.0, "content": "th generator of ", "type": "text"}, {"bbox": [330, 637, 334, 645], "score": 0.89, "content": "t", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [335, 635, 339, 650], "score": 1.0, "content": ".", "type": "text"}], "index": 33}], "index": 29.5, "bbox_fs": [61, 533, 538, 650]}, {"type": "text", "bbox": [63, 648, 537, 676], "lines": [{"bbox": [63, 649, 536, 663], "spans": [{"bbox": [63, 649, 536, 663], "score": 1.0, "content": "Now, it should be obvious that we get help from the construction method for new connections", "type": "text"}], "index": 34}, {"bbox": [63, 664, 478, 678], "spans": [{"bbox": [63, 664, 478, 678], "score": 1.0, "content": "introduced in [10]. Before we do this we recall an important notation used there.", "type": "text"}], "index": 35}], "index": 34.5, "bbox_fs": [63, 649, 536, 678]}]}
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[{"type": "title", "bbox": [63, 10, 294, 29], "content": "7 Denseness of the Strata", "index": 0}, {"type": "text", "bbox": [62, 40, 538, 98], "content": "The next theorem we want to prove is that the set is not only open, but also dense in . This assertion does – in contrast to the slice theorem and the openness of the strata – not follow from the general theory of transformation groups. We have to show this directly on the level of .", "index": 1}, {"type": "text", "bbox": [63, 98, 421, 114], "content": "As we will see in a moment, the next proposition will be very helpful.", "index": 2}, {"type": "text", "bbox": [63, 121, 538, 168], "content": "Proposition 7.1 Let and be finitely many graphs. Then there is for any an with and for all .", "index": 3}, {"type": "text", "bbox": [62, 176, 150, 191], "content": "Namely, we have", "index": 4}, {"type": "text", "bbox": [62, 199, 322, 216], "content": "Corollary 7.2 is dense in for all .", "index": 5}, {"type": "text", "bbox": [62, 226, 538, 336], "content": "Proof Let . We have to show that any neighbourhood of contains an having type . It is sufficient to prove this assertion for all graphs and all with open and for all with finite , beca use any general open contains such a set. Now let and be chosen as just described. Due to Proposition 7.1 above there exists an with and for all , i.e. with and for all , thus, .", "index": 6}, {"type": "text", "bbox": [63, 359, 411, 374], "content": "Along with the proposition about the openness of the strata we get", "index": 7}, {"type": "text", "bbox": [62, 382, 444, 399], "content": "Corollary 7.3 For all the closure of w.r.t. is equal to .", "index": 8}, {"type": "text", "bbox": [63, 410, 538, 491], "content": "Proof Denote the closure of w.r.t. by . Due to the denseness of in we have . Since the closure is compatible with the relative topology, we have , i.e. . But, due to Corollary 6.3, itself is closed in . Hence, . qed", "index": 9}, {"type": "title", "bbox": [63, 507, 320, 524], "content": "7.1 How to Prove Proposition 7.1?", "index": 10}, {"type": "text", "bbox": [62, 531, 538, 647], "content": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we get help from the finiteness lemma for centralizers. Namely, let be chosen such that . is finitely generated as well. Thus, we have to construct a connection whose type is determined by and the generators of . For this we use the induction on the number of generators of . In conclusion, we have to construct inductively from new connections , such that coincides with at least along the paths that pass or that lie in the graphs . But, at the same time, there has to exist a path , such that equals the th generator of .", "index": 11}, {"type": "text", "bbox": [63, 648, 537, 676], "content": "Now, it should be obvious that we get help from the construction method for new connections introduced in [10]. Before we do this we recall an important notation used there.", "index": 12}]
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[{"bbox": [65, 14, 293, 29], "content": "7 Denseness of the Strata", "parent_index": 0, "line_index": 0}, {"bbox": [62, 43, 537, 57], "content": "The next theorem we want to prove is that the set is not only open, but also dense in", "parent_index": 1, "line_index": 0}, {"bbox": [63, 56, 537, 73], "content": ". This assertion does – in contrast to the slice theorem and the openness of the strata –", "parent_index": 1, "line_index": 1}, {"bbox": [62, 72, 537, 87], "content": "not follow from the general theory of transformation groups. We have to show this directly", "parent_index": 1, "line_index": 2}, {"bbox": [62, 86, 154, 100], "content": "on the level of .", "parent_index": 1, "line_index": 3}, {"bbox": [63, 101, 420, 115], "content": "As we will see in a moment, the next proposition will be very helpful.", "parent_index": 2, "line_index": 0}, {"bbox": [62, 124, 383, 141], "content": "Proposition 7.1 Let and be finitely many graphs.", "parent_index": 3, "line_index": 0}, {"bbox": [162, 139, 537, 154], "content": "Then there is for any an with and", "parent_index": 3, "line_index": 1}, {"bbox": [164, 153, 296, 168], "content": "for all .", "parent_index": 3, "line_index": 2}, {"bbox": [62, 178, 150, 193], "content": "Namely, we have", "parent_index": 4, "line_index": 0}, {"bbox": [63, 202, 322, 217], "content": "Corollary 7.2 is dense in for all .", "parent_index": 5, "line_index": 0}, {"bbox": [61, 228, 538, 245], "content": "Proof Let . We have to show that any neighbourhood of contains an", "parent_index": 6, "line_index": 0}, {"bbox": [106, 243, 539, 262], "content": "having type . It is sufficient to prove this assertion for all graphs and all", "parent_index": 6, "line_index": 1}, {"bbox": [106, 257, 539, 277], "content": "with open and for all with finite ,", "parent_index": 6, "line_index": 2}, {"bbox": [104, 274, 355, 292], "content": "beca use any general open contains such a set.", "parent_index": 6, "line_index": 3}, {"bbox": [105, 289, 538, 305], "content": "Now let and be chosen as just described. Due to Proposition 7.1 above there", "parent_index": 6, "line_index": 4}, {"bbox": [105, 300, 538, 319], "content": "exists an with and for all , i.e. with", "parent_index": 6, "line_index": 5}, {"bbox": [106, 316, 538, 336], "content": "and for all , thus, .", "parent_index": 6, "line_index": 6}, {"bbox": [63, 361, 410, 376], "content": "Along with the proposition about the openness of the strata we get", "parent_index": 7, "line_index": 0}, {"bbox": [62, 384, 443, 401], "content": "Corollary 7.3 For all the closure of w.r.t. is equal to .", "parent_index": 8, "line_index": 0}, {"bbox": [62, 413, 334, 428], "content": "Proof Denote the closure of w.r.t. by .", "parent_index": 9, "line_index": 0}, {"bbox": [104, 425, 540, 447], "content": "Due to the denseness of in we have . Since the closure is", "parent_index": 9, "line_index": 1}, {"bbox": [105, 443, 538, 463], "content": "compatible with the relative topology, we have ,", "parent_index": 9, "line_index": 2}, {"bbox": [105, 461, 538, 478], "content": "i.e. . But, due to Corollary 6.3, itself is closed in .", "parent_index": 9, "line_index": 3}, {"bbox": [106, 477, 537, 492], "content": "Hence, . qed", "parent_index": 9, "line_index": 4}, {"bbox": [63, 510, 318, 524], "content": "7.1 How to Prove Proposition 7.1?", "parent_index": 10, "line_index": 0}, {"bbox": [62, 533, 537, 548], "content": "Which ideas will the proof of Proposition 7.1 be based on? As in the last two sections we", "parent_index": 11, "line_index": 0}, {"bbox": [62, 548, 537, 561], "content": "get help from the finiteness lemma for centralizers. Namely, let be chosen such that", "parent_index": 11, "line_index": 1}, {"bbox": [63, 562, 538, 577], "content": ". is finitely generated as well. Thus, we have to", "parent_index": 11, "line_index": 2}, {"bbox": [61, 577, 538, 591], "content": "construct a connection whose type is determined by and the generators of . For this", "parent_index": 11, "line_index": 3}, {"bbox": [63, 592, 537, 605], "content": "we use the induction on the number of generators of . In conclusion, we have to construct", "parent_index": 11, "line_index": 4}, {"bbox": [62, 605, 537, 619], "content": "inductively from new connections , such that coincides with at least along the", "parent_index": 11, "line_index": 5}, {"bbox": [62, 620, 538, 635], "content": "paths that pass or that lie in the graphs . But, at the same time, there has to exist a", "parent_index": 11, "line_index": 6}, {"bbox": [62, 635, 339, 650], "content": "path , such that equals the th generator of .", "parent_index": 11, "line_index": 7}, {"bbox": [63, 649, 536, 663], "content": "Now, it should be obvious that we get help from the construction method for new connections", "parent_index": 12, "line_index": 0}, {"bbox": [63, 664, 478, 678], "content": "introduced in [10]. Before we do this we recall an important notation used there.", "parent_index": 12, "line_index": 1}]
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[]
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[{"bbox": [332, 43, 351, 56], "content": "\\overline{{A}}_{=t}", "parent_index": 1, "subtype": "inline"}, {"bbox": [63, 58, 83, 72], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "parent_index": 1, "subtype": "inline"}, {"bbox": [140, 87, 149, 97], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [184, 126, 218, 137], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [244, 128, 255, 138], "content": "\\Gamma_{i}", "parent_index": 3, "subtype": "inline"}, {"bbox": [283, 140, 348, 154], "content": "t\\,\\geq\\,\\mathrm{Typ}(\\overline{{A}})", "parent_index": 3, "subtype": "inline"}, {"bbox": [370, 139, 412, 151], "content": "\\overline{{A}}^{\\prime}\\,\\in\\,\\overline{{A}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [444, 139, 512, 154], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})\\;=\\;t", "parent_index": 3, "subtype": "inline"}, {"bbox": [164, 154, 249, 168], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "parent_index": 3, "subtype": "inline"}, {"bbox": [287, 156, 291, 165], "content": "i", "parent_index": 3, "subtype": "inline"}, {"bbox": [150, 203, 171, 216], "content": "\\overline{{A}}_{=t}", "parent_index": 5, "subtype": "inline"}, {"bbox": [232, 203, 252, 217], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "parent_index": 5, "subtype": "inline"}, {"bbox": [289, 205, 318, 214], "content": "t\\in\\mathcal T", "parent_index": 5, "subtype": "inline"}, {"bbox": [128, 231, 202, 244], "content": "\\overline{{A}}\\in\\overline{{A}}_{\\leq t}\\subseteq\\overline{{A}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [435, 232, 445, 241], "content": "U", "parent_index": 6, "subtype": "inline"}, {"bbox": [463, 231, 473, 241], "content": "\\overline{{A}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [106, 245, 118, 257], "content": "\\overline{{A}}^{\\prime}", "parent_index": 6, "subtype": "inline"}, {"bbox": [190, 249, 194, 257], "content": "t", "parent_index": 6, "subtype": "inline"}, {"bbox": [483, 248, 494, 258], "content": "\\Gamma_{i}", "parent_index": 6, "subtype": "inline"}, {"bbox": [106, 261, 187, 275], "content": "\\begin{array}{r}{U=\\bigcap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})}\\end{array}", "parent_index": 6, "subtype": "inline"}, {"bbox": [245, 260, 315, 273], "content": "W_{i}\\subseteq\\mathbf{G}^{\\#\\mathbf{E}(\\Gamma_{i})}", "parent_index": 6, "subtype": "inline"}, {"bbox": [342, 260, 404, 274], "content": "\\pi_{\\Gamma_{i}}(\\overline{{A}})\\in W_{i}", "parent_index": 6, "subtype": "inline"}, {"bbox": [442, 263, 466, 271], "content": "i\\in I", "parent_index": 6, "subtype": "inline"}, {"bbox": [527, 263, 533, 271], "content": "I", "parent_index": 6, "subtype": "inline"}, {"bbox": [240, 277, 250, 286], "content": "U", "parent_index": 6, "subtype": "inline"}, {"bbox": [151, 290, 162, 302], "content": "\\Gamma_{i}", "parent_index": 6, "subtype": "inline"}, {"bbox": [189, 291, 199, 300], "content": "U", "parent_index": 6, "subtype": "inline"}, {"bbox": [154, 303, 190, 315], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [218, 303, 335, 317], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t\\geq\\mathrm{Typ}(\\overline{{A}})", "parent_index": 6, "subtype": "inline"}, {"bbox": [360, 303, 446, 317], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})", "parent_index": 6, "subtype": "inline"}, {"bbox": [482, 307, 487, 315], "content": "i", "parent_index": 6, "subtype": "inline"}, {"bbox": [106, 319, 154, 332], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}_{=t}", "parent_index": 6, "subtype": "inline"}, {"bbox": [181, 318, 344, 336], "content": "\\overline{{A}}^{\\prime}\\in\\pi_{\\Gamma_{i}}^{-1}\\Big(\\pi_{\\Gamma_{i}}\\big(\\{\\overline{{A}}\\}\\big)\\Big)\\subseteq\\pi_{\\Gamma_{i}}^{-1}(W_{i})", "parent_index": 6, "subtype": "inline"}, {"bbox": [382, 321, 387, 331], "content": "i", "parent_index": 6, "subtype": "inline"}, {"bbox": [423, 319, 533, 335], "content": "\\overline{{A}}^{\\prime}\\in\\cap_{i}\\pi_{\\Gamma_{i}}^{-1}(W_{i})=U", "parent_index": 6, "subtype": "inline"}, {"bbox": [188, 388, 217, 397], "content": "t\\in\\mathcal T", "parent_index": 8, "subtype": "inline"}, {"bbox": [293, 385, 313, 398], "content": "\\overline{{\\mathcal{A}}}_{=t}", "parent_index": 8, "subtype": "inline"}, {"bbox": [348, 385, 359, 397], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [419, 385, 439, 400], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}", "parent_index": 8, "subtype": "inline"}, {"bbox": [219, 415, 228, 424], "content": "F", "parent_index": 9, "subtype": "inline"}, {"bbox": [265, 415, 274, 424], "content": "E", "parent_index": 9, "subtype": "inline"}, {"bbox": [294, 414, 332, 427], "content": "\\operatorname{Cl}_{E}(F)", "parent_index": 9, "subtype": "inline"}, {"bbox": [232, 428, 252, 440], "content": "\\overline{{A}}_{=t}", "parent_index": 9, "subtype": "inline"}, {"bbox": [269, 428, 289, 442], "content": "\\overline{{\\mathbf{\\mathcal{A}}}}_{\\leq t}", "parent_index": 9, "subtype": "inline"}, {"bbox": [337, 427, 430, 444], "content": "\\mathrm{Cl}_{\\overline{{A}}_{\\leq t}}(\\overline{{A}}_{=t})=\\overline{{A}}_{\\leq t}", "parent_index": 9, "subtype": "inline"}, {"bbox": [346, 444, 533, 461], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}=\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}_{\\leq t}}(\\overline{{\\mathcal{A}}}_{=t})=\\overline{{\\mathcal{A}}}_{\\leq t}\\cap\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "parent_index": 9, "subtype": "inline"}, {"bbox": [129, 462, 216, 476], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\,\\subseteq\\,\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "parent_index": 9, "subtype": "inline"}, {"bbox": [368, 462, 427, 477], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\overline{{\\mathcal{A}}}_{=t}", "parent_index": 9, "subtype": "inline"}, {"bbox": [523, 462, 533, 473], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 9, "subtype": "inline"}, {"bbox": [144, 477, 228, 491], "content": "\\overline{{\\mathcal{A}}}_{\\leq t}\\supseteq\\mathrm{Cl}_{\\overline{{\\mathcal{A}}}}(\\overline{{\\mathcal{A}}}_{=t})", "parent_index": 9, "subtype": "inline"}, {"bbox": [389, 550, 431, 560], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [63, 562, 232, 576], "content": "\\mathrm{Typ}(\\overline{{A}})=[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\varphi_{\\alpha}(\\overline{{A}}))]", "parent_index": 11, "subtype": "inline"}, {"bbox": [240, 562, 299, 576], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "parent_index": 11, "subtype": "inline"}, {"bbox": [333, 577, 366, 591], "content": "\\varphi_{\\alpha}(\\overline{{A}})", "parent_index": 11, "subtype": "inline"}, {"bbox": [483, 580, 487, 588], "content": "t", "parent_index": 11, "subtype": "inline"}, {"bbox": [339, 594, 343, 602], "content": "t", "parent_index": 11, "subtype": "inline"}, {"bbox": [151, 606, 160, 616], "content": "\\overline{{A}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [252, 606, 265, 618], "content": "{\\overline{{A}}}_{i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [325, 606, 348, 619], "content": "\\overline{{A}}_{i-1}", "parent_index": 11, "subtype": "inline"}, {"bbox": [429, 606, 441, 618], "content": "{\\overline{{A}}}_{i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [147, 625, 156, 631], "content": "\\alpha", "parent_index": 11, "subtype": "inline"}, {"bbox": [291, 622, 302, 632], "content": "\\Gamma_{i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [90, 640, 96, 645], "content": "{e}", "parent_index": 11, "subtype": "inline"}, {"bbox": [154, 636, 186, 650], "content": "h_{\\overline{{A}}_{i}}(e)", "parent_index": 11, "subtype": "inline"}, {"bbox": [245, 637, 249, 645], "content": "i", "parent_index": 11, "subtype": "inline"}, {"bbox": [330, 637, 334, 645], "content": "t", "parent_index": 11, "subtype": "inline"}]
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[]
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Definition 7.1 Let $\gamma_{1},\gamma_{2}\in\mathcal{P}$ .
We say that $\gamma_{1}$ and $\gamma_{2}$ have the same initial segment (shortly: $\gamma_{1}$ ↑↑ $\gamma_{2}$ ) iff there exist $0<\delta_{1},\delta_{2}\leq1$ such that $\gamma_{1}\mid_{[0,\delta_{1}]}$ and $\gamma_{2}\mid_{[0,\delta_{2}]}$ coincide up to the parametrization.
We say analogously that the final segment of $\gamma_{1}$ coincides with the initial segment of $\gamma_{2}$ (shortly: $\gamma_{1}\downarrow\uparrow\gamma_{2}$ ) iff there exist $0\,<\,\delta_{1},\delta_{2}\,\leq\,1$ such that $\gamma_{1}^{-1}~|_{[0,\delta_{1}]}$ and $\gamma_{2}\mid_{[0,\delta_{2}]}$ coincide up to the parametrization.
Iff the corresponding relations are not fulfilled, we write $\gamma_{1}$ ↑↑ $\gamma_{2}$ and $\gamma_{1}\neq\gamma_{2}$ , respectively.
Finally, we recall the decomposition lemma.
Lemma 7.4 Let $x\in M$ be a point. Any $\gamma\in\mathcal{P}$ can be written (up to parametrization) as a product $\Pi\,\gamma_{i}$ with $\gamma_{i}\in\mathcal{P}$ , such that • int $\gamma_{i}\cap\{x\}=\emptyset$ or • int $\gamma_{i}=\{x\}$ .
# 7.2 Successive Magnifying of the Types
In order to prove Proposition 7.1 we need the following lemma for magnifying the types.
Hereby, we will use explicitly the construction of a new connection $\overline{{A}}^{\prime}$ from $\overline{{A}}$ as given in [10].
Lemma 7.5 Let $\Gamma_{i}$ be finitely many graphs, ${\overline{{A}}}\in{\overline{{A}}}$ and $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ be a finite set of paths with $Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$ . Furthermore, let $g\in\mathbf G$ be arbitrary. Then there is an $\overline{{A}}^{\prime}\in\overline{{A}}$ , such that: • $h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$ , • $\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})=\pi_{\Gamma_{i}}(\overline{{{A}}})$ for all $i$ , • $h_{\overline{{{A}}}^{\prime}}(e)=g$ for an $e\in{\mathcal{H}}{\mathcal{G}}$ and • $\bar{Z}(\mathbf{H}_{\overline{{A}}^{\prime}})=Z(\{g\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$ .
Proof 1. Let $m^{\prime}\in M$ be some point that is neither contained in the images of $\Gamma_{i}$ nor in that of $\alpha$ , and join $m$ with $m^{\prime}$ by some path $\gamma$ . Now let $e^{\prime}$ be some closed path in $M$ with base point $m^{\prime}$ and without self-intersections, such that
$$
\begin{array}{r}{\operatorname{m}e^{\prime}\cap\left(\operatorname{int}\gamma\cup\operatorname{im}\left(\alpha\right)\cup\bigcup\operatorname{im}\left(\Gamma_{i}\right)\right)\right)=\emptyset.}\end{array}
$$
Obviously, there exists such an $e^{\prime}$ because $M$ is supposed to be at least twodimensional. Set $e:=\gamma\,e^{\prime}\,\gamma^{-1}\in\mathcal{H}\mathcal{G}$ and $g^{\prime}:=h_{\overline{{{A}}}}(\gamma)^{-1}g h_{\overline{{{A}}}}(\gamma)$ . Finally, define a connection $\overline{{A}}^{\prime}$ for $\overline{{A}}$ , $e^{\prime}$ and $g^{\prime}$ as follows:
2. Construction of $\overline{{A}}^{\prime}$
• Let $\delta\in\mathcal{P}$ be for the moment a ”genuine” path (i.e., not an equivalence class) that does not contain the initial point $e^{\prime}(0)\,\equiv\,m^{\prime}$ of $e^{\prime}$ as an inner point. Explicitly we have int $\delta\cap\{e^{\prime}(0)\}=\emptyset$ . Define $h_{\overline{{{A}}}^{\prime}}(\delta):=\left\{\!\!\begin{array}{r l r}{{g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\delta)\,h_{\overline{{{A}}}}(e^{\prime})\,g^{\prime-1},}}&{{\mathrm{for~}\delta\,\uparrow\uparrow e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\delta)\,}}&{{\mathrm{for~}\delta\,\uparrow\uparrow e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{h_{\overline{{{A}}}}(\delta)\,h_{\overline{{{A}}}}(e^{\prime})\,g^{\prime-1},}}&{{\mathrm{for~}\delta\,\#\,e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{h_{\overline{{{A}}}}(\delta)\,}}&{{\mathrm{else}}}\end{array}\!\!\right..$ For every trivial path $\delta$ set $h_{\overline{{A}}^{\prime}}(\delta)=e_{\mathbf{G}}$ .
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<html><body>
<p data-bbox="62 14 230 29">Definition 7.1 Let $\gamma_{1},\gamma_{2}\in\mathcal{P}$ . </p>
<p data-bbox="60 17 539 146">We say that $\gamma_{1}$ and $\gamma_{2}$ have the same initial segment (shortly: $\gamma_{1}$ ↑↑ $\gamma_{2}$ ) iff there exist $0<\delta_{1},\delta_{2}\leq1$ such that $\gamma_{1}\mid_{[0,\delta_{1}]}$ and $\gamma_{2}\mid_{[0,\delta_{2}]}$ coincide up to the parametrization.
We say analogously that the final segment of $\gamma_{1}$ coincides with the initial segment of $\gamma_{2}$ (shortly: $\gamma_{1}\downarrow\uparrow\gamma_{2}$ ) iff there exist $0\,<\,\delta_{1},\delta_{2}\,\leq\,1$ such that $\gamma_{1}^{-1}~|_{[0,\delta_{1}]}$ and $\gamma_{2}\mid_{[0,\delta_{2}]}$ coincide up to the parametrization.
Iff the corresponding relations are not fulfilled, we write $\gamma_{1}$ ↑↑ $\gamma_{2}$ and $\gamma_{1}\neq\gamma_{2}$ , respectively. </p>
<p data-bbox="62 154 289 169">Finally, we recall the decomposition lemma. </p>
<p data-bbox="63 178 537 238">Lemma 7.4 Let $x\in M$ be a point. Any $\gamma\in\mathcal{P}$ can be written (up to parametrization) as a product $\Pi\,\gamma_{i}$ with $\gamma_{i}\in\mathcal{P}$ , such that • int $\gamma_{i}\cap\{x\}=\emptyset$ or • int $\gamma_{i}=\{x\}$ . </p>
<h1 data-bbox="63 254 353 272">7.2 Successive Magnifying of the Types </h1>
<p data-bbox="63 279 537 309">In order to prove Proposition 7.1 we need the following lemma for magnifying the types.
Hereby, we will use explicitly the construction of a new connection $\overline{{A}}^{\prime}$ from $\overline{{A}}$ as given in [10]. </p>
<p data-bbox="61 317 537 425">Lemma 7.5 Let $\Gamma_{i}$ be finitely many graphs, ${\overline{{A}}}\in{\overline{{A}}}$ and $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ be a finite set of paths with $Z(\mathbf{H}_{\overline{{A}}})=Z(h_{\overline{{A}}}(\alpha))$ . Furthermore, let $g\in\mathbf G$ be arbitrary. Then there is an $\overline{{A}}^{\prime}\in\overline{{A}}$ , such that: • $h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$ , • $\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})=\pi_{\Gamma_{i}}(\overline{{{A}}})$ for all $i$ , • $h_{\overline{{{A}}}^{\prime}}(e)=g$ for an $e\in{\mathcal{H}}{\mathcal{G}}$ and • $\bar{Z}(\mathbf{H}_{\overline{{A}}^{\prime}})=Z(\{g\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$ . </p>
<p data-bbox="62 435 537 478">Proof 1. Let $m^{\prime}\in M$ be some point that is neither contained in the images of $\Gamma_{i}$ nor in that of $\alpha$ , and join $m$ with $m^{\prime}$ by some path $\gamma$ . Now let $e^{\prime}$ be some closed path in $M$ with base point $m^{\prime}$ and without self-intersections, such that </p>
<div class="equation" data-bbox="227 480 438 500">$$
\begin{array}{r}{\operatorname{m}e^{\prime}\cap\left(\operatorname{int}\gamma\cup\operatorname{im}\left(\alpha\right)\cup\bigcup\operatorname{im}\left(\Gamma_{i}\right)\right)\right)=\emptyset.}\end{array}
$$</div>
<p data-bbox="119 500 537 544">Obviously, there exists such an $e^{\prime}$ because $M$ is supposed to be at least twodimensional. Set $e:=\gamma\,e^{\prime}\,\gamma^{-1}\in\mathcal{H}\mathcal{G}$ and $g^{\prime}:=h_{\overline{{{A}}}}(\gamma)^{-1}g h_{\overline{{{A}}}}(\gamma)$ . Finally, define a connection $\overline{{A}}^{\prime}$ for $\overline{{A}}$ , $e^{\prime}$ and $g^{\prime}$ as follows: </p>
<p data-bbox="105 545 222 558">2. Construction of $\overline{{A}}^{\prime}$ </p>
<p data-bbox="123 559 537 688">• Let $\delta\in\mathcal{P}$ be for the moment a ”genuine” path (i.e., not an equivalence class) that does not contain the initial point $e^{\prime}(0)\,\equiv\,m^{\prime}$ of $e^{\prime}$ as an inner point. Explicitly we have int $\delta\cap\{e^{\prime}(0)\}=\emptyset$ . Define $h_{\overline{{{A}}}^{\prime}}(\delta):=\left\{\!\!\begin{array}{r l r}{{g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\delta)\,h_{\overline{{{A}}}}(e^{\prime})\,g^{\prime-1},}}&{{\mathrm{for~}\delta\,\uparrow\uparrow e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\delta)\,}}&{{\mathrm{for~}\delta\,\uparrow\uparrow e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{h_{\overline{{{A}}}}(\delta)\,h_{\overline{{{A}}}}(e^{\prime})\,g^{\prime-1},}}&{{\mathrm{for~}\delta\,\#\,e^{\prime}\mathrm{~and~}\delta\downarrow\uparrow e^{\prime}}}\\ {{h_{\overline{{{A}}}}(\delta)\,}}&{{\mathrm{else}}}\end{array}\!\!\right..$ For every trivial path $\delta$ set $h_{\overline{{A}}^{\prime}}(\delta)=e_{\mathbf{G}}$ . </p>
</body></html>
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0001008v1
| 12 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "Definition 7.1 Let $\\gamma_{1},\\gamma_{2}\\in\\mathcal{P}$ . ", "page_idx": 12}, {"type": "text", "text": "We say that $\\gamma_{1}$ and $\\gamma_{2}$ have the same initial segment (shortly: $\\gamma_{1}$ ↑↑ $\\gamma_{2}$ ) iff there exist $0<\\delta_{1},\\delta_{2}\\leq1$ such that $\\gamma_{1}\\mid_{[0,\\delta_{1}]}$ and $\\gamma_{2}\\mid_{[0,\\delta_{2}]}$ coincide up to the parametrization. \nWe say analogously that the final segment of $\\gamma_{1}$ coincides with the initial segment of $\\gamma_{2}$ (shortly: $\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}$ ) iff there exist $0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1$ such that $\\gamma_{1}^{-1}~|_{[0,\\delta_{1}]}$ and $\\gamma_{2}\\mid_{[0,\\delta_{2}]}$ coincide up to the parametrization. \nIff the corresponding relations are not fulfilled, we write $\\gamma_{1}$ ↑↑ $\\gamma_{2}$ and $\\gamma_{1}\\neq\\gamma_{2}$ , respectively. ", "page_idx": 12}, {"type": "text", "text": "Finally, we recall the decomposition lemma. ", "page_idx": 12}, {"type": "text", "text": "Lemma 7.4 Let $x\\in M$ be a point. Any $\\gamma\\in\\mathcal{P}$ can be written (up to parametrization) as a product $\\Pi\\,\\gamma_{i}$ with $\\gamma_{i}\\in\\mathcal{P}$ , such that • int $\\gamma_{i}\\cap\\{x\\}=\\emptyset$ or • int $\\gamma_{i}=\\{x\\}$ . ", "page_idx": 12}, {"type": "text", "text": "7.2 Successive Magnifying of the Types ", "text_level": 1, "page_idx": 12}, {"type": "text", "text": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types. \nHereby, we will use explicitly the construction of a new connection $\\overline{{A}}^{\\prime}$ from $\\overline{{A}}$ as given in [10]. ", "page_idx": 12}, {"type": "text", "text": "Lemma 7.5 Let $\\Gamma_{i}$ be finitely many graphs, ${\\overline{{A}}}\\in{\\overline{{A}}}$ and $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ be a finite set of paths with $Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))$ . Furthermore, let $g\\in\\mathbf G$ be arbitrary. Then there is an $\\overline{{A}}^{\\prime}\\in\\overline{{A}}$ , such that: • $h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$ , • $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$ for all $i$ , • $h_{\\overline{{{A}}}^{\\prime}}(e)=g$ for an $e\\in{\\mathcal{H}}{\\mathcal{G}}$ and • $\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$ . ", "page_idx": 12}, {"type": "text", "text": "Proof 1. Let $m^{\\prime}\\in M$ be some point that is neither contained in the images of $\\Gamma_{i}$ nor in that of $\\alpha$ , and join $m$ with $m^{\\prime}$ by some path $\\gamma$ . Now let $e^{\\prime}$ be some closed path in $M$ with base point $m^{\\prime}$ and without self-intersections, such that ", "page_idx": 12}, {"type": "equation", "text": "$$\n\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}\n$$", "text_format": "latex", "page_idx": 12}, {"type": "text", "text": "Obviously, there exists such an $e^{\\prime}$ because $M$ is supposed to be at least twodimensional. Set $e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}$ and $g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)$ . Finally, define a connection $\\overline{{A}}^{\\prime}$ for $\\overline{{A}}$ , $e^{\\prime}$ and $g^{\\prime}$ as follows: ", "page_idx": 12}, {"type": "text", "text": "2. Construction of $\\overline{{A}}^{\\prime}$ ", "page_idx": 12}, {"type": "text", "text": "• Let $\\delta\\in\\mathcal{P}$ be for the moment a ”genuine” path (i.e., not an equivalence class) that does not contain the initial point $e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}$ of $e^{\\prime}$ as an inner point. Explicitly we have int $\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset$ . Define $h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..$ For every trivial path $\\delta$ set $h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}$ . ", "page_idx": 12}]
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"width": 38}, {"bbox": [375, 44, 401, 63], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [401, 46, 439, 60], "score": 0.91, "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [440, 44, 538, 63], "score": 1.0, "content": " coincide up to the", "type": "text"}], "index": 2}, {"bbox": [152, 60, 239, 74], "spans": [{"bbox": [152, 60, 239, 74], "score": 1.0, "content": "parametrization.", "type": "text"}], "index": 3}, {"bbox": [155, 75, 537, 89], "spans": [{"bbox": [155, 75, 392, 89], "score": 1.0, "content": "We say analogously that the final segment of ", "type": "text"}, {"bbox": [392, 80, 403, 88], "score": 0.87, "content": "\\gamma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [403, 75, 537, 89], "score": 1.0, "content": " coincides with the initial", "type": "text"}], "index": 4}, {"bbox": [153, 89, 537, 103], "spans": [{"bbox": [153, 89, 213, 103], "score": 1.0, "content": "segment of ", "type": "text"}, {"bbox": [214, 92, 225, 102], "score": 0.79, "content": "\\gamma_{2}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [226, 89, 280, 103], "score": 1.0, "content": " (shortly: ", "type": "text"}, {"bbox": [280, 91, 324, 102], "score": 0.68, "content": "\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [325, 89, 407, 103], "score": 1.0, "content": ") iff there exist ", "type": "text"}, {"bbox": [407, 89, 483, 102], "score": 0.92, "content": "0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [483, 89, 537, 103], "score": 1.0, "content": " such that", "type": "text"}], "index": 5}, {"bbox": [153, 103, 449, 119], "spans": [{"bbox": [153, 103, 198, 118], "score": 0.89, "content": "\\gamma_{1}^{-1}~|_{[0,\\delta_{1}]}", "type": "inline_equation", "height": 15, "width": 45}, {"bbox": [199, 104, 224, 119], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [224, 104, 262, 118], "score": 0.91, "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [262, 104, 449, 119], "score": 1.0, "content": " coincide up to the parametrization.", "type": "text"}], "index": 6}, {"bbox": [152, 118, 538, 133], "spans": [{"bbox": [152, 118, 468, 133], "score": 1.0, "content": "Iff the corresponding relations are not fulfilled, we write ", "type": "text"}, {"bbox": [469, 119, 481, 131], "score": 0.82, "content": "\\gamma_{1}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [482, 118, 498, 133], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [498, 119, 511, 131], "score": 0.83, "content": "\\gamma_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [511, 118, 538, 133], "score": 1.0, "content": " and", "type": "text"}], "index": 7}, {"bbox": [153, 133, 263, 146], "spans": [{"bbox": [153, 133, 194, 146], "score": 0.63, "content": "\\gamma_{1}\\neq\\gamma_{2}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [195, 133, 263, 146], "score": 1.0, "content": ", respectively.", "type": "text"}], "index": 8}], "index": 4.5}, {"type": "text", "bbox": [62, 154, 289, 169], "lines": [{"bbox": [63, 157, 288, 171], "spans": [{"bbox": [63, 157, 288, 171], "score": 1.0, "content": "Finally, we recall the decomposition lemma.", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [63, 178, 537, 238], "lines": [{"bbox": [62, 181, 537, 196], "spans": [{"bbox": [62, 181, 159, 196], "score": 1.0, "content": "Lemma 7.4 Let ", "type": "text"}, {"bbox": [159, 182, 195, 192], "score": 0.87, "content": "x\\in M", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [195, 181, 285, 196], "score": 1.0, "content": " be a point. Any ", "type": "text"}, {"bbox": [285, 183, 316, 194], "score": 0.94, "content": "\\gamma\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [317, 181, 537, 196], "score": 1.0, "content": " can be written (up to parametrization) as", "type": "text"}], "index": 10}, {"bbox": [137, 196, 334, 210], "spans": [{"bbox": [137, 196, 191, 210], "score": 1.0, "content": "a product", "type": "text"}, {"bbox": [192, 196, 213, 209], "score": 0.9, "content": "\\Pi\\,\\gamma_{i}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [214, 196, 243, 210], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [243, 198, 277, 209], "score": 0.92, "content": "\\gamma_{i}\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [277, 196, 334, 210], "score": 1.0, "content": ", such that", "type": "text"}], "index": 11}, {"bbox": [137, 210, 252, 225], "spans": [{"bbox": [137, 210, 171, 225], "score": 1.0, "content": "• int", "type": "text"}, {"bbox": [171, 210, 235, 224], "score": 0.75, "content": "\\gamma_{i}\\cap\\{x\\}=\\emptyset", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [235, 210, 252, 225], "score": 1.0, "content": "or", "type": "text"}], "index": 12}, {"bbox": [137, 224, 219, 240], "spans": [{"bbox": [137, 224, 172, 240], "score": 1.0, "content": "• int ", "type": "text"}, {"bbox": [172, 226, 216, 239], "score": 0.81, "content": "\\gamma_{i}=\\{x\\}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [216, 224, 219, 240], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "title", "bbox": [63, 254, 353, 272], "lines": [{"bbox": [63, 258, 351, 273], "spans": [{"bbox": [63, 258, 95, 272], "score": 1.0, "content": "7.2", "type": "text"}, {"bbox": [97, 258, 351, 273], "score": 1.0, "content": "Successive Magnifying of the Types", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [63, 279, 537, 309], "lines": [{"bbox": [63, 282, 535, 297], "spans": [{"bbox": [63, 282, 535, 297], "score": 1.0, "content": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types.", "type": "text"}], "index": 15}, {"bbox": [62, 295, 536, 311], "spans": [{"bbox": [62, 295, 403, 311], "score": 1.0, "content": "Hereby, we will use explicitly the construction of a new connection ", "type": "text"}, {"bbox": [403, 295, 415, 307], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [415, 295, 445, 311], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [445, 296, 454, 307], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [454, 295, 536, 311], "score": 1.0, "content": " as given in [10].", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "text", "bbox": [61, 317, 537, 425], "lines": [{"bbox": [61, 319, 537, 336], "spans": [{"bbox": [61, 319, 159, 336], "score": 1.0, "content": "Lemma 7.5 Let ", "type": "text"}, {"bbox": [160, 322, 171, 333], "score": 0.88, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [171, 319, 305, 336], "score": 1.0, "content": " be finitely many graphs, ", "type": "text"}, {"bbox": [306, 320, 341, 332], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [341, 319, 368, 336], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [369, 322, 413, 333], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [414, 319, 537, 336], "score": 1.0, "content": " be a finite set of paths", "type": "text"}], "index": 17}, {"bbox": [137, 334, 464, 351], "spans": [{"bbox": [137, 334, 165, 351], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [165, 335, 266, 349], "score": 0.92, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [266, 334, 362, 351], "score": 1.0, "content": ". Furthermore, let ", "type": "text"}, {"bbox": [362, 337, 394, 348], "score": 0.92, "content": "g\\in\\mathbf G", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [394, 334, 464, 351], "score": 1.0, "content": " be arbitrary.", "type": "text"}], "index": 18}, {"bbox": [137, 349, 323, 365], "spans": [{"bbox": [137, 349, 226, 365], "score": 1.0, "content": "Then there is an ", "type": "text"}, {"bbox": [227, 349, 263, 362], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [263, 349, 323, 365], "score": 1.0, "content": ", such that:", "type": "text"}], "index": 19}, {"bbox": [137, 365, 242, 381], "spans": [{"bbox": [137, 365, 154, 381], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [155, 365, 238, 379], "score": 0.74, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 14, "width": 83}, {"bbox": [239, 365, 242, 381], "score": 1.0, "content": ",", "type": "text"}], "index": 20}, {"bbox": [136, 380, 289, 397], "spans": [{"bbox": [136, 380, 154, 397], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [155, 380, 241, 396], "score": 0.81, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 16, "width": 86}, {"bbox": [241, 380, 278, 397], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [279, 383, 283, 393], "score": 0.59, "content": "i", "type": "inline_equation", "height": 10, "width": 4}, {"bbox": [284, 380, 289, 397], "score": 1.0, "content": ",", "type": "text"}], "index": 21}, {"bbox": [136, 396, 309, 410], "spans": [{"bbox": [136, 396, 154, 410], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [155, 397, 208, 410], "score": 0.73, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [209, 397, 245, 410], "score": 1.0, "content": " for an ", "type": "text"}, {"bbox": [245, 396, 284, 408], "score": 0.51, "content": "e\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [285, 396, 309, 410], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [137, 410, 294, 428], "spans": [{"bbox": [137, 410, 154, 428], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [155, 410, 290, 426], "score": 0.75, "content": "\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 16, "width": 135}, {"bbox": [290, 410, 294, 428], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 20}, {"type": "text", "bbox": [62, 435, 537, 478], "lines": [{"bbox": [62, 437, 537, 452], "spans": [{"bbox": [62, 437, 147, 452], "score": 1.0, "content": "Proof 1. Let ", "type": "text"}, {"bbox": [147, 439, 190, 449], "score": 0.91, "content": "m^{\\prime}\\in M", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [190, 437, 489, 452], "score": 1.0, "content": " be some point that is neither contained in the images of ", "type": "text"}, {"bbox": [490, 440, 501, 451], "score": 0.92, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [501, 437, 537, 452], "score": 1.0, "content": " nor in", "type": "text"}], "index": 24}, {"bbox": [125, 452, 538, 467], "spans": [{"bbox": [125, 452, 165, 467], "score": 1.0, "content": "that of ", "type": "text"}, {"bbox": [165, 457, 174, 463], "score": 0.78, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [175, 452, 227, 467], "score": 1.0, "content": ", and join ", "type": "text"}, {"bbox": [228, 457, 239, 463], "score": 0.82, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [239, 452, 269, 467], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [269, 453, 282, 463], "score": 0.87, "content": "m^{\\prime}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [283, 452, 359, 467], "score": 1.0, "content": " by some path ", "type": "text"}, {"bbox": [360, 457, 367, 465], "score": 0.88, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [367, 452, 419, 467], "score": 1.0, "content": ". Now let ", "type": "text"}, {"bbox": [420, 454, 428, 463], "score": 0.91, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 452, 538, 467], "score": 1.0, "content": " be some closed path", "type": "text"}], "index": 25}, {"bbox": [126, 468, 464, 480], "spans": [{"bbox": [126, 468, 140, 480], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [140, 469, 153, 477], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [153, 468, 239, 480], "score": 1.0, "content": " with base point ", "type": "text"}, {"bbox": [239, 468, 253, 477], "score": 0.9, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [253, 468, 464, 480], "score": 1.0, "content": " and without self-intersections, such that", "type": "text"}], "index": 26}], "index": 25}, {"type": "interline_equation", "bbox": [227, 480, 438, 500], "lines": [{"bbox": [227, 480, 438, 500], "spans": [{"bbox": [227, 480, 438, 500], "score": 0.74, "content": "\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [119, 500, 537, 544], "lines": [{"bbox": [126, 501, 536, 517], "spans": [{"bbox": [126, 501, 295, 517], "score": 1.0, "content": "Obviously, there exists such an ", "type": "text"}, {"bbox": [295, 504, 304, 513], "score": 0.89, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [304, 501, 353, 517], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [353, 504, 366, 513], "score": 0.89, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [367, 501, 536, 517], "score": 1.0, "content": " is supposed to be at least two-", "type": "text"}], "index": 28}, {"bbox": [126, 516, 445, 532], "spans": [{"bbox": [126, 516, 215, 532], "score": 1.0, "content": "dimensional. Set ", "type": "text"}, {"bbox": [215, 517, 311, 530], "score": 0.93, "content": "e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [312, 516, 337, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 517, 442, 531], "score": 0.93, "content": "g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)", "type": "inline_equation", "height": 14, "width": 104}, {"bbox": [442, 516, 445, 532], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 531, 421, 546], "spans": [{"bbox": [125, 531, 270, 546], "score": 1.0, "content": "Finally, define a connection ", "type": "text"}, {"bbox": [270, 531, 282, 543], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [282, 531, 303, 546], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 533, 313, 543], "score": 0.85, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [313, 531, 319, 546], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [320, 534, 328, 543], "score": 0.85, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [329, 531, 354, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 534, 363, 545], "score": 0.91, "content": "g^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [364, 531, 421, 546], "score": 1.0, "content": " as follows:", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [105, 545, 222, 558], "lines": [{"bbox": [104, 546, 221, 559], "spans": [{"bbox": [104, 546, 209, 559], "score": 1.0, "content": "2. Construction of ", "type": "text"}, {"bbox": [210, 546, 221, 557], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}], "index": 31}], "index": 31}, {"type": "text", "bbox": [123, 559, 537, 688], "lines": [{"bbox": [127, 561, 536, 577], "spans": [{"bbox": [127, 561, 163, 577], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [163, 563, 193, 573], "score": 0.92, "content": "\\delta\\in\\mathcal{P}", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [193, 561, 536, 577], "score": 1.0, "content": " be for the moment a ”genuine” path (i.e., not an equivalence class)", "type": "text"}], "index": 32}, {"bbox": [142, 575, 536, 591], "spans": [{"bbox": [142, 575, 350, 591], "score": 1.0, "content": "that does not contain the initial point ", "type": "text"}, {"bbox": [351, 576, 408, 589], "score": 0.92, "content": "e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [408, 575, 427, 591], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [428, 576, 436, 586], "score": 0.86, "content": "e^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [437, 575, 536, 591], "score": 1.0, "content": " as an inner point.", "type": "text"}], "index": 33}, {"bbox": [144, 590, 376, 605], "spans": [{"bbox": [144, 590, 257, 605], "score": 1.0, "content": "Explicitly we have int ", "type": "text"}, {"bbox": [257, 591, 333, 604], "score": 0.92, "content": "\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [334, 590, 376, 605], "score": 1.0, "content": ". Define", "type": "text"}], "index": 34}, {"bbox": [142, 604, 481, 673], "spans": [{"bbox": [142, 604, 481, 673], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..", "type": "inline_equation"}], "index": 35}, {"bbox": [140, 670, 350, 690], "spans": [{"bbox": [140, 670, 257, 690], "score": 1.0, "content": "For every trivial path ", "type": "text"}, {"bbox": [257, 675, 263, 684], "score": 0.84, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [263, 670, 284, 690], "score": 1.0, "content": " set ", "type": "text"}, {"bbox": [284, 674, 345, 688], "score": 0.86, "content": "h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}", "type": "inline_equation", "height": 14, "width": 61}, {"bbox": [346, 670, 350, 690], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [227, 480, 438, 500], "lines": [{"bbox": [227, 480, 438, 500], "spans": [{"bbox": [227, 480, 438, 500], "score": 0.74, "content": "\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}", "type": "interline_equation"}], "index": 27}], "index": 27}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [62, 14, 230, 29], "lines": [{"bbox": [61, 16, 230, 32], "spans": [{"bbox": [61, 16, 174, 31], "score": 1.0, "content": "Definition 7.1 Let ", "type": "text"}, {"bbox": [174, 19, 226, 30], "score": 0.93, "content": "\\gamma_{1},\\gamma_{2}\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 52}, {"bbox": [226, 16, 230, 32], "score": 1.0, "content": ".", "type": "text"}], "index": 0}], "index": 0, "bbox_fs": [61, 16, 230, 32]}, {"type": "list", "bbox": [60, 17, 539, 146], "lines": [{"bbox": [152, 31, 539, 47], "spans": [{"bbox": [152, 31, 219, 47], "score": 1.0, "content": "We say that ", "type": "text"}, {"bbox": [219, 37, 230, 45], "score": 0.9, "content": "\\gamma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [231, 31, 257, 47], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [257, 37, 268, 45], "score": 0.91, "content": "\\gamma_{2}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [268, 31, 475, 47], "score": 1.0, "content": " have the same initial segment (shortly: ", "type": "text"}, {"bbox": [475, 32, 488, 45], "score": 0.76, "content": "\\gamma_{1}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [488, 31, 504, 47], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [505, 33, 518, 45], "score": 0.79, "content": "\\gamma_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [518, 31, 539, 47], "score": 1.0, "content": ") iff", "type": "text"}], "index": 1}, {"bbox": [151, 44, 538, 63], "spans": [{"bbox": [151, 44, 211, 63], "score": 1.0, "content": "there exist ", "type": "text"}, {"bbox": [211, 47, 281, 59], "score": 0.93, "content": "0<\\delta_{1},\\delta_{2}\\leq1", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [282, 44, 336, 63], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [337, 47, 375, 60], "score": 0.92, "content": "\\gamma_{1}\\mid_{[0,\\delta_{1}]}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [375, 44, 401, 63], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [401, 46, 439, 60], "score": 0.91, "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [440, 44, 538, 63], "score": 1.0, "content": " coincide up to the", "type": "text"}], "index": 2}, {"bbox": [152, 60, 239, 74], "spans": [{"bbox": [152, 60, 239, 74], "score": 1.0, "content": "parametrization.", "type": "text"}], "index": 3, "is_list_end_line": true}, {"bbox": [155, 75, 537, 89], "spans": [{"bbox": [155, 75, 392, 89], "score": 1.0, "content": "We say analogously that the final segment of ", "type": "text"}, {"bbox": [392, 80, 403, 88], "score": 0.87, "content": "\\gamma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [403, 75, 537, 89], "score": 1.0, "content": " coincides with the initial", "type": "text"}], "index": 4, "is_list_start_line": true}, {"bbox": [153, 89, 537, 103], "spans": [{"bbox": [153, 89, 213, 103], "score": 1.0, "content": "segment of ", "type": "text"}, {"bbox": [214, 92, 225, 102], "score": 0.79, "content": "\\gamma_{2}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [226, 89, 280, 103], "score": 1.0, "content": " (shortly: ", "type": "text"}, {"bbox": [280, 91, 324, 102], "score": 0.68, "content": "\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [325, 89, 407, 103], "score": 1.0, "content": ") iff there exist ", "type": "text"}, {"bbox": [407, 89, 483, 102], "score": 0.92, "content": "0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [483, 89, 537, 103], "score": 1.0, "content": " such that", "type": "text"}], "index": 5}, {"bbox": [153, 103, 449, 119], "spans": [{"bbox": [153, 103, 198, 118], "score": 0.89, "content": "\\gamma_{1}^{-1}~|_{[0,\\delta_{1}]}", "type": "inline_equation", "height": 15, "width": 45}, {"bbox": [199, 104, 224, 119], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [224, 104, 262, 118], "score": 0.91, "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [262, 104, 449, 119], "score": 1.0, "content": " coincide up to the parametrization.", "type": "text"}], "index": 6, "is_list_end_line": true}, {"bbox": [152, 118, 538, 133], "spans": [{"bbox": [152, 118, 468, 133], "score": 1.0, "content": "Iff the corresponding relations are not fulfilled, we write ", "type": "text"}, {"bbox": [469, 119, 481, 131], "score": 0.82, "content": "\\gamma_{1}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [482, 118, 498, 133], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [498, 119, 511, 131], "score": 0.83, "content": "\\gamma_{2}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [511, 118, 538, 133], "score": 1.0, "content": " and", "type": "text"}], "index": 7, "is_list_start_line": true}, {"bbox": [153, 133, 263, 146], "spans": [{"bbox": [153, 133, 194, 146], "score": 0.63, "content": "\\gamma_{1}\\neq\\gamma_{2}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [195, 133, 263, 146], "score": 1.0, "content": ", respectively.", "type": "text"}], "index": 8, "is_list_end_line": true}], "index": 4.5, "bbox_fs": [151, 31, 539, 146]}, {"type": "text", "bbox": [62, 154, 289, 169], "lines": [{"bbox": [63, 157, 288, 171], "spans": [{"bbox": [63, 157, 288, 171], "score": 1.0, "content": "Finally, we recall the decomposition lemma.", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [63, 157, 288, 171]}, {"type": "text", "bbox": [63, 178, 537, 238], "lines": [{"bbox": [62, 181, 537, 196], "spans": [{"bbox": [62, 181, 159, 196], "score": 1.0, "content": "Lemma 7.4 Let ", "type": "text"}, {"bbox": [159, 182, 195, 192], "score": 0.87, "content": "x\\in M", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [195, 181, 285, 196], "score": 1.0, "content": " be a point. Any ", "type": "text"}, {"bbox": [285, 183, 316, 194], "score": 0.94, "content": "\\gamma\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [317, 181, 537, 196], "score": 1.0, "content": " can be written (up to parametrization) as", "type": "text"}], "index": 10}, {"bbox": [137, 196, 334, 210], "spans": [{"bbox": [137, 196, 191, 210], "score": 1.0, "content": "a product", "type": "text"}, {"bbox": [192, 196, 213, 209], "score": 0.9, "content": "\\Pi\\,\\gamma_{i}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [214, 196, 243, 210], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [243, 198, 277, 209], "score": 0.92, "content": "\\gamma_{i}\\in\\mathcal{P}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [277, 196, 334, 210], "score": 1.0, "content": ", such that", "type": "text"}], "index": 11}, {"bbox": [137, 210, 252, 225], "spans": [{"bbox": [137, 210, 171, 225], "score": 1.0, "content": "• int", "type": "text"}, {"bbox": [171, 210, 235, 224], "score": 0.75, "content": "\\gamma_{i}\\cap\\{x\\}=\\emptyset", "type": "inline_equation", "height": 14, "width": 64}, {"bbox": [235, 210, 252, 225], "score": 1.0, "content": "or", "type": "text"}], "index": 12}, {"bbox": [137, 224, 219, 240], "spans": [{"bbox": [137, 224, 172, 240], "score": 1.0, "content": "• int ", "type": "text"}, {"bbox": [172, 226, 216, 239], "score": 0.81, "content": "\\gamma_{i}=\\{x\\}", "type": "inline_equation", "height": 13, "width": 44}, {"bbox": [216, 224, 219, 240], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5, "bbox_fs": [62, 181, 537, 240]}, {"type": "title", "bbox": [63, 254, 353, 272], "lines": [{"bbox": [63, 258, 351, 273], "spans": [{"bbox": [63, 258, 95, 272], "score": 1.0, "content": "7.2", "type": "text"}, {"bbox": [97, 258, 351, 273], "score": 1.0, "content": "Successive Magnifying of the Types", "type": "text"}], "index": 14}], "index": 14}, {"type": "list", "bbox": [63, 279, 537, 309], "lines": [{"bbox": [63, 282, 535, 297], "spans": [{"bbox": [63, 282, 535, 297], "score": 1.0, "content": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types.", "type": "text"}], "index": 15, "is_list_end_line": true}, {"bbox": [62, 295, 536, 311], "spans": [{"bbox": [62, 295, 403, 311], "score": 1.0, "content": "Hereby, we will use explicitly the construction of a new connection ", "type": "text"}, {"bbox": [403, 295, 415, 307], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [415, 295, 445, 311], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [445, 296, 454, 307], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [454, 295, 536, 311], "score": 1.0, "content": " as given in [10].", "type": "text"}], "index": 16, "is_list_start_line": true, "is_list_end_line": true}], "index": 15.5, "bbox_fs": [62, 282, 536, 311]}, {"type": "text", "bbox": [61, 317, 537, 425], "lines": [{"bbox": [61, 319, 537, 336], "spans": [{"bbox": [61, 319, 159, 336], "score": 1.0, "content": "Lemma 7.5 Let ", "type": "text"}, {"bbox": [160, 322, 171, 333], "score": 0.88, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [171, 319, 305, 336], "score": 1.0, "content": " be finitely many graphs, ", "type": "text"}, {"bbox": [306, 320, 341, 332], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [341, 319, 368, 336], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [369, 322, 413, 333], "score": 0.93, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [414, 319, 537, 336], "score": 1.0, "content": " be a finite set of paths", "type": "text"}], "index": 17}, {"bbox": [137, 334, 464, 351], "spans": [{"bbox": [137, 334, 165, 351], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [165, 335, 266, 349], "score": 0.92, "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [266, 334, 362, 351], "score": 1.0, "content": ". Furthermore, let ", "type": "text"}, {"bbox": [362, 337, 394, 348], "score": 0.92, "content": "g\\in\\mathbf G", "type": "inline_equation", "height": 11, "width": 32}, {"bbox": [394, 334, 464, 351], "score": 1.0, "content": " be arbitrary.", "type": "text"}], "index": 18}, {"bbox": [137, 349, 323, 365], "spans": [{"bbox": [137, 349, 226, 365], "score": 1.0, "content": "Then there is an ", "type": "text"}, {"bbox": [227, 349, 263, 362], "score": 0.91, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [263, 349, 323, 365], "score": 1.0, "content": ", such that:", "type": "text"}], "index": 19}, {"bbox": [137, 365, 242, 381], "spans": [{"bbox": [137, 365, 154, 381], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [155, 365, 238, 379], "score": 0.74, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 14, "width": 83}, {"bbox": [239, 365, 242, 381], "score": 1.0, "content": ",", "type": "text"}], "index": 20}, {"bbox": [136, 380, 289, 397], "spans": [{"bbox": [136, 380, 154, 397], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [155, 380, 241, 396], "score": 0.81, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 16, "width": 86}, {"bbox": [241, 380, 278, 397], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [279, 383, 283, 393], "score": 0.59, "content": "i", "type": "inline_equation", "height": 10, "width": 4}, {"bbox": [284, 380, 289, 397], "score": 1.0, "content": ",", "type": "text"}], "index": 21}, {"bbox": [136, 396, 309, 410], "spans": [{"bbox": [136, 396, 154, 410], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [155, 397, 208, 410], "score": 0.73, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "type": "inline_equation", "height": 13, "width": 53}, {"bbox": [209, 397, 245, 410], "score": 1.0, "content": " for an ", "type": "text"}, {"bbox": [245, 396, 284, 408], "score": 0.51, "content": "e\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [285, 396, 309, 410], "score": 1.0, "content": " and", "type": "text"}], "index": 22}, {"bbox": [137, 410, 294, 428], "spans": [{"bbox": [137, 410, 154, 428], "score": 1.0, "content": "•", "type": "text"}, {"bbox": [155, 410, 290, 426], "score": 0.75, "content": "\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 16, "width": 135}, {"bbox": [290, 410, 294, 428], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 20, "bbox_fs": [61, 319, 537, 428]}, {"type": "text", "bbox": [62, 435, 537, 478], "lines": [{"bbox": [62, 437, 537, 452], "spans": [{"bbox": [62, 437, 147, 452], "score": 1.0, "content": "Proof 1. Let ", "type": "text"}, {"bbox": [147, 439, 190, 449], "score": 0.91, "content": "m^{\\prime}\\in M", "type": "inline_equation", "height": 10, "width": 43}, {"bbox": [190, 437, 489, 452], "score": 1.0, "content": " be some point that is neither contained in the images of ", "type": "text"}, {"bbox": [490, 440, 501, 451], "score": 0.92, "content": "\\Gamma_{i}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [501, 437, 537, 452], "score": 1.0, "content": " nor in", "type": "text"}], "index": 24}, {"bbox": [125, 452, 538, 467], "spans": [{"bbox": [125, 452, 165, 467], "score": 1.0, "content": "that of ", "type": "text"}, {"bbox": [165, 457, 174, 463], "score": 0.78, "content": "\\alpha", "type": "inline_equation", "height": 6, "width": 9}, {"bbox": [175, 452, 227, 467], "score": 1.0, "content": ", and join ", "type": "text"}, {"bbox": [228, 457, 239, 463], "score": 0.82, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [239, 452, 269, 467], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [269, 453, 282, 463], "score": 0.87, "content": "m^{\\prime}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [283, 452, 359, 467], "score": 1.0, "content": " by some path ", "type": "text"}, {"bbox": [360, 457, 367, 465], "score": 0.88, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [367, 452, 419, 467], "score": 1.0, "content": ". Now let ", "type": "text"}, {"bbox": [420, 454, 428, 463], "score": 0.91, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [428, 452, 538, 467], "score": 1.0, "content": " be some closed path", "type": "text"}], "index": 25}, {"bbox": [126, 468, 464, 480], "spans": [{"bbox": [126, 468, 140, 480], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [140, 469, 153, 477], "score": 0.91, "content": "M", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [153, 468, 239, 480], "score": 1.0, "content": " with base point ", "type": "text"}, {"bbox": [239, 468, 253, 477], "score": 0.9, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [253, 468, 464, 480], "score": 1.0, "content": " and without self-intersections, such that", "type": "text"}], "index": 26}], "index": 25, "bbox_fs": [62, 437, 538, 480]}, {"type": "interline_equation", "bbox": [227, 480, 438, 500], "lines": [{"bbox": [227, 480, 438, 500], "spans": [{"bbox": [227, 480, 438, 500], "score": 0.74, "content": "\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}", "type": "interline_equation"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [119, 500, 537, 544], "lines": [{"bbox": [126, 501, 536, 517], "spans": [{"bbox": [126, 501, 295, 517], "score": 1.0, "content": "Obviously, there exists such an ", "type": "text"}, {"bbox": [295, 504, 304, 513], "score": 0.89, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [304, 501, 353, 517], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [353, 504, 366, 513], "score": 0.89, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [367, 501, 536, 517], "score": 1.0, "content": " is supposed to be at least two-", "type": "text"}], "index": 28}, {"bbox": [126, 516, 445, 532], "spans": [{"bbox": [126, 516, 215, 532], "score": 1.0, "content": "dimensional. Set ", "type": "text"}, {"bbox": [215, 517, 311, 530], "score": 0.93, "content": "e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [312, 516, 337, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [338, 517, 442, 531], "score": 0.93, "content": "g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)", "type": "inline_equation", "height": 14, "width": 104}, {"bbox": [442, 516, 445, 532], "score": 1.0, "content": ".", "type": "text"}], "index": 29}, {"bbox": [125, 531, 421, 546], "spans": [{"bbox": [125, 531, 270, 546], "score": 1.0, "content": "Finally, define a connection ", "type": "text"}, {"bbox": [270, 531, 282, 543], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [282, 531, 303, 546], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [304, 533, 313, 543], "score": 0.85, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [313, 531, 319, 546], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [320, 534, 328, 543], "score": 0.85, "content": "e^{\\prime}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [329, 531, 354, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [354, 534, 363, 545], "score": 0.91, "content": "g^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [364, 531, 421, 546], "score": 1.0, "content": " as follows:", "type": "text"}], "index": 30}], "index": 29, "bbox_fs": [125, 501, 536, 546]}, {"type": "text", "bbox": [105, 545, 222, 558], "lines": [{"bbox": [104, 546, 221, 559], "spans": [{"bbox": [104, 546, 209, 559], "score": 1.0, "content": "2. Construction of ", "type": "text"}, {"bbox": [210, 546, 221, 557], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 11, "width": 11}], "index": 31}], "index": 31, "bbox_fs": [104, 546, 221, 559]}, {"type": "text", "bbox": [123, 559, 537, 688], "lines": [{"bbox": [127, 561, 536, 577], "spans": [{"bbox": [127, 561, 163, 577], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [163, 563, 193, 573], "score": 0.92, "content": "\\delta\\in\\mathcal{P}", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [193, 561, 536, 577], "score": 1.0, "content": " be for the moment a ”genuine” path (i.e., not an equivalence class)", "type": "text"}], "index": 32}, {"bbox": [142, 575, 536, 591], "spans": [{"bbox": [142, 575, 350, 591], "score": 1.0, "content": "that does not contain the initial point ", "type": "text"}, {"bbox": [351, 576, 408, 589], "score": 0.92, "content": "e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [408, 575, 427, 591], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [428, 576, 436, 586], "score": 0.86, "content": "e^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [437, 575, 536, 591], "score": 1.0, "content": " as an inner point.", "type": "text"}], "index": 33}, {"bbox": [144, 590, 376, 605], "spans": [{"bbox": [144, 590, 257, 605], "score": 1.0, "content": "Explicitly we have int ", "type": "text"}, {"bbox": [257, 591, 333, 604], "score": 0.92, "content": "\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [334, 590, 376, 605], "score": 1.0, "content": ". Define", "type": "text"}], "index": 34}, {"bbox": [142, 604, 481, 673], "spans": [{"bbox": [142, 604, 481, 673], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..", "type": "inline_equation"}], "index": 35}, {"bbox": [140, 670, 350, 690], "spans": [{"bbox": [140, 670, 257, 690], "score": 1.0, "content": "For every trivial path ", "type": "text"}, {"bbox": [257, 675, 263, 684], "score": 0.84, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [263, 670, 284, 690], "score": 1.0, "content": " set ", "type": "text"}, {"bbox": [284, 674, 345, 688], "score": 0.86, "content": "h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}", "type": "inline_equation", "height": 14, "width": 61}, {"bbox": [346, 670, 350, 690], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34, "bbox_fs": [127, 561, 536, 690]}]}
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[{"type": "text", "bbox": [62, 14, 230, 29], "content": "Definition 7.1 Let .", "index": 0}, {"type": "list", "bbox": [60, 17, 539, 146], "content": "", "index": 1}, {"type": "text", "bbox": [62, 154, 289, 169], "content": "Finally, we recall the decomposition lemma.", "index": 2}, {"type": "text", "bbox": [63, 178, 537, 238], "content": "Lemma 7.4 Let be a point. Any can be written (up to parametrization) as a product with , such that • int or • int .", "index": 3}, {"type": "title", "bbox": [63, 254, 353, 272], "content": "7.2 Successive Magnifying of the Types", "index": 4}, {"type": "list", "bbox": [63, 279, 537, 309], "content": "", "index": 5}, {"type": "text", "bbox": [61, 317, 537, 425], "content": "Lemma 7.5 Let be finitely many graphs, and be a finite set of paths with . Furthermore, let be arbitrary. Then there is an , such that: • , • for all , • for an and • .", "index": 6}, {"type": "text", "bbox": [62, 435, 537, 478], "content": "Proof 1. Let be some point that is neither contained in the images of nor in that of , and join with by some path . Now let be some closed path in with base point and without self-intersections, such that", "index": 7}, {"type": "interline_equation", "bbox": [227, 480, 438, 500], "content": "", "index": 8}, {"type": "text", "bbox": [119, 500, 537, 544], "content": "Obviously, there exists such an because is supposed to be at least two- dimensional. Set and . Finally, define a connection for , and as follows:", "index": 9}, {"type": "text", "bbox": [105, 545, 222, 558], "content": "2. Construction of", "index": 10}, {"type": "text", "bbox": [123, 559, 537, 688], "content": "• Let be for the moment a ”genuine” path (i.e., not an equivalence class) that does not contain the initial point of as an inner point. Explicitly we have int . Define For every trivial path set .", "index": 11}]
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[{"bbox": [61, 16, 230, 32], "content": "Definition 7.1 Let .", "parent_index": 0, "line_index": 0}, {"bbox": [152, 31, 539, 47], "content": "We say that and have the same initial segment (shortly: ↑↑ ) iff", "parent_index": 1, "line_index": 0}, {"bbox": [151, 44, 538, 63], "content": "there exist such that and coincide up to the", "parent_index": 1, "line_index": 1}, {"bbox": [152, 60, 239, 74], "content": "parametrization.", "parent_index": 1, "line_index": 2}, {"bbox": [155, 75, 537, 89], "content": "We say analogously that the final segment of coincides with the initial", "parent_index": 1, "line_index": 3}, {"bbox": [153, 89, 537, 103], "content": "segment of (shortly: ) iff there exist such that", "parent_index": 1, "line_index": 4}, {"bbox": [153, 103, 449, 119], "content": "and coincide up to the parametrization.", "parent_index": 1, "line_index": 5}, {"bbox": [152, 118, 538, 133], "content": "Iff the corresponding relations are not fulfilled, we write ↑↑ and", "parent_index": 1, "line_index": 6}, {"bbox": [153, 133, 263, 146], "content": ", respectively.", "parent_index": 1, "line_index": 7}, {"bbox": [63, 157, 288, 171], "content": "Finally, we recall the decomposition lemma.", "parent_index": 2, "line_index": 0}, {"bbox": [62, 181, 537, 196], "content": "Lemma 7.4 Let be a point. Any can be written (up to parametrization) as", "parent_index": 3, "line_index": 0}, {"bbox": [137, 196, 334, 210], "content": "a product with , such that", "parent_index": 3, "line_index": 1}, {"bbox": [137, 210, 252, 225], "content": "• int or", "parent_index": 3, "line_index": 2}, {"bbox": [137, 224, 219, 240], "content": "• int .", "parent_index": 3, "line_index": 3}, {"bbox": [63, 258, 351, 273], "content": "7.2 Successive Magnifying of the Types", "parent_index": 4, "line_index": 0}, {"bbox": [63, 282, 535, 297], "content": "In order to prove Proposition 7.1 we need the following lemma for magnifying the types.", "parent_index": 5, "line_index": 0}, {"bbox": [62, 295, 536, 311], "content": "Hereby, we will use explicitly the construction of a new connection from as given in [10].", "parent_index": 5, "line_index": 1}, {"bbox": [61, 319, 537, 336], "content": "Lemma 7.5 Let be finitely many graphs, and be a finite set of paths", "parent_index": 6, "line_index": 0}, {"bbox": [137, 334, 464, 351], "content": "with . Furthermore, let be arbitrary.", "parent_index": 6, "line_index": 1}, {"bbox": [137, 349, 323, 365], "content": "Then there is an , such that:", "parent_index": 6, "line_index": 2}, {"bbox": [137, 365, 242, 381], "content": "• ,", "parent_index": 6, "line_index": 3}, {"bbox": [136, 380, 289, 397], "content": "• for all ,", "parent_index": 6, "line_index": 4}, {"bbox": [136, 396, 309, 410], "content": "• for an and", "parent_index": 6, "line_index": 5}, {"bbox": [137, 410, 294, 428], "content": "• .", "parent_index": 6, "line_index": 6}, {"bbox": [62, 437, 537, 452], "content": "Proof 1. Let be some point that is neither contained in the images of nor in", "parent_index": 7, "line_index": 0}, {"bbox": [125, 452, 538, 467], "content": "that of , and join with by some path . Now let be some closed path", "parent_index": 7, "line_index": 1}, {"bbox": [126, 468, 464, 480], "content": "in with base point and without self-intersections, such that", "parent_index": 7, "line_index": 2}, {"bbox": [126, 501, 536, 517], "content": "Obviously, there exists such an because is supposed to be at least two-", "parent_index": 9, "line_index": 0}, {"bbox": [126, 516, 445, 532], "content": "dimensional. Set and .", "parent_index": 9, "line_index": 1}, {"bbox": [125, 531, 421, 546], "content": "Finally, define a connection for , and as follows:", "parent_index": 9, "line_index": 2}, {"bbox": [104, 546, 221, 559], "content": "2. Construction of", "parent_index": 10, "line_index": 0}, {"bbox": [127, 561, 536, 577], "content": "• Let be for the moment a ”genuine” path (i.e., not an equivalence class)", "parent_index": 11, "line_index": 0}, {"bbox": [142, 575, 536, 591], "content": "that does not contain the initial point of as an inner point.", "parent_index": 11, "line_index": 1}, {"bbox": [144, 590, 376, 605], "content": "Explicitly we have int . Define", "parent_index": 11, "line_index": 2}, {"bbox": [142, 604, 481, 673], "content": "", "parent_index": 11, "line_index": 3}, {"bbox": [140, 670, 350, 690], "content": "For every trivial path set .", "parent_index": 11, "line_index": 4}]
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[]
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[{"bbox": [174, 19, 226, 30], "content": "\\gamma_{1},\\gamma_{2}\\in\\mathcal{P}", "parent_index": 0, "subtype": "inline"}, {"bbox": [219, 37, 230, 45], "content": "\\gamma_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [257, 37, 268, 45], "content": "\\gamma_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [475, 32, 488, 45], "content": "\\gamma_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [505, 33, 518, 45], "content": "\\gamma_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [211, 47, 281, 59], "content": "0<\\delta_{1},\\delta_{2}\\leq1", "parent_index": 1, "subtype": "inline"}, {"bbox": [337, 47, 375, 60], "content": "\\gamma_{1}\\mid_{[0,\\delta_{1}]}", "parent_index": 1, "subtype": "inline"}, {"bbox": [401, 46, 439, 60], "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "parent_index": 1, "subtype": "inline"}, {"bbox": [392, 80, 403, 88], "content": "\\gamma_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [214, 92, 225, 102], "content": "\\gamma_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [280, 91, 324, 102], "content": "\\gamma_{1}\\downarrow\\uparrow\\gamma_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [407, 89, 483, 102], "content": "0\\,<\\,\\delta_{1},\\delta_{2}\\,\\leq\\,1", "parent_index": 1, "subtype": "inline"}, {"bbox": [153, 103, 198, 118], "content": "\\gamma_{1}^{-1}~|_{[0,\\delta_{1}]}", "parent_index": 1, "subtype": "inline"}, {"bbox": [224, 104, 262, 118], "content": "\\gamma_{2}\\mid_{[0,\\delta_{2}]}", "parent_index": 1, "subtype": "inline"}, {"bbox": [469, 119, 481, 131], "content": "\\gamma_{1}", "parent_index": 1, "subtype": "inline"}, {"bbox": [498, 119, 511, 131], "content": "\\gamma_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [153, 133, 194, 146], "content": "\\gamma_{1}\\neq\\gamma_{2}", "parent_index": 1, "subtype": "inline"}, {"bbox": [159, 182, 195, 192], "content": "x\\in M", "parent_index": 3, "subtype": "inline"}, {"bbox": [285, 183, 316, 194], "content": "\\gamma\\in\\mathcal{P}", "parent_index": 3, "subtype": "inline"}, {"bbox": [192, 196, 213, 209], "content": "\\Pi\\,\\gamma_{i}", "parent_index": 3, "subtype": "inline"}, {"bbox": [243, 198, 277, 209], "content": "\\gamma_{i}\\in\\mathcal{P}", "parent_index": 3, "subtype": "inline"}, {"bbox": [171, 210, 235, 224], "content": "\\gamma_{i}\\cap\\{x\\}=\\emptyset", "parent_index": 3, "subtype": "inline"}, {"bbox": [172, 226, 216, 239], "content": "\\gamma_{i}=\\{x\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [403, 295, 415, 307], "content": "\\overline{{A}}^{\\prime}", "parent_index": 5, "subtype": "inline"}, {"bbox": [445, 296, 454, 307], "content": "\\overline{{A}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [160, 322, 171, 333], "content": "\\Gamma_{i}", "parent_index": 6, "subtype": "inline"}, {"bbox": [306, 320, 341, 332], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [369, 322, 413, 333], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [165, 335, 266, 349], "content": "Z(\\mathbf{H}_{\\overline{{A}}})=Z(h_{\\overline{{A}}}(\\alpha))", "parent_index": 6, "subtype": "inline"}, {"bbox": [362, 337, 394, 348], "content": "g\\in\\mathbf G", "parent_index": 6, "subtype": "inline"}, {"bbox": [227, 349, 263, 362], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [155, 365, 238, 379], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "parent_index": 6, "subtype": "inline"}, {"bbox": [155, 380, 241, 396], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "parent_index": 6, "subtype": "inline"}, {"bbox": [279, 383, 283, 393], "content": "i", "parent_index": 6, "subtype": "inline"}, {"bbox": [155, 397, 208, 410], "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "parent_index": 6, "subtype": "inline"}, {"bbox": [245, 396, 284, 408], "content": "e\\in{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [155, 410, 290, 426], "content": "\\bar{Z}(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "parent_index": 6, "subtype": "inline"}, {"bbox": [147, 439, 190, 449], "content": "m^{\\prime}\\in M", "parent_index": 7, "subtype": "inline"}, {"bbox": [490, 440, 501, 451], "content": "\\Gamma_{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [165, 457, 174, 463], "content": "\\alpha", "parent_index": 7, "subtype": "inline"}, {"bbox": [228, 457, 239, 463], "content": "m", "parent_index": 7, "subtype": "inline"}, {"bbox": [269, 453, 282, 463], "content": "m^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [360, 457, 367, 465], "content": "\\gamma", "parent_index": 7, "subtype": "inline"}, {"bbox": [420, 454, 428, 463], "content": "e^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [140, 469, 153, 477], "content": "M", "parent_index": 7, "subtype": "inline"}, {"bbox": [239, 468, 253, 477], "content": "m^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [227, 480, 438, 500], "content": "\\begin{array}{r}{\\operatorname{m}e^{\\prime}\\cap\\left(\\operatorname{int}\\gamma\\cup\\operatorname{im}\\left(\\alpha\\right)\\cup\\bigcup\\operatorname{im}\\left(\\Gamma_{i}\\right)\\right)\\right)=\\emptyset.}\\end{array}", "parent_index": 8, "subtype": "interline"}, {"bbox": [295, 504, 304, 513], "content": "e^{\\prime}", "parent_index": 9, "subtype": "inline"}, {"bbox": [353, 504, 366, 513], "content": "M", "parent_index": 9, "subtype": "inline"}, {"bbox": [215, 517, 311, 530], "content": "e:=\\gamma\\,e^{\\prime}\\,\\gamma^{-1}\\in\\mathcal{H}\\mathcal{G}", "parent_index": 9, "subtype": "inline"}, {"bbox": [338, 517, 442, 531], "content": "g^{\\prime}:=h_{\\overline{{{A}}}}(\\gamma)^{-1}g h_{\\overline{{{A}}}}(\\gamma)", "parent_index": 9, "subtype": "inline"}, {"bbox": [270, 531, 282, 543], "content": "\\overline{{A}}^{\\prime}", "parent_index": 9, "subtype": "inline"}, {"bbox": [304, 533, 313, 543], "content": "\\overline{{A}}", "parent_index": 9, "subtype": "inline"}, {"bbox": [320, 534, 328, 543], "content": "e^{\\prime}", "parent_index": 9, "subtype": "inline"}, {"bbox": [354, 534, 363, 545], "content": "g^{\\prime}", "parent_index": 9, "subtype": "inline"}, {"bbox": [210, 546, 221, 557], "content": "\\overline{{A}}^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [163, 563, 193, 573], "content": "\\delta\\in\\mathcal{P}", "parent_index": 11, "subtype": "inline"}, {"bbox": [351, 576, 408, 589], "content": "e^{\\prime}(0)\\,\\equiv\\,m^{\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [428, 576, 436, 586], "content": "e^{\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [257, 591, 333, 604], "content": "\\delta\\cap\\{e^{\\prime}(0)\\}=\\emptyset", "parent_index": 11, "subtype": "inline"}, {"bbox": [142, 604, 481, 673], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\delta):=\\left\\{\\!\\!\\begin{array}{r l r}{{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{for~}\\delta\\,\\uparrow\\uparrow e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,h_{\\overline{{{A}}}}(e^{\\prime})\\,g^{\\prime-1},}}&{{\\mathrm{for~}\\delta\\,\\#\\,e^{\\prime}\\mathrm{~and~}\\delta\\downarrow\\uparrow e^{\\prime}}}\\\\ {{h_{\\overline{{{A}}}}(\\delta)\\,}}&{{\\mathrm{else}}}\\end{array}\\!\\!\\right..", "parent_index": 11, "subtype": "inline"}, {"bbox": [257, 675, 263, 684], "content": "\\delta", "parent_index": 11, "subtype": "inline"}, {"bbox": [284, 674, 345, 688], "content": "h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}", "parent_index": 11, "subtype": "inline"}]
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[]
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• Now, let $\delta\ \in\ {\mathcal{P}}$ be an arbitrary path. Decompose $\delta$ into a finite product $\Pi\,\delta_{i}$ due to Lemma 7.4 such that no $\delta_{i}$ contains the point $e^{\prime}(0)$ in the interior supposed $\delta_{i}$ is not trivial. Here, set $h_{\overline{{A}}^{\prime}}(\delta):=\Pi\,h_{\overline{{A}}^{\prime}}(\delta_{i})$ .
We know from [10] that $\overline{{A}}^{\prime}$ is indeed a connectio n.
3. The assertion $\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})\;=\;\pi_{\Gamma_{i}}(\overline{{{A}}})$ for all $i$ is an immediate consequence of the construction because im $(\Gamma_{i})\cap\operatorname{int}e^{\prime}=\varnothing$ . As well, we get $h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$ .
4. Moreover, from (4), the fact that $e^{\prime}$ has no self-intersections and the definition of $\overline{{A}}^{\prime}$ we get $h_{\overline{{{A}}}^{\prime}}(\gamma)=h_{\overline{{{A}}}}(\gamma)$ and so
$$
h_{\overline{{{A}}}^{\prime}}(e)=h_{\overline{{{A}}}^{\prime}}(\gamma)\;h_{\overline{{{A}}}^{\prime}}(e^{\prime})\;h_{\overline{{{A}}}^{\prime}}(\gamma^{-1})=h_{\overline{{{A}}}}(\gamma)\;g^{\prime}\;h_{\overline{{{A}}}}(\gamma)^{-1}=g.
$$
5. We have $Z(\mathbf{H}_{\overline{{{A}}}^{\prime}})=Z(\{g\}\cup\mathbf{H}_{\overline{{{A}}}})$ .
Let $f\in\cal Z(\mathbf{H}_{\overline{{\cal A}}^{\prime}})$ , i.e. $f\;h_{\overline{{{A}}}^{\prime}}(\alpha)=h_{\overline{{{A}}}^{\prime}}(\alpha)\;f$ for all $\alpha\in{\mathcal{H}}{\mathcal{G}}$ .
• From $h_{\overline{{{A}}}^{\prime}}(e)=g$ follows $f g=g f$ , i.e. $f\in Z(\{g\})$ . From im e′ ∩ im $({\pmb{\alpha}})=\emptyset$ follows $h_{\overline{{A}}}(\alpha_{i})=h_{\overline{{A}}^{\prime}}(\alpha_{i})$ , i.e. $f\,\in\,Z(h_{\overline{{A}}}(\alpha_{i}))$ for all $i$ .
Let $\alpha^{\prime}$ be a path from $m^{\prime}$ to $m^{\prime}$ , such that int $\alpha^{\prime}\cap\{m^{\prime}\}=\emptyset$ or int $\alpha^{\prime}=$ $\{m^{\prime}\}$ . Set $\alpha:=\gamma\,\alpha^{\prime}\,\gamma^{-1}$ . Then by construction we have
$$
\begin{array}{l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}^{\prime}}(\gamma)\;h_{\overline{{{A}}}^{\prime}}(\alpha^{\prime})\;h_{\overline{{{A}}}^{\prime}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}^{\prime}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(\gamma)^{-1}.}}\end{array}
$$
There are four cases:
$-\mathrm{~\boldmath~\alpha~}\alpha^{\prime}$ ↑↑ $e^{\prime}$ and $\alpha^{\prime}\,\#\,e^{\prime}$ :
$$
\begin{array}{l l l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(\gamma)^{-1}}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma\;\alpha^{\prime}\;\gamma^{-1})}}\\ {{}}&{{=}}&{{h_{\overline{{{A}}}}(\alpha).}}&{{}}&{{}}\end{array}
$$
$-\mathrm{~\boldmath~\alpha~}\alpha^{\prime}$ ↑↑ $e^{\prime}$ and $\alpha^{\prime}\,\#\,e^{\prime}$ :
$$
\begin{array}{l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{g\,h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\alpha^{\prime})\,h_{\overline{{{A}}}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{g\,h_{\overline{{{A}}}}(\gamma e^{\prime-1}\alpha^{\prime}\gamma^{-1}).}}\end{array}
$$
$\alpha^{\prime}$ ↑↑ $e^{\prime}$ and $\alpha^{\prime}\downarrow\uparrow e^{\prime}$ :
$$
\begin{array}{l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(e^{\prime})\;(g^{\prime})^{-1}h_{\overline{{{A}}}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(e^{\prime})h_{\overline{{{A}}}}(\gamma)^{-1}\;g^{-1}}}\\ {{}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma\alpha^{\prime}e^{\prime}\gamma^{-1})\;g^{-1}.}}\end{array}
$$
$\alpha^{\prime}$ ↑↑ $e^{\prime}$ and $\alpha^{\prime}\downarrow\uparrow e^{\prime}$ :
$$
\begin{array}{l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(e^{\prime})\;(g^{\prime})^{-1}\,h_{\overline{{{A}}}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{g\,h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(e^{\prime})^{-1}\;h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(e^{\prime})\;h_{\overline{{{A}}}}(\gamma)^{-1}\;g^{-1}}}\\ {{}}&{{=}}&{{g\,h_{\overline{{{A}}}}(\gamma e^{\prime-1}\alpha^{\prime}e^{\prime}\gamma^{-1})\;g^{-1}.}}\end{array}
$$
Thus, in each case we get $f\in Z(\{h_{\overline{{A}}^{\prime}}(\alpha)\})$ .
Now, let $\alpha\in{\mathcal{H}}{\mathcal{G}}$ be arbitrary and $\alpha^{\prime}:=\gamma^{-1}\alpha\gamma$ .
By the Decomposition Lemma 7.4 there is a decomposition $\alpha^{\prime}\ =$ $\Pi\,\alpha_{i}^{\prime}$ with int $\alpha_{i}^{\prime}\cap\{m^{\prime}\}\ =\ \emptyset$ or int $\alpha_{i}^{\prime}\ =\ \{m^{\prime}\}$ for all $i$ . Thus, $\alpha\,=\,\gamma\bigl(\Pi\,\alpha_{i}^{\prime}\bigr)\gamma^{-1}\,=\,\Pi\bigl(\gamma\alpha_{i}^{\prime}\gamma^{-1}\bigr)$ . Using the result just proven we get $f\in Z\big(\big\{h_{\overline{{A}}^{\prime}}\big(\Pi\big(\gamma\alpha_{i}^{\prime}\gamma^{-1}\big)\big)\big\}\big)=Z(\{h_{\overline{{A}}^{\prime}}(\alpha)\})$ .
|
<html><body>
<p data-bbox="125 14 537 59">• Now, let $\delta\ \in\ {\mathcal{P}}$ be an arbitrary path. Decompose $\delta$ into a finite product $\Pi\,\delta_{i}$ due to Lemma 7.4 such that no $\delta_{i}$ contains the point $e^{\prime}(0)$ in the interior supposed $\delta_{i}$ is not trivial. Here, set $h_{\overline{{A}}^{\prime}}(\delta):=\Pi\,h_{\overline{{A}}^{\prime}}(\delta_{i})$ . </p>
<p data-bbox="126 59 384 74">We know from [10] that $\overline{{A}}^{\prime}$ is indeed a connectio n. </p>
<p data-bbox="106 75 538 104">3. The assertion $\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})\;=\;\pi_{\Gamma_{i}}(\overline{{{A}}})$ for all $i$ is an immediate consequence of the construction because im $(\Gamma_{i})\cap\operatorname{int}e^{\prime}=\varnothing$ . As well, we get $h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$ . </p>
<p data-bbox="105 105 539 132">4. Moreover, from (4), the fact that $e^{\prime}$ has no self-intersections and the definition of $\overline{{A}}^{\prime}$ we get $h_{\overline{{{A}}}^{\prime}}(\gamma)=h_{\overline{{{A}}}}(\gamma)$ and so </p>
<div class="equation" data-bbox="187 134 475 149">$$
h_{\overline{{{A}}}^{\prime}}(e)=h_{\overline{{{A}}}^{\prime}}(\gamma)\;h_{\overline{{{A}}}^{\prime}}(e^{\prime})\;h_{\overline{{{A}}}^{\prime}}(\gamma^{-1})=h_{\overline{{{A}}}}(\gamma)\;g^{\prime}\;h_{\overline{{{A}}}}(\gamma)^{-1}=g.
$$</div>
<p data-bbox="105 147 296 162">5. We have $Z(\mathbf{H}_{\overline{{{A}}}^{\prime}})=Z(\{g\}\cup\mathbf{H}_{\overline{{{A}}}})$ . </p>
<p data-bbox="147 162 536 219">Let $f\in\cal Z(\mathbf{H}_{\overline{{\cal A}}^{\prime}})$ , i.e. $f\;h_{\overline{{{A}}}^{\prime}}(\alpha)=h_{\overline{{{A}}}^{\prime}}(\alpha)\;f$ for all $\alpha\in{\mathcal{H}}{\mathcal{G}}$ .
• From $h_{\overline{{{A}}}^{\prime}}(e)=g$ follows $f g=g f$ , i.e. $f\in Z(\{g\})$ . From im e′ ∩ im $({\pmb{\alpha}})=\emptyset$ follows $h_{\overline{{A}}}(\alpha_{i})=h_{\overline{{A}}^{\prime}}(\alpha_{i})$ , i.e. $f\,\in\,Z(h_{\overline{{A}}}(\alpha_{i}))$ for all $i$ . </p>
<p data-bbox="157 249 538 277">Let $\alpha^{\prime}$ be a path from $m^{\prime}$ to $m^{\prime}$ , such that int $\alpha^{\prime}\cap\{m^{\prime}\}=\emptyset$ or int $\alpha^{\prime}=$ $\{m^{\prime}\}$ . Set $\alpha:=\gamma\,\alpha^{\prime}\,\gamma^{-1}$ . Then by construction we have </p>
<div class="equation" data-bbox="265 282 444 315">$$
\begin{array}{l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}^{\prime}}(\gamma)\;h_{\overline{{{A}}}^{\prime}}(\alpha^{\prime})\;h_{\overline{{{A}}}^{\prime}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}^{\prime}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(\gamma)^{-1}.}}\end{array}
$$</div>
<p data-bbox="174 316 282 329">There are four cases: </p>
<p data-bbox="174 330 301 344">$-\mathrm{~\boldmath~\alpha~}\alpha^{\prime}$ ↑↑ $e^{\prime}$ and $\alpha^{\prime}\,\#\,e^{\prime}$ : </p>
<div class="equation" data-bbox="233 349 497 381">$$
\begin{array}{l l l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(\gamma)^{-1}}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma\;\alpha^{\prime}\;\gamma^{-1})}}\\ {{}}&{{=}}&{{h_{\overline{{{A}}}}(\alpha).}}&{{}}&{{}}\end{array}
$$</div>
<p data-bbox="173 383 302 398">$-\mathrm{~\boldmath~\alpha~}\alpha^{\prime}$ ↑↑ $e^{\prime}$ and $\alpha^{\prime}\,\#\,e^{\prime}$ : </p>
<div class="equation" data-bbox="250 402 480 451">$$
\begin{array}{l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{g\,h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\alpha^{\prime})\,h_{\overline{{{A}}}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{g\,h_{\overline{{{A}}}}(\gamma e^{\prime-1}\alpha^{\prime}\gamma^{-1}).}}\end{array}
$$</div>
<p data-bbox="173 452 302 467">$\alpha^{\prime}$ ↑↑ $e^{\prime}$ and $\alpha^{\prime}\downarrow\uparrow e^{\prime}$ : </p>
<div class="equation" data-bbox="247 471 483 519">$$
\begin{array}{l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(e^{\prime})\;(g^{\prime})^{-1}h_{\overline{{{A}}}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(e^{\prime})h_{\overline{{{A}}}}(\gamma)^{-1}\;g^{-1}}}\\ {{}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma\alpha^{\prime}e^{\prime}\gamma^{-1})\;g^{-1}.}}\end{array}
$$</div>
<p data-bbox="193 519 302 534">$\alpha^{\prime}$ ↑↑ $e^{\prime}$ and $\alpha^{\prime}\downarrow\uparrow e^{\prime}$ : </p>
<div class="equation" data-bbox="217 539 513 588">$$
\begin{array}{l l l}{{h_{\overline{{{A}}}^{\prime}}(\alpha)}}&{{=}}&{{h_{\overline{{{A}}}}(\gamma)\;g^{\prime}\,h_{\overline{{{A}}}}(e^{\prime})^{-1}\,h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(e^{\prime})\;(g^{\prime})^{-1}\,h_{\overline{{{A}}}}(\gamma)^{-1}}}\\ {{}}&{{=}}&{{g\,h_{\overline{{{A}}}}(\gamma)\;h_{\overline{{{A}}}}(e^{\prime})^{-1}\;h_{\overline{{{A}}}}(\alpha^{\prime})\;h_{\overline{{{A}}}}(e^{\prime})\;h_{\overline{{{A}}}}(\gamma)^{-1}\;g^{-1}}}\\ {{}}&{{=}}&{{g\,h_{\overline{{{A}}}}(\gamma e^{\prime-1}\alpha^{\prime}e^{\prime}\gamma^{-1})\;g^{-1}.}}\end{array}
$$</div>
<p data-bbox="168 589 401 604">Thus, in each case we get $f\in Z(\{h_{\overline{{A}}^{\prime}}(\alpha)\})$ . </p>
<p data-bbox="158 605 420 618">Now, let $\alpha\in{\mathcal{H}}{\mathcal{G}}$ be arbitrary and $\alpha^{\prime}:=\gamma^{-1}\alpha\gamma$ . </p>
<p data-bbox="171 619 537 686">By the Decomposition Lemma 7.4 there is a decomposition $\alpha^{\prime}\ =$ $\Pi\,\alpha_{i}^{\prime}$ with int $\alpha_{i}^{\prime}\cap\{m^{\prime}\}\ =\ \emptyset$ or int $\alpha_{i}^{\prime}\ =\ \{m^{\prime}\}$ for all $i$ . Thus, $\alpha\,=\,\gamma\bigl(\Pi\,\alpha_{i}^{\prime}\bigr)\gamma^{-1}\,=\,\Pi\bigl(\gamma\alpha_{i}^{\prime}\gamma^{-1}\bigr)$ . Using the result just proven we get $f\in Z\big(\big\{h_{\overline{{A}}^{\prime}}\big(\Pi\big(\gamma\alpha_{i}^{\prime}\gamma^{-1}\big)\big)\big\}\big)=Z(\{h_{\overline{{A}}^{\prime}}(\alpha)\})$ . </p>
</body></html>
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0001008v1
| 13 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "• Now, let $\\delta\\ \\in\\ {\\mathcal{P}}$ be an arbitrary path. Decompose $\\delta$ into a finite product $\\Pi\\,\\delta_{i}$ due to Lemma 7.4 such that no $\\delta_{i}$ contains the point $e^{\\prime}(0)$ in the interior supposed $\\delta_{i}$ is not trivial. Here, set $h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})$ . ", "page_idx": 13}, {"type": "text", "text": "We know from [10] that $\\overline{{A}}^{\\prime}$ is indeed a connectio n. ", "page_idx": 13}, {"type": "text", "text": "3. The assertion $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$ for all $i$ is an immediate consequence of the construction because im $(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing$ . As well, we get $h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$ . ", "page_idx": 13}, {"type": "text", "text": "4. Moreover, from (4), the fact that $e^{\\prime}$ has no self-intersections and the definition of $\\overline{{A}}^{\\prime}$ we get $h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)$ and so ", "page_idx": 13}, {"type": "equation", "text": "$$\nh_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "5. We have $Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})$ . ", "page_idx": 13}, {"type": "text", "text": "Let $f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})$ , i.e. $f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f$ for all $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ . \n• From $h_{\\overline{{{A}}}^{\\prime}}(e)=g$ follows $f g=g f$ , i.e. $f\\in Z(\\{g\\})$ . From im e′ ∩ im $({\\pmb{\\alpha}})=\\emptyset$ follows $h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})$ , i.e. $f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))$ for all $i$ . ", "page_idx": 13}, {"type": "text", "text": "Let $\\alpha^{\\prime}$ be a path from $m^{\\prime}$ to $m^{\\prime}$ , such that int $\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset$ or int $\\alpha^{\\prime}=$ $\\{m^{\\prime}\\}$ . Set $\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}$ . Then by construction we have ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "There are four cases: ", "page_idx": 13}, {"type": "text", "text": "$-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}$ ↑↑ $e^{\\prime}$ and $\\alpha^{\\prime}\\,\\#\\,e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}$ ↑↑ $e^{\\prime}$ and $\\alpha^{\\prime}\\,\\#\\,e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$\\alpha^{\\prime}$ ↑↑ $e^{\\prime}$ and $\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$\\alpha^{\\prime}$ ↑↑ $e^{\\prime}$ and $\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\n\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "Thus, in each case we get $f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})$ . ", "page_idx": 13}, {"type": "text", "text": "Now, let $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ be arbitrary and $\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma$ . ", "page_idx": 13}, {"type": "text", "text": "By the Decomposition Lemma 7.4 there is a decomposition $\\alpha^{\\prime}\\ =$ $\\Pi\\,\\alpha_{i}^{\\prime}$ with int $\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset$ or int $\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}$ for all $i$ . Thus, $\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)$ . Using the result just proven we get $f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})$ . ", "page_idx": 13}]
|
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{"preproc_blocks": [{"type": "text", "bbox": [125, 14, 537, 59], "lines": [{"bbox": [128, 16, 536, 32], "spans": [{"bbox": [128, 16, 192, 32], "score": 1.0, "content": "• Now, let ", "type": "text"}, {"bbox": [193, 19, 227, 28], "score": 0.93, "content": "\\delta\\ \\in\\ {\\mathcal{P}}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [227, 16, 417, 32], "score": 1.0, "content": " be an arbitrary path. Decompose ", "type": "text"}, {"bbox": [417, 19, 423, 28], "score": 0.89, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [423, 16, 536, 32], "score": 1.0, "content": " into a finite product", "type": "text"}], "index": 0}, {"bbox": [143, 31, 537, 47], "spans": [{"bbox": [143, 33, 163, 44], "score": 0.93, "content": "\\Pi\\,\\delta_{i}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [163, 31, 330, 47], "score": 1.0, "content": " due to Lemma 7.4 such that no ", "type": "text"}, {"bbox": [330, 33, 339, 44], "score": 0.91, "content": "\\delta_{i}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [339, 31, 438, 47], "score": 1.0, "content": " contains the point ", "type": "text"}, {"bbox": [438, 33, 461, 45], "score": 0.94, "content": "e^{\\prime}(0)", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [462, 31, 537, 47], "score": 1.0, "content": " in the interior", "type": "text"}], "index": 1}, {"bbox": [143, 45, 426, 61], "spans": [{"bbox": [143, 45, 193, 61], "score": 1.0, "content": "supposed ", "type": "text"}, {"bbox": [194, 48, 202, 58], "score": 0.92, "content": "\\delta_{i}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [203, 45, 326, 61], "score": 1.0, "content": " is not trivial. Here, set ", "type": "text"}, {"bbox": [327, 47, 423, 61], "score": 0.95, "content": "h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})", "type": "inline_equation", "height": 14, "width": 96}, {"bbox": [423, 45, 426, 61], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [126, 59, 384, 74], "lines": [{"bbox": [125, 60, 385, 76], "spans": [{"bbox": [125, 60, 252, 76], "score": 1.0, "content": "We know from [10] that ", "type": "text"}, {"bbox": [252, 61, 263, 73], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [264, 60, 385, 76], "score": 1.0, "content": " is indeed a connectio n.", "type": "text"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [106, 75, 538, 104], "lines": [{"bbox": [104, 74, 538, 93], "spans": [{"bbox": [104, 74, 202, 93], "score": 1.0, "content": "3. The assertion ", "type": "text"}, {"bbox": [202, 76, 294, 91], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 15, "width": 92}, {"bbox": [294, 74, 336, 93], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [337, 79, 341, 87], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [341, 74, 538, 93], "score": 1.0, "content": " is an immediate consequence of the", "type": "text"}], "index": 4}, {"bbox": [126, 90, 508, 106], "spans": [{"bbox": [126, 90, 252, 106], "score": 1.0, "content": "construction because im ", "type": "text"}, {"bbox": [253, 92, 333, 105], "score": 0.71, "content": "(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [333, 90, 421, 106], "score": 1.0, "content": ". As well, we get ", "type": "text"}, {"bbox": [422, 92, 504, 106], "score": 0.93, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [504, 90, 508, 106], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 4.5}, {"type": "text", "bbox": [105, 105, 539, 132], "lines": [{"bbox": [105, 105, 539, 120], "spans": [{"bbox": [105, 105, 297, 120], "score": 1.0, "content": "4. Moreover, from (4), the fact that ", "type": "text"}, {"bbox": [297, 107, 305, 117], "score": 0.89, "content": "e^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [306, 105, 539, 120], "score": 1.0, "content": " has no self-intersections and the definition of", "type": "text"}], "index": 6}, {"bbox": [126, 118, 295, 136], "spans": [{"bbox": [126, 119, 137, 131], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [138, 118, 178, 136], "score": 1.0, "content": " we get ", "type": "text"}, {"bbox": [178, 122, 255, 135], "score": 0.94, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [256, 118, 295, 136], "score": 1.0, "content": " and so", "type": "text"}], "index": 7}], "index": 6.5}, {"type": "interline_equation", "bbox": [187, 134, 475, 149], "lines": [{"bbox": [187, 134, 475, 149], "spans": [{"bbox": [187, 134, 475, 149], "score": 0.77, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [105, 147, 296, 162], "lines": [{"bbox": [105, 147, 296, 164], "spans": [{"bbox": [105, 147, 173, 163], "score": 1.0, "content": "5. We have ", "type": "text"}, {"bbox": [173, 150, 294, 164], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})", "type": "inline_equation", "height": 14, "width": 121}, {"bbox": [294, 147, 296, 163], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [147, 162, 536, 219], "lines": [{"bbox": [156, 161, 450, 180], "spans": [{"bbox": [156, 161, 178, 180], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [179, 165, 239, 178], "score": 0.91, "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [239, 161, 265, 180], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [266, 165, 367, 178], "score": 0.92, "content": "f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f", "type": "inline_equation", "height": 13, "width": 101}, {"bbox": [367, 161, 405, 180], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [405, 165, 445, 175], "score": 0.87, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [446, 161, 450, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [157, 177, 432, 193], "spans": [{"bbox": [157, 177, 205, 192], "score": 1.0, "content": "• From ", "type": "text"}, {"bbox": [205, 178, 258, 193], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [258, 177, 300, 192], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [300, 180, 342, 191], "score": 0.92, "content": "f g=g f", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [343, 177, 369, 192], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [369, 178, 427, 191], "score": 0.9, "content": "f\\in Z(\\{g\\})", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [428, 177, 432, 192], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [166, 191, 536, 208], "spans": [{"bbox": [166, 191, 260, 208], "score": 1.0, "content": "From im e′ ∩ im ", "type": "text"}, {"bbox": [261, 194, 302, 206], "score": 0.52, "content": "({\\pmb{\\alpha}})=\\emptyset", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [302, 191, 344, 208], "score": 1.0, "content": "follows ", "type": "text"}, {"bbox": [345, 193, 433, 207], "score": 0.92, "content": "h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})", "type": "inline_equation", "height": 14, "width": 88}, {"bbox": [433, 191, 460, 208], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [461, 193, 536, 207], "score": 0.91, "content": "f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))", "type": "inline_equation", "height": 14, "width": 75}], "index": 12}, {"bbox": [173, 207, 217, 219], "spans": [{"bbox": [173, 207, 208, 219], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [209, 209, 213, 217], "score": 0.87, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [213, 207, 217, 219], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "text", "bbox": [157, 249, 538, 277], "lines": [{"bbox": [171, 250, 538, 264], "spans": [{"bbox": [171, 250, 195, 264], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [195, 252, 206, 261], "score": 0.89, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [206, 250, 289, 264], "score": 1.0, "content": " be a path from ", "type": "text"}, {"bbox": [290, 252, 303, 261], "score": 0.89, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 250, 320, 264], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [321, 252, 334, 261], "score": 0.9, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [335, 250, 408, 264], "score": 1.0, "content": ", such that int", "type": "text"}, {"bbox": [408, 251, 479, 264], "score": 0.71, "content": "\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [480, 250, 513, 264], "score": 1.0, "content": "or int ", "type": "text"}, {"bbox": [513, 251, 538, 263], "score": 0.57, "content": "\\alpha^{\\prime}=", "type": "inline_equation", "height": 12, "width": 25}], "index": 14}, {"bbox": [174, 265, 460, 279], "spans": [{"bbox": [174, 266, 200, 279], "score": 0.91, "content": "\\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [200, 265, 227, 279], "score": 1.0, "content": ". Set ", "type": "text"}, {"bbox": [228, 265, 295, 278], "score": 0.93, "content": "\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [295, 265, 460, 279], "score": 1.0, "content": ". Then by construction we have", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [265, 282, 444, 315], "lines": [{"bbox": [265, 282, 444, 315], "spans": [{"bbox": [265, 282, 444, 315], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [174, 316, 282, 329], "lines": [{"bbox": [174, 317, 283, 330], "spans": [{"bbox": [174, 317, 283, 330], "score": 1.0, "content": "There are four cases:", "type": "text"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [174, 330, 301, 344], "lines": [{"bbox": [174, 331, 300, 345], "spans": [{"bbox": [174, 331, 206, 345], "score": 0.59, "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "type": "inline_equation", "height": 14, "width": 32}, {"bbox": [207, 332, 222, 345], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [223, 331, 232, 344], "score": 0.76, "content": "e^{\\prime}", "type": "inline_equation", "height": 13, "width": 9}, {"bbox": [233, 332, 258, 345], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 331, 295, 345], "score": 0.52, "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [296, 332, 300, 345], "score": 1.0, "content": ":", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [233, 349, 497, 381], "lines": [{"bbox": [233, 349, 497, 381], "spans": [{"bbox": [233, 349, 497, 381], "score": 0.91, "content": "\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [173, 383, 302, 398], "lines": [{"bbox": [173, 384, 300, 398], "spans": [{"bbox": [173, 384, 206, 398], "score": 0.55, "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [207, 385, 222, 398], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [223, 385, 232, 397], "score": 0.82, "content": "e^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [233, 385, 258, 398], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 385, 296, 398], "score": 0.8, "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [296, 385, 300, 398], "score": 1.0, "content": ":", "type": "text"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [250, 402, 480, 451], "lines": [{"bbox": [250, 402, 480, 451], "spans": [{"bbox": [250, 402, 480, 451], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [173, 452, 302, 467], "lines": [{"bbox": [194, 453, 300, 467], "spans": [{"bbox": [194, 454, 206, 466], "score": 0.75, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [206, 454, 222, 467], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [223, 454, 232, 465], "score": 0.83, "content": "e^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [233, 454, 258, 467], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 453, 296, 467], "score": 0.77, "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [296, 454, 300, 467], "score": 1.0, "content": ":", "type": "text"}], "index": 22}], "index": 22}, {"type": "interline_equation", "bbox": [247, 471, 483, 519], "lines": [{"bbox": [247, 471, 483, 519], "spans": [{"bbox": [247, 471, 483, 519], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [193, 519, 302, 534], "lines": [{"bbox": [194, 521, 299, 535], "spans": [{"bbox": [194, 522, 206, 534], "score": 0.72, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [207, 522, 222, 534], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [223, 522, 232, 533], "score": 0.84, "content": "e^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [233, 522, 258, 534], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 521, 295, 535], "score": 0.71, "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [296, 522, 299, 534], "score": 1.0, "content": ":", "type": "text"}], "index": 24}], "index": 24}, {"type": "interline_equation", "bbox": [217, 539, 513, 588], "lines": [{"bbox": [217, 539, 513, 588], "spans": [{"bbox": [217, 539, 513, 588], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [168, 589, 401, 604], "lines": [{"bbox": [174, 590, 397, 606], "spans": [{"bbox": [174, 590, 308, 606], "score": 1.0, "content": "Thus, in each case we get ", "type": "text"}, {"bbox": [309, 591, 394, 606], "score": 0.9, "content": "f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "type": "inline_equation", "height": 15, "width": 85}, {"bbox": [394, 590, 397, 606], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [158, 605, 420, 618], "lines": [{"bbox": [172, 604, 419, 620], "spans": [{"bbox": [172, 604, 221, 620], "score": 1.0, "content": "Now, let ", "type": "text"}, {"bbox": [221, 605, 262, 618], "score": 0.89, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [262, 604, 353, 620], "score": 1.0, "content": " be arbitrary and ", "type": "text"}, {"bbox": [354, 606, 416, 619], "score": 0.93, "content": "\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma", "type": "inline_equation", "height": 13, "width": 62}, {"bbox": [416, 604, 419, 620], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 27}, {"type": "text", "bbox": [171, 619, 537, 686], "lines": [{"bbox": [174, 620, 536, 634], "spans": [{"bbox": [174, 620, 507, 634], "score": 1.0, "content": "By the Decomposition Lemma 7.4 there is a decomposition ", "type": "text"}, {"bbox": [508, 621, 536, 633], "score": 0.83, "content": "\\alpha^{\\prime}\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 28}, {"bbox": [174, 635, 536, 648], "spans": [{"bbox": [174, 635, 198, 648], "score": 0.85, "content": "\\Pi\\,\\alpha_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [198, 635, 251, 648], "score": 1.0, "content": " with int ", "type": "text"}, {"bbox": [252, 635, 336, 648], "score": 0.9, "content": "\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [337, 635, 378, 648], "score": 1.0, "content": "or int ", "type": "text"}, {"bbox": [378, 635, 440, 648], "score": 0.88, "content": "\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 62}, {"bbox": [440, 635, 485, 648], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [486, 637, 490, 645], "score": 0.81, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [490, 635, 536, 648], "score": 1.0, "content": ". Thus,", "type": "text"}], "index": 29}, {"bbox": [174, 647, 538, 667], "spans": [{"bbox": [174, 649, 337, 667], "score": 0.93, "content": "\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)", "type": "inline_equation", "height": 18, "width": 163}, {"bbox": [338, 647, 538, 666], "score": 1.0, "content": ". Using the result just proven we get", "type": "text"}], "index": 30}, {"bbox": [174, 668, 398, 686], "spans": [{"bbox": [174, 668, 395, 686], "score": 0.86, "content": "f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "type": "inline_equation", "height": 18, "width": 221}, {"bbox": [395, 668, 398, 685], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 29.5}], "layout_bboxes": [], "page_idx": 13, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [187, 134, 475, 149], "lines": [{"bbox": [187, 134, 475, 149], "spans": [{"bbox": [187, 134, 475, 149], "score": 0.77, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [265, 282, 444, 315], "lines": [{"bbox": [265, 282, 444, 315], "spans": [{"bbox": [265, 282, 444, 315], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [233, 349, 497, 381], "lines": [{"bbox": [233, 349, 497, 381], "spans": [{"bbox": [233, 349, 497, 381], "score": 0.91, "content": "\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [250, 402, 480, 451], "lines": [{"bbox": [250, 402, 480, 451], "spans": [{"bbox": [250, 402, 480, 451], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [247, 471, 483, 519], "lines": [{"bbox": [247, 471, 483, 519], "spans": [{"bbox": [247, 471, 483, 519], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "interline_equation", "bbox": [217, 539, 513, 588], "lines": [{"bbox": [217, 539, 513, 588], "spans": [{"bbox": [217, 539, 513, 588], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 25}], "index": 25}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 307, 715], "lines": [{"bbox": [292, 705, 308, 718], "spans": [{"bbox": [292, 705, 308, 718], "score": 1.0, "content": "14", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [125, 14, 537, 59], "lines": [{"bbox": [128, 16, 536, 32], "spans": [{"bbox": [128, 16, 192, 32], "score": 1.0, "content": "• Now, let ", "type": "text"}, {"bbox": [193, 19, 227, 28], "score": 0.93, "content": "\\delta\\ \\in\\ {\\mathcal{P}}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [227, 16, 417, 32], "score": 1.0, "content": " be an arbitrary path. Decompose ", "type": "text"}, {"bbox": [417, 19, 423, 28], "score": 0.89, "content": "\\delta", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [423, 16, 536, 32], "score": 1.0, "content": " into a finite product", "type": "text"}], "index": 0}, {"bbox": [143, 31, 537, 47], "spans": [{"bbox": [143, 33, 163, 44], "score": 0.93, "content": "\\Pi\\,\\delta_{i}", "type": "inline_equation", "height": 11, "width": 20}, {"bbox": [163, 31, 330, 47], "score": 1.0, "content": " due to Lemma 7.4 such that no ", "type": "text"}, {"bbox": [330, 33, 339, 44], "score": 0.91, "content": "\\delta_{i}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [339, 31, 438, 47], "score": 1.0, "content": " contains the point ", "type": "text"}, {"bbox": [438, 33, 461, 45], "score": 0.94, "content": "e^{\\prime}(0)", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [462, 31, 537, 47], "score": 1.0, "content": " in the interior", "type": "text"}], "index": 1}, {"bbox": [143, 45, 426, 61], "spans": [{"bbox": [143, 45, 193, 61], "score": 1.0, "content": "supposed ", "type": "text"}, {"bbox": [194, 48, 202, 58], "score": 0.92, "content": "\\delta_{i}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [203, 45, 326, 61], "score": 1.0, "content": " is not trivial. Here, set ", "type": "text"}, {"bbox": [327, 47, 423, 61], "score": 0.95, "content": "h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})", "type": "inline_equation", "height": 14, "width": 96}, {"bbox": [423, 45, 426, 61], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [128, 16, 537, 61]}, {"type": "text", "bbox": [126, 59, 384, 74], "lines": [{"bbox": [125, 60, 385, 76], "spans": [{"bbox": [125, 60, 252, 76], "score": 1.0, "content": "We know from [10] that ", "type": "text"}, {"bbox": [252, 61, 263, 73], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [264, 60, 385, 76], "score": 1.0, "content": " is indeed a connectio n.", "type": "text"}], "index": 3}], "index": 3, "bbox_fs": [125, 60, 385, 76]}, {"type": "text", "bbox": [106, 75, 538, 104], "lines": [{"bbox": [104, 74, 538, 93], "spans": [{"bbox": [104, 74, 202, 93], "score": 1.0, "content": "3. The assertion ", "type": "text"}, {"bbox": [202, 76, 294, 91], "score": 0.94, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 15, "width": 92}, {"bbox": [294, 74, 336, 93], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [337, 79, 341, 87], "score": 0.88, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [341, 74, 538, 93], "score": 1.0, "content": " is an immediate consequence of the", "type": "text"}], "index": 4}, {"bbox": [126, 90, 508, 106], "spans": [{"bbox": [126, 90, 252, 106], "score": 1.0, "content": "construction because im ", "type": "text"}, {"bbox": [253, 92, 333, 105], "score": 0.71, "content": "(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [333, 90, 421, 106], "score": 1.0, "content": ". As well, we get ", "type": "text"}, {"bbox": [422, 92, 504, 106], "score": 0.93, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [504, 90, 508, 106], "score": 1.0, "content": ".", "type": "text"}], "index": 5}], "index": 4.5, "bbox_fs": [104, 74, 538, 106]}, {"type": "text", "bbox": [105, 105, 539, 132], "lines": [{"bbox": [105, 105, 539, 120], "spans": [{"bbox": [105, 105, 297, 120], "score": 1.0, "content": "4. Moreover, from (4), the fact that ", "type": "text"}, {"bbox": [297, 107, 305, 117], "score": 0.89, "content": "e^{\\prime}", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [306, 105, 539, 120], "score": 1.0, "content": " has no self-intersections and the definition of", "type": "text"}], "index": 6}, {"bbox": [126, 118, 295, 136], "spans": [{"bbox": [126, 119, 137, 131], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [138, 118, 178, 136], "score": 1.0, "content": " we get ", "type": "text"}, {"bbox": [178, 122, 255, 135], "score": 0.94, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [256, 118, 295, 136], "score": 1.0, "content": " and so", "type": "text"}], "index": 7}], "index": 6.5, "bbox_fs": [105, 105, 539, 136]}, {"type": "interline_equation", "bbox": [187, 134, 475, 149], "lines": [{"bbox": [187, 134, 475, 149], "spans": [{"bbox": [187, 134, 475, 149], "score": 0.77, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.", "type": "interline_equation"}], "index": 8}], "index": 8}, {"type": "text", "bbox": [105, 147, 296, 162], "lines": [{"bbox": [105, 147, 296, 164], "spans": [{"bbox": [105, 147, 173, 163], "score": 1.0, "content": "5. We have ", "type": "text"}, {"bbox": [173, 150, 294, 164], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})", "type": "inline_equation", "height": 14, "width": 121}, {"bbox": [294, 147, 296, 163], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [105, 147, 296, 164]}, {"type": "list", "bbox": [147, 162, 536, 219], "lines": [{"bbox": [156, 161, 450, 180], "spans": [{"bbox": [156, 161, 178, 180], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [179, 165, 239, 178], "score": 0.91, "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [239, 161, 265, 180], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [266, 165, 367, 178], "score": 0.92, "content": "f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f", "type": "inline_equation", "height": 13, "width": 101}, {"bbox": [367, 161, 405, 180], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [405, 165, 445, 175], "score": 0.87, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [446, 161, 450, 180], "score": 1.0, "content": ".", "type": "text"}], "index": 10, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [157, 177, 432, 193], "spans": [{"bbox": [157, 177, 205, 192], "score": 1.0, "content": "• From ", "type": "text"}, {"bbox": [205, 178, 258, 193], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "type": "inline_equation", "height": 15, "width": 53}, {"bbox": [258, 177, 300, 192], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [300, 180, 342, 191], "score": 0.92, "content": "f g=g f", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [343, 177, 369, 192], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [369, 178, 427, 191], "score": 0.9, "content": "f\\in Z(\\{g\\})", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [428, 177, 432, 192], "score": 1.0, "content": ".", "type": "text"}], "index": 11, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [166, 191, 536, 208], "spans": [{"bbox": [166, 191, 260, 208], "score": 1.0, "content": "From im e′ ∩ im ", "type": "text"}, {"bbox": [261, 194, 302, 206], "score": 0.52, "content": "({\\pmb{\\alpha}})=\\emptyset", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [302, 191, 344, 208], "score": 1.0, "content": "follows ", "type": "text"}, {"bbox": [345, 193, 433, 207], "score": 0.92, "content": "h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})", "type": "inline_equation", "height": 14, "width": 88}, {"bbox": [433, 191, 460, 208], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [461, 193, 536, 207], "score": 0.91, "content": "f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))", "type": "inline_equation", "height": 14, "width": 75}], "index": 12}, {"bbox": [173, 207, 217, 219], "spans": [{"bbox": [173, 207, 208, 219], "score": 1.0, "content": "for all ", "type": "text"}, {"bbox": [209, 209, 213, 217], "score": 0.87, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [213, 207, 217, 219], "score": 1.0, "content": ".", "type": "text"}], "index": 13, "is_list_end_line": true}], "index": 11.5, "bbox_fs": [156, 161, 536, 219]}, {"type": "text", "bbox": [157, 249, 538, 277], "lines": [{"bbox": [171, 250, 538, 264], "spans": [{"bbox": [171, 250, 195, 264], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [195, 252, 206, 261], "score": 0.89, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [206, 250, 289, 264], "score": 1.0, "content": " be a path from ", "type": "text"}, {"bbox": [290, 252, 303, 261], "score": 0.89, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [303, 250, 320, 264], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [321, 252, 334, 261], "score": 0.9, "content": "m^{\\prime}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [335, 250, 408, 264], "score": 1.0, "content": ", such that int", "type": "text"}, {"bbox": [408, 251, 479, 264], "score": 0.71, "content": "\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [480, 250, 513, 264], "score": 1.0, "content": "or int ", "type": "text"}, {"bbox": [513, 251, 538, 263], "score": 0.57, "content": "\\alpha^{\\prime}=", "type": "inline_equation", "height": 12, "width": 25}], "index": 14}, {"bbox": [174, 265, 460, 279], "spans": [{"bbox": [174, 266, 200, 279], "score": 0.91, "content": "\\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [200, 265, 227, 279], "score": 1.0, "content": ". Set ", "type": "text"}, {"bbox": [228, 265, 295, 278], "score": 0.93, "content": "\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [295, 265, 460, 279], "score": 1.0, "content": ". Then by construction we have", "type": "text"}], "index": 15}], "index": 14.5, "bbox_fs": [171, 250, 538, 279]}, {"type": "interline_equation", "bbox": [265, 282, 444, 315], "lines": [{"bbox": [265, 282, 444, 315], "spans": [{"bbox": [265, 282, 444, 315], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [174, 316, 282, 329], "lines": [{"bbox": [174, 317, 283, 330], "spans": [{"bbox": [174, 317, 283, 330], "score": 1.0, "content": "There are four cases:", "type": "text"}], "index": 17}], "index": 17, "bbox_fs": [174, 317, 283, 330]}, {"type": "text", "bbox": [174, 330, 301, 344], "lines": [{"bbox": [174, 331, 300, 345], "spans": [{"bbox": [174, 331, 206, 345], "score": 0.59, "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "type": "inline_equation", "height": 14, "width": 32}, {"bbox": [207, 332, 222, 345], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [223, 331, 232, 344], "score": 0.76, "content": "e^{\\prime}", "type": "inline_equation", "height": 13, "width": 9}, {"bbox": [233, 332, 258, 345], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 331, 295, 345], "score": 0.52, "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [296, 332, 300, 345], "score": 1.0, "content": ":", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [174, 331, 300, 345]}, {"type": "interline_equation", "bbox": [233, 349, 497, 381], "lines": [{"bbox": [233, 349, 497, 381], "spans": [{"bbox": [233, 349, 497, 381], "score": 0.91, "content": "\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [173, 383, 302, 398], "lines": [{"bbox": [173, 384, 300, 398], "spans": [{"bbox": [173, 384, 206, 398], "score": 0.55, "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "type": "inline_equation", "height": 14, "width": 33}, {"bbox": [207, 385, 222, 398], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [223, 385, 232, 397], "score": 0.82, "content": "e^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [233, 385, 258, 398], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 385, 296, 398], "score": 0.8, "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [296, 385, 300, 398], "score": 1.0, "content": ":", "type": "text"}], "index": 20}], "index": 20, "bbox_fs": [173, 384, 300, 398]}, {"type": "interline_equation", "bbox": [250, 402, 480, 451], "lines": [{"bbox": [250, 402, 480, 451], "spans": [{"bbox": [250, 402, 480, 451], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [173, 452, 302, 467], "lines": [{"bbox": [194, 453, 300, 467], "spans": [{"bbox": [194, 454, 206, 466], "score": 0.75, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [206, 454, 222, 467], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [223, 454, 232, 465], "score": 0.83, "content": "e^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [233, 454, 258, 467], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 453, 296, 467], "score": 0.77, "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "type": "inline_equation", "height": 14, "width": 38}, {"bbox": [296, 454, 300, 467], "score": 1.0, "content": ":", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [194, 453, 300, 467]}, {"type": "interline_equation", "bbox": [247, 471, 483, 519], "lines": [{"bbox": [247, 471, 483, 519], "spans": [{"bbox": [247, 471, 483, 519], "score": 0.93, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 23}], "index": 23}, {"type": "text", "bbox": [193, 519, 302, 534], "lines": [{"bbox": [194, 521, 299, 535], "spans": [{"bbox": [194, 522, 206, 534], "score": 0.72, "content": "\\alpha^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [207, 522, 222, 534], "score": 1.0, "content": " ↑↑", "type": "text"}, {"bbox": [223, 522, 232, 533], "score": 0.84, "content": "e^{\\prime}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [233, 522, 258, 534], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [258, 521, 295, 535], "score": 0.71, "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [296, 522, 299, 534], "score": 1.0, "content": ":", "type": "text"}], "index": 24}], "index": 24, "bbox_fs": [194, 521, 299, 535]}, {"type": "interline_equation", "bbox": [217, 539, 513, 588], "lines": [{"bbox": [217, 539, 513, 588], "spans": [{"bbox": [217, 539, 513, 588], "score": 0.92, "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "type": "interline_equation"}], "index": 25}], "index": 25}, {"type": "text", "bbox": [168, 589, 401, 604], "lines": [{"bbox": [174, 590, 397, 606], "spans": [{"bbox": [174, 590, 308, 606], "score": 1.0, "content": "Thus, in each case we get ", "type": "text"}, {"bbox": [309, 591, 394, 606], "score": 0.9, "content": "f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "type": "inline_equation", "height": 15, "width": 85}, {"bbox": [394, 590, 397, 606], "score": 1.0, "content": ".", "type": "text"}], "index": 26}], "index": 26, "bbox_fs": [174, 590, 397, 606]}, {"type": "text", "bbox": [158, 605, 420, 618], "lines": [{"bbox": [172, 604, 419, 620], "spans": [{"bbox": [172, 604, 221, 620], "score": 1.0, "content": "Now, let ", "type": "text"}, {"bbox": [221, 605, 262, 618], "score": 0.89, "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [262, 604, 353, 620], "score": 1.0, "content": " be arbitrary and ", "type": "text"}, {"bbox": [354, 606, 416, 619], "score": 0.93, "content": "\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma", "type": "inline_equation", "height": 13, "width": 62}, {"bbox": [416, 604, 419, 620], "score": 1.0, "content": ".", "type": "text"}], "index": 27}], "index": 27, "bbox_fs": [172, 604, 419, 620]}, {"type": "text", "bbox": [171, 619, 537, 686], "lines": [{"bbox": [174, 620, 536, 634], "spans": [{"bbox": [174, 620, 507, 634], "score": 1.0, "content": "By the Decomposition Lemma 7.4 there is a decomposition ", "type": "text"}, {"bbox": [508, 621, 536, 633], "score": 0.83, "content": "\\alpha^{\\prime}\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 28}, {"bbox": [174, 635, 536, 648], "spans": [{"bbox": [174, 635, 198, 648], "score": 0.85, "content": "\\Pi\\,\\alpha_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [198, 635, 251, 648], "score": 1.0, "content": " with int ", "type": "text"}, {"bbox": [252, 635, 336, 648], "score": 0.9, "content": "\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [337, 635, 378, 648], "score": 1.0, "content": "or int ", "type": "text"}, {"bbox": [378, 635, 440, 648], "score": 0.88, "content": "\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}", "type": "inline_equation", "height": 13, "width": 62}, {"bbox": [440, 635, 485, 648], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [486, 637, 490, 645], "score": 0.81, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [490, 635, 536, 648], "score": 1.0, "content": ". Thus,", "type": "text"}], "index": 29}, {"bbox": [174, 647, 538, 667], "spans": [{"bbox": [174, 649, 337, 667], "score": 0.93, "content": "\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)", "type": "inline_equation", "height": 18, "width": 163}, {"bbox": [338, 647, 538, 666], "score": 1.0, "content": ". Using the result just proven we get", "type": "text"}], "index": 30}, {"bbox": [174, 668, 398, 686], "spans": [{"bbox": [174, 668, 395, 686], "score": 0.86, "content": "f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "type": "inline_equation", "height": 18, "width": 221}, {"bbox": [395, 668, 398, 685], "score": 1.0, "content": ".", "type": "text"}], "index": 31}], "index": 29.5, "bbox_fs": [174, 620, 538, 686]}]}
|
[{"type": "text", "bbox": [125, 14, 537, 59], "content": "• Now, let be an arbitrary path. Decompose into a finite product due to Lemma 7.4 such that no contains the point in the interior supposed is not trivial. Here, set .", "index": 0}, {"type": "text", "bbox": [126, 59, 384, 74], "content": "We know from [10] that is indeed a connectio n.", "index": 1}, {"type": "text", "bbox": [106, 75, 538, 104], "content": "3. The assertion for all is an immediate consequence of the construction because im . As well, we get .", "index": 2}, {"type": "text", "bbox": [105, 105, 539, 132], "content": "4. Moreover, from (4), the fact that has no self-intersections and the definition of we get and so", "index": 3}, {"type": "interline_equation", "bbox": [187, 134, 475, 149], "content": "", "index": 4}, {"type": "text", "bbox": [105, 147, 296, 162], "content": "5. We have .", "index": 5}, {"type": "list", "bbox": [147, 162, 536, 219], "content": "", "index": 6}, {"type": "text", "bbox": [157, 249, 538, 277], "content": "Let be a path from to , such that int or int . Set . Then by construction we have", "index": 7}, {"type": "interline_equation", "bbox": [265, 282, 444, 315], "content": "", "index": 8}, {"type": "text", "bbox": [174, 316, 282, 329], "content": "There are four cases:", "index": 9}, {"type": "text", "bbox": [174, 330, 301, 344], "content": "↑↑ and :", "index": 10}, {"type": "interline_equation", "bbox": [233, 349, 497, 381], "content": "", "index": 11}, {"type": "text", "bbox": [173, 383, 302, 398], "content": "↑↑ and :", "index": 12}, {"type": "interline_equation", "bbox": [250, 402, 480, 451], "content": "", "index": 13}, {"type": "text", "bbox": [173, 452, 302, 467], "content": "↑↑ and :", "index": 14}, {"type": "interline_equation", "bbox": [247, 471, 483, 519], "content": "", "index": 15}, {"type": "text", "bbox": [193, 519, 302, 534], "content": "↑↑ and :", "index": 16}, {"type": "interline_equation", "bbox": [217, 539, 513, 588], "content": "", "index": 17}, {"type": "text", "bbox": [168, 589, 401, 604], "content": "Thus, in each case we get .", "index": 18}, {"type": "text", "bbox": [158, 605, 420, 618], "content": "Now, let be arbitrary and .", "index": 19}, {"type": "text", "bbox": [171, 619, 537, 686], "content": "By the Decomposition Lemma 7.4 there is a decomposition with int or int for all . Thus, . Using the result just proven we get .", "index": 20}]
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[{"bbox": [128, 16, 536, 32], "content": "• Now, let be an arbitrary path. Decompose into a finite product", "parent_index": 0, "line_index": 0}, {"bbox": [143, 31, 537, 47], "content": "due to Lemma 7.4 such that no contains the point in the interior", "parent_index": 0, "line_index": 1}, {"bbox": [143, 45, 426, 61], "content": "supposed is not trivial. Here, set .", "parent_index": 0, "line_index": 2}, {"bbox": [125, 60, 385, 76], "content": "We know from [10] that is indeed a connectio n.", "parent_index": 1, "line_index": 0}, {"bbox": [104, 74, 538, 93], "content": "3. The assertion for all is an immediate consequence of the", "parent_index": 2, "line_index": 0}, {"bbox": [126, 90, 508, 106], "content": "construction because im . As well, we get .", "parent_index": 2, "line_index": 1}, {"bbox": [105, 105, 539, 120], "content": "4. Moreover, from (4), the fact that has no self-intersections and the definition of", "parent_index": 3, "line_index": 0}, {"bbox": [126, 118, 295, 136], "content": "we get and so", "parent_index": 3, "line_index": 1}, {"bbox": [105, 147, 296, 164], "content": "5. We have .", "parent_index": 5, "line_index": 0}, {"bbox": [156, 161, 450, 180], "content": "Let , i.e. for all .", "parent_index": 6, "line_index": 0}, {"bbox": [157, 177, 432, 193], "content": "• From follows , i.e. .", "parent_index": 6, "line_index": 1}, {"bbox": [166, 191, 536, 208], "content": "From im e′ ∩ im follows , i.e.", "parent_index": 6, "line_index": 2}, {"bbox": [173, 207, 217, 219], "content": "for all .", "parent_index": 6, "line_index": 3}, {"bbox": [171, 250, 538, 264], "content": "Let be a path from to , such that int or int", "parent_index": 7, "line_index": 0}, {"bbox": [174, 265, 460, 279], "content": ". Set . Then by construction we have", "parent_index": 7, "line_index": 1}, {"bbox": [174, 317, 283, 330], "content": "There are four cases:", "parent_index": 9, "line_index": 0}, {"bbox": [174, 331, 300, 345], "content": "↑↑ and :", "parent_index": 10, "line_index": 0}, {"bbox": [173, 384, 300, 398], "content": "↑↑ and :", "parent_index": 12, "line_index": 0}, {"bbox": [194, 453, 300, 467], "content": "↑↑ and :", "parent_index": 14, "line_index": 0}, {"bbox": [194, 521, 299, 535], "content": "↑↑ and :", "parent_index": 16, "line_index": 0}, {"bbox": [174, 590, 397, 606], "content": "Thus, in each case we get .", "parent_index": 18, "line_index": 0}, {"bbox": [172, 604, 419, 620], "content": "Now, let be arbitrary and .", "parent_index": 19, "line_index": 0}, {"bbox": [174, 620, 536, 634], "content": "By the Decomposition Lemma 7.4 there is a decomposition", "parent_index": 20, "line_index": 0}, {"bbox": [174, 635, 536, 648], "content": "with int or int for all . Thus,", "parent_index": 20, "line_index": 1}, {"bbox": [174, 647, 538, 667], "content": ". Using the result just proven we get", "parent_index": 20, "line_index": 2}, {"bbox": [174, 668, 398, 686], "content": ".", "parent_index": 20, "line_index": 3}]
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[]
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[{"bbox": [193, 19, 227, 28], "content": "\\delta\\ \\in\\ {\\mathcal{P}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [417, 19, 423, 28], "content": "\\delta", "parent_index": 0, "subtype": "inline"}, {"bbox": [143, 33, 163, 44], "content": "\\Pi\\,\\delta_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [330, 33, 339, 44], "content": "\\delta_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [438, 33, 461, 45], "content": "e^{\\prime}(0)", "parent_index": 0, "subtype": "inline"}, {"bbox": [194, 48, 202, 58], "content": "\\delta_{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [327, 47, 423, 61], "content": "h_{\\overline{{A}}^{\\prime}}(\\delta):=\\Pi\\,h_{\\overline{{A}}^{\\prime}}(\\delta_{i})", "parent_index": 0, "subtype": "inline"}, {"bbox": [252, 61, 263, 73], "content": "\\overline{{A}}^{\\prime}", "parent_index": 1, "subtype": "inline"}, {"bbox": [202, 76, 294, 91], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})\\;=\\;\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "parent_index": 2, "subtype": "inline"}, {"bbox": [337, 79, 341, 87], "content": "i", "parent_index": 2, "subtype": "inline"}, {"bbox": [253, 92, 333, 105], "content": "(\\Gamma_{i})\\cap\\operatorname{int}e^{\\prime}=\\varnothing", "parent_index": 2, "subtype": "inline"}, {"bbox": [422, 92, 504, 106], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "parent_index": 2, "subtype": "inline"}, {"bbox": [297, 107, 305, 117], "content": "e^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [126, 119, 137, 131], "content": "\\overline{{A}}^{\\prime}", "parent_index": 3, "subtype": "inline"}, {"bbox": [178, 122, 255, 135], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\gamma)=h_{\\overline{{{A}}}}(\\gamma)", "parent_index": 3, "subtype": "inline"}, {"bbox": [187, 134, 475, 149], "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(e^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma^{-1})=h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}=g.", "parent_index": 4, "subtype": "interline"}, {"bbox": [173, 150, 294, 164], "content": "Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})", "parent_index": 5, "subtype": "inline"}, {"bbox": [179, 165, 239, 178], "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "parent_index": 6, "subtype": "inline"}, {"bbox": [266, 165, 367, 178], "content": "f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f", "parent_index": 6, "subtype": "inline"}, {"bbox": [405, 165, 445, 175], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 6, "subtype": "inline"}, {"bbox": [205, 178, 258, 193], "content": "h_{\\overline{{{A}}}^{\\prime}}(e)=g", "parent_index": 6, "subtype": "inline"}, {"bbox": [300, 180, 342, 191], "content": "f g=g f", "parent_index": 6, "subtype": "inline"}, {"bbox": [369, 178, 427, 191], "content": "f\\in Z(\\{g\\})", "parent_index": 6, "subtype": "inline"}, {"bbox": [261, 194, 302, 206], "content": "({\\pmb{\\alpha}})=\\emptyset", "parent_index": 6, "subtype": "inline"}, {"bbox": [345, 193, 433, 207], "content": "h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})", "parent_index": 6, "subtype": "inline"}, {"bbox": [461, 193, 536, 207], "content": "f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))", "parent_index": 6, "subtype": "inline"}, {"bbox": [209, 209, 213, 217], "content": "i", "parent_index": 6, "subtype": "inline"}, {"bbox": [195, 252, 206, 261], "content": "\\alpha^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [290, 252, 303, 261], "content": "m^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [321, 252, 334, 261], "content": "m^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [408, 251, 479, 264], "content": "\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset", "parent_index": 7, "subtype": "inline"}, {"bbox": [513, 251, 538, 263], "content": "\\alpha^{\\prime}=", "parent_index": 7, "subtype": "inline"}, {"bbox": [174, 266, 200, 279], "content": "\\{m^{\\prime}\\}", "parent_index": 7, "subtype": "inline"}, {"bbox": [228, 265, 295, 278], "content": "\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}", "parent_index": 7, "subtype": "inline"}, {"bbox": [265, 282, 444, 315], "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}^{\\prime}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}^{\\prime}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}.}}\\end{array}", "parent_index": 8, "subtype": "interline"}, {"bbox": [174, 331, 206, 345], "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [223, 331, 232, 344], "content": "e^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [258, 331, 295, 345], "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "parent_index": 10, "subtype": "inline"}, {"bbox": [233, 349, 497, 381], "content": "\\begin{array}{l l l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\;\\alpha^{\\prime}\\;\\gamma^{-1})}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\alpha).}}&{{}}&{{}}\\end{array}", "parent_index": 11, "subtype": "interline"}, {"bbox": [173, 384, 206, 398], "content": "-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [223, 385, 232, 397], "content": "e^{\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [258, 385, 296, 398], "content": "\\alpha^{\\prime}\\,\\#\\,e^{\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [250, 402, 480, 451], "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}\\gamma^{-1}).}}\\end{array}", "parent_index": 13, "subtype": "interline"}, {"bbox": [194, 454, 206, 466], "content": "\\alpha^{\\prime}", "parent_index": 14, "subtype": "inline"}, {"bbox": [223, 454, 232, 465], "content": "e^{\\prime}", "parent_index": 14, "subtype": "inline"}, {"bbox": [258, 453, 296, 467], "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "parent_index": 14, "subtype": "inline"}, {"bbox": [247, 471, 483, 519], "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "parent_index": 15, "subtype": "interline"}, {"bbox": [194, 522, 206, 534], "content": "\\alpha^{\\prime}", "parent_index": 16, "subtype": "inline"}, {"bbox": [223, 522, 232, 533], "content": "e^{\\prime}", "parent_index": 16, "subtype": "inline"}, {"bbox": [258, 521, 295, 535], "content": "\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}", "parent_index": 16, "subtype": "inline"}, {"bbox": [217, 539, 513, 588], "content": "\\begin{array}{l l l}{{h_{\\overline{{{A}}}^{\\prime}}(\\alpha)}}&{{=}}&{{h_{\\overline{{{A}}}}(\\gamma)\\;g^{\\prime}\\,h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;(g^{\\prime})^{-1}\\,h_{\\overline{{{A}}}}(\\gamma)^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma)\\;h_{\\overline{{{A}}}}(e^{\\prime})^{-1}\\;h_{\\overline{{{A}}}}(\\alpha^{\\prime})\\;h_{\\overline{{{A}}}}(e^{\\prime})\\;h_{\\overline{{{A}}}}(\\gamma)^{-1}\\;g^{-1}}}\\\\ {{}}&{{=}}&{{g\\,h_{\\overline{{{A}}}}(\\gamma e^{\\prime-1}\\alpha^{\\prime}e^{\\prime}\\gamma^{-1})\\;g^{-1}.}}\\end{array}", "parent_index": 17, "subtype": "interline"}, {"bbox": [309, 591, 394, 606], "content": "f\\in Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "parent_index": 18, "subtype": "inline"}, {"bbox": [221, 605, 262, 618], "content": "\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 19, "subtype": "inline"}, {"bbox": [354, 606, 416, 619], "content": "\\alpha^{\\prime}:=\\gamma^{-1}\\alpha\\gamma", "parent_index": 19, "subtype": "inline"}, {"bbox": [508, 621, 536, 633], "content": "\\alpha^{\\prime}\\ =", "parent_index": 20, "subtype": "inline"}, {"bbox": [174, 635, 198, 648], "content": "\\Pi\\,\\alpha_{i}^{\\prime}", "parent_index": 20, "subtype": "inline"}, {"bbox": [252, 635, 336, 648], "content": "\\alpha_{i}^{\\prime}\\cap\\{m^{\\prime}\\}\\ =\\ \\emptyset", "parent_index": 20, "subtype": "inline"}, {"bbox": [378, 635, 440, 648], "content": "\\alpha_{i}^{\\prime}\\ =\\ \\{m^{\\prime}\\}", "parent_index": 20, "subtype": "inline"}, {"bbox": [486, 637, 490, 645], "content": "i", "parent_index": 20, "subtype": "inline"}, {"bbox": [174, 649, 337, 667], "content": "\\alpha\\,=\\,\\gamma\\bigl(\\Pi\\,\\alpha_{i}^{\\prime}\\bigr)\\gamma^{-1}\\,=\\,\\Pi\\bigl(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\bigr)", "parent_index": 20, "subtype": "inline"}, {"bbox": [174, 668, 395, 686], "content": "f\\in Z\\big(\\big\\{h_{\\overline{{A}}^{\\prime}}\\big(\\Pi\\big(\\gamma\\alpha_{i}^{\\prime}\\gamma^{-1}\\big)\\big)\\big\\}\\big)=Z(\\{h_{\\overline{{A}}^{\\prime}}(\\alpha)\\})", "parent_index": 20, "subtype": "inline"}]
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[]
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Thus, $f\in\cal Z(\mathbf{H}_{\overline{{\cal A}}^{\prime}})$ . Due to the definition of $\pmb{x}$ we have $Z(\mathbf{H}_{\overline{{A}}^{\prime}})=Z(\{g\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$ .
# 7.3 Construction of Arbitrary Types
Finally, we can now prove the desired proposition.
# Proof Proposition 7.1
• Let $t\in\mathcal T$ and $t\geq\mathrm{Typ}(\overline{{A}})$ . Then there exist a Howe subgroup $V^{\prime}\subseteq\mathbf{G}$ with $t=$ $\left[V^{\prime}\right]$ and a $g\in\mathbf G$ , such that $Z(\mathbf{H}_{\overline{{A}}})\supseteq g^{-1}V^{\prime}g=:V$ . Since $V$ is a Howe subgroup, we have $Z(Z(V))\,=\,V$ and so by Lemma 4.1 there exist certain $u_{0},\dotsc,u_{k}\in$ $Z(V)\subseteq\mathbf{G}$ , such that $V=Z(Z(V))=Z(\{u_{0},\dots,u_{k}\})$ .
• Now let $Z(\mathbf{H}_{\overline{{A}}})\,=\,Z(h_{\overline{{A}}}(\alpha))$ with an appropriate $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ as in Corollary 4.2. Because of $V\subseteq Z(\mathbf{H}_{\overline{{A}}})$ we have $V=V\cap Z(\mathbf{H}_{\overline{{A}}})=Z(\{u_{0},\dots,u_{k}\})\cap Z(h_{\overline{{A}}}(\alpha))=$ $Z(\left\{u_{0},\dots,u_{k}\right\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$ .
• We now use inductively Lemma 7.5. Let $\overline{{A}}_{0}:=\overline{{A}}$ and $\alpha_{0}:=\alpha$ . Construct for all $j=0,\dots,k$ a connection $\overline{{A}}_{j+1}$ and an $e_{j}\in{\mathcal{H}}{\mathcal{G}}$ from $\overline{{A}}_{j}$ and $\alpha_{j}$ by that lemma, such that $\pi_{\Gamma_{i}}(\overline{{{A}}}_{j+1})=\pi_{\Gamma_{i}}(\overline{{{A}}}_{j})$ for all $i$ , $h_{\overline{{A}}_{j+1}}(\pmb{\alpha}_{j})=h_{\overline{{A}}_{j}}(\pmb{\alpha}_{j})$ , $h_{\overline{{A}}_{j+1}}(e_{j})=u_{j}$ and $Z(\mathbf{H}_{\overline{{A}}_{j+1}})=Z(\{u_{j}\}\cup h_{\overline{{A}}_{j}}(\pmb{\alpha}_{j}))$ . Setting $\alpha_{j+1}:=\alpha_{j}\cup\{e_{j}\}$ we get $Z(\mathbf{H}_{\overline{{A}}_{j+1}})=Z(\{u_{j}\}\cup h_{\overline{{A}}_{j}}(\alpha_{j}))=Z(h_{\overline{{A}}_{j+1}}(\alpha_{j+1})).$ Finally, we define $\overline{{A}}^{\prime}:=\overline{{A}}_{k+1}$ . Now, we get $\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})=\pi_{\Gamma_{i}}(\overline{{{A}}})$ for all $i$ , $h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$ and $h_{\overline{{A}}^{\prime}}(e_{j})=u_{j}$ . Thus, $\begin{array}{l l l}{{Z({\bf H}_{\overline{{{A}}}^{\prime}})}}&{{=}}&{{Z(h_{\overline{{{A}}}^{\prime}}(\alpha_{k+1}))}}\\ {{}}&{{=}}&{{Z(h_{\overline{{{A}}}^{\prime}}(\{e_{0},\ldots,e_{k}\}\cup h_{\overline{{{A}}}^{\prime}}(\alpha)))}}\\ {{}}&{{=}}&{{Z(\{u_{0},\ldots,u_{k}\}\cup h_{\overline{{{A}}}}(\alpha))}}\\ {{}}&{{=}}&{{V,}}\end{array}$ i.e., $\mathrm{Typ}({\overline{{A}}}^{\prime})=[V]=t$ . qed
The proposition just proven has a further immediate consequence.
Corollary 7.6 $\overline{{A}}_{=t}$ is non-empty for all $t\in\mathcal T$ .
Proof Let $\overline{{A}}$ be the trivial connection, i.e. $h_{\overline{{A}}}(\alpha)=e_{\mathbf{G}}$ for all $\alpha\in\mathcal{P}$ . The type of $\overline{{A}}$ is $[\mathbf G]$ , thus minimal, i.e. we have $t\geq\mathrm{Typ}(\overline{{A}})$ for all $t\in\mathcal T$ . By means of Proposition 7.1 there is an $\overline{{A}}^{\prime}\in\overline{{A}}$ with $\mathrm{Typ}(\overline{{A}}^{\prime})=t$ . qed
This corollary solves the problem which gauge orbit types exist for generalized connections.
Theorem 7.7 The set of all gauge orbit types on $\overline{{\mathcal{A}}}$ is the set of all conjugacy classes of Howe subgroups of $\mathbf{G}$ .
Furthermore we have
Corollary 7.8 Let $\Gamma$ be some graph. Then $\pi_{\Gamma}(\overline{{A}}_{=t_{\mathrm{max}}})\:=\:\pi_{\Gamma}(\overline{{A}})$ . In other words: $\pi_{\Gamma}$ is surjective even on the generic connections.
Proof $\pi_{\Gamma}$ is surjective on $\overline{{\mathcal{A}}}$ as proven in [10]. By Proposition 7.1 there is now an $\overline{{A}}^{\prime}$ with $\mathrm{Typ}(\overline{{A}}^{\prime})=t_{\mathrm{max}}$ and $\pi_{\Gamma}(\overline{{A}}^{\prime})=\pi_{\Gamma}(\overline{{A}})$ . qed
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<p data-bbox="124 14 448 45">Thus, $f\in\cal Z(\mathbf{H}_{\overline{{\cal A}}^{\prime}})$ . Due to the definition of $\pmb{x}$ we have $Z(\mathbf{H}_{\overline{{A}}^{\prime}})=Z(\{g\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$ . </p>
<h1 data-bbox="62 61 333 78">7.3 Construction of Arbitrary Types </h1>
<p data-bbox="62 85 322 100">Finally, we can now prove the desired proposition. </p>
<h1 data-bbox="62 109 198 123">Proof Proposition 7.1 </h1>
<p data-bbox="106 124 538 417">• Let $t\in\mathcal T$ and $t\geq\mathrm{Typ}(\overline{{A}})$ . Then there exist a Howe subgroup $V^{\prime}\subseteq\mathbf{G}$ with $t=$ $\left[V^{\prime}\right]$ and a $g\in\mathbf G$ , such that $Z(\mathbf{H}_{\overline{{A}}})\supseteq g^{-1}V^{\prime}g=:V$ . Since $V$ is a Howe subgroup, we have $Z(Z(V))\,=\,V$ and so by Lemma 4.1 there exist certain $u_{0},\dotsc,u_{k}\in$ $Z(V)\subseteq\mathbf{G}$ , such that $V=Z(Z(V))=Z(\{u_{0},\dots,u_{k}\})$ .
• Now let $Z(\mathbf{H}_{\overline{{A}}})\,=\,Z(h_{\overline{{A}}}(\alpha))$ with an appropriate $\alpha\subseteq{\mathcal{H}}{\mathcal{G}}$ as in Corollary 4.2. Because of $V\subseteq Z(\mathbf{H}_{\overline{{A}}})$ we have $V=V\cap Z(\mathbf{H}_{\overline{{A}}})=Z(\{u_{0},\dots,u_{k}\})\cap Z(h_{\overline{{A}}}(\alpha))=$ $Z(\left\{u_{0},\dots,u_{k}\right\}\cup h_{\overline{{A}}}(\pmb{\alpha}))$ .
• We now use inductively Lemma 7.5. Let $\overline{{A}}_{0}:=\overline{{A}}$ and $\alpha_{0}:=\alpha$ . Construct for all $j=0,\dots,k$ a connection $\overline{{A}}_{j+1}$ and an $e_{j}\in{\mathcal{H}}{\mathcal{G}}$ from $\overline{{A}}_{j}$ and $\alpha_{j}$ by that lemma, such that $\pi_{\Gamma_{i}}(\overline{{{A}}}_{j+1})=\pi_{\Gamma_{i}}(\overline{{{A}}}_{j})$ for all $i$ , $h_{\overline{{A}}_{j+1}}(\pmb{\alpha}_{j})=h_{\overline{{A}}_{j}}(\pmb{\alpha}_{j})$ , $h_{\overline{{A}}_{j+1}}(e_{j})=u_{j}$ and $Z(\mathbf{H}_{\overline{{A}}_{j+1}})=Z(\{u_{j}\}\cup h_{\overline{{A}}_{j}}(\pmb{\alpha}_{j}))$ . Setting $\alpha_{j+1}:=\alpha_{j}\cup\{e_{j}\}$ we get $Z(\mathbf{H}_{\overline{{A}}_{j+1}})=Z(\{u_{j}\}\cup h_{\overline{{A}}_{j}}(\alpha_{j}))=Z(h_{\overline{{A}}_{j+1}}(\alpha_{j+1})).$ Finally, we define $\overline{{A}}^{\prime}:=\overline{{A}}_{k+1}$ . Now, we get $\pi_{\Gamma_{i}}(\overline{{{A}}}^{\prime})=\pi_{\Gamma_{i}}(\overline{{{A}}})$ for all $i$ , $h_{\overline{{{A}}}^{\prime}}(\pmb{\alpha})=h_{\overline{{{A}}}}(\pmb{\alpha})$ and $h_{\overline{{A}}^{\prime}}(e_{j})=u_{j}$ . Thus, $\begin{array}{l l l}{{Z({\bf H}_{\overline{{{A}}}^{\prime}})}}&{{=}}&{{Z(h_{\overline{{{A}}}^{\prime}}(\alpha_{k+1}))}}\\ {{}}&{{=}}&{{Z(h_{\overline{{{A}}}^{\prime}}(\{e_{0},\ldots,e_{k}\}\cup h_{\overline{{{A}}}^{\prime}}(\alpha)))}}\\ {{}}&{{=}}&{{Z(\{u_{0},\ldots,u_{k}\}\cup h_{\overline{{{A}}}}(\alpha))}}\\ {{}}&{{=}}&{{V,}}\end{array}$ i.e., $\mathrm{Typ}({\overline{{A}}}^{\prime})=[V]=t$ . qed </p>
<p data-bbox="64 426 404 441">The proposition just proven has a further immediate consequence. </p>
<p data-bbox="63 447 311 464">Corollary 7.6 $\overline{{A}}_{=t}$ is non-empty for all $t\in\mathcal T$ . </p>
<p data-bbox="63 474 537 520">Proof Let $\overline{{A}}$ be the trivial connection, i.e. $h_{\overline{{A}}}(\alpha)=e_{\mathbf{G}}$ for all $\alpha\in\mathcal{P}$ . The type of $\overline{{A}}$ is $[\mathbf G]$ , thus minimal, i.e. we have $t\geq\mathrm{Typ}(\overline{{A}})$ for all $t\in\mathcal T$ . By means of Proposition 7.1 there is an $\overline{{A}}^{\prime}\in\overline{{A}}$ with $\mathrm{Typ}(\overline{{A}}^{\prime})=t$ . qed </p>
<p data-bbox="63 529 535 545">This corollary solves the problem which gauge orbit types exist for generalized connections. </p>
<p data-bbox="63 552 538 583">Theorem 7.7 The set of all gauge orbit types on $\overline{{\mathcal{A}}}$ is the set of all conjugacy classes of Howe subgroups of $\mathbf{G}$ . </p>
<p data-bbox="62 591 172 605">Furthermore we have </p>
<p data-bbox="63 613 538 644">Corollary 7.8 Let $\Gamma$ be some graph. Then $\pi_{\Gamma}(\overline{{A}}_{=t_{\mathrm{max}}})\:=\:\pi_{\Gamma}(\overline{{A}})$ . In other words: $\pi_{\Gamma}$ is surjective even on the generic connections. </p>
<p data-bbox="63 653 538 686">Proof $\pi_{\Gamma}$ is surjective on $\overline{{\mathcal{A}}}$ as proven in [10]. By Proposition 7.1 there is now an $\overline{{A}}^{\prime}$ with $\mathrm{Typ}(\overline{{A}}^{\prime})=t_{\mathrm{max}}$ and $\pi_{\Gamma}(\overline{{A}}^{\prime})=\pi_{\Gamma}(\overline{{A}})$ . qed </p>
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0001008v1
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[{"type": "text", "text": "Thus, $f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})$ . Due to the definition of $\\pmb{x}$ we have $Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$ . ", "page_idx": 14}, {"type": "text", "text": "7.3 Construction of Arbitrary Types ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "Finally, we can now prove the desired proposition. ", "page_idx": 14}, {"type": "text", "text": "Proof Proposition 7.1 ", "text_level": 1, "page_idx": 14}, {"type": "text", "text": "• Let $t\\in\\mathcal T$ and $t\\geq\\mathrm{Typ}(\\overline{{A}})$ . Then there exist a Howe subgroup $V^{\\prime}\\subseteq\\mathbf{G}$ with $t=$ $\\left[V^{\\prime}\\right]$ and a $g\\in\\mathbf G$ , such that $Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V$ . Since $V$ is a Howe subgroup, we have $Z(Z(V))\\,=\\,V$ and so by Lemma 4.1 there exist certain $u_{0},\\dotsc,u_{k}\\in$ $Z(V)\\subseteq\\mathbf{G}$ , such that $V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})$ . \n• Now let $Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))$ with an appropriate $\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}$ as in Corollary 4.2. Because of $V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})$ we have $V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=$ $Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))$ . \n• We now use inductively Lemma 7.5. Let $\\overline{{A}}_{0}:=\\overline{{A}}$ and $\\alpha_{0}:=\\alpha$ . Construct for all $j=0,\\dots,k$ a connection $\\overline{{A}}_{j+1}$ and an $e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}$ from $\\overline{{A}}_{j}$ and $\\alpha_{j}$ by that lemma, such that $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})$ for all $i$ , $h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})$ , $h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}$ and $Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))$ . Setting $\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}$ we get $Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).$ Finally, we define $\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}$ . Now, we get $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$ for all $i$ , $h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$ and $h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}$ . Thus, $\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}$ i.e., $\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t$ . qed ", "page_idx": 14}, {"type": "text", "text": "The proposition just proven has a further immediate consequence. ", "page_idx": 14}, {"type": "text", "text": "Corollary 7.6 $\\overline{{A}}_{=t}$ is non-empty for all $t\\in\\mathcal T$ . ", "page_idx": 14}, {"type": "text", "text": "Proof Let $\\overline{{A}}$ be the trivial connection, i.e. $h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}$ for all $\\alpha\\in\\mathcal{P}$ . The type of $\\overline{{A}}$ is $[\\mathbf G]$ , thus minimal, i.e. we have $t\\geq\\mathrm{Typ}(\\overline{{A}})$ for all $t\\in\\mathcal T$ . By means of Proposition 7.1 there is an $\\overline{{A}}^{\\prime}\\in\\overline{{A}}$ with $\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t$ . qed ", "page_idx": 14}, {"type": "text", "text": "This corollary solves the problem which gauge orbit types exist for generalized connections. ", "page_idx": 14}, {"type": "text", "text": "Theorem 7.7 The set of all gauge orbit types on $\\overline{{\\mathcal{A}}}$ is the set of all conjugacy classes of Howe subgroups of $\\mathbf{G}$ . ", "page_idx": 14}, {"type": "text", "text": "Furthermore we have ", "page_idx": 14}, {"type": "text", "text": "Corollary 7.8 Let $\\Gamma$ be some graph. Then $\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})$ . In other words: $\\pi_{\\Gamma}$ is surjective even on the generic connections. ", "page_idx": 14}, {"type": "text", "text": "Proof $\\pi_{\\Gamma}$ is surjective on $\\overline{{\\mathcal{A}}}$ as proven in [10]. By Proposition 7.1 there is now an $\\overline{{A}}^{\\prime}$ with $\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}$ and $\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})$ . qed ", "page_idx": 14}]
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{"preproc_blocks": [{"type": "text", "bbox": [124, 14, 448, 45], "lines": [{"bbox": [157, 15, 254, 32], "spans": [{"bbox": [157, 15, 190, 32], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [191, 18, 251, 32], "score": 0.94, "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "type": "inline_equation", "height": 14, "width": 60}, {"bbox": [251, 15, 254, 32], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [124, 31, 446, 46], "spans": [{"bbox": [124, 31, 250, 46], "score": 1.0, "content": "Due to the definition of ", "type": "text"}, {"bbox": [251, 37, 259, 42], "score": 0.89, "content": "\\pmb{x}", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [260, 31, 307, 46], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [308, 33, 442, 46], "score": 0.92, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [443, 31, 446, 46], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "title", "bbox": [62, 61, 333, 78], "lines": [{"bbox": [63, 64, 331, 79], "spans": [{"bbox": [63, 65, 86, 78], "score": 1.0, "content": "7.3", "type": "text"}, {"bbox": [98, 64, 331, 79], "score": 1.0, "content": "Construction of Arbitrary Types", "type": "text"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [62, 85, 322, 100], "lines": [{"bbox": [63, 88, 321, 102], "spans": [{"bbox": [63, 88, 321, 102], "score": 1.0, "content": "Finally, we can now prove the desired proposition.", "type": "text"}], "index": 3}], "index": 3}, {"type": "title", "bbox": [62, 109, 198, 123], "lines": [{"bbox": [63, 112, 196, 124], "spans": [{"bbox": [63, 112, 196, 124], "score": 1.0, "content": "Proof Proposition 7.1", "type": "text"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [106, 124, 538, 417], "lines": [{"bbox": [106, 126, 538, 140], "spans": [{"bbox": [106, 126, 144, 140], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [144, 128, 173, 137], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [174, 126, 199, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [200, 126, 259, 140], "score": 0.92, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [259, 126, 448, 140], "score": 1.0, "content": ". Then there exist a Howe subgroup ", "type": "text"}, {"bbox": [449, 127, 488, 138], "score": 0.92, "content": "V^{\\prime}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [489, 126, 518, 140], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [519, 128, 538, 138], "score": 0.8, "content": "t=", "type": "inline_equation", "height": 10, "width": 19}], "index": 5}, {"bbox": [123, 139, 537, 156], "spans": [{"bbox": [123, 141, 142, 154], "score": 0.9, "content": "\\left[V^{\\prime}\\right]", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [142, 139, 177, 156], "score": 1.0, "content": " and a ", "type": "text"}, {"bbox": [177, 142, 208, 153], "score": 0.91, "content": "g\\in\\mathbf G", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [209, 139, 266, 156], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [266, 141, 382, 154], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [383, 139, 420, 156], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [420, 141, 430, 151], "score": 0.75, "content": "V", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [430, 139, 537, 156], "score": 1.0, "content": " is a Howe subgroup,", "type": "text"}], "index": 6}, {"bbox": [123, 155, 537, 170], "spans": [{"bbox": [123, 155, 169, 170], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [169, 156, 245, 169], "score": 0.94, "content": "Z(Z(V))\\,=\\,V", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [246, 155, 470, 170], "score": 1.0, "content": " and so by Lemma 4.1 there exist certain ", "type": "text"}, {"bbox": [470, 156, 537, 168], "score": 0.84, "content": "u_{0},\\dotsc,u_{k}\\in", "type": "inline_equation", "height": 12, "width": 67}], "index": 7}, {"bbox": [123, 169, 408, 185], "spans": [{"bbox": [123, 171, 177, 183], "score": 0.93, "content": "Z(V)\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [177, 169, 235, 184], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [236, 170, 403, 183], "score": 0.93, "content": "V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})", "type": "inline_equation", "height": 13, "width": 167}, {"bbox": [404, 169, 408, 185], "score": 1.0, "content": ".", "type": "text"}], "index": 8}, {"bbox": [108, 183, 537, 199], "spans": [{"bbox": [108, 183, 168, 199], "score": 1.0, "content": "• Now let ", "type": "text"}, {"bbox": [168, 185, 272, 198], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 13, "width": 104}, {"bbox": [272, 183, 385, 199], "score": 1.0, "content": " with an appropriate ", "type": "text"}, {"bbox": [385, 184, 431, 196], "score": 0.88, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [432, 183, 537, 199], "score": 1.0, "content": " as in Corollary 4.2.", "type": "text"}], "index": 9}, {"bbox": [122, 197, 539, 214], "spans": [{"bbox": [122, 197, 179, 214], "score": 1.0, "content": "Because of ", "type": "text"}, {"bbox": [180, 199, 240, 212], "score": 0.93, "content": "V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [241, 197, 286, 214], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [286, 199, 539, 212], "score": 0.86, "content": "V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=", "type": "inline_equation", "height": 13, "width": 253}], "index": 10}, {"bbox": [123, 212, 254, 228], "spans": [{"bbox": [123, 213, 249, 226], "score": 0.91, "content": "Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [249, 212, 254, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [109, 226, 538, 241], "spans": [{"bbox": [109, 226, 333, 241], "score": 1.0, "content": "• We now use inductively Lemma 7.5. Let ", "type": "text"}, {"bbox": [333, 226, 375, 240], "score": 0.92, "content": "\\overline{{A}}_{0}:=\\overline{{A}}", "type": "inline_equation", "height": 14, "width": 42}, {"bbox": [375, 226, 401, 241], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [401, 229, 443, 240], "score": 0.8, "content": "\\alpha_{0}:=\\alpha", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [443, 226, 538, 241], "score": 1.0, "content": ". Construct for all", "type": "text"}], "index": 12}, {"bbox": [123, 241, 538, 257], "spans": [{"bbox": [123, 243, 185, 254], "score": 0.89, "content": "j=0,\\dots,k", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [186, 241, 257, 257], "score": 1.0, "content": " a connection ", "type": "text"}, {"bbox": [257, 241, 282, 255], "score": 0.91, "content": "\\overline{{A}}_{j+1}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [282, 241, 324, 257], "score": 1.0, "content": " and an ", "type": "text"}, {"bbox": [324, 241, 368, 255], "score": 0.89, "content": "e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [369, 241, 399, 257], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [399, 241, 413, 255], "score": 0.91, "content": "\\overline{{A}}_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [413, 241, 439, 257], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [439, 246, 453, 255], "score": 0.79, "content": "\\alpha_{j}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [453, 241, 538, 257], "score": 1.0, "content": " by that lemma,", "type": "text"}], "index": 13}, {"bbox": [119, 253, 540, 276], "spans": [{"bbox": [119, 253, 174, 276], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [175, 255, 278, 270], "score": 0.91, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})", "type": "inline_equation", "height": 15, "width": 103}, {"bbox": [278, 253, 315, 276], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [316, 256, 320, 267], "score": 0.67, "content": "i", "type": "inline_equation", "height": 11, "width": 4}, {"bbox": [321, 253, 327, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [327, 256, 433, 272], "score": 0.89, "content": "h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})", "type": "inline_equation", "height": 16, "width": 106}, {"bbox": [433, 253, 439, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [440, 257, 513, 272], "score": 0.87, "content": "h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}", "type": "inline_equation", "height": 15, "width": 73}, {"bbox": [514, 253, 540, 276], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [123, 271, 286, 288], "spans": [{"bbox": [123, 272, 281, 288], "score": 0.91, "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))", "type": "inline_equation", "height": 16, "width": 158}, {"bbox": [281, 271, 286, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 15}, {"bbox": [120, 285, 535, 307], "spans": [{"bbox": [120, 285, 162, 307], "score": 1.0, "content": "Setting ", "type": "text"}, {"bbox": [162, 288, 249, 301], "score": 0.9, "content": "\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [249, 285, 284, 307], "score": 1.0, "content": " we get", "type": "text"}, {"bbox": [285, 286, 535, 304], "score": 0.82, "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).", "type": "inline_equation", "height": 18, "width": 250}], "index": 16}, {"bbox": [122, 300, 278, 322], "spans": [{"bbox": [122, 300, 216, 322], "score": 1.0, "content": "Finally, we define ", "type": "text"}, {"bbox": [216, 303, 272, 318], "score": 0.89, "content": "\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [272, 300, 278, 322], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [120, 317, 534, 338], "spans": [{"bbox": [120, 317, 189, 338], "score": 1.0, "content": "Now, we get ", "type": "text"}, {"bbox": [189, 318, 275, 334], "score": 0.92, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 16, "width": 86}, {"bbox": [276, 317, 312, 338], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [312, 321, 317, 331], "score": 0.48, "content": "i", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [317, 317, 323, 338], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [324, 319, 406, 335], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 16, "width": 82}, {"bbox": [407, 317, 432, 338], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [432, 321, 495, 335], "score": 0.93, "content": "h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [495, 317, 534, 338], "score": 1.0, "content": ". Thus,", "type": "text"}], "index": 18}, {"bbox": [218, 338, 441, 397], "spans": [{"bbox": [218, 338, 441, 397], "score": 0.93, "content": "\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}", "type": "inline_equation"}], "index": 19}, {"bbox": [122, 402, 539, 420], "spans": [{"bbox": [122, 402, 145, 419], "score": 1.0, "content": "i.e., ", "type": "text"}, {"bbox": [145, 403, 239, 419], "score": 0.92, "content": "\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t", "type": "inline_equation", "height": 16, "width": 94}, {"bbox": [239, 402, 243, 419], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 405, 539, 420], "score": 1.0, "content": "qed", "type": "text"}], "index": 20}], "index": 12.5}, {"type": "text", "bbox": [64, 426, 404, 441], "lines": [{"bbox": [63, 428, 402, 442], "spans": [{"bbox": [63, 428, 402, 442], "score": 1.0, "content": "The proposition just proven has a further immediate consequence.", "type": "text"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [63, 447, 311, 464], "lines": [{"bbox": [64, 451, 311, 464], "spans": [{"bbox": [64, 451, 150, 464], "score": 1.0, "content": "Corollary 7.6", "type": "text"}, {"bbox": [151, 452, 171, 464], "score": 0.91, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [171, 451, 278, 464], "score": 1.0, "content": " is non-empty for all ", "type": "text"}, {"bbox": [279, 453, 307, 462], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [308, 451, 311, 464], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [63, 474, 537, 520], "lines": [{"bbox": [61, 475, 537, 493], "spans": [{"bbox": [61, 475, 127, 493], "score": 1.0, "content": "Proof Let ", "type": "text"}, {"bbox": [127, 477, 136, 488], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [136, 475, 291, 493], "score": 1.0, "content": " be the trivial connection, i.e. ", "type": "text"}, {"bbox": [291, 479, 351, 492], "score": 0.92, "content": "h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [351, 475, 388, 493], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [388, 480, 420, 489], "score": 0.93, "content": "\\alpha\\in\\mathcal{P}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [420, 475, 491, 493], "score": 1.0, "content": ". The type of ", "type": "text"}, {"bbox": [491, 478, 500, 488], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [501, 475, 515, 493], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [515, 479, 533, 491], "score": 0.54, "content": "[\\mathbf G]", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [533, 475, 537, 493], "score": 1.0, "content": ",", "type": "text"}], "index": 23}, {"bbox": [105, 491, 537, 506], "spans": [{"bbox": [105, 492, 248, 506], "score": 1.0, "content": "thus minimal, i.e. we have ", "type": "text"}, {"bbox": [248, 491, 309, 505], "score": 0.85, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 61}, {"bbox": [309, 492, 348, 506], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [348, 494, 379, 503], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [379, 492, 537, 506], "score": 1.0, "content": ". By means of Proposition 7.1", "type": "text"}], "index": 24}, {"bbox": [105, 504, 537, 520], "spans": [{"bbox": [105, 504, 164, 520], "score": 1.0, "content": "there is an ", "type": "text"}, {"bbox": [164, 505, 200, 517], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [200, 504, 230, 520], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [231, 505, 292, 520], "score": 0.9, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t", "type": "inline_equation", "height": 15, "width": 61}, {"bbox": [293, 504, 297, 520], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 507, 537, 520], "score": 1.0, "content": "qed", "type": "text"}], "index": 25}], "index": 24}, {"type": "text", "bbox": [63, 529, 535, 545], "lines": [{"bbox": [61, 531, 533, 549], "spans": [{"bbox": [61, 531, 533, 549], "score": 1.0, "content": "This corollary solves the problem which gauge orbit types exist for generalized connections.", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [63, 552, 538, 583], "lines": [{"bbox": [62, 555, 538, 570], "spans": [{"bbox": [62, 555, 335, 570], "score": 1.0, "content": "Theorem 7.7 The set of all gauge orbit types on ", "type": "text"}, {"bbox": [335, 556, 345, 566], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [345, 555, 538, 570], "score": 1.0, "content": " is the set of all conjugacy classes of", "type": "text"}], "index": 27}, {"bbox": [147, 569, 263, 584], "spans": [{"bbox": [147, 569, 248, 584], "score": 1.0, "content": "Howe subgroups of ", "type": "text"}, {"bbox": [248, 572, 259, 581], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [259, 569, 263, 584], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27.5}, {"type": "text", "bbox": [62, 591, 172, 605], "lines": [{"bbox": [63, 593, 172, 606], "spans": [{"bbox": [63, 593, 172, 606], "score": 1.0, "content": "Furthermore we have", "type": "text"}], "index": 29}], "index": 29}, {"type": "text", "bbox": [63, 613, 538, 644], "lines": [{"bbox": [63, 615, 539, 633], "spans": [{"bbox": [63, 615, 172, 633], "score": 1.0, "content": "Corollary 7.8 Let ", "type": "text"}, {"bbox": [172, 619, 180, 627], "score": 0.85, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [181, 615, 305, 633], "score": 1.0, "content": " be some graph. Then ", "type": "text"}, {"bbox": [305, 617, 411, 631], "score": 0.93, "content": "\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 106}, {"bbox": [411, 615, 510, 633], "score": 1.0, "content": ". In other words: ", "type": "text"}, {"bbox": [510, 622, 523, 629], "score": 0.89, "content": "\\pi_{\\Gamma}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [523, 615, 539, 633], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [150, 633, 369, 646], "spans": [{"bbox": [150, 633, 369, 646], "score": 1.0, "content": "surjective even on the generic connections.", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [63, 653, 538, 686], "lines": [{"bbox": [61, 656, 538, 672], "spans": [{"bbox": [61, 656, 106, 672], "score": 1.0, "content": "Proof", "type": "text"}, {"bbox": [106, 662, 119, 669], "score": 0.86, "content": "\\pi_{\\Gamma}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [119, 656, 204, 672], "score": 1.0, "content": " is surjective on ", "type": "text"}, {"bbox": [204, 658, 214, 668], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [214, 656, 497, 672], "score": 1.0, "content": " as proven in [10]. By Proposition 7.1 there is now an ", "type": "text"}, {"bbox": [497, 656, 509, 668], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [509, 656, 538, 672], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [106, 669, 539, 687], "spans": [{"bbox": [106, 671, 185, 685], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}", "type": "inline_equation", "height": 14, "width": 79}, {"bbox": [185, 669, 210, 687], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [211, 671, 290, 686], "score": 0.94, "content": "\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 79}, {"bbox": [290, 669, 294, 687], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [512, 672, 539, 687], "score": 1.0, "content": "qed", "type": "text"}], "index": 33}], "index": 32.5}], "layout_bboxes": [], "page_idx": 14, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [291, 705, 307, 717], "spans": [{"bbox": [291, 705, 307, 717], "score": 1.0, "content": "15", "type": "text"}]}]}, {"type": "discarded", "bbox": [514, 30, 537, 43], "lines": [{"bbox": [513, 31, 539, 46], "spans": [{"bbox": [513, 31, 539, 46], "score": 1.0, "content": "qed", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [124, 14, 448, 45], "lines": [{"bbox": [157, 15, 254, 32], "spans": [{"bbox": [157, 15, 190, 32], "score": 1.0, "content": "Thus, ", "type": "text"}, {"bbox": [191, 18, 251, 32], "score": 0.94, "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "type": "inline_equation", "height": 14, "width": 60}, {"bbox": [251, 15, 254, 32], "score": 1.0, "content": ".", "type": "text"}], "index": 0}, {"bbox": [124, 31, 446, 46], "spans": [{"bbox": [124, 31, 250, 46], "score": 1.0, "content": "Due to the definition of ", "type": "text"}, {"bbox": [251, 37, 259, 42], "score": 0.89, "content": "\\pmb{x}", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [260, 31, 307, 46], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [308, 33, 442, 46], "score": 0.92, "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 13, "width": 134}, {"bbox": [443, 31, 446, 46], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [124, 15, 446, 46]}, {"type": "title", "bbox": [62, 61, 333, 78], "lines": [{"bbox": [63, 64, 331, 79], "spans": [{"bbox": [63, 65, 86, 78], "score": 1.0, "content": "7.3", "type": "text"}, {"bbox": [98, 64, 331, 79], "score": 1.0, "content": "Construction of Arbitrary Types", "type": "text"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [62, 85, 322, 100], "lines": [{"bbox": [63, 88, 321, 102], "spans": [{"bbox": [63, 88, 321, 102], "score": 1.0, "content": "Finally, we can now prove the desired proposition.", "type": "text"}], "index": 3}], "index": 3, "bbox_fs": [63, 88, 321, 102]}, {"type": "title", "bbox": [62, 109, 198, 123], "lines": [{"bbox": [63, 112, 196, 124], "spans": [{"bbox": [63, 112, 196, 124], "score": 1.0, "content": "Proof Proposition 7.1", "type": "text"}], "index": 4}], "index": 4}, {"type": "list", "bbox": [106, 124, 538, 417], "lines": [{"bbox": [106, 126, 538, 140], "spans": [{"bbox": [106, 126, 144, 140], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [144, 128, 173, 137], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [174, 126, 199, 140], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [200, 126, 259, 140], "score": 0.92, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 59}, {"bbox": [259, 126, 448, 140], "score": 1.0, "content": ". Then there exist a Howe subgroup ", "type": "text"}, {"bbox": [449, 127, 488, 138], "score": 0.92, "content": "V^{\\prime}\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [489, 126, 518, 140], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [519, 128, 538, 138], "score": 0.8, "content": "t=", "type": "inline_equation", "height": 10, "width": 19}], "index": 5, "is_list_start_line": true}, {"bbox": [123, 139, 537, 156], "spans": [{"bbox": [123, 141, 142, 154], "score": 0.9, "content": "\\left[V^{\\prime}\\right]", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [142, 139, 177, 156], "score": 1.0, "content": " and a ", "type": "text"}, {"bbox": [177, 142, 208, 153], "score": 0.91, "content": "g\\in\\mathbf G", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [209, 139, 266, 156], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [266, 141, 382, 154], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [383, 139, 420, 156], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [420, 141, 430, 151], "score": 0.75, "content": "V", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [430, 139, 537, 156], "score": 1.0, "content": " is a Howe subgroup,", "type": "text"}], "index": 6}, {"bbox": [123, 155, 537, 170], "spans": [{"bbox": [123, 155, 169, 170], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [169, 156, 245, 169], "score": 0.94, "content": "Z(Z(V))\\,=\\,V", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [246, 155, 470, 170], "score": 1.0, "content": " and so by Lemma 4.1 there exist certain ", "type": "text"}, {"bbox": [470, 156, 537, 168], "score": 0.84, "content": "u_{0},\\dotsc,u_{k}\\in", "type": "inline_equation", "height": 12, "width": 67}], "index": 7}, {"bbox": [123, 169, 408, 185], "spans": [{"bbox": [123, 171, 177, 183], "score": 0.93, "content": "Z(V)\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [177, 169, 235, 184], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [236, 170, 403, 183], "score": 0.93, "content": "V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})", "type": "inline_equation", "height": 13, "width": 167}, {"bbox": [404, 169, 408, 185], "score": 1.0, "content": ".", "type": "text"}], "index": 8, "is_list_end_line": true}, {"bbox": [108, 183, 537, 199], "spans": [{"bbox": [108, 183, 168, 199], "score": 1.0, "content": "• Now let ", "type": "text"}, {"bbox": [168, 185, 272, 198], "score": 0.93, "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "type": "inline_equation", "height": 13, "width": 104}, {"bbox": [272, 183, 385, 199], "score": 1.0, "content": " with an appropriate ", "type": "text"}, {"bbox": [385, 184, 431, 196], "score": 0.88, "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 12, "width": 46}, {"bbox": [432, 183, 537, 199], "score": 1.0, "content": " as in Corollary 4.2.", "type": "text"}], "index": 9, "is_list_start_line": true}, {"bbox": [122, 197, 539, 214], "spans": [{"bbox": [122, 197, 179, 214], "score": 1.0, "content": "Because of ", "type": "text"}, {"bbox": [180, 199, 240, 212], "score": 0.93, "content": "V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [241, 197, 286, 214], "score": 1.0, "content": " we have ", "type": "text"}, {"bbox": [286, 199, 539, 212], "score": 0.86, "content": "V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=", "type": "inline_equation", "height": 13, "width": 253}], "index": 10}, {"bbox": [123, 212, 254, 228], "spans": [{"bbox": [123, 213, 249, 226], "score": 0.91, "content": "Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "type": "inline_equation", "height": 13, "width": 126}, {"bbox": [249, 212, 254, 228], "score": 1.0, "content": ".", "type": "text"}], "index": 11, "is_list_end_line": true}, {"bbox": [109, 226, 538, 241], "spans": [{"bbox": [109, 226, 333, 241], "score": 1.0, "content": "• We now use inductively Lemma 7.5. Let ", "type": "text"}, {"bbox": [333, 226, 375, 240], "score": 0.92, "content": "\\overline{{A}}_{0}:=\\overline{{A}}", "type": "inline_equation", "height": 14, "width": 42}, {"bbox": [375, 226, 401, 241], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [401, 229, 443, 240], "score": 0.8, "content": "\\alpha_{0}:=\\alpha", "type": "inline_equation", "height": 11, "width": 42}, {"bbox": [443, 226, 538, 241], "score": 1.0, "content": ". Construct for all", "type": "text"}], "index": 12, "is_list_start_line": true}, {"bbox": [123, 241, 538, 257], "spans": [{"bbox": [123, 243, 185, 254], "score": 0.89, "content": "j=0,\\dots,k", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [186, 241, 257, 257], "score": 1.0, "content": " a connection ", "type": "text"}, {"bbox": [257, 241, 282, 255], "score": 0.91, "content": "\\overline{{A}}_{j+1}", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [282, 241, 324, 257], "score": 1.0, "content": " and an ", "type": "text"}, {"bbox": [324, 241, 368, 255], "score": 0.89, "content": "e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}", "type": "inline_equation", "height": 14, "width": 44}, {"bbox": [369, 241, 399, 257], "score": 1.0, "content": " from ", "type": "text"}, {"bbox": [399, 241, 413, 255], "score": 0.91, "content": "\\overline{{A}}_{j}", "type": "inline_equation", "height": 14, "width": 14}, {"bbox": [413, 241, 439, 257], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [439, 246, 453, 255], "score": 0.79, "content": "\\alpha_{j}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [453, 241, 538, 257], "score": 1.0, "content": " by that lemma,", "type": "text"}], "index": 13}, {"bbox": [119, 253, 540, 276], "spans": [{"bbox": [119, 253, 174, 276], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [175, 255, 278, 270], "score": 0.91, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})", "type": "inline_equation", "height": 15, "width": 103}, {"bbox": [278, 253, 315, 276], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [316, 256, 320, 267], "score": 0.67, "content": "i", "type": "inline_equation", "height": 11, "width": 4}, {"bbox": [321, 253, 327, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [327, 256, 433, 272], "score": 0.89, "content": "h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})", "type": "inline_equation", "height": 16, "width": 106}, {"bbox": [433, 253, 439, 276], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [440, 257, 513, 272], "score": 0.87, "content": "h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}", "type": "inline_equation", "height": 15, "width": 73}, {"bbox": [514, 253, 540, 276], "score": 1.0, "content": " and", "type": "text"}], "index": 14}, {"bbox": [123, 271, 286, 288], "spans": [{"bbox": [123, 272, 281, 288], "score": 0.91, "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))", "type": "inline_equation", "height": 16, "width": 158}, {"bbox": [281, 271, 286, 287], "score": 1.0, "content": ".", "type": "text"}], "index": 15, "is_list_end_line": true}, {"bbox": [120, 285, 535, 307], "spans": [{"bbox": [120, 285, 162, 307], "score": 1.0, "content": "Setting ", "type": "text"}, {"bbox": [162, 288, 249, 301], "score": 0.9, "content": "\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [249, 285, 284, 307], "score": 1.0, "content": " we get", "type": "text"}, {"bbox": [285, 286, 535, 304], "score": 0.82, "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).", "type": "inline_equation", "height": 18, "width": 250}], "index": 16}, {"bbox": [122, 300, 278, 322], "spans": [{"bbox": [122, 300, 216, 322], "score": 1.0, "content": "Finally, we define ", "type": "text"}, {"bbox": [216, 303, 272, 318], "score": 0.89, "content": "\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}", "type": "inline_equation", "height": 15, "width": 56}, {"bbox": [272, 300, 278, 322], "score": 1.0, "content": ".", "type": "text"}], "index": 17, "is_list_end_line": true}, {"bbox": [120, 317, 534, 338], "spans": [{"bbox": [120, 317, 189, 338], "score": 1.0, "content": "Now, we get ", "type": "text"}, {"bbox": [189, 318, 275, 334], "score": 0.92, "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "type": "inline_equation", "height": 16, "width": 86}, {"bbox": [276, 317, 312, 338], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [312, 321, 317, 331], "score": 0.48, "content": "i", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [317, 317, 323, 338], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [324, 319, 406, 335], "score": 0.9, "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "type": "inline_equation", "height": 16, "width": 82}, {"bbox": [407, 317, 432, 338], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [432, 321, 495, 335], "score": 0.93, "content": "h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}", "type": "inline_equation", "height": 14, "width": 63}, {"bbox": [495, 317, 534, 338], "score": 1.0, "content": ". Thus,", "type": "text"}], "index": 18}, {"bbox": [218, 338, 441, 397], "spans": [{"bbox": [218, 338, 441, 397], "score": 0.93, "content": "\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}", "type": "inline_equation"}], "index": 19, "is_list_end_line": true}, {"bbox": [122, 402, 539, 420], "spans": [{"bbox": [122, 402, 145, 419], "score": 1.0, "content": "i.e., ", "type": "text"}, {"bbox": [145, 403, 239, 419], "score": 0.92, "content": "\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t", "type": "inline_equation", "height": 16, "width": 94}, {"bbox": [239, 402, 243, 419], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 405, 539, 420], "score": 1.0, "content": "qed", "type": "text"}], "index": 20}], "index": 12.5, "bbox_fs": [106, 126, 540, 420]}, {"type": "text", "bbox": [64, 426, 404, 441], "lines": [{"bbox": [63, 428, 402, 442], "spans": [{"bbox": [63, 428, 402, 442], "score": 1.0, "content": "The proposition just proven has a further immediate consequence.", "type": "text"}], "index": 21}], "index": 21, "bbox_fs": [63, 428, 402, 442]}, {"type": "text", "bbox": [63, 447, 311, 464], "lines": [{"bbox": [64, 451, 311, 464], "spans": [{"bbox": [64, 451, 150, 464], "score": 1.0, "content": "Corollary 7.6", "type": "text"}, {"bbox": [151, 452, 171, 464], "score": 0.91, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [171, 451, 278, 464], "score": 1.0, "content": " is non-empty for all ", "type": "text"}, {"bbox": [279, 453, 307, 462], "score": 0.91, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [308, 451, 311, 464], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [64, 451, 311, 464]}, {"type": "text", "bbox": [63, 474, 537, 520], "lines": [{"bbox": [61, 475, 537, 493], "spans": [{"bbox": [61, 475, 127, 493], "score": 1.0, "content": "Proof Let ", "type": "text"}, {"bbox": [127, 477, 136, 488], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [136, 475, 291, 493], "score": 1.0, "content": " be the trivial connection, i.e. ", "type": "text"}, {"bbox": [291, 479, 351, 492], "score": 0.92, "content": "h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [351, 475, 388, 493], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [388, 480, 420, 489], "score": 0.93, "content": "\\alpha\\in\\mathcal{P}", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [420, 475, 491, 493], "score": 1.0, "content": ". The type of ", "type": "text"}, {"bbox": [491, 478, 500, 488], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [501, 475, 515, 493], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [515, 479, 533, 491], "score": 0.54, "content": "[\\mathbf G]", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [533, 475, 537, 493], "score": 1.0, "content": ",", "type": "text"}], "index": 23}, {"bbox": [105, 491, 537, 506], "spans": [{"bbox": [105, 492, 248, 506], "score": 1.0, "content": "thus minimal, i.e. we have ", "type": "text"}, {"bbox": [248, 491, 309, 505], "score": 0.85, "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 61}, {"bbox": [309, 492, 348, 506], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [348, 494, 379, 503], "score": 0.92, "content": "t\\in\\mathcal T", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [379, 492, 537, 506], "score": 1.0, "content": ". By means of Proposition 7.1", "type": "text"}], "index": 24}, {"bbox": [105, 504, 537, 520], "spans": [{"bbox": [105, 504, 164, 520], "score": 1.0, "content": "there is an ", "type": "text"}, {"bbox": [164, 505, 200, 517], "score": 0.93, "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "type": "inline_equation", "height": 12, "width": 36}, {"bbox": [200, 504, 230, 520], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [231, 505, 292, 520], "score": 0.9, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t", "type": "inline_equation", "height": 15, "width": 61}, {"bbox": [293, 504, 297, 520], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 507, 537, 520], "score": 1.0, "content": "qed", "type": "text"}], "index": 25}], "index": 24, "bbox_fs": [61, 475, 537, 520]}, {"type": "text", "bbox": [63, 529, 535, 545], "lines": [{"bbox": [61, 531, 533, 549], "spans": [{"bbox": [61, 531, 533, 549], "score": 1.0, "content": "This corollary solves the problem which gauge orbit types exist for generalized connections.", "type": "text"}], "index": 26}], "index": 26, "bbox_fs": [61, 531, 533, 549]}, {"type": "text", "bbox": [63, 552, 538, 583], "lines": [{"bbox": [62, 555, 538, 570], "spans": [{"bbox": [62, 555, 335, 570], "score": 1.0, "content": "Theorem 7.7 The set of all gauge orbit types on ", "type": "text"}, {"bbox": [335, 556, 345, 566], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [345, 555, 538, 570], "score": 1.0, "content": " is the set of all conjugacy classes of", "type": "text"}], "index": 27}, {"bbox": [147, 569, 263, 584], "spans": [{"bbox": [147, 569, 248, 584], "score": 1.0, "content": "Howe subgroups of ", "type": "text"}, {"bbox": [248, 572, 259, 581], "score": 0.9, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [259, 569, 263, 584], "score": 1.0, "content": ".", "type": "text"}], "index": 28}], "index": 27.5, "bbox_fs": [62, 555, 538, 584]}, {"type": "text", "bbox": [62, 591, 172, 605], "lines": [{"bbox": [63, 593, 172, 606], "spans": [{"bbox": [63, 593, 172, 606], "score": 1.0, "content": "Furthermore we have", "type": "text"}], "index": 29}], "index": 29, "bbox_fs": [63, 593, 172, 606]}, {"type": "text", "bbox": [63, 613, 538, 644], "lines": [{"bbox": [63, 615, 539, 633], "spans": [{"bbox": [63, 615, 172, 633], "score": 1.0, "content": "Corollary 7.8 Let ", "type": "text"}, {"bbox": [172, 619, 180, 627], "score": 0.85, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [181, 615, 305, 633], "score": 1.0, "content": " be some graph. Then ", "type": "text"}, {"bbox": [305, 617, 411, 631], "score": 0.93, "content": "\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})", "type": "inline_equation", "height": 14, "width": 106}, {"bbox": [411, 615, 510, 633], "score": 1.0, "content": ". In other words: ", "type": "text"}, {"bbox": [510, 622, 523, 629], "score": 0.89, "content": "\\pi_{\\Gamma}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [523, 615, 539, 633], "score": 1.0, "content": " is", "type": "text"}], "index": 30}, {"bbox": [150, 633, 369, 646], "spans": [{"bbox": [150, 633, 369, 646], "score": 1.0, "content": "surjective even on the generic connections.", "type": "text"}], "index": 31}], "index": 30.5, "bbox_fs": [63, 615, 539, 646]}, {"type": "text", "bbox": [63, 653, 538, 686], "lines": [{"bbox": [61, 656, 538, 672], "spans": [{"bbox": [61, 656, 106, 672], "score": 1.0, "content": "Proof", "type": "text"}, {"bbox": [106, 662, 119, 669], "score": 0.86, "content": "\\pi_{\\Gamma}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [119, 656, 204, 672], "score": 1.0, "content": " is surjective on ", "type": "text"}, {"bbox": [204, 658, 214, 668], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [214, 656, 497, 672], "score": 1.0, "content": " as proven in [10]. By Proposition 7.1 there is now an ", "type": "text"}, {"bbox": [497, 656, 509, 668], "score": 0.9, "content": "\\overline{{A}}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [509, 656, 538, 672], "score": 1.0, "content": " with", "type": "text"}], "index": 32}, {"bbox": [106, 669, 539, 687], "spans": [{"bbox": [106, 671, 185, 685], "score": 0.92, "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}", "type": "inline_equation", "height": 14, "width": 79}, {"bbox": [185, 669, 210, 687], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [211, 671, 290, 686], "score": 0.94, "content": "\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})", "type": "inline_equation", "height": 15, "width": 79}, {"bbox": [290, 669, 294, 687], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [512, 672, 539, 687], "score": 1.0, "content": "qed", "type": "text"}], "index": 33}], "index": 32.5, "bbox_fs": [61, 656, 539, 687]}]}
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[{"type": "text", "bbox": [124, 14, 448, 45], "content": "Thus, . Due to the definition of we have .", "index": 0}, {"type": "title", "bbox": [62, 61, 333, 78], "content": "7.3 Construction of Arbitrary Types", "index": 1}, {"type": "text", "bbox": [62, 85, 322, 100], "content": "Finally, we can now prove the desired proposition.", "index": 2}, {"type": "title", "bbox": [62, 109, 198, 123], "content": "Proof Proposition 7.1", "index": 3}, {"type": "list", "bbox": [106, 124, 538, 417], "content": "", "index": 4}, {"type": "text", "bbox": [64, 426, 404, 441], "content": "The proposition just proven has a further immediate consequence.", "index": 5}, {"type": "text", "bbox": [63, 447, 311, 464], "content": "Corollary 7.6 is non-empty for all .", "index": 6}, {"type": "text", "bbox": [63, 474, 537, 520], "content": "Proof Let be the trivial connection, i.e. for all . The type of is , thus minimal, i.e. we have for all . By means of Proposition 7.1 there is an with . qed", "index": 7}, {"type": "text", "bbox": [63, 529, 535, 545], "content": "This corollary solves the problem which gauge orbit types exist for generalized connections.", "index": 8}, {"type": "text", "bbox": [63, 552, 538, 583], "content": "Theorem 7.7 The set of all gauge orbit types on is the set of all conjugacy classes of Howe subgroups of .", "index": 9}, {"type": "text", "bbox": [62, 591, 172, 605], "content": "Furthermore we have", "index": 10}, {"type": "text", "bbox": [63, 613, 538, 644], "content": "Corollary 7.8 Let be some graph. Then . In other words: is surjective even on the generic connections.", "index": 11}, {"type": "text", "bbox": [63, 653, 538, 686], "content": "Proof is surjective on as proven in [10]. By Proposition 7.1 there is now an with and . qed", "index": 12}]
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[{"bbox": [157, 15, 254, 32], "content": "Thus, .", "parent_index": 0, "line_index": 0}, {"bbox": [124, 31, 446, 46], "content": "Due to the definition of we have .", "parent_index": 0, "line_index": 1}, {"bbox": [63, 64, 331, 79], "content": "7.3 Construction of Arbitrary Types", "parent_index": 1, "line_index": 0}, {"bbox": [63, 88, 321, 102], "content": "Finally, we can now prove the desired proposition.", "parent_index": 2, "line_index": 0}, {"bbox": [63, 112, 196, 124], "content": "Proof Proposition 7.1", "parent_index": 3, "line_index": 0}, {"bbox": [106, 126, 538, 140], "content": "• Let and . Then there exist a Howe subgroup with", "parent_index": 4, "line_index": 0}, {"bbox": [123, 139, 537, 156], "content": "and a , such that . Since is a Howe subgroup,", "parent_index": 4, "line_index": 1}, {"bbox": [123, 155, 537, 170], "content": "we have and so by Lemma 4.1 there exist certain", "parent_index": 4, "line_index": 2}, {"bbox": [123, 169, 408, 185], "content": ", such that .", "parent_index": 4, "line_index": 3}, {"bbox": [108, 183, 537, 199], "content": "• Now let with an appropriate as in Corollary 4.2.", "parent_index": 4, "line_index": 4}, {"bbox": [122, 197, 539, 214], "content": "Because of we have", "parent_index": 4, "line_index": 5}, {"bbox": [123, 212, 254, 228], "content": ".", "parent_index": 4, "line_index": 6}, {"bbox": [109, 226, 538, 241], "content": "• We now use inductively Lemma 7.5. Let and . Construct for all", "parent_index": 4, "line_index": 7}, {"bbox": [123, 241, 538, 257], "content": "a connection and an from and by that lemma,", "parent_index": 4, "line_index": 8}, {"bbox": [119, 253, 540, 276], "content": "such that for all , , and", "parent_index": 4, "line_index": 9}, {"bbox": [123, 271, 286, 288], "content": ".", "parent_index": 4, "line_index": 10}, {"bbox": [120, 285, 535, 307], "content": "Setting we get", "parent_index": 4, "line_index": 11}, {"bbox": [122, 300, 278, 322], "content": "Finally, we define .", "parent_index": 4, "line_index": 12}, {"bbox": [120, 317, 534, 338], "content": "Now, we get for all , and . Thus,", "parent_index": 4, "line_index": 13}, {"bbox": [218, 338, 441, 397], "content": "", "parent_index": 4, "line_index": 14}, {"bbox": [122, 402, 539, 420], "content": "i.e., . qed", "parent_index": 4, "line_index": 15}, {"bbox": [63, 428, 402, 442], "content": "The proposition just proven has a further immediate consequence.", "parent_index": 5, "line_index": 0}, {"bbox": [64, 451, 311, 464], "content": "Corollary 7.6 is non-empty for all .", "parent_index": 6, "line_index": 0}, {"bbox": [61, 475, 537, 493], "content": "Proof Let be the trivial connection, i.e. for all . The type of is ,", "parent_index": 7, "line_index": 0}, {"bbox": [105, 491, 537, 506], "content": "thus minimal, i.e. we have for all . By means of Proposition 7.1", "parent_index": 7, "line_index": 1}, {"bbox": [105, 504, 537, 520], "content": "there is an with . qed", "parent_index": 7, "line_index": 2}, {"bbox": [61, 531, 533, 549], "content": "This corollary solves the problem which gauge orbit types exist for generalized connections.", "parent_index": 8, "line_index": 0}, {"bbox": [62, 555, 538, 570], "content": "Theorem 7.7 The set of all gauge orbit types on is the set of all conjugacy classes of", "parent_index": 9, "line_index": 0}, {"bbox": [147, 569, 263, 584], "content": "Howe subgroups of .", "parent_index": 9, "line_index": 1}, {"bbox": [63, 593, 172, 606], "content": "Furthermore we have", "parent_index": 10, "line_index": 0}, {"bbox": [63, 615, 539, 633], "content": "Corollary 7.8 Let be some graph. Then . In other words: is", "parent_index": 11, "line_index": 0}, {"bbox": [150, 633, 369, 646], "content": "surjective even on the generic connections.", "parent_index": 11, "line_index": 1}, {"bbox": [61, 656, 538, 672], "content": "Proof is surjective on as proven in [10]. By Proposition 7.1 there is now an with", "parent_index": 12, "line_index": 0}, {"bbox": [106, 669, 539, 687], "content": "and . qed", "parent_index": 12, "line_index": 1}]
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[]
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[{"bbox": [191, 18, 251, 32], "content": "f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})", "parent_index": 0, "subtype": "inline"}, {"bbox": [251, 37, 259, 42], "content": "\\pmb{x}", "parent_index": 0, "subtype": "inline"}, {"bbox": [308, 33, 442, 46], "content": "Z(\\mathbf{H}_{\\overline{{A}}^{\\prime}})=Z(\\{g\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "parent_index": 0, "subtype": "inline"}, {"bbox": [144, 128, 173, 137], "content": "t\\in\\mathcal T", "parent_index": 4, "subtype": "inline"}, {"bbox": [200, 126, 259, 140], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "parent_index": 4, "subtype": "inline"}, {"bbox": [449, 127, 488, 138], "content": "V^{\\prime}\\subseteq\\mathbf{G}", "parent_index": 4, "subtype": "inline"}, {"bbox": [519, 128, 538, 138], "content": "t=", "parent_index": 4, "subtype": "inline"}, {"bbox": [123, 141, 142, 154], "content": "\\left[V^{\\prime}\\right]", "parent_index": 4, "subtype": "inline"}, {"bbox": [177, 142, 208, 153], "content": "g\\in\\mathbf G", "parent_index": 4, "subtype": "inline"}, {"bbox": [266, 141, 382, 154], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\supseteq g^{-1}V^{\\prime}g=:V", "parent_index": 4, "subtype": "inline"}, {"bbox": [420, 141, 430, 151], "content": "V", "parent_index": 4, "subtype": "inline"}, {"bbox": [169, 156, 245, 169], "content": "Z(Z(V))\\,=\\,V", "parent_index": 4, "subtype": "inline"}, {"bbox": [470, 156, 537, 168], "content": "u_{0},\\dotsc,u_{k}\\in", "parent_index": 4, "subtype": "inline"}, {"bbox": [123, 171, 177, 183], "content": "Z(V)\\subseteq\\mathbf{G}", "parent_index": 4, "subtype": "inline"}, {"bbox": [236, 170, 403, 183], "content": "V=Z(Z(V))=Z(\\{u_{0},\\dots,u_{k}\\})", "parent_index": 4, "subtype": "inline"}, {"bbox": [168, 185, 272, 198], "content": "Z(\\mathbf{H}_{\\overline{{A}}})\\,=\\,Z(h_{\\overline{{A}}}(\\alpha))", "parent_index": 4, "subtype": "inline"}, {"bbox": [385, 184, 431, 196], "content": "\\alpha\\subseteq{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [180, 199, 240, 212], "content": "V\\subseteq Z(\\mathbf{H}_{\\overline{{A}}})", "parent_index": 4, "subtype": "inline"}, {"bbox": [286, 199, 539, 212], "content": "V=V\\cap Z(\\mathbf{H}_{\\overline{{A}}})=Z(\\{u_{0},\\dots,u_{k}\\})\\cap Z(h_{\\overline{{A}}}(\\alpha))=", "parent_index": 4, "subtype": "inline"}, {"bbox": [123, 213, 249, 226], "content": "Z(\\left\\{u_{0},\\dots,u_{k}\\right\\}\\cup h_{\\overline{{A}}}(\\pmb{\\alpha}))", "parent_index": 4, "subtype": "inline"}, {"bbox": [333, 226, 375, 240], "content": "\\overline{{A}}_{0}:=\\overline{{A}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [401, 229, 443, 240], "content": "\\alpha_{0}:=\\alpha", "parent_index": 4, "subtype": "inline"}, {"bbox": [123, 243, 185, 254], "content": "j=0,\\dots,k", "parent_index": 4, "subtype": "inline"}, {"bbox": [257, 241, 282, 255], "content": "\\overline{{A}}_{j+1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [324, 241, 368, 255], "content": "e_{j}\\in{\\mathcal{H}}{\\mathcal{G}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [399, 241, 413, 255], "content": "\\overline{{A}}_{j}", "parent_index": 4, "subtype": "inline"}, {"bbox": [439, 246, 453, 255], "content": "\\alpha_{j}", "parent_index": 4, "subtype": "inline"}, {"bbox": [175, 255, 278, 270], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j+1})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}}_{j})", "parent_index": 4, "subtype": "inline"}, {"bbox": [316, 256, 320, 267], "content": "i", "parent_index": 4, "subtype": "inline"}, {"bbox": [327, 256, 433, 272], "content": "h_{\\overline{{A}}_{j+1}}(\\pmb{\\alpha}_{j})=h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j})", "parent_index": 4, "subtype": "inline"}, {"bbox": [440, 257, 513, 272], "content": "h_{\\overline{{A}}_{j+1}}(e_{j})=u_{j}", "parent_index": 4, "subtype": "inline"}, {"bbox": [123, 272, 281, 288], "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\pmb{\\alpha}_{j}))", "parent_index": 4, "subtype": "inline"}, {"bbox": [162, 288, 249, 301], "content": "\\alpha_{j+1}:=\\alpha_{j}\\cup\\{e_{j}\\}", "parent_index": 4, "subtype": "inline"}, {"bbox": [285, 286, 535, 304], "content": "Z(\\mathbf{H}_{\\overline{{A}}_{j+1}})=Z(\\{u_{j}\\}\\cup h_{\\overline{{A}}_{j}}(\\alpha_{j}))=Z(h_{\\overline{{A}}_{j+1}}(\\alpha_{j+1})).", "parent_index": 4, "subtype": "inline"}, {"bbox": [216, 303, 272, 318], "content": "\\overline{{A}}^{\\prime}:=\\overline{{A}}_{k+1}", "parent_index": 4, "subtype": "inline"}, {"bbox": [189, 318, 275, 334], "content": "\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})", "parent_index": 4, "subtype": "inline"}, {"bbox": [312, 321, 317, 331], "content": "i", "parent_index": 4, "subtype": "inline"}, {"bbox": [324, 319, 406, 335], "content": "h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})", "parent_index": 4, "subtype": "inline"}, {"bbox": [432, 321, 495, 335], "content": "h_{\\overline{{A}}^{\\prime}}(e_{j})=u_{j}", "parent_index": 4, "subtype": "inline"}, {"bbox": [218, 338, 441, 397], "content": "\\begin{array}{l l l}{{Z({\\bf H}_{\\overline{{{A}}}^{\\prime}})}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\alpha_{k+1}))}}\\\\ {{}}&{{=}}&{{Z(h_{\\overline{{{A}}}^{\\prime}}(\\{e_{0},\\ldots,e_{k}\\}\\cup h_{\\overline{{{A}}}^{\\prime}}(\\alpha)))}}\\\\ {{}}&{{=}}&{{Z(\\{u_{0},\\ldots,u_{k}\\}\\cup h_{\\overline{{{A}}}}(\\alpha))}}\\\\ {{}}&{{=}}&{{V,}}\\end{array}", "parent_index": 4, "subtype": "inline"}, {"bbox": [145, 403, 239, 419], "content": "\\mathrm{Typ}({\\overline{{A}}}^{\\prime})=[V]=t", "parent_index": 4, "subtype": "inline"}, {"bbox": [151, 452, 171, 464], "content": "\\overline{{A}}_{=t}", "parent_index": 6, "subtype": "inline"}, {"bbox": [279, 453, 307, 462], "content": "t\\in\\mathcal T", "parent_index": 6, "subtype": "inline"}, {"bbox": [127, 477, 136, 488], "content": "\\overline{{A}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [291, 479, 351, 492], "content": "h_{\\overline{{A}}}(\\alpha)=e_{\\mathbf{G}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [388, 480, 420, 489], "content": "\\alpha\\in\\mathcal{P}", "parent_index": 7, "subtype": "inline"}, {"bbox": [491, 478, 500, 488], "content": "\\overline{{A}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [515, 479, 533, 491], "content": "[\\mathbf G]", "parent_index": 7, "subtype": "inline"}, {"bbox": [248, 491, 309, 505], "content": "t\\geq\\mathrm{Typ}(\\overline{{A}})", "parent_index": 7, "subtype": "inline"}, {"bbox": [348, 494, 379, 503], "content": "t\\in\\mathcal T", "parent_index": 7, "subtype": "inline"}, {"bbox": [164, 505, 200, 517], "content": "\\overline{{A}}^{\\prime}\\in\\overline{{A}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [231, 505, 292, 520], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t", "parent_index": 7, "subtype": "inline"}, {"bbox": [335, 556, 345, 566], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 9, "subtype": "inline"}, {"bbox": [248, 572, 259, 581], "content": "\\mathbf{G}", "parent_index": 9, "subtype": "inline"}, {"bbox": [172, 619, 180, 627], "content": "\\Gamma", "parent_index": 11, "subtype": "inline"}, {"bbox": [305, 617, 411, 631], "content": "\\pi_{\\Gamma}(\\overline{{A}}_{=t_{\\mathrm{max}}})\\:=\\:\\pi_{\\Gamma}(\\overline{{A}})", "parent_index": 11, "subtype": "inline"}, {"bbox": [510, 622, 523, 629], "content": "\\pi_{\\Gamma}", "parent_index": 11, "subtype": "inline"}, {"bbox": [106, 662, 119, 669], "content": "\\pi_{\\Gamma}", "parent_index": 12, "subtype": "inline"}, {"bbox": [204, 658, 214, 668], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 12, "subtype": "inline"}, {"bbox": [497, 656, 509, 668], "content": "\\overline{{A}}^{\\prime}", "parent_index": 12, "subtype": "inline"}, {"bbox": [106, 671, 185, 685], "content": "\\mathrm{Typ}(\\overline{{A}}^{\\prime})=t_{\\mathrm{max}}", "parent_index": 12, "subtype": "inline"}, {"bbox": [211, 671, 290, 686], "content": "\\pi_{\\Gamma}(\\overline{{A}}^{\\prime})=\\pi_{\\Gamma}(\\overline{{A}})", "parent_index": 12, "subtype": "inline"}]
|
[]
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# 8 Stratification of $\overline{{\mathcal{A}}}$
First we recall the general definition of a stratification [12].
Definition 8.1 A countable family $\boldsymbol{S}$ of non-empty subsets of a topological space $X$ is called stratification of $X$ iff $\boldsymbol{S}$ is a covering for $X$ and for all $U,V\in S$ we have • $U\cap V\neq\emptyset\Longrightarrow U=V$ , • $\overline{{U}}\cap V\neq\emptyset\Longrightarrow\overline{{U}}\supseteq V$ and • $\overline{{{U}}}\cap V\neq\varnothing\Longrightarrow\overline{{{V}}}\cap(U\cup V)=V$ . The elements of such a stratification $\mathcal{S}$ are called strata. A stratification is called topologically regular iff for all $U,V\in S$ $U\neq V$ and $\overline{{U}}\cap V\neq\emptyset\Longrightarrow\overline{{V}}\cap U=\emptyset$ .
Theorem 8.1 ${\cal S}:=\{\overline{{{\cal A}}}_{=t}\mid t\in{\cal T}\}$ is a topologically regular stratification of $\overline{{\mathcal{A}}}$ . Analogously, $\{(\overline{{A}}/\overline{{\mathcal{G}}})_{=t}\ |\ t\ \in\ T\}$ is a topologically regular stratification of $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ .
oof • Obviously, $_S$ is a covering of $\overline{{\mathcal{A}}}$ . • For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe subgroups of $\mathbf{G}$ , is at most countable (cf. [12]). • Moreover, from $\overline{{\mathcal{A}}}_{=t_{1}}\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$ immediately follows $\overline{{A}}_{=t_{1}}=\overline{{A}}_{=t_{2}}$ . • Due to Corollary 7.3 we have5 $\operatorname{Cl}(\overline{{A}}_{=t_{1}})=\overline{{A}}_{\leq t_{1}}$ , i.e. from $\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{1}})\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$ follows $t_{2}\leq t_{1}$ and thus $\operatorname{Cl}({\overline{{A}}}_{=t_{1}})\supseteq{\overline{{A}}}_{=t_{2}}$ . • Analogously we get $\operatorname{Cl}(\overline{{A}}_{=t_{2}})\cap(\overline{{A}}_{=t_{1}}\cup\overline{{A}}_{=t_{2}})=\overline{{A}}_{\leq t_{2}}\cap(\overline{{A}}_{=t_{1}}\cup\overline{{A}}_{=t_{2}})=\overline{{A}}_{=t_{2}}$ . • As well, from $\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{1}})\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$ and $\overline{{A}}_{=t_{1}}\neq\overline{{A}}_{=t_{2}}$ follows $t_{1}>t_{2}$ , i.e. $\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{2}})\cap$ $\overline{{A}}_{=t_{1}}=\emptyset$ . Consequently, $\boldsymbol{S}$ is a topologically regular stratification of $\overline{{\mathcal{A}}}$ . qed
For a regular stratification it would be required that each stratum carries the structure of a manifold that is compatible with the topology of the total space. In contrast to the case of the classical gauge orbit space [12], this is not fulfilled for generalized connections.
# 9 Non-complete Connections
We shall round off that paper with the proof that the set of the so-called non-complete connections is contained in a set of measure zero. This section actually stands a little bit separated from the context because it is the only section that is not only algebraic and topological, but also measure theoretical.
Definition 9.1 Let ${\overline{{A}}}\in{\overline{{A}}}$ be a connection. 1. $\overline{{A}}$ is called complete $\Longleftrightarrow\mathbf{H}_{\overline{{A}}}=\mathbf{G}$ . 2. $\overline{{A}}$ is called almost complete $\Longleftrightarrow\overline{{\mathbf{H}_{\overline{{A}}}}}=\mathbf{G}$ . 3. $\overline{{A}}$ is called non-complete $\Longleftrightarrow\overline{{\mathbf{H}_{\overline{{A}}}}}\neq\mathbf{G}$ .
Obviously, we have
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<html><body>
<h1 data-bbox="61 12 248 33">8 Stratification of $\overline{{\mathcal{A}}}$ </h1>
<p data-bbox="62 43 371 59">First we recall the general definition of a stratification [12]. </p>
<p data-bbox="61 68 537 185">Definition 8.1 A countable family $\boldsymbol{S}$ of non-empty subsets of a topological space $X$ is called stratification of $X$ iff $\boldsymbol{S}$ is a covering for $X$ and for all $U,V\in S$ we have • $U\cap V\neq\emptyset\Longrightarrow U=V$ , • $\overline{{U}}\cap V\neq\emptyset\Longrightarrow\overline{{U}}\supseteq V$ and • $\overline{{{U}}}\cap V\neq\varnothing\Longrightarrow\overline{{{V}}}\cap(U\cup V)=V$ . The elements of such a stratification $\mathcal{S}$ are called strata. A stratification is called topologically regular iff for all $U,V\in S$ $U\neq V$ and $\overline{{U}}\cap V\neq\emptyset\Longrightarrow\overline{{V}}\cap U=\emptyset$ . </p>
<p data-bbox="62 195 538 241">Theorem 8.1 ${\cal S}:=\{\overline{{{\cal A}}}_{=t}\mid t\in{\cal T}\}$ is a topologically regular stratification of $\overline{{\mathcal{A}}}$ . Analogously, $\{(\overline{{A}}/\overline{{\mathcal{G}}})_{=t}\ |\ t\ \in\ T\}$ is a topologically regular stratification of $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ . </p>
<p data-bbox="81 252 538 398">oof • Obviously, $_S$ is a covering of $\overline{{\mathcal{A}}}$ . • For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe subgroups of $\mathbf{G}$ , is at most countable (cf. [12]). • Moreover, from $\overline{{\mathcal{A}}}_{=t_{1}}\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$ immediately follows $\overline{{A}}_{=t_{1}}=\overline{{A}}_{=t_{2}}$ . • Due to Corollary 7.3 we have5 $\operatorname{Cl}(\overline{{A}}_{=t_{1}})=\overline{{A}}_{\leq t_{1}}$ , i.e. from $\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{1}})\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$ follows $t_{2}\leq t_{1}$ and thus $\operatorname{Cl}({\overline{{A}}}_{=t_{1}})\supseteq{\overline{{A}}}_{=t_{2}}$ . • Analogously we get $\operatorname{Cl}(\overline{{A}}_{=t_{2}})\cap(\overline{{A}}_{=t_{1}}\cup\overline{{A}}_{=t_{2}})=\overline{{A}}_{\leq t_{2}}\cap(\overline{{A}}_{=t_{1}}\cup\overline{{A}}_{=t_{2}})=\overline{{A}}_{=t_{2}}$ . • As well, from $\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{1}})\cap\overline{{\mathcal{A}}}_{=t_{2}}\neq\emptyset$ and $\overline{{A}}_{=t_{1}}\neq\overline{{A}}_{=t_{2}}$ follows $t_{1}>t_{2}$ , i.e. $\mathrm{Cl}(\overline{{\mathcal{A}}}_{=t_{2}})\cap$ $\overline{{A}}_{=t_{1}}=\emptyset$ . Consequently, $\boldsymbol{S}$ is a topologically regular stratification of $\overline{{\mathcal{A}}}$ . qed </p>
<p data-bbox="62 410 538 455">For a regular stratification it would be required that each stratum carries the structure of a manifold that is compatible with the topology of the total space. In contrast to the case of the classical gauge orbit space [12], this is not fulfilled for generalized connections. </p>
<h1 data-bbox="62 475 322 495">9 Non-complete Connections </h1>
<p data-bbox="62 505 538 564">We shall round off that paper with the proof that the set of the so-called non-complete connections is contained in a set of measure zero. This section actually stands a little bit separated from the context because it is the only section that is not only algebraic and topological, but also measure theoretical. </p>
<p data-bbox="61 572 405 632">Definition 9.1 Let ${\overline{{A}}}\in{\overline{{A}}}$ be a connection. 1. $\overline{{A}}$ is called complete $\Longleftrightarrow\mathbf{H}_{\overline{{A}}}=\mathbf{G}$ . 2. $\overline{{A}}$ is called almost complete $\Longleftrightarrow\overline{{\mathbf{H}_{\overline{{A}}}}}=\mathbf{G}$ . 3. $\overline{{A}}$ is called non-complete $\Longleftrightarrow\overline{{\mathbf{H}_{\overline{{A}}}}}\neq\mathbf{G}$ . </p>
<p data-bbox="62 642 163 656">Obviously, we have </p>
</body></html>
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0001008v1
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[{"type": "text", "text": "8 Stratification of $\\overline{{\\mathcal{A}}}$ ", "text_level": 1, "page_idx": 15}, {"type": "text", "text": "First we recall the general definition of a stratification [12]. ", "page_idx": 15}, {"type": "text", "text": "Definition 8.1 A countable family $\\boldsymbol{S}$ of non-empty subsets of a topological space $X$ is called stratification of $X$ iff $\\boldsymbol{S}$ is a covering for $X$ and for all $U,V\\in S$ we have • $U\\cap V\\neq\\emptyset\\Longrightarrow U=V$ , • $\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{U}}\\supseteq V$ and • $\\overline{{{U}}}\\cap V\\neq\\varnothing\\Longrightarrow\\overline{{{V}}}\\cap(U\\cup V)=V$ . The elements of such a stratification $\\mathcal{S}$ are called strata. A stratification is called topologically regular iff for all $U,V\\in S$ $U\\neq V$ and $\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{V}}\\cap U=\\emptyset$ . ", "page_idx": 15}, {"type": "text", "text": "Theorem 8.1 ${\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}$ is a topologically regular stratification of $\\overline{{\\mathcal{A}}}$ . Analogously, $\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ |\\ t\\ \\in\\ T\\}$ is a topologically regular stratification of $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ . ", "page_idx": 15}, {"type": "text", "text": "oof • Obviously, $_S$ is a covering of $\\overline{{\\mathcal{A}}}$ . • For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe subgroups of $\\mathbf{G}$ , is at most countable (cf. [12]). • Moreover, from $\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$ immediately follows $\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}$ . • Due to Corollary 7.3 we have5 $\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}$ , i.e. from $\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$ follows $t_{2}\\leq t_{1}$ and thus $\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}$ . • Analogously we get $\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}$ . • As well, from $\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset$ and $\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}$ follows $t_{1}>t_{2}$ , i.e. $\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap$ $\\overline{{A}}_{=t_{1}}=\\emptyset$ . Consequently, $\\boldsymbol{S}$ is a topologically regular stratification of $\\overline{{\\mathcal{A}}}$ . qed ", "page_idx": 15}, {"type": "text", "text": "For a regular stratification it would be required that each stratum carries the structure of a manifold that is compatible with the topology of the total space. In contrast to the case of the classical gauge orbit space [12], this is not fulfilled for generalized connections. ", "page_idx": 15}, {"type": "text", "text": "9 Non-complete Connections ", "text_level": 1, "page_idx": 15}, {"type": "text", "text": "We shall round off that paper with the proof that the set of the so-called non-complete connections is contained in a set of measure zero. This section actually stands a little bit separated from the context because it is the only section that is not only algebraic and topological, but also measure theoretical. ", "page_idx": 15}, {"type": "text", "text": "Definition 9.1 Let ${\\overline{{A}}}\\in{\\overline{{A}}}$ be a connection. 1. $\\overline{{A}}$ is called complete $\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}$ . 2. $\\overline{{A}}$ is called almost complete $\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}$ . 3. $\\overline{{A}}$ is called non-complete $\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}$ . ", "page_idx": 15}, {"type": "text", "text": "Obviously, we have ", "page_idx": 15}]
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8.1", "type": "text"}, {"bbox": [147, 199, 245, 213], "score": 0.91, "content": "{\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [246, 199, 461, 214], "score": 1.0, "content": " is a topologically regular stratification of ", "type": "text"}, {"bbox": [461, 200, 471, 210], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [472, 199, 475, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [146, 212, 539, 230], "spans": [{"bbox": [146, 212, 217, 230], "score": 1.0, "content": "Analogously, ", "type": "text"}, {"bbox": [217, 213, 318, 228], "score": 0.91, "content": "\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ |\\ t\\ \\in\\ T\\}", "type": "inline_equation", "height": 15, "width": 101}, {"bbox": [318, 212, 539, 230], "score": 1.0, "content": " is a topologically regular stratification of", "type": "text"}], "index": 11}, {"bbox": [148, 227, 176, 243], "spans": [{"bbox": [148, 228, 172, 242], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [172, 227, 176, 243], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [81, 252, 538, 398], "lines": [{"bbox": [78, 255, 286, 269], "spans": [{"bbox": [78, 255, 180, 269], "score": 1.0, "content": "oof • Obviously, ", "type": "text"}, {"bbox": [180, 258, 189, 266], "score": 0.88, "content": "_S", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [189, 255, 273, 269], "score": 1.0, "content": " is a covering of ", "type": "text"}, {"bbox": [273, 256, 283, 266], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [283, 255, 286, 269], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [104, 270, 538, 285], "spans": [{"bbox": [104, 270, 538, 285], "score": 1.0, "content": "• For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe", "type": "text"}], "index": 14}, {"bbox": [122, 284, 366, 300], "spans": [{"bbox": [122, 284, 191, 300], "score": 1.0, "content": "subgroups of ", "type": "text"}, {"bbox": [192, 286, 203, 295], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [203, 284, 366, 300], "score": 1.0, "content": ", is at most countable (cf. [12]).", "type": "text"}], "index": 15}, {"bbox": [103, 296, 465, 316], "spans": [{"bbox": [103, 296, 205, 316], "score": 1.0, "content": "• Moreover, from ", "type": "text"}, {"bbox": [205, 299, 289, 312], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [289, 296, 396, 316], "score": 1.0, "content": "immediately follows ", "type": "text"}, {"bbox": [397, 299, 461, 312], "score": 0.92, "content": "\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [462, 296, 465, 316], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [104, 312, 536, 329], "spans": [{"bbox": [104, 312, 284, 329], "score": 1.0, "content": "• Due to Corollary 7.3 we have5 ", "type": "text"}, {"bbox": [285, 313, 371, 327], "score": 0.92, "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 312, 428, 329], "score": 1.0, "content": ", i.e. from ", "type": "text"}, {"bbox": [429, 313, 536, 327], "score": 0.9, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 14, "width": 107}], "index": 17}, {"bbox": [121, 325, 338, 345], "spans": [{"bbox": [121, 325, 161, 345], "score": 1.0, "content": "follows ", "type": "text"}, {"bbox": [161, 331, 195, 340], "score": 0.91, "content": "t_{2}\\leq t_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [196, 325, 248, 345], "score": 1.0, "content": " and thus ", "type": "text"}, {"bbox": [248, 328, 333, 342], "score": 0.91, "content": "\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [333, 325, 338, 345], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [104, 340, 523, 359], "spans": [{"bbox": [104, 340, 226, 359], "score": 1.0, "content": "• Analogously we get ", "type": "text"}, {"bbox": [226, 342, 518, 356], "score": 0.88, "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 14, "width": 292}, {"bbox": [519, 340, 523, 359], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [104, 356, 536, 373], "spans": [{"bbox": [104, 356, 192, 373], "score": 1.0, "content": "• As well, from ", "type": "text"}, {"bbox": [192, 356, 293, 370], "score": 0.91, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [294, 356, 317, 373], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [318, 357, 381, 370], "score": 0.93, "content": "\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [381, 356, 421, 373], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [421, 358, 456, 370], "score": 0.88, "content": "t_{1}>t_{2}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [456, 356, 481, 373], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [481, 356, 536, 370], "score": 0.91, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap", "type": "inline_equation", "height": 14, "width": 55}], "index": 20}, {"bbox": [123, 369, 174, 387], "spans": [{"bbox": [123, 371, 169, 385], "score": 0.93, "content": "\\overline{{A}}_{=t_{1}}=\\emptyset", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [169, 369, 174, 387], "score": 1.0, "content": ".", "type": "text"}], "index": 21}, {"bbox": [105, 384, 539, 401], "spans": [{"bbox": [105, 384, 181, 399], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [181, 388, 190, 396], "score": 0.89, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [190, 384, 405, 399], "score": 1.0, "content": " is a topologically regular stratification of ", "type": "text"}, {"bbox": [406, 386, 416, 397], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [416, 384, 420, 399], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 385, 539, 401], "score": 1.0, "content": "qed", "type": "text"}], "index": 22}], "index": 17.5}, {"type": "text", "bbox": [62, 410, 538, 455], "lines": [{"bbox": [62, 414, 537, 428], "spans": [{"bbox": [62, 414, 537, 428], "score": 1.0, "content": "For a regular stratification it would be required that each stratum carries the structure of a", "type": "text"}], "index": 23}, {"bbox": [62, 428, 539, 443], "spans": [{"bbox": [62, 428, 539, 443], "score": 1.0, "content": "manifold that is compatible with the topology of the total space. In contrast to the case of", "type": "text"}], "index": 24}, {"bbox": [63, 442, 485, 456], "spans": [{"bbox": [63, 442, 485, 456], "score": 1.0, "content": "the classical gauge orbit space [12], this is not fulfilled for generalized connections.", "type": "text"}], "index": 25}], "index": 24}, {"type": "title", "bbox": [62, 475, 322, 495], "lines": [{"bbox": [63, 478, 321, 496], "spans": [{"bbox": [63, 480, 74, 493], "score": 1.0, "content": "9", "type": "text"}, {"bbox": [90, 478, 321, 496], "score": 1.0, "content": "Non-complete Connections", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [62, 505, 538, 564], "lines": [{"bbox": [62, 507, 537, 523], "spans": [{"bbox": [62, 507, 537, 523], "score": 1.0, "content": "We shall round off that paper with the proof that the set of the so-called non-complete", "type": "text"}], "index": 27}, {"bbox": [62, 523, 537, 536], "spans": [{"bbox": [62, 523, 537, 536], "score": 1.0, "content": "connections is contained in a set of measure zero. This section actually stands a little bit", "type": "text"}], "index": 28}, {"bbox": [62, 538, 538, 552], "spans": [{"bbox": [62, 538, 538, 552], "score": 1.0, "content": "separated from the context because it is the only section that is not only algebraic and", "type": "text"}], "index": 29}, {"bbox": [63, 552, 274, 566], "spans": [{"bbox": [63, 552, 274, 566], "score": 1.0, "content": "topological, but also measure theoretical.", "type": "text"}], "index": 30}], "index": 28.5}, {"type": "text", "bbox": [61, 572, 405, 632], "lines": [{"bbox": [62, 575, 295, 590], "spans": [{"bbox": [62, 575, 174, 590], "score": 1.0, "content": "Definition 9.1 Let ", "type": "text"}, {"bbox": [174, 576, 208, 587], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [208, 575, 295, 590], "score": 1.0, "content": " be a connection.", "type": "text"}], "index": 31}, {"bbox": [152, 589, 362, 605], "spans": [{"bbox": [152, 589, 173, 605], "score": 1.0, "content": "1.", "type": "text"}, {"bbox": [173, 591, 182, 601], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [183, 589, 287, 605], "score": 1.0, "content": " is called complete ", "type": "text"}, {"bbox": [288, 592, 357, 604], "score": 0.87, "content": "\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [358, 589, 362, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [150, 603, 404, 619], "spans": [{"bbox": [150, 603, 173, 619], "score": 1.0, "content": "2.", "type": "text"}, {"bbox": [173, 605, 182, 616], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [183, 604, 330, 619], "score": 1.0, "content": " is called almost complete ", "type": "text"}, {"bbox": [330, 605, 401, 619], "score": 0.9, "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 71}, {"bbox": [401, 604, 404, 619], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [151, 618, 388, 635], "spans": [{"bbox": [151, 618, 173, 635], "score": 1.0, "content": "3.", "type": "text"}, {"bbox": [173, 619, 182, 630], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [183, 618, 313, 635], "score": 1.0, "content": " is called non-complete ", "type": "text"}, {"bbox": [313, 619, 383, 633], "score": 0.9, "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 70}, {"bbox": [384, 618, 388, 635], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 32.5}, {"type": "text", "bbox": [62, 642, 163, 656], "lines": [{"bbox": [63, 644, 162, 657], "spans": [{"bbox": [63, 644, 162, 657], "score": 1.0, "content": "Obviously, we have", "type": "text"}], "index": 35}], "index": 35}], "layout_bboxes": [], "page_idx": 15, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [76, 662, 309, 676], "lines": [{"bbox": [80, 663, 309, 678], "spans": [{"bbox": [80, 666, 107, 677], "score": 0.63, "content": "\\mathrm{Cl}(U)", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [108, 663, 234, 678], "score": 1.0, "content": " denotes again the closure of ", "type": "text"}, {"bbox": [235, 667, 243, 674], "score": 0.89, "content": "U", "type": "inline_equation", "height": 7, "width": 8}, {"bbox": [243, 663, 297, 678], "score": 1.0, "content": ", here w.r.t. 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1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [152, 142, 447, 158], "spans": [{"bbox": [152, 142, 343, 158], "score": 1.0, "content": "The elements of such a stratification ", "type": "text"}, {"bbox": [344, 143, 353, 154], "score": 0.81, "content": "\\mathcal{S}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [353, 142, 447, 158], "score": 1.0, "content": " are called strata.", "type": "text"}], "index": 7}, {"bbox": [153, 157, 498, 172], "spans": [{"bbox": [153, 157, 453, 172], "score": 1.0, "content": "A stratification is called topologically regular iff for all ", "type": "text"}, {"bbox": [453, 159, 498, 171], "score": 0.91, "content": "U,V\\in S", "type": "inline_equation", "height": 12, "width": 45}], "index": 8}, {"bbox": [245, 171, 445, 186], "spans": [{"bbox": [245, 171, 281, 185], "score": 0.91, "content": "U\\neq V", "type": "inline_equation", "height": 14, "width": 36}, {"bbox": [281, 172, 306, 186], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [306, 171, 441, 185], "score": 0.94, "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{V}}\\cap U=\\emptyset", "type": "inline_equation", "height": 14, "width": 135}, {"bbox": [442, 172, 445, 186], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 5.5, "bbox_fs": [61, 70, 537, 186]}, {"type": "text", "bbox": [62, 195, 538, 241], "lines": [{"bbox": [62, 199, 475, 214], "spans": [{"bbox": [62, 199, 147, 214], "score": 1.0, "content": "Theorem 8.1", "type": "text"}, {"bbox": [147, 199, 245, 213], "score": 0.91, "content": "{\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}", "type": "inline_equation", "height": 14, "width": 98}, {"bbox": [246, 199, 461, 214], "score": 1.0, "content": " is a topologically regular stratification of ", "type": "text"}, {"bbox": [461, 200, 471, 210], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [472, 199, 475, 214], "score": 1.0, "content": ".", "type": "text"}], "index": 10}, {"bbox": [146, 212, 539, 230], "spans": [{"bbox": [146, 212, 217, 230], "score": 1.0, "content": "Analogously, ", "type": "text"}, {"bbox": [217, 213, 318, 228], "score": 0.91, "content": "\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ |\\ t\\ \\in\\ T\\}", "type": "inline_equation", "height": 15, "width": 101}, {"bbox": [318, 212, 539, 230], "score": 1.0, "content": " is a topologically regular stratification of", "type": "text"}], "index": 11}, {"bbox": [148, 227, 176, 243], "spans": [{"bbox": [148, 228, 172, 242], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [172, 227, 176, 243], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [62, 199, 539, 243]}, {"type": "text", "bbox": [81, 252, 538, 398], "lines": [{"bbox": [78, 255, 286, 269], "spans": [{"bbox": [78, 255, 180, 269], "score": 1.0, "content": "oof • Obviously, ", "type": "text"}, {"bbox": [180, 258, 189, 266], "score": 0.88, "content": "_S", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [189, 255, 273, 269], "score": 1.0, "content": " is a covering of ", "type": "text"}, {"bbox": [273, 256, 283, 266], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [283, 255, 286, 269], "score": 1.0, "content": ".", "type": "text"}], "index": 13}, {"bbox": [104, 270, 538, 285], "spans": [{"bbox": [104, 270, 538, 285], "score": 1.0, "content": "• For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe", "type": "text"}], "index": 14}, {"bbox": [122, 284, 366, 300], "spans": [{"bbox": [122, 284, 191, 300], "score": 1.0, "content": "subgroups of ", "type": "text"}, {"bbox": [192, 286, 203, 295], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [203, 284, 366, 300], "score": 1.0, "content": ", is at most countable (cf. [12]).", "type": "text"}], "index": 15}, {"bbox": [103, 296, 465, 316], "spans": [{"bbox": [103, 296, 205, 316], "score": 1.0, "content": "• Moreover, from ", "type": "text"}, {"bbox": [205, 299, 289, 312], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [289, 296, 396, 316], "score": 1.0, "content": "immediately follows ", "type": "text"}, {"bbox": [397, 299, 461, 312], "score": 0.92, "content": "\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 13, "width": 64}, {"bbox": [462, 296, 465, 316], "score": 1.0, "content": ".", "type": "text"}], "index": 16}, {"bbox": [104, 312, 536, 329], "spans": [{"bbox": [104, 312, 284, 329], "score": 1.0, "content": "• Due to Corollary 7.3 we have5 ", "type": "text"}, {"bbox": [285, 313, 371, 327], "score": 0.92, "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 312, 428, 329], "score": 1.0, "content": ", i.e. from ", "type": "text"}, {"bbox": [429, 313, 536, 327], "score": 0.9, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 14, "width": 107}], "index": 17}, {"bbox": [121, 325, 338, 345], "spans": [{"bbox": [121, 325, 161, 345], "score": 1.0, "content": "follows ", "type": "text"}, {"bbox": [161, 331, 195, 340], "score": 0.91, "content": "t_{2}\\leq t_{1}", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [196, 325, 248, 345], "score": 1.0, "content": " and thus ", "type": "text"}, {"bbox": [248, 328, 333, 342], "score": 0.91, "content": "\\operatorname{Cl}({\\overline{{A}}}_{=t_{1}})\\supseteq{\\overline{{A}}}_{=t_{2}}", "type": "inline_equation", "height": 14, "width": 85}, {"bbox": [333, 325, 338, 345], "score": 1.0, "content": ".", "type": "text"}], "index": 18}, {"bbox": [104, 340, 523, 359], "spans": [{"bbox": [104, 340, 226, 359], "score": 1.0, "content": "• Analogously we get ", "type": "text"}, {"bbox": [226, 342, 518, 356], "score": 0.88, "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{2}})\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{\\leq t_{2}}\\cap(\\overline{{A}}_{=t_{1}}\\cup\\overline{{A}}_{=t_{2}})=\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 14, "width": 292}, {"bbox": [519, 340, 523, 359], "score": 1.0, "content": ".", "type": "text"}], "index": 19}, {"bbox": [104, 356, 536, 373], "spans": [{"bbox": [104, 356, 192, 373], "score": 1.0, "content": "• As well, from ", "type": "text"}, {"bbox": [192, 356, 293, 370], "score": 0.91, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{1}})\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "type": "inline_equation", "height": 14, "width": 101}, {"bbox": [294, 356, 317, 373], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [318, 357, 381, 370], "score": 0.93, "content": "\\overline{{A}}_{=t_{1}}\\neq\\overline{{A}}_{=t_{2}}", "type": "inline_equation", "height": 13, "width": 63}, {"bbox": [381, 356, 421, 373], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [421, 358, 456, 370], "score": 0.88, "content": "t_{1}>t_{2}", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [456, 356, 481, 373], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [481, 356, 536, 370], "score": 0.91, "content": "\\mathrm{Cl}(\\overline{{\\mathcal{A}}}_{=t_{2}})\\cap", "type": "inline_equation", "height": 14, "width": 55}], "index": 20}, {"bbox": [123, 369, 174, 387], "spans": [{"bbox": [123, 371, 169, 385], "score": 0.93, "content": "\\overline{{A}}_{=t_{1}}=\\emptyset", "type": "inline_equation", "height": 14, "width": 46}, {"bbox": [169, 369, 174, 387], "score": 1.0, "content": ".", "type": "text"}], "index": 21}, {"bbox": [105, 384, 539, 401], "spans": [{"bbox": [105, 384, 181, 399], "score": 1.0, "content": "Consequently, ", "type": "text"}, {"bbox": [181, 388, 190, 396], "score": 0.89, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [190, 384, 405, 399], "score": 1.0, "content": " is a topologically regular stratification of ", "type": "text"}, {"bbox": [406, 386, 416, 397], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [416, 384, 420, 399], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 385, 539, 401], "score": 1.0, "content": "qed", "type": "text"}], "index": 22}], "index": 17.5, "bbox_fs": [78, 255, 539, 401]}, {"type": "text", "bbox": [62, 410, 538, 455], "lines": [{"bbox": [62, 414, 537, 428], "spans": [{"bbox": [62, 414, 537, 428], "score": 1.0, "content": "For a regular stratification it would be required that each stratum carries the structure of a", "type": "text"}], "index": 23}, {"bbox": [62, 428, 539, 443], "spans": [{"bbox": [62, 428, 539, 443], "score": 1.0, "content": "manifold that is compatible with the topology of the total space. In contrast to the case of", "type": "text"}], "index": 24}, {"bbox": [63, 442, 485, 456], "spans": [{"bbox": [63, 442, 485, 456], "score": 1.0, "content": "the classical gauge orbit space [12], this is not fulfilled for generalized connections.", "type": "text"}], "index": 25}], "index": 24, "bbox_fs": [62, 414, 539, 456]}, {"type": "title", "bbox": [62, 475, 322, 495], "lines": [{"bbox": [63, 478, 321, 496], "spans": [{"bbox": [63, 480, 74, 493], "score": 1.0, "content": "9", "type": "text"}, {"bbox": [90, 478, 321, 496], "score": 1.0, "content": "Non-complete Connections", "type": "text"}], "index": 26}], "index": 26}, {"type": "text", "bbox": [62, 505, 538, 564], "lines": [{"bbox": [62, 507, 537, 523], "spans": [{"bbox": [62, 507, 537, 523], "score": 1.0, "content": "We shall round off that paper with the proof that the set of the so-called non-complete", "type": "text"}], "index": 27}, {"bbox": [62, 523, 537, 536], "spans": [{"bbox": [62, 523, 537, 536], "score": 1.0, "content": "connections is contained in a set of measure zero. This section actually stands a little bit", "type": "text"}], "index": 28}, {"bbox": [62, 538, 538, 552], "spans": [{"bbox": [62, 538, 538, 552], "score": 1.0, "content": "separated from the context because it is the only section that is not only algebraic and", "type": "text"}], "index": 29}, {"bbox": [63, 552, 274, 566], "spans": [{"bbox": [63, 552, 274, 566], "score": 1.0, "content": "topological, but also measure theoretical.", "type": "text"}], "index": 30}], "index": 28.5, "bbox_fs": [62, 507, 538, 566]}, {"type": "text", "bbox": [61, 572, 405, 632], "lines": [{"bbox": [62, 575, 295, 590], "spans": [{"bbox": [62, 575, 174, 590], "score": 1.0, "content": "Definition 9.1 Let ", "type": "text"}, {"bbox": [174, 576, 208, 587], "score": 0.93, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [208, 575, 295, 590], "score": 1.0, "content": " be a connection.", "type": "text"}], "index": 31}, {"bbox": [152, 589, 362, 605], "spans": [{"bbox": [152, 589, 173, 605], "score": 1.0, "content": "1.", "type": "text"}, {"bbox": [173, 591, 182, 601], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [183, 589, 287, 605], "score": 1.0, "content": " is called complete ", "type": "text"}, {"bbox": [288, 592, 357, 604], "score": 0.87, "content": "\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}", "type": "inline_equation", "height": 12, "width": 69}, {"bbox": [358, 589, 362, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [150, 603, 404, 619], "spans": [{"bbox": [150, 603, 173, 619], "score": 1.0, "content": "2.", "type": "text"}, {"bbox": [173, 605, 182, 616], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [183, 604, 330, 619], "score": 1.0, "content": " is called almost complete ", "type": "text"}, {"bbox": [330, 605, 401, 619], "score": 0.9, "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 71}, {"bbox": [401, 604, 404, 619], "score": 1.0, "content": ".", "type": "text"}], "index": 33}, {"bbox": [151, 618, 388, 635], "spans": [{"bbox": [151, 618, 173, 635], "score": 1.0, "content": "3.", "type": "text"}, {"bbox": [173, 619, 182, 630], "score": 0.89, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [183, 618, 313, 635], "score": 1.0, "content": " is called non-complete ", "type": "text"}, {"bbox": [313, 619, 383, 633], "score": 0.9, "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}", "type": "inline_equation", "height": 14, "width": 70}, {"bbox": [384, 618, 388, 635], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 32.5, "bbox_fs": [62, 575, 404, 635]}, {"type": "text", "bbox": [62, 642, 163, 656], "lines": [{"bbox": [63, 644, 162, 657], "spans": [{"bbox": [63, 644, 162, 657], "score": 1.0, "content": "Obviously, we have", "type": "text"}], "index": 35}], "index": 35, "bbox_fs": [63, 644, 162, 657]}]}
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[{"type": "title", "bbox": [61, 12, 248, 33], "content": "8 Stratification of", "index": 0}, {"type": "text", "bbox": [62, 43, 371, 59], "content": "First we recall the general definition of a stratification [12].", "index": 1}, {"type": "text", "bbox": [61, 68, 537, 185], "content": "Definition 8.1 A countable family of non-empty subsets of a topological space is called stratification of iff is a covering for and for all we have • , • and • . The elements of such a stratification are called strata. A stratification is called topologically regular iff for all and .", "index": 2}, {"type": "text", "bbox": [62, 195, 538, 241], "content": "Theorem 8.1 is a topologically regular stratification of . Analogously, is a topologically regular stratification of .", "index": 3}, {"type": "text", "bbox": [81, 252, 538, 398], "content": "oof • Obviously, is a covering of . • For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe subgroups of , is at most countable (cf. [12]). • Moreover, from immediately follows . • Due to Corollary 7.3 we have5 , i.e. from follows and thus . • Analogously we get . • As well, from and follows , i.e. . Consequently, is a topologically regular stratification of . qed", "index": 4}, {"type": "text", "bbox": [62, 410, 538, 455], "content": "For a regular stratification it would be required that each stratum carries the structure of a manifold that is compatible with the topology of the total space. In contrast to the case of the classical gauge orbit space [12], this is not fulfilled for generalized connections.", "index": 5}, {"type": "title", "bbox": [62, 475, 322, 495], "content": "9 Non-complete Connections", "index": 6}, {"type": "text", "bbox": [62, 505, 538, 564], "content": "We shall round off that paper with the proof that the set of the so-called non-complete connections is contained in a set of measure zero. This section actually stands a little bit separated from the context because it is the only section that is not only algebraic and topological, but also measure theoretical.", "index": 7}, {"type": "text", "bbox": [61, 572, 405, 632], "content": "Definition 9.1 Let be a connection. 1. is called complete . 2. is called almost complete . 3. is called non-complete .", "index": 8}, {"type": "text", "bbox": [62, 642, 163, 656], "content": "Obviously, we have", "index": 9}]
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[{"bbox": [64, 16, 245, 32], "content": "8 Stratification of", "parent_index": 0, "line_index": 0}, {"bbox": [62, 46, 366, 61], "content": "First we recall the general definition of a stratification [12].", "parent_index": 1, "line_index": 0}, {"bbox": [61, 70, 537, 87], "content": "Definition 8.1 A countable family of non-empty subsets of a topological space is called", "parent_index": 2, "line_index": 0}, {"bbox": [152, 85, 534, 100], "content": "stratification of iff is a covering for and for all we have", "parent_index": 2, "line_index": 1}, {"bbox": [151, 100, 291, 114], "content": "• ,", "parent_index": 2, "line_index": 2}, {"bbox": [152, 111, 311, 129], "content": "• and", "parent_index": 2, "line_index": 3}, {"bbox": [152, 127, 345, 143], "content": "• .", "parent_index": 2, "line_index": 4}, {"bbox": [152, 142, 447, 158], "content": "The elements of such a stratification are called strata.", "parent_index": 2, "line_index": 5}, {"bbox": [153, 157, 498, 172], "content": "A stratification is called topologically regular iff for all", "parent_index": 2, "line_index": 6}, {"bbox": [245, 171, 445, 186], "content": "and .", "parent_index": 2, "line_index": 7}, {"bbox": [62, 199, 475, 214], "content": "Theorem 8.1 is a topologically regular stratification of .", "parent_index": 3, "line_index": 0}, {"bbox": [146, 212, 539, 230], "content": "Analogously, is a topologically regular stratification of", "parent_index": 3, "line_index": 1}, {"bbox": [148, 227, 176, 243], "content": ".", "parent_index": 3, "line_index": 2}, {"bbox": [78, 255, 286, 269], "content": "oof • Obviously, is a covering of .", "parent_index": 4, "line_index": 0}, {"bbox": [104, 270, 538, 285], "content": "• For a compact Lie group the set of all types, i.e. all conjugacy classes of Howe", "parent_index": 4, "line_index": 1}, {"bbox": [122, 284, 366, 300], "content": "subgroups of , is at most countable (cf. [12]).", "parent_index": 4, "line_index": 2}, {"bbox": [103, 296, 465, 316], "content": "• Moreover, from immediately follows .", "parent_index": 4, "line_index": 3}, {"bbox": [104, 312, 536, 329], "content": "• Due to Corollary 7.3 we have5 , i.e. from", "parent_index": 4, "line_index": 4}, {"bbox": [121, 325, 338, 345], "content": "follows and thus .", "parent_index": 4, "line_index": 5}, {"bbox": [104, 340, 523, 359], "content": "• Analogously we get .", "parent_index": 4, "line_index": 6}, {"bbox": [104, 356, 536, 373], "content": "• As well, from and follows , i.e.", "parent_index": 4, "line_index": 7}, {"bbox": [123, 369, 174, 387], "content": ".", "parent_index": 4, "line_index": 8}, {"bbox": [105, 384, 539, 401], "content": "Consequently, is a topologically regular stratification of . qed", "parent_index": 4, "line_index": 9}, {"bbox": [62, 414, 537, 428], "content": "For a regular stratification it would be required that each stratum carries the structure of a", "parent_index": 5, "line_index": 0}, {"bbox": [62, 428, 539, 443], "content": "manifold that is compatible with the topology of the total space. In contrast to the case of", "parent_index": 5, "line_index": 1}, {"bbox": [63, 442, 485, 456], "content": "the classical gauge orbit space [12], this is not fulfilled for generalized connections.", "parent_index": 5, "line_index": 2}, {"bbox": [63, 478, 321, 496], "content": "9 Non-complete Connections", "parent_index": 6, "line_index": 0}, {"bbox": [62, 507, 537, 523], "content": "We shall round off that paper with the proof that the set of the so-called non-complete", "parent_index": 7, "line_index": 0}, {"bbox": [62, 523, 537, 536], "content": "connections is contained in a set of measure zero. This section actually stands a little bit", "parent_index": 7, "line_index": 1}, {"bbox": [62, 538, 538, 552], "content": "separated from the context because it is the only section that is not only algebraic and", "parent_index": 7, "line_index": 2}, {"bbox": [63, 552, 274, 566], "content": "topological, but also measure theoretical.", "parent_index": 7, "line_index": 3}, {"bbox": [62, 575, 295, 590], "content": "Definition 9.1 Let be a connection.", "parent_index": 8, "line_index": 0}, {"bbox": [152, 589, 362, 605], "content": "1. is called complete .", "parent_index": 8, "line_index": 1}, {"bbox": [150, 603, 404, 619], "content": "2. is called almost complete .", "parent_index": 8, "line_index": 2}, {"bbox": [151, 618, 388, 635], "content": "3. is called non-complete .", "parent_index": 8, "line_index": 3}, {"bbox": [63, 644, 162, 657], "content": "Obviously, we have", "parent_index": 9, "line_index": 0}]
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[]
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"content": "\\overline{{{U}}}\\cap V\\neq\\varnothing\\Longrightarrow\\overline{{{V}}}\\cap(U\\cup V)=V", "parent_index": 2, "subtype": "inline"}, {"bbox": [344, 143, 353, 154], "content": "\\mathcal{S}", "parent_index": 2, "subtype": "inline"}, {"bbox": [453, 159, 498, 171], "content": "U,V\\in S", "parent_index": 2, "subtype": "inline"}, {"bbox": [245, 171, 281, 185], "content": "U\\neq V", "parent_index": 2, "subtype": "inline"}, {"bbox": [306, 171, 441, 185], "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{V}}\\cap U=\\emptyset", "parent_index": 2, "subtype": "inline"}, {"bbox": [147, 199, 245, 213], "content": "{\\cal S}:=\\{\\overline{{{\\cal A}}}_{=t}\\mid t\\in{\\cal T}\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [461, 200, 471, 210], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [217, 213, 318, 228], "content": "\\{(\\overline{{A}}/\\overline{{\\mathcal{G}}})_{=t}\\ |\\ t\\ \\in\\ T\\}", "parent_index": 3, "subtype": "inline"}, {"bbox": [148, 228, 172, 242], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "parent_index": 3, "subtype": "inline"}, {"bbox": [180, 258, 189, 266], "content": "_S", "parent_index": 4, "subtype": "inline"}, {"bbox": [273, 256, 283, 266], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [192, 286, 203, 295], "content": "\\mathbf{G}", "parent_index": 4, "subtype": "inline"}, {"bbox": [205, 299, 289, 312], "content": "\\overline{{\\mathcal{A}}}_{=t_{1}}\\cap\\overline{{\\mathcal{A}}}_{=t_{2}}\\neq\\emptyset", "parent_index": 4, "subtype": "inline"}, {"bbox": [397, 299, 461, 312], "content": "\\overline{{A}}_{=t_{1}}=\\overline{{A}}_{=t_{2}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [285, 313, 371, 327], "content": "\\operatorname{Cl}(\\overline{{A}}_{=t_{1}})=\\overline{{A}}_{\\leq t_{1}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [429, 313, 536, 327], "content": 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"\\Longleftrightarrow\\mathbf{H}_{\\overline{{A}}}=\\mathbf{G}", "parent_index": 8, "subtype": "inline"}, {"bbox": [173, 605, 182, 616], "content": "\\overline{{A}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [330, 605, 401, 619], "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}=\\mathbf{G}", "parent_index": 8, "subtype": "inline"}, {"bbox": [173, 619, 182, 630], "content": "\\overline{{A}}", "parent_index": 8, "subtype": "inline"}, {"bbox": [313, 619, 383, 633], "content": "\\Longleftrightarrow\\overline{{\\mathbf{H}_{\\overline{{A}}}}}\\neq\\mathbf{G}", "parent_index": 8, "subtype": "inline"}]
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Lemma 9.1 If ${\overline{{A}}}\in{\overline{{A}}}$ is complete (almost complete, non-complete), so $\overline{{A}}\circ\overline{{g}}$ is complete (almost complete, non-complete) for all ${\overline{{g}}}\in{\overline{{g}}}$ .
Thus, the total information about the completeness of a connection is already contained in its gauge orbit. Now, to the main assertion of this section.
Proposition 9.2 Let $N:=\{\overline{{A}}\in\overline{{A}}\mid\overline{{A}}$ non-complete}. Then $N$ is contained in a set of $\mu_{0}$ -measure zero whereas $\mu_{0}$ is the induced Haar measure on $\overline{{\mathcal{A}}}$ . [2, 6, 10]
Since $N$ is gauge invariant, we have
Corollary 9.3 Let $[N]:=\{[\overline{{A}}]\in\overline{{A}}/\overline{{\mathcal{G}}}\mid\overline{{A}}$ non-complete}. Then $[N]$ is contained in a set of $\mu_{0}$ -measure zero.
For the proof of the proposition we still need the follow
Lemma 9.4 Let $U\subseteq\mathbf{G}$ be measurable with $\mu_{\mathrm{Haar}}(U)\,>\,0$ and $N_{U}\;:=\;\{\overline{{{A}}}\,\in\,\overline{{{A}}}\;\vert\;\mathbf{H}_{\overline{{{A}}}}\subseteq$ $\mathbf{G}\setminus U\}$ . Then $N_{U}$ is contained in a set of $\mu_{0}$ -measure zero.
Proof • Let $k\ \in\ \mathbb{N}$ and $\Gamma_{k}$ be some connected graph with one vertex $m$ and $k$ edges $\alpha_{1},\ldots,\alpha_{k}\in\mathcal{H}\mathcal{G}$ .6 Furthermore, let $\pi_{k}:\overline{{\mathcal{A}}}\;\;\longrightarrow$ $\mathbf{G}^{k}$ . $\begin{array}{r}{A\;\;\longmapsto\;\;(h_{\overline{{A}}}(\alpha_{1}),\dots,h_{\overline{{A}}}(\alpha_{k}))}\end{array}$ • Denote now by $N_{k,U}:=\pi_{k}^{-1}((\mathbf G\backslash U)^{k})$ the set of all connections whose holonomies on $\Gamma_{k}$ are not contained in $U$ . Per constructionem we have $N_{U}\subseteq N_{k,U}$ . • Since the characteristic function $\chi_{N_{k,U}}$ for $N_{k,U}$ is obviously a cylindrical function, we get
$$
\begin{array}{r c l}{\mu_{0}(N_{k,U})}&{=}&{\displaystyle\int_{\overline{{\mathcal{A}}}}\chi_{N_{k,U}}\;d\mu_{0}\ =\ \int_{\overline{{\mathcal{A}}}}\pi_{k}^{*}(\chi_{(\mathbf{G}\backslash U)^{k}})\;d\mu_{0}}\\ &{=}&{\displaystyle\int_{\mathbf{G}^{k}}\chi_{(\mathbf{G}\backslash U)^{k}}\;d\mu_{\mathrm{Haar}}^{k}\ =\ \left[\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)\right]^{k}.}\end{array}
$$
• From $N_{U}\subseteq N_{k,U}$ for all $k$ follows $N_{U}\subseteq\bigcap_{k}N_{k,U}$ . But, $\mu_{0}(\bigcap_{k}N_{k,U})\leq\mu_{0}(N_{k,U})=$ $\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)^{k}$ for all $k$ , i.e. $\mu_{0}\bigl(\bigcap_{k}N_{k,U}\bigr)=0$ , because $\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)=1\!-\!\mu_{\mathrm{Haar}}(U)<$ 1. qed
# Proof Proposition 9.2
• Let $(\epsilon_{k})_{k\in\mathbb{N}}$ be some null sequence. Furthermore, let $\{U_{k,i}\}_{i}$ be for each $k$ a finite covering of $\mathbf{G}$ by open sets $U_{k,i}$ whose respective diameters are smaller than $\epsilon_{k}$ . Now define $N^{\prime}:=\cup_{k}\mathopen{}\left(\cup_{i}\,N_{U_{k,i}}\right)$ . Since $U_{k,i}$ is open and $\mathbf{G}$ is compact, $U_{k,i}$ is measureable with $\mu_{\mathrm{Haar}}(U_{k,i})\,>\,0$ . Due to Lemma 9.4 we have $N_{U_{k,i}}\ \subseteq\ N_{U_{k,i}}^{*}$ with $\mu_{0}(N_{U_{k,i}}^{*})\,=\,0$ for all $k,i$ ; thus $N^{\prime}\subseteq N^{*}:=\cup_{k}\bigl(\cup_{i}N_{U_{k,i}}^{*}\bigr)$ with $\mu_{0}(N^{\ast})=0$ . We are left to show $N\subseteq N^{\prime}$ . Let ${\overline{{A}}}\in N$ . Then there is an open $U\subseteq\mathbf{G}$ with $\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}\setminus U$ . Now let $m\in U$ . Then $\epsilon:=\mathrm{dist}(m,\partial U)>0$ . Choose $k$ such that $\epsilon_{k}<\epsilon$ . Then choose a $U_{k,i}$ with $m\in U_{k,i}$ . We get for all $x\in U_{k,i}$ : $d(x,m)\leq$ diam $U_{k,i}<\epsilon_{k}<\epsilon$ , i.e. $x\in U$ . Consequently, $U_{k,i}\subseteq U$ and thus $\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}\setminus U_{k,i}$ , i.e. ${\overline{{A}}}\in N^{\prime}$ . qed
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<html><body>
<p data-bbox="63 12 538 44">Lemma 9.1 If ${\overline{{A}}}\in{\overline{{A}}}$ is complete (almost complete, non-complete), so $\overline{{A}}\circ\overline{{g}}$ is complete (almost complete, non-complete) for all ${\overline{{g}}}\in{\overline{{g}}}$ . </p>
<p data-bbox="62 50 538 79">Thus, the total information about the completeness of a connection is already contained in its gauge orbit. Now, to the main assertion of this section. </p>
<p data-bbox="63 83 538 115">Proposition 9.2 Let $N:=\{\overline{{A}}\in\overline{{A}}\mid\overline{{A}}$ non-complete}. Then $N$ is contained in a set of $\mu_{0}$ -measure zero whereas $\mu_{0}$ is the induced Haar measure on $\overline{{\mathcal{A}}}$ . [2, 6, 10] </p>
<p data-bbox="62 120 248 136">Since $N$ is gauge invariant, we have </p>
<p data-bbox="63 139 538 171">Corollary 9.3 Let $[N]:=\{[\overline{{A}}]\in\overline{{A}}/\overline{{\mathcal{G}}}\mid\overline{{A}}$ non-complete}. Then $[N]$ is contained in a set of $\mu_{0}$ -measure zero. </p>
<p data-bbox="62 176 348 191">For the proof of the proposition we still need the follow </p>
<p data-bbox="63 196 538 241">Lemma 9.4 Let $U\subseteq\mathbf{G}$ be measurable with $\mu_{\mathrm{Haar}}(U)\,>\,0$ and $N_{U}\;:=\;\{\overline{{{A}}}\,\in\,\overline{{{A}}}\;\vert\;\mathbf{H}_{\overline{{{A}}}}\subseteq$ $\mathbf{G}\setminus U\}$ . Then $N_{U}$ is contained in a set of $\mu_{0}$ -measure zero. </p>
<p data-bbox="65 248 538 351">Proof • Let $k\ \in\ \mathbb{N}$ and $\Gamma_{k}$ be some connected graph with one vertex $m$ and $k$ edges $\alpha_{1},\ldots,\alpha_{k}\in\mathcal{H}\mathcal{G}$ .6 Furthermore, let $\pi_{k}:\overline{{\mathcal{A}}}\;\;\longrightarrow$ $\mathbf{G}^{k}$ . $\begin{array}{r}{A\;\;\longmapsto\;\;(h_{\overline{{A}}}(\alpha_{1}),\dots,h_{\overline{{A}}}(\alpha_{k}))}\end{array}$ • Denote now by $N_{k,U}:=\pi_{k}^{-1}((\mathbf G\backslash U)^{k})$ the set of all connections whose holonomies on $\Gamma_{k}$ are not contained in $U$ . Per constructionem we have $N_{U}\subseteq N_{k,U}$ . • Since the characteristic function $\chi_{N_{k,U}}$ for $N_{k,U}$ is obviously a cylindrical function, we get </p>
<div class="equation" data-bbox="191 355 466 411">$$
\begin{array}{r c l}{\mu_{0}(N_{k,U})}&{=}&{\displaystyle\int_{\overline{{\mathcal{A}}}}\chi_{N_{k,U}}\;d\mu_{0}\ =\ \int_{\overline{{\mathcal{A}}}}\pi_{k}^{*}(\chi_{(\mathbf{G}\backslash U)^{k}})\;d\mu_{0}}\\ &{=}&{\displaystyle\int_{\mathbf{G}^{k}}\chi_{(\mathbf{G}\backslash U)^{k}}\;d\mu_{\mathrm{Haar}}^{k}\ =\ \left[\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)\right]^{k}.}\end{array}
$$</div>
<p data-bbox="106 411 537 455">• From $N_{U}\subseteq N_{k,U}$ for all $k$ follows $N_{U}\subseteq\bigcap_{k}N_{k,U}$ . But, $\mu_{0}(\bigcap_{k}N_{k,U})\leq\mu_{0}(N_{k,U})=$ $\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)^{k}$ for all $k$ , i.e. $\mu_{0}\bigl(\bigcap_{k}N_{k,U}\bigr)=0$ , because $\mu_{\mathrm{Haar}}(\mathbf{G}\backslash U)=1\!-\!\mu_{\mathrm{Haar}}(U)<$ 1. qed </p>
<h1 data-bbox="61 463 198 477">Proof Proposition 9.2 </h1>
<p data-bbox="106 478 539 646">• Let $(\epsilon_{k})_{k\in\mathbb{N}}$ be some null sequence. Furthermore, let $\{U_{k,i}\}_{i}$ be for each $k$ a finite covering of $\mathbf{G}$ by open sets $U_{k,i}$ whose respective diameters are smaller than $\epsilon_{k}$ . Now define $N^{\prime}:=\cup_{k}\mathopen{}\left(\cup_{i}\,N_{U_{k,i}}\right)$ . Since $U_{k,i}$ is open and $\mathbf{G}$ is compact, $U_{k,i}$ is measureable with $\mu_{\mathrm{Haar}}(U_{k,i})\,>\,0$ . Due to Lemma 9.4 we have $N_{U_{k,i}}\ \subseteq\ N_{U_{k,i}}^{*}$ with $\mu_{0}(N_{U_{k,i}}^{*})\,=\,0$ for all $k,i$ ; thus $N^{\prime}\subseteq N^{*}:=\cup_{k}\bigl(\cup_{i}N_{U_{k,i}}^{*}\bigr)$ with $\mu_{0}(N^{\ast})=0$ . We are left to show $N\subseteq N^{\prime}$ . Let ${\overline{{A}}}\in N$ . Then there is an open $U\subseteq\mathbf{G}$ with $\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}\setminus U$ . Now let $m\in U$ . Then $\epsilon:=\mathrm{dist}(m,\partial U)>0$ . Choose $k$ such that $\epsilon_{k}<\epsilon$ . Then choose a $U_{k,i}$ with $m\in U_{k,i}$ . We get for all $x\in U_{k,i}$ : $d(x,m)\leq$ diam $U_{k,i}<\epsilon_{k}<\epsilon$ , i.e. $x\in U$ . Consequently, $U_{k,i}\subseteq U$ and thus $\mathbf{H}_{\overline{{A}}}\subseteq\mathbf{G}\setminus U_{k,i}$ , i.e. ${\overline{{A}}}\in N^{\prime}$ . qed </p>
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0001008v1
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[{"type": "text", "text": "Lemma 9.1 If ${\\overline{{A}}}\\in{\\overline{{A}}}$ is complete (almost complete, non-complete), so $\\overline{{A}}\\circ\\overline{{g}}$ is complete (almost complete, non-complete) for all ${\\overline{{g}}}\\in{\\overline{{g}}}$ . ", "page_idx": 16}, {"type": "text", "text": "Thus, the total information about the completeness of a connection is already contained in its gauge orbit. Now, to the main assertion of this section. ", "page_idx": 16}, {"type": "text", "text": "Proposition 9.2 Let $N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}$ non-complete}. Then $N$ is contained in a set of $\\mu_{0}$ -measure zero whereas $\\mu_{0}$ is the induced Haar measure on $\\overline{{\\mathcal{A}}}$ . [2, 6, 10] ", "page_idx": 16}, {"type": "text", "text": "Since $N$ is gauge invariant, we have ", "page_idx": 16}, {"type": "text", "text": "Corollary 9.3 Let $[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}$ non-complete}. Then $[N]$ is contained in a set of $\\mu_{0}$ -measure zero. ", "page_idx": 16}, {"type": "text", "text": "For the proof of the proposition we still need the follow ", "page_idx": 16}, {"type": "text", "text": "Lemma 9.4 Let $U\\subseteq\\mathbf{G}$ be measurable with $\\mu_{\\mathrm{Haar}}(U)\\,>\\,0$ and $N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq$ $\\mathbf{G}\\setminus U\\}$ . Then $N_{U}$ is contained in a set of $\\mu_{0}$ -measure zero. ", "page_idx": 16}, {"type": "text", "text": "Proof • Let $k\\ \\in\\ \\mathbb{N}$ and $\\Gamma_{k}$ be some connected graph with one vertex $m$ and $k$ edges $\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}$ .6 Furthermore, let $\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow$ $\\mathbf{G}^{k}$ . $\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}$ • Denote now by $N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})$ the set of all connections whose holonomies on $\\Gamma_{k}$ are not contained in $U$ . Per constructionem we have $N_{U}\\subseteq N_{k,U}$ . • Since the characteristic function $\\chi_{N_{k,U}}$ for $N_{k,U}$ is obviously a cylindrical function, we get ", "page_idx": 16}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{*}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}\n$$", "text_format": "latex", "page_idx": 16}, {"type": "text", "text": "• From $N_{U}\\subseteq N_{k,U}$ for all $k$ follows $N_{U}\\subseteq\\bigcap_{k}N_{k,U}$ . But, $\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=$ $\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}$ for all $k$ , i.e. $\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0$ , because $\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<$ 1. qed ", "page_idx": 16}, {"type": "text", "text": "Proof Proposition 9.2 ", "text_level": 1, "page_idx": 16}, {"type": "text", "text": "• Let $(\\epsilon_{k})_{k\\in\\mathbb{N}}$ be some null sequence. Furthermore, let $\\{U_{k,i}\\}_{i}$ be for each $k$ a finite covering of $\\mathbf{G}$ by open sets $U_{k,i}$ whose respective diameters are smaller than $\\epsilon_{k}$ . Now define $N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)$ . Since $U_{k,i}$ is open and $\\mathbf{G}$ is compact, $U_{k,i}$ is measureable with $\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0$ . Due to Lemma 9.4 we have $N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{*}$ with $\\mu_{0}(N_{U_{k,i}}^{*})\\,=\\,0$ for all $k,i$ ; thus $N^{\\prime}\\subseteq N^{*}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{*}\\bigr)$ with $\\mu_{0}(N^{\\ast})=0$ . We are left to show $N\\subseteq N^{\\prime}$ . Let ${\\overline{{A}}}\\in N$ . Then there is an open $U\\subseteq\\mathbf{G}$ with $\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U$ . Now let $m\\in U$ . Then $\\epsilon:=\\mathrm{dist}(m,\\partial U)>0$ . Choose $k$ such that $\\epsilon_{k}<\\epsilon$ . Then choose a $U_{k,i}$ with $m\\in U_{k,i}$ . We get for all $x\\in U_{k,i}$ : $d(x,m)\\leq$ diam $U_{k,i}<\\epsilon_{k}<\\epsilon$ , i.e. $x\\in U$ . Consequently, $U_{k,i}\\subseteq U$ and thus $\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}$ , i.e. ${\\overline{{A}}}\\in N^{\\prime}$ . qed ", "page_idx": 16}]
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{"preproc_blocks": [{"type": "text", "bbox": [63, 12, 538, 44], "lines": [{"bbox": [61, 15, 537, 33], "spans": [{"bbox": [61, 15, 151, 33], "score": 1.0, "content": "Lemma 9.1 If ", "type": "text"}, {"bbox": [151, 17, 186, 28], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [187, 15, 446, 33], "score": 1.0, "content": " is complete (almost complete, non-complete), so ", "type": "text"}, {"bbox": [446, 17, 474, 30], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{g}}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [474, 15, 537, 33], "score": 1.0, "content": " is complete", "type": "text"}], "index": 0}, {"bbox": [138, 31, 377, 46], "spans": [{"bbox": [138, 31, 344, 46], "score": 1.0, "content": "(almost complete, non-complete) for all ", "type": "text"}, {"bbox": [344, 32, 373, 45], "score": 0.94, "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [374, 31, 377, 46], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [62, 50, 538, 79], "lines": [{"bbox": [63, 52, 537, 66], "spans": [{"bbox": [63, 52, 537, 66], "score": 1.0, "content": "Thus, the total information about the completeness of a connection is already contained in", "type": "text"}], "index": 2}, {"bbox": [62, 67, 364, 82], "spans": [{"bbox": [62, 67, 364, 82], "score": 1.0, "content": "its gauge orbit. Now, to the main assertion of this section.", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "text", "bbox": [63, 83, 538, 115], "lines": [{"bbox": [61, 86, 539, 103], "spans": [{"bbox": [61, 86, 185, 103], "score": 1.0, "content": "Proposition 9.2 Let ", "type": "text"}, {"bbox": [186, 88, 282, 101], "score": 0.9, "content": "N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [283, 86, 399, 103], "score": 1.0, "content": " non-complete}. Then ", "type": "text"}, {"bbox": [400, 90, 411, 98], "score": 0.91, "content": "N", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [411, 86, 539, 103], "score": 1.0, "content": " is contained in a set of", "type": "text"}], "index": 4}, {"bbox": [164, 102, 537, 117], "spans": [{"bbox": [164, 107, 175, 115], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [176, 102, 293, 117], "score": 1.0, "content": "-measure zero whereas ", "type": "text"}, {"bbox": [293, 107, 305, 115], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [306, 102, 474, 117], "score": 1.0, "content": " is the induced Haar measure on ", "type": "text"}, {"bbox": [474, 102, 484, 113], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [484, 102, 537, 117], "score": 1.0, "content": ". [2, 6, 10]", "type": "text"}], "index": 5}], "index": 4.5}, {"type": "text", "bbox": [62, 120, 248, 136], "lines": [{"bbox": [63, 124, 246, 136], "spans": [{"bbox": [63, 124, 93, 136], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [93, 125, 104, 133], "score": 0.91, "content": "N", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [105, 124, 246, 136], "score": 1.0, "content": " is gauge invariant, we have", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [63, 139, 538, 171], "lines": [{"bbox": [64, 143, 539, 159], "spans": [{"bbox": [64, 143, 171, 159], "score": 1.0, "content": "Corollary 9.3 Let ", "type": "text"}, {"bbox": [172, 144, 285, 157], "score": 0.87, "content": "[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [286, 143, 400, 159], "score": 1.0, "content": " non-complete}. Then ", "type": "text"}, {"bbox": [401, 145, 418, 157], "score": 0.92, "content": "[N]", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [418, 143, 539, 159], "score": 1.0, "content": " is contained in a set of", "type": "text"}], "index": 7}, {"bbox": [151, 160, 237, 173], "spans": [{"bbox": [151, 163, 163, 171], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [163, 160, 237, 173], "score": 1.0, "content": "-measure zero.", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [62, 176, 348, 191], "lines": [{"bbox": [63, 180, 349, 192], "spans": [{"bbox": [63, 180, 349, 192], "score": 1.0, "content": "For the proof of the proposition we still need the follow", "type": "text"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [63, 196, 538, 241], "lines": [{"bbox": [61, 197, 538, 216], "spans": [{"bbox": [61, 197, 160, 216], "score": 1.0, "content": "Lemma 9.4 Let ", "type": "text"}, {"bbox": [160, 201, 200, 212], "score": 0.93, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [200, 197, 311, 216], "score": 1.0, "content": " be measurable with ", "type": "text"}, {"bbox": [311, 201, 381, 213], "score": 0.93, "content": "\\mu_{\\mathrm{Haar}}(U)\\,>\\,0", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [381, 197, 409, 216], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [409, 200, 538, 213], "score": 0.91, "content": "N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq", "type": "inline_equation", "height": 13, "width": 129}], "index": 10}, {"bbox": [138, 213, 181, 229], "spans": [{"bbox": [138, 215, 176, 228], "score": 0.93, "content": "\\mathbf{G}\\setminus U\\}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [176, 213, 181, 229], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [138, 228, 396, 243], "spans": [{"bbox": [138, 228, 169, 243], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 230, 186, 241], "score": 0.91, "content": "N_{U}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [186, 228, 309, 243], "score": 1.0, "content": " is contained in a set of ", "type": "text"}, {"bbox": [309, 234, 321, 241], "score": 0.89, "content": "\\mu_{0}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [321, 228, 396, 243], "score": 1.0, "content": "-measure zero.", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [65, 248, 538, 351], "lines": [{"bbox": [63, 251, 537, 266], "spans": [{"bbox": [63, 251, 145, 266], "score": 1.0, "content": "Proof • Let ", "type": "text"}, {"bbox": [145, 252, 181, 262], "score": 0.87, "content": "k\\ \\in\\ \\mathbb{N}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [182, 251, 210, 266], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 253, 223, 263], "score": 0.9, "content": "\\Gamma_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [223, 251, 456, 266], "score": 1.0, "content": " be some connected graph with one vertex ", "type": "text"}, {"bbox": [456, 256, 467, 262], "score": 0.8, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [467, 251, 496, 266], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [496, 253, 503, 262], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [504, 251, 537, 266], "score": 1.0, "content": " edges", "type": "text"}], "index": 13}, {"bbox": [123, 264, 467, 281], "spans": [{"bbox": [123, 267, 209, 279], "score": 0.88, "content": "\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [209, 264, 325, 281], "score": 1.0, "content": ".6 Furthermore, let", "type": "text"}, {"bbox": [325, 266, 393, 278], "score": 0.28, "content": "\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [447, 266, 463, 276], "score": 0.8, "content": "\\mathbf{G}^{k}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [463, 264, 467, 279], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [356, 281, 510, 294], "spans": [{"bbox": [356, 281, 510, 294], "score": 0.54, "content": "\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}", "type": "inline_equation", "height": 13, "width": 154}], "index": 15}, {"bbox": [105, 295, 538, 309], "spans": [{"bbox": [105, 295, 201, 309], "score": 1.0, "content": "• Denote now by ", "type": "text"}, {"bbox": [202, 296, 314, 309], "score": 0.93, "content": "N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [315, 295, 538, 309], "score": 1.0, "content": " the set of all connections whose holonomies", "type": "text"}], "index": 16}, {"bbox": [120, 308, 489, 326], "spans": [{"bbox": [120, 308, 139, 326], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [139, 312, 152, 322], "score": 0.91, "content": "\\Gamma_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [152, 308, 262, 326], "score": 1.0, "content": " are not contained in ", "type": "text"}, {"bbox": [263, 312, 272, 321], "score": 0.88, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [272, 308, 427, 326], "score": 1.0, "content": ". Per constructionem we have ", "type": "text"}, {"bbox": [428, 312, 484, 324], "score": 0.94, "content": "N_{U}\\subseteq N_{k,U}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [484, 308, 489, 326], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [110, 322, 538, 342], "spans": [{"bbox": [110, 322, 289, 342], "score": 1.0, "content": "• Since the characteristic function ", "type": "text"}, {"bbox": [289, 329, 316, 340], "score": 0.91, "content": "\\chi_{N_{k,U}}", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [316, 322, 336, 342], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 326, 360, 338], "score": 0.93, "content": "N_{k,U}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [360, 322, 538, 342], "score": 1.0, "content": " is obviously a cylindrical function,", "type": "text"}], "index": 18}, {"bbox": [122, 340, 158, 353], "spans": [{"bbox": [122, 340, 158, 353], "score": 1.0, "content": "we get", "type": "text"}], "index": 19}], "index": 16}, {"type": "interline_equation", "bbox": [191, 355, 466, 411], "lines": [{"bbox": [191, 355, 466, 411], "spans": [{"bbox": [191, 355, 466, 411], "score": 0.92, "content": "\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{*}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [106, 411, 537, 455], "lines": [{"bbox": [106, 413, 538, 429], "spans": [{"bbox": [106, 413, 154, 429], "score": 1.0, "content": "• From ", "type": "text"}, {"bbox": [154, 416, 210, 428], "score": 0.93, "content": "N_{U}\\subseteq N_{k,U}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [210, 414, 248, 429], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [248, 416, 254, 425], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [255, 414, 296, 429], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [297, 416, 368, 428], "score": 0.92, "content": "N_{U}\\subseteq\\bigcap_{k}N_{k,U}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [369, 414, 403, 429], "score": 1.0, "content": ". But, ", "type": "text"}, {"bbox": [403, 415, 538, 428], "score": 0.92, "content": "\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=", "type": "inline_equation", "height": 13, "width": 135}], "index": 21}, {"bbox": [123, 427, 537, 444], "spans": [{"bbox": [123, 429, 189, 442], "score": 0.92, "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [190, 427, 224, 444], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [224, 430, 231, 439], "score": 0.85, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 427, 256, 444], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [256, 429, 338, 442], "score": 0.93, "content": "\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0", "type": "inline_equation", "height": 13, "width": 82}, {"bbox": [339, 427, 387, 444], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [387, 429, 537, 442], "score": 0.91, "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<", "type": "inline_equation", "height": 13, "width": 150}], "index": 22}, {"bbox": [123, 443, 539, 457], "spans": [{"bbox": [123, 444, 133, 455], "score": 1.0, "content": "1.", "type": "text"}, {"bbox": [513, 443, 539, 457], "score": 1.0, "content": "qed", "type": "text"}], "index": 23}], "index": 22}, {"type": "title", "bbox": [61, 463, 198, 477], "lines": [{"bbox": [63, 465, 197, 478], "spans": [{"bbox": [63, 465, 197, 478], "score": 1.0, "content": "Proof Proposition 9.2", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [106, 478, 539, 646], "lines": [{"bbox": [107, 479, 538, 494], "spans": [{"bbox": [107, 479, 144, 494], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [144, 480, 180, 493], "score": 0.92, "content": "(\\epsilon_{k})_{k\\in\\mathbb{N}}", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [180, 479, 392, 494], "score": 1.0, "content": " be some null sequence. Furthermore, let ", "type": "text"}, {"bbox": [392, 481, 426, 493], "score": 0.94, "content": "\\{U_{k,i}\\}_{i}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [426, 479, 489, 494], "score": 1.0, "content": " be for each ", "type": "text"}, {"bbox": [489, 482, 496, 490], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [497, 479, 538, 494], "score": 1.0, "content": " a finite", "type": "text"}], "index": 25}, {"bbox": [123, 495, 537, 509], "spans": [{"bbox": [123, 495, 183, 509], "score": 1.0, "content": "covering of ", "type": "text"}, {"bbox": [183, 496, 194, 505], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [194, 495, 267, 509], "score": 1.0, "content": " by open sets ", "type": "text"}, {"bbox": [267, 496, 286, 508], "score": 0.92, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [286, 495, 522, 509], "score": 1.0, "content": " whose respective diameters are smaller than ", "type": "text"}, {"bbox": [523, 499, 533, 506], "score": 0.88, "content": "\\epsilon_{k}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [533, 495, 537, 509], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [120, 506, 285, 527], "spans": [{"bbox": [120, 506, 183, 527], "score": 1.0, "content": "Now define ", "type": "text"}, {"bbox": [183, 509, 280, 527], "score": 0.95, "content": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)", "type": "inline_equation", "height": 18, "width": 97}, {"bbox": [281, 506, 285, 527], "score": 1.0, "content": ".", "type": "text"}], "index": 27}, {"bbox": [115, 526, 537, 541], "spans": [{"bbox": [115, 526, 154, 541], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [155, 528, 173, 540], "score": 0.93, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [173, 526, 243, 541], "score": 1.0, "content": " is open and ", "type": "text"}, {"bbox": [244, 528, 254, 537], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 526, 323, 541], "score": 1.0, "content": " is compact, ", "type": "text"}, {"bbox": [323, 528, 342, 540], "score": 0.92, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [342, 526, 454, 541], "score": 1.0, "content": " is measureable with ", "type": "text"}, {"bbox": [454, 528, 533, 540], "score": 0.96, "content": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [533, 526, 537, 541], "score": 1.0, "content": ".", "type": "text"}], "index": 28}, {"bbox": [120, 538, 539, 559], "spans": [{"bbox": [120, 538, 273, 559], "score": 1.0, "content": "Due to Lemma 9.4 we have ", "type": "text"}, {"bbox": [273, 543, 343, 557], "score": 0.93, "content": "N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{*}", "type": "inline_equation", "height": 14, "width": 70}, {"bbox": [344, 538, 376, 559], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [376, 542, 448, 557], "score": 0.94, "content": "\\mu_{0}(N_{U_{k,i}}^{*})\\,=\\,0", "type": "inline_equation", "height": 15, "width": 72}, {"bbox": [449, 538, 489, 559], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [489, 542, 505, 554], "score": 0.9, "content": "k,i", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [506, 538, 539, 559], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [123, 555, 345, 578], "spans": [{"bbox": [123, 558, 251, 576], "score": 0.94, "content": "N^{\\prime}\\subseteq N^{*}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{*}\\bigr)", "type": "inline_equation", "height": 18, "width": 128}, {"bbox": [252, 555, 281, 578], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [281, 560, 340, 573], "score": 0.94, "content": "\\mu_{0}(N^{\\ast})=0", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [340, 555, 345, 578], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [118, 575, 271, 588], "spans": [{"bbox": [118, 575, 226, 588], "score": 1.0, "content": "We are left to show ", "type": "text"}, {"bbox": [227, 577, 267, 588], "score": 0.92, "content": "N\\subseteq N^{\\prime}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [267, 575, 271, 588], "score": 1.0, "content": ".", "type": "text"}], "index": 31}, {"bbox": [120, 587, 439, 605], "spans": [{"bbox": [120, 587, 144, 605], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [144, 590, 178, 601], "score": 0.93, "content": "{\\overline{{A}}}\\in N", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [178, 587, 303, 605], "score": 1.0, "content": ". Then there is an open ", "type": "text"}, {"bbox": [303, 591, 339, 602], "score": 0.92, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [339, 587, 369, 605], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [369, 591, 434, 603], "score": 0.93, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [434, 587, 439, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [122, 603, 537, 619], "spans": [{"bbox": [122, 603, 167, 619], "score": 1.0, "content": "Now let ", "type": "text"}, {"bbox": [168, 606, 204, 615], "score": 0.93, "content": "m\\in U", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [204, 603, 244, 619], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [244, 605, 353, 617], "score": 0.92, "content": "\\epsilon:=\\mathrm{dist}(m,\\partial U)>0", "type": "inline_equation", "height": 12, "width": 109}, {"bbox": [353, 603, 403, 619], "score": 1.0, "content": ". Choose ", "type": "text"}, {"bbox": [403, 606, 410, 615], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [411, 603, 467, 619], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [467, 607, 500, 616], "score": 0.84, "content": "\\epsilon_{k}<\\epsilon", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [500, 603, 537, 619], "score": 1.0, "content": ". Then", "type": "text"}], "index": 33}, {"bbox": [122, 617, 537, 634], "spans": [{"bbox": [122, 617, 168, 634], "score": 1.0, "content": "choose a ", "type": "text"}, {"bbox": [169, 621, 187, 632], "score": 0.94, "content": "U_{k,i}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [187, 617, 215, 634], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [216, 620, 259, 632], "score": 0.93, "content": "m\\in U_{k,i}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [259, 617, 338, 634], "score": 1.0, "content": ". We get for all ", "type": "text"}, {"bbox": [338, 621, 378, 632], "score": 0.92, "content": "x\\in U_{k,i}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [378, 617, 385, 634], "score": 1.0, "content": ": ", "type": "text"}, {"bbox": [385, 619, 436, 632], "score": 0.9, "content": "d(x,m)\\leq", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [436, 617, 468, 634], "score": 1.0, "content": "diam ", "type": "text"}, {"bbox": [468, 620, 533, 632], "score": 0.89, "content": "U_{k,i}<\\epsilon_{k}<\\epsilon", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [533, 617, 537, 634], "score": 1.0, "content": ",", "type": "text"}], "index": 34}, {"bbox": [121, 632, 537, 648], "spans": [{"bbox": [121, 632, 142, 648], "score": 1.0, "content": "i.e. ", "type": "text"}, {"bbox": [143, 635, 174, 644], "score": 0.92, "content": "x\\in U", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [174, 632, 255, 648], "score": 1.0, "content": ". Consequently, ", "type": "text"}, {"bbox": [256, 635, 299, 646], "score": 0.94, "content": "U_{k,i}\\subseteq U", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [300, 632, 349, 648], "score": 1.0, "content": " and thus ", "type": "text"}, {"bbox": [350, 634, 420, 647], "score": 0.94, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [421, 632, 446, 648], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [447, 633, 484, 644], "score": 0.92, "content": "{\\overline{{A}}}\\in N^{\\prime}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [484, 632, 491, 648], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 633, 537, 647], "score": 1.0, "content": "qed", "type": "text"}], "index": 35}], "index": 30}], "layout_bboxes": [], "page_idx": 16, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [191, 355, 466, 411], "lines": [{"bbox": [191, 355, 466, 411], "spans": [{"bbox": [191, 355, 466, 411], "score": 0.92, "content": "\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{*}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20}], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "17", "type": "text"}]}]}, {"type": "discarded", "bbox": [62, 650, 537, 687], "lines": [{"bbox": [74, 649, 536, 668], "spans": [{"bbox": [74, 649, 235, 668], "score": 1.0, "content": "6Such a graph does indeed exist for ", "type": "text"}, {"bbox": [235, 654, 283, 663], "score": 0.9, "content": "\\dim M\\geq2", "type": "inline_equation", "height": 9, "width": 48}, {"bbox": [284, 649, 372, 668], "score": 1.0, "content": ". For instance, take ", "type": "text"}, {"bbox": [372, 654, 378, 661], "score": 0.9, "content": "k", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [378, 649, 411, 668], "score": 1.0, "content": " circles ", "type": "text"}, {"bbox": [411, 654, 423, 663], "score": 0.91, "content": "K_{i}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [423, 649, 494, 668], "score": 1.0, "content": " with centers in ", "type": "text"}, {"bbox": [495, 653, 536, 665], "score": 0.93, "content": "\\textstyle({\\frac{1}{i}},0,\\dots)", "type": "inline_equation", "height": 12, "width": 41}]}, {"bbox": [62, 664, 537, 678], "spans": [{"bbox": [62, 664, 104, 678], "score": 1.0, "content": "and radii", "type": "text"}, {"bbox": [104, 664, 111, 677], "score": 0.88, "content": "\\textstyle{\\frac{1}{i}}", "type": "inline_equation", "height": 13, "width": 7}, {"bbox": [111, 664, 338, 678], "score": 1.0, "content": ". By means of an appropriate chart mapping around ", "type": "text"}, {"bbox": [338, 669, 347, 673], "score": 0.89, "content": "m", "type": "inline_equation", "height": 4, "width": 9}, {"bbox": [347, 664, 537, 678], "score": 1.0, "content": " these circles define a graph with the desired", "type": "text"}]}, {"bbox": [62, 677, 110, 689], "spans": [{"bbox": [62, 677, 110, 689], "score": 1.0, "content": "properties.", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [63, 12, 538, 44], "lines": [{"bbox": [61, 15, 537, 33], "spans": [{"bbox": [61, 15, 151, 33], "score": 1.0, "content": "Lemma 9.1 If ", "type": "text"}, {"bbox": [151, 17, 186, 28], "score": 0.94, "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [187, 15, 446, 33], "score": 1.0, "content": " is complete (almost complete, non-complete), so ", "type": "text"}, {"bbox": [446, 17, 474, 30], "score": 0.94, "content": "\\overline{{A}}\\circ\\overline{{g}}", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [474, 15, 537, 33], "score": 1.0, "content": " is complete", "type": "text"}], "index": 0}, {"bbox": [138, 31, 377, 46], "spans": [{"bbox": [138, 31, 344, 46], "score": 1.0, "content": "(almost complete, non-complete) for all ", "type": "text"}, {"bbox": [344, 32, 373, 45], "score": 0.94, "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [374, 31, 377, 46], "score": 1.0, "content": ".", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [61, 15, 537, 46]}, {"type": "text", "bbox": [62, 50, 538, 79], "lines": [{"bbox": [63, 52, 537, 66], "spans": [{"bbox": [63, 52, 537, 66], "score": 1.0, "content": "Thus, the total information about the completeness of a connection is already contained in", "type": "text"}], "index": 2}, {"bbox": [62, 67, 364, 82], "spans": [{"bbox": [62, 67, 364, 82], "score": 1.0, "content": "its gauge orbit. Now, to the main assertion of this section.", "type": "text"}], "index": 3}], "index": 2.5, "bbox_fs": [62, 52, 537, 82]}, {"type": "text", "bbox": [63, 83, 538, 115], "lines": [{"bbox": [61, 86, 539, 103], "spans": [{"bbox": [61, 86, 185, 103], "score": 1.0, "content": "Proposition 9.2 Let ", "type": "text"}, {"bbox": [186, 88, 282, 101], "score": 0.9, "content": "N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [283, 86, 399, 103], "score": 1.0, "content": " non-complete}. Then ", "type": "text"}, {"bbox": [400, 90, 411, 98], "score": 0.91, "content": "N", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [411, 86, 539, 103], "score": 1.0, "content": " is contained in a set of", "type": "text"}], "index": 4}, {"bbox": [164, 102, 537, 117], "spans": [{"bbox": [164, 107, 175, 115], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [176, 102, 293, 117], "score": 1.0, "content": "-measure zero whereas ", "type": "text"}, {"bbox": [293, 107, 305, 115], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [306, 102, 474, 117], "score": 1.0, "content": " is the induced Haar measure on ", "type": "text"}, {"bbox": [474, 102, 484, 113], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [484, 102, 537, 117], "score": 1.0, "content": ". [2, 6, 10]", "type": "text"}], "index": 5}], "index": 4.5, "bbox_fs": [61, 86, 539, 117]}, {"type": "text", "bbox": [62, 120, 248, 136], "lines": [{"bbox": [63, 124, 246, 136], "spans": [{"bbox": [63, 124, 93, 136], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [93, 125, 104, 133], "score": 0.91, "content": "N", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [105, 124, 246, 136], "score": 1.0, "content": " is gauge invariant, we have", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [63, 124, 246, 136]}, {"type": "text", "bbox": [63, 139, 538, 171], "lines": [{"bbox": [64, 143, 539, 159], "spans": [{"bbox": [64, 143, 171, 159], "score": 1.0, "content": "Corollary 9.3 Let ", "type": "text"}, {"bbox": [172, 144, 285, 157], "score": 0.87, "content": "[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}", "type": "inline_equation", "height": 13, "width": 113}, {"bbox": [286, 143, 400, 159], "score": 1.0, "content": " non-complete}. Then ", "type": "text"}, {"bbox": [401, 145, 418, 157], "score": 0.92, "content": "[N]", "type": "inline_equation", "height": 12, "width": 17}, {"bbox": [418, 143, 539, 159], "score": 1.0, "content": " is contained in a set of", "type": "text"}], "index": 7}, {"bbox": [151, 160, 237, 173], "spans": [{"bbox": [151, 163, 163, 171], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [163, 160, 237, 173], "score": 1.0, "content": "-measure zero.", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [64, 143, 539, 173]}, {"type": "text", "bbox": [62, 176, 348, 191], "lines": [{"bbox": [63, 180, 349, 192], "spans": [{"bbox": [63, 180, 349, 192], "score": 1.0, "content": "For the proof of the proposition we still need the follow", "type": "text"}], "index": 9}], "index": 9, "bbox_fs": [63, 180, 349, 192]}, {"type": "text", "bbox": [63, 196, 538, 241], "lines": [{"bbox": [61, 197, 538, 216], "spans": [{"bbox": [61, 197, 160, 216], "score": 1.0, "content": "Lemma 9.4 Let ", "type": "text"}, {"bbox": [160, 201, 200, 212], "score": 0.93, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [200, 197, 311, 216], "score": 1.0, "content": " be measurable with ", "type": "text"}, {"bbox": [311, 201, 381, 213], "score": 0.93, "content": "\\mu_{\\mathrm{Haar}}(U)\\,>\\,0", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [381, 197, 409, 216], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [409, 200, 538, 213], "score": 0.91, "content": "N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq", "type": "inline_equation", "height": 13, "width": 129}], "index": 10}, {"bbox": [138, 213, 181, 229], "spans": [{"bbox": [138, 215, 176, 228], "score": 0.93, "content": "\\mathbf{G}\\setminus U\\}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [176, 213, 181, 229], "score": 1.0, "content": ".", "type": "text"}], "index": 11}, {"bbox": [138, 228, 396, 243], "spans": [{"bbox": [138, 228, 169, 243], "score": 1.0, "content": "Then ", "type": "text"}, {"bbox": [169, 230, 186, 241], "score": 0.91, "content": "N_{U}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [186, 228, 309, 243], "score": 1.0, "content": " is contained in a set of ", "type": "text"}, {"bbox": [309, 234, 321, 241], "score": 0.89, "content": "\\mu_{0}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [321, 228, 396, 243], "score": 1.0, "content": "-measure zero.", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [61, 197, 538, 243]}, {"type": "text", "bbox": [65, 248, 538, 351], "lines": [{"bbox": [63, 251, 537, 266], "spans": [{"bbox": [63, 251, 145, 266], "score": 1.0, "content": "Proof • Let ", "type": "text"}, {"bbox": [145, 252, 181, 262], "score": 0.87, "content": "k\\ \\in\\ \\mathbb{N}", "type": "inline_equation", "height": 10, "width": 36}, {"bbox": [182, 251, 210, 266], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [210, 253, 223, 263], "score": 0.9, "content": "\\Gamma_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [223, 251, 456, 266], "score": 1.0, "content": " be some connected graph with one vertex ", "type": "text"}, {"bbox": [456, 256, 467, 262], "score": 0.8, "content": "m", "type": "inline_equation", "height": 6, "width": 11}, {"bbox": [467, 251, 496, 266], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [496, 253, 503, 262], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [504, 251, 537, 266], "score": 1.0, "content": " edges", "type": "text"}], "index": 13}, {"bbox": [123, 264, 467, 281], "spans": [{"bbox": [123, 267, 209, 279], "score": 0.88, "content": "\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}", "type": "inline_equation", "height": 12, "width": 86}, {"bbox": [209, 264, 325, 281], "score": 1.0, "content": ".6 Furthermore, let", "type": "text"}, {"bbox": [325, 266, 393, 278], "score": 0.28, "content": "\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [447, 266, 463, 276], "score": 0.8, "content": "\\mathbf{G}^{k}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [463, 264, 467, 279], "score": 1.0, "content": ".", "type": "text"}], "index": 14}, {"bbox": [356, 281, 510, 294], "spans": [{"bbox": [356, 281, 510, 294], "score": 0.54, "content": "\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}", "type": "inline_equation", "height": 13, "width": 154}], "index": 15}, {"bbox": [105, 295, 538, 309], "spans": [{"bbox": [105, 295, 201, 309], "score": 1.0, "content": "• Denote now by ", "type": "text"}, {"bbox": [202, 296, 314, 309], "score": 0.93, "content": "N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})", "type": "inline_equation", "height": 13, "width": 112}, {"bbox": [315, 295, 538, 309], "score": 1.0, "content": " the set of all connections whose holonomies", "type": "text"}], "index": 16}, {"bbox": [120, 308, 489, 326], "spans": [{"bbox": [120, 308, 139, 326], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [139, 312, 152, 322], "score": 0.91, "content": "\\Gamma_{k}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [152, 308, 262, 326], "score": 1.0, "content": " are not contained in ", "type": "text"}, {"bbox": [263, 312, 272, 321], "score": 0.88, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [272, 308, 427, 326], "score": 1.0, "content": ". Per constructionem we have ", "type": "text"}, {"bbox": [428, 312, 484, 324], "score": 0.94, "content": "N_{U}\\subseteq N_{k,U}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [484, 308, 489, 326], "score": 1.0, "content": ".", "type": "text"}], "index": 17}, {"bbox": [110, 322, 538, 342], "spans": [{"bbox": [110, 322, 289, 342], "score": 1.0, "content": "• Since the characteristic function ", "type": "text"}, {"bbox": [289, 329, 316, 340], "score": 0.91, "content": "\\chi_{N_{k,U}}", "type": "inline_equation", "height": 11, "width": 27}, {"bbox": [316, 322, 336, 342], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [336, 326, 360, 338], "score": 0.93, "content": "N_{k,U}", "type": "inline_equation", "height": 12, "width": 24}, {"bbox": [360, 322, 538, 342], "score": 1.0, "content": " is obviously a cylindrical function,", "type": "text"}], "index": 18}, {"bbox": [122, 340, 158, 353], "spans": [{"bbox": [122, 340, 158, 353], "score": 1.0, "content": "we get", "type": "text"}], "index": 19}], "index": 16, "bbox_fs": [63, 251, 538, 353]}, {"type": "interline_equation", "bbox": [191, 355, 466, 411], "lines": [{"bbox": [191, 355, 466, 411], "spans": [{"bbox": [191, 355, 466, 411], "score": 0.92, "content": "\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{*}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [106, 411, 537, 455], "lines": [{"bbox": [106, 413, 538, 429], "spans": [{"bbox": [106, 413, 154, 429], "score": 1.0, "content": "• From ", "type": "text"}, {"bbox": [154, 416, 210, 428], "score": 0.93, "content": "N_{U}\\subseteq N_{k,U}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [210, 414, 248, 429], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [248, 416, 254, 425], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [255, 414, 296, 429], "score": 1.0, "content": " follows ", "type": "text"}, {"bbox": [297, 416, 368, 428], "score": 0.92, "content": "N_{U}\\subseteq\\bigcap_{k}N_{k,U}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [369, 414, 403, 429], "score": 1.0, "content": ". But, ", "type": "text"}, {"bbox": [403, 415, 538, 428], "score": 0.92, "content": "\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=", "type": "inline_equation", "height": 13, "width": 135}], "index": 21}, {"bbox": [123, 427, 537, 444], "spans": [{"bbox": [123, 429, 189, 442], "score": 0.92, "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}", "type": "inline_equation", "height": 13, "width": 66}, {"bbox": [190, 427, 224, 444], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [224, 430, 231, 439], "score": 0.85, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [231, 427, 256, 444], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [256, 429, 338, 442], "score": 0.93, "content": "\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0", "type": "inline_equation", "height": 13, "width": 82}, {"bbox": [339, 427, 387, 444], "score": 1.0, "content": ", because ", "type": "text"}, {"bbox": [387, 429, 537, 442], "score": 0.91, "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<", "type": "inline_equation", "height": 13, "width": 150}], "index": 22}, {"bbox": [123, 443, 539, 457], "spans": [{"bbox": [123, 444, 133, 455], "score": 1.0, "content": "1.", "type": "text"}, {"bbox": [513, 443, 539, 457], "score": 1.0, "content": "qed", "type": "text"}], "index": 23}], "index": 22, "bbox_fs": [106, 413, 539, 457]}, {"type": "title", "bbox": [61, 463, 198, 477], "lines": [{"bbox": [63, 465, 197, 478], "spans": [{"bbox": [63, 465, 197, 478], "score": 1.0, "content": "Proof Proposition 9.2", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [106, 478, 539, 646], "lines": [{"bbox": [107, 479, 538, 494], "spans": [{"bbox": [107, 479, 144, 494], "score": 1.0, "content": "• Let ", "type": "text"}, {"bbox": [144, 480, 180, 493], "score": 0.92, "content": "(\\epsilon_{k})_{k\\in\\mathbb{N}}", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [180, 479, 392, 494], "score": 1.0, "content": " be some null sequence. Furthermore, let ", "type": "text"}, {"bbox": [392, 481, 426, 493], "score": 0.94, "content": "\\{U_{k,i}\\}_{i}", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [426, 479, 489, 494], "score": 1.0, "content": " be for each ", "type": "text"}, {"bbox": [489, 482, 496, 490], "score": 0.89, "content": "k", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [497, 479, 538, 494], "score": 1.0, "content": " a finite", "type": "text"}], "index": 25}, {"bbox": [123, 495, 537, 509], "spans": [{"bbox": [123, 495, 183, 509], "score": 1.0, "content": "covering of ", "type": "text"}, {"bbox": [183, 496, 194, 505], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [194, 495, 267, 509], "score": 1.0, "content": " by open sets ", "type": "text"}, {"bbox": [267, 496, 286, 508], "score": 0.92, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [286, 495, 522, 509], "score": 1.0, "content": " whose respective diameters are smaller than ", "type": "text"}, {"bbox": [523, 499, 533, 506], "score": 0.88, "content": "\\epsilon_{k}", "type": "inline_equation", "height": 7, "width": 10}, {"bbox": [533, 495, 537, 509], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [120, 506, 285, 527], "spans": [{"bbox": [120, 506, 183, 527], "score": 1.0, "content": "Now define ", "type": "text"}, {"bbox": [183, 509, 280, 527], "score": 0.95, "content": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)", "type": "inline_equation", "height": 18, "width": 97}, {"bbox": [281, 506, 285, 527], "score": 1.0, "content": ".", "type": "text"}], "index": 27}, {"bbox": [115, 526, 537, 541], "spans": [{"bbox": [115, 526, 154, 541], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [155, 528, 173, 540], "score": 0.93, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 18}, {"bbox": [173, 526, 243, 541], "score": 1.0, "content": " is open and ", "type": "text"}, {"bbox": [244, 528, 254, 537], "score": 0.87, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 526, 323, 541], "score": 1.0, "content": " is compact, ", "type": "text"}, {"bbox": [323, 528, 342, 540], "score": 0.92, "content": "U_{k,i}", "type": "inline_equation", "height": 12, "width": 19}, {"bbox": [342, 526, 454, 541], "score": 1.0, "content": " is measureable with ", "type": "text"}, {"bbox": [454, 528, 533, 540], "score": 0.96, "content": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [533, 526, 537, 541], "score": 1.0, "content": ".", "type": "text"}], "index": 28}, {"bbox": [120, 538, 539, 559], "spans": [{"bbox": [120, 538, 273, 559], "score": 1.0, "content": "Due to Lemma 9.4 we have ", "type": "text"}, {"bbox": [273, 543, 343, 557], "score": 0.93, "content": "N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{*}", "type": "inline_equation", "height": 14, "width": 70}, {"bbox": [344, 538, 376, 559], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [376, 542, 448, 557], "score": 0.94, "content": "\\mu_{0}(N_{U_{k,i}}^{*})\\,=\\,0", "type": "inline_equation", "height": 15, "width": 72}, {"bbox": [449, 538, 489, 559], "score": 1.0, "content": " for all ", "type": "text"}, {"bbox": [489, 542, 505, 554], "score": 0.9, "content": "k,i", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [506, 538, 539, 559], "score": 1.0, "content": "; thus", "type": "text"}], "index": 29}, {"bbox": [123, 555, 345, 578], "spans": [{"bbox": [123, 558, 251, 576], "score": 0.94, "content": "N^{\\prime}\\subseteq N^{*}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{*}\\bigr)", "type": "inline_equation", "height": 18, "width": 128}, {"bbox": [252, 555, 281, 578], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [281, 560, 340, 573], "score": 0.94, "content": "\\mu_{0}(N^{\\ast})=0", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [340, 555, 345, 578], "score": 1.0, "content": ".", "type": "text"}], "index": 30}, {"bbox": [118, 575, 271, 588], "spans": [{"bbox": [118, 575, 226, 588], "score": 1.0, "content": "We are left to show ", "type": "text"}, {"bbox": [227, 577, 267, 588], "score": 0.92, "content": "N\\subseteq N^{\\prime}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [267, 575, 271, 588], "score": 1.0, "content": ".", "type": "text"}], "index": 31}, {"bbox": [120, 587, 439, 605], "spans": [{"bbox": [120, 587, 144, 605], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [144, 590, 178, 601], "score": 0.93, "content": "{\\overline{{A}}}\\in N", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [178, 587, 303, 605], "score": 1.0, "content": ". Then there is an open ", "type": "text"}, {"bbox": [303, 591, 339, 602], "score": 0.92, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 36}, {"bbox": [339, 587, 369, 605], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [369, 591, 434, 603], "score": 0.93, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [434, 587, 439, 605], "score": 1.0, "content": ".", "type": "text"}], "index": 32}, {"bbox": [122, 603, 537, 619], "spans": [{"bbox": [122, 603, 167, 619], "score": 1.0, "content": "Now let ", "type": "text"}, {"bbox": [168, 606, 204, 615], "score": 0.93, "content": "m\\in U", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [204, 603, 244, 619], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [244, 605, 353, 617], "score": 0.92, "content": "\\epsilon:=\\mathrm{dist}(m,\\partial U)>0", "type": "inline_equation", "height": 12, "width": 109}, {"bbox": [353, 603, 403, 619], "score": 1.0, "content": ". Choose ", "type": "text"}, {"bbox": [403, 606, 410, 615], "score": 0.88, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [411, 603, 467, 619], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [467, 607, 500, 616], "score": 0.84, "content": "\\epsilon_{k}<\\epsilon", "type": "inline_equation", "height": 9, "width": 33}, {"bbox": [500, 603, 537, 619], "score": 1.0, "content": ". Then", "type": "text"}], "index": 33}, {"bbox": [122, 617, 537, 634], "spans": [{"bbox": [122, 617, 168, 634], "score": 1.0, "content": "choose a ", "type": "text"}, {"bbox": [169, 621, 187, 632], "score": 0.94, "content": "U_{k,i}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [187, 617, 215, 634], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [216, 620, 259, 632], "score": 0.93, "content": "m\\in U_{k,i}", "type": "inline_equation", "height": 12, "width": 43}, {"bbox": [259, 617, 338, 634], "score": 1.0, "content": ". We get for all ", "type": "text"}, {"bbox": [338, 621, 378, 632], "score": 0.92, "content": "x\\in U_{k,i}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [378, 617, 385, 634], "score": 1.0, "content": ": ", "type": "text"}, {"bbox": [385, 619, 436, 632], "score": 0.9, "content": "d(x,m)\\leq", "type": "inline_equation", "height": 13, "width": 51}, {"bbox": [436, 617, 468, 634], "score": 1.0, "content": "diam ", "type": "text"}, {"bbox": [468, 620, 533, 632], "score": 0.89, "content": "U_{k,i}<\\epsilon_{k}<\\epsilon", "type": "inline_equation", "height": 12, "width": 65}, {"bbox": [533, 617, 537, 634], "score": 1.0, "content": ",", "type": "text"}], "index": 34}, {"bbox": [121, 632, 537, 648], "spans": [{"bbox": [121, 632, 142, 648], "score": 1.0, "content": "i.e. ", "type": "text"}, {"bbox": [143, 635, 174, 644], "score": 0.92, "content": "x\\in U", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [174, 632, 255, 648], "score": 1.0, "content": ". Consequently, ", "type": "text"}, {"bbox": [256, 635, 299, 646], "score": 0.94, "content": "U_{k,i}\\subseteq U", "type": "inline_equation", "height": 11, "width": 43}, {"bbox": [300, 632, 349, 648], "score": 1.0, "content": " and thus ", "type": "text"}, {"bbox": [350, 634, 420, 647], "score": 0.94, "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}", "type": "inline_equation", "height": 13, "width": 70}, {"bbox": [421, 632, 446, 648], "score": 1.0, "content": ", i.e. ", "type": "text"}, {"bbox": [447, 633, 484, 644], "score": 0.92, "content": "{\\overline{{A}}}\\in N^{\\prime}", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [484, 632, 491, 648], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [513, 633, 537, 647], "score": 1.0, "content": "qed", "type": "text"}], "index": 35}], "index": 30, "bbox_fs": [107, 479, 539, 648]}]}
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[{"type": "text", "bbox": [63, 12, 538, 44], "content": "Lemma 9.1 If is complete (almost complete, non-complete), so is complete (almost complete, non-complete) for all .", "index": 0}, {"type": "text", "bbox": [62, 50, 538, 79], "content": "Thus, the total information about the completeness of a connection is already contained in its gauge orbit. Now, to the main assertion of this section.", "index": 1}, {"type": "text", "bbox": [63, 83, 538, 115], "content": "Proposition 9.2 Let non-complete}. Then is contained in a set of -measure zero whereas is the induced Haar measure on . [2, 6, 10]", "index": 2}, {"type": "text", "bbox": [62, 120, 248, 136], "content": "Since is gauge invariant, we have", "index": 3}, {"type": "text", "bbox": [63, 139, 538, 171], "content": "Corollary 9.3 Let non-complete}. Then is contained in a set of -measure zero.", "index": 4}, {"type": "text", "bbox": [62, 176, 348, 191], "content": "For the proof of the proposition we still need the follow", "index": 5}, {"type": "text", "bbox": [63, 196, 538, 241], "content": "Lemma 9.4 Let be measurable with and . Then is contained in a set of -measure zero.", "index": 6}, {"type": "text", "bbox": [65, 248, 538, 351], "content": "Proof • Let and be some connected graph with one vertex and edges .6 Furthermore, let . • Denote now by the set of all connections whose holonomies on are not contained in . Per constructionem we have . • Since the characteristic function for is obviously a cylindrical function, we get", "index": 7}, {"type": "interline_equation", "bbox": [191, 355, 466, 411], "content": "", "index": 8}, {"type": "text", "bbox": [106, 411, 537, 455], "content": "• From for all follows . But, for all , i.e. , because 1. qed", "index": 9}, {"type": "title", "bbox": [61, 463, 198, 477], "content": "Proof Proposition 9.2", "index": 10}, {"type": "text", "bbox": [106, 478, 539, 646], "content": "• Let be some null sequence. Furthermore, let be for each a finite covering of by open sets whose respective diameters are smaller than . Now define . Since is open and is compact, is measureable with . Due to Lemma 9.4 we have with for all ; thus with . We are left to show . Let . Then there is an open with . Now let . Then . Choose such that . Then choose a with . We get for all : diam , i.e. . Consequently, and thus , i.e. . qed", "index": 11}]
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[{"bbox": [61, 15, 537, 33], "content": "Lemma 9.1 If is complete (almost complete, non-complete), so is complete", "parent_index": 0, "line_index": 0}, {"bbox": [138, 31, 377, 46], "content": "(almost complete, non-complete) for all .", "parent_index": 0, "line_index": 1}, {"bbox": [63, 52, 537, 66], "content": "Thus, the total information about the completeness of a connection is already contained in", "parent_index": 1, "line_index": 0}, {"bbox": [62, 67, 364, 82], "content": "its gauge orbit. Now, to the main assertion of this section.", "parent_index": 1, "line_index": 1}, {"bbox": [61, 86, 539, 103], "content": "Proposition 9.2 Let non-complete}. Then is contained in a set of", "parent_index": 2, "line_index": 0}, {"bbox": [164, 102, 537, 117], "content": "-measure zero whereas is the induced Haar measure on . [2, 6, 10]", "parent_index": 2, "line_index": 1}, {"bbox": [63, 124, 246, 136], "content": "Since is gauge invariant, we have", "parent_index": 3, "line_index": 0}, {"bbox": [64, 143, 539, 159], "content": "Corollary 9.3 Let non-complete}. Then is contained in a set of", "parent_index": 4, "line_index": 0}, {"bbox": [151, 160, 237, 173], "content": "-measure zero.", "parent_index": 4, "line_index": 1}, {"bbox": [63, 180, 349, 192], "content": "For the proof of the proposition we still need the follow", "parent_index": 5, "line_index": 0}, {"bbox": [61, 197, 538, 216], "content": "Lemma 9.4 Let be measurable with and", "parent_index": 6, "line_index": 0}, {"bbox": [138, 213, 181, 229], "content": ".", "parent_index": 6, "line_index": 1}, {"bbox": [138, 228, 396, 243], "content": "Then is contained in a set of -measure zero.", "parent_index": 6, "line_index": 2}, {"bbox": [63, 251, 537, 266], "content": "Proof • Let and be some connected graph with one vertex and edges", "parent_index": 7, "line_index": 0}, {"bbox": [123, 264, 467, 281], "content": ".6 Furthermore, let .", "parent_index": 7, "line_index": 1}, {"bbox": [356, 281, 510, 294], "content": "", "parent_index": 7, "line_index": 2}, {"bbox": [105, 295, 538, 309], "content": "• Denote now by the set of all connections whose holonomies", "parent_index": 7, "line_index": 3}, {"bbox": [120, 308, 489, 326], "content": "on are not contained in . Per constructionem we have .", "parent_index": 7, "line_index": 4}, {"bbox": [110, 322, 538, 342], "content": "• Since the characteristic function for is obviously a cylindrical function,", "parent_index": 7, "line_index": 5}, {"bbox": [122, 340, 158, 353], "content": "we get", "parent_index": 7, "line_index": 6}, {"bbox": [106, 413, 538, 429], "content": "• From for all follows . But,", "parent_index": 9, "line_index": 0}, {"bbox": [123, 427, 537, 444], "content": "for all , i.e. , because", "parent_index": 9, "line_index": 1}, {"bbox": [123, 443, 539, 457], "content": "1. qed", "parent_index": 9, "line_index": 2}, {"bbox": [63, 465, 197, 478], "content": "Proof Proposition 9.2", "parent_index": 10, "line_index": 0}, {"bbox": [107, 479, 538, 494], "content": "• Let be some null sequence. Furthermore, let be for each a finite", "parent_index": 11, "line_index": 0}, {"bbox": [123, 495, 537, 509], "content": "covering of by open sets whose respective diameters are smaller than .", "parent_index": 11, "line_index": 1}, {"bbox": [120, 506, 285, 527], "content": "Now define .", "parent_index": 11, "line_index": 2}, {"bbox": [115, 526, 537, 541], "content": "Since is open and is compact, is measureable with .", "parent_index": 11, "line_index": 3}, {"bbox": [120, 538, 539, 559], "content": "Due to Lemma 9.4 we have with for all ; thus", "parent_index": 11, "line_index": 4}, {"bbox": [123, 555, 345, 578], "content": "with .", "parent_index": 11, "line_index": 5}, {"bbox": [118, 575, 271, 588], "content": "We are left to show .", "parent_index": 11, "line_index": 6}, {"bbox": [120, 587, 439, 605], "content": "Let . Then there is an open with .", "parent_index": 11, "line_index": 7}, {"bbox": [122, 603, 537, 619], "content": "Now let . Then . Choose such that . Then", "parent_index": 11, "line_index": 8}, {"bbox": [122, 617, 537, 634], "content": "choose a with . We get for all : diam ,", "parent_index": 11, "line_index": 9}, {"bbox": [121, 632, 537, 648], "content": "i.e. . Consequently, and thus , i.e. . qed", "parent_index": 11, "line_index": 10}]
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[]
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[{"bbox": [151, 17, 186, 28], "content": "{\\overline{{A}}}\\in{\\overline{{A}}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [446, 17, 474, 30], "content": "\\overline{{A}}\\circ\\overline{{g}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [344, 32, 373, 45], "content": "{\\overline{{g}}}\\in{\\overline{{g}}}", "parent_index": 0, "subtype": "inline"}, {"bbox": [186, 88, 282, 101], "content": "N:=\\{\\overline{{A}}\\in\\overline{{A}}\\mid\\overline{{A}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [400, 90, 411, 98], "content": "N", "parent_index": 2, "subtype": "inline"}, {"bbox": [164, 107, 175, 115], "content": "\\mu_{0}", "parent_index": 2, "subtype": "inline"}, {"bbox": [293, 107, 305, 115], "content": "\\mu_{0}", "parent_index": 2, "subtype": "inline"}, {"bbox": [474, 102, 484, 113], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [93, 125, 104, 133], "content": "N", "parent_index": 3, "subtype": "inline"}, {"bbox": [172, 144, 285, 157], "content": "[N]:=\\{[\\overline{{A}}]\\in\\overline{{A}}/\\overline{{\\mathcal{G}}}\\mid\\overline{{A}}", "parent_index": 4, "subtype": "inline"}, {"bbox": [401, 145, 418, 157], "content": "[N]", "parent_index": 4, "subtype": "inline"}, {"bbox": [151, 163, 163, 171], "content": "\\mu_{0}", "parent_index": 4, "subtype": "inline"}, {"bbox": [160, 201, 200, 212], "content": "U\\subseteq\\mathbf{G}", "parent_index": 6, "subtype": "inline"}, {"bbox": [311, 201, 381, 213], "content": "\\mu_{\\mathrm{Haar}}(U)\\,>\\,0", "parent_index": 6, "subtype": "inline"}, {"bbox": [409, 200, 538, 213], "content": "N_{U}\\;:=\\;\\{\\overline{{{A}}}\\,\\in\\,\\overline{{{A}}}\\;\\vert\\;\\mathbf{H}_{\\overline{{{A}}}}\\subseteq", "parent_index": 6, "subtype": "inline"}, {"bbox": [138, 215, 176, 228], "content": "\\mathbf{G}\\setminus U\\}", "parent_index": 6, "subtype": "inline"}, {"bbox": [169, 230, 186, 241], "content": "N_{U}", "parent_index": 6, "subtype": "inline"}, {"bbox": [309, 234, 321, 241], "content": "\\mu_{0}", "parent_index": 6, "subtype": "inline"}, {"bbox": [145, 252, 181, 262], "content": "k\\ \\in\\ \\mathbb{N}", "parent_index": 7, "subtype": "inline"}, {"bbox": [210, 253, 223, 263], "content": "\\Gamma_{k}", "parent_index": 7, "subtype": "inline"}, {"bbox": [456, 256, 467, 262], "content": "m", "parent_index": 7, "subtype": "inline"}, {"bbox": [496, 253, 503, 262], "content": "k", "parent_index": 7, "subtype": "inline"}, {"bbox": [123, 267, 209, 279], "content": "\\alpha_{1},\\ldots,\\alpha_{k}\\in\\mathcal{H}\\mathcal{G}", "parent_index": 7, "subtype": "inline"}, {"bbox": [325, 266, 393, 278], "content": "\\pi_{k}:\\overline{{\\mathcal{A}}}\\;\\;\\longrightarrow", "parent_index": 7, "subtype": "inline"}, {"bbox": [447, 266, 463, 276], "content": "\\mathbf{G}^{k}", "parent_index": 7, "subtype": "inline"}, {"bbox": [356, 281, 510, 294], "content": "\\begin{array}{r}{A\\;\\;\\longmapsto\\;\\;(h_{\\overline{{A}}}(\\alpha_{1}),\\dots,h_{\\overline{{A}}}(\\alpha_{k}))}\\end{array}", "parent_index": 7, "subtype": "inline"}, {"bbox": [202, 296, 314, 309], "content": "N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})", "parent_index": 7, "subtype": "inline"}, {"bbox": [139, 312, 152, 322], "content": "\\Gamma_{k}", "parent_index": 7, "subtype": "inline"}, {"bbox": [263, 312, 272, 321], "content": "U", "parent_index": 7, "subtype": "inline"}, {"bbox": [428, 312, 484, 324], "content": "N_{U}\\subseteq N_{k,U}", "parent_index": 7, "subtype": "inline"}, {"bbox": [289, 329, 316, 340], "content": "\\chi_{N_{k,U}}", "parent_index": 7, "subtype": "inline"}, {"bbox": [336, 326, 360, 338], "content": "N_{k,U}", "parent_index": 7, "subtype": "inline"}, {"bbox": [191, 355, 466, 411], "content": "\\begin{array}{r c l}{\\mu_{0}(N_{k,U})}&{=}&{\\displaystyle\\int_{\\overline{{\\mathcal{A}}}}\\chi_{N_{k,U}}\\;d\\mu_{0}\\ =\\ \\int_{\\overline{{\\mathcal{A}}}}\\pi_{k}^{*}(\\chi_{(\\mathbf{G}\\backslash U)^{k}})\\;d\\mu_{0}}\\\\ &{=}&{\\displaystyle\\int_{\\mathbf{G}^{k}}\\chi_{(\\mathbf{G}\\backslash U)^{k}}\\;d\\mu_{\\mathrm{Haar}}^{k}\\ =\\ \\left[\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)\\right]^{k}.}\\end{array}", "parent_index": 8, "subtype": "interline"}, {"bbox": [154, 416, 210, 428], "content": "N_{U}\\subseteq N_{k,U}", "parent_index": 9, "subtype": "inline"}, {"bbox": [248, 416, 254, 425], "content": "k", "parent_index": 9, "subtype": "inline"}, {"bbox": [297, 416, 368, 428], "content": "N_{U}\\subseteq\\bigcap_{k}N_{k,U}", "parent_index": 9, "subtype": "inline"}, {"bbox": [403, 415, 538, 428], "content": "\\mu_{0}(\\bigcap_{k}N_{k,U})\\leq\\mu_{0}(N_{k,U})=", "parent_index": 9, "subtype": "inline"}, {"bbox": [123, 429, 189, 442], "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)^{k}", "parent_index": 9, "subtype": "inline"}, {"bbox": [224, 430, 231, 439], "content": "k", "parent_index": 9, "subtype": "inline"}, {"bbox": [256, 429, 338, 442], "content": "\\mu_{0}\\bigl(\\bigcap_{k}N_{k,U}\\bigr)=0", "parent_index": 9, "subtype": "inline"}, {"bbox": [387, 429, 537, 442], "content": "\\mu_{\\mathrm{Haar}}(\\mathbf{G}\\backslash U)=1\\!-\\!\\mu_{\\mathrm{Haar}}(U)<", "parent_index": 9, "subtype": "inline"}, {"bbox": [144, 480, 180, 493], "content": "(\\epsilon_{k})_{k\\in\\mathbb{N}}", "parent_index": 11, "subtype": "inline"}, {"bbox": [392, 481, 426, 493], "content": "\\{U_{k,i}\\}_{i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [489, 482, 496, 490], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [183, 496, 194, 505], "content": "\\mathbf{G}", "parent_index": 11, "subtype": "inline"}, {"bbox": [267, 496, 286, 508], "content": "U_{k,i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [523, 499, 533, 506], "content": "\\epsilon_{k}", "parent_index": 11, "subtype": "inline"}, {"bbox": [183, 509, 280, 527], "content": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)", "parent_index": 11, "subtype": "inline"}, {"bbox": [155, 528, 173, 540], "content": "U_{k,i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [244, 528, 254, 537], "content": "\\mathbf{G}", "parent_index": 11, "subtype": "inline"}, {"bbox": [323, 528, 342, 540], "content": "U_{k,i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [454, 528, 533, 540], "content": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0", "parent_index": 11, "subtype": "inline"}, {"bbox": [273, 543, 343, 557], "content": "N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{*}", "parent_index": 11, "subtype": "inline"}, {"bbox": [376, 542, 448, 557], "content": "\\mu_{0}(N_{U_{k,i}}^{*})\\,=\\,0", "parent_index": 11, "subtype": "inline"}, {"bbox": [489, 542, 505, 554], "content": "k,i", "parent_index": 11, "subtype": "inline"}, {"bbox": [123, 558, 251, 576], "content": "N^{\\prime}\\subseteq N^{*}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{*}\\bigr)", "parent_index": 11, "subtype": "inline"}, {"bbox": [281, 560, 340, 573], "content": "\\mu_{0}(N^{\\ast})=0", "parent_index": 11, "subtype": "inline"}, {"bbox": [227, 577, 267, 588], "content": "N\\subseteq N^{\\prime}", "parent_index": 11, "subtype": "inline"}, {"bbox": [144, 590, 178, 601], "content": "{\\overline{{A}}}\\in N", "parent_index": 11, "subtype": "inline"}, {"bbox": [303, 591, 339, 602], "content": "U\\subseteq\\mathbf{G}", "parent_index": 11, "subtype": "inline"}, {"bbox": [369, 591, 434, 603], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U", "parent_index": 11, "subtype": "inline"}, {"bbox": [168, 606, 204, 615], "content": "m\\in U", "parent_index": 11, "subtype": "inline"}, {"bbox": [244, 605, 353, 617], "content": "\\epsilon:=\\mathrm{dist}(m,\\partial U)>0", "parent_index": 11, "subtype": "inline"}, {"bbox": [403, 606, 410, 615], "content": "k", "parent_index": 11, "subtype": "inline"}, {"bbox": [467, 607, 500, 616], "content": "\\epsilon_{k}<\\epsilon", "parent_index": 11, "subtype": "inline"}, {"bbox": [169, 621, 187, 632], "content": "U_{k,i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [216, 620, 259, 632], "content": "m\\in U_{k,i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [338, 621, 378, 632], "content": "x\\in U_{k,i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [385, 619, 436, 632], "content": "d(x,m)\\leq", "parent_index": 11, "subtype": "inline"}, {"bbox": [468, 620, 533, 632], "content": "U_{k,i}<\\epsilon_{k}<\\epsilon", "parent_index": 11, "subtype": "inline"}, {"bbox": [143, 635, 174, 644], "content": "x\\in U", "parent_index": 11, "subtype": "inline"}, {"bbox": [256, 635, 299, 646], "content": "U_{k,i}\\subseteq U", "parent_index": 11, "subtype": "inline"}, {"bbox": [350, 634, 420, 647], "content": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [447, 633, 484, 644], "content": "{\\overline{{A}}}\\in N^{\\prime}", "parent_index": 11, "subtype": "inline"}]
|
[]
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Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1.
Proof Every almost complete connection $\overline{{A}}$ has type $[Z(\mathbf{H}_{\overline{{A}}})]=[Z(\mathbf{G})]=t_{\operatorname*{max}}$ . (Observe that the centralizer of a set $U\subseteq\mathbf{G}$ equals that of the closure $\overline{U}$ .) Since $\overline{{A}}_{=t_{\mathrm{{max}}}}$ is open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion. qed
The last assertion is very important: It justifies the definition of the natural induced Haar measure on $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ (cf. [2, 10]). Actually, there were (at least) two different possibilities for this. Namely, let $X$ be some general topological space equipped with a measure $\mu$ and let $G$ be some topological group acting on $X$ . The problem now is to find a natural measure $\mu_{G}$ on the orbit space $X/G$ . On the one hand, one could simply define $\mu_{G}(U):=\mu(\pi^{-1}(U))$ for all measurable $U\subseteq X/G$ . ( $\pi:X\longrightarrow X/G$ is the canonical projection.) But, on the other hand, one also could stratify the orbit space. For instance, in the easiest case we could have $X=X/G\times G$ . In general, one gets (roughly speaking) $X=\cup\big(V/G\times_{\mathit{G}_{V}}\big\backslash\;G\big)$ whereas $\bigcup V$ on . ow one naively defines $X$ $G_{V}$ , $V$ $\begin{array}{r}{\mu_{G}(U)\;:=\;\sum_{V}\frac{\mu(\pi^{-1}(U)\cap V)}{\mu_{G,V}(G/G_{V})}\;:=\;\sum_{V}\mu\Big(\pi^{-1}(U)\cap V\Big)\mu_{V}(G_{V})}\end{array}$ where $\mu_{V}$ measures the ”size” of the stabilizer $G_{V}$ in $G$ . This second variant is nothing but the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi determinant) $\frac{d\mu}{d\mu_{G}}$ . In contrast to the first method, here the orbit space and not the total space is regarded to be primary. For a uniform distribution of the measure over all points of the total space the image measure on the orbit space needs no longer be uniformly distributed; the orbits are weighted by size. But, for the second method the uniformity is maintained. In other words, the gauge freedom does not play any role when the Faddeev-Popov method is used.
Nevertheless, we see in our concrete case of $\pi_{\overline{{{A}}}/\overline{{{\mathcal{G}}}}}\,:\,\overline{{{A}}}\,\longrightarrow\,\overline{{{A}}}/\overline{{{\mathcal{G}}}}$ that both methods are equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of $\mu_{0}$ -measure zero). This follows immediately from the slice theorem and the corollary above that the generic connections have total measure 1.
# 10 Summary and Discussion
In the present paper and its predecessor [9] we gained a lot of information about the structure of the generalized gauge orbit space within the Ashtekar framework. The most important tool was the theory of compact transformation groups on topological spaces. This enabled us to investigate the action of the group of generalized gauge transforms on the space of generalized connections. Our considerations were guided by the results of Kondracki and Rogulski [12] about the structure of the classical gauge orbit space for Sobolev connections. The methods used there are however fundamentally different from ours. Within the Ashtekar approach most of the proofs are purely algebraic or topological; in the classical case the methods are especially based on the theory of fiber bundles, i.e. analysis and differential geometry.
In a preceding paper [9] we proved that the $\mathcal{G}$ -stabilizer $\mathbf{B}(\overline{{A}})$ of a connection $\overline{{A}}$ is isomorphic to the $\mathbf{G}$ -centralizer $Z(\mathbf{H}_{\overline{{A}}})$ of the holonomy group of $\overline{{A}}$ . Furthermore, two connections have conjugate $\overline{{g}}$ -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the type of a generalized connection can be defined equivalently both by the $\overline{{g}}$ -conjugacy class of $\mathbf{B}(\overline{{A}})$ (as known from the general theory of transformation groups) and by the G-conjugacy class of $Z(\mathbf{H}_{\overline{{A}}})$ . This is a significant difference to the classical case.
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<p data-bbox="63 14 537 43">Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1. </p>
<p data-bbox="63 55 538 114">Proof Every almost complete connection $\overline{{A}}$ has type $[Z(\mathbf{H}_{\overline{{A}}})]=[Z(\mathbf{G})]=t_{\operatorname*{max}}$ . (Observe that the centralizer of a set $U\subseteq\mathbf{G}$ equals that of the closure $\overline{U}$ .) Since $\overline{{A}}_{=t_{\mathrm{{max}}}}$ is open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion. qed </p>
<p data-bbox="63 126 537 389">The last assertion is very important: It justifies the definition of the natural induced Haar measure on $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ (cf. [2, 10]). Actually, there were (at least) two different possibilities for this. Namely, let $X$ be some general topological space equipped with a measure $\mu$ and let $G$ be some topological group acting on $X$ . The problem now is to find a natural measure $\mu_{G}$ on the orbit space $X/G$ . On the one hand, one could simply define $\mu_{G}(U):=\mu(\pi^{-1}(U))$ for all measurable $U\subseteq X/G$ . ( $\pi:X\longrightarrow X/G$ is the canonical projection.) But, on the other hand, one also could stratify the orbit space. For instance, in the easiest case we could have $X=X/G\times G$ . In general, one gets (roughly speaking) $X=\cup\big(V/G\times_{\mathit{G}_{V}}\big\backslash\;G\big)$ whereas $\bigcup V$ on . ow one naively defines $X$ $G_{V}$ , $V$ $\begin{array}{r}{\mu_{G}(U)\;:=\;\sum_{V}\frac{\mu(\pi^{-1}(U)\cap V)}{\mu_{G,V}(G/G_{V})}\;:=\;\sum_{V}\mu\Big(\pi^{-1}(U)\cap V\Big)\mu_{V}(G_{V})}\end{array}$ where $\mu_{V}$ measures the ”size” of the stabilizer $G_{V}$ in $G$ . This second variant is nothing but the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi determinant) $\frac{d\mu}{d\mu_{G}}$ . In contrast to the first method, here the orbit space and not the total space is regarded to be primary. For a uniform distribution of the measure over all points of the total space the image measure on the orbit space needs no longer be uniformly distributed; the orbits are weighted by size. But, for the second method the uniformity is maintained. In other words, the gauge freedom does not play any role when the Faddeev-Popov method is used. </p>
<p data-bbox="63 389 538 447">Nevertheless, we see in our concrete case of $\pi_{\overline{{{A}}}/\overline{{{\mathcal{G}}}}}\,:\,\overline{{{A}}}\,\longrightarrow\,\overline{{{A}}}/\overline{{{\mathcal{G}}}}$ that both methods are equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of $\mu_{0}$ -measure zero). This follows immediately from the slice theorem and the corollary above that the generic connections have total measure 1. </p>
<h1 data-bbox="63 468 316 488">10 Summary and Discussion </h1>
<p data-bbox="63 498 538 628">In the present paper and its predecessor [9] we gained a lot of information about the structure of the generalized gauge orbit space within the Ashtekar framework. The most important tool was the theory of compact transformation groups on topological spaces. This enabled us to investigate the action of the group of generalized gauge transforms on the space of generalized connections. Our considerations were guided by the results of Kondracki and Rogulski [12] about the structure of the classical gauge orbit space for Sobolev connections. The methods used there are however fundamentally different from ours. Within the Ashtekar approach most of the proofs are purely algebraic or topological; in the classical case the methods are especially based on the theory of fiber bundles, i.e. analysis and differential geometry. </p>
<p data-bbox="64 630 538 687">In a preceding paper [9] we proved that the $\mathcal{G}$ -stabilizer $\mathbf{B}(\overline{{A}})$ of a connection $\overline{{A}}$ is isomorphic to the $\mathbf{G}$ -centralizer $Z(\mathbf{H}_{\overline{{A}}})$ of the holonomy group of $\overline{{A}}$ . Furthermore, two connections have conjugate $\overline{{g}}$ -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the type of a generalized connection can be defined equivalently both by the $\overline{{g}}$ -conjugacy class of $\mathbf{B}(\overline{{A}})$ (as known from the general theory of transformation groups) and by the G-conjugacy class of $Z(\mathbf{H}_{\overline{{A}}})$ . This is a significant difference to the classical case. </p>
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[{"type": "text", "text": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1. ", "page_idx": 17}, {"type": "text", "text": "Proof Every almost complete connection $\\overline{{A}}$ has type $[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\mathbf{G})]=t_{\\operatorname*{max}}$ . (Observe that the centralizer of a set $U\\subseteq\\mathbf{G}$ equals that of the closure $\\overline{U}$ .) Since $\\overline{{A}}_{=t_{\\mathrm{{max}}}}$ is open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion. qed ", "page_idx": 17}, {"type": "text", "text": "The last assertion is very important: It justifies the definition of the natural induced Haar measure on $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ (cf. [2, 10]). Actually, there were (at least) two different possibilities for this. Namely, let $X$ be some general topological space equipped with a measure $\\mu$ and let $G$ be some topological group acting on $X$ . The problem now is to find a natural measure $\\mu_{G}$ on the orbit space $X/G$ . On the one hand, one could simply define $\\mu_{G}(U):=\\mu(\\pi^{-1}(U))$ for all measurable $U\\subseteq X/G$ . ( $\\pi:X\\longrightarrow X/G$ is the canonical projection.) But, on the other hand, one also could stratify the orbit space. For instance, in the easiest case we could have $X=X/G\\times G$ . In general, one gets (roughly speaking) $X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)$ whereas $\\bigcup V$ on . ow one naively defines $X$ $G_{V}$ , $V$ $\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}$ where $\\mu_{V}$ measures the ”size” of the stabilizer $G_{V}$ in $G$ . This second variant is nothing but the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi determinant) $\\frac{d\\mu}{d\\mu_{G}}$ . In contrast to the first method, here the orbit space and not the total space is regarded to be primary. For a uniform distribution of the measure over all points of the total space the image measure on the orbit space needs no longer be uniformly distributed; the orbits are weighted by size. But, for the second method the uniformity is maintained. In other words, the gauge freedom does not play any role when the Faddeev-Popov method is used. ", "page_idx": 17}, {"type": "text", "text": "Nevertheless, we see in our concrete case of $\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}$ that both methods are equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of $\\mu_{0}$ -measure zero). This follows immediately from the slice theorem and the corollary above that the generic connections have total measure 1. ", "page_idx": 17}, {"type": "text", "text": "10 Summary and Discussion ", "text_level": 1, "page_idx": 17}, {"type": "text", "text": "In the present paper and its predecessor [9] we gained a lot of information about the structure of the generalized gauge orbit space within the Ashtekar framework. The most important tool was the theory of compact transformation groups on topological spaces. This enabled us to investigate the action of the group of generalized gauge transforms on the space of generalized connections. Our considerations were guided by the results of Kondracki and Rogulski [12] about the structure of the classical gauge orbit space for Sobolev connections. The methods used there are however fundamentally different from ours. Within the Ashtekar approach most of the proofs are purely algebraic or topological; in the classical case the methods are especially based on the theory of fiber bundles, i.e. analysis and differential geometry. ", "page_idx": 17}, {"type": "text", "text": "In a preceding paper [9] we proved that the $\\mathcal{G}$ -stabilizer $\\mathbf{B}(\\overline{{A}})$ of a connection $\\overline{{A}}$ is isomorphic to the $\\mathbf{G}$ -centralizer $Z(\\mathbf{H}_{\\overline{{A}}})$ of the holonomy group of $\\overline{{A}}$ . Furthermore, two connections have conjugate $\\overline{{g}}$ -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the type of a generalized connection can be defined equivalently both by the $\\overline{{g}}$ -conjugacy class of $\\mathbf{B}(\\overline{{A}})$ (as known from the general theory of transformation groups) and by the G-conjugacy class of $Z(\\mathbf{H}_{\\overline{{A}}})$ . This is a significant difference to the classical case. ", "page_idx": 17}]
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(Observe", "type": "text"}], "index": 2}, {"bbox": [105, 70, 539, 89], "spans": [{"bbox": [105, 70, 254, 89], "score": 1.0, "content": "that the centralizer of a set ", "type": "text"}, {"bbox": [254, 75, 293, 86], "score": 0.93, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [293, 70, 434, 89], "score": 1.0, "content": " equals that of the closure ", "type": "text"}, {"bbox": [434, 73, 444, 84], "score": 0.86, "content": "\\overline{U}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [444, 70, 489, 89], "score": 1.0, "content": ".) Since ", "type": "text"}, {"bbox": [489, 73, 523, 86], "score": 0.93, "content": "\\overline{{A}}_{=t_{\\mathrm{{max}}}}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [524, 70, 539, 89], "score": 1.0, "content": " is", "type": "text"}], "index": 3}, {"bbox": [106, 88, 537, 101], "spans": [{"bbox": [106, 88, 537, 101], "score": 1.0, "content": "open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion.", "type": "text"}], "index": 4}, {"bbox": [512, 102, 539, 116], "spans": [{"bbox": [512, 102, 539, 116], "score": 1.0, "content": "qed", "type": "text"}], "index": 5}], "index": 3.5}, {"type": "text", "bbox": [63, 126, 537, 389], "lines": [{"bbox": [61, 128, 537, 145], "spans": [{"bbox": [61, 128, 537, 145], "score": 1.0, "content": "The last assertion is very important: It justifies the definition of the natural induced Haar", "type": "text"}], "index": 6}, {"bbox": [62, 143, 537, 159], "spans": [{"bbox": [62, 143, 126, 159], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [127, 144, 150, 158], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [150, 143, 537, 159], "score": 1.0, "content": " (cf. [2, 10]). Actually, there were (at least) two different possibilities for", "type": "text"}], "index": 7}, {"bbox": [61, 158, 536, 174], "spans": [{"bbox": [61, 158, 152, 174], "score": 1.0, "content": "this. Namely, let ", "type": "text"}, {"bbox": [153, 160, 163, 169], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [164, 158, 475, 174], "score": 1.0, "content": " be some general topological space equipped with a measure ", "type": "text"}, {"bbox": [476, 163, 483, 171], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [483, 158, 526, 174], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [527, 160, 536, 169], "score": 0.9, "content": "G", "type": "inline_equation", "height": 9, "width": 9}], "index": 8}, {"bbox": [60, 171, 536, 189], "spans": [{"bbox": [60, 171, 255, 189], "score": 1.0, "content": "be some topological group acting on ", "type": "text"}, {"bbox": [255, 174, 267, 183], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [267, 171, 521, 189], "score": 1.0, "content": ". The problem now is to find a natural measure ", "type": "text"}, {"bbox": [522, 177, 536, 186], "score": 0.9, "content": "\\mu_{G}", "type": "inline_equation", "height": 9, "width": 14}], "index": 9}, {"bbox": [61, 187, 537, 201], "spans": [{"bbox": [61, 187, 159, 201], "score": 1.0, "content": "on the orbit space ", "type": "text"}, {"bbox": [160, 188, 185, 200], "score": 0.93, "content": "X/G", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [186, 187, 412, 201], "score": 1.0, "content": ". On the one hand, one could simply define ", "type": "text"}, {"bbox": [413, 188, 518, 201], "score": 0.93, "content": "\\mu_{G}(U):=\\mu(\\pi^{-1}(U))", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [518, 187, 537, 201], "score": 1.0, "content": " for", "type": "text"}], "index": 10}, {"bbox": [61, 200, 537, 216], "spans": [{"bbox": [61, 200, 141, 216], "score": 1.0, "content": "all measurable ", "type": "text"}, {"bbox": [141, 203, 193, 215], "score": 0.94, "content": "U\\subseteq X/G", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [194, 200, 207, 216], "score": 1.0, "content": ". (", "type": "text"}, {"bbox": [207, 203, 289, 215], "score": 0.89, "content": "\\pi:X\\longrightarrow X/G", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [290, 200, 537, 216], "score": 1.0, "content": " is the canonical projection.) But, on the other", "type": "text"}], "index": 11}, {"bbox": [61, 216, 537, 230], "spans": [{"bbox": [61, 216, 537, 230], "score": 1.0, "content": "hand, one also could stratify the orbit space. For instance, in the easiest case we could have", "type": "text"}], "index": 12}, {"bbox": [63, 228, 536, 247], "spans": [{"bbox": [63, 231, 137, 244], "score": 0.95, "content": "X=X/G\\times G", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [138, 229, 348, 247], "score": 1.0, "content": ". In general, one gets (roughly speaking) ", "type": "text"}, {"bbox": [348, 228, 468, 247], "score": 0.94, "content": "X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)", "type": "inline_equation", "height": 19, "width": 120}, {"bbox": [469, 229, 515, 247], "score": 1.0, "content": "whereas", "type": "text"}, {"bbox": [515, 232, 536, 243], "score": 0.91, "content": "\\bigcup V", "type": "inline_equation", "height": 11, "width": 21}], "index": 13}, {"bbox": [54, 245, 538, 288], "spans": [{"bbox": [54, 245, 80, 288], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [91, 245, 97, 288], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [112, 245, 236, 288], "score": 1.0, "content": "ow one naively defines ", "type": "text"}, {"bbox": [292, 247, 303, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 247, 348, 258], "score": 0.9, "content": "G_{V}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [534, 245, 538, 288], "score": 1.0, "content": ",", "type": "text"}], "index": 15}, {"bbox": [81, 258, 533, 277], "spans": [{"bbox": [81, 262, 91, 271], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 258, 533, 277], "score": 0.93, "content": "\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}", "type": "inline_equation", "height": 19, "width": 297}], "index": 14}, {"bbox": [63, 276, 537, 291], "spans": [{"bbox": [63, 276, 97, 291], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [97, 281, 111, 289], "score": 0.9, "content": "\\mu_{V}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [112, 276, 304, 291], "score": 1.0, "content": " measures the ”size” of the stabilizer ", "type": "text"}, {"bbox": [305, 278, 321, 289], "score": 0.9, "content": "G_{V}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [322, 276, 339, 291], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [340, 278, 349, 287], "score": 0.87, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [349, 276, 537, 291], "score": 1.0, "content": ". This second variant is nothing but", "type": "text"}], "index": 16}, {"bbox": [61, 291, 538, 306], "spans": [{"bbox": [61, 291, 538, 306], "score": 1.0, "content": "the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi", "type": "text"}], "index": 17}, {"bbox": [62, 301, 540, 322], "spans": [{"bbox": [62, 301, 135, 322], "score": 1.0, "content": "determinant)", "type": "text"}, {"bbox": [135, 304, 153, 322], "score": 0.93, "content": "\\frac{d\\mu}{d\\mu_{G}}", "type": "inline_equation", "height": 18, "width": 18}, {"bbox": [153, 301, 540, 322], "score": 1.0, "content": ". In contrast to the first method, here the orbit space and not the total", "type": "text"}], "index": 18}, {"bbox": [61, 320, 538, 334], "spans": [{"bbox": [61, 320, 538, 334], "score": 1.0, "content": "space is regarded to be primary. For a uniform distribution of the measure over all points of", "type": "text"}], "index": 19}, {"bbox": [61, 334, 538, 350], "spans": [{"bbox": [61, 334, 538, 350], "score": 1.0, "content": "the total space the image measure on the orbit space needs no longer be uniformly distributed;", "type": "text"}], "index": 20}, {"bbox": [63, 349, 537, 362], "spans": [{"bbox": [63, 349, 537, 362], "score": 1.0, "content": "the orbits are weighted by size. But, for the second method the uniformity is maintained. In", "type": "text"}], "index": 21}, {"bbox": [62, 362, 538, 378], "spans": [{"bbox": [62, 362, 538, 378], "score": 0.988952100276947, "content": "other words, the gauge freedom does not play any role when the Faddeev-Popov method is", "type": "text"}], "index": 22}, {"bbox": [63, 379, 90, 391], "spans": [{"bbox": [63, 379, 90, 391], "score": 1.0, "content": "used.", "type": "text"}], "index": 23}], "index": 14.5}, {"type": "text", "bbox": [63, 389, 538, 447], "lines": [{"bbox": [61, 390, 539, 408], "spans": [{"bbox": [61, 390, 303, 408], "score": 1.0, "content": "Nevertheless, we see in our concrete case of ", "type": "text"}, {"bbox": [304, 392, 409, 408], "score": 0.94, "content": "\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}", "type": "inline_equation", "height": 16, "width": 105}, {"bbox": [409, 390, 539, 408], "score": 1.0, "content": " that both methods are", "type": "text"}], "index": 24}, {"bbox": [64, 407, 538, 420], "spans": [{"bbox": [64, 407, 538, 420], "score": 1.0, "content": "equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of", "type": "text"}], "index": 25}, {"bbox": [63, 421, 538, 435], "spans": [{"bbox": [63, 426, 75, 434], "score": 0.91, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [75, 421, 538, 435], "score": 1.0, "content": "-measure zero). This follows immediately from the slice theorem and the corollary above", "type": "text"}], "index": 26}, {"bbox": [63, 436, 323, 448], "spans": [{"bbox": [63, 436, 323, 448], "score": 1.0, "content": "that the generic connections have total measure 1.", "type": "text"}], "index": 27}], "index": 25.5}, {"type": "title", "bbox": [63, 468, 316, 488], "lines": [{"bbox": [63, 471, 316, 488], "spans": [{"bbox": [63, 472, 84, 487], "score": 1.0, "content": "10", "type": "text"}, {"bbox": [100, 471, 316, 488], "score": 1.0, "content": "Summary and Discussion", "type": "text"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [63, 498, 538, 628], "lines": [{"bbox": [60, 500, 538, 517], "spans": [{"bbox": [60, 500, 538, 517], "score": 1.0, "content": "In the present paper and its predecessor [9] we gained a lot of information about the structure", "type": "text"}], "index": 29}, {"bbox": [62, 515, 538, 530], "spans": [{"bbox": [62, 515, 538, 530], "score": 1.0, "content": "of the generalized gauge orbit space within the Ashtekar framework. The most important tool", "type": "text"}], "index": 30}, {"bbox": [63, 531, 538, 545], "spans": [{"bbox": [63, 531, 538, 545], "score": 1.0, "content": "was the theory of compact transformation groups on topological spaces. This enabled us to", "type": "text"}], "index": 31}, {"bbox": [63, 546, 537, 559], "spans": [{"bbox": [63, 546, 537, 559], "score": 1.0, "content": "investigate the action of the group of generalized gauge transforms on the space of generalized", "type": "text"}], "index": 32}, {"bbox": [62, 559, 537, 574], "spans": [{"bbox": [62, 559, 537, 574], "score": 1.0, "content": "connections. Our considerations were guided by the results of Kondracki and Rogulski [12]", "type": "text"}], "index": 33}, {"bbox": [63, 574, 538, 588], "spans": [{"bbox": [63, 574, 538, 588], "score": 1.0, "content": "about the structure of the classical gauge orbit space for Sobolev connections. The methods", "type": "text"}], "index": 34}, {"bbox": [63, 589, 537, 602], "spans": [{"bbox": [63, 589, 537, 602], "score": 1.0, "content": "used there are however fundamentally different from ours. Within the Ashtekar approach", "type": "text"}], "index": 35}, {"bbox": [63, 603, 538, 616], "spans": [{"bbox": [63, 603, 538, 616], "score": 1.0, "content": "most of the proofs are purely algebraic or topological; in the classical case the methods are", "type": "text"}], "index": 36}, {"bbox": [63, 617, 504, 632], "spans": [{"bbox": [63, 617, 504, 632], "score": 1.0, "content": "especially based on the theory of fiber bundles, i.e. analysis and differential geometry.", "type": "text"}], "index": 37}], "index": 33}, {"type": "text", "bbox": [64, 630, 538, 687], "lines": [{"bbox": [62, 631, 536, 646], "spans": [{"bbox": [62, 631, 284, 646], "score": 1.0, "content": "In a preceding paper [9] we proved that the", "type": "text"}, {"bbox": [285, 632, 293, 643], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [293, 631, 346, 646], "score": 1.0, "content": "-stabilizer ", "type": "text"}, {"bbox": [346, 632, 374, 645], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [374, 631, 457, 646], "score": 1.0, "content": " of a connection ", "type": "text"}, {"bbox": [457, 632, 466, 642], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [466, 631, 536, 646], "score": 1.0, "content": " is isomorphic", "type": "text"}], "index": 38}, {"bbox": [61, 645, 538, 661], "spans": [{"bbox": [61, 645, 97, 661], "score": 1.0, "content": "to the ", "type": "text"}, {"bbox": [97, 648, 108, 657], "score": 0.58, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [108, 645, 168, 661], "score": 1.0, "content": "-centralizer ", "type": "text"}, {"bbox": [168, 647, 204, 660], "score": 0.95, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [204, 645, 339, 661], "score": 1.0, "content": " of the holonomy group of ", "type": "text"}, {"bbox": [340, 646, 349, 657], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [349, 645, 538, 661], "score": 1.0, "content": ". Furthermore, two connections have", "type": "text"}], "index": 39}, {"bbox": [63, 659, 537, 675], "spans": [{"bbox": [63, 659, 117, 675], "score": 1.0, "content": "conjugate ", "type": "text"}, {"bbox": [117, 660, 125, 672], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [125, 659, 537, 675], "score": 1.0, "content": "-stabilizers if and only if their holonomy centralizers are conjugate. Thus, the", "type": "text"}], "index": 40}, {"bbox": [63, 675, 538, 689], "spans": [{"bbox": [63, 675, 433, 689], "score": 1.0, "content": "type of a generalized connection can be defined equivalently both by the ", "type": "text"}, {"bbox": [433, 675, 441, 687], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [442, 675, 538, 689], "score": 1.0, "content": "-conjugacy class of", "type": "text"}], "index": 41}], "index": 39.5}], "layout_bboxes": [], "page_idx": 17, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [294, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "18", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [63, 14, 537, 43], "lines": [{"bbox": [63, 17, 537, 32], "spans": [{"bbox": [63, 17, 537, 32], "score": 1.0, "content": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has", "type": "text"}], "index": 0}, {"bbox": [149, 33, 223, 46], "spans": [{"bbox": [149, 33, 223, 46], "score": 1.0, "content": "µ0-measure 1.", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [63, 17, 537, 46]}, {"type": "text", "bbox": [63, 55, 538, 114], "lines": [{"bbox": [61, 57, 538, 74], "spans": [{"bbox": [61, 57, 287, 74], "score": 1.0, "content": "Proof Every almost complete connection ", "type": "text"}, {"bbox": [288, 59, 297, 69], "score": 0.9, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [297, 57, 348, 74], "score": 1.0, "content": " has type ", "type": "text"}, {"bbox": [348, 60, 481, 73], "score": 0.91, "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]=[Z(\\mathbf{G})]=t_{\\operatorname*{max}}", "type": "inline_equation", "height": 13, "width": 133}, {"bbox": [481, 57, 538, 74], "score": 1.0, "content": ". (Observe", "type": "text"}], "index": 2}, {"bbox": [105, 70, 539, 89], "spans": [{"bbox": [105, 70, 254, 89], "score": 1.0, "content": "that the centralizer of a set ", "type": "text"}, {"bbox": [254, 75, 293, 86], "score": 0.93, "content": "U\\subseteq\\mathbf{G}", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [293, 70, 434, 89], "score": 1.0, "content": " equals that of the closure ", "type": "text"}, {"bbox": [434, 73, 444, 84], "score": 0.86, "content": "\\overline{U}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [444, 70, 489, 89], "score": 1.0, "content": ".) Since ", "type": "text"}, {"bbox": [489, 73, 523, 86], "score": 0.93, "content": "\\overline{{A}}_{=t_{\\mathrm{{max}}}}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [524, 70, 539, 89], "score": 1.0, "content": " is", "type": "text"}], "index": 3}, {"bbox": [106, 88, 537, 101], "spans": [{"bbox": [106, 88, 537, 101], "score": 1.0, "content": "open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion.", "type": "text"}], "index": 4}, {"bbox": [512, 102, 539, 116], "spans": [{"bbox": [512, 102, 539, 116], "score": 1.0, "content": "qed", "type": "text"}], "index": 5}], "index": 3.5, "bbox_fs": [61, 57, 539, 116]}, {"type": "text", "bbox": [63, 126, 537, 389], "lines": [{"bbox": [61, 128, 537, 145], "spans": [{"bbox": [61, 128, 537, 145], "score": 1.0, "content": "The last assertion is very important: It justifies the definition of the natural induced Haar", "type": "text"}], "index": 6}, {"bbox": [62, 143, 537, 159], "spans": [{"bbox": [62, 143, 126, 159], "score": 1.0, "content": "measure on ", "type": "text"}, {"bbox": [127, 144, 150, 158], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 23}, {"bbox": [150, 143, 537, 159], "score": 1.0, "content": " (cf. [2, 10]). Actually, there were (at least) two different possibilities for", "type": "text"}], "index": 7}, {"bbox": [61, 158, 536, 174], "spans": [{"bbox": [61, 158, 152, 174], "score": 1.0, "content": "this. Namely, let ", "type": "text"}, {"bbox": [153, 160, 163, 169], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [164, 158, 475, 174], "score": 1.0, "content": " be some general topological space equipped with a measure ", "type": "text"}, {"bbox": [476, 163, 483, 171], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [483, 158, 526, 174], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [527, 160, 536, 169], "score": 0.9, "content": "G", "type": "inline_equation", "height": 9, "width": 9}], "index": 8}, {"bbox": [60, 171, 536, 189], "spans": [{"bbox": [60, 171, 255, 189], "score": 1.0, "content": "be some topological group acting on ", "type": "text"}, {"bbox": [255, 174, 267, 183], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [267, 171, 521, 189], "score": 1.0, "content": ". The problem now is to find a natural measure ", "type": "text"}, {"bbox": [522, 177, 536, 186], "score": 0.9, "content": "\\mu_{G}", "type": "inline_equation", "height": 9, "width": 14}], "index": 9}, {"bbox": [61, 187, 537, 201], "spans": [{"bbox": [61, 187, 159, 201], "score": 1.0, "content": "on the orbit space ", "type": "text"}, {"bbox": [160, 188, 185, 200], "score": 0.93, "content": "X/G", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [186, 187, 412, 201], "score": 1.0, "content": ". On the one hand, one could simply define ", "type": "text"}, {"bbox": [413, 188, 518, 201], "score": 0.93, "content": "\\mu_{G}(U):=\\mu(\\pi^{-1}(U))", "type": "inline_equation", "height": 13, "width": 105}, {"bbox": [518, 187, 537, 201], "score": 1.0, "content": " for", "type": "text"}], "index": 10}, {"bbox": [61, 200, 537, 216], "spans": [{"bbox": [61, 200, 141, 216], "score": 1.0, "content": "all measurable ", "type": "text"}, {"bbox": [141, 203, 193, 215], "score": 0.94, "content": "U\\subseteq X/G", "type": "inline_equation", "height": 12, "width": 52}, {"bbox": [194, 200, 207, 216], "score": 1.0, "content": ". (", "type": "text"}, {"bbox": [207, 203, 289, 215], "score": 0.89, "content": "\\pi:X\\longrightarrow X/G", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [290, 200, 537, 216], "score": 1.0, "content": " is the canonical projection.) But, on the other", "type": "text"}], "index": 11}, {"bbox": [61, 216, 537, 230], "spans": [{"bbox": [61, 216, 537, 230], "score": 1.0, "content": "hand, one also could stratify the orbit space. For instance, in the easiest case we could have", "type": "text"}], "index": 12}, {"bbox": [63, 228, 536, 247], "spans": [{"bbox": [63, 231, 137, 244], "score": 0.95, "content": "X=X/G\\times G", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [138, 229, 348, 247], "score": 1.0, "content": ". In general, one gets (roughly speaking) ", "type": "text"}, {"bbox": [348, 228, 468, 247], "score": 0.94, "content": "X=\\cup\\big(V/G\\times_{\\mathit{G}_{V}}\\big\\backslash\\;G\\big)", "type": "inline_equation", "height": 19, "width": 120}, {"bbox": [469, 229, 515, 247], "score": 1.0, "content": "whereas", "type": "text"}, {"bbox": [515, 232, 536, 243], "score": 0.91, "content": "\\bigcup V", "type": "inline_equation", "height": 11, "width": 21}], "index": 13}, {"bbox": [54, 245, 538, 288], "spans": [{"bbox": [54, 245, 80, 288], "score": 1.0, "content": "on ", "type": "text"}, {"bbox": [91, 245, 97, 288], "score": 1.0, "content": ".", "type": "text"}, {"bbox": [112, 245, 236, 288], "score": 1.0, "content": "ow one naively defines ", "type": "text"}, {"bbox": [292, 247, 303, 255], "score": 0.9, "content": "X", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [331, 247, 348, 258], "score": 0.9, "content": "G_{V}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [534, 245, 538, 288], "score": 1.0, "content": ",", "type": "text"}], "index": 15}, {"bbox": [81, 258, 533, 277], "spans": [{"bbox": [81, 262, 91, 271], "score": 0.9, "content": "V", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [236, 258, 533, 277], "score": 0.93, "content": "\\begin{array}{r}{\\mu_{G}(U)\\;:=\\;\\sum_{V}\\frac{\\mu(\\pi^{-1}(U)\\cap V)}{\\mu_{G,V}(G/G_{V})}\\;:=\\;\\sum_{V}\\mu\\Big(\\pi^{-1}(U)\\cap V\\Big)\\mu_{V}(G_{V})}\\end{array}", "type": "inline_equation", "height": 19, "width": 297}], "index": 14}, {"bbox": [63, 276, 537, 291], "spans": [{"bbox": [63, 276, 97, 291], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [97, 281, 111, 289], "score": 0.9, "content": "\\mu_{V}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [112, 276, 304, 291], "score": 1.0, "content": " measures the ”size” of the stabilizer ", "type": "text"}, {"bbox": [305, 278, 321, 289], "score": 0.9, "content": "G_{V}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [322, 276, 339, 291], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [340, 278, 349, 287], "score": 0.87, "content": "G", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [349, 276, 537, 291], "score": 1.0, "content": ". This second variant is nothing but", "type": "text"}], "index": 16}, {"bbox": [61, 291, 538, 306], "spans": [{"bbox": [61, 291, 538, 306], "score": 1.0, "content": "the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi", "type": "text"}], "index": 17}, {"bbox": [62, 301, 540, 322], "spans": [{"bbox": [62, 301, 135, 322], "score": 1.0, "content": "determinant)", "type": "text"}, {"bbox": [135, 304, 153, 322], "score": 0.93, "content": "\\frac{d\\mu}{d\\mu_{G}}", "type": "inline_equation", "height": 18, "width": 18}, {"bbox": [153, 301, 540, 322], "score": 1.0, "content": ". In contrast to the first method, here the orbit space and not the total", "type": "text"}], "index": 18}, {"bbox": [61, 320, 538, 334], "spans": [{"bbox": [61, 320, 538, 334], "score": 1.0, "content": "space is regarded to be primary. For a uniform distribution of the measure over all points of", "type": "text"}], "index": 19}, {"bbox": [61, 334, 538, 350], "spans": [{"bbox": [61, 334, 538, 350], "score": 1.0, "content": "the total space the image measure on the orbit space needs no longer be uniformly distributed;", "type": "text"}], "index": 20}, {"bbox": [63, 349, 537, 362], "spans": [{"bbox": [63, 349, 537, 362], "score": 1.0, "content": "the orbits are weighted by size. But, for the second method the uniformity is maintained. In", "type": "text"}], "index": 21}, {"bbox": [62, 362, 538, 378], "spans": [{"bbox": [62, 362, 538, 378], "score": 0.988952100276947, "content": "other words, the gauge freedom does not play any role when the Faddeev-Popov method is", "type": "text"}], "index": 22}, {"bbox": [63, 379, 90, 391], "spans": [{"bbox": [63, 379, 90, 391], "score": 1.0, "content": "used.", "type": "text"}], "index": 23}], "index": 14.5, "bbox_fs": [54, 128, 540, 391]}, {"type": "text", "bbox": [63, 389, 538, 447], "lines": [{"bbox": [61, 390, 539, 408], "spans": [{"bbox": [61, 390, 303, 408], "score": 1.0, "content": "Nevertheless, we see in our concrete case of ", "type": "text"}, {"bbox": [304, 392, 409, 408], "score": 0.94, "content": "\\pi_{\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}}\\,:\\,\\overline{{{A}}}\\,\\longrightarrow\\,\\overline{{{A}}}/\\overline{{{\\mathcal{G}}}}", "type": "inline_equation", "height": 16, "width": 105}, {"bbox": [409, 390, 539, 408], "score": 1.0, "content": " that both methods are", "type": "text"}], "index": 24}, {"bbox": [64, 407, 538, 420], "spans": [{"bbox": [64, 407, 538, 420], "score": 1.0, "content": "equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of", "type": "text"}], "index": 25}, {"bbox": [63, 421, 538, 435], "spans": [{"bbox": [63, 426, 75, 434], "score": 0.91, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [75, 421, 538, 435], "score": 1.0, "content": "-measure zero). This follows immediately from the slice theorem and the corollary above", "type": "text"}], "index": 26}, {"bbox": [63, 436, 323, 448], "spans": [{"bbox": [63, 436, 323, 448], "score": 1.0, "content": "that the generic connections have total measure 1.", "type": "text"}], "index": 27}], "index": 25.5, "bbox_fs": [61, 390, 539, 448]}, {"type": "title", "bbox": [63, 468, 316, 488], "lines": [{"bbox": [63, 471, 316, 488], "spans": [{"bbox": [63, 472, 84, 487], "score": 1.0, "content": "10", "type": "text"}, {"bbox": [100, 471, 316, 488], "score": 1.0, "content": "Summary and Discussion", "type": "text"}], "index": 28}], "index": 28}, {"type": "text", "bbox": [63, 498, 538, 628], "lines": [{"bbox": [60, 500, 538, 517], "spans": [{"bbox": [60, 500, 538, 517], "score": 1.0, "content": "In the present paper and its predecessor [9] we gained a lot of information about the structure", "type": "text"}], "index": 29}, {"bbox": [62, 515, 538, 530], "spans": [{"bbox": [62, 515, 538, 530], "score": 1.0, "content": "of the generalized gauge orbit space within the Ashtekar framework. The most important tool", "type": "text"}], "index": 30}, {"bbox": [63, 531, 538, 545], "spans": [{"bbox": [63, 531, 538, 545], "score": 1.0, "content": "was the theory of compact transformation groups on topological spaces. This enabled us to", "type": "text"}], "index": 31}, {"bbox": [63, 546, 537, 559], "spans": [{"bbox": [63, 546, 537, 559], "score": 1.0, "content": "investigate the action of the group of generalized gauge transforms on the space of generalized", "type": "text"}], "index": 32}, {"bbox": [62, 559, 537, 574], "spans": [{"bbox": [62, 559, 537, 574], "score": 1.0, "content": "connections. Our considerations were guided by the results of Kondracki and Rogulski [12]", "type": "text"}], "index": 33}, {"bbox": [63, 574, 538, 588], "spans": [{"bbox": [63, 574, 538, 588], "score": 1.0, "content": "about the structure of the classical gauge orbit space for Sobolev connections. The methods", "type": "text"}], "index": 34}, {"bbox": [63, 589, 537, 602], "spans": [{"bbox": [63, 589, 537, 602], "score": 1.0, "content": "used there are however fundamentally different from ours. Within the Ashtekar approach", "type": "text"}], "index": 35}, {"bbox": [63, 603, 538, 616], "spans": [{"bbox": [63, 603, 538, 616], "score": 1.0, "content": "most of the proofs are purely algebraic or topological; in the classical case the methods are", "type": "text"}], "index": 36}, {"bbox": [63, 617, 504, 632], "spans": [{"bbox": [63, 617, 504, 632], "score": 1.0, "content": "especially based on the theory of fiber bundles, i.e. analysis and differential geometry.", "type": "text"}], "index": 37}], "index": 33, "bbox_fs": [60, 500, 538, 632]}, {"type": "text", "bbox": [64, 630, 538, 687], "lines": [{"bbox": [62, 631, 536, 646], "spans": [{"bbox": [62, 631, 284, 646], "score": 1.0, "content": "In a preceding paper [9] we proved that the", "type": "text"}, {"bbox": [285, 632, 293, 643], "score": 0.9, "content": "\\mathcal{G}", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [293, 631, 346, 646], "score": 1.0, "content": "-stabilizer ", "type": "text"}, {"bbox": [346, 632, 374, 645], "score": 0.94, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [374, 631, 457, 646], "score": 1.0, "content": " of a connection ", "type": "text"}, {"bbox": [457, 632, 466, 642], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [466, 631, 536, 646], "score": 1.0, "content": " is isomorphic", "type": "text"}], "index": 38}, {"bbox": [61, 645, 538, 661], "spans": [{"bbox": [61, 645, 97, 661], "score": 1.0, "content": "to the ", "type": "text"}, {"bbox": [97, 648, 108, 657], "score": 0.58, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [108, 645, 168, 661], "score": 1.0, "content": "-centralizer ", "type": "text"}, {"bbox": [168, 647, 204, 660], "score": 0.95, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 13, "width": 36}, {"bbox": [204, 645, 339, 661], "score": 1.0, "content": " of the holonomy group of ", "type": "text"}, {"bbox": [340, 646, 349, 657], "score": 0.91, "content": "\\overline{{A}}", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [349, 645, 538, 661], "score": 1.0, "content": ". Furthermore, two connections have", "type": "text"}], "index": 39}, {"bbox": [63, 659, 537, 675], "spans": [{"bbox": [63, 659, 117, 675], "score": 1.0, "content": "conjugate ", "type": "text"}, {"bbox": [117, 660, 125, 672], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [125, 659, 537, 675], "score": 1.0, "content": "-stabilizers if and only if their holonomy centralizers are conjugate. Thus, the", "type": "text"}], "index": 40}, {"bbox": [63, 675, 538, 689], "spans": [{"bbox": [63, 675, 433, 689], "score": 1.0, "content": "type of a generalized connection can be defined equivalently both by the ", "type": "text"}, {"bbox": [433, 675, 441, 687], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [442, 675, 538, 689], "score": 1.0, "content": "-conjugacy class of", "type": "text"}], "index": 41}, {"bbox": [63, 16, 536, 33], "spans": [{"bbox": [63, 17, 91, 30], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28, "cross_page": true}, {"bbox": [91, 16, 536, 33], "score": 1.0, "content": " (as known from the general theory of transformation groups) and by the G-conjugacy", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [62, 31, 409, 46], "spans": [{"bbox": [62, 31, 104, 46], "score": 1.0, "content": "class of ", "type": "text", "cross_page": true}, {"bbox": [104, 33, 139, 45], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 12, "width": 35, "cross_page": true}, {"bbox": [140, 31, 409, 46], "score": 1.0, "content": ". This is a significant difference to the classical case.", "type": "text", "cross_page": true}], "index": 1}], "index": 39.5, "bbox_fs": [61, 631, 538, 689]}]}
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[{"type": "text", "bbox": [63, 14, 537, 43], "content": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1.", "index": 0}, {"type": "text", "bbox": [63, 55, 538, 114], "content": "Proof Every almost complete connection has type . (Observe that the centralizer of a set equals that of the closure .) Since is open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion. qed", "index": 1}, {"type": "text", "bbox": [63, 126, 537, 389], "content": "The last assertion is very important: It justifies the definition of the natural induced Haar measure on (cf. [2, 10]). Actually, there were (at least) two different possibilities for this. Namely, let be some general topological space equipped with a measure and let be some topological group acting on . The problem now is to find a natural measure on the orbit space . On the one hand, one could simply define for all measurable . ( is the canonical projection.) But, on the other hand, one also could stratify the orbit space. For instance, in the easiest case we could have . In general, one gets (roughly speaking) whereas on . ow one naively defines , where measures the ”size” of the stabilizer in . This second variant is nothing but the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi determinant) . In contrast to the first method, here the orbit space and not the total space is regarded to be primary. For a uniform distribution of the measure over all points of the total space the image measure on the orbit space needs no longer be uniformly distributed; the orbits are weighted by size. But, for the second method the uniformity is maintained. In other words, the gauge freedom does not play any role when the Faddeev-Popov method is used.", "index": 2}, {"type": "text", "bbox": [63, 389, 538, 447], "content": "Nevertheless, we see in our concrete case of that both methods are equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of -measure zero). This follows immediately from the slice theorem and the corollary above that the generic connections have total measure 1.", "index": 3}, {"type": "title", "bbox": [63, 468, 316, 488], "content": "10 Summary and Discussion", "index": 4}, {"type": "text", "bbox": [63, 498, 538, 628], "content": "In the present paper and its predecessor [9] we gained a lot of information about the structure of the generalized gauge orbit space within the Ashtekar framework. The most important tool was the theory of compact transformation groups on topological spaces. This enabled us to investigate the action of the group of generalized gauge transforms on the space of generalized connections. Our considerations were guided by the results of Kondracki and Rogulski [12] about the structure of the classical gauge orbit space for Sobolev connections. The methods used there are however fundamentally different from ours. Within the Ashtekar approach most of the proofs are purely algebraic or topological; in the classical case the methods are especially based on the theory of fiber bundles, i.e. analysis and differential geometry.", "index": 5}, {"type": "text", "bbox": [64, 630, 538, 687], "content": "In a preceding paper [9] we proved that the -stabilizer of a connection is isomorphic to the -centralizer of the holonomy group of . Furthermore, two connections have conjugate -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the type of a generalized connection can be defined equivalently both by the -conjugacy class of (as known from the general theory of transformation groups) and by the G-conjugacy class of . This is a significant difference to the classical case.", "index": 6}]
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[{"bbox": [63, 17, 537, 32], "content": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has", "parent_index": 0, "line_index": 0}, {"bbox": [149, 33, 223, 46], "content": "µ0-measure 1.", "parent_index": 0, "line_index": 1}, {"bbox": [61, 57, 538, 74], "content": "Proof Every almost complete connection has type . (Observe", "parent_index": 1, "line_index": 0}, {"bbox": [105, 70, 539, 89], "content": "that the centralizer of a set equals that of the closure .) Since is", "parent_index": 1, "line_index": 1}, {"bbox": [106, 88, 537, 101], "content": "open due to Proposition 6.1, thus measurable, Proposition 9.2 yields the assertion.", "parent_index": 1, "line_index": 2}, {"bbox": [512, 102, 539, 116], "content": "qed", "parent_index": 1, "line_index": 3}, {"bbox": [61, 128, 537, 145], "content": "The last assertion is very important: It justifies the definition of the natural induced Haar", "parent_index": 2, "line_index": 0}, {"bbox": [62, 143, 537, 159], "content": "measure on (cf. [2, 10]). Actually, there were (at least) two different possibilities for", "parent_index": 2, "line_index": 1}, {"bbox": [61, 158, 536, 174], "content": "this. Namely, let be some general topological space equipped with a measure and let", "parent_index": 2, "line_index": 2}, {"bbox": [60, 171, 536, 189], "content": "be some topological group acting on . The problem now is to find a natural measure", "parent_index": 2, "line_index": 3}, {"bbox": [61, 187, 537, 201], "content": "on the orbit space . On the one hand, one could simply define for", "parent_index": 2, "line_index": 4}, {"bbox": [61, 200, 537, 216], "content": "all measurable . ( is the canonical projection.) But, on the other", "parent_index": 2, "line_index": 5}, {"bbox": [61, 216, 537, 230], "content": "hand, one also could stratify the orbit space. For instance, in the easiest case we could have", "parent_index": 2, "line_index": 6}, {"bbox": [63, 228, 536, 247], "content": ". In general, one gets (roughly speaking) whereas", "parent_index": 2, "line_index": 7}, {"bbox": [54, 245, 538, 288], "content": "on . ow one naively defines ,", "parent_index": 2, "line_index": 8}, {"bbox": [81, 258, 533, 277], "content": "", "parent_index": 2, "line_index": 9}, {"bbox": [63, 276, 537, 291], "content": "where measures the ”size” of the stabilizer in . This second variant is nothing but", "parent_index": 2, "line_index": 10}, {"bbox": [61, 291, 538, 306], "content": "the transformation of the measures using the Faddeev-Popov determinant (i.e. the Jacobi", "parent_index": 2, "line_index": 11}, {"bbox": [62, 301, 540, 322], "content": "determinant) . In contrast to the first method, here the orbit space and not the total", "parent_index": 2, "line_index": 12}, {"bbox": [61, 320, 538, 334], "content": "space is regarded to be primary. For a uniform distribution of the measure over all points of", "parent_index": 2, "line_index": 13}, {"bbox": [61, 334, 538, 350], "content": "the total space the image measure on the orbit space needs no longer be uniformly distributed;", "parent_index": 2, "line_index": 14}, {"bbox": [63, 349, 537, 362], "content": "the orbits are weighted by size. But, for the second method the uniformity is maintained. In", "parent_index": 2, "line_index": 15}, {"bbox": [62, 362, 538, 378], "content": "other words, the gauge freedom does not play any role when the Faddeev-Popov method is", "parent_index": 2, "line_index": 16}, {"bbox": [63, 379, 90, 391], "content": "used.", "parent_index": 2, "line_index": 17}, {"bbox": [61, 390, 539, 408], "content": "Nevertheless, we see in our concrete case of that both methods are", "parent_index": 3, "line_index": 0}, {"bbox": [64, 407, 538, 420], "content": "equivalent because the Faddeev-Popov determinant is equal to 1 (at least outside a set of", "parent_index": 3, "line_index": 1}, {"bbox": [63, 421, 538, 435], "content": "-measure zero). This follows immediately from the slice theorem and the corollary above", "parent_index": 3, "line_index": 2}, {"bbox": [63, 436, 323, 448], "content": "that the generic connections have total measure 1.", "parent_index": 3, "line_index": 3}, {"bbox": [63, 471, 316, 488], "content": "10 Summary and Discussion", "parent_index": 4, "line_index": 0}, {"bbox": [60, 500, 538, 517], "content": "In the present paper and its predecessor [9] we gained a lot of information about the structure", "parent_index": 5, "line_index": 0}, {"bbox": [62, 515, 538, 530], "content": "of the generalized gauge orbit space within the Ashtekar framework. The most important tool", "parent_index": 5, "line_index": 1}, {"bbox": [63, 531, 538, 545], "content": "was the theory of compact transformation groups on topological spaces. This enabled us to", "parent_index": 5, "line_index": 2}, {"bbox": [63, 546, 537, 559], "content": "investigate the action of the group of generalized gauge transforms on the space of generalized", "parent_index": 5, "line_index": 3}, {"bbox": [62, 559, 537, 574], "content": "connections. Our considerations were guided by the results of Kondracki and Rogulski [12]", "parent_index": 5, "line_index": 4}, {"bbox": [63, 574, 538, 588], "content": "about the structure of the classical gauge orbit space for Sobolev connections. The methods", "parent_index": 5, "line_index": 5}, {"bbox": [63, 589, 537, 602], "content": "used there are however fundamentally different from ours. Within the Ashtekar approach", "parent_index": 5, "line_index": 6}, {"bbox": [63, 603, 538, 616], "content": "most of the proofs are purely algebraic or topological; in the classical case the methods are", "parent_index": 5, "line_index": 7}, {"bbox": [63, 617, 504, 632], "content": "especially based on the theory of fiber bundles, i.e. analysis and differential geometry.", "parent_index": 5, "line_index": 8}, {"bbox": [62, 631, 536, 646], "content": "In a preceding paper [9] we proved that the -stabilizer of a connection is isomorphic", "parent_index": 6, "line_index": 0}, {"bbox": [61, 645, 538, 661], "content": "to the -centralizer of the holonomy group of . Furthermore, two connections have", "parent_index": 6, "line_index": 1}, {"bbox": [63, 659, 537, 675], "content": "conjugate -stabilizers if and only if their holonomy centralizers are conjugate. Thus, the", "parent_index": 6, "line_index": 2}, {"bbox": [63, 675, 538, 689], "content": "type of a generalized connection can be defined equivalently both by the -conjugacy class of", "parent_index": 6, "line_index": 3}, {"bbox": [63, 16, 536, 33], "content": "(as known from the general theory of transformation groups) and by the G-conjugacy", "parent_index": 6, "line_index": 4}, {"bbox": [62, 31, 409, 46], "content": "class of . This is a significant difference to the classical case.", "parent_index": 6, "line_index": 5}]
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[]
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The reduction of our problem from structures in $\overline{{g}}$ to those in $\mathbf{G}$ was the crucial idea in the present paper. Since stabilizers in compact groups are even generated by a finite number of elements, we could model the gauge orbit type $[Z(\mathbf{H}_{\overline{{A}}})]$ on a finite-dimensional space. Using an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on $\overline{{\mathcal{A}}}$ . This is the main result of our paper. Collecting connections of one and the same type we got the so-called strata whose openness was an immediate consequence of the slice theorem. In the next step we showed that the natural ordering on the set of the types encodes the topological properties of the strata. More precisely, we proved that the closure of a stratum contains (besides the stratum itself) exactly the union of all strata having a smaller type. This implied that this decomposition of $\overline{{\mathcal{A}}}$ is a topologically regular stratification.
All these results hold in the classical case as well. This is very remarkable because our proofs used partially completely different ideas. However, two results of this paper go beyond the classical theorems. First, we were able to determine the full set of all gauge orbit types occurring in $\overline{{\mathcal{A}}}$ . This set is known for Sobolev connections – to the best of our knowlegde – only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem completely for $S U(n)$ -bundels $P$ over two-, three- and four-dimensional manifolds [18]. The main problem in the Sobolev case is the non-triviality of the bundle $P$ . This can exclude orbit types that occur in the trivial bundle $M\times S U(n)$ . But, this problem is irrelevant for the Ashtekar framework: Every regular connection in every G-bundle over $M$ is contained in $\overline{{{A}}}$ [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in the Ashtekar framework there is a well-defined natural measure on $\overline{{\mathcal{A}}}$ . Using this we could show that the generic stratum has the total measure one; this is not true in the classical case. The proposition above implies now that the Faddeev-Popov determinant for the transformation from $\overline{{\mathcal{A}}}$ to $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ is equal to 1. This, on the other hand, justifies the definition of the induced Haar measure on $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ by projecting the corresponding measure for $\overline{{\mathcal{A}}}$ which has been discussed in detail in section 9.
Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost) completely to the Ashtekar program. We emphasize that all assertions are valid for each compact structure group – both in the analytical and in the $C^{r}$ -smooth case.
What could be next steps in this area? An important – and in this paper completely ignored – item is the physical interpretation of the gained knowledge. So we will conclude our paper with a few ideas that could link mathematics and physics:
Topology
What is the topological structure of the strata? Are they connected or is $\overline{{\mathcal{A}}}$ connected itself (at least for connected $\mathbf{G}$ )? Is $\overline{{A}}_{=t}$ globally trivial over $(\overline{{\cal{A}}}/\overline{{\cal{G}}})_{=t}$ , at least for the generic stratum with $t=t_{\mathrm{max}}$ ? What sections do exist in these bundles, i.e. what gauge fixings do exist in $\overline{{\mathcal{A}}}$ ?
These problems are closely related to the so-called Gribov problem, the non-existence of global gauge fixings for classical connections in principal fiber bundles with compact, noncommutative structure group (see, e.g., [19]). From this lots of difficulties result for the quantization of such a Yang-Mills theory that are not circumvented up to now.
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<p data-bbox="63 44 538 187">The reduction of our problem from structures in $\overline{{g}}$ to those in $\mathbf{G}$ was the crucial idea in the present paper. Since stabilizers in compact groups are even generated by a finite number of elements, we could model the gauge orbit type $[Z(\mathbf{H}_{\overline{{A}}})]$ on a finite-dimensional space. Using an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on $\overline{{\mathcal{A}}}$ . This is the main result of our paper. Collecting connections of one and the same type we got the so-called strata whose openness was an immediate consequence of the slice theorem. In the next step we showed that the natural ordering on the set of the types encodes the topological properties of the strata. More precisely, we proved that the closure of a stratum contains (besides the stratum itself) exactly the union of all strata having a smaller type. This implied that this decomposition of $\overline{{\mathcal{A}}}$ is a topologically regular stratification. </p>
<p data-bbox="63 189 538 419">All these results hold in the classical case as well. This is very remarkable because our proofs used partially completely different ideas. However, two results of this paper go beyond the classical theorems. First, we were able to determine the full set of all gauge orbit types occurring in $\overline{{\mathcal{A}}}$ . This set is known for Sobolev connections – to the best of our knowlegde – only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem completely for $S U(n)$ -bundels $P$ over two-, three- and four-dimensional manifolds [18]. The main problem in the Sobolev case is the non-triviality of the bundle $P$ . This can exclude orbit types that occur in the trivial bundle $M\times S U(n)$ . But, this problem is irrelevant for the Ashtekar framework: Every regular connection in every G-bundle over $M$ is contained in $\overline{{{A}}}$ [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in the Ashtekar framework there is a well-defined natural measure on $\overline{{\mathcal{A}}}$ . Using this we could show that the generic stratum has the total measure one; this is not true in the classical case. The proposition above implies now that the Faddeev-Popov determinant for the transformation from $\overline{{\mathcal{A}}}$ to $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ is equal to 1. This, on the other hand, justifies the definition of the induced Haar measure on $\overline{{\mathcal{A}}}/\overline{{\mathcal{G}}}$ by projecting the corresponding measure for $\overline{{\mathcal{A}}}$ which has been discussed in detail in section 9. </p>
<p data-bbox="64 419 537 462">Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost) completely to the Ashtekar program. We emphasize that all assertions are valid for each compact structure group – both in the analytical and in the $C^{r}$ -smooth case. </p>
<p data-bbox="63 473 537 516">What could be next steps in this area? An important – and in this paper completely ignored – item is the physical interpretation of the gained knowledge. So we will conclude our paper with a few ideas that could link mathematics and physics: </p>
<p data-bbox="64 518 538 647">Topology
What is the topological structure of the strata? Are they connected or is $\overline{{\mathcal{A}}}$ connected itself (at least for connected $\mathbf{G}$ )? Is $\overline{{A}}_{=t}$ globally trivial over $(\overline{{\cal{A}}}/\overline{{\cal{G}}})_{=t}$ , at least for the generic stratum with $t=t_{\mathrm{max}}$ ? What sections do exist in these bundles, i.e. what gauge fixings do exist in $\overline{{\mathcal{A}}}$ ?
These problems are closely related to the so-called Gribov problem, the non-existence of global gauge fixings for classical connections in principal fiber bundles with compact, noncommutative structure group (see, e.g., [19]). From this lots of difficulties result for the quantization of such a Yang-Mills theory that are not circumvented up to now. </p>
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0001008v1
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[{"type": "text", "text": "", "page_idx": 18}, {"type": "text", "text": "The reduction of our problem from structures in $\\overline{{g}}$ to those in $\\mathbf{G}$ was the crucial idea in the present paper. Since stabilizers in compact groups are even generated by a finite number of elements, we could model the gauge orbit type $[Z(\\mathbf{H}_{\\overline{{A}}})]$ on a finite-dimensional space. Using an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on $\\overline{{\\mathcal{A}}}$ . This is the main result of our paper. Collecting connections of one and the same type we got the so-called strata whose openness was an immediate consequence of the slice theorem. In the next step we showed that the natural ordering on the set of the types encodes the topological properties of the strata. More precisely, we proved that the closure of a stratum contains (besides the stratum itself) exactly the union of all strata having a smaller type. This implied that this decomposition of $\\overline{{\\mathcal{A}}}$ is a topologically regular stratification. ", "page_idx": 18}, {"type": "text", "text": "All these results hold in the classical case as well. This is very remarkable because our proofs used partially completely different ideas. However, two results of this paper go beyond the classical theorems. First, we were able to determine the full set of all gauge orbit types occurring in $\\overline{{\\mathcal{A}}}$ . This set is known for Sobolev connections – to the best of our knowlegde – only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem completely for $S U(n)$ -bundels $P$ over two-, three- and four-dimensional manifolds [18]. The main problem in the Sobolev case is the non-triviality of the bundle $P$ . This can exclude orbit types that occur in the trivial bundle $M\\times S U(n)$ . But, this problem is irrelevant for the Ashtekar framework: Every regular connection in every G-bundle over $M$ is contained in $\\overline{{{A}}}$ [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in the Ashtekar framework there is a well-defined natural measure on $\\overline{{\\mathcal{A}}}$ . Using this we could show that the generic stratum has the total measure one; this is not true in the classical case. The proposition above implies now that the Faddeev-Popov determinant for the transformation from $\\overline{{\\mathcal{A}}}$ to $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ is equal to 1. This, on the other hand, justifies the definition of the induced Haar measure on $\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}$ by projecting the corresponding measure for $\\overline{{\\mathcal{A}}}$ which has been discussed in detail in section 9. ", "page_idx": 18}, {"type": "text", "text": "Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost) completely to the Ashtekar program. We emphasize that all assertions are valid for each compact structure group – both in the analytical and in the $C^{r}$ -smooth case. ", "page_idx": 18}, {"type": "text", "text": "What could be next steps in this area? An important – and in this paper completely ignored – item is the physical interpretation of the gained knowledge. So we will conclude our paper with a few ideas that could link mathematics and physics: ", "page_idx": 18}, {"type": "text", "text": "Topology \nWhat is the topological structure of the strata? Are they connected or is $\\overline{{\\mathcal{A}}}$ connected itself (at least for connected $\\mathbf{G}$ )? Is $\\overline{{A}}_{=t}$ globally trivial over $(\\overline{{\\cal{A}}}/\\overline{{\\cal{G}}})_{=t}$ , at least for the generic stratum with $t=t_{\\mathrm{max}}$ ? What sections do exist in these bundles, i.e. what gauge fixings do exist in $\\overline{{\\mathcal{A}}}$ ? \nThese problems are closely related to the so-called Gribov problem, the non-existence of global gauge fixings for classical connections in principal fiber bundles with compact, noncommutative structure group (see, e.g., [19]). From this lots of difficulties result for the quantization of such a Yang-Mills theory that are not circumvented up to now. ", "page_idx": 18}]
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{"preproc_blocks": [{"type": "text", "bbox": [63, 15, 537, 43], "lines": [{"bbox": [63, 16, 536, 33], "spans": [{"bbox": [63, 17, 91, 30], "score": 0.93, "content": "\\mathbf{B}(\\overline{{A}})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [91, 16, 536, 33], "score": 1.0, "content": " (as known from the general theory of transformation groups) and by the G-conjugacy", "type": "text"}], "index": 0}, {"bbox": [62, 31, 409, 46], "spans": [{"bbox": [62, 31, 104, 46], "score": 1.0, "content": "class of ", "type": "text"}, {"bbox": [104, 33, 139, 45], "score": 0.94, "content": "Z(\\mathbf{H}_{\\overline{{A}}})", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [140, 31, 409, 46], "score": 1.0, "content": ". This is a significant difference to the classical case.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [63, 44, 538, 187], "lines": [{"bbox": [61, 45, 538, 60], "spans": [{"bbox": [61, 45, 315, 60], "score": 1.0, "content": "The reduction of our problem from structures in ", "type": "text"}, {"bbox": [315, 46, 323, 58], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [324, 45, 385, 60], "score": 1.0, "content": " to those in ", "type": "text"}, {"bbox": [386, 48, 397, 56], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [397, 45, 538, 60], "score": 1.0, "content": " was the crucial idea in the", "type": "text"}], "index": 2}, {"bbox": [61, 60, 539, 75], "spans": [{"bbox": [61, 60, 539, 75], "score": 1.0, "content": "present paper. Since stabilizers in compact groups are even generated by a finite number of", "type": "text"}], "index": 3}, {"bbox": [61, 74, 538, 90], "spans": [{"bbox": [61, 74, 305, 90], "score": 1.0, "content": "elements, we could model the gauge orbit type ", "type": "text"}, {"bbox": [305, 76, 347, 88], "score": 0.94, "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [347, 74, 538, 90], "score": 1.0, "content": " on a finite-dimensional space. Using", "type": "text"}], "index": 4}, {"bbox": [61, 89, 538, 103], "spans": [{"bbox": [61, 89, 523, 103], "score": 1.0, "content": "an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on ", "type": "text"}, {"bbox": [523, 90, 533, 100], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [533, 89, 538, 103], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [61, 103, 538, 119], "spans": [{"bbox": [61, 103, 538, 119], "score": 1.0, "content": "This is the main result of our paper. Collecting connections of one and the same type we", "type": "text"}], "index": 6}, {"bbox": [61, 118, 537, 133], "spans": [{"bbox": [61, 118, 537, 133], "score": 1.0, "content": "got the so-called strata whose openness was an immediate consequence of the slice theorem.", "type": "text"}], "index": 7}, {"bbox": [62, 133, 537, 147], "spans": [{"bbox": [62, 133, 537, 147], "score": 1.0, "content": "In the next step we showed that the natural ordering on the set of the types encodes the", "type": "text"}], "index": 8}, {"bbox": [63, 147, 536, 162], "spans": [{"bbox": [63, 147, 536, 162], "score": 1.0, "content": "topological properties of the strata. More precisely, we proved that the closure of a stratum", "type": "text"}], "index": 9}, {"bbox": [62, 162, 537, 176], "spans": [{"bbox": [62, 162, 537, 176], "score": 1.0, "content": "contains (besides the stratum itself) exactly the union of all strata having a smaller type.", "type": "text"}], "index": 10}, {"bbox": [63, 176, 482, 190], "spans": [{"bbox": [63, 176, 270, 190], "score": 1.0, "content": "This implied that this decomposition of ", "type": "text"}, {"bbox": [270, 176, 280, 187], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [280, 176, 482, 190], "score": 1.0, "content": " is a topologically regular stratification.", "type": "text"}], "index": 11}], "index": 6.5}, {"type": "text", "bbox": [63, 189, 538, 419], "lines": [{"bbox": [63, 190, 537, 205], "spans": [{"bbox": [63, 190, 537, 205], "score": 1.0, "content": "All these results hold in the classical case as well. This is very remarkable because our proofs", "type": "text"}], "index": 12}, {"bbox": [63, 206, 537, 219], "spans": [{"bbox": [63, 206, 537, 219], "score": 1.0, "content": "used partially completely different ideas. However, two results of this paper go beyond the", "type": "text"}], "index": 13}, {"bbox": [63, 220, 536, 233], "spans": [{"bbox": [63, 220, 536, 233], "score": 1.0, "content": "classical theorems. First, we were able to determine the full set of all gauge orbit types", "type": "text"}], "index": 14}, {"bbox": [64, 234, 537, 247], "spans": [{"bbox": [64, 234, 129, 247], "score": 1.0, "content": "occurring in ", "type": "text"}, {"bbox": [130, 234, 140, 244], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [140, 234, 537, 247], "score": 1.0, "content": ". This set is known for Sobolev connections – to the best of our knowlegde", "type": "text"}], "index": 15}, {"bbox": [64, 249, 536, 262], "spans": [{"bbox": [64, 249, 536, 262], "score": 1.0, "content": "– only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem", "type": "text"}], "index": 16}, {"bbox": [64, 263, 537, 276], "spans": [{"bbox": [64, 263, 140, 276], "score": 1.0, "content": "completely for ", "type": "text"}, {"bbox": [140, 264, 174, 276], "score": 0.95, "content": "S U(n)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [174, 263, 220, 276], "score": 1.0, "content": "-bundels ", "type": "text"}, {"bbox": [221, 264, 230, 273], "score": 0.91, "content": "P", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [230, 263, 537, 276], "score": 1.0, "content": " over two-, three- and four-dimensional manifolds [18]. The", "type": "text"}], "index": 17}, {"bbox": [63, 278, 537, 291], "spans": [{"bbox": [63, 278, 426, 291], "score": 1.0, "content": "main problem in the Sobolev case is the non-triviality of the bundle ", "type": "text"}, {"bbox": [427, 279, 436, 288], "score": 0.9, "content": "P", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [437, 278, 537, 291], "score": 1.0, "content": ". This can exclude", "type": "text"}], "index": 18}, {"bbox": [62, 291, 538, 306], "spans": [{"bbox": [62, 291, 289, 306], "score": 1.0, "content": "orbit types that occur in the trivial bundle ", "type": "text"}, {"bbox": [290, 293, 351, 305], "score": 0.95, "content": "M\\times S U(n)", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [351, 291, 538, 306], "score": 1.0, "content": ". But, this problem is irrelevant for", "type": "text"}], "index": 19}, {"bbox": [61, 305, 538, 320], "spans": [{"bbox": [61, 305, 445, 320], "score": 1.0, "content": "the Ashtekar framework: Every regular connection in every G-bundle over ", "type": "text"}, {"bbox": [446, 308, 458, 316], "score": 0.92, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [459, 305, 538, 320], "score": 1.0, "content": " is contained in", "type": "text"}], "index": 20}, {"bbox": [63, 319, 538, 335], "spans": [{"bbox": [63, 321, 73, 331], "score": 0.86, "content": "\\overline{{{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [73, 319, 538, 335], "score": 1.0, "content": " [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in", "type": "text"}], "index": 21}, {"bbox": [62, 335, 537, 348], "spans": [{"bbox": [62, 335, 414, 348], "score": 1.0, "content": "the Ashtekar framework there is a well-defined natural measure on ", "type": "text"}, {"bbox": [414, 335, 424, 345], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [424, 335, 537, 348], "score": 1.0, "content": ". Using this we could", "type": "text"}], "index": 22}, {"bbox": [61, 349, 538, 364], "spans": [{"bbox": [61, 349, 538, 364], "score": 1.0, "content": "show that the generic stratum has the total measure one; this is not true in the classical", "type": "text"}], "index": 23}, {"bbox": [63, 365, 537, 378], "spans": [{"bbox": [63, 365, 537, 378], "score": 1.0, "content": "case. The proposition above implies now that the Faddeev-Popov determinant for the trans-", "type": "text"}], "index": 24}, {"bbox": [63, 378, 538, 392], "spans": [{"bbox": [63, 379, 145, 392], "score": 1.0, "content": "formation from ", "type": "text"}, {"bbox": [145, 378, 155, 389], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [156, 379, 174, 392], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [174, 378, 198, 392], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [198, 379, 538, 392], "score": 1.0, "content": " is equal to 1. This, on the other hand, justifies the definition of", "type": "text"}], "index": 25}, {"bbox": [61, 392, 538, 408], "spans": [{"bbox": [61, 392, 216, 408], "score": 1.0, "content": "the induced Haar measure on ", "type": "text"}, {"bbox": [216, 393, 240, 406], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [240, 392, 472, 408], "score": 1.0, "content": " by projecting the corresponding measure for ", "type": "text"}, {"bbox": [472, 393, 482, 403], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [482, 392, 538, 408], "score": 1.0, "content": " which has", "type": "text"}], "index": 26}, {"bbox": [63, 408, 252, 420], "spans": [{"bbox": [63, 408, 252, 420], "score": 1.0, "content": "been discussed in detail in section 9.", "type": "text"}], "index": 27}], "index": 19.5}, {"type": "text", "bbox": [64, 419, 537, 462], "lines": [{"bbox": [63, 421, 537, 435], "spans": [{"bbox": [63, 421, 537, 435], "score": 1.0, "content": "Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost)", "type": "text"}], "index": 28}, {"bbox": [63, 436, 538, 450], "spans": [{"bbox": [63, 436, 538, 450], "score": 1.0, "content": "completely to the Ashtekar program. We emphasize that all assertions are valid for each", "type": "text"}], "index": 29}, {"bbox": [63, 451, 456, 466], "spans": [{"bbox": [63, 451, 371, 466], "score": 1.0, "content": "compact structure group – both in the analytical and in the ", "type": "text"}, {"bbox": [371, 452, 385, 461], "score": 0.92, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [385, 451, 456, 466], "score": 1.0, "content": "-smooth case.", "type": "text"}], "index": 30}], "index": 29}, {"type": "text", "bbox": [63, 473, 537, 516], "lines": [{"bbox": [63, 475, 537, 489], "spans": [{"bbox": [63, 475, 537, 489], "score": 1.0, "content": "What could be next steps in this area? An important – and in this paper completely ignored", "type": "text"}], "index": 31}, {"bbox": [63, 489, 537, 504], "spans": [{"bbox": [63, 489, 537, 504], "score": 1.0, "content": "– item is the physical interpretation of the gained knowledge. So we will conclude our paper", "type": "text"}], "index": 32}, {"bbox": [63, 504, 362, 518], "spans": [{"bbox": [63, 504, 362, 518], "score": 1.0, "content": "with a few ideas that could link mathematics and physics:", "type": "text"}], "index": 33}], "index": 32}, {"type": "text", "bbox": [64, 518, 538, 647], "lines": [{"bbox": [79, 518, 127, 534], "spans": [{"bbox": [79, 518, 127, 534], "score": 1.0, "content": "Topology", "type": "text"}], "index": 34}, {"bbox": [79, 533, 538, 547], "spans": [{"bbox": [79, 533, 470, 547], "score": 1.0, "content": "What is the topological structure of the strata? Are they connected or is ", "type": "text"}, {"bbox": [471, 533, 480, 543], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [481, 533, 538, 547], "score": 1.0, "content": " connected", "type": "text"}], "index": 35}, {"bbox": [78, 547, 537, 561], "spans": [{"bbox": [78, 547, 232, 561], "score": 1.0, "content": "itself (at least for connected ", "type": "text"}, {"bbox": [232, 549, 243, 558], "score": 0.77, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 547, 274, 561], "score": 1.0, "content": ")? Is ", "type": "text"}, {"bbox": [274, 547, 295, 560], "score": 0.93, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [295, 547, 406, 561], "score": 1.0, "content": " globally trivial over ", "type": "text"}, {"bbox": [406, 547, 449, 561], "score": 0.94, "content": "(\\overline{{\\cal{A}}}/\\overline{{\\cal{G}}})_{=t}", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [449, 547, 537, 561], "score": 1.0, "content": ", at least for the", "type": "text"}], "index": 36}, {"bbox": [78, 561, 539, 578], "spans": [{"bbox": [78, 561, 191, 578], "score": 1.0, "content": "generic stratum with ", "type": "text"}, {"bbox": [191, 564, 233, 574], "score": 0.91, "content": "t=t_{\\mathrm{max}}", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [233, 561, 539, 578], "score": 1.0, "content": "? What sections do exist in these bundles, i.e. what gauge", "type": "text"}], "index": 37}, {"bbox": [78, 576, 191, 590], "spans": [{"bbox": [78, 576, 173, 590], "score": 1.0, "content": "fixings do exist in ", "type": "text"}, {"bbox": [174, 577, 184, 587], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [185, 576, 191, 590], "score": 1.0, "content": "?", "type": "text"}], "index": 38}, {"bbox": [79, 590, 538, 605], "spans": [{"bbox": [79, 590, 538, 605], "score": 1.0, "content": "These problems are closely related to the so-called Gribov problem, the non-existence of", "type": "text"}], "index": 39}, {"bbox": [79, 605, 538, 620], "spans": [{"bbox": [79, 605, 538, 620], "score": 1.0, "content": "global gauge fixings for classical connections in principal fiber bundles with compact, non-", "type": "text"}], "index": 40}, {"bbox": [79, 619, 538, 635], "spans": [{"bbox": [79, 619, 538, 635], "score": 1.0, "content": "commutative structure group (see, e.g., [19]). From this lots of difficulties result for the", "type": "text"}], "index": 41}, {"bbox": [79, 635, 486, 649], "spans": [{"bbox": [79, 635, 486, 649], "score": 1.0, "content": "quantization of such a Yang-Mills theory that are not circumvented up to now.", "type": "text"}], "index": 42}], "index": 38}], "layout_bboxes": [], "page_idx": 18, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "19", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [63, 15, 537, 43], "lines": [], "index": 0.5, "bbox_fs": [62, 16, 536, 46], "lines_deleted": true}, {"type": "text", "bbox": [63, 44, 538, 187], "lines": [{"bbox": [61, 45, 538, 60], "spans": [{"bbox": [61, 45, 315, 60], "score": 1.0, "content": "The reduction of our problem from structures in ", "type": "text"}, {"bbox": [315, 46, 323, 58], "score": 0.9, "content": "\\overline{{g}}", "type": "inline_equation", "height": 12, "width": 8}, {"bbox": [324, 45, 385, 60], "score": 1.0, "content": " to those in ", "type": "text"}, {"bbox": [386, 48, 397, 56], "score": 0.88, "content": "\\mathbf{G}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [397, 45, 538, 60], "score": 1.0, "content": " was the crucial idea in the", "type": "text"}], "index": 2}, {"bbox": [61, 60, 539, 75], "spans": [{"bbox": [61, 60, 539, 75], "score": 1.0, "content": "present paper. Since stabilizers in compact groups are even generated by a finite number of", "type": "text"}], "index": 3}, {"bbox": [61, 74, 538, 90], "spans": [{"bbox": [61, 74, 305, 90], "score": 1.0, "content": "elements, we could model the gauge orbit type ", "type": "text"}, {"bbox": [305, 76, 347, 88], "score": 0.94, "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]", "type": "inline_equation", "height": 12, "width": 42}, {"bbox": [347, 74, 538, 90], "score": 1.0, "content": " on a finite-dimensional space. Using", "type": "text"}], "index": 4}, {"bbox": [61, 89, 538, 103], "spans": [{"bbox": [61, 89, 523, 103], "score": 1.0, "content": "an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on ", "type": "text"}, {"bbox": [523, 90, 533, 100], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [533, 89, 538, 103], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [61, 103, 538, 119], "spans": [{"bbox": [61, 103, 538, 119], "score": 1.0, "content": "This is the main result of our paper. Collecting connections of one and the same type we", "type": "text"}], "index": 6}, {"bbox": [61, 118, 537, 133], "spans": [{"bbox": [61, 118, 537, 133], "score": 1.0, "content": "got the so-called strata whose openness was an immediate consequence of the slice theorem.", "type": "text"}], "index": 7}, {"bbox": [62, 133, 537, 147], "spans": [{"bbox": [62, 133, 537, 147], "score": 1.0, "content": "In the next step we showed that the natural ordering on the set of the types encodes the", "type": "text"}], "index": 8}, {"bbox": [63, 147, 536, 162], "spans": [{"bbox": [63, 147, 536, 162], "score": 1.0, "content": "topological properties of the strata. More precisely, we proved that the closure of a stratum", "type": "text"}], "index": 9}, {"bbox": [62, 162, 537, 176], "spans": [{"bbox": [62, 162, 537, 176], "score": 1.0, "content": "contains (besides the stratum itself) exactly the union of all strata having a smaller type.", "type": "text"}], "index": 10}, {"bbox": [63, 176, 482, 190], "spans": [{"bbox": [63, 176, 270, 190], "score": 1.0, "content": "This implied that this decomposition of ", "type": "text"}, {"bbox": [270, 176, 280, 187], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [280, 176, 482, 190], "score": 1.0, "content": " is a topologically regular stratification.", "type": "text"}], "index": 11}], "index": 6.5, "bbox_fs": [61, 45, 539, 190]}, {"type": "text", "bbox": [63, 189, 538, 419], "lines": [{"bbox": [63, 190, 537, 205], "spans": [{"bbox": [63, 190, 537, 205], "score": 1.0, "content": "All these results hold in the classical case as well. This is very remarkable because our proofs", "type": "text"}], "index": 12}, {"bbox": [63, 206, 537, 219], "spans": [{"bbox": [63, 206, 537, 219], "score": 1.0, "content": "used partially completely different ideas. However, two results of this paper go beyond the", "type": "text"}], "index": 13}, {"bbox": [63, 220, 536, 233], "spans": [{"bbox": [63, 220, 536, 233], "score": 1.0, "content": "classical theorems. First, we were able to determine the full set of all gauge orbit types", "type": "text"}], "index": 14}, {"bbox": [64, 234, 537, 247], "spans": [{"bbox": [64, 234, 129, 247], "score": 1.0, "content": "occurring in ", "type": "text"}, {"bbox": [130, 234, 140, 244], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [140, 234, 537, 247], "score": 1.0, "content": ". This set is known for Sobolev connections – to the best of our knowlegde", "type": "text"}], "index": 15}, {"bbox": [64, 249, 536, 262], "spans": [{"bbox": [64, 249, 536, 262], "score": 1.0, "content": "– only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem", "type": "text"}], "index": 16}, {"bbox": [64, 263, 537, 276], "spans": [{"bbox": [64, 263, 140, 276], "score": 1.0, "content": "completely for ", "type": "text"}, {"bbox": [140, 264, 174, 276], "score": 0.95, "content": "S U(n)", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [174, 263, 220, 276], "score": 1.0, "content": "-bundels ", "type": "text"}, {"bbox": [221, 264, 230, 273], "score": 0.91, "content": "P", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [230, 263, 537, 276], "score": 1.0, "content": " over two-, three- and four-dimensional manifolds [18]. The", "type": "text"}], "index": 17}, {"bbox": [63, 278, 537, 291], "spans": [{"bbox": [63, 278, 426, 291], "score": 1.0, "content": "main problem in the Sobolev case is the non-triviality of the bundle ", "type": "text"}, {"bbox": [427, 279, 436, 288], "score": 0.9, "content": "P", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [437, 278, 537, 291], "score": 1.0, "content": ". This can exclude", "type": "text"}], "index": 18}, {"bbox": [62, 291, 538, 306], "spans": [{"bbox": [62, 291, 289, 306], "score": 1.0, "content": "orbit types that occur in the trivial bundle ", "type": "text"}, {"bbox": [290, 293, 351, 305], "score": 0.95, "content": "M\\times S U(n)", "type": "inline_equation", "height": 12, "width": 61}, {"bbox": [351, 291, 538, 306], "score": 1.0, "content": ". But, this problem is irrelevant for", "type": "text"}], "index": 19}, {"bbox": [61, 305, 538, 320], "spans": [{"bbox": [61, 305, 445, 320], "score": 1.0, "content": "the Ashtekar framework: Every regular connection in every G-bundle over ", "type": "text"}, {"bbox": [446, 308, 458, 316], "score": 0.92, "content": "M", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [459, 305, 538, 320], "score": 1.0, "content": " is contained in", "type": "text"}], "index": 20}, {"bbox": [63, 319, 538, 335], "spans": [{"bbox": [63, 321, 73, 331], "score": 0.86, "content": "\\overline{{{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [73, 319, 538, 335], "score": 1.0, "content": " [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in", "type": "text"}], "index": 21}, {"bbox": [62, 335, 537, 348], "spans": [{"bbox": [62, 335, 414, 348], "score": 1.0, "content": "the Ashtekar framework there is a well-defined natural measure on ", "type": "text"}, {"bbox": [414, 335, 424, 345], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [424, 335, 537, 348], "score": 1.0, "content": ". Using this we could", "type": "text"}], "index": 22}, {"bbox": [61, 349, 538, 364], "spans": [{"bbox": [61, 349, 538, 364], "score": 1.0, "content": "show that the generic stratum has the total measure one; this is not true in the classical", "type": "text"}], "index": 23}, {"bbox": [63, 365, 537, 378], "spans": [{"bbox": [63, 365, 537, 378], "score": 1.0, "content": "case. The proposition above implies now that the Faddeev-Popov determinant for the trans-", "type": "text"}], "index": 24}, {"bbox": [63, 378, 538, 392], "spans": [{"bbox": [63, 379, 145, 392], "score": 1.0, "content": "formation from ", "type": "text"}, {"bbox": [145, 378, 155, 389], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [156, 379, 174, 392], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [174, 378, 198, 392], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 14, "width": 24}, {"bbox": [198, 379, 538, 392], "score": 1.0, "content": " is equal to 1. This, on the other hand, justifies the definition of", "type": "text"}], "index": 25}, {"bbox": [61, 392, 538, 408], "spans": [{"bbox": [61, 392, 216, 408], "score": 1.0, "content": "the induced Haar measure on ", "type": "text"}, {"bbox": [216, 393, 240, 406], "score": 0.94, "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 13, "width": 24}, {"bbox": [240, 392, 472, 408], "score": 1.0, "content": " by projecting the corresponding measure for ", "type": "text"}, {"bbox": [472, 393, 482, 403], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [482, 392, 538, 408], "score": 1.0, "content": " which has", "type": "text"}], "index": 26}, {"bbox": [63, 408, 252, 420], "spans": [{"bbox": [63, 408, 252, 420], "score": 1.0, "content": "been discussed in detail in section 9.", "type": "text"}], "index": 27}], "index": 19.5, "bbox_fs": [61, 190, 538, 420]}, {"type": "text", "bbox": [64, 419, 537, 462], "lines": [{"bbox": [63, 421, 537, 435], "spans": [{"bbox": [63, 421, 537, 435], "score": 1.0, "content": "Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost)", "type": "text"}], "index": 28}, {"bbox": [63, 436, 538, 450], "spans": [{"bbox": [63, 436, 538, 450], "score": 1.0, "content": "completely to the Ashtekar program. We emphasize that all assertions are valid for each", "type": "text"}], "index": 29}, {"bbox": [63, 451, 456, 466], "spans": [{"bbox": [63, 451, 371, 466], "score": 1.0, "content": "compact structure group – both in the analytical and in the ", "type": "text"}, {"bbox": [371, 452, 385, 461], "score": 0.92, "content": "C^{r}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [385, 451, 456, 466], "score": 1.0, "content": "-smooth case.", "type": "text"}], "index": 30}], "index": 29, "bbox_fs": [63, 421, 538, 466]}, {"type": "text", "bbox": [63, 473, 537, 516], "lines": [{"bbox": [63, 475, 537, 489], "spans": [{"bbox": [63, 475, 537, 489], "score": 1.0, "content": "What could be next steps in this area? An important – and in this paper completely ignored", "type": "text"}], "index": 31}, {"bbox": [63, 489, 537, 504], "spans": [{"bbox": [63, 489, 537, 504], "score": 1.0, "content": "– item is the physical interpretation of the gained knowledge. So we will conclude our paper", "type": "text"}], "index": 32}, {"bbox": [63, 504, 362, 518], "spans": [{"bbox": [63, 504, 362, 518], "score": 1.0, "content": "with a few ideas that could link mathematics and physics:", "type": "text"}], "index": 33}], "index": 32, "bbox_fs": [63, 475, 537, 518]}, {"type": "list", "bbox": [64, 518, 538, 647], "lines": [{"bbox": [79, 518, 127, 534], "spans": [{"bbox": [79, 518, 127, 534], "score": 1.0, "content": "Topology", "type": "text"}], "index": 34, "is_list_end_line": true}, {"bbox": [79, 533, 538, 547], "spans": [{"bbox": [79, 533, 470, 547], "score": 1.0, "content": "What is the topological structure of the strata? Are they connected or is ", "type": "text"}, {"bbox": [471, 533, 480, 543], "score": 0.89, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [481, 533, 538, 547], "score": 1.0, "content": " connected", "type": "text"}], "index": 35, "is_list_start_line": true}, {"bbox": [78, 547, 537, 561], "spans": [{"bbox": [78, 547, 232, 561], "score": 1.0, "content": "itself (at least for connected ", "type": "text"}, {"bbox": [232, 549, 243, 558], "score": 0.77, "content": "\\mathbf{G}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 547, 274, 561], "score": 1.0, "content": ")? Is ", "type": "text"}, {"bbox": [274, 547, 295, 560], "score": 0.93, "content": "\\overline{{A}}_{=t}", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [295, 547, 406, 561], "score": 1.0, "content": " globally trivial over ", "type": "text"}, {"bbox": [406, 547, 449, 561], "score": 0.94, "content": "(\\overline{{\\cal{A}}}/\\overline{{\\cal{G}}})_{=t}", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [449, 547, 537, 561], "score": 1.0, "content": ", at least for the", "type": "text"}], "index": 36}, {"bbox": [78, 561, 539, 578], "spans": [{"bbox": [78, 561, 191, 578], "score": 1.0, "content": "generic stratum with ", "type": "text"}, {"bbox": [191, 564, 233, 574], "score": 0.91, "content": "t=t_{\\mathrm{max}}", "type": "inline_equation", "height": 10, "width": 42}, {"bbox": [233, 561, 539, 578], "score": 1.0, "content": "? What sections do exist in these bundles, i.e. what gauge", "type": "text"}], "index": 37}, {"bbox": [78, 576, 191, 590], "spans": [{"bbox": [78, 576, 173, 590], "score": 1.0, "content": "fixings do exist in ", "type": "text"}, {"bbox": [174, 577, 184, 587], "score": 0.88, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [185, 576, 191, 590], "score": 1.0, "content": "?", "type": "text"}], "index": 38, "is_list_end_line": true}, {"bbox": [79, 590, 538, 605], "spans": [{"bbox": [79, 590, 538, 605], "score": 1.0, "content": "These problems are closely related to the so-called Gribov problem, the non-existence of", "type": "text"}], "index": 39, "is_list_start_line": true}, {"bbox": [79, 605, 538, 620], "spans": [{"bbox": [79, 605, 538, 620], "score": 1.0, "content": "global gauge fixings for classical connections in principal fiber bundles with compact, non-", "type": "text"}], "index": 40}, {"bbox": [79, 619, 538, 635], "spans": [{"bbox": [79, 619, 538, 635], "score": 1.0, "content": "commutative structure group (see, e.g., [19]). From this lots of difficulties result for the", "type": "text"}], "index": 41}, {"bbox": [79, 635, 486, 649], "spans": [{"bbox": [79, 635, 486, 649], "score": 1.0, "content": "quantization of such a Yang-Mills theory that are not circumvented up to now.", "type": "text"}], "index": 42, "is_list_end_line": true}], "index": 38, "bbox_fs": [78, 518, 539, 649]}]}
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[{"type": "text", "bbox": [63, 15, 537, 43], "content": "", "index": 0}, {"type": "text", "bbox": [63, 44, 538, 187], "content": "The reduction of our problem from structures in to those in was the crucial idea in the present paper. Since stabilizers in compact groups are even generated by a finite number of elements, we could model the gauge orbit type on a finite-dimensional space. Using an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on . This is the main result of our paper. Collecting connections of one and the same type we got the so-called strata whose openness was an immediate consequence of the slice theorem. In the next step we showed that the natural ordering on the set of the types encodes the topological properties of the strata. More precisely, we proved that the closure of a stratum contains (besides the stratum itself) exactly the union of all strata having a smaller type. This implied that this decomposition of is a topologically regular stratification.", "index": 1}, {"type": "text", "bbox": [63, 189, 538, 419], "content": "All these results hold in the classical case as well. This is very remarkable because our proofs used partially completely different ideas. However, two results of this paper go beyond the classical theorems. First, we were able to determine the full set of all gauge orbit types occurring in . This set is known for Sobolev connections – to the best of our knowlegde – only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem completely for -bundels over two-, three- and four-dimensional manifolds [18]. The main problem in the Sobolev case is the non-triviality of the bundle . This can exclude orbit types that occur in the trivial bundle . But, this problem is irrelevant for the Ashtekar framework: Every regular connection in every G-bundle over is contained in [2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in the Ashtekar framework there is a well-defined natural measure on . Using this we could show that the generic stratum has the total measure one; this is not true in the classical case. The proposition above implies now that the Faddeev-Popov determinant for the trans- formation from to is equal to 1. This, on the other hand, justifies the definition of the induced Haar measure on by projecting the corresponding measure for which has been discussed in detail in section 9.", "index": 2}, {"type": "text", "bbox": [64, 419, 537, 462], "content": "Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost) completely to the Ashtekar program. We emphasize that all assertions are valid for each compact structure group – both in the analytical and in the -smooth case.", "index": 3}, {"type": "text", "bbox": [63, 473, 537, 516], "content": "What could be next steps in this area? An important – and in this paper completely ignored – item is the physical interpretation of the gained knowledge. So we will conclude our paper with a few ideas that could link mathematics and physics:", "index": 4}, {"type": "list", "bbox": [64, 518, 538, 647], "content": "", "index": 5}]
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[{"bbox": [61, 45, 538, 60], "content": "The reduction of our problem from structures in to those in was the crucial idea in the", "parent_index": 1, "line_index": 0}, {"bbox": [61, 60, 539, 75], "content": "present paper. Since stabilizers in compact groups are even generated by a finite number of", "parent_index": 1, "line_index": 1}, {"bbox": [61, 74, 538, 90], "content": "elements, we could model the gauge orbit type on a finite-dimensional space. Using", "parent_index": 1, "line_index": 2}, {"bbox": [61, 89, 538, 103], "content": "an appropriate mapping we lifted the corresponding slice theorem to a slice theorem on .", "parent_index": 1, "line_index": 3}, {"bbox": [61, 103, 538, 119], "content": "This is the main result of our paper. Collecting connections of one and the same type we", "parent_index": 1, "line_index": 4}, {"bbox": [61, 118, 537, 133], "content": "got the so-called strata whose openness was an immediate consequence of the slice theorem.", "parent_index": 1, "line_index": 5}, {"bbox": [62, 133, 537, 147], "content": "In the next step we showed that the natural ordering on the set of the types encodes the", "parent_index": 1, "line_index": 6}, {"bbox": [63, 147, 536, 162], "content": "topological properties of the strata. More precisely, we proved that the closure of a stratum", "parent_index": 1, "line_index": 7}, {"bbox": [62, 162, 537, 176], "content": "contains (besides the stratum itself) exactly the union of all strata having a smaller type.", "parent_index": 1, "line_index": 8}, {"bbox": [63, 176, 482, 190], "content": "This implied that this decomposition of is a topologically regular stratification.", "parent_index": 1, "line_index": 9}, {"bbox": [63, 190, 537, 205], "content": "All these results hold in the classical case as well. This is very remarkable because our proofs", "parent_index": 2, "line_index": 0}, {"bbox": [63, 206, 537, 219], "content": "used partially completely different ideas. However, two results of this paper go beyond the", "parent_index": 2, "line_index": 1}, {"bbox": [63, 220, 536, 233], "content": "classical theorems. First, we were able to determine the full set of all gauge orbit types", "parent_index": 2, "line_index": 2}, {"bbox": [64, 234, 537, 247], "content": "occurring in . This set is known for Sobolev connections – to the best of our knowlegde", "parent_index": 2, "line_index": 3}, {"bbox": [64, 249, 536, 262], "content": "– only for certain bundles. Recently, Rudolph, Schmidt and Volobuev solved this problem", "parent_index": 2, "line_index": 4}, {"bbox": [64, 263, 537, 276], "content": "completely for -bundels over two-, three- and four-dimensional manifolds [18]. The", "parent_index": 2, "line_index": 5}, {"bbox": [63, 278, 537, 291], "content": "main problem in the Sobolev case is the non-triviality of the bundle . This can exclude", "parent_index": 2, "line_index": 6}, {"bbox": [62, 291, 538, 306], "content": "orbit types that occur in the trivial bundle . But, this problem is irrelevant for", "parent_index": 2, "line_index": 7}, {"bbox": [61, 305, 538, 320], "content": "the Ashtekar framework: Every regular connection in every G-bundle over is contained in", "parent_index": 2, "line_index": 8}, {"bbox": [63, 319, 538, 335], "content": "[2]. This means, in a certain sense, we only have to deal with trivial bundles. Second, in", "parent_index": 2, "line_index": 9}, {"bbox": [62, 335, 537, 348], "content": "the Ashtekar framework there is a well-defined natural measure on . Using this we could", "parent_index": 2, "line_index": 10}, {"bbox": [61, 349, 538, 364], "content": "show that the generic stratum has the total measure one; this is not true in the classical", "parent_index": 2, "line_index": 11}, {"bbox": [63, 365, 537, 378], "content": "case. The proposition above implies now that the Faddeev-Popov determinant for the trans-", "parent_index": 2, "line_index": 12}, {"bbox": [63, 378, 538, 392], "content": "formation from to is equal to 1. This, on the other hand, justifies the definition of", "parent_index": 2, "line_index": 13}, {"bbox": [61, 392, 538, 408], "content": "the induced Haar measure on by projecting the corresponding measure for which has", "parent_index": 2, "line_index": 14}, {"bbox": [63, 408, 252, 420], "content": "been discussed in detail in section 9.", "parent_index": 2, "line_index": 15}, {"bbox": [63, 421, 537, 435], "content": "Hence, we were able to ”transfer” the classical theory of strata in a certain sense (almost)", "parent_index": 3, "line_index": 0}, {"bbox": [63, 436, 538, 450], "content": "completely to the Ashtekar program. We emphasize that all assertions are valid for each", "parent_index": 3, "line_index": 1}, {"bbox": [63, 451, 456, 466], "content": "compact structure group – both in the analytical and in the -smooth case.", "parent_index": 3, "line_index": 2}, {"bbox": [63, 475, 537, 489], "content": "What could be next steps in this area? An important – and in this paper completely ignored", "parent_index": 4, "line_index": 0}, {"bbox": [63, 489, 537, 504], "content": "– item is the physical interpretation of the gained knowledge. So we will conclude our paper", "parent_index": 4, "line_index": 1}, {"bbox": [63, 504, 362, 518], "content": "with a few ideas that could link mathematics and physics:", "parent_index": 4, "line_index": 2}, {"bbox": [79, 518, 127, 534], "content": "Topology", "parent_index": 5, "line_index": 0}, {"bbox": [79, 533, 538, 547], "content": "What is the topological structure of the strata? Are they connected or is connected", "parent_index": 5, "line_index": 1}, {"bbox": [78, 547, 537, 561], "content": "itself (at least for connected )? Is globally trivial over , at least for the", "parent_index": 5, "line_index": 2}, {"bbox": [78, 561, 539, 578], "content": "generic stratum with ? What sections do exist in these bundles, i.e. what gauge", "parent_index": 5, "line_index": 3}, {"bbox": [78, 576, 191, 590], "content": "fixings do exist in ?", "parent_index": 5, "line_index": 4}, {"bbox": [79, 590, 538, 605], "content": "These problems are closely related to the so-called Gribov problem, the non-existence of", "parent_index": 5, "line_index": 5}, {"bbox": [79, 605, 538, 620], "content": "global gauge fixings for classical connections in principal fiber bundles with compact, non-", "parent_index": 5, "line_index": 6}, {"bbox": [79, 619, 538, 635], "content": "commutative structure group (see, e.g., [19]). From this lots of difficulties result for the", "parent_index": 5, "line_index": 7}, {"bbox": [79, 635, 486, 649], "content": "quantization of such a Yang-Mills theory that are not circumvented up to now.", "parent_index": 5, "line_index": 8}]
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[]
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[{"bbox": [315, 46, 323, 58], "content": "\\overline{{g}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [386, 48, 397, 56], "content": "\\mathbf{G}", "parent_index": 1, "subtype": "inline"}, {"bbox": [305, 76, 347, 88], "content": "[Z(\\mathbf{H}_{\\overline{{A}}})]", "parent_index": 1, "subtype": "inline"}, {"bbox": [523, 90, 533, 100], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [270, 176, 280, 187], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 1, "subtype": "inline"}, {"bbox": [130, 234, 140, 244], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [140, 264, 174, 276], "content": "S U(n)", "parent_index": 2, "subtype": "inline"}, {"bbox": [221, 264, 230, 273], "content": "P", "parent_index": 2, "subtype": "inline"}, {"bbox": [427, 279, 436, 288], "content": "P", "parent_index": 2, "subtype": "inline"}, {"bbox": [290, 293, 351, 305], "content": "M\\times S U(n)", "parent_index": 2, "subtype": "inline"}, {"bbox": [446, 308, 458, 316], "content": "M", "parent_index": 2, "subtype": "inline"}, {"bbox": [63, 321, 73, 331], "content": "\\overline{{{A}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [414, 335, 424, 345], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [145, 378, 155, 389], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [174, 378, 198, 392], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [216, 393, 240, 406], "content": "\\overline{{\\mathcal{A}}}/\\overline{{\\mathcal{G}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [472, 393, 482, 403], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 2, "subtype": "inline"}, {"bbox": [371, 452, 385, 461], "content": "C^{r}", "parent_index": 3, "subtype": "inline"}, {"bbox": [471, 533, 480, 543], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [232, 549, 243, 558], "content": "\\mathbf{G}", "parent_index": 5, "subtype": "inline"}, {"bbox": [274, 547, 295, 560], "content": "\\overline{{A}}_{=t}", "parent_index": 5, "subtype": "inline"}, {"bbox": [406, 547, 449, 561], "content": "(\\overline{{\\cal{A}}}/\\overline{{\\cal{G}}})_{=t}", "parent_index": 5, "subtype": "inline"}, {"bbox": [191, 564, 233, 574], "content": "t=t_{\\mathrm{max}}", "parent_index": 5, "subtype": "inline"}, {"bbox": [174, 577, 184, 587], "content": "\\overline{{\\mathcal{A}}}", "parent_index": 5, "subtype": "inline"}]
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[]
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Algebraic topology Is there a meaningful, i.e. especially non-trivial cohomology theory on $\overline{{\mathcal{A}}}$ ?7 Is it possible to construct this way characteristic classes or even topological invariants?
How are arbitrary measures distributed over single strata? In other words: What properties do measures have that are defined by the choice of a measure on each single stratum? This is extremely interesting, in particular, from the physical point of view because the choice of a $\mu_{0}$ -absolutely continuous measure $\mu$ on $\overline{{\mathcal{A}}}$ corresponds to the choice of an action functional $S$ on $\overline{{\mathcal{A}}}$ by $\textstyle{\int}\!\!{\overline{{A}}}\,f\;d\mu\,=\,{\int}\!\!{\overline{{A}}}\,f\;e^{-S}\;d\mu_{0}$ . According to Lebesgue’s decomposition theorem all measures whose support is not fully contained in the generic stratum have singular parts.
Finally, we have to stress that the present paper only investigates the case of pure gauge theories. Of course, this is physically not satisfying. Therefore the next goal should be the inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the aspects considered in the present paper did not play any role in Thiemann's paper.
# Acknowledgements
I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me to hope for the existence of a slice theorem on $\overline{{\mathcal{A}}}$ . Finally, I thank the Max-Planck-Institut fir Mathematik in den Naturwissenschaften for its generous promotion.
# References
[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity and nonabelian gauge theories. Class. Quant. Grav., 9:1433–1468, 1992.
[2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy $C^{*}$ algebras. In Knots and Quantum Gravity, edited by John C. Baez (Oxford Lecture Series in Mathematics and its Applications), Oxford University Press, Oxford, 1994.
[3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995.
[4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integration for gauge theories. J. Math. Phys., 36:2170–2191, 1995.
[5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thiemann. $S U(N)$ quantum Yang-Mills theory in two-dimensions: A complete solution. J. Math. Phys., 38:5453–5482, 1997.
[6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. J. Funct. Anal., 150:1–26, 1997.
[7] Glen E. Bredon. Introduction to Compact Transformation Groups. Academic Press, Inc., New York, 1972.
[8] N. Burbaki. Gruppy i algebry Li, Gl. IX (Kompaktnye vewestvennye gruppy Li). Izdatelьstvo «Mir», Moskva, 1986.
[9] Christian Fleischhack. Gauge Orbit Types for Generalized Connections. MIS-Preprint 2/2000, math-ph/0001006.
[10] Christian Fleischhack. Hyphs and the Ashtekar-Lewandowski Measure. MIS-Preprint 3/2000, math-ph/0001007.
[11] Christian Fleischhack. A new type of loop independence and $S U(N)$ quantum Yang-Mills theory in two dimensions. J. Math. Phys., 41:76–102, 2000.
[12] Witold Kondracki and Jan Rogulski. On the stratification of the orbit space for the action of automorphisms on connections (Dissertationes mathematicae 250). Warszawa, 1985.
[13] Witold Kondracki and Pawel Sadowski. Geometric structure on the orbit space of gauge connections. J. Geom. Phys., 3:421–434, 1986.
[14] Jerzy Lewandowski. Group of loops, holonomy maps, path bundle and path connection. Class. Quant. Grav., 10:879–904, 1993.
[15] Donald Marolf and Jose M. Mourao. On the support of the Ashtekar-Lewandowski measure. Commun. Math. Phys., 170:583–606, 1995.
[16] P. K. Mitter. Geometry of the space of gauge orbits and the Yang-Mills dynamical system. Lectures given at Carg´ese Summer Inst. on Recent Developments in Gauge Theories, Carg\`ese, 1979.
[17] Alan D. Rendall. Comment on a paper of Ashtekar and Isham. Class. Quant. Grav., 10:605–608, 1993.
[18] Gerd Rudolph, Matthias Schmidt, and Igor Volobuev. Classification of gauge orbit types for $S U(n)$ gauge theories (in preparation).
[19] I. M. Singer. Some remarks on the Gribov ambiguity. Commun. Math. Phys., 60:7–12, 1978.
[20] T. Thiemann. Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories. Class. Quant. Grav., 15:1487–1512, 1998.
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<html><body>
<p data-bbox="67 15 536 57">Algebraic topology Is there a meaningful, i.e. especially non-trivial cohomology theory on $\overline{{\mathcal{A}}}$ ?7 Is it possible to construct this way characteristic classes or even topological invariants? </p>
<p data-bbox="74 71 537 172">How are arbitrary measures distributed over single strata? In other words: What properties do measures have that are defined by the choice of a measure on each single stratum? This is extremely interesting, in particular, from the physical point of view because the choice of a $\mu_{0}$ -absolutely continuous measure $\mu$ on $\overline{{\mathcal{A}}}$ corresponds to the choice of an action functional $S$ on $\overline{{\mathcal{A}}}$ by $\textstyle{\int}\!\!{\overline{{A}}}\,f\;d\mu\,=\,{\int}\!\!{\overline{{A}}}\,f\;e^{-S}\;d\mu_{0}$ . According to Lebesgue’s decomposition theorem all measures whose support is not fully contained in the generic stratum have singular parts. </p>
<p data-bbox="62 174 538 231">Finally, we have to stress that the present paper only investigates the case of pure gauge theories. Of course, this is physically not satisfying. Therefore the next goal should be the inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the aspects considered in the present paper did not play any role in Thiemann's paper. </p>
<h1 data-bbox="63 252 226 272">Acknowledgements </h1>
<p data-bbox="63 282 538 355">I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me to hope for the existence of a slice theorem on $\overline{{\mathcal{A}}}$ . Finally, I thank the Max-Planck-Institut fir Mathematik in den Naturwissenschaften for its generous promotion. </p>
<h1 data-bbox="63 376 155 395">References </h1>
<p data-bbox="66 405 538 669">[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity and nonabelian gauge theories. Class. Quant. Grav., 9:1433–1468, 1992.
[2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy $C^{*}$ algebras. In Knots and Quantum Gravity, edited by John C. Baez (Oxford Lecture Series in Mathematics and its Applications), Oxford University Press, Oxford, 1994.
[3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995.
[4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integration for gauge theories. J. Math. Phys., 36:2170–2191, 1995.
[5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thiemann. $S U(N)$ quantum Yang-Mills theory in two-dimensions: A complete solution. J. Math. Phys., 38:5453–5482, 1997.
[6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. J. Funct. Anal., 150:1–26, 1997.
[7] Glen E. Bredon. Introduction to Compact Transformation Groups. Academic Press, Inc., New York, 1972.
[8] N. Burbaki. Gruppy i algebry Li, Gl. IX (Kompaktnye vewestvennye gruppy Li). Izdatelьstvo «Mir», Moskva, 1986.
[9] Christian Fleischhack. Gauge Orbit Types for Generalized Connections. MIS-Preprint 2/2000, math-ph/0001006.
[10] Christian Fleischhack. Hyphs and the Ashtekar-Lewandowski Measure. MIS-Preprint 3/2000, math-ph/0001007.
[11] Christian Fleischhack. A new type of loop independence and $S U(N)$ quantum Yang-Mills theory in two dimensions. J. Math. Phys., 41:76–102, 2000.
[12] Witold Kondracki and Jan Rogulski. On the stratification of the orbit space for the action of automorphisms on connections (Dissertationes mathematicae 250). Warszawa, 1985.
[13] Witold Kondracki and Pawel Sadowski. Geometric structure on the orbit space of gauge connections. J. Geom. Phys., 3:421–434, 1986.
[14] Jerzy Lewandowski. Group of loops, holonomy maps, path bundle and path connection. Class. Quant. Grav., 10:879–904, 1993.
[15] Donald Marolf and Jose M. Mourao. On the support of the Ashtekar-Lewandowski measure. Commun. Math. Phys., 170:583–606, 1995.
[16] P. K. Mitter. Geometry of the space of gauge orbits and the Yang-Mills dynamical system. Lectures given at Carg´ese Summer Inst. on Recent Developments in Gauge Theories, Carg\`ese, 1979.
[17] Alan D. Rendall. Comment on a paper of Ashtekar and Isham. Class. Quant. Grav., 10:605–608, 1993.
[18] Gerd Rudolph, Matthias Schmidt, and Igor Volobuev. Classification of gauge orbit types for $S U(n)$ gauge theories (in preparation).
[19] I. M. Singer. Some remarks on the Gribov ambiguity. Commun. Math. Phys., 60:7–12, 1978.
[20] T. Thiemann. Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories. Class. Quant. Grav., 15:1487–1512, 1998. </p>
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[{"type": "text", "text": "Algebraic topology Is there a meaningful, i.e. especially non-trivial cohomology theory on $\\overline{{\\mathcal{A}}}$ ?7 Is it possible to construct this way characteristic classes or even topological invariants? ", "page_idx": 19}, {"type": "text", "text": "How are arbitrary measures distributed over single strata? In other words: What properties do measures have that are defined by the choice of a measure on each single stratum? This is extremely interesting, in particular, from the physical point of view because the choice of a $\\mu_{0}$ -absolutely continuous measure $\\mu$ on $\\overline{{\\mathcal{A}}}$ corresponds to the choice of an action functional $S$ on $\\overline{{\\mathcal{A}}}$ by $\\textstyle{\\int}\\!\\!{\\overline{{A}}}\\,f\\;d\\mu\\,=\\,{\\int}\\!\\!{\\overline{{A}}}\\,f\\;e^{-S}\\;d\\mu_{0}$ . According to Lebesgue’s decomposition theorem all measures whose support is not fully contained in the generic stratum have singular parts. ", "page_idx": 19}, {"type": "text", "text": "Finally, we have to stress that the present paper only investigates the case of pure gauge theories. Of course, this is physically not satisfying. Therefore the next goal should be the inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the aspects considered in the present paper did not play any role in Thiemann's paper. ", "page_idx": 19}, {"type": "text", "text": "Acknowledgements ", "text_level": 1, "page_idx": 19}, {"type": "text", "text": "I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me to hope for the existence of a slice theorem on $\\overline{{\\mathcal{A}}}$ . Finally, I thank the Max-Planck-Institut fir Mathematik in den Naturwissenschaften for its generous promotion. ", "page_idx": 19}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 19}, {"type": "text", "text": "[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity and nonabelian gauge theories. Class. Quant. Grav., 9:1433–1468, 1992. \n[2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy $C^{*}$ algebras. In Knots and Quantum Gravity, edited by John C. Baez (Oxford Lecture Series in Mathematics and its Applications), Oxford University Press, Oxford, 1994. \n[3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connections via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995. \n[4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integration for gauge theories. J. Math. Phys., 36:2170–2191, 1995. \n[5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thiemann. $S U(N)$ quantum Yang-Mills theory in two-dimensions: A complete solution. J. Math. Phys., 38:5453–5482, 1997. \n[6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. J. Funct. Anal., 150:1–26, 1997. \n[7] Glen E. Bredon. Introduction to Compact Transformation Groups. Academic Press, Inc., New York, 1972. \n[8] N. Burbaki. Gruppy i algebry Li, Gl. IX (Kompaktnye vewestvennye gruppy Li). Izdatelьstvo «Mir», Moskva, 1986. \n[9] Christian Fleischhack. Gauge Orbit Types for Generalized Connections. MIS-Preprint 2/2000, math-ph/0001006. \n[10] Christian Fleischhack. Hyphs and the Ashtekar-Lewandowski Measure. MIS-Preprint 3/2000, math-ph/0001007. \n[11] Christian Fleischhack. A new type of loop independence and $S U(N)$ quantum Yang-Mills theory in two dimensions. J. Math. Phys., 41:76–102, 2000. \n[12] Witold Kondracki and Jan Rogulski. On the stratification of the orbit space for the action of automorphisms on connections (Dissertationes mathematicae 250). Warszawa, 1985. \n[13] Witold Kondracki and Pawel Sadowski. Geometric structure on the orbit space of gauge connections. J. Geom. Phys., 3:421–434, 1986. \n[14] Jerzy Lewandowski. Group of loops, holonomy maps, path bundle and path connection. Class. Quant. Grav., 10:879–904, 1993. \n[15] Donald Marolf and Jose M. Mourao. On the support of the Ashtekar-Lewandowski measure. Commun. Math. Phys., 170:583–606, 1995. \n[16] P. K. Mitter. Geometry of the space of gauge orbits and the Yang-Mills dynamical system. Lectures given at Carg´ese Summer Inst. on Recent Developments in Gauge Theories, Carg\\`ese, 1979. \n[17] Alan D. Rendall. Comment on a paper of Ashtekar and Isham. Class. Quant. Grav., 10:605–608, 1993. \n[18] Gerd Rudolph, Matthias Schmidt, and Igor Volobuev. Classification of gauge orbit types for $S U(n)$ gauge theories (in preparation). \n[19] I. M. Singer. Some remarks on the Gribov ambiguity. Commun. Math. Phys., 60:7–12, 1978. \n[20] T. Thiemann. Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories. Class. Quant. Grav., 15:1487–1512, 1998. ", "page_idx": 19}]
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In other words: What proper-", "type": "text"}], "index": 3}, {"bbox": [79, 89, 536, 103], "spans": [{"bbox": [79, 89, 536, 103], "score": 1.0, "content": "ties do measures have that are defined by the choice of a measure on each single stratum?", "type": "text"}], "index": 4}, {"bbox": [79, 104, 538, 119], "spans": [{"bbox": [79, 104, 538, 119], "score": 1.0, "content": "This is extremely interesting, in particular, from the physical point of view because the", "type": "text"}], "index": 5}, {"bbox": [78, 117, 537, 133], "spans": [{"bbox": [78, 117, 135, 133], "score": 1.0, "content": "choice of a ", "type": "text"}, {"bbox": [135, 123, 147, 131], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [147, 117, 308, 133], "score": 1.0, "content": "-absolutely continuous measure ", "type": "text"}, {"bbox": [308, 123, 315, 131], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [316, 117, 334, 133], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [334, 118, 344, 129], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [344, 117, 537, 133], "score": 1.0, "content": " corresponds to the choice of an action", "type": "text"}], "index": 6}, {"bbox": [77, 131, 538, 149], "spans": [{"bbox": [77, 131, 136, 149], "score": 1.0, "content": "functional ", "type": "text"}, {"bbox": [136, 135, 144, 144], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [144, 131, 165, 149], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [166, 133, 176, 144], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [176, 131, 198, 149], "score": 1.0, "content": " by", "type": "text"}, {"bbox": [198, 133, 319, 147], "score": 0.91, "content": "\\textstyle{\\int}\\!\\!{\\overline{{A}}}\\,f\\;d\\mu\\,=\\,{\\int}\\!\\!{\\overline{{A}}}\\,f\\;e^{-S}\\;d\\mu_{0}", "type": "inline_equation", "height": 14, "width": 121}, {"bbox": [320, 131, 538, 149], "score": 1.0, "content": ". According to Lebesgue’s decomposition", "type": "text"}], "index": 7}, {"bbox": [78, 146, 538, 162], "spans": [{"bbox": [78, 146, 538, 162], "score": 1.0, "content": "theorem all measures whose support is not fully contained in the generic stratum have", "type": "text"}], "index": 8}, {"bbox": [78, 162, 154, 176], "spans": [{"bbox": [78, 162, 154, 176], "score": 1.0, "content": "singular parts.", "type": "text"}], "index": 9}], "index": 6}, {"type": "text", "bbox": [62, 174, 538, 231], "lines": [{"bbox": [61, 175, 537, 191], "spans": [{"bbox": [61, 175, 537, 191], "score": 1.0, "content": "Finally, we have to stress that the present paper only investigates the case of pure gauge", "type": "text"}], "index": 10}, {"bbox": [62, 191, 537, 205], "spans": [{"bbox": [62, 191, 537, 205], "score": 1.0, "content": "theories. Of course, this is physically not satisfying. Therefore the next goal should be the", "type": "text"}], "index": 11}, {"bbox": [62, 204, 538, 219], "spans": [{"bbox": [62, 204, 538, 219], "score": 1.0, "content": "inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the", "type": "text"}], "index": 12}, {"bbox": [62, 218, 491, 235], "spans": [{"bbox": [62, 218, 491, 235], "score": 0.9957560300827026, "content": "aspects considered in the present paper did not play any role in Thiemann's paper.", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "title", "bbox": [63, 252, 226, 272], "lines": [{"bbox": [64, 255, 225, 272], "spans": [{"bbox": [64, 255, 225, 272], "score": 1.0, "content": "Acknowledgements", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [63, 282, 538, 355], "lines": [{"bbox": [60, 284, 537, 301], "spans": [{"bbox": [60, 284, 537, 301], "score": 1.0, "content": "I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote", "type": "text"}], "index": 15}, {"bbox": [63, 301, 537, 315], "spans": [{"bbox": [63, 301, 537, 315], "score": 1.0, "content": "my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the", "type": "text"}], "index": 16}, {"bbox": [61, 313, 538, 330], "spans": [{"bbox": [61, 313, 538, 330], "score": 1.0, "content": "drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me", "type": "text"}], "index": 17}, {"bbox": [63, 329, 536, 342], "spans": [{"bbox": [63, 330, 307, 342], "score": 1.0, "content": "to hope for the existence of a slice theorem on ", "type": "text"}, {"bbox": [307, 329, 317, 339], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [317, 330, 536, 342], "score": 1.0, "content": ". Finally, I thank the Max-Planck-Institut", "type": "text"}], "index": 18}, {"bbox": [62, 343, 430, 358], "spans": [{"bbox": [62, 343, 430, 358], "score": 0.9836291074752808, "content": "fir Mathematik in den Naturwissenschaften for its generous promotion.", "type": "text"}], "index": 19}], "index": 17}, {"type": "title", "bbox": [63, 376, 155, 395], "lines": [{"bbox": [63, 379, 156, 397], "spans": [{"bbox": [63, 379, 156, 397], "score": 1.0, "content": "References", "type": "text"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [66, 405, 538, 669], "lines": [{"bbox": [69, 409, 536, 425], "spans": [{"bbox": [69, 409, 536, 425], "score": 1.0, "content": "[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity", "type": "text"}], "index": 21}, {"bbox": [86, 424, 457, 439], "spans": [{"bbox": [86, 424, 457, 439], "score": 1.0, "content": "and nonabelian gauge theories. Class. Quant. 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Functional integration on spaces of connections. J.", "type": "text"}], "index": 33}, {"bbox": [88, 647, 241, 662], "spans": [{"bbox": [88, 647, 241, 662], "score": 1.0, "content": "Funct. Anal., 150:1–26, 1997.", "type": "text"}], "index": 34}], "index": 27.5}], "layout_bboxes": [], "page_idx": 19, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "20", "type": "text"}]}]}, {"type": "discarded", "bbox": [75, 666, 306, 678], "lines": [{"bbox": [76, 666, 306, 682], "spans": [{"bbox": [76, 666, 306, 682], "score": 1.0, "content": "7First abstract attempts can be found, e.g., in [4, 3].", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [67, 15, 536, 57], "lines": [{"bbox": [73, 16, 174, 32], "spans": [{"bbox": [73, 16, 174, 32], "score": 1.0, "content": "Algebraic topology", "type": "text"}], "index": 0}, {"bbox": [77, 30, 537, 46], "spans": [{"bbox": [77, 30, 445, 46], "score": 1.0, "content": "Is there a meaningful, i.e. especially non-trivial cohomology theory on ", "type": "text"}, {"bbox": [445, 32, 456, 42], "score": 0.86, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 11}, {"bbox": [456, 30, 537, 46], "score": 1.0, "content": "?7 Is it possible", "type": "text"}], "index": 1}, {"bbox": [79, 46, 459, 60], "spans": [{"bbox": [79, 46, 459, 60], "score": 1.0, "content": "to construct this way characteristic classes or even topological invariants?", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [73, 16, 537, 60]}, {"type": "text", "bbox": [74, 71, 537, 172], "lines": [{"bbox": [78, 74, 537, 89], "spans": [{"bbox": [78, 74, 537, 89], "score": 1.0, "content": "How are arbitrary measures distributed over single strata? In other words: What proper-", "type": "text"}], "index": 3}, {"bbox": [79, 89, 536, 103], "spans": [{"bbox": [79, 89, 536, 103], "score": 1.0, "content": "ties do measures have that are defined by the choice of a measure on each single stratum?", "type": "text"}], "index": 4}, {"bbox": [79, 104, 538, 119], "spans": [{"bbox": [79, 104, 538, 119], "score": 1.0, "content": "This is extremely interesting, in particular, from the physical point of view because the", "type": "text"}], "index": 5}, {"bbox": [78, 117, 537, 133], "spans": [{"bbox": [78, 117, 135, 133], "score": 1.0, "content": "choice of a ", "type": "text"}, {"bbox": [135, 123, 147, 131], "score": 0.9, "content": "\\mu_{0}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [147, 117, 308, 133], "score": 1.0, "content": "-absolutely continuous measure ", "type": "text"}, {"bbox": [308, 123, 315, 131], "score": 0.89, "content": "\\mu", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [316, 117, 334, 133], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [334, 118, 344, 129], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [344, 117, 537, 133], "score": 1.0, "content": " corresponds to the choice of an action", "type": "text"}], "index": 6}, {"bbox": [77, 131, 538, 149], "spans": [{"bbox": [77, 131, 136, 149], "score": 1.0, "content": "functional ", "type": "text"}, {"bbox": [136, 135, 144, 144], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [144, 131, 165, 149], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [166, 133, 176, 144], "score": 0.9, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [176, 131, 198, 149], "score": 1.0, "content": " by", "type": "text"}, {"bbox": [198, 133, 319, 147], "score": 0.91, "content": "\\textstyle{\\int}\\!\\!{\\overline{{A}}}\\,f\\;d\\mu\\,=\\,{\\int}\\!\\!{\\overline{{A}}}\\,f\\;e^{-S}\\;d\\mu_{0}", "type": "inline_equation", "height": 14, "width": 121}, {"bbox": [320, 131, 538, 149], "score": 1.0, "content": ". According to Lebesgue’s decomposition", "type": "text"}], "index": 7}, {"bbox": [78, 146, 538, 162], "spans": [{"bbox": [78, 146, 538, 162], "score": 1.0, "content": "theorem all measures whose support is not fully contained in the generic stratum have", "type": "text"}], "index": 8}, {"bbox": [78, 162, 154, 176], "spans": [{"bbox": [78, 162, 154, 176], "score": 1.0, "content": "singular parts.", "type": "text"}], "index": 9}], "index": 6, "bbox_fs": [77, 74, 538, 176]}, {"type": "text", "bbox": [62, 174, 538, 231], "lines": [{"bbox": [61, 175, 537, 191], "spans": [{"bbox": [61, 175, 537, 191], "score": 1.0, "content": "Finally, we have to stress that the present paper only investigates the case of pure gauge", "type": "text"}], "index": 10}, {"bbox": [62, 191, 537, 205], "spans": [{"bbox": [62, 191, 537, 205], "score": 1.0, "content": "theories. Of course, this is physically not satisfying. Therefore the next goal should be the", "type": "text"}], "index": 11}, {"bbox": [62, 204, 538, 219], "spans": [{"bbox": [62, 204, 538, 219], "score": 1.0, "content": "inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the", "type": "text"}], "index": 12}, {"bbox": [62, 218, 491, 235], "spans": [{"bbox": [62, 218, 491, 235], "score": 0.9957560300827026, "content": "aspects considered in the present paper did not play any role in Thiemann's paper.", "type": "text"}], "index": 13}], "index": 11.5, "bbox_fs": [61, 175, 538, 235]}, {"type": "title", "bbox": [63, 252, 226, 272], "lines": [{"bbox": [64, 255, 225, 272], "spans": [{"bbox": [64, 255, 225, 272], "score": 1.0, "content": "Acknowledgements", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [63, 282, 538, 355], "lines": [{"bbox": [60, 284, 537, 301], "spans": [{"bbox": [60, 284, 537, 301], "score": 1.0, "content": "I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote", "type": "text"}], "index": 15}, {"bbox": [63, 301, 537, 315], "spans": [{"bbox": [63, 301, 537, 315], "score": 1.0, "content": "my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the", "type": "text"}], "index": 16}, {"bbox": [61, 313, 538, 330], "spans": [{"bbox": [61, 313, 538, 330], "score": 1.0, "content": "drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me", "type": "text"}], "index": 17}, {"bbox": [63, 329, 536, 342], "spans": [{"bbox": [63, 330, 307, 342], "score": 1.0, "content": "to hope for the existence of a slice theorem on ", "type": "text"}, {"bbox": [307, 329, 317, 339], "score": 0.91, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [317, 330, 536, 342], "score": 1.0, "content": ". Finally, I thank the Max-Planck-Institut", "type": "text"}], "index": 18}, {"bbox": [62, 343, 430, 358], "spans": [{"bbox": [62, 343, 430, 358], "score": 0.9836291074752808, "content": "fir Mathematik in den Naturwissenschaften for its generous promotion.", "type": "text"}], "index": 19}], "index": 17, "bbox_fs": [60, 284, 538, 358]}, {"type": "title", "bbox": [63, 376, 155, 395], "lines": [{"bbox": [63, 379, 156, 397], "spans": [{"bbox": [63, 379, 156, 397], "score": 1.0, "content": "References", "type": "text"}], "index": 20}], "index": 20}, {"type": "list", "bbox": [66, 405, 538, 669], "lines": [{"bbox": [69, 409, 536, 425], "spans": [{"bbox": [69, 409, 536, 425], "score": 1.0, "content": "[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity", "type": "text"}], "index": 21, "is_list_start_line": true}, {"bbox": [86, 424, 457, 439], "spans": [{"bbox": [86, 424, 457, 439], "score": 1.0, "content": "and nonabelian gauge theories. Class. Quant. 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[{"type": "text", "bbox": [67, 15, 536, 57], "content": "Algebraic topology Is there a meaningful, i.e. especially non-trivial cohomology theory on ?7 Is it possible to construct this way characteristic classes or even topological invariants?", "index": 0}, {"type": "text", "bbox": [74, 71, 537, 172], "content": "How are arbitrary measures distributed over single strata? In other words: What proper- ties do measures have that are defined by the choice of a measure on each single stratum? This is extremely interesting, in particular, from the physical point of view because the choice of a -absolutely continuous measure on corresponds to the choice of an action functional on by . According to Lebesgue’s decomposition theorem all measures whose support is not fully contained in the generic stratum have singular parts.", "index": 1}, {"type": "text", "bbox": [62, 174, 538, 231], "content": "Finally, we have to stress that the present paper only investigates the case of pure gauge theories. Of course, this is physically not satisfying. Therefore the next goal should be the inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the aspects considered in the present paper did not play any role in Thiemann's paper.", "index": 2}, {"type": "title", "bbox": [63, 252, 226, 272], "content": "Acknowledgements", "index": 3}, {"type": "text", "bbox": [63, 282, 538, 355], "content": "I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me to hope for the existence of a slice theorem on . Finally, I thank the Max-Planck-Institut fir Mathematik in den Naturwissenschaften for its generous promotion.", "index": 4}, {"type": "title", "bbox": [63, 376, 155, 395], "content": "References", "index": 5}, {"type": "list", "bbox": [66, 405, 538, 669], "content": "", "index": 6}]
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[{"bbox": [73, 16, 174, 32], "content": "Algebraic topology", "parent_index": 0, "line_index": 0}, {"bbox": [77, 30, 537, 46], "content": "Is there a meaningful, i.e. especially non-trivial cohomology theory on ?7 Is it possible", "parent_index": 0, "line_index": 1}, {"bbox": [79, 46, 459, 60], "content": "to construct this way characteristic classes or even topological invariants?", "parent_index": 0, "line_index": 2}, {"bbox": [78, 74, 537, 89], "content": "How are arbitrary measures distributed over single strata? In other words: What proper-", "parent_index": 1, "line_index": 0}, {"bbox": [79, 89, 536, 103], "content": "ties do measures have that are defined by the choice of a measure on each single stratum?", "parent_index": 1, "line_index": 1}, {"bbox": [79, 104, 538, 119], "content": "This is extremely interesting, in particular, from the physical point of view because the", "parent_index": 1, "line_index": 2}, {"bbox": [78, 117, 537, 133], "content": "choice of a -absolutely continuous measure on corresponds to the choice of an action", "parent_index": 1, "line_index": 3}, {"bbox": [77, 131, 538, 149], "content": "functional on by . According to Lebesgue’s decomposition", "parent_index": 1, "line_index": 4}, {"bbox": [78, 146, 538, 162], "content": "theorem all measures whose support is not fully contained in the generic stratum have", "parent_index": 1, "line_index": 5}, {"bbox": [78, 162, 154, 176], "content": "singular parts.", "parent_index": 1, "line_index": 6}, {"bbox": [61, 175, 537, 191], "content": "Finally, we have to stress that the present paper only investigates the case of pure gauge", "parent_index": 2, "line_index": 0}, {"bbox": [62, 191, 537, 205], "content": "theories. Of course, this is physically not satisfying. Therefore the next goal should be the", "parent_index": 2, "line_index": 1}, {"bbox": [62, 204, 538, 219], "content": "inclusion of matter fields. A first step has already been done by Thiemann [20] whereas the", "parent_index": 2, "line_index": 2}, {"bbox": [62, 218, 491, 235], "content": "aspects considered in the present paper did not play any role in Thiemann's paper.", "parent_index": 2, "line_index": 3}, {"bbox": [64, 255, 225, 272], "content": "Acknowledgements", "parent_index": 3, "line_index": 0}, {"bbox": [60, 284, 537, 301], "content": "I am very grateful to Gerd Rudolph and Eberhard Zeidler for their great support while I wrote", "parent_index": 4, "line_index": 0}, {"bbox": [63, 301, 537, 315], "content": "my diploma thesis and the present paper. Additionally, I thank Gerd Rudolph for reading the", "parent_index": 4, "line_index": 1}, {"bbox": [61, 313, 538, 330], "content": "drafts. Moreover, I am grateful to Domenico Giulini and Matthias Schmidt for convincing me", "parent_index": 4, "line_index": 2}, {"bbox": [63, 329, 536, 342], "content": "to hope for the existence of a slice theorem on . Finally, I thank the Max-Planck-Institut", "parent_index": 4, "line_index": 3}, {"bbox": [62, 343, 430, 358], "content": "fir Mathematik in den Naturwissenschaften for its generous promotion.", "parent_index": 4, "line_index": 4}, {"bbox": [63, 379, 156, 397], "content": "References", "parent_index": 5, "line_index": 0}, {"bbox": [69, 409, 536, 425], "content": "[1] Abhay Ashtekar and C. J. Isham. Representations of the holonomy algebras of gravity", "parent_index": 6, "line_index": 0}, {"bbox": [86, 424, 457, 439], "content": "and nonabelian gauge theories. Class. Quant. Grav., 9:1433–1468, 1992.", "parent_index": 6, "line_index": 1}, {"bbox": [68, 448, 537, 464], "content": "[2] Abhay Ashtekar and Jerzy Lewandowski. Representation theory of analytic holonomy", "parent_index": 6, "line_index": 2}, {"bbox": [87, 463, 538, 478], "content": "algebras. In Knots and Quantum Gravity, edited by John C. 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# Invariance Theorems for Supersymmetric Yang-Mills Theories
Savdeep Sethi $^*1$ and Mark Stern†2
∗ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA † Department of Mathematics, Duke University, Durham, NC 27706, USA
We consider quantum mechanical Yang-Mills theories with eight supercharges and a $S p i n(5)\times S U(2)_{R}$ flavor symmetry. We show that all normalizable ground states in these gauge theories are invariant under this flavor symmetry. This includes, as a special case, all bound states of D0-branes and D4-branes. As a consequence, all bound states of D0-branes are invariant under the $S p i n(9)$ flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique.
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<html><body>
<h1 data-bbox="118 185 493 237">Invariance Theorems for Supersymmetric Yang-Mills Theories </h1>
<p data-bbox="212 281 394 298">Savdeep Sethi $^*1$ and Mark Stern†2 </p>
<p data-bbox="78 309 532 324">∗ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA † Department of Mathematics, Duke University, Durham, NC 27706, USA </p>
<p data-bbox="71 398 542 509">We consider quantum mechanical Yang-Mills theories with eight supercharges and a $S p i n(5)\times S U(2)_{R}$ flavor symmetry. We show that all normalizable ground states in these gauge theories are invariant under this flavor symmetry. This includes, as a special case, all bound states of D0-branes and D4-branes. As a consequence, all bound states of D0-branes are invariant under the $S p i n(9)$ flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique. </p>
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# 1. Introduction
The existence of normalizable vacua in supersymetric Yang-Mills theories is a question that arises in many different contexts in string theory and field theory. Index arguments can be used to determine whether any vacua exist, but not exactly how many vacua. An index only counts the difference between the number of bosonic and fermionic vacua. To count the actual number of vacua, we need more information such as how the vacua transform under the global symmetries of the theory.
In this paper, we consider quantum mechanical Yang-Mills theories with eight supercharges and an $S p i n(5)\times S U(2)_{R}$ symmetry. We take our theories to be dimensional reductions of $d=6$ N=1 Yang-Mills theories coupled to matter. The question of normalizable ground states in these models arises in the study of bound states of D0-branes and D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to bind using $L^{2}$ index arguments [3] generalized to theories without a gap. Other examples from string theory involve D0-branes moving on orbifolds [4], and the question of counting H-monopoles in the heterotic string [5,3].
In the following section, we describe the field content and symmetries of these gauge theories. We then show that all normalizable ground states in these theories must be invariant under the $S U(2)_{R}$ symmetry. The argument we give is suggested by recent work on the $L^{2}$ -cohomology of hyperKahler spaces by Hitchin [6]. Our result should have implications for defining and computing the $L^{2}$ -cohomology of instanton moduli spaces. Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind under consideration. For example, the moduli space of $U(N)$ instantons in $\mathbb{R}^{4}$ appears as the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these spaces can be singular, their embedding into quantum mechanical gauge theory provides a natural regularization of the singularities. Heuristically, the wavefunction for a state corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb branch. It would be interesting to explore this connection further.
There is a second $R$ -symmetry in these theories which comes from the dimensional reduction of the Lorentz group. For reductions of $d\,=\,6$ N=1 Yang-Mills theories, this is a $S p i n(5)$ symmetry. Using basically the same argument as in the case of the $S U(2)_{R}$ symmetry, we show that all normalizable ground states in these theories are invariant under this $S p i n(5)$ symmetry. For reductions of $d=10$ N=1 Yang-Mills theories [7], the $R$ -symmetry group is $S p i n(9)$ . It is quite straightforward to argue that as a consequence of the $S U(2)_{R}\times S p i n(5)$ invariance theorem, all ground states in these theories with sixteen supercharges must be invariant under the $S p i n(9)$ symmetry.
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<html><body>
<h1 data-bbox="71 71 165 86">1. Introduction </h1>
<p data-bbox="70 98 542 210">The existence of normalizable vacua in supersymetric Yang-Mills theories is a question that arises in many different contexts in string theory and field theory. Index arguments can be used to determine whether any vacua exist, but not exactly how many vacua. An index only counts the difference between the number of bosonic and fermionic vacua. To count the actual number of vacua, we need more information such as how the vacua transform under the global symmetries of the theory. </p>
<p data-bbox="70 214 542 366">In this paper, we consider quantum mechanical Yang-Mills theories with eight supercharges and an $S p i n(5)\times S U(2)_{R}$ symmetry. We take our theories to be dimensional reductions of $d=6$ N=1 Yang-Mills theories coupled to matter. The question of normalizable ground states in these models arises in the study of bound states of D0-branes and D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to bind using $L^{2}$ index arguments [3] generalized to theories without a gap. Other examples from string theory involve D0-branes moving on orbifolds [4], and the question of counting H-monopoles in the heterotic string [5,3]. </p>
<p data-bbox="70 369 542 599">In the following section, we describe the field content and symmetries of these gauge theories. We then show that all normalizable ground states in these theories must be invariant under the $S U(2)_{R}$ symmetry. The argument we give is suggested by recent work on the $L^{2}$ -cohomology of hyperKahler spaces by Hitchin [6]. Our result should have implications for defining and computing the $L^{2}$ -cohomology of instanton moduli spaces. Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind under consideration. For example, the moduli space of $U(N)$ instantons in $\mathbb{R}^{4}$ appears as the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these spaces can be singular, their embedding into quantum mechanical gauge theory provides a natural regularization of the singularities. Heuristically, the wavefunction for a state corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb branch. It would be interesting to explore this connection further. </p>
<p data-bbox="70 603 542 716">There is a second $R$ -symmetry in these theories which comes from the dimensional reduction of the Lorentz group. For reductions of $d\,=\,6$ N=1 Yang-Mills theories, this is a $S p i n(5)$ symmetry. Using basically the same argument as in the case of the $S U(2)_{R}$ symmetry, we show that all normalizable ground states in these theories are invariant under this $S p i n(5)$ symmetry. For reductions of $d=10$ N=1 Yang-Mills theories [7], the $R$ -symmetry group is $S p i n(9)$ . It is quite straightforward to argue that as a consequence of the $S U(2)_{R}\times S p i n(5)$ invariance theorem, all ground states in these theories with sixteen supercharges must be invariant under the $S p i n(9)$ symmetry. </p>
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0001189v2
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Index arguments", "type": "text"}], "index": 2}, {"bbox": [71, 140, 540, 154], "spans": [{"bbox": [71, 140, 540, 154], "score": 1.0, "content": "can be used to determine whether any vacua exist, but not exactly how many vacua.", "type": "text"}], "index": 3}, {"bbox": [72, 160, 540, 173], "spans": [{"bbox": [72, 160, 540, 173], "score": 1.0, "content": "An index only counts the difference between the number of bosonic and fermionic vacua.", "type": "text"}], "index": 4}, {"bbox": [71, 179, 540, 192], "spans": [{"bbox": [71, 179, 540, 192], "score": 1.0, "content": "To count the actual number of vacua, we need more information such as how the vacua", "type": "text"}], "index": 5}, {"bbox": [70, 198, 350, 213], "spans": [{"bbox": [70, 198, 350, 213], "score": 1.0, "content": "transform under the global symmetries of the theory.", "type": "text"}], "index": 6}], "index": 3.5}, {"type": "text", "bbox": [70, 214, 542, 366], "lines": [{"bbox": [93, 217, 539, 233], "spans": [{"bbox": [93, 217, 539, 233], "score": 1.0, "content": "In this paper, we consider quantum mechanical Yang-Mills theories with eight su-", "type": "text"}], "index": 7}, {"bbox": [70, 237, 540, 252], "spans": [{"bbox": [70, 237, 171, 252], "score": 1.0, "content": "percharges and an ", "type": "text"}, {"bbox": [171, 238, 267, 250], "score": 0.94, "content": "S p i n(5)\\times S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [268, 237, 540, 252], "score": 1.0, "content": " symmetry. We take our theories to be dimensional", "type": "text"}], "index": 8}, {"bbox": [71, 257, 540, 271], "spans": [{"bbox": [71, 257, 143, 271], "score": 1.0, "content": "reductions of ", "type": "text"}, {"bbox": [143, 258, 172, 267], "score": 0.92, "content": "d=6", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [172, 257, 540, 271], "score": 1.0, "content": " N=1 Yang-Mills theories coupled to matter. The question of normal-", "type": "text"}], "index": 9}, {"bbox": [70, 276, 541, 290], "spans": [{"bbox": [70, 276, 541, 290], "score": 1.0, "content": "izable ground states in these models arises in the study of bound states of D0-branes and", "type": "text"}], "index": 10}, {"bbox": [70, 293, 542, 311], "spans": [{"bbox": [70, 293, 542, 311], "score": 1.0, "content": "D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to", "type": "text"}], "index": 11}, {"bbox": [71, 313, 540, 330], "spans": [{"bbox": [71, 313, 129, 330], "score": 1.0, "content": "bind using ", "type": "text"}, {"bbox": [129, 315, 143, 325], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [144, 313, 540, 330], "score": 1.0, "content": " index arguments [3] generalized to theories without a gap. Other examples", "type": "text"}], "index": 12}, {"bbox": [70, 332, 541, 350], "spans": [{"bbox": [70, 332, 541, 350], "score": 1.0, "content": "from string theory involve D0-branes moving on orbifolds [4], and the question of counting", "type": "text"}], "index": 13}, {"bbox": [70, 352, 288, 369], "spans": [{"bbox": [70, 352, 288, 369], "score": 1.0, "content": "H-monopoles in the heterotic string [5,3].", "type": "text"}], "index": 14}], "index": 10.5}, {"type": "text", "bbox": [70, 369, 542, 599], "lines": [{"bbox": [94, 372, 541, 388], "spans": [{"bbox": [94, 372, 541, 388], "score": 1.0, "content": "In the following section, we describe the field content and symmetries of these gauge", "type": "text"}], "index": 15}, {"bbox": [72, 393, 541, 406], "spans": [{"bbox": [72, 393, 541, 406], "score": 1.0, "content": "theories. We then show that all normalizable ground states in these theories must be", "type": "text"}], "index": 16}, {"bbox": [71, 412, 541, 426], "spans": [{"bbox": [71, 412, 181, 426], "score": 1.0, "content": "invariant under the ", "type": "text"}, {"bbox": [181, 412, 222, 425], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [222, 412, 541, 426], "score": 1.0, "content": " symmetry. The argument we give is suggested by recent", "type": "text"}], "index": 17}, {"bbox": [71, 431, 541, 446], "spans": [{"bbox": [71, 431, 137, 446], "score": 1.0, "content": "work on the ", "type": "text"}, {"bbox": [137, 431, 151, 441], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [151, 431, 541, 446], "score": 0.9890029430389404, "content": "-cohomology of hyperKahler spaces by Hitchin [6]. Our result should have", "type": "text"}], "index": 18}, {"bbox": [71, 451, 539, 465], "spans": [{"bbox": [71, 451, 309, 465], "score": 1.0, "content": "implications for defining and computing the ", "type": "text"}, {"bbox": [310, 451, 324, 461], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [324, 451, 539, 465], "score": 1.0, "content": "-cohomology of instanton moduli spaces.", "type": "text"}], "index": 19}, {"bbox": [70, 470, 541, 486], "spans": [{"bbox": [70, 470, 541, 486], "score": 1.0, "content": "Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind", "type": "text"}], "index": 20}, {"bbox": [70, 488, 542, 505], "spans": [{"bbox": [70, 488, 359, 505], "score": 1.0, "content": "under consideration. For example, the moduli space of ", "type": "text"}, {"bbox": [360, 490, 389, 503], "score": 0.95, "content": "U(N)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [390, 488, 464, 505], "score": 1.0, "content": " instantons in ", "type": "text"}, {"bbox": [464, 488, 480, 500], "score": 0.92, "content": "\\mathbb{R}^{4}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [481, 488, 542, 505], "score": 1.0, "content": " appears as", "type": "text"}], "index": 21}, {"bbox": [71, 509, 541, 524], "spans": [{"bbox": [71, 509, 541, 524], "score": 1.0, "content": "the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these", "type": "text"}], "index": 22}, {"bbox": [70, 529, 541, 543], "spans": [{"bbox": [70, 529, 541, 543], "score": 1.0, "content": "spaces can be singular, their embedding into quantum mechanical gauge theory provides", "type": "text"}], "index": 23}, {"bbox": [70, 547, 541, 561], "spans": [{"bbox": [70, 547, 541, 561], "score": 1.0, "content": "a natural regularization of the singularities. Heuristically, the wavefunction for a state", "type": "text"}], "index": 24}, {"bbox": [70, 565, 541, 582], "spans": [{"bbox": [70, 565, 541, 582], "score": 1.0, "content": "corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb", "type": "text"}], "index": 25}, {"bbox": [70, 585, 418, 601], "spans": [{"bbox": [70, 585, 418, 601], "score": 1.0, "content": "branch. It would be interesting to explore this connection further.", "type": "text"}], "index": 26}], "index": 20.5}, {"type": "text", "bbox": [70, 603, 542, 716], "lines": [{"bbox": [95, 606, 541, 621], "spans": [{"bbox": [95, 606, 194, 621], "score": 1.0, "content": "There is a second ", "type": "text"}, {"bbox": [194, 608, 203, 617], "score": 0.91, "content": "R", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [204, 606, 541, 621], "score": 1.0, "content": "-symmetry in these theories which comes from the dimensional", "type": "text"}], "index": 27}, {"bbox": [70, 624, 541, 639], "spans": [{"bbox": [70, 624, 344, 639], "score": 1.0, "content": "reduction of the Lorentz group. For reductions of ", "type": "text"}, {"bbox": [344, 627, 376, 636], "score": 0.92, "content": "d\\,=\\,6", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [376, 624, 541, 639], "score": 1.0, "content": " N=1 Yang-Mills theories, this", "type": "text"}], "index": 28}, {"bbox": [69, 645, 539, 659], "spans": [{"bbox": [69, 645, 93, 659], "score": 1.0, "content": "is a ", "type": "text"}, {"bbox": [94, 645, 135, 658], "score": 0.92, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [135, 645, 498, 659], "score": 1.0, "content": " symmetry. Using basically the same argument as in the case of the ", "type": "text"}, {"bbox": [499, 645, 539, 658], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}], "index": 29}, {"bbox": [70, 664, 541, 678], "spans": [{"bbox": [70, 664, 541, 678], "score": 1.0, "content": "symmetry, we show that all normalizable ground states in these theories are invariant", "type": "text"}], "index": 30}, {"bbox": [70, 683, 542, 699], "spans": [{"bbox": [70, 683, 129, 699], "score": 1.0, "content": "under this ", "type": "text"}, {"bbox": [129, 684, 170, 697], "score": 0.92, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [171, 683, 328, 699], "score": 1.0, "content": " symmetry. For reductions of ", "type": "text"}, {"bbox": [329, 685, 364, 694], "score": 0.91, "content": "d=10", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [365, 683, 542, 699], "score": 1.0, "content": " N=1 Yang-Mills theories [7], the", "type": "text"}], "index": 31}, {"bbox": [71, 703, 542, 719], "spans": [{"bbox": [71, 705, 81, 713], "score": 0.9, "content": "R", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [81, 703, 184, 719], "score": 1.0, "content": "-symmetry group is ", "type": "text"}, {"bbox": [184, 704, 225, 716], "score": 0.91, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [225, 703, 542, 719], "score": 1.0, "content": ". It is quite straightforward to argue that as a consequence of", "type": "text"}], "index": 32}], "index": 29.5}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [71, 71, 165, 86], "lines": [{"bbox": [71, 74, 165, 85], "spans": [{"bbox": [71, 74, 165, 85], "score": 1.0, "content": "1. Introduction", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [70, 98, 542, 210], "lines": [{"bbox": [94, 100, 540, 116], "spans": [{"bbox": [94, 100, 540, 116], "score": 1.0, "content": "The existence of normalizable vacua in supersymetric Yang-Mills theories is a question", "type": "text"}], "index": 1}, {"bbox": [70, 120, 541, 136], "spans": [{"bbox": [70, 120, 541, 136], "score": 1.0, "content": "that arises in many different contexts in string theory and field theory. Index arguments", "type": "text"}], "index": 2}, {"bbox": [71, 140, 540, 154], "spans": [{"bbox": [71, 140, 540, 154], "score": 1.0, "content": "can be used to determine whether any vacua exist, but not exactly how many vacua.", "type": "text"}], "index": 3}, {"bbox": [72, 160, 540, 173], "spans": [{"bbox": [72, 160, 540, 173], "score": 1.0, "content": "An index only counts the difference between the number of bosonic and fermionic vacua.", "type": "text"}], "index": 4}, {"bbox": [71, 179, 540, 192], "spans": [{"bbox": [71, 179, 540, 192], "score": 1.0, "content": "To count the actual number of vacua, we need more information such as how the vacua", "type": "text"}], "index": 5}, {"bbox": [70, 198, 350, 213], "spans": [{"bbox": [70, 198, 350, 213], "score": 1.0, "content": "transform under the global symmetries of the theory.", "type": "text"}], "index": 6}], "index": 3.5, "bbox_fs": [70, 100, 541, 213]}, {"type": "text", "bbox": [70, 214, 542, 366], "lines": [{"bbox": [93, 217, 539, 233], "spans": [{"bbox": [93, 217, 539, 233], "score": 1.0, "content": "In this paper, we consider quantum mechanical Yang-Mills theories with eight su-", "type": "text"}], "index": 7}, {"bbox": [70, 237, 540, 252], "spans": [{"bbox": [70, 237, 171, 252], "score": 1.0, "content": "percharges and an ", "type": "text"}, {"bbox": [171, 238, 267, 250], "score": 0.94, "content": "S p i n(5)\\times S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [268, 237, 540, 252], "score": 1.0, "content": " symmetry. We take our theories to be dimensional", "type": "text"}], "index": 8}, {"bbox": [71, 257, 540, 271], "spans": [{"bbox": [71, 257, 143, 271], "score": 1.0, "content": "reductions of ", "type": "text"}, {"bbox": [143, 258, 172, 267], "score": 0.92, "content": "d=6", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [172, 257, 540, 271], "score": 1.0, "content": " N=1 Yang-Mills theories coupled to matter. The question of normal-", "type": "text"}], "index": 9}, {"bbox": [70, 276, 541, 290], "spans": [{"bbox": [70, 276, 541, 290], "score": 1.0, "content": "izable ground states in these models arises in the study of bound states of D0-branes and", "type": "text"}], "index": 10}, {"bbox": [70, 293, 542, 311], "spans": [{"bbox": [70, 293, 542, 311], "score": 1.0, "content": "D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to", "type": "text"}], "index": 11}, {"bbox": [71, 313, 540, 330], "spans": [{"bbox": [71, 313, 129, 330], "score": 1.0, "content": "bind using ", "type": "text"}, {"bbox": [129, 315, 143, 325], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [144, 313, 540, 330], "score": 1.0, "content": " index arguments [3] generalized to theories without a gap. Other examples", "type": "text"}], "index": 12}, {"bbox": [70, 332, 541, 350], "spans": [{"bbox": [70, 332, 541, 350], "score": 1.0, "content": "from string theory involve D0-branes moving on orbifolds [4], and the question of counting", "type": "text"}], "index": 13}, {"bbox": [70, 352, 288, 369], "spans": [{"bbox": [70, 352, 288, 369], "score": 1.0, "content": "H-monopoles in the heterotic string [5,3].", "type": "text"}], "index": 14}], "index": 10.5, "bbox_fs": [70, 217, 542, 369]}, {"type": "text", "bbox": [70, 369, 542, 599], "lines": [{"bbox": [94, 372, 541, 388], "spans": [{"bbox": [94, 372, 541, 388], "score": 1.0, "content": "In the following section, we describe the field content and symmetries of these gauge", "type": "text"}], "index": 15}, {"bbox": [72, 393, 541, 406], "spans": [{"bbox": [72, 393, 541, 406], "score": 1.0, "content": "theories. We then show that all normalizable ground states in these theories must be", "type": "text"}], "index": 16}, {"bbox": [71, 412, 541, 426], "spans": [{"bbox": [71, 412, 181, 426], "score": 1.0, "content": "invariant under the ", "type": "text"}, {"bbox": [181, 412, 222, 425], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [222, 412, 541, 426], "score": 1.0, "content": " symmetry. The argument we give is suggested by recent", "type": "text"}], "index": 17}, {"bbox": [71, 431, 541, 446], "spans": [{"bbox": [71, 431, 137, 446], "score": 1.0, "content": "work on the ", "type": "text"}, {"bbox": [137, 431, 151, 441], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [151, 431, 541, 446], "score": 0.9890029430389404, "content": "-cohomology of hyperKahler spaces by Hitchin [6]. Our result should have", "type": "text"}], "index": 18}, {"bbox": [71, 451, 539, 465], "spans": [{"bbox": [71, 451, 309, 465], "score": 1.0, "content": "implications for defining and computing the ", "type": "text"}, {"bbox": [310, 451, 324, 461], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [324, 451, 539, 465], "score": 1.0, "content": "-cohomology of instanton moduli spaces.", "type": "text"}], "index": 19}, {"bbox": [70, 470, 541, 486], "spans": [{"bbox": [70, 470, 541, 486], "score": 1.0, "content": "Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind", "type": "text"}], "index": 20}, {"bbox": [70, 488, 542, 505], "spans": [{"bbox": [70, 488, 359, 505], "score": 1.0, "content": "under consideration. For example, the moduli space of ", "type": "text"}, {"bbox": [360, 490, 389, 503], "score": 0.95, "content": "U(N)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [390, 488, 464, 505], "score": 1.0, "content": " instantons in ", "type": "text"}, {"bbox": [464, 488, 480, 500], "score": 0.92, "content": "\\mathbb{R}^{4}", "type": "inline_equation", "height": 12, "width": 16}, {"bbox": [481, 488, 542, 505], "score": 1.0, "content": " appears as", "type": "text"}], "index": 21}, {"bbox": [71, 509, 541, 524], "spans": [{"bbox": [71, 509, 541, 524], "score": 1.0, "content": "the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these", "type": "text"}], "index": 22}, {"bbox": [70, 529, 541, 543], "spans": [{"bbox": [70, 529, 541, 543], "score": 1.0, "content": "spaces can be singular, their embedding into quantum mechanical gauge theory provides", "type": "text"}], "index": 23}, {"bbox": [70, 547, 541, 561], "spans": [{"bbox": [70, 547, 541, 561], "score": 1.0, "content": "a natural regularization of the singularities. Heuristically, the wavefunction for a state", "type": "text"}], "index": 24}, {"bbox": [70, 565, 541, 582], "spans": [{"bbox": [70, 565, 541, 582], "score": 1.0, "content": "corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb", "type": "text"}], "index": 25}, {"bbox": [70, 585, 418, 601], "spans": [{"bbox": [70, 585, 418, 601], "score": 1.0, "content": "branch. It would be interesting to explore this connection further.", "type": "text"}], "index": 26}], "index": 20.5, "bbox_fs": [70, 372, 542, 601]}, {"type": "text", "bbox": [70, 603, 542, 716], "lines": [{"bbox": [95, 606, 541, 621], "spans": [{"bbox": [95, 606, 194, 621], "score": 1.0, "content": "There is a second ", "type": "text"}, {"bbox": [194, 608, 203, 617], "score": 0.91, "content": "R", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [204, 606, 541, 621], "score": 1.0, "content": "-symmetry in these theories which comes from the dimensional", "type": "text"}], "index": 27}, {"bbox": [70, 624, 541, 639], "spans": [{"bbox": [70, 624, 344, 639], "score": 1.0, "content": "reduction of the Lorentz group. For reductions of ", "type": "text"}, {"bbox": [344, 627, 376, 636], "score": 0.92, "content": "d\\,=\\,6", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [376, 624, 541, 639], "score": 1.0, "content": " N=1 Yang-Mills theories, this", "type": "text"}], "index": 28}, {"bbox": [69, 645, 539, 659], "spans": [{"bbox": [69, 645, 93, 659], "score": 1.0, "content": "is a ", "type": "text"}, {"bbox": [94, 645, 135, 658], "score": 0.92, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [135, 645, 498, 659], "score": 1.0, "content": " symmetry. Using basically the same argument as in the case of the ", "type": "text"}, {"bbox": [499, 645, 539, 658], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}], "index": 29}, {"bbox": [70, 664, 541, 678], "spans": [{"bbox": [70, 664, 541, 678], "score": 1.0, "content": "symmetry, we show that all normalizable ground states in these theories are invariant", "type": "text"}], "index": 30}, {"bbox": [70, 683, 542, 699], "spans": [{"bbox": [70, 683, 129, 699], "score": 1.0, "content": "under this ", "type": "text"}, {"bbox": [129, 684, 170, 697], "score": 0.92, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [171, 683, 328, 699], "score": 1.0, "content": " symmetry. For reductions of ", "type": "text"}, {"bbox": [329, 685, 364, 694], "score": 0.91, "content": "d=10", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [365, 683, 542, 699], "score": 1.0, "content": " N=1 Yang-Mills theories [7], the", "type": "text"}], "index": 31}, {"bbox": [71, 703, 542, 719], "spans": [{"bbox": [71, 705, 81, 713], "score": 0.9, "content": "R", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [81, 703, 184, 719], "score": 1.0, "content": "-symmetry group is ", "type": "text"}, {"bbox": [184, 704, 225, 716], "score": 0.91, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [225, 703, 542, 719], "score": 1.0, "content": ". It is quite straightforward to argue that as a consequence of", "type": "text"}], "index": 32}, {"bbox": [71, 73, 541, 89], "spans": [{"bbox": [71, 73, 91, 89], "score": 1.0, "content": "the ", "type": "text", "cross_page": true}, {"bbox": [91, 75, 187, 87], "score": 0.94, "content": "S U(2)_{R}\\times S p i n(5)", "type": "inline_equation", "height": 12, "width": 96, "cross_page": true}, {"bbox": [187, 73, 541, 89], "score": 1.0, "content": " invariance theorem, all ground states in these theories with sixteen", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [70, 92, 392, 109], "spans": [{"bbox": [70, 92, 292, 109], "score": 1.0, "content": "supercharges must be invariant under the ", "type": "text", "cross_page": true}, {"bbox": [293, 94, 333, 107], "score": 0.87, "content": "S p i n(9)", "type": "inline_equation", "height": 13, "width": 40, "cross_page": true}, {"bbox": [334, 92, 392, 109], "score": 1.0, "content": " symmetry.", "type": "text", "cross_page": true}], "index": 1}], "index": 29.5, "bbox_fs": [69, 606, 542, 719]}]}
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[{"type": "title", "bbox": [71, 71, 165, 86], "content": "1. Introduction", "index": 0}, {"type": "text", "bbox": [70, 98, 542, 210], "content": "The existence of normalizable vacua in supersymetric Yang-Mills theories is a question that arises in many different contexts in string theory and field theory. Index arguments can be used to determine whether any vacua exist, but not exactly how many vacua. An index only counts the difference between the number of bosonic and fermionic vacua. To count the actual number of vacua, we need more information such as how the vacua transform under the global symmetries of the theory.", "index": 1}, {"type": "text", "bbox": [70, 214, 542, 366], "content": "In this paper, we consider quantum mechanical Yang-Mills theories with eight su- percharges and an symmetry. We take our theories to be dimensional reductions of N=1 Yang-Mills theories coupled to matter. The question of normal- izable ground states in these models arises in the study of bound states of D0-branes and D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to bind using index arguments [3] generalized to theories without a gap. Other examples from string theory involve D0-branes moving on orbifolds [4], and the question of counting H-monopoles in the heterotic string [5,3].", "index": 2}, {"type": "text", "bbox": [70, 369, 542, 599], "content": "In the following section, we describe the field content and symmetries of these gauge theories. We then show that all normalizable ground states in these theories must be invariant under the symmetry. The argument we give is suggested by recent work on the -cohomology of hyperKahler spaces by Hitchin [6]. Our result should have implications for defining and computing the -cohomology of instanton moduli spaces. Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind under consideration. For example, the moduli space of instantons in appears as the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these spaces can be singular, their embedding into quantum mechanical gauge theory provides a natural regularization of the singularities. Heuristically, the wavefunction for a state corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb branch. It would be interesting to explore this connection further.", "index": 3}, {"type": "text", "bbox": [70, 603, 542, 716], "content": "There is a second -symmetry in these theories which comes from the dimensional reduction of the Lorentz group. For reductions of N=1 Yang-Mills theories, this is a symmetry. Using basically the same argument as in the case of the symmetry, we show that all normalizable ground states in these theories are invariant under this symmetry. For reductions of N=1 Yang-Mills theories [7], the -symmetry group is . It is quite straightforward to argue that as a consequence of the invariance theorem, all ground states in these theories with sixteen supercharges must be invariant under the symmetry.", "index": 4}]
|
[{"bbox": [71, 74, 165, 85], "content": "1. Introduction", "parent_index": 0, "line_index": 0}, {"bbox": [94, 100, 540, 116], "content": "The existence of normalizable vacua in supersymetric Yang-Mills theories is a question", "parent_index": 1, "line_index": 0}, {"bbox": [70, 120, 541, 136], "content": "that arises in many different contexts in string theory and field theory. Index arguments", "parent_index": 1, "line_index": 1}, {"bbox": [71, 140, 540, 154], "content": "can be used to determine whether any vacua exist, but not exactly how many vacua.", "parent_index": 1, "line_index": 2}, {"bbox": [72, 160, 540, 173], "content": "An index only counts the difference between the number of bosonic and fermionic vacua.", "parent_index": 1, "line_index": 3}, {"bbox": [71, 179, 540, 192], "content": "To count the actual number of vacua, we need more information such as how the vacua", "parent_index": 1, "line_index": 4}, {"bbox": [70, 198, 350, 213], "content": "transform under the global symmetries of the theory.", "parent_index": 1, "line_index": 5}, {"bbox": [93, 217, 539, 233], "content": "In this paper, we consider quantum mechanical Yang-Mills theories with eight su-", "parent_index": 2, "line_index": 0}, {"bbox": [70, 237, 540, 252], "content": "percharges and an symmetry. We take our theories to be dimensional", "parent_index": 2, "line_index": 1}, {"bbox": [71, 257, 540, 271], "content": "reductions of N=1 Yang-Mills theories coupled to matter. The question of normal-", "parent_index": 2, "line_index": 2}, {"bbox": [70, 276, 541, 290], "content": "izable ground states in these models arises in the study of bound states of D0-branes and", "parent_index": 2, "line_index": 3}, {"bbox": [70, 293, 542, 311], "content": "D4-branes [1,2]; for example, a single D0-brane and a single D4-brane can be shown to", "parent_index": 2, "line_index": 4}, {"bbox": [71, 313, 540, 330], "content": "bind using index arguments [3] generalized to theories without a gap. Other examples", "parent_index": 2, "line_index": 5}, {"bbox": [70, 332, 541, 350], "content": "from string theory involve D0-branes moving on orbifolds [4], and the question of counting", "parent_index": 2, "line_index": 6}, {"bbox": [70, 352, 288, 369], "content": "H-monopoles in the heterotic string [5,3].", "parent_index": 2, "line_index": 7}, {"bbox": [94, 372, 541, 388], "content": "In the following section, we describe the field content and symmetries of these gauge", "parent_index": 3, "line_index": 0}, {"bbox": [72, 393, 541, 406], "content": "theories. We then show that all normalizable ground states in these theories must be", "parent_index": 3, "line_index": 1}, {"bbox": [71, 412, 541, 426], "content": "invariant under the symmetry. The argument we give is suggested by recent", "parent_index": 3, "line_index": 2}, {"bbox": [71, 431, 541, 446], "content": "work on the -cohomology of hyperKahler spaces by Hitchin [6]. Our result should have", "parent_index": 3, "line_index": 3}, {"bbox": [71, 451, 539, 465], "content": "implications for defining and computing the -cohomology of instanton moduli spaces.", "parent_index": 3, "line_index": 4}, {"bbox": [70, 470, 541, 486], "content": "Certain instanton moduli spaces appear as Higgs branches in gauge theories of the kind", "parent_index": 3, "line_index": 5}, {"bbox": [70, 488, 542, 505], "content": "under consideration. For example, the moduli space of instantons in appears as", "parent_index": 3, "line_index": 6}, {"bbox": [71, 509, 541, 524], "content": "the Higgs branch of the quantum mechanics describing D0-D4 systems. Although these", "parent_index": 3, "line_index": 7}, {"bbox": [70, 529, 541, 543], "content": "spaces can be singular, their embedding into quantum mechanical gauge theory provides", "parent_index": 3, "line_index": 8}, {"bbox": [70, 547, 541, 561], "content": "a natural regularization of the singularities. Heuristically, the wavefunction for a state", "parent_index": 3, "line_index": 9}, {"bbox": [70, 565, 541, 582], "content": "corresponding to a form on the Higgs branch is smoothed out by leaking onto the Coulomb", "parent_index": 3, "line_index": 10}, {"bbox": [70, 585, 418, 601], "content": "branch. It would be interesting to explore this connection further.", "parent_index": 3, "line_index": 11}, {"bbox": [95, 606, 541, 621], "content": "There is a second -symmetry in these theories which comes from the dimensional", "parent_index": 4, "line_index": 0}, {"bbox": [70, 624, 541, 639], "content": "reduction of the Lorentz group. For reductions of N=1 Yang-Mills theories, this", "parent_index": 4, "line_index": 1}, {"bbox": [69, 645, 539, 659], "content": "is a symmetry. Using basically the same argument as in the case of the", "parent_index": 4, "line_index": 2}, {"bbox": [70, 664, 541, 678], "content": "symmetry, we show that all normalizable ground states in these theories are invariant", "parent_index": 4, "line_index": 3}, {"bbox": [70, 683, 542, 699], "content": "under this symmetry. For reductions of N=1 Yang-Mills theories [7], the", "parent_index": 4, "line_index": 4}, {"bbox": [71, 703, 542, 719], "content": "-symmetry group is . It is quite straightforward to argue that as a consequence of", "parent_index": 4, "line_index": 5}, {"bbox": [71, 73, 541, 89], "content": "the invariance theorem, all ground states in these theories with sixteen", "parent_index": 4, "line_index": 6}, {"bbox": [70, 92, 392, 109], "content": "supercharges must be invariant under the symmetry.", "parent_index": 4, "line_index": 7}]
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[]
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[{"bbox": [171, 238, 267, 250], "content": "S p i n(5)\\times S U(2)_{R}", "parent_index": 2, "subtype": "inline"}, {"bbox": [143, 258, 172, 267], "content": "d=6", "parent_index": 2, "subtype": "inline"}, {"bbox": [129, 315, 143, 325], "content": "L^{2}", "parent_index": 2, "subtype": "inline"}, {"bbox": [181, 412, 222, 425], "content": "S U(2)_{R}", "parent_index": 3, "subtype": "inline"}, {"bbox": [137, 431, 151, 441], "content": "L^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [310, 451, 324, 461], "content": "L^{2}", "parent_index": 3, "subtype": "inline"}, {"bbox": [360, 490, 389, 503], "content": "U(N)", "parent_index": 3, "subtype": "inline"}, {"bbox": [464, 488, 480, 500], "content": "\\mathbb{R}^{4}", "parent_index": 3, "subtype": "inline"}, {"bbox": [194, 608, 203, 617], "content": "R", "parent_index": 4, "subtype": "inline"}, {"bbox": [344, 627, 376, 636], "content": "d\\,=\\,6", "parent_index": 4, "subtype": "inline"}, {"bbox": [94, 645, 135, 658], "content": "S p i n(5)", "parent_index": 4, "subtype": "inline"}, {"bbox": [499, 645, 539, 658], "content": "S U(2)_{R}", "parent_index": 4, "subtype": "inline"}, {"bbox": [129, 684, 170, 697], "content": "S p i n(5)", "parent_index": 4, "subtype": "inline"}, {"bbox": [329, 685, 364, 694], "content": "d=10", "parent_index": 4, "subtype": "inline"}, {"bbox": [71, 705, 81, 713], "content": "R", "parent_index": 4, "subtype": "inline"}, {"bbox": [184, 704, 225, 716], "content": "S p i n(9)", "parent_index": 4, "subtype": "inline"}, {"bbox": [91, 75, 187, 87], "content": "S U(2)_{R}\\times S p i n(5)", "parent_index": 4, "subtype": "inline"}, {"bbox": [293, 94, 333, 107], "content": "S p i n(9)", "parent_index": 4, "subtype": "inline"}]
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[]
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We can couple these invariance theorems with results from $L^{2}$ index theory [8,9]. The $L^{2}$ index for the non-Fredholm theory $^{1}$ of two D0-branes is proven to be one [8]. We also know that the $L^{2}$ index for the theory of a single D0-brane and a single D4-brane is one [3]. Our invariance results imply that all bound states in these theories are bosonic, and therefore unique. These results can also be combined with other interesting but heuristic attempts to study the $L^{2}$ index by either deforming the Yang-Mills theory [10,11], or by using insights from string theory [12] to compute the bulk and defect terms. The bulk terms for various Yang-Mills theories have been directly computed in [13,14,15]. There have also been a number of comments on the implications of invariance for the asymptotic form of particular bound state wavefunctions [16,17].
# 2. The Field Content and Symmetries
# 2.1. The vector multiplet supercharge
The argument we wish to make requires reasonably little explicit knowledge of the gauge theory. There is a $S p i n(5)\times S U(2)_{R}$ symmetry which commutes with the Hamiltonian $H$ . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars $x^{\mu}$ with $\mu=1,\dots,5$ transforming in the ${\bf(5,1)}$ of the symmetry group. These scalars transform in the adjoint representation of the gauge group $G$ . Let $p^{\mu}$ be the associated canonical momenta obeying,
$$
[x_{A}^{\mu},p_{B}^{\nu}]=i\delta^{\mu\nu}\delta_{A B},
$$
where the subscript $A$ is a group index.
Associated to these bosons are eight real fermions $\lambda_{a}$ where $a=1,\dotsc,8$ transforming in the $(\mathbf{4},\mathbf{2})$ representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the $(\mathbf{4},\mathbf{2})$ representation. These fermions obey the usual quantization relation,
$$
\left\{\lambda_{a A},\lambda_{b B}\right\}=\delta_{a b}\delta_{A B}.
$$
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<html><body>
<p data-bbox="70 110 542 302">We can couple these invariance theorems with results from $L^{2}$ index theory [8,9]. The $L^{2}$ index for the non-Fredholm theory $^{1}$ of two D0-branes is proven to be one [8]. We also know that the $L^{2}$ index for the theory of a single D0-brane and a single D4-brane is one [3]. Our invariance results imply that all bound states in these theories are bosonic, and therefore unique. These results can also be combined with other interesting but heuristic attempts to study the $L^{2}$ index by either deforming the Yang-Mills theory [10,11], or by using insights from string theory [12] to compute the bulk and defect terms. The bulk terms for various Yang-Mills theories have been directly computed in [13,14,15]. There have also been a number of comments on the implications of invariance for the asymptotic form of particular bound state wavefunctions [16,17]. </p>
<h1 data-bbox="71 335 303 351">2. The Field Content and Symmetries </h1>
<h1 data-bbox="71 363 267 378">2.1. The vector multiplet supercharge </h1>
<p data-bbox="70 390 542 504">The argument we wish to make requires reasonably little explicit knowledge of the gauge theory. There is a $S p i n(5)\times S U(2)_{R}$ symmetry which commutes with the Hamiltonian $H$ . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars $x^{\mu}$ with $\mu=1,\dots,5$ transforming in the ${\bf(5,1)}$ of the symmetry group. These scalars transform in the adjoint representation of the gauge group $G$ . Let $p^{\mu}$ be the associated canonical momenta obeying, </p>
<div class="equation" data-bbox="253 524 357 539">$$
[x_{A}^{\mu},p_{B}^{\nu}]=i\delta^{\mu\nu}\delta_{A B},
$$</div>
<p data-bbox="70 557 280 572">where the subscript $A$ is a group index. </p>
<p data-bbox="70 577 542 651">Associated to these bosons are eight real fermions $\lambda_{a}$ where $a=1,\dotsc,8$ transforming in the $(\mathbf{4},\mathbf{2})$ representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the $(\mathbf{4},\mathbf{2})$ representation. These fermions obey the usual quantization relation, </p>
<div class="equation" data-bbox="249 671 363 686">$$
\left\{\lambda_{a A},\lambda_{b B}\right\}=\delta_{a b}\delta_{A B}.
$$</div>
</body></html>
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0001189v2
| 2 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "", "page_idx": 2}, {"type": "text", "text": "We can couple these invariance theorems with results from $L^{2}$ index theory [8,9]. The $L^{2}$ index for the non-Fredholm theory $^{1}$ of two D0-branes is proven to be one [8]. We also know that the $L^{2}$ index for the theory of a single D0-brane and a single D4-brane is one [3]. Our invariance results imply that all bound states in these theories are bosonic, and therefore unique. These results can also be combined with other interesting but heuristic attempts to study the $L^{2}$ index by either deforming the Yang-Mills theory [10,11], or by using insights from string theory [12] to compute the bulk and defect terms. The bulk terms for various Yang-Mills theories have been directly computed in [13,14,15]. There have also been a number of comments on the implications of invariance for the asymptotic form of particular bound state wavefunctions [16,17]. ", "page_idx": 2}, {"type": "text", "text": "2. The Field Content and Symmetries ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "2.1. The vector multiplet supercharge ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "The argument we wish to make requires reasonably little explicit knowledge of the gauge theory. There is a $S p i n(5)\\times S U(2)_{R}$ symmetry which commutes with the Hamiltonian $H$ . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars $x^{\\mu}$ with $\\mu=1,\\dots,5$ transforming in the ${\\bf(5,1)}$ of the symmetry group. These scalars transform in the adjoint representation of the gauge group $G$ . Let $p^{\\mu}$ be the associated canonical momenta obeying, ", "page_idx": 2}, {"type": "equation", "text": "$$\n[x_{A}^{\\mu},p_{B}^{\\nu}]=i\\delta^{\\mu\\nu}\\delta_{A B},\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where the subscript $A$ is a group index. ", "page_idx": 2}, {"type": "text", "text": "Associated to these bosons are eight real fermions $\\lambda_{a}$ where $a=1,\\dotsc,8$ transforming in the $(\\mathbf{4},\\mathbf{2})$ representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the $(\\mathbf{4},\\mathbf{2})$ representation. These fermions obey the usual quantization relation, ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\left\\{\\lambda_{a A},\\lambda_{b B}\\right\\}=\\delta_{a b}\\delta_{A B}.\n$$", "text_format": "latex", "page_idx": 2}]
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{"preproc_blocks": [{"type": "text", "bbox": [70, 70, 541, 105], "lines": [{"bbox": [71, 73, 541, 89], "spans": [{"bbox": [71, 73, 91, 89], "score": 1.0, "content": "the ", "type": "text"}, {"bbox": [91, 75, 187, 87], "score": 0.94, "content": "S U(2)_{R}\\times S p i n(5)", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [187, 73, 541, 89], "score": 1.0, "content": " invariance theorem, all ground states in these theories with sixteen", "type": "text"}], "index": 0}, {"bbox": [70, 92, 392, 109], "spans": [{"bbox": [70, 92, 292, 109], "score": 1.0, "content": "supercharges must be invariant under the ", "type": "text"}, {"bbox": [293, 94, 333, 107], "score": 0.87, "content": "S p i n(9)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [334, 92, 392, 109], "score": 1.0, "content": " symmetry.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [70, 110, 542, 302], "lines": [{"bbox": [95, 113, 540, 128], "spans": [{"bbox": [95, 113, 402, 128], "score": 1.0, "content": "We can couple these invariance theorems with results from ", "type": "text"}, {"bbox": [402, 113, 416, 123], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [416, 113, 540, 128], "score": 1.0, "content": " index theory [8,9]. The", "type": "text"}], "index": 2}, {"bbox": [71, 131, 540, 147], "spans": [{"bbox": [71, 133, 85, 143], "score": 0.91, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [85, 131, 270, 147], "score": 1.0, "content": " index for the non-Fredholm theory", "type": "text"}, {"bbox": [271, 133, 276, 143], "score": 0.29, "content": "^{1}", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [276, 131, 540, 147], "score": 1.0, "content": " of two D0-branes is proven to be one [8]. We also", "type": "text"}], "index": 3}, {"bbox": [70, 151, 541, 168], "spans": [{"bbox": [70, 151, 150, 168], "score": 1.0, "content": "know that the ", "type": "text"}, {"bbox": [150, 153, 164, 163], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [164, 151, 541, 168], "score": 1.0, "content": " index for the theory of a single D0-brane and a single D4-brane is one", "type": "text"}], "index": 4}, {"bbox": [71, 172, 540, 185], "spans": [{"bbox": [71, 172, 540, 185], "score": 1.0, "content": "[3]. Our invariance results imply that all bound states in these theories are bosonic, and", "type": "text"}], "index": 5}, {"bbox": [71, 192, 540, 205], "spans": [{"bbox": [71, 192, 540, 205], "score": 1.0, "content": "therefore unique. These results can also be combined with other interesting but heuristic", "type": "text"}], "index": 6}, {"bbox": [71, 211, 541, 226], "spans": [{"bbox": [71, 211, 191, 226], "score": 1.0, "content": "attempts to study the ", "type": "text"}, {"bbox": [192, 211, 205, 221], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [205, 211, 541, 226], "score": 1.0, "content": " index by either deforming the Yang-Mills theory [10,11], or by", "type": "text"}], "index": 7}, {"bbox": [71, 230, 541, 245], "spans": [{"bbox": [71, 230, 541, 245], "score": 1.0, "content": "using insights from string theory [12] to compute the bulk and defect terms. The bulk", "type": "text"}], "index": 8}, {"bbox": [71, 250, 540, 264], "spans": [{"bbox": [71, 250, 540, 264], "score": 1.0, "content": "terms for various Yang-Mills theories have been directly computed in [13,14,15]. There", "type": "text"}], "index": 9}, {"bbox": [70, 268, 540, 284], "spans": [{"bbox": [70, 268, 540, 284], "score": 1.0, "content": "have also been a number of comments on the implications of invariance for the asymptotic", "type": "text"}], "index": 10}, {"bbox": [71, 289, 349, 304], "spans": [{"bbox": [71, 289, 349, 304], "score": 1.0, "content": "form of particular bound state wavefunctions [16,17].", "type": "text"}], "index": 11}], "index": 6.5}, {"type": "title", "bbox": [71, 335, 303, 351], "lines": [{"bbox": [71, 339, 302, 352], "spans": [{"bbox": [71, 339, 302, 352], "score": 1.0, "content": "2. The Field Content and Symmetries", "type": "text"}], "index": 12}], "index": 12}, {"type": "title", "bbox": [71, 363, 267, 378], "lines": [{"bbox": [71, 365, 268, 380], "spans": [{"bbox": [71, 365, 268, 380], "score": 1.0, "content": "2.1. The vector multiplet supercharge", "type": "text"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [70, 390, 542, 504], "lines": [{"bbox": [94, 392, 541, 409], "spans": [{"bbox": [94, 392, 541, 409], "score": 1.0, "content": "The argument we wish to make requires reasonably little explicit knowledge of the", "type": "text"}], "index": 14}, {"bbox": [70, 412, 541, 429], "spans": [{"bbox": [70, 412, 206, 429], "score": 1.0, "content": "gauge theory. There is a ", "type": "text"}, {"bbox": [206, 414, 303, 426], "score": 0.95, "content": "S p i n(5)\\times S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [303, 412, 541, 429], "score": 1.0, "content": " symmetry which commutes with the Hamil-", "type": "text"}], "index": 15}, {"bbox": [70, 431, 542, 449], "spans": [{"bbox": [70, 431, 110, 449], "score": 1.0, "content": "tonian ", "type": "text"}, {"bbox": [110, 434, 121, 443], "score": 0.91, "content": "H", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 431, 542, 449], "score": 1.0, "content": ". Since we are considering a gauge theory, we must have at least one vector", "type": "text"}], "index": 16}, {"bbox": [70, 452, 541, 468], "spans": [{"bbox": [70, 452, 252, 468], "score": 1.0, "content": "multiplet. It contains five scalars ", "type": "text"}, {"bbox": [253, 454, 266, 462], "score": 0.91, "content": "x^{\\mu}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [266, 452, 299, 468], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [299, 454, 363, 465], "score": 0.94, "content": "\\mu=1,\\dots,5", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [363, 452, 475, 468], "score": 1.0, "content": " transforming in the ", "type": "text"}, {"bbox": [475, 453, 504, 465], "score": 0.93, "content": "{\\bf(5,1)}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [504, 452, 541, 468], "score": 1.0, "content": " of the", "type": "text"}], "index": 17}, {"bbox": [70, 471, 540, 487], "spans": [{"bbox": [70, 471, 540, 487], "score": 1.0, "content": "symmetry group. 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These fermions are also in the adjoint", "type": "text"}], "index": 23}, {"bbox": [70, 618, 540, 636], "spans": [{"bbox": [70, 618, 511, 636], "score": 1.0, "content": "representation of the gauge group. The eight supercharges also transform in the ", "type": "text"}, {"bbox": [511, 620, 540, 632], "score": 0.78, "content": "(\\mathbf{4},\\mathbf{2})", "type": "inline_equation", "height": 12, "width": 29}], "index": 24}, {"bbox": [70, 637, 430, 654], "spans": [{"bbox": [70, 637, 430, 654], "score": 1.0, "content": "representation. 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The", "type": "text"}], "index": 2}, {"bbox": [71, 131, 540, 147], "spans": [{"bbox": [71, 133, 85, 143], "score": 0.91, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [85, 131, 270, 147], "score": 1.0, "content": " index for the non-Fredholm theory", "type": "text"}, {"bbox": [271, 133, 276, 143], "score": 0.29, "content": "^{1}", "type": "inline_equation", "height": 10, "width": 5}, {"bbox": [276, 131, 540, 147], "score": 1.0, "content": " of two D0-branes is proven to be one [8]. We also", "type": "text"}], "index": 3}, {"bbox": [70, 151, 541, 168], "spans": [{"bbox": [70, 151, 150, 168], "score": 1.0, "content": "know that the ", "type": "text"}, {"bbox": [150, 153, 164, 163], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [164, 151, 541, 168], "score": 1.0, "content": " index for the theory of a single D0-brane and a single D4-brane is one", "type": "text"}], "index": 4}, {"bbox": [71, 172, 540, 185], "spans": [{"bbox": [71, 172, 540, 185], "score": 1.0, "content": "[3]. Our invariance results imply that all bound states in these theories are bosonic, and", "type": "text"}], "index": 5}, {"bbox": [71, 192, 540, 205], "spans": [{"bbox": [71, 192, 540, 205], "score": 1.0, "content": "therefore unique. These results can also be combined with other interesting but heuristic", "type": "text"}], "index": 6}, {"bbox": [71, 211, 541, 226], "spans": [{"bbox": [71, 211, 191, 226], "score": 1.0, "content": "attempts to study the ", "type": "text"}, {"bbox": [192, 211, 205, 221], "score": 0.92, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [205, 211, 541, 226], "score": 1.0, "content": " index by either deforming the Yang-Mills theory [10,11], or by", "type": "text"}], "index": 7}, {"bbox": [71, 230, 541, 245], "spans": [{"bbox": [71, 230, 541, 245], "score": 1.0, "content": "using insights from string theory [12] to compute the bulk and defect terms. The bulk", "type": "text"}], "index": 8}, {"bbox": [71, 250, 540, 264], "spans": [{"bbox": [71, 250, 540, 264], "score": 1.0, "content": "terms for various Yang-Mills theories have been directly computed in [13,14,15]. There", "type": "text"}], "index": 9}, {"bbox": [70, 268, 540, 284], "spans": [{"bbox": [70, 268, 540, 284], "score": 1.0, "content": "have also been a number of comments on the implications of invariance for the asymptotic", "type": "text"}], "index": 10}, {"bbox": [71, 289, 349, 304], "spans": [{"bbox": [71, 289, 349, 304], "score": 1.0, "content": "form of particular bound state wavefunctions [16,17].", "type": "text"}], "index": 11}], "index": 6.5, "bbox_fs": [70, 113, 541, 304]}, {"type": "title", "bbox": [71, 335, 303, 351], "lines": [{"bbox": [71, 339, 302, 352], "spans": [{"bbox": [71, 339, 302, 352], "score": 1.0, "content": "2. The Field Content and Symmetries", "type": "text"}], "index": 12}], "index": 12}, {"type": "title", "bbox": [71, 363, 267, 378], "lines": [{"bbox": [71, 365, 268, 380], "spans": [{"bbox": [71, 365, 268, 380], "score": 1.0, "content": "2.1. The vector multiplet supercharge", "type": "text"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [70, 390, 542, 504], "lines": [{"bbox": [94, 392, 541, 409], "spans": [{"bbox": [94, 392, 541, 409], "score": 1.0, "content": "The argument we wish to make requires reasonably little explicit knowledge of the", "type": "text"}], "index": 14}, {"bbox": [70, 412, 541, 429], "spans": [{"bbox": [70, 412, 206, 429], "score": 1.0, "content": "gauge theory. There is a ", "type": "text"}, {"bbox": [206, 414, 303, 426], "score": 0.95, "content": "S p i n(5)\\times S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 97}, {"bbox": [303, 412, 541, 429], "score": 1.0, "content": " symmetry which commutes with the Hamil-", "type": "text"}], "index": 15}, {"bbox": [70, 431, 542, 449], "spans": [{"bbox": [70, 431, 110, 449], "score": 1.0, "content": "tonian ", "type": "text"}, {"bbox": [110, 434, 121, 443], "score": 0.91, "content": "H", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [122, 431, 542, 449], "score": 1.0, "content": ". Since we are considering a gauge theory, we must have at least one vector", "type": "text"}], "index": 16}, {"bbox": [70, 452, 541, 468], "spans": [{"bbox": [70, 452, 252, 468], "score": 1.0, "content": "multiplet. It contains five scalars ", "type": "text"}, {"bbox": [253, 454, 266, 462], "score": 0.91, "content": "x^{\\mu}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [266, 452, 299, 468], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [299, 454, 363, 465], "score": 0.94, "content": "\\mu=1,\\dots,5", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [363, 452, 475, 468], "score": 1.0, "content": " transforming in the ", "type": "text"}, {"bbox": [475, 453, 504, 465], "score": 0.93, "content": "{\\bf(5,1)}", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [504, 452, 541, 468], "score": 1.0, "content": " of the", "type": "text"}], "index": 17}, {"bbox": [70, 471, 540, 487], "spans": [{"bbox": [70, 471, 540, 487], "score": 1.0, "content": "symmetry group. 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These fermions are also in the adjoint", "type": "text"}], "index": 23}, {"bbox": [70, 618, 540, 636], "spans": [{"bbox": [70, 618, 511, 636], "score": 1.0, "content": "representation of the gauge group. The eight supercharges also transform in the ", "type": "text"}, {"bbox": [511, 620, 540, 632], "score": 0.78, "content": "(\\mathbf{4},\\mathbf{2})", "type": "inline_equation", "height": 12, "width": 29}], "index": 24}, {"bbox": [70, 637, 430, 654], "spans": [{"bbox": [70, 637, 430, 654], "score": 1.0, "content": "representation. These fermions obey the usual quantization relation,", "type": "text"}], "index": 25}], "index": 23.5, "bbox_fs": [70, 578, 541, 654]}, {"type": "interline_equation", "bbox": [249, 671, 363, 686], "lines": [{"bbox": [249, 671, 363, 686], "spans": [{"bbox": [249, 671, 363, 686], "score": 0.92, "content": "\\left\\{\\lambda_{a A},\\lambda_{b B}\\right\\}=\\delta_{a b}\\delta_{A B}.", "type": "interline_equation"}], "index": 26}], "index": 26}]}
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[{"bbox": [95, 113, 540, 128], "content": "We can couple these invariance theorems with results from index theory [8,9]. The", "parent_index": 1, "line_index": 0}, {"bbox": [71, 131, 540, 147], "content": "index for the non-Fredholm theory of two D0-branes is proven to be one [8]. We also", "parent_index": 1, "line_index": 1}, {"bbox": [70, 151, 541, 168], "content": "know that the index for the theory of a single D0-brane and a single D4-brane is one", "parent_index": 1, "line_index": 2}, {"bbox": [71, 172, 540, 185], "content": "[3]. Our invariance results imply that all bound states in these theories are bosonic, and", "parent_index": 1, "line_index": 3}, {"bbox": [71, 192, 540, 205], "content": "therefore unique. These results can also be combined with other interesting but heuristic", "parent_index": 1, "line_index": 4}, {"bbox": [71, 211, 541, 226], "content": "attempts to study the index by either deforming the Yang-Mills theory [10,11], or by", "parent_index": 1, "line_index": 5}, {"bbox": [71, 230, 541, 245], "content": "using insights from string theory [12] to compute the bulk and defect terms. The bulk", "parent_index": 1, "line_index": 6}, {"bbox": [71, 250, 540, 264], "content": "terms for various Yang-Mills theories have been directly computed in [13,14,15]. There", "parent_index": 1, "line_index": 7}, {"bbox": [70, 268, 540, 284], "content": "have also been a number of comments on the implications of invariance for the asymptotic", "parent_index": 1, "line_index": 8}, {"bbox": [71, 289, 349, 304], "content": "form of particular bound state wavefunctions [16,17].", "parent_index": 1, "line_index": 9}, {"bbox": [71, 339, 302, 352], "content": "2. The Field Content and Symmetries", "parent_index": 2, "line_index": 0}, {"bbox": [71, 365, 268, 380], "content": "2.1. The vector multiplet supercharge", "parent_index": 3, "line_index": 0}, {"bbox": [94, 392, 541, 409], "content": "The argument we wish to make requires reasonably little explicit knowledge of the", "parent_index": 4, "line_index": 0}, {"bbox": [70, 412, 541, 429], "content": "gauge theory. There is a symmetry which commutes with the Hamil-", "parent_index": 4, "line_index": 1}, {"bbox": [70, 431, 542, 449], "content": "tonian . Since we are considering a gauge theory, we must have at least one vector", "parent_index": 4, "line_index": 2}, {"bbox": [70, 452, 541, 468], "content": "multiplet. It contains five scalars with transforming in the of the", "parent_index": 4, "line_index": 3}, {"bbox": [70, 471, 540, 487], "content": "symmetry group. These scalars transform in the adjoint representation of the gauge group", "parent_index": 4, "line_index": 4}, {"bbox": [71, 489, 370, 508], "content": ". Let be the associated canonical momenta obeying,", "parent_index": 4, "line_index": 5}, {"bbox": [72, 560, 277, 573], "content": "where the subscript is a group index.", "parent_index": 6, "line_index": 0}, {"bbox": [95, 578, 541, 596], "content": "Associated to these bosons are eight real fermions where transforming", "parent_index": 7, "line_index": 0}, {"bbox": [70, 598, 540, 615], "content": "in the representation of the symmetry group. These fermions are also in the adjoint", "parent_index": 7, "line_index": 1}, {"bbox": [70, 618, 540, 636], "content": "representation of the gauge group. The eight supercharges also transform in the", "parent_index": 7, "line_index": 2}, {"bbox": [70, 637, 430, 654], "content": "representation. These fermions obey the usual quantization relation,", "parent_index": 7, "line_index": 3}]
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[]
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[]
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Let $\gamma^{\mu}$ be hermitian real gamma matrices which obey,
$$
\{\gamma^{\mu},\gamma^{\nu}\}=2\delta^{\mu\nu}.
$$
Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action.
The supercharge takes the form,
$$
Q_{a}^{v}=(\gamma^{\mu}p_{A}^{\mu}\lambda_{A})_{a}+\frac{1}{2}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}+D_{a b A}\lambda_{b A},
$$
where $f_{A B C}$ are the structure constants and $\gamma^{\mu\nu}\,=\,(1/2)(\gamma^{\mu}\gamma^{\nu}\,-\,\gamma^{\nu}\gamma^{\mu})$ . The real antisymmetric matrix $D$ does not involve momenta. The $D$ -term transforms in the $\left(\mathbf{1},\mathbf{3}\right)$ representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of $D$ is not important for our argument. In general, there can be many vector multiplets. In that case, the terms in the supercharge (2.4) generalize in an obvious way.
# 2.2. The hypermultiplet supercharge
A hypermultiplet contains four real scalars which we can package into a quaternion $q$ with components $q^{i}$ where $i=1,2,3,4$ . This field transforms as $(\mathbf{1},\mathbf{2})$ under the symmetry group, and in some representation ${\cal L}^{\prime}$ of the gauge group. We again introduce canonical momenta $p_{i}$ satisfying the usual commutation relations. Now $S U(2)_{R}\,\sim\,S p(1)_{R}$ is the group of unit quaternions. We choose $S U(2)_{R}$ to act on a hypermultiplet $q$ by right multiplication by a unit quaternion. The gauge symmetry commutes with the $S U(2)_{R}$ symmetry and acts by left multiplication on $q$ . See Appendix A for a more detailed discussion.
The superpartner to $q$ is a real fermion $\psi_{a}$ with $a=1,\dotsc,8$ satisfying,
$$
\left\{\psi_{a}^{R},\psi_{b S}\right\}=\delta_{a b}\delta_{S}^{R}.
$$
These fermions transform in the $(4,1)$ representation, and the $R,S$ subscripts index the ${\cal T}$ representation of $G$ . For ${\boldsymbol{n}}$ hypermultiplets, the gauge group $G$ acts via a subgroup of the $S p(n)_{L}$ symmetry. In terms of the $s^{j}$ operators given in Appendix A, the hypermultiplet charge takes the form
$$
Q_{a}^{h}=s_{a b}^{j}\psi_{b}\,p_{j}+I_{a b}\psi_{b}.
$$
|
<html><body>
<p data-bbox="69 70 356 86">Let $\gamma^{\mu}$ be hermitian real gamma matrices which obey, </p>
<div class="equation" data-bbox="262 106 349 121">$$
\{\gamma^{\mu},\gamma^{\nu}\}=2\delta^{\mu\nu}.
$$</div>
<p data-bbox="70 138 542 173">Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action. </p>
<p data-bbox="93 177 266 192">The supercharge takes the form, </p>
<div class="equation" data-bbox="162 205 448 232">$$
Q_{a}^{v}=(\gamma^{\mu}p_{A}^{\mu}\lambda_{A})_{a}+\frac{1}{2}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}+D_{a b A}\lambda_{b A},
$$</div>
<p data-bbox="69 243 542 356">where $f_{A B C}$ are the structure constants and $\gamma^{\mu\nu}\,=\,(1/2)(\gamma^{\mu}\gamma^{\nu}\,-\,\gamma^{\nu}\gamma^{\mu})$ . The real antisymmetric matrix $D$ does not involve momenta. The $D$ -term transforms in the $\left(\mathbf{1},\mathbf{3}\right)$ representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of $D$ is not important for our argument. In general, there can be many vector multiplets. In that case, the terms in the supercharge (2.4) generalize in an obvious way. </p>
<h1 data-bbox="71 375 261 390">2.2. The hypermultiplet supercharge </h1>
<p data-bbox="69 401 542 551">A hypermultiplet contains four real scalars which we can package into a quaternion $q$ with components $q^{i}$ where $i=1,2,3,4$ . This field transforms as $(\mathbf{1},\mathbf{2})$ under the symmetry group, and in some representation ${\cal L}^{\prime}$ of the gauge group. We again introduce canonical momenta $p_{i}$ satisfying the usual commutation relations. Now $S U(2)_{R}\,\sim\,S p(1)_{R}$ is the group of unit quaternions. We choose $S U(2)_{R}$ to act on a hypermultiplet $q$ by right multiplication by a unit quaternion. The gauge symmetry commutes with the $S U(2)_{R}$ symmetry and acts by left multiplication on $q$ . See Appendix A for a more detailed discussion. </p>
<p data-bbox="94 555 469 571">The superpartner to $q$ is a real fermion $\psi_{a}$ with $a=1,\dotsc,8$ satisfying, </p>
<div class="equation" data-bbox="254 589 358 606">$$
\left\{\psi_{a}^{R},\psi_{b S}\right\}=\delta_{a b}\delta_{S}^{R}.
$$</div>
<p data-bbox="70 622 541 696">These fermions transform in the $(4,1)$ representation, and the $R,S$ subscripts index the ${\cal T}$ representation of $G$ . For ${\boldsymbol{n}}$ hypermultiplets, the gauge group $G$ acts via a subgroup of the $S p(n)_{L}$ symmetry. In terms of the $s^{j}$ operators given in Appendix A, the hypermultiplet charge takes the form </p>
<div class="equation" data-bbox="246 699 366 717">$$
Q_{a}^{h}=s_{a b}^{j}\psi_{b}\,p_{j}+I_{a b}\psi_{b}.
$$</div>
</body></html>
|
0001189v2
| 3 | 612 | 792 | 1,275 | 1,650 |
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This field transforms as ", "type": "text"}, {"bbox": [403, 424, 431, 437], "score": 0.93, "content": "(\\mathbf{1},\\mathbf{2})", "type": "inline_equation", "height": 13, "width": 28}, {"bbox": [431, 423, 540, 438], "score": 1.0, "content": " under the symmetry", "type": "text"}], "index": 14}, {"bbox": [70, 442, 541, 457], "spans": [{"bbox": [70, 442, 257, 457], "score": 1.0, "content": "group, and in some representation ", "type": "text"}, {"bbox": [258, 444, 267, 453], "score": 0.91, "content": "{\\cal L}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [267, 442, 541, 457], "score": 1.0, "content": " of the gauge group. 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We choose ", "type": "text"}, {"bbox": [286, 482, 327, 495], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [328, 481, 484, 496], "score": 1.0, "content": " to act on a hypermultiplet ", "type": "text"}, {"bbox": [484, 486, 491, 494], "score": 0.87, "content": "q", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [491, 481, 541, 496], "score": 1.0, "content": " by right", "type": "text"}], "index": 17}, {"bbox": [71, 501, 539, 515], "spans": [{"bbox": [71, 501, 498, 515], "score": 1.0, "content": "multiplication by a unit quaternion. The gauge symmetry commutes with the ", "type": "text"}, {"bbox": [499, 501, 539, 514], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}], "index": 18}, {"bbox": [70, 519, 541, 534], "spans": [{"bbox": [70, 519, 319, 534], "score": 1.0, "content": "symmetry and acts by left multiplication on ", "type": "text"}, {"bbox": [319, 524, 325, 532], "score": 0.86, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [326, 519, 541, 534], "score": 1.0, "content": ". See Appendix A for a more detailed", "type": "text"}], "index": 19}, {"bbox": [71, 540, 127, 553], "spans": [{"bbox": [71, 540, 127, 553], "score": 1.0, "content": "discussion.", "type": "text"}], "index": 20}], "index": 16.5}, {"type": "text", "bbox": [94, 555, 469, 571], "lines": [{"bbox": [94, 558, 466, 573], "spans": [{"bbox": [94, 558, 205, 573], "score": 1.0, "content": "The superpartner to ", "type": "text"}, {"bbox": [205, 563, 211, 571], "score": 0.88, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [212, 558, 303, 573], "score": 1.0, "content": " is a real fermion ", "type": "text"}, {"bbox": [304, 560, 317, 571], "score": 0.92, "content": "\\psi_{a}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [318, 558, 348, 573], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [349, 560, 410, 571], "score": 0.93, "content": "a=1,\\dotsc,8", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [410, 558, 466, 573], "score": 1.0, "content": " satisfying,", "type": "text"}], "index": 21}], "index": 21}, {"type": "interline_equation", "bbox": [254, 589, 358, 606], "lines": [{"bbox": [254, 589, 358, 606], "spans": [{"bbox": [254, 589, 358, 606], "score": 0.93, "content": "\\left\\{\\psi_{a}^{R},\\psi_{b S}\\right\\}=\\delta_{a b}\\delta_{S}^{R}.", "type": "interline_equation"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [70, 622, 541, 696], "lines": [{"bbox": [70, 624, 540, 641], "spans": [{"bbox": [70, 624, 241, 641], "score": 1.0, "content": "These fermions transform in the ", "type": "text"}, {"bbox": [241, 627, 270, 639], "score": 0.93, "content": "(4,1)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [270, 624, 397, 641], "score": 1.0, "content": " representation, and the ", "type": "text"}, {"bbox": [397, 627, 420, 639], "score": 0.93, "content": "R,S", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [420, 624, 530, 641], "score": 1.0, "content": " subscripts index the ", "type": "text"}, {"bbox": [531, 628, 540, 636], "score": 0.9, "content": "{\\cal T}", "type": "inline_equation", "height": 8, "width": 9}], "index": 23}, {"bbox": [69, 645, 541, 660], "spans": [{"bbox": [69, 645, 163, 660], "score": 1.0, "content": "representation of ", "type": "text"}, {"bbox": [163, 646, 173, 655], "score": 0.91, "content": "G", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [173, 645, 202, 660], "score": 1.0, "content": ". For ", "type": "text"}, {"bbox": [202, 650, 209, 655], "score": 0.89, "content": "{\\boldsymbol{n}}", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [210, 645, 390, 660], "score": 1.0, "content": " hypermultiplets, the gauge group ", "type": "text"}, {"bbox": [390, 646, 400, 655], "score": 0.9, "content": "G", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [400, 645, 541, 660], "score": 1.0, "content": " acts via a subgroup of the", "type": "text"}], "index": 24}, {"bbox": [71, 663, 540, 681], "spans": [{"bbox": [71, 665, 109, 678], "score": 0.95, "content": "S p(n)_{L}", "type": "inline_equation", "height": 13, "width": 38}, {"bbox": [109, 663, 256, 681], "score": 1.0, "content": " symmetry. 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In terms of the ", "type": "text"}, {"bbox": [257, 664, 267, 675], "score": 0.92, "content": "s^{j}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [267, 663, 540, 681], "score": 1.0, "content": " operators given in Appendix A, the hypermultiplet", "type": "text"}], "index": 25}, {"bbox": [70, 683, 186, 699], "spans": [{"bbox": [70, 683, 186, 699], "score": 1.0, "content": "charge takes the form", "type": "text"}], "index": 26}], "index": 24.5, "bbox_fs": [69, 624, 541, 699]}, {"type": "interline_equation", "bbox": [246, 699, 366, 717], "lines": [{"bbox": [246, 699, 366, 717], "spans": [{"bbox": [246, 699, 366, 717], "score": 0.93, "content": "Q_{a}^{h}=s_{a b}^{j}\\psi_{b}\\,p_{j}+I_{a b}\\psi_{b}.", "type": "interline_equation"}], "index": 27}], "index": 27}]}
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[{"bbox": [92, 76, 106, 87], "content": "\\gamma^{\\mu}", "parent_index": 0, "subtype": "inline"}, {"bbox": [262, 106, 349, 121], "content": "\\{\\gamma^{\\mu},\\gamma^{\\nu}\\}=2\\delta^{\\mu\\nu}.", "parent_index": 1, "subtype": "interline"}, {"bbox": [162, 205, 448, 232], "content": "Q_{a}^{v}=(\\gamma^{\\mu}p_{A}^{\\mu}\\lambda_{A})_{a}+\\frac{1}{2}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}+D_{a b A}\\lambda_{b A},", "parent_index": 4, "subtype": "interline"}, {"bbox": [107, 249, 136, 260], "content": "f_{A B C}", "parent_index": 5, "subtype": "inline"}, {"bbox": [313, 248, 457, 261], "content": "\\gamma^{\\mu\\nu}\\,=\\,(1/2)(\\gamma^{\\mu}\\gamma^{\\nu}\\,-\\,\\gamma^{\\nu}\\gamma^{\\mu})", "parent_index": 5, "subtype": "inline"}, {"bbox": [171, 268, 182, 277], "content": "D", "parent_index": 5, "subtype": "inline"}, {"bbox": [367, 268, 378, 277], "content": "D", "parent_index": 5, "subtype": "inline"}, {"bbox": [511, 267, 540, 280], "content": "\\left(\\mathbf{1},\\mathbf{3}\\right)", "parent_index": 5, "subtype": "inline"}, {"bbox": [215, 307, 225, 316], "content": "D", "parent_index": 5, "subtype": "inline"}, {"bbox": [533, 409, 540, 417], "content": "q", "parent_index": 7, "subtype": "inline"}, {"bbox": [163, 424, 172, 436], "content": "q^{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [210, 425, 270, 436], "content": "i=1,2,3,4", "parent_index": 7, "subtype": "inline"}, {"bbox": [403, 424, 431, 437], "content": "(\\mathbf{1},\\mathbf{2})", "parent_index": 7, "subtype": "inline"}, {"bbox": [258, 444, 267, 453], "content": "{\\cal L}^{\\prime}", "parent_index": 7, "subtype": "inline"}, {"bbox": [124, 467, 135, 475], "content": "p_{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [407, 463, 505, 475], "content": "S U(2)_{R}\\,\\sim\\,S p(1)_{R}", "parent_index": 7, "subtype": "inline"}, {"bbox": [286, 482, 327, 495], "content": "S U(2)_{R}", "parent_index": 7, "subtype": "inline"}, {"bbox": [484, 486, 491, 494], "content": "q", "parent_index": 7, "subtype": "inline"}, {"bbox": [499, 501, 539, 514], "content": "S U(2)_{R}", "parent_index": 7, "subtype": "inline"}, {"bbox": [319, 524, 325, 532], "content": "q", "parent_index": 7, "subtype": "inline"}, {"bbox": [205, 563, 211, 571], "content": "q", "parent_index": 8, "subtype": "inline"}, {"bbox": [304, 560, 317, 571], "content": "\\psi_{a}", "parent_index": 8, "subtype": "inline"}, {"bbox": [349, 560, 410, 571], "content": "a=1,\\dotsc,8", "parent_index": 8, "subtype": "inline"}, {"bbox": [254, 589, 358, 606], "content": "\\left\\{\\psi_{a}^{R},\\psi_{b S}\\right\\}=\\delta_{a b}\\delta_{S}^{R}.", "parent_index": 9, "subtype": "interline"}, {"bbox": [241, 627, 270, 639], "content": "(4,1)", "parent_index": 10, "subtype": "inline"}, {"bbox": [397, 627, 420, 639], "content": "R,S", "parent_index": 10, "subtype": "inline"}, {"bbox": [531, 628, 540, 636], "content": "{\\cal T}", "parent_index": 10, "subtype": "inline"}, {"bbox": [163, 646, 173, 655], "content": "G", "parent_index": 10, "subtype": "inline"}, {"bbox": [202, 650, 209, 655], "content": "{\\boldsymbol{n}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [390, 646, 400, 655], "content": "G", "parent_index": 10, "subtype": "inline"}, {"bbox": [71, 665, 109, 678], "content": "S p(n)_{L}", "parent_index": 10, "subtype": "inline"}, {"bbox": [257, 664, 267, 675], "content": "s^{j}", "parent_index": 10, "subtype": "inline"}, {"bbox": [246, 699, 366, 717], "content": "Q_{a}^{h}=s_{a b}^{j}\\psi_{b}\\,p_{j}+I_{a b}\\psi_{b}.", "parent_index": 11, "subtype": "interline"}]
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[]
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We have lumped all the interactions into the non-derivative operator $I$ which transforms in the 2 of $S U(2)_{R}$ . We also need to note that $I$ is proportional to $x^{\mu}\gamma^{\mu}$ with a proportionality constant that commutes with the $S p i n(5)$ generators. We have also suppressed gauge indices. Note that since the $s^{j}$ implement right multiplication by a quaternion, they commute with $\gamma^{\mu}$ . Again, there can be many hypermultiplets in different representations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes in a straightforward way. The full Hermitian supercharge is the sum of the vector and hypermultiplet supercharges,
$$
Q_{a}=Q_{a}^{v}+Q_{a}^{h}.
$$
# 2.3. The $S U(2)_{R}$ currents
The three generators of $S U(2)_{R}$ correspond to right multiplication by $I,J,K$ and are given in terms of the gauge invariant rotation generators,
$$
W_{i j}=q_{i}p_{j}-q_{j}p_{i}.
$$
Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three $S U(2)_{R}$ generators by ${\tilde{s}}^{i}$ :
$$
\begin{array}{l}{{\displaystyle\tilde{s}^{2}=W_{12}-W_{34}+\frac{i}{2}\,\lambda s^{2}\lambda}}\\ {{\displaystyle\tilde{s}^{3}=W_{13}+W_{24}+\frac{i}{2}\,\lambda s^{3}\lambda}}\\ {{\displaystyle\tilde{s}^{4}=W_{14}-W_{23}+\frac{i}{2}\,\lambda s^{4}\lambda.}}\end{array}
$$
As they should, these generators act on the bosons of the hypermultiplet and the fermions of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is straightforward: we simply need to sum the contributions to the three currents (2.8) from each multiplet.
# 2.4. The Spin(5) currents
The ten generators of $S p i n(5)$ act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by:
$$
T^{\mu\nu}=x^{\mu}p^{\nu}-x^{\nu}p^{\mu}-\frac{i}{4}\gamma_{a b}^{\mu\nu}\left(\lambda_{a}\lambda_{b}+\psi_{a}\psi_{b}\right).
$$
Adding either more vector multiplets or more hypermultiplets is again straightforward.
|
<html><body>
<p data-bbox="69 69 542 218">We have lumped all the interactions into the non-derivative operator $I$ which transforms in the 2 of $S U(2)_{R}$ . We also need to note that $I$ is proportional to $x^{\mu}\gamma^{\mu}$ with a proportionality constant that commutes with the $S p i n(5)$ generators. We have also suppressed gauge indices. Note that since the $s^{j}$ implement right multiplication by a quaternion, they commute with $\gamma^{\mu}$ . Again, there can be many hypermultiplets in different representations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes in a straightforward way. The full Hermitian supercharge is the sum of the vector and hypermultiplet supercharges, </p>
<div class="equation" data-bbox="265 224 346 239">$$
Q_{a}=Q_{a}^{v}+Q_{a}^{h}.
$$</div>
<h1 data-bbox="71 254 211 270">2.3. The $S U(2)_{R}$ currents </h1>
<p data-bbox="70 280 541 315">The three generators of $S U(2)_{R}$ correspond to right multiplication by $I,J,K$ and are given in terms of the gauge invariant rotation generators, </p>
<div class="equation" data-bbox="258 333 354 348">$$
W_{i j}=q_{i}p_{j}-q_{j}p_{i}.
$$</div>
<p data-bbox="70 363 541 397">Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three $S U(2)_{R}$ generators by ${\tilde{s}}^{i}$ : </p>
<div class="equation" data-bbox="235 409 375 497">$$
\begin{array}{l}{{\displaystyle\tilde{s}^{2}=W_{12}-W_{34}+\frac{i}{2}\,\lambda s^{2}\lambda}}\\ {{\displaystyle\tilde{s}^{3}=W_{13}+W_{24}+\frac{i}{2}\,\lambda s^{3}\lambda}}\\ {{\displaystyle\tilde{s}^{4}=W_{14}-W_{23}+\frac{i}{2}\,\lambda s^{4}\lambda.}}\end{array}
$$</div>
<p data-bbox="69 502 542 575">As they should, these generators act on the bosons of the hypermultiplet and the fermions of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is straightforward: we simply need to sum the contributions to the three currents (2.8) from each multiplet. </p>
<h1 data-bbox="71 592 211 608">2.4. The Spin(5) currents </h1>
<p data-bbox="69 618 541 653">The ten generators of $S p i n(5)$ act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by: </p>
<div class="equation" data-bbox="192 663 419 690">$$
T^{\mu\nu}=x^{\mu}p^{\nu}-x^{\nu}p^{\mu}-\frac{i}{4}\gamma_{a b}^{\mu\nu}\left(\lambda_{a}\lambda_{b}+\psi_{a}\psi_{b}\right).
$$</div>
<p data-bbox="70 700 528 716">Adding either more vector multiplets or more hypermultiplets is again straightforward. </p>
</body></html>
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0001189v2
| 4 | 612 | 792 | 1,275 | 1,650 |
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[{"type": "text", "bbox": [69, 69, 542, 218], "content": "We have lumped all the interactions into the non-derivative operator which transforms in the 2 of . We also need to note that is proportional to with a propor- tionality constant that commutes with the generators. We have also suppressed gauge indices. Note that since the implement right multiplication by a quaternion, they commute with . Again, there can be many hypermultiplets in different represen- tations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes in a straightforward way. The full Hermitian supercharge is the sum of the vector and hypermultiplet supercharges,", "index": 0}, {"type": "interline_equation", "bbox": [265, 224, 346, 239], "content": "", "index": 1}, {"type": "title", "bbox": [71, 254, 211, 270], "content": "2.3. The currents", "index": 2}, {"type": "text", "bbox": [70, 280, 541, 315], "content": "The three generators of correspond to right multiplication by and are given in terms of the gauge invariant rotation generators,", "index": 3}, {"type": "interline_equation", "bbox": [258, 333, 354, 348], "content": "", "index": 4}, {"type": "text", "bbox": [70, 363, 541, 397], "content": "Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three generators by :", "index": 5}, {"type": "interline_equation", "bbox": [235, 409, 375, 497], "content": "", "index": 6}, {"type": "text", "bbox": [69, 502, 542, 575], "content": "As they should, these generators act on the bosons of the hypermultiplet and the fermions of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is straightforward: we simply need to sum the contributions to the three currents (2.8) from each multiplet.", "index": 7}, {"type": "title", "bbox": [71, 592, 211, 608], "content": "2.4. The Spin(5) currents", "index": 8}, {"type": "text", "bbox": [69, 618, 541, 653], "content": "The ten generators of act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by:", "index": 9}, {"type": "interline_equation", "bbox": [192, 663, 419, 690], "content": "", "index": 10}, {"type": "text", "bbox": [70, 700, 528, 716], "content": "Adding either more vector multiplets or more hypermultiplets is again straightforward.", "index": 11}]
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[{"bbox": [71, 73, 540, 88], "content": "We have lumped all the interactions into the non-derivative operator which transforms", "parent_index": 0, "line_index": 0}, {"bbox": [70, 92, 540, 108], "content": "in the 2 of . We also need to note that is proportional to with a propor-", "parent_index": 0, "line_index": 1}, {"bbox": [70, 112, 541, 127], "content": "tionality constant that commutes with the generators. We have also suppressed", "parent_index": 0, "line_index": 2}, {"bbox": [70, 130, 540, 146], "content": "gauge indices. Note that since the implement right multiplication by a quaternion,", "parent_index": 0, "line_index": 3}, {"bbox": [71, 149, 539, 166], "content": "they commute with . Again, there can be many hypermultiplets in different represen-", "parent_index": 0, "line_index": 4}, {"bbox": [70, 168, 541, 184], "content": "tations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes", "parent_index": 0, "line_index": 5}, {"bbox": [70, 187, 541, 202], "content": "in a straightforward way. The full Hermitian supercharge is the sum of the vector and", "parent_index": 0, "line_index": 6}, {"bbox": [70, 204, 223, 223], "content": "hypermultiplet supercharges,", "parent_index": 0, "line_index": 7}, {"bbox": [71, 255, 210, 273], "content": "2.3. The currents", "parent_index": 2, "line_index": 0}, {"bbox": [93, 282, 542, 300], "content": "The three generators of correspond to right multiplication by and are", "parent_index": 3, "line_index": 0}, {"bbox": [71, 304, 372, 317], "content": "given in terms of the gauge invariant rotation generators,", "parent_index": 3, "line_index": 1}, {"bbox": [72, 366, 541, 381], "content": "Again here and in the subsequent discussion, we generally suppress gauge indices. In", "parent_index": 5, "line_index": 0}, {"bbox": [70, 383, 459, 400], "content": "accord with prior notation, we denote the three generators by :", "parent_index": 5, "line_index": 1}, {"bbox": [70, 505, 541, 521], "content": "As they should, these generators act on the bosons of the hypermultiplet and the fermions", "parent_index": 7, "line_index": 0}, {"bbox": [70, 524, 541, 540], "content": "of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is", "parent_index": 7, "line_index": 1}, {"bbox": [71, 544, 540, 558], "content": "straightforward: we simply need to sum the contributions to the three currents (2.8) from", "parent_index": 7, "line_index": 2}, {"bbox": [72, 563, 149, 578], "content": "each multiplet.", "parent_index": 7, "line_index": 3}, {"bbox": [72, 595, 210, 608], "content": "2.4. The Spin(5) currents", "parent_index": 8, "line_index": 0}, {"bbox": [94, 619, 541, 637], "content": "The ten generators of act on the bosons of the vector multiplet and all fermions", "parent_index": 9, "line_index": 0}, {"bbox": [71, 641, 307, 654], "content": "in the problem. The generators are given by:", "parent_index": 9, "line_index": 1}, {"bbox": [70, 702, 528, 718], "content": "Adding either more vector multiplets or more hypermultiplets is again straightforward.", "parent_index": 11, "line_index": 0}]
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[]
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[{"bbox": [438, 75, 445, 84], "content": "I", "parent_index": 0, "subtype": "inline"}, {"bbox": [132, 94, 173, 106], "content": "S U(2)_{R}", "parent_index": 0, "subtype": "inline"}, {"bbox": [324, 95, 331, 104], "content": "I", "parent_index": 0, "subtype": "inline"}, {"bbox": [432, 95, 458, 106], "content": "x^{\\mu}\\gamma^{\\mu}", "parent_index": 0, "subtype": "inline"}, {"bbox": [300, 113, 341, 125], "content": "S p i n(5)", "parent_index": 0, "subtype": "inline"}, {"bbox": [266, 131, 276, 141], "content": "s^{j}", "parent_index": 0, "subtype": "inline"}, {"bbox": [178, 152, 192, 163], "content": "\\gamma^{\\mu}", "parent_index": 0, "subtype": "inline"}, {"bbox": [265, 224, 346, 239], "content": "Q_{a}=Q_{a}^{v}+Q_{a}^{h}.", "parent_index": 1, "subtype": "interline"}, {"bbox": [121, 259, 162, 271], "content": "S U(2)_{R}", "parent_index": 2, "subtype": "inline"}, {"bbox": [221, 285, 262, 297], "content": "S U(2)_{R}", "parent_index": 3, "subtype": "inline"}, {"bbox": [462, 285, 497, 297], "content": "I,J,K", "parent_index": 3, "subtype": "inline"}, {"bbox": [258, 333, 354, 348], "content": "W_{i j}=q_{i}p_{j}-q_{j}p_{i}.", "parent_index": 4, "subtype": "interline"}, {"bbox": [324, 385, 365, 398], "content": "S U(2)_{R}", "parent_index": 5, "subtype": "inline"}, {"bbox": [443, 385, 453, 395], "content": "{\\tilde{s}}^{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [235, 409, 375, 497], "content": "\\begin{array}{l}{{\\displaystyle\\tilde{s}^{2}=W_{12}-W_{34}+\\frac{i}{2}\\,\\lambda s^{2}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{3}=W_{13}+W_{24}+\\frac{i}{2}\\,\\lambda s^{3}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{4}=W_{14}-W_{23}+\\frac{i}{2}\\,\\lambda s^{4}\\lambda.}}\\end{array}", "parent_index": 6, "subtype": "interline"}, {"bbox": [207, 622, 248, 634], "content": "S p i n(5)", "parent_index": 9, "subtype": "inline"}, {"bbox": [192, 663, 419, 690], "content": "T^{\\mu\\nu}=x^{\\mu}p^{\\nu}-x^{\\nu}p^{\\mu}-\\frac{i}{4}\\gamma_{a b}^{\\mu\\nu}\\left(\\lambda_{a}\\lambda_{b}+\\psi_{a}\\psi_{b}\\right).", "parent_index": 10, "subtype": "interline"}]
|
[]
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|
# 3. An Invariance Argument for the $S U(2)_{R}$ Symmetry
3.1. Relating the $S U(2)_{R}$ currents to the supercharge
A key point in the argument is a relation between the supercharge and the $S U(2)_{R}$ currents. For some choice of $v_{a}^{i}$ , we want to show that:
$$
\tilde{s}^{i}=\sum_{a}\;\{Q_{a},v_{a}^{i}\}.
$$
Let us start with the vector multiplet. We take a candidate gauge singlet,
$$
\left(v_{1}\right)_{a}^{i}=\left(s^{i}\gamma^{\nu}\lambda\right)_{a}x^{\nu}.
$$
First note that this choice anti-commutes with $Q^{h}$ because $\lambda$ anti-commutes with $\psi$ . It also anti-commutes with the $\cal{D}$ -term in (2.4). To see this, we compute:
$$
\sum_{a}\left\{D_{a b}\lambda_{b},(v_{1})_{a}^{i}\right\}=x_{A}^{\nu}\mathrm{tr}\left(s^{i}\gamma^{\nu}D_{A}^{T}\right),
$$
However, we can immediately see that (3.3) vanishes by noting that the operator $s^{i}\gamma^{\nu}D^{T}$ does not contain a singlet under $S p i n(5)$ . The trace of the operator therefore vanishes. Our choice for $v_{1}$ anti-commutes with ${\textstyle\frac{1}{2}}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}$ for the same reason: the resulting trace does not contain a singlet of $S p i n(5)$ .
What remains is the following anti-commutator which is not hard to compute,
$$
\sum_{a}\left\{(\gamma^{\mu}p^{\mu}\lambda)_{a}\,,(v_{1})_{a}^{i}\right\}\sim i\,\lambda s^{i}\lambda.
$$
The exact proportionality constant does not matter for this argument. The important point is that we can use (3.2) to generate the terms in the $S U(2)_{R}$ currents which act on vector multiplets.
For the hypermultiplet, we take the following candidate gauge singlet:
$$
(v_{2})_{a}^{i}=\left(s^{i}s^{l}\psi\right)_{a}q^{l}.
$$
Note that $v_{2}$ anti-commutes with $Q^{v}$ because $\lambda$ anti-commutes with $\psi$ . It is also not too hard to argue that the anti-commutator of $v_{2}$ with the interaction term $I$ in (2.6) must vanish. We see that,
$$
\sum_{a}\left\{I_{a b}\psi_{b},(v_{2})_{a}^{i}\right\}\sim q^{l}\mathrm{tr}\left(s^{i}s^{l}I\right),
$$
|
<html><body>
<h1 data-bbox="70 70 396 86">3. An Invariance Argument for the $S U(2)_{R}$ Symmetry </h1>
<p data-bbox="71 97 352 113">3.1. Relating the $S U(2)_{R}$ currents to the supercharge </p>
<p data-bbox="69 123 541 158">A key point in the argument is a relation between the supercharge and the $S U(2)_{R}$ currents. For some choice of $v_{a}^{i}$ , we want to show that: </p>
<div class="equation" data-bbox="257 174 354 204">$$
\tilde{s}^{i}=\sum_{a}\;\{Q_{a},v_{a}^{i}\}.
$$</div>
<p data-bbox="70 213 460 230">Let us start with the vector multiplet. We take a candidate gauge singlet, </p>
<div class="equation" data-bbox="252 248 360 266">$$
\left(v_{1}\right)_{a}^{i}=\left(s^{i}\gamma^{\nu}\lambda\right)_{a}x^{\nu}.
$$</div>
<p data-bbox="69 279 540 315">First note that this choice anti-commutes with $Q^{h}$ because $\lambda$ anti-commutes with $\psi$ . It also anti-commutes with the $\cal{D}$ -term in (2.4). To see this, we compute: </p>
<div class="equation" data-bbox="207 329 403 358">$$
\sum_{a}\left\{D_{a b}\lambda_{b},(v_{1})_{a}^{i}\right\}=x_{A}^{\nu}\mathrm{tr}\left(s^{i}\gamma^{\nu}D_{A}^{T}\right),
$$</div>
<p data-bbox="69 369 542 443">However, we can immediately see that (3.3) vanishes by noting that the operator $s^{i}\gamma^{\nu}D^{T}$ does not contain a singlet under $S p i n(5)$ . The trace of the operator therefore vanishes. Our choice for $v_{1}$ anti-commutes with ${\textstyle\frac{1}{2}}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}$ for the same reason: the resulting trace does not contain a singlet of $S p i n(5)$ . </p>
<p data-bbox="93 446 507 463">What remains is the following anti-commutator which is not hard to compute, </p>
<div class="equation" data-bbox="222 474 388 504">$$
\sum_{a}\left\{(\gamma^{\mu}p^{\mu}\lambda)_{a}\,,(v_{1})_{a}^{i}\right\}\sim i\,\lambda s^{i}\lambda.
$$</div>
<p data-bbox="69 512 541 567">The exact proportionality constant does not matter for this argument. The important point is that we can use (3.2) to generate the terms in the $S U(2)_{R}$ currents which act on vector multiplets. </p>
<p data-bbox="93 569 465 586">For the hypermultiplet, we take the following candidate gauge singlet: </p>
<div class="equation" data-bbox="254 604 357 622">$$
(v_{2})_{a}^{i}=\left(s^{i}s^{l}\psi\right)_{a}q^{l}.
$$</div>
<p data-bbox="70 636 541 688">Note that $v_{2}$ anti-commutes with $Q^{v}$ because $\lambda$ anti-commutes with $\psi$ . It is also not too hard to argue that the anti-commutator of $v_{2}$ with the interaction term $I$ in (2.6) must vanish. We see that, </p>
<div class="equation" data-bbox="219 688 391 718">$$
\sum_{a}\left\{I_{a b}\psi_{b},(v_{2})_{a}^{i}\right\}\sim q^{l}\mathrm{tr}\left(s^{i}s^{l}I\right),
$$</div>
</body></html>
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0001189v2
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{"preproc_blocks": [{"type": "title", "bbox": [70, 70, 396, 86], "lines": [{"bbox": [70, 73, 395, 88], "spans": [{"bbox": [70, 73, 287, 88], "score": 1.0, "content": "3. An Invariance Argument for the ", "type": "text"}, {"bbox": [288, 75, 329, 87], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [329, 73, 395, 88], "score": 1.0, "content": " Symmetry", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [71, 97, 352, 113], "lines": [{"bbox": [72, 100, 349, 114], "spans": [{"bbox": [72, 100, 163, 113], "score": 1.0, "content": "3.1. Relating the ", "type": "text"}, {"bbox": [163, 101, 204, 114], "score": 0.93, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [204, 100, 349, 113], "score": 1.0, "content": " currents to the supercharge", "type": "text"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [69, 123, 541, 158], "lines": [{"bbox": [95, 125, 539, 142], "spans": [{"bbox": [95, 125, 498, 142], "score": 1.0, "content": "A key point in the argument is a relation between the supercharge and the ", "type": "text"}, {"bbox": [499, 127, 539, 140], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}], "index": 2}, {"bbox": [69, 142, 361, 163], "spans": [{"bbox": [69, 142, 222, 163], "score": 1.0, "content": "currents. For some choice of ", "type": "text"}, {"bbox": [223, 146, 235, 159], "score": 0.92, "content": "v_{a}^{i}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [235, 142, 361, 163], "score": 1.0, "content": ", we want to show that:", "type": "text"}], "index": 3}], "index": 2.5}, {"type": "interline_equation", "bbox": [257, 174, 354, 204], "lines": [{"bbox": [257, 174, 354, 204], "spans": [{"bbox": [257, 174, 354, 204], "score": 0.94, "content": "\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [70, 213, 460, 230], "lines": [{"bbox": [69, 215, 459, 232], "spans": [{"bbox": [69, 215, 459, 232], "score": 1.0, "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [252, 248, 360, 266], "lines": [{"bbox": [252, 248, 360, 266], "spans": [{"bbox": [252, 248, 360, 266], "score": 0.93, "content": "\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [69, 279, 540, 315], "lines": [{"bbox": [70, 282, 541, 298], "spans": [{"bbox": [70, 282, 325, 298], "score": 1.0, "content": "First note that this choice anti-commutes with ", "type": "text"}, {"bbox": [325, 283, 340, 296], "score": 0.92, "content": "Q^{h}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [341, 282, 390, 298], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [390, 285, 398, 294], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [398, 282, 511, 298], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [511, 285, 520, 296], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [520, 282, 541, 298], "score": 1.0, "content": ". It", "type": "text"}], "index": 7}, {"bbox": [70, 302, 443, 317], "spans": [{"bbox": [70, 302, 223, 317], "score": 1.0, "content": "also anti-commutes with the ", "type": "text"}, {"bbox": [223, 304, 234, 313], "score": 0.91, "content": "\\cal{D}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [234, 302, 443, 317], "score": 1.0, "content": "-term in (2.4). To see this, we compute:", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "interline_equation", "bbox": [207, 329, 403, 358], "lines": [{"bbox": [207, 329, 403, 358], "spans": [{"bbox": [207, 329, 403, 358], "score": 0.92, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [69, 369, 542, 443], "lines": [{"bbox": [69, 371, 539, 389], "spans": [{"bbox": [69, 371, 499, 389], "score": 1.0, "content": "However, we can immediately see that (3.3) vanishes by noting that the operator ", "type": "text"}, {"bbox": [499, 373, 539, 386], "score": 0.95, "content": "s^{i}\\gamma^{\\nu}D^{T}", "type": "inline_equation", "height": 13, "width": 40}], "index": 10}, {"bbox": [70, 392, 540, 408], "spans": [{"bbox": [70, 392, 237, 408], "score": 1.0, "content": "does not contain a singlet under ", "type": "text"}, {"bbox": [237, 392, 278, 406], "score": 0.8, "content": "S p i n(5)", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [279, 392, 540, 408], "score": 1.0, "content": ". The trace of the operator therefore vanishes. Our", "type": "text"}], "index": 11}, {"bbox": [69, 409, 542, 430], "spans": [{"bbox": [69, 409, 124, 430], "score": 1.0, "content": "choice for ", "type": "text"}, {"bbox": [124, 416, 135, 424], "score": 0.9, "content": "v_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [135, 409, 245, 430], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [245, 410, 362, 426], "score": 0.92, "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "type": "inline_equation", "height": 16, "width": 117}, {"bbox": [362, 409, 542, 430], "score": 1.0, "content": " for the same reason: the resulting", "type": "text"}], "index": 12}, {"bbox": [70, 429, 298, 446], "spans": [{"bbox": [70, 429, 253, 446], "score": 1.0, "content": "trace does not contain a singlet of ", "type": "text"}, {"bbox": [253, 430, 294, 444], "score": 0.89, "content": "S p i n(5)", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [294, 429, 298, 446], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5}, {"type": "text", "bbox": [93, 446, 507, 463], "lines": [{"bbox": [95, 450, 506, 463], "spans": [{"bbox": [95, 450, 506, 463], "score": 1.0, "content": "What remains is the following anti-commutator which is not hard to compute,", "type": "text"}], "index": 14}], "index": 14}, {"type": "interline_equation", "bbox": [222, 474, 388, 504], "lines": [{"bbox": [222, 474, 388, 504], "spans": [{"bbox": [222, 474, 388, 504], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [69, 512, 541, 567], "lines": [{"bbox": [70, 516, 540, 532], "spans": [{"bbox": [70, 516, 540, 532], "score": 1.0, "content": "The exact proportionality constant does not matter for this argument. The important", "type": "text"}], "index": 16}, {"bbox": [70, 535, 540, 550], "spans": [{"bbox": [70, 535, 381, 550], "score": 1.0, "content": "point is that we can use (3.2) to generate the terms in the ", "type": "text"}, {"bbox": [381, 535, 422, 549], "score": 0.93, "content": "S U(2)_{R}", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [423, 535, 540, 550], "score": 1.0, "content": " currents which act on", "type": "text"}], "index": 17}, {"bbox": [71, 555, 163, 568], "spans": [{"bbox": [71, 555, 163, 568], "score": 1.0, "content": "vector multiplets.", "type": "text"}], "index": 18}], "index": 17}, {"type": "text", "bbox": [93, 569, 465, 586], "lines": [{"bbox": [95, 573, 462, 587], "spans": [{"bbox": [95, 573, 462, 587], "score": 1.0, "content": "For the hypermultiplet, we take the following candidate gauge singlet:", "type": "text"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [254, 604, 357, 622], "lines": [{"bbox": [254, 604, 357, 622], "spans": [{"bbox": [254, 604, 357, 622], "score": 0.93, "content": "(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 636, 541, 688], "lines": [{"bbox": [69, 637, 541, 655], "spans": [{"bbox": [69, 637, 126, 655], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [127, 644, 138, 651], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [138, 637, 250, 655], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [250, 641, 265, 652], "score": 0.92, "content": "Q^{v}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [266, 637, 314, 655], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [314, 641, 321, 650], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 637, 433, 655], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [433, 641, 442, 652], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [442, 637, 541, 655], "score": 1.0, "content": ". It is also not too", "type": "text"}], "index": 21}, {"bbox": [71, 659, 540, 673], "spans": [{"bbox": [71, 659, 303, 673], "score": 1.0, "content": "hard to argue that the anti-commutator of ", "type": "text"}, {"bbox": [303, 663, 315, 671], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [315, 659, 458, 673], "score": 1.0, "content": " with the interaction term ", "type": "text"}, {"bbox": [459, 660, 465, 669], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [466, 659, 540, 673], "score": 1.0, "content": " in (2.6) must", "type": "text"}], "index": 22}, {"bbox": [72, 677, 179, 691], "spans": [{"bbox": [72, 677, 179, 691], "score": 1.0, "content": "vanish. We see that,", "type": "text"}], "index": 23}], "index": 22}, {"type": "interline_equation", "bbox": [219, 688, 391, 718], "lines": [{"bbox": [219, 688, 391, 718], "spans": [{"bbox": [219, 688, 391, 718], "score": 0.93, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),", "type": "interline_equation"}], "index": 24}], "index": 24}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [257, 174, 354, 204], "lines": [{"bbox": [257, 174, 354, 204], "spans": [{"bbox": [257, 174, 354, 204], "score": 0.94, "content": "\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [252, 248, 360, 266], "lines": [{"bbox": [252, 248, 360, 266], "spans": [{"bbox": [252, 248, 360, 266], "score": 0.93, "content": "\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [207, 329, 403, 358], "lines": [{"bbox": [207, 329, 403, 358], "spans": [{"bbox": [207, 329, 403, 358], "score": 0.92, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [222, 474, 388, 504], "lines": [{"bbox": [222, 474, 388, 504], "spans": [{"bbox": [222, 474, 388, 504], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [254, 604, 357, 622], "lines": [{"bbox": [254, 604, 357, 622], "spans": [{"bbox": [254, 604, 357, 622], "score": 0.93, "content": "(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "interline_equation", "bbox": [219, 688, 391, 718], "lines": [{"bbox": [219, 688, 391, 718], "spans": [{"bbox": [219, 688, 391, 718], "score": 0.93, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),", "type": "interline_equation"}], "index": 24}], "index": 24}], "discarded_blocks": [{"type": "discarded", "bbox": [300, 730, 311, 742], "lines": [{"bbox": [302, 732, 309, 744], "spans": [{"bbox": [302, 732, 309, 744], "score": 1.0, "content": "5", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [70, 70, 396, 86], "lines": [{"bbox": [70, 73, 395, 88], "spans": [{"bbox": [70, 73, 287, 88], "score": 1.0, "content": "3. An Invariance Argument for the ", "type": "text"}, {"bbox": [288, 75, 329, 87], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [329, 73, 395, 88], "score": 1.0, "content": " Symmetry", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [71, 97, 352, 113], "lines": [{"bbox": [72, 100, 349, 114], "spans": [{"bbox": [72, 100, 163, 113], "score": 1.0, "content": "3.1. Relating the ", "type": "text"}, {"bbox": [163, 101, 204, 114], "score": 0.93, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [204, 100, 349, 113], "score": 1.0, "content": " currents to the supercharge", "type": "text"}], "index": 1}], "index": 1, "bbox_fs": [72, 100, 349, 114]}, {"type": "text", "bbox": [69, 123, 541, 158], "lines": [{"bbox": [95, 125, 539, 142], "spans": [{"bbox": [95, 125, 498, 142], "score": 1.0, "content": "A key point in the argument is a relation between the supercharge and the ", "type": "text"}, {"bbox": [499, 127, 539, 140], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}], "index": 2}, {"bbox": [69, 142, 361, 163], "spans": [{"bbox": [69, 142, 222, 163], "score": 1.0, "content": "currents. For some choice of ", "type": "text"}, {"bbox": [223, 146, 235, 159], "score": 0.92, "content": "v_{a}^{i}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [235, 142, 361, 163], "score": 1.0, "content": ", we want to show that:", "type": "text"}], "index": 3}], "index": 2.5, "bbox_fs": [69, 125, 539, 163]}, {"type": "interline_equation", "bbox": [257, 174, 354, 204], "lines": [{"bbox": [257, 174, 354, 204], "spans": [{"bbox": [257, 174, 354, 204], "score": 0.94, "content": "\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [70, 213, 460, 230], "lines": [{"bbox": [69, 215, 459, 232], "spans": [{"bbox": [69, 215, 459, 232], "score": 1.0, "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "type": "text"}], "index": 5}], "index": 5, "bbox_fs": [69, 215, 459, 232]}, {"type": "interline_equation", "bbox": [252, 248, 360, 266], "lines": [{"bbox": [252, 248, 360, 266], "spans": [{"bbox": [252, 248, 360, 266], "score": 0.93, "content": "\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [69, 279, 540, 315], "lines": [{"bbox": [70, 282, 541, 298], "spans": [{"bbox": [70, 282, 325, 298], "score": 1.0, "content": "First note that this choice anti-commutes with ", "type": "text"}, {"bbox": [325, 283, 340, 296], "score": 0.92, "content": "Q^{h}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [341, 282, 390, 298], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [390, 285, 398, 294], "score": 0.88, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [398, 282, 511, 298], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [511, 285, 520, 296], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [520, 282, 541, 298], "score": 1.0, "content": ". It", "type": "text"}], "index": 7}, {"bbox": [70, 302, 443, 317], "spans": [{"bbox": [70, 302, 223, 317], "score": 1.0, "content": "also anti-commutes with the ", "type": "text"}, {"bbox": [223, 304, 234, 313], "score": 0.91, "content": "\\cal{D}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [234, 302, 443, 317], "score": 1.0, "content": "-term in (2.4). To see this, we compute:", "type": "text"}], "index": 8}], "index": 7.5, "bbox_fs": [70, 282, 541, 317]}, {"type": "interline_equation", "bbox": [207, 329, 403, 358], "lines": [{"bbox": [207, 329, 403, 358], "spans": [{"bbox": [207, 329, 403, 358], "score": 0.92, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [69, 369, 542, 443], "lines": [{"bbox": [69, 371, 539, 389], "spans": [{"bbox": [69, 371, 499, 389], "score": 1.0, "content": "However, we can immediately see that (3.3) vanishes by noting that the operator ", "type": "text"}, {"bbox": [499, 373, 539, 386], "score": 0.95, "content": "s^{i}\\gamma^{\\nu}D^{T}", "type": "inline_equation", "height": 13, "width": 40}], "index": 10}, {"bbox": [70, 392, 540, 408], "spans": [{"bbox": [70, 392, 237, 408], "score": 1.0, "content": "does not contain a singlet under ", "type": "text"}, {"bbox": [237, 392, 278, 406], "score": 0.8, "content": "S p i n(5)", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [279, 392, 540, 408], "score": 1.0, "content": ". The trace of the operator therefore vanishes. Our", "type": "text"}], "index": 11}, {"bbox": [69, 409, 542, 430], "spans": [{"bbox": [69, 409, 124, 430], "score": 1.0, "content": "choice for ", "type": "text"}, {"bbox": [124, 416, 135, 424], "score": 0.9, "content": "v_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [135, 409, 245, 430], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [245, 410, 362, 426], "score": 0.92, "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "type": "inline_equation", "height": 16, "width": 117}, {"bbox": [362, 409, 542, 430], "score": 1.0, "content": " for the same reason: the resulting", "type": "text"}], "index": 12}, {"bbox": [70, 429, 298, 446], "spans": [{"bbox": [70, 429, 253, 446], "score": 1.0, "content": "trace does not contain a singlet of ", "type": "text"}, {"bbox": [253, 430, 294, 444], "score": 0.89, "content": "S p i n(5)", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [294, 429, 298, 446], "score": 1.0, "content": ".", "type": "text"}], "index": 13}], "index": 11.5, "bbox_fs": [69, 371, 542, 446]}, {"type": "text", "bbox": [93, 446, 507, 463], "lines": [{"bbox": [95, 450, 506, 463], "spans": [{"bbox": [95, 450, 506, 463], "score": 1.0, "content": "What remains is the following anti-commutator which is not hard to compute,", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [95, 450, 506, 463]}, {"type": "interline_equation", "bbox": [222, 474, 388, 504], "lines": [{"bbox": [222, 474, 388, 504], "spans": [{"bbox": [222, 474, 388, 504], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [69, 512, 541, 567], "lines": [{"bbox": [70, 516, 540, 532], "spans": [{"bbox": [70, 516, 540, 532], "score": 1.0, "content": "The exact proportionality constant does not matter for this argument. The important", "type": "text"}], "index": 16}, {"bbox": [70, 535, 540, 550], "spans": [{"bbox": [70, 535, 381, 550], "score": 1.0, "content": "point is that we can use (3.2) to generate the terms in the ", "type": "text"}, {"bbox": [381, 535, 422, 549], "score": 0.93, "content": "S U(2)_{R}", "type": "inline_equation", "height": 14, "width": 41}, {"bbox": [423, 535, 540, 550], "score": 1.0, "content": " currents which act on", "type": "text"}], "index": 17}, {"bbox": [71, 555, 163, 568], "spans": [{"bbox": [71, 555, 163, 568], "score": 1.0, "content": "vector multiplets.", "type": "text"}], "index": 18}], "index": 17, "bbox_fs": [70, 516, 540, 568]}, {"type": "text", "bbox": [93, 569, 465, 586], "lines": [{"bbox": [95, 573, 462, 587], "spans": [{"bbox": [95, 573, 462, 587], "score": 1.0, "content": "For the hypermultiplet, we take the following candidate gauge singlet:", "type": "text"}], "index": 19}], "index": 19, "bbox_fs": [95, 573, 462, 587]}, {"type": "interline_equation", "bbox": [254, 604, 357, 622], "lines": [{"bbox": [254, 604, 357, 622], "spans": [{"bbox": [254, 604, 357, 622], "score": 0.93, "content": "(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [70, 636, 541, 688], "lines": [{"bbox": [69, 637, 541, 655], "spans": [{"bbox": [69, 637, 126, 655], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [127, 644, 138, 651], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [138, 637, 250, 655], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [250, 641, 265, 652], "score": 0.92, "content": "Q^{v}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [266, 637, 314, 655], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [314, 641, 321, 650], "score": 0.89, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [322, 637, 433, 655], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [433, 641, 442, 652], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [442, 637, 541, 655], "score": 1.0, "content": ". It is also not too", "type": "text"}], "index": 21}, {"bbox": [71, 659, 540, 673], "spans": [{"bbox": [71, 659, 303, 673], "score": 1.0, "content": "hard to argue that the anti-commutator of ", "type": "text"}, {"bbox": [303, 663, 315, 671], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [315, 659, 458, 673], "score": 1.0, "content": " with the interaction term ", "type": "text"}, {"bbox": [459, 660, 465, 669], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [466, 659, 540, 673], "score": 1.0, "content": " in (2.6) must", "type": "text"}], "index": 22}, {"bbox": [72, 677, 179, 691], "spans": [{"bbox": [72, 677, 179, 691], "score": 1.0, "content": "vanish. We see that,", "type": "text"}], "index": 23}], "index": 22, "bbox_fs": [69, 637, 541, 691]}, {"type": "interline_equation", "bbox": [219, 688, 391, 718], "lines": [{"bbox": [219, 688, 391, 718], "spans": [{"bbox": [219, 688, 391, 718], "score": 0.93, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),", "type": "interline_equation"}], "index": 24}], "index": 24}]}
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[{"type": "title", "bbox": [70, 70, 396, 86], "content": "3. An Invariance Argument for the Symmetry", "index": 0}, {"type": "text", "bbox": [71, 97, 352, 113], "content": "3.1. Relating the currents to the supercharge", "index": 1}, {"type": "text", "bbox": [69, 123, 541, 158], "content": "A key point in the argument is a relation between the supercharge and the currents. For some choice of , we want to show that:", "index": 2}, {"type": "interline_equation", "bbox": [257, 174, 354, 204], "content": "", "index": 3}, {"type": "text", "bbox": [70, 213, 460, 230], "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "index": 4}, {"type": "interline_equation", "bbox": [252, 248, 360, 266], "content": "", "index": 5}, {"type": "text", "bbox": [69, 279, 540, 315], "content": "First note that this choice anti-commutes with because anti-commutes with . It also anti-commutes with the -term in (2.4). To see this, we compute:", "index": 6}, {"type": "interline_equation", "bbox": [207, 329, 403, 358], "content": "", "index": 7}, {"type": "text", "bbox": [69, 369, 542, 443], "content": "However, we can immediately see that (3.3) vanishes by noting that the operator does not contain a singlet under . The trace of the operator therefore vanishes. Our choice for anti-commutes with for the same reason: the resulting trace does not contain a singlet of .", "index": 8}, {"type": "text", "bbox": [93, 446, 507, 463], "content": "What remains is the following anti-commutator which is not hard to compute,", "index": 9}, {"type": "interline_equation", "bbox": [222, 474, 388, 504], "content": "", "index": 10}, {"type": "text", "bbox": [69, 512, 541, 567], "content": "The exact proportionality constant does not matter for this argument. The important point is that we can use (3.2) to generate the terms in the currents which act on vector multiplets.", "index": 11}, {"type": "text", "bbox": [93, 569, 465, 586], "content": "For the hypermultiplet, we take the following candidate gauge singlet:", "index": 12}, {"type": "interline_equation", "bbox": [254, 604, 357, 622], "content": "", "index": 13}, {"type": "text", "bbox": [70, 636, 541, 688], "content": "Note that anti-commutes with because anti-commutes with . It is also not too hard to argue that the anti-commutator of with the interaction term in (2.6) must vanish. We see that,", "index": 14}, {"type": "interline_equation", "bbox": [219, 688, 391, 718], "content": "", "index": 15}]
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[{"bbox": [70, 73, 395, 88], "content": "3. An Invariance Argument for the Symmetry", "parent_index": 0, "line_index": 0}, {"bbox": [72, 100, 349, 114], "content": "3.1. Relating the currents to the supercharge", "parent_index": 1, "line_index": 0}, {"bbox": [95, 125, 539, 142], "content": "A key point in the argument is a relation between the supercharge and the", "parent_index": 2, "line_index": 0}, {"bbox": [69, 142, 361, 163], "content": "currents. For some choice of , we want to show that:", "parent_index": 2, "line_index": 1}, {"bbox": [69, 215, 459, 232], "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "parent_index": 4, "line_index": 0}, {"bbox": [70, 282, 541, 298], "content": "First note that this choice anti-commutes with because anti-commutes with . It", "parent_index": 6, "line_index": 0}, {"bbox": [70, 302, 443, 317], "content": "also anti-commutes with the -term in (2.4). To see this, we compute:", "parent_index": 6, "line_index": 1}, {"bbox": [69, 371, 539, 389], "content": "However, we can immediately see that (3.3) vanishes by noting that the operator", "parent_index": 8, "line_index": 0}, {"bbox": [70, 392, 540, 408], "content": "does not contain a singlet under . The trace of the operator therefore vanishes. Our", "parent_index": 8, "line_index": 1}, {"bbox": [69, 409, 542, 430], "content": "choice for anti-commutes with for the same reason: the resulting", "parent_index": 8, "line_index": 2}, {"bbox": [70, 429, 298, 446], "content": "trace does not contain a singlet of .", "parent_index": 8, "line_index": 3}, {"bbox": [95, 450, 506, 463], "content": "What remains is the following anti-commutator which is not hard to compute,", "parent_index": 9, "line_index": 0}, {"bbox": [70, 516, 540, 532], "content": "The exact proportionality constant does not matter for this argument. The important", "parent_index": 11, "line_index": 0}, {"bbox": [70, 535, 540, 550], "content": "point is that we can use (3.2) to generate the terms in the currents which act on", "parent_index": 11, "line_index": 1}, {"bbox": [71, 555, 163, 568], "content": "vector multiplets.", "parent_index": 11, "line_index": 2}, {"bbox": [95, 573, 462, 587], "content": "For the hypermultiplet, we take the following candidate gauge singlet:", "parent_index": 12, "line_index": 0}, {"bbox": [69, 637, 541, 655], "content": "Note that anti-commutes with because anti-commutes with . It is also not too", "parent_index": 14, "line_index": 0}, {"bbox": [71, 659, 540, 673], "content": "hard to argue that the anti-commutator of with the interaction term in (2.6) must", "parent_index": 14, "line_index": 1}, {"bbox": [72, 677, 179, 691], "content": "vanish. We see that,", "parent_index": 14, "line_index": 2}]
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[]
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[{"bbox": [288, 75, 329, 87], "content": "S U(2)_{R}", "parent_index": 0, "subtype": "inline"}, {"bbox": [163, 101, 204, 114], "content": "S U(2)_{R}", "parent_index": 1, "subtype": "inline"}, {"bbox": [499, 127, 539, 140], "content": "S U(2)_{R}", "parent_index": 2, "subtype": "inline"}, {"bbox": [223, 146, 235, 159], "content": "v_{a}^{i}", "parent_index": 2, "subtype": "inline"}, {"bbox": [257, 174, 354, 204], "content": "\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.", "parent_index": 3, "subtype": "interline"}, {"bbox": [252, 248, 360, 266], "content": "\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.", "parent_index": 5, "subtype": "interline"}, {"bbox": [325, 283, 340, 296], "content": "Q^{h}", "parent_index": 6, "subtype": "inline"}, {"bbox": [390, 285, 398, 294], "content": "\\lambda", "parent_index": 6, "subtype": "inline"}, {"bbox": [511, 285, 520, 296], "content": "\\psi", "parent_index": 6, "subtype": "inline"}, {"bbox": [223, 304, 234, 313], "content": "\\cal{D}", "parent_index": 6, "subtype": "inline"}, {"bbox": [207, 329, 403, 358], "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),", "parent_index": 7, "subtype": "interline"}, {"bbox": [499, 373, 539, 386], "content": "s^{i}\\gamma^{\\nu}D^{T}", "parent_index": 8, "subtype": "inline"}, {"bbox": [237, 392, 278, 406], "content": "S p i n(5)", "parent_index": 8, "subtype": "inline"}, {"bbox": [124, 416, 135, 424], "content": "v_{1}", "parent_index": 8, "subtype": "inline"}, {"bbox": [245, 410, 362, 426], "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "parent_index": 8, "subtype": "inline"}, {"bbox": [253, 430, 294, 444], "content": "S p i n(5)", "parent_index": 8, "subtype": "inline"}, {"bbox": [222, 474, 388, 504], "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.", "parent_index": 10, "subtype": "interline"}, {"bbox": [381, 535, 422, 549], "content": "S U(2)_{R}", "parent_index": 11, "subtype": "inline"}, {"bbox": [254, 604, 357, 622], "content": "(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.", "parent_index": 13, "subtype": "interline"}, {"bbox": [127, 644, 138, 651], "content": "v_{2}", "parent_index": 14, "subtype": "inline"}, {"bbox": [250, 641, 265, 652], "content": "Q^{v}", "parent_index": 14, "subtype": "inline"}, {"bbox": [314, 641, 321, 650], "content": "\\lambda", "parent_index": 14, "subtype": "inline"}, {"bbox": [433, 641, 442, 652], "content": "\\psi", "parent_index": 14, "subtype": "inline"}, {"bbox": [303, 663, 315, 671], "content": "v_{2}", "parent_index": 14, "subtype": "inline"}, {"bbox": [459, 660, 465, 669], "content": "I", "parent_index": 14, "subtype": "inline"}, {"bbox": [219, 688, 391, 718], "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),", "parent_index": 15, "subtype": "interline"}]
|
[]
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but $s^{i}s^{l}I$ does not contain a singlet under the $S p i n(5)$ action on fermions because $I$ is proportional to $\gamma^{\mu}$ so the trace vanishes.
Again what remains is the anti-commutator,
$$
\sum_{a}\left\{s_{a b}^{j}\psi_{b}\,p_{j},(v_{2})_{a}^{i}\right\}.
$$
It is easy to check that the $\psi\psi$ terms in the anti-commutator vanish because,
$$
\sum_{k}\,\psi\{s^{k}\}^{T}s^{i}s^{k}\psi=0.
$$
With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8) up to an overall non-vanishing constant. We therefore conclude that for appropriately chosen constants $\alpha_{1}$ and $\alpha_{2}$ , the choice
$$
v_{a}^{i}=\alpha_{1}(v_{1})_{a}^{i}+\alpha_{2}(v_{2})_{a}^{i}
$$
satisfies (3.1).
3.2. Rotating a ground state
We assume there exists a normalizable ground state $\Psi$ which is not a singlet under $S U(2)_{R}$ . Under some $S U(2)_{R}$ rotation, we obtain another non-trivial $L^{2}$ zero-energy state. What does $L^{2}$ imply? Let us collectively denote all the bosonic coordinates $x$ and $q$ by $y^{i}$ where $i=1,\dots,D$ . Normalizability requires that,
$$
<\Psi,\Psi>=\int d^{D}y\,\Psi^{\dag}(y^{i})\,\Psi(y^{i})<\infty.
$$
For some $\tilde{s}^{i}$ , the state ${\tilde{s}}^{i}\Psi$ is a non-trivial ground state. It satisfies the relation,
$$
Q_{a}\left(\tilde{s}^{i}\Psi\right)=Q_{a}\Psi=0,
$$
for each $a$ by definition of a ground state. Using (3.1), we find that
$$
\begin{array}{r c l}{{}}&{{}}&{{\tilde{s}^{i}\Psi=\displaystyle\sum_{a}\,\left\{Q_{a},v_{a}^{i}\right\}\Psi,}}\\ {{}}&{{}}&{{=\displaystyle\sum_{a}Q_{a}\left(v_{a}^{i}\Psi\right).}}\end{array}
$$
|
<html><body>
<p data-bbox="69 70 541 105">but $s^{i}s^{l}I$ does not contain a singlet under the $S p i n(5)$ action on fermions because $I$ is proportional to $\gamma^{\mu}$ so the trace vanishes. </p>
<p data-bbox="93 110 330 125">Again what remains is the anti-commutator, </p>
<div class="equation" data-bbox="249 141 362 172">$$
\sum_{a}\left\{s_{a b}^{j}\psi_{b}\,p_{j},(v_{2})_{a}^{i}\right\}.
$$</div>
<p data-bbox="70 183 477 199">It is easy to check that the $\psi\psi$ terms in the anti-commutator vanish because, </p>
<div class="equation" data-bbox="246 217 364 247">$$
\sum_{k}\,\psi\{s^{k}\}^{T}s^{i}s^{k}\psi=0.
$$</div>
<p data-bbox="70 258 541 313">With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8) up to an overall non-vanishing constant. We therefore conclude that for appropriately chosen constants $\alpha_{1}$ and $\alpha_{2}$ , the choice </p>
<div class="equation" data-bbox="245 332 366 348">$$
v_{a}^{i}=\alpha_{1}(v_{1})_{a}^{i}+\alpha_{2}(v_{2})_{a}^{i}
$$</div>
<p data-bbox="70 366 145 381">satisfies (3.1). </p>
<p data-bbox="71 401 222 416">3.2. Rotating a ground state </p>
<p data-bbox="70 427 542 502">We assume there exists a normalizable ground state $\Psi$ which is not a singlet under $S U(2)_{R}$ . Under some $S U(2)_{R}$ rotation, we obtain another non-trivial $L^{2}$ zero-energy state. What does $L^{2}$ imply? Let us collectively denote all the bosonic coordinates $x$ and $q$ by $y^{i}$ where $i=1,\dots,D$ . Normalizability requires that, </p>
<div class="equation" data-bbox="207 516 403 545">$$
<\Psi,\Psi>=\int d^{D}y\,\Psi^{\dag}(y^{i})\,\Psi(y^{i})<\infty.
$$</div>
<p data-bbox="69 557 490 573">For some $\tilde{s}^{i}$ , the state ${\tilde{s}}^{i}\Psi$ is a non-trivial ground state. It satisfies the relation, </p>
<div class="equation" data-bbox="249 592 362 609">$$
Q_{a}\left(\tilde{s}^{i}\Psi\right)=Q_{a}\Psi=0,
$$</div>
<p data-bbox="69 626 426 641">for each $a$ by definition of a ground state. Using (3.1), we find that </p>
<div class="equation" data-bbox="247 656 363 718">$$
\begin{array}{r c l}{{}}&{{}}&{{\tilde{s}^{i}\Psi=\displaystyle\sum_{a}\,\left\{Q_{a},v_{a}^{i}\right\}\Psi,}}\\ {{}}&{{}}&{{=\displaystyle\sum_{a}Q_{a}\left(v_{a}^{i}\Psi\right).}}\end{array}
$$</div>
</body></html>
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0001189v2
| 6 | 612 | 792 | 1,275 | 1,650 |
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Normalizability requires that,", "type": "text"}], "index": 15}], "index": 13.5, "bbox_fs": [70, 430, 540, 505]}, {"type": "interline_equation", "bbox": [207, 516, 403, 545], "lines": [{"bbox": [207, 516, 403, 545], "spans": [{"bbox": [207, 516, 403, 545], "score": 0.94, "content": "<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty.", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [69, 557, 490, 573], "lines": [{"bbox": [70, 558, 490, 576], "spans": [{"bbox": [70, 558, 122, 576], "score": 1.0, "content": "For some ", "type": "text"}, {"bbox": [123, 560, 132, 570], "score": 0.9, "content": "\\tilde{s}^{i}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [133, 558, 189, 576], "score": 1.0, "content": ", the state ", "type": "text"}, {"bbox": [190, 560, 208, 570], "score": 0.92, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 18}, {"bbox": [209, 558, 490, 576], "score": 1.0, "content": " is a non-trivial ground state. It satisfies the relation,", "type": "text"}], "index": 17}], "index": 17, "bbox_fs": [70, 558, 490, 576]}, {"type": "interline_equation", "bbox": [249, 592, 362, 609], "lines": [{"bbox": [249, 592, 362, 609], "spans": [{"bbox": [249, 592, 362, 609], "score": 0.93, "content": "Q_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [69, 626, 426, 641], "lines": [{"bbox": [70, 628, 425, 642], "spans": [{"bbox": [70, 628, 116, 642], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [117, 633, 123, 639], "score": 0.89, "content": "a", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [123, 628, 425, 642], "score": 1.0, "content": " by definition of a ground state. Using (3.1), we find that", "type": "text"}], "index": 19}], "index": 19, "bbox_fs": [70, 628, 425, 642]}, {"type": "interline_equation", "bbox": [247, 656, 363, 718], "lines": [{"bbox": [247, 656, 363, 718], "spans": [{"bbox": [247, 656, 363, 718], "score": 0.92, "content": "\\begin{array}{r c l}{{}}&{{}}&{{\\tilde{s}^{i}\\Psi=\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi,}}\\\\ {{}}&{{}}&{{=\\displaystyle\\sum_{a}Q_{a}\\left(v_{a}^{i}\\Psi\\right).}}\\end{array}", "type": "interline_equation"}], "index": 20}], "index": 20}]}
|
[{"type": "text", "bbox": [69, 70, 541, 105], "content": "but does not contain a singlet under the action on fermions because is proportional to so the trace vanishes.", "index": 0}, {"type": "text", "bbox": [93, 110, 330, 125], "content": "Again what remains is the anti-commutator,", "index": 1}, {"type": "interline_equation", "bbox": [249, 141, 362, 172], "content": "", "index": 2}, {"type": "text", "bbox": [70, 183, 477, 199], "content": "It is easy to check that the terms in the anti-commutator vanish because,", "index": 3}, {"type": "interline_equation", "bbox": [246, 217, 364, 247], "content": "", "index": 4}, {"type": "text", "bbox": [70, 258, 541, 313], "content": "With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8) up to an overall non-vanishing constant. We therefore conclude that for appropriately chosen constants and , the choice", "index": 5}, {"type": "interline_equation", "bbox": [245, 332, 366, 348], "content": "", "index": 6}, {"type": "text", "bbox": [70, 366, 145, 381], "content": "satisfies (3.1).", "index": 7}, {"type": "text", "bbox": [71, 401, 222, 416], "content": "3.2. Rotating a ground state", "index": 8}, {"type": "text", "bbox": [70, 427, 542, 502], "content": "We assume there exists a normalizable ground state which is not a singlet under . Under some rotation, we obtain another non-trivial zero-energy state. What does imply? Let us collectively denote all the bosonic coordinates and by where . Normalizability requires that,", "index": 9}, {"type": "interline_equation", "bbox": [207, 516, 403, 545], "content": "", "index": 10}, {"type": "text", "bbox": [69, 557, 490, 573], "content": "For some , the state is a non-trivial ground state. It satisfies the relation,", "index": 11}, {"type": "interline_equation", "bbox": [249, 592, 362, 609], "content": "", "index": 12}, {"type": "text", "bbox": [69, 626, 426, 641], "content": "for each by definition of a ground state. Using (3.1), we find that", "index": 13}, {"type": "interline_equation", "bbox": [247, 656, 363, 718], "content": "", "index": 14}]
|
[{"bbox": [69, 71, 542, 90], "content": "but does not contain a singlet under the action on fermions because is", "parent_index": 0, "line_index": 0}, {"bbox": [72, 94, 284, 108], "content": "proportional to so the trace vanishes.", "parent_index": 0, "line_index": 1}, {"bbox": [96, 113, 329, 127], "content": "Again what remains is the anti-commutator,", "parent_index": 1, "line_index": 0}, {"bbox": [69, 185, 477, 202], "content": "It is easy to check that the terms in the anti-commutator vanish because,", "parent_index": 3, "line_index": 0}, {"bbox": [70, 261, 540, 277], "content": "With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8)", "parent_index": 5, "line_index": 0}, {"bbox": [69, 280, 540, 297], "content": "up to an overall non-vanishing constant. We therefore conclude that for appropriately", "parent_index": 5, "line_index": 1}, {"bbox": [71, 301, 277, 315], "content": "chosen constants and , the choice", "parent_index": 5, "line_index": 2}, {"bbox": [70, 368, 144, 384], "content": "satisfies (3.1).", "parent_index": 7, "line_index": 0}, {"bbox": [72, 404, 221, 417], "content": "3.2. Rotating a ground state", "parent_index": 8, "line_index": 0}, {"bbox": [95, 430, 540, 445], "content": "We assume there exists a normalizable ground state which is not a singlet under", "parent_index": 9, "line_index": 0}, {"bbox": [71, 449, 540, 465], "content": ". Under some rotation, we obtain another non-trivial zero-energy state.", "parent_index": 9, "line_index": 1}, {"bbox": [70, 468, 539, 485], "content": "What does imply? Let us collectively denote all the bosonic coordinates and by", "parent_index": 9, "line_index": 2}, {"bbox": [70, 489, 333, 505], "content": "where . Normalizability requires that,", "parent_index": 9, "line_index": 3}, {"bbox": [70, 558, 490, 576], "content": "For some , the state is a non-trivial ground state. It satisfies the relation,", "parent_index": 11, "line_index": 0}, {"bbox": [70, 628, 425, 642], "content": "for each by definition of a ground state. Using (3.1), we find that", "parent_index": 13, "line_index": 0}]
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[]
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[{"bbox": [94, 74, 120, 84], "content": "s^{i}s^{l}I", "parent_index": 0, "subtype": "inline"}, {"bbox": [325, 75, 366, 87], "content": "S p i n(5)", "parent_index": 0, "subtype": "inline"}, {"bbox": [520, 75, 527, 84], "content": "I", "parent_index": 0, "subtype": "inline"}, {"bbox": [154, 95, 168, 106], "content": "\\gamma^{\\mu}", "parent_index": 0, "subtype": "inline"}, {"bbox": [249, 141, 362, 172], "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}.", "parent_index": 2, "subtype": "interline"}, {"bbox": [216, 188, 232, 199], "content": "\\psi\\psi", "parent_index": 3, "subtype": "inline"}, {"bbox": [246, 217, 364, 247], "content": "\\sum_{k}\\,\\psi\\{s^{k}\\}^{T}s^{i}s^{k}\\psi=0.", "parent_index": 4, "subtype": "interline"}, {"bbox": [163, 306, 175, 313], "content": "\\alpha_{1}", "parent_index": 5, "subtype": "inline"}, {"bbox": [203, 306, 216, 313], "content": "\\alpha_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [245, 332, 366, 348], "content": "v_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}", "parent_index": 6, "subtype": "interline"}, {"bbox": [376, 433, 386, 441], "content": "\\Psi", "parent_index": 9, "subtype": "inline"}, {"bbox": [71, 451, 112, 464], "content": "S U(2)_{R}", "parent_index": 9, "subtype": "inline"}, {"bbox": [185, 451, 225, 464], "content": "S U(2)_{R}", "parent_index": 9, "subtype": "inline"}, {"bbox": [431, 450, 445, 461], "content": "L^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [131, 470, 145, 480], "content": "L^{2}", "parent_index": 9, "subtype": "inline"}, {"bbox": [469, 473, 476, 480], "content": "x", "parent_index": 9, "subtype": "inline"}, {"bbox": [503, 475, 509, 483], "content": "q", "parent_index": 9, "subtype": "inline"}, {"bbox": [529, 470, 539, 483], "content": "y^{i}", "parent_index": 9, "subtype": "inline"}, {"bbox": [106, 492, 169, 502], "content": "i=1,\\dots,D", "parent_index": 9, "subtype": "inline"}, {"bbox": [207, 516, 403, 545], "content": "<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty.", "parent_index": 10, "subtype": "interline"}, {"bbox": [123, 560, 132, 570], "content": "\\tilde{s}^{i}", "parent_index": 11, "subtype": "inline"}, {"bbox": [190, 560, 208, 570], "content": "{\\tilde{s}}^{i}\\Psi", "parent_index": 11, "subtype": "inline"}, {"bbox": [249, 592, 362, 609], "content": "Q_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,", "parent_index": 12, "subtype": "interline"}, {"bbox": [117, 633, 123, 639], "content": "a", "parent_index": 13, "subtype": "inline"}, {"bbox": [247, 656, 363, 718], "content": "\\begin{array}{r c l}{{}}&{{}}&{{\\tilde{s}^{i}\\Psi=\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi,}}\\\\ {{}}&{{}}&{{=\\displaystyle\\sum_{a}Q_{a}\\left(v_{a}^{i}\\Psi\\right).}}\\end{array}", "parent_index": 14, "subtype": "interline"}]
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[]
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The new ground state looks $Q$ -trivial. To show that it really is physically trivial, we need to check that it has zero norm. Since $Q$ is Hermitian and kills ${\tilde{s}}^{i}\Psi$ , the norm of ${\tilde{s}}^{i}\Psi$ vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo [18,6]: in terms of $y=|y^{i}|$ , we can cutoff of the integral using a smooth bump function $\rho_{R}(y)$ which vanishes for $y>2R$ , satisfies $|d\rho_{R}|<4/R$ and is one for $y<R$ ,
$$
<\tilde{s}^{i}\Psi,\tilde{s}^{i}\Psi>=\operatorname*{lim}_{R\rightarrow\infty}<\rho_{R}(y)\tilde{s}^{i}\Psi,\tilde{s}^{i}\Psi>.
$$
Using (3.9) and (3.10), we see that
$$
\begin{array}{r}{{<\tilde{s}^{i}\Psi,\tilde{s}^{i}\Psi>\,=\,\operatorname*{lim}_{R\rightarrow\infty}\,<\rho_{R}(y)\tilde{s}^{i}\Psi,\displaystyle\sum_{a}\,\left\{Q_{a},v_{a}^{i}\right\}\Psi>},}\\ {{=\displaystyle\operatorname*{lim}_{R\rightarrow\infty}\sum_{a}\,<\left[Q_{a},\rho_{R}(y)\right]\tilde{s}^{i}\Psi,v_{a}^{i}\Psi>.}}\end{array}
$$
We see that $[Q_{a},\rho_{R}(y)]$ is $O(1/y)$ and vanishes for $y<R$ and $y>2R$ . Since $v_{i}^{a}$ is $O(y)$ at worst, the right hand side of (3.11) vanishes. The $S U(2)_{R}$ symmetry therefore acts trivially on all normalizable ground states.
# 4. Invariance Under the $S p i n(5)$ Symmetry
4.1. Relating the Spin(5) currents to the supercharge
We want to use essentially the same argument as in the $S U(2)_{R}$ case. For some choice of $v_{a}^{\mu\nu}$ , we want to show that:
$$
T^{\mu\nu}=\sum_{a}\,\{Q_{a},v_{a}^{\mu\nu}\}.
$$
Let us start with the vector multiplet. We take a candidate gauge singlet,
$$
(v_{1})_{a}^{\mu\nu}=\{\gamma^{\mu}x^{\nu}-\gamma^{\nu}x^{\mu}\}_{a b}\,\lambda_{b}.
$$
Again this choice anti-commutes with $Q^{h}$ because $\lambda$ anti-commutes with $\psi$ . The anticommutator with ${\textstyle\frac{1}{2}}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}$ results in a trace of three gamma matrices and so vanishes. It also anti-commutes with the $\cal{D}$ -term in (2.4). To see this, we compute:
$$
\sum_{a}\left\{D_{a b}\lambda_{b},(v_{1})_{a}^{\mu\nu}\right\}=D_{a b}\left\{\gamma^{\mu}x^{\nu}-\gamma^{\nu}x^{\mu}\right\}_{a b}.
$$
However, this combination does not contain a singlet under $S p i n(5)$ so (4.3) vanishes.
|
<html><body>
<p data-bbox="69 69 542 163">The new ground state looks $Q$ -trivial. To show that it really is physically trivial, we need to check that it has zero norm. Since $Q$ is Hermitian and kills ${\tilde{s}}^{i}\Psi$ , the norm of ${\tilde{s}}^{i}\Psi$ vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo [18,6]: in terms of $y=|y^{i}|$ , we can cutoff of the integral using a smooth bump function $\rho_{R}(y)$ which vanishes for $y>2R$ , satisfies $|d\rho_{R}|<4/R$ and is one for $y<R$ , </p>
<div class="equation" data-bbox="197 179 414 200">$$
<\tilde{s}^{i}\Psi,\tilde{s}^{i}\Psi>=\operatorname*{lim}_{R\rightarrow\infty}<\rho_{R}(y)\tilde{s}^{i}\Psi,\tilde{s}^{i}\Psi>.
$$</div>
<p data-bbox="70 209 256 225">Using (3.9) and (3.10), we see that </p>
<div class="equation" data-bbox="168 237 442 301">$$
\begin{array}{r}{{<\tilde{s}^{i}\Psi,\tilde{s}^{i}\Psi>\,=\,\operatorname*{lim}_{R\rightarrow\infty}\,<\rho_{R}(y)\tilde{s}^{i}\Psi,\displaystyle\sum_{a}\,\left\{Q_{a},v_{a}^{i}\right\}\Psi>},}\\ {{=\displaystyle\operatorname*{lim}_{R\rightarrow\infty}\sum_{a}\,<\left[Q_{a},\rho_{R}(y)\right]\tilde{s}^{i}\Psi,v_{a}^{i}\Psi>.}}\end{array}
$$</div>
<p data-bbox="70 308 541 362">We see that $[Q_{a},\rho_{R}(y)]$ is $O(1/y)$ and vanishes for $y<R$ and $y>2R$ . Since $v_{i}^{a}$ is $O(y)$ at worst, the right hand side of (3.11) vanishes. The $S U(2)_{R}$ symmetry therefore acts trivially on all normalizable ground states. </p>
<h1 data-bbox="72 392 330 408">4. Invariance Under the $S p i n(5)$ Symmetry </h1>
<p data-bbox="72 419 352 434">4.1. Relating the Spin(5) currents to the supercharge </p>
<p data-bbox="69 444 542 478">We want to use essentially the same argument as in the $S U(2)_{R}$ case. For some choice of $v_{a}^{\mu\nu}$ , we want to show that: </p>
<div class="equation" data-bbox="249 493 362 523">$$
T^{\mu\nu}=\sum_{a}\,\{Q_{a},v_{a}^{\mu\nu}\}.
$$</div>
<p data-bbox="70 532 460 548">Let us start with the vector multiplet. We take a candidate gauge singlet, </p>
<div class="equation" data-bbox="228 565 384 580">$$
(v_{1})_{a}^{\mu\nu}=\{\gamma^{\mu}x^{\nu}-\gamma^{\nu}x^{\mu}\}_{a b}\,\lambda_{b}.
$$</div>
<p data-bbox="69 594 542 649">Again this choice anti-commutes with $Q^{h}$ because $\lambda$ anti-commutes with $\psi$ . The anticommutator with ${\textstyle\frac{1}{2}}f_{A B C}\left(\gamma^{\mu\nu}\lambda_{A}x_{B}^{\mu}x_{C}^{\nu}\right)_{a}$ results in a trace of three gamma matrices and so vanishes. It also anti-commutes with the $\cal{D}$ -term in (2.4). To see this, we compute: </p>
<div class="equation" data-bbox="188 661 423 690">$$
\sum_{a}\left\{D_{a b}\lambda_{b},(v_{1})_{a}^{\mu\nu}\right\}=D_{a b}\left\{\gamma^{\mu}x^{\nu}-\gamma^{\nu}x^{\mu}\right\}_{a b}.
$$</div>
<p data-bbox="70 700 521 716">However, this combination does not contain a singlet under $S p i n(5)$ so (4.3) vanishes. </p>
</body></html>
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0001189v2
| 7 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "The new ground state looks $Q$ -trivial. To show that it really is physically trivial, we need to check that it has zero norm. Since $Q$ is Hermitian and kills ${\\tilde{s}}^{i}\\Psi$ , the norm of ${\\tilde{s}}^{i}\\Psi$ vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo [18,6]: in terms of $y=|y^{i}|$ , we can cutoff of the integral using a smooth bump function $\\rho_{R}(y)$ which vanishes for $y>2R$ , satisfies $|d\\rho_{R}|<4/R$ and is one for $y<R$ , ", "page_idx": 7}, {"type": "equation", "text": "$$\n<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname*{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Using (3.9) and (3.10), we see that ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname*{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname*{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "We see that $[Q_{a},\\rho_{R}(y)]$ is $O(1/y)$ and vanishes for $y<R$ and $y>2R$ . Since $v_{i}^{a}$ is $O(y)$ at worst, the right hand side of (3.11) vanishes. The $S U(2)_{R}$ symmetry therefore acts trivially on all normalizable ground states. ", "page_idx": 7}, {"type": "text", "text": "4. Invariance Under the $S p i n(5)$ Symmetry ", "text_level": 1, "page_idx": 7}, {"type": "text", "text": "4.1. Relating the Spin(5) currents to the supercharge ", "page_idx": 7}, {"type": "text", "text": "We want to use essentially the same argument as in the $S U(2)_{R}$ case. For some choice of $v_{a}^{\\mu\\nu}$ , we want to show that: ", "page_idx": 7}, {"type": "equation", "text": "$$\nT^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Let us start with the vector multiplet. We take a candidate gauge singlet, ", "page_idx": 7}, {"type": "equation", "text": "$$\n(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "Again this choice anti-commutes with $Q^{h}$ because $\\lambda$ anti-commutes with $\\psi$ . The anticommutator with ${\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}$ results in a trace of three gamma matrices and so vanishes. It also anti-commutes with the $\\cal{D}$ -term in (2.4). To see this, we compute: ", "page_idx": 7}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.\n$$", "text_format": "latex", "page_idx": 7}, {"type": "text", "text": "However, this combination does not contain a singlet under $S p i n(5)$ so (4.3) vanishes. ", "page_idx": 7}]
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{"preproc_blocks": [{"type": "text", "bbox": [69, 69, 542, 163], "lines": [{"bbox": [70, 72, 542, 88], "spans": [{"bbox": [70, 72, 220, 88], "score": 1.0, "content": "The new ground state looks ", "type": "text"}, {"bbox": [221, 75, 230, 87], "score": 0.9, "content": "Q", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [230, 72, 542, 88], "score": 1.0, "content": "-trivial. To show that it really is physically trivial, we need", "type": "text"}], "index": 0}, {"bbox": [71, 93, 540, 107], "spans": [{"bbox": [71, 93, 284, 107], "score": 1.0, "content": "to check that it has zero norm. Since ", "type": "text"}, {"bbox": [285, 95, 294, 106], "score": 0.9, "content": "Q", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [295, 93, 422, 107], "score": 1.0, "content": " is Hermitian and kills ", "type": "text"}, {"bbox": [423, 93, 442, 103], "score": 0.9, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [442, 93, 520, 107], "score": 1.0, "content": ", the norm of ", "type": "text"}, {"bbox": [520, 93, 540, 103], "score": 0.92, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 20}], "index": 1}, {"bbox": [71, 111, 541, 127], "spans": [{"bbox": [71, 111, 541, 127], "score": 1.0, "content": "vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo", "type": "text"}], "index": 2}, {"bbox": [71, 130, 540, 145], "spans": [{"bbox": [71, 130, 172, 145], "score": 1.0, "content": "[18,6]: in terms of ", "type": "text"}, {"bbox": [172, 131, 214, 144], "score": 0.94, "content": "y=|y^{i}|", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [214, 130, 540, 145], "score": 1.0, "content": ", we can cutoff of the integral using a smooth bump function", "type": "text"}], "index": 3}, {"bbox": [71, 150, 473, 165], "spans": [{"bbox": [71, 151, 101, 163], "score": 0.94, "content": "\\rho_{R}(y)", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [101, 150, 204, 165], "score": 1.0, "content": " which vanishes for ", "type": "text"}, {"bbox": [204, 151, 242, 163], "score": 0.92, "content": "y>2R", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [243, 150, 293, 165], "score": 1.0, "content": ", satisfies ", "type": "text"}, {"bbox": [293, 151, 357, 163], "score": 0.95, "content": "|d\\rho_{R}|<4/R", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [358, 150, 436, 165], "score": 1.0, "content": " and is one for ", "type": "text"}, {"bbox": [437, 151, 468, 162], "score": 0.93, "content": "y<R", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [469, 150, 473, 165], "score": 1.0, "content": ",", "type": "text"}], "index": 4}], "index": 2}, {"type": "interline_equation", "bbox": [197, 179, 414, 200], "lines": [{"bbox": [197, 179, 414, 200], "spans": [{"bbox": [197, 179, 414, 200], "score": 0.9, "content": "<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname*{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [70, 209, 256, 225], "lines": [{"bbox": [72, 212, 254, 227], "spans": [{"bbox": [72, 212, 254, 227], "score": 1.0, "content": "Using (3.9) and (3.10), we see that", "type": "text"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [168, 237, 442, 301], "lines": [{"bbox": [168, 237, 442, 301], "spans": [{"bbox": [168, 237, 442, 301], "score": 0.93, "content": "\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname*{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname*{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 308, 541, 362], "lines": [{"bbox": [70, 311, 540, 329], "spans": [{"bbox": [70, 311, 138, 329], "score": 1.0, "content": "We see that ", "type": "text"}, {"bbox": [138, 313, 195, 326], "score": 0.94, "content": "[Q_{a},\\rho_{R}(y)]", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [196, 311, 212, 329], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [212, 313, 249, 326], "score": 0.94, "content": "O(1/y)", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [250, 311, 343, 329], "score": 1.0, "content": " and vanishes for ", "type": "text"}, {"bbox": [343, 314, 377, 325], "score": 0.93, "content": "y<R", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [377, 311, 404, 329], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 314, 444, 325], "score": 0.92, "content": "y>2R", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [444, 311, 485, 329], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [485, 314, 497, 325], "score": 0.91, "content": "v_{i}^{a}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [497, 311, 514, 329], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [514, 313, 540, 326], "score": 0.95, "content": "O(y)", "type": "inline_equation", "height": 13, "width": 26}], "index": 8}, {"bbox": [70, 331, 541, 347], "spans": [{"bbox": [70, 331, 363, 347], "score": 1.0, "content": "at worst, the right hand side of (3.11) vanishes. The ", "type": "text"}, {"bbox": [364, 332, 405, 344], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [406, 331, 541, 347], "score": 1.0, "content": " symmetry therefore acts", "type": "text"}], "index": 9}, {"bbox": [71, 351, 294, 365], "spans": [{"bbox": [71, 351, 294, 365], "score": 1.0, "content": "trivially on all normalizable ground states.", "type": "text"}], "index": 10}], "index": 9}, {"type": "title", "bbox": [72, 392, 330, 408], "lines": [{"bbox": [71, 395, 328, 410], "spans": [{"bbox": [71, 395, 220, 409], "score": 1.0, "content": "4. Invariance Under the ", "type": "text"}, {"bbox": [221, 397, 262, 410], "score": 0.39, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [262, 395, 328, 409], "score": 1.0, "content": " Symmetry", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [72, 419, 352, 434], "lines": [{"bbox": [72, 421, 351, 435], "spans": [{"bbox": [72, 421, 351, 435], "score": 1.0, "content": "4.1. Relating the Spin(5) currents to the supercharge", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 444, 542, 478], "lines": [{"bbox": [94, 446, 541, 462], "spans": [{"bbox": [94, 446, 384, 462], "score": 1.0, "content": "We want to use essentially the same argument as in the ", "type": "text"}, {"bbox": [384, 448, 425, 461], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [425, 446, 541, 462], "score": 1.0, "content": " case. For some choice", "type": "text"}], "index": 13}, {"bbox": [69, 464, 230, 483], "spans": [{"bbox": [69, 464, 84, 483], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [85, 468, 103, 479], "score": 0.93, "content": "v_{a}^{\\mu\\nu}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [104, 464, 230, 483], "score": 1.0, "content": ", we want to show that:", "type": "text"}], "index": 14}], "index": 13.5}, {"type": "interline_equation", "bbox": [249, 493, 362, 523], "lines": [{"bbox": [249, 493, 362, 523], "spans": [{"bbox": [249, 493, 362, 523], "score": 0.94, "content": "T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [70, 532, 460, 548], "lines": [{"bbox": [70, 534, 459, 550], "spans": [{"bbox": [70, 534, 459, 550], "score": 1.0, "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "type": "text"}], "index": 16}], "index": 16}, {"type": "interline_equation", "bbox": [228, 565, 384, 580], "lines": [{"bbox": [228, 565, 384, 580], "spans": [{"bbox": [228, 565, 384, 580], "score": 0.92, "content": "(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [69, 594, 542, 649], "lines": [{"bbox": [72, 597, 541, 613], "spans": [{"bbox": [72, 597, 279, 613], "score": 1.0, "content": "Again this choice anti-commutes with ", "type": "text"}, {"bbox": [279, 598, 295, 611], "score": 0.92, "content": "Q^{h}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [295, 597, 345, 613], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [345, 600, 353, 609], "score": 0.87, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [353, 597, 468, 613], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [468, 600, 477, 611], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [477, 597, 541, 613], "score": 1.0, "content": ". The anti-", "type": "text"}], "index": 18}, {"bbox": [69, 614, 543, 636], "spans": [{"bbox": [69, 614, 167, 636], "score": 1.0, "content": "commutator with ", "type": "text"}, {"bbox": [167, 616, 284, 631], "score": 0.92, "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "type": "inline_equation", "height": 15, "width": 117}, {"bbox": [285, 614, 543, 636], "score": 1.0, "content": " results in a trace of three gamma matrices and", "type": "text"}], "index": 19}, {"bbox": [71, 636, 522, 650], "spans": [{"bbox": [71, 636, 302, 650], "score": 1.0, "content": "so vanishes. It also anti-commutes with the ", "type": "text"}, {"bbox": [303, 638, 313, 646], "score": 0.9, "content": "\\cal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [313, 636, 522, 650], "score": 1.0, "content": "-term in (2.4). To see this, we compute:", "type": "text"}], "index": 20}], "index": 19}, {"type": "interline_equation", "bbox": [188, 661, 423, 690], "lines": [{"bbox": [188, 661, 423, 690], "spans": [{"bbox": [188, 661, 423, 690], "score": 0.93, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [70, 700, 521, 716], "lines": [{"bbox": [70, 701, 521, 719], "spans": [{"bbox": [70, 701, 386, 719], "score": 1.0, "content": "However, this combination does not contain a singlet under ", "type": "text"}, {"bbox": [387, 704, 427, 716], "score": 0.89, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [428, 701, 521, 719], "score": 1.0, "content": " so (4.3) vanishes.", "type": "text"}], "index": 22}], "index": 22}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [197, 179, 414, 200], "lines": [{"bbox": [197, 179, 414, 200], "spans": [{"bbox": [197, 179, 414, 200], "score": 0.9, "content": "<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname*{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [168, 237, 442, 301], "lines": [{"bbox": [168, 237, 442, 301], "spans": [{"bbox": [168, 237, 442, 301], "score": 0.93, "content": "\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname*{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname*{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [249, 493, 362, 523], "lines": [{"bbox": [249, 493, 362, 523], "spans": [{"bbox": [249, 493, 362, 523], "score": 0.94, "content": "T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "interline_equation", "bbox": [228, 565, 384, 580], "lines": [{"bbox": [228, 565, 384, 580], "spans": [{"bbox": [228, 565, 384, 580], "score": 0.92, "content": "(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [188, 661, 423, 690], "lines": [{"bbox": [188, 661, 423, 690], "spans": [{"bbox": [188, 661, 423, 690], "score": 0.93, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.", "type": "interline_equation"}], "index": 21}], "index": 21}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 69, 542, 163], "lines": [{"bbox": [70, 72, 542, 88], "spans": [{"bbox": [70, 72, 220, 88], "score": 1.0, "content": "The new ground state looks ", "type": "text"}, {"bbox": [221, 75, 230, 87], "score": 0.9, "content": "Q", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [230, 72, 542, 88], "score": 1.0, "content": "-trivial. To show that it really is physically trivial, we need", "type": "text"}], "index": 0}, {"bbox": [71, 93, 540, 107], "spans": [{"bbox": [71, 93, 284, 107], "score": 1.0, "content": "to check that it has zero norm. Since ", "type": "text"}, {"bbox": [285, 95, 294, 106], "score": 0.9, "content": "Q", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [295, 93, 422, 107], "score": 1.0, "content": " is Hermitian and kills ", "type": "text"}, {"bbox": [423, 93, 442, 103], "score": 0.9, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 19}, {"bbox": [442, 93, 520, 107], "score": 1.0, "content": ", the norm of ", "type": "text"}, {"bbox": [520, 93, 540, 103], "score": 0.92, "content": "{\\tilde{s}}^{i}\\Psi", "type": "inline_equation", "height": 10, "width": 20}], "index": 1}, {"bbox": [71, 111, 541, 127], "spans": [{"bbox": [71, 111, 541, 127], "score": 1.0, "content": "vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo", "type": "text"}], "index": 2}, {"bbox": [71, 130, 540, 145], "spans": [{"bbox": [71, 130, 172, 145], "score": 1.0, "content": "[18,6]: in terms of ", "type": "text"}, {"bbox": [172, 131, 214, 144], "score": 0.94, "content": "y=|y^{i}|", "type": "inline_equation", "height": 13, "width": 42}, {"bbox": [214, 130, 540, 145], "score": 1.0, "content": ", we can cutoff of the integral using a smooth bump function", "type": "text"}], "index": 3}, {"bbox": [71, 150, 473, 165], "spans": [{"bbox": [71, 151, 101, 163], "score": 0.94, "content": "\\rho_{R}(y)", "type": "inline_equation", "height": 12, "width": 30}, {"bbox": [101, 150, 204, 165], "score": 1.0, "content": " which vanishes for ", "type": "text"}, {"bbox": [204, 151, 242, 163], "score": 0.92, "content": "y>2R", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [243, 150, 293, 165], "score": 1.0, "content": ", satisfies ", "type": "text"}, {"bbox": [293, 151, 357, 163], "score": 0.95, "content": "|d\\rho_{R}|<4/R", "type": "inline_equation", "height": 12, "width": 64}, {"bbox": [358, 150, 436, 165], "score": 1.0, "content": " and is one for ", "type": "text"}, {"bbox": [437, 151, 468, 162], "score": 0.93, "content": "y<R", "type": "inline_equation", "height": 11, "width": 31}, {"bbox": [469, 150, 473, 165], "score": 1.0, "content": ",", "type": "text"}], "index": 4}], "index": 2, "bbox_fs": [70, 72, 542, 165]}, {"type": "interline_equation", "bbox": [197, 179, 414, 200], "lines": [{"bbox": [197, 179, 414, 200], "spans": [{"bbox": [197, 179, 414, 200], "score": 0.9, "content": "<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname*{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [70, 209, 256, 225], "lines": [{"bbox": [72, 212, 254, 227], "spans": [{"bbox": [72, 212, 254, 227], "score": 1.0, "content": "Using (3.9) and (3.10), we see that", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [72, 212, 254, 227]}, {"type": "interline_equation", "bbox": [168, 237, 442, 301], "lines": [{"bbox": [168, 237, 442, 301], "spans": [{"bbox": [168, 237, 442, 301], "score": 0.93, "content": "\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname*{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname*{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 308, 541, 362], "lines": [{"bbox": [70, 311, 540, 329], "spans": [{"bbox": [70, 311, 138, 329], "score": 1.0, "content": "We see that ", "type": "text"}, {"bbox": [138, 313, 195, 326], "score": 0.94, "content": "[Q_{a},\\rho_{R}(y)]", "type": "inline_equation", "height": 13, "width": 57}, {"bbox": [196, 311, 212, 329], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [212, 313, 249, 326], "score": 0.94, "content": "O(1/y)", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [250, 311, 343, 329], "score": 1.0, "content": " and vanishes for ", "type": "text"}, {"bbox": [343, 314, 377, 325], "score": 0.93, "content": "y<R", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [377, 311, 404, 329], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 314, 444, 325], "score": 0.92, "content": "y>2R", "type": "inline_equation", "height": 11, "width": 39}, {"bbox": [444, 311, 485, 329], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [485, 314, 497, 325], "score": 0.91, "content": "v_{i}^{a}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [497, 311, 514, 329], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [514, 313, 540, 326], "score": 0.95, "content": "O(y)", "type": "inline_equation", "height": 13, "width": 26}], "index": 8}, {"bbox": [70, 331, 541, 347], "spans": [{"bbox": [70, 331, 363, 347], "score": 1.0, "content": "at worst, the right hand side of (3.11) vanishes. The ", "type": "text"}, {"bbox": [364, 332, 405, 344], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [406, 331, 541, 347], "score": 1.0, "content": " symmetry therefore acts", "type": "text"}], "index": 9}, {"bbox": [71, 351, 294, 365], "spans": [{"bbox": [71, 351, 294, 365], "score": 1.0, "content": "trivially on all normalizable ground states.", "type": "text"}], "index": 10}], "index": 9, "bbox_fs": [70, 311, 541, 365]}, {"type": "title", "bbox": [72, 392, 330, 408], "lines": [{"bbox": [71, 395, 328, 410], "spans": [{"bbox": [71, 395, 220, 409], "score": 1.0, "content": "4. Invariance Under the ", "type": "text"}, {"bbox": [221, 397, 262, 410], "score": 0.39, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [262, 395, 328, 409], "score": 1.0, "content": " Symmetry", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [72, 419, 352, 434], "lines": [{"bbox": [72, 421, 351, 435], "spans": [{"bbox": [72, 421, 351, 435], "score": 1.0, "content": "4.1. Relating the Spin(5) currents to the supercharge", "type": "text"}], "index": 12}], "index": 12, "bbox_fs": [72, 421, 351, 435]}, {"type": "text", "bbox": [69, 444, 542, 478], "lines": [{"bbox": [94, 446, 541, 462], "spans": [{"bbox": [94, 446, 384, 462], "score": 1.0, "content": "We want to use essentially the same argument as in the ", "type": "text"}, {"bbox": [384, 448, 425, 461], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [425, 446, 541, 462], "score": 1.0, "content": " case. For some choice", "type": "text"}], "index": 13}, {"bbox": [69, 464, 230, 483], "spans": [{"bbox": [69, 464, 84, 483], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [85, 468, 103, 479], "score": 0.93, "content": "v_{a}^{\\mu\\nu}", "type": "inline_equation", "height": 11, "width": 18}, {"bbox": [104, 464, 230, 483], "score": 1.0, "content": ", we want to show that:", "type": "text"}], "index": 14}], "index": 13.5, "bbox_fs": [69, 446, 541, 483]}, {"type": "interline_equation", "bbox": [249, 493, 362, 523], "lines": [{"bbox": [249, 493, 362, 523], "spans": [{"bbox": [249, 493, 362, 523], "score": 0.94, "content": "T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.", "type": "interline_equation"}], "index": 15}], "index": 15}, {"type": "text", "bbox": [70, 532, 460, 548], "lines": [{"bbox": [70, 534, 459, 550], "spans": [{"bbox": [70, 534, 459, 550], "score": 1.0, "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "type": "text"}], "index": 16}], "index": 16, "bbox_fs": [70, 534, 459, 550]}, {"type": "interline_equation", "bbox": [228, 565, 384, 580], "lines": [{"bbox": [228, 565, 384, 580], "spans": [{"bbox": [228, 565, 384, 580], "score": 0.92, "content": "(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [69, 594, 542, 649], "lines": [{"bbox": [72, 597, 541, 613], "spans": [{"bbox": [72, 597, 279, 613], "score": 1.0, "content": "Again this choice anti-commutes with ", "type": "text"}, {"bbox": [279, 598, 295, 611], "score": 0.92, "content": "Q^{h}", "type": "inline_equation", "height": 13, "width": 16}, {"bbox": [295, 597, 345, 613], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [345, 600, 353, 609], "score": 0.87, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [353, 597, 468, 613], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [468, 600, 477, 611], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 9}, {"bbox": [477, 597, 541, 613], "score": 1.0, "content": ". The anti-", "type": "text"}], "index": 18}, {"bbox": [69, 614, 543, 636], "spans": [{"bbox": [69, 614, 167, 636], "score": 1.0, "content": "commutator with ", "type": "text"}, {"bbox": [167, 616, 284, 631], "score": 0.92, "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "type": "inline_equation", "height": 15, "width": 117}, {"bbox": [285, 614, 543, 636], "score": 1.0, "content": " results in a trace of three gamma matrices and", "type": "text"}], "index": 19}, {"bbox": [71, 636, 522, 650], "spans": [{"bbox": [71, 636, 302, 650], "score": 1.0, "content": "so vanishes. It also anti-commutes with the ", "type": "text"}, {"bbox": [303, 638, 313, 646], "score": 0.9, "content": "\\cal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [313, 636, 522, 650], "score": 1.0, "content": "-term in (2.4). To see this, we compute:", "type": "text"}], "index": 20}], "index": 19, "bbox_fs": [69, 597, 543, 650]}, {"type": "interline_equation", "bbox": [188, 661, 423, 690], "lines": [{"bbox": [188, 661, 423, 690], "spans": [{"bbox": [188, 661, 423, 690], "score": 0.93, "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.", "type": "interline_equation"}], "index": 21}], "index": 21}, {"type": "text", "bbox": [70, 700, 521, 716], "lines": [{"bbox": [70, 701, 521, 719], "spans": [{"bbox": [70, 701, 386, 719], "score": 1.0, "content": "However, this combination does not contain a singlet under ", "type": "text"}, {"bbox": [387, 704, 427, 716], "score": 0.89, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [428, 701, 521, 719], "score": 1.0, "content": " so (4.3) vanishes.", "type": "text"}], "index": 22}], "index": 22, "bbox_fs": [70, 701, 521, 719]}]}
|
[{"type": "text", "bbox": [69, 69, 542, 163], "content": "The new ground state looks -trivial. To show that it really is physically trivial, we need to check that it has zero norm. Since is Hermitian and kills , the norm of vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo [18,6]: in terms of , we can cutoff of the integral using a smooth bump function which vanishes for , satisfies and is one for ,", "index": 0}, {"type": "interline_equation", "bbox": [197, 179, 414, 200], "content": "", "index": 1}, {"type": "text", "bbox": [70, 209, 256, 225], "content": "Using (3.9) and (3.10), we see that", "index": 2}, {"type": "interline_equation", "bbox": [168, 237, 442, 301], "content": "", "index": 3}, {"type": "text", "bbox": [70, 308, 541, 362], "content": "We see that is and vanishes for and . Since is at worst, the right hand side of (3.11) vanishes. The symmetry therefore acts trivially on all normalizable ground states.", "index": 4}, {"type": "title", "bbox": [72, 392, 330, 408], "content": "4. Invariance Under the Symmetry", "index": 5}, {"type": "text", "bbox": [72, 419, 352, 434], "content": "4.1. Relating the Spin(5) currents to the supercharge", "index": 6}, {"type": "text", "bbox": [69, 444, 542, 478], "content": "We want to use essentially the same argument as in the case. For some choice of , we want to show that:", "index": 7}, {"type": "interline_equation", "bbox": [249, 493, 362, 523], "content": "", "index": 8}, {"type": "text", "bbox": [70, 532, 460, 548], "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "index": 9}, {"type": "interline_equation", "bbox": [228, 565, 384, 580], "content": "", "index": 10}, {"type": "text", "bbox": [69, 594, 542, 649], "content": "Again this choice anti-commutes with because anti-commutes with . The anti- commutator with results in a trace of three gamma matrices and so vanishes. It also anti-commutes with the -term in (2.4). To see this, we compute:", "index": 11}, {"type": "interline_equation", "bbox": [188, 661, 423, 690], "content": "", "index": 12}, {"type": "text", "bbox": [70, 700, 521, 716], "content": "However, this combination does not contain a singlet under so (4.3) vanishes.", "index": 13}]
|
[{"bbox": [70, 72, 542, 88], "content": "The new ground state looks -trivial. To show that it really is physically trivial, we need", "parent_index": 0, "line_index": 0}, {"bbox": [71, 93, 540, 107], "content": "to check that it has zero norm. Since is Hermitian and kills , the norm of", "parent_index": 0, "line_index": 1}, {"bbox": [71, 111, 541, 127], "content": "vanishes if we can integrate by parts. To integrate by parts, we argue as in Jost and Zuo", "parent_index": 0, "line_index": 2}, {"bbox": [71, 130, 540, 145], "content": "[18,6]: in terms of , we can cutoff of the integral using a smooth bump function", "parent_index": 0, "line_index": 3}, {"bbox": [71, 150, 473, 165], "content": "which vanishes for , satisfies and is one for ,", "parent_index": 0, "line_index": 4}, {"bbox": [72, 212, 254, 227], "content": "Using (3.9) and (3.10), we see that", "parent_index": 2, "line_index": 0}, {"bbox": [70, 311, 540, 329], "content": "We see that is and vanishes for and . Since is", "parent_index": 4, "line_index": 0}, {"bbox": [70, 331, 541, 347], "content": "at worst, the right hand side of (3.11) vanishes. The symmetry therefore acts", "parent_index": 4, "line_index": 1}, {"bbox": [71, 351, 294, 365], "content": "trivially on all normalizable ground states.", "parent_index": 4, "line_index": 2}, {"bbox": [71, 395, 328, 410], "content": "4. Invariance Under the Symmetry", "parent_index": 5, "line_index": 0}, {"bbox": [72, 421, 351, 435], "content": "4.1. Relating the Spin(5) currents to the supercharge", "parent_index": 6, "line_index": 0}, {"bbox": [94, 446, 541, 462], "content": "We want to use essentially the same argument as in the case. For some choice", "parent_index": 7, "line_index": 0}, {"bbox": [69, 464, 230, 483], "content": "of , we want to show that:", "parent_index": 7, "line_index": 1}, {"bbox": [70, 534, 459, 550], "content": "Let us start with the vector multiplet. We take a candidate gauge singlet,", "parent_index": 9, "line_index": 0}, {"bbox": [72, 597, 541, 613], "content": "Again this choice anti-commutes with because anti-commutes with . The anti-", "parent_index": 11, "line_index": 0}, {"bbox": [69, 614, 543, 636], "content": "commutator with results in a trace of three gamma matrices and", "parent_index": 11, "line_index": 1}, {"bbox": [71, 636, 522, 650], "content": "so vanishes. It also anti-commutes with the -term in (2.4). To see this, we compute:", "parent_index": 11, "line_index": 2}, {"bbox": [70, 701, 521, 719], "content": "However, this combination does not contain a singlet under so (4.3) vanishes.", "parent_index": 13, "line_index": 0}]
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[]
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[{"bbox": [221, 75, 230, 87], "content": "Q", "parent_index": 0, "subtype": "inline"}, {"bbox": [285, 95, 294, 106], "content": "Q", "parent_index": 0, "subtype": "inline"}, {"bbox": [423, 93, 442, 103], "content": "{\\tilde{s}}^{i}\\Psi", "parent_index": 0, "subtype": "inline"}, {"bbox": [520, 93, 540, 103], "content": "{\\tilde{s}}^{i}\\Psi", "parent_index": 0, "subtype": "inline"}, {"bbox": [172, 131, 214, 144], "content": "y=|y^{i}|", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 151, 101, 163], "content": "\\rho_{R}(y)", "parent_index": 0, "subtype": "inline"}, {"bbox": [204, 151, 242, 163], "content": "y>2R", "parent_index": 0, "subtype": "inline"}, {"bbox": [293, 151, 357, 163], "content": "|d\\rho_{R}|<4/R", "parent_index": 0, "subtype": "inline"}, {"bbox": [437, 151, 468, 162], "content": "y<R", "parent_index": 0, "subtype": "inline"}, {"bbox": [197, 179, 414, 200], "content": "<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname*{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.", "parent_index": 1, "subtype": "interline"}, {"bbox": [168, 237, 442, 301], "content": "\\begin{array}{r}{{<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>\\,=\\,\\operatorname*{lim}_{R\\rightarrow\\infty}\\,<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi>},}\\\\ {{=\\displaystyle\\operatorname*{lim}_{R\\rightarrow\\infty}\\sum_{a}\\,<\\left[Q_{a},\\rho_{R}(y)\\right]\\tilde{s}^{i}\\Psi,v_{a}^{i}\\Psi>.}}\\end{array}", "parent_index": 3, "subtype": "interline"}, {"bbox": [138, 313, 195, 326], "content": "[Q_{a},\\rho_{R}(y)]", "parent_index": 4, "subtype": "inline"}, {"bbox": [212, 313, 249, 326], "content": "O(1/y)", "parent_index": 4, "subtype": "inline"}, {"bbox": [343, 314, 377, 325], "content": "y<R", "parent_index": 4, "subtype": "inline"}, {"bbox": [405, 314, 444, 325], "content": "y>2R", "parent_index": 4, "subtype": "inline"}, {"bbox": [485, 314, 497, 325], "content": "v_{i}^{a}", "parent_index": 4, "subtype": "inline"}, {"bbox": [514, 313, 540, 326], "content": "O(y)", "parent_index": 4, "subtype": "inline"}, {"bbox": [364, 332, 405, 344], "content": "S U(2)_{R}", "parent_index": 4, "subtype": "inline"}, {"bbox": [221, 397, 262, 410], "content": "S p i n(5)", "parent_index": 5, "subtype": "inline"}, {"bbox": [384, 448, 425, 461], "content": "S U(2)_{R}", "parent_index": 7, "subtype": "inline"}, {"bbox": [85, 468, 103, 479], "content": "v_{a}^{\\mu\\nu}", "parent_index": 7, "subtype": "inline"}, {"bbox": [249, 493, 362, 523], "content": "T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.", "parent_index": 8, "subtype": "interline"}, {"bbox": [228, 565, 384, 580], "content": "(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.", "parent_index": 10, "subtype": "interline"}, {"bbox": [279, 598, 295, 611], "content": "Q^{h}", "parent_index": 11, "subtype": "inline"}, {"bbox": [345, 600, 353, 609], "content": "\\lambda", "parent_index": 11, "subtype": "inline"}, {"bbox": [468, 600, 477, 611], "content": "\\psi", "parent_index": 11, "subtype": "inline"}, {"bbox": [167, 616, 284, 631], "content": "{\\textstyle\\frac{1}{2}}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}", "parent_index": 11, "subtype": "inline"}, {"bbox": [303, 638, 313, 646], "content": "\\cal{D}", "parent_index": 11, "subtype": "inline"}, {"bbox": [188, 661, 423, 690], "content": "\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{\\mu\\nu}\\right\\}=D_{a b}\\left\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\right\\}_{a b}.", "parent_index": 12, "subtype": "interline"}, {"bbox": [387, 704, 427, 716], "content": "S p i n(5)", "parent_index": 13, "subtype": "inline"}]
|
[]
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|
We are left with the following anti-commutator which we need to compute quite carefully,
$$
\sum_{a}\left\{(\gamma^{\mu}p^{\mu}\lambda)_{a}\,,(v_{1})_{a}^{\mu\nu}\right\}=8\,(x^{\nu}p^{\mu}-x^{\mu}p^{\nu})+2i\lambda\gamma^{\mu\nu}\lambda.
$$
This computation is sensitive to the size of the $\gamma$ matrix. We obtain precisely the right ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from six dimensions. We would not obtain the right ratio had we considered a theory reduced from ten dimensions with a $S p i n(9)$ symmetry. Again, we can use (4.2) to generate the terms in the $S p i n(5)$ currents which act on vector multiplets.
For the hypermultiplet, we take the following choice:
$$
(v_{2})_{a}^{\mu\nu}=\left(\gamma^{\mu\nu}s^{i}\psi\right)_{a}q^{i}.
$$
Again $v_{2}$ anti-commutes with $Q^{v}$ because $\lambda$ anti-commutes with $\psi$ . In much the same way as before, we can argue that the anti-commutator of $v_{2}$ with the interaction term $I$ in (2.6) must vanish. We see that,
$$
\sum_{a}\left\{I_{a b}\psi_{b},(v_{2})_{a}^{\mu\nu}\right\}\sim q^{i}\mathrm{tr}\left(I\gamma^{\mu\nu}s^{i}\right),
$$
but $I\gamma^{\mu\nu}s^{i}$ again does not contain a singlet under $S p i n(5)$ so the trace vanishes.
The remaining anti-commutator involves the kinetic term in the hypermultiplet charge,
$$
\sum_{a}\left\{s_{a b}^{j}\psi_{b}\,p_{j},(v_{2})_{a}^{\mu\nu}\right\}=-i\psi\gamma^{\mu\nu}\psi.
$$
Again we conclude that for appropriately chosen constants $\alpha_{1}$ and $\alpha_{2}$ , the choice
$$
v_{a}^{\mu\nu}=\alpha_{1}(v_{1})_{a}^{\mu\nu}+\alpha_{2}(v_{2})_{a}^{\mu\nu}
$$
satisfies (4.1). A straightforward repeat of the argument given in section ${\it3.2}$ then implies that the $S p i n(5)$ symmetry acts trivially on all normalizable ground states.
# 4.2. Theories with sixteen supercharges
For theories obtained by reduction from ten dimensions, the previous argument does not apply directly to the $S p i n(9)$ symmetry for reasons mentioned earlier. These theories contain scalars $y^{i}$ where $i\,=\,1,...\,,9$ transforming in the adjoint representation of the gauge group. The superpartners to these scalars are real fermions $\eta_{\alpha}$ where $\alpha=1,\ldots,16$ also in the adjoint representation.
|
<html><body>
<p data-bbox="69 70 540 101">We are left with the following anti-commutator which we need to compute quite carefully, </p>
<div class="equation" data-bbox="165 103 445 133">$$
\sum_{a}\left\{(\gamma^{\mu}p^{\mu}\lambda)_{a}\,,(v_{1})_{a}^{\mu\nu}\right\}=8\,(x^{\nu}p^{\mu}-x^{\mu}p^{\nu})+2i\lambda\gamma^{\mu\nu}\lambda.
$$</div>
<p data-bbox="69 135 542 227">This computation is sensitive to the size of the $\gamma$ matrix. We obtain precisely the right ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from six dimensions. We would not obtain the right ratio had we considered a theory reduced from ten dimensions with a $S p i n(9)$ symmetry. Again, we can use (4.2) to generate the terms in the $S p i n(5)$ currents which act on vector multiplets. </p>
<p data-bbox="93 229 373 245">For the hypermultiplet, we take the following choice: </p>
<div class="equation" data-bbox="247 262 365 279">$$
(v_{2})_{a}^{\mu\nu}=\left(\gamma^{\mu\nu}s^{i}\psi\right)_{a}q^{i}.
$$</div>
<p data-bbox="69 292 541 345">Again $v_{2}$ anti-commutes with $Q^{v}$ because $\lambda$ anti-commutes with $\psi$ . In much the same way as before, we can argue that the anti-commutator of $v_{2}$ with the interaction term $I$ in (2.6) must vanish. We see that, </p>
<div class="equation" data-bbox="212 356 398 386">$$
\sum_{a}\left\{I_{a b}\psi_{b},(v_{2})_{a}^{\mu\nu}\right\}\sim q^{i}\mathrm{tr}\left(I\gamma^{\mu\nu}s^{i}\right),
$$</div>
<p data-bbox="70 396 492 411">but $I\gamma^{\mu\nu}s^{i}$ again does not contain a singlet under $S p i n(5)$ so the trace vanishes. </p>
<p data-bbox="70 415 540 447">The remaining anti-commutator involves the kinetic term in the hypermultiplet charge, </p>
<div class="equation" data-bbox="213 449 398 480">$$
\sum_{a}\left\{s_{a b}^{j}\psi_{b}\,p_{j},(v_{2})_{a}^{\mu\nu}\right\}=-i\psi\gamma^{\mu\nu}\psi.
$$</div>
<p data-bbox="70 484 498 501">Again we conclude that for appropriately chosen constants $\alpha_{1}$ and $\alpha_{2}$ , the choice </p>
<div class="equation" data-bbox="235 517 375 532">$$
v_{a}^{\mu\nu}=\alpha_{1}(v_{1})_{a}^{\mu\nu}+\alpha_{2}(v_{2})_{a}^{\mu\nu}
$$</div>
<p data-bbox="70 547 541 582">satisfies (4.1). A straightforward repeat of the argument given in section ${\it3.2}$ then implies that the $S p i n(5)$ symmetry acts trivially on all normalizable ground states. </p>
<h1 data-bbox="70 598 278 614">4.2. Theories with sixteen supercharges </h1>
<p data-bbox="70 624 542 716">For theories obtained by reduction from ten dimensions, the previous argument does not apply directly to the $S p i n(9)$ symmetry for reasons mentioned earlier. These theories contain scalars $y^{i}$ where $i\,=\,1,...\,,9$ transforming in the adjoint representation of the gauge group. The superpartners to these scalars are real fermions $\eta_{\alpha}$ where $\alpha=1,\ldots,16$ also in the adjoint representation. </p>
</body></html>
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0001189v2
| 8 | 612 | 792 | 1,275 | 1,650 |
[{"type": "text", "text": "We are left with the following anti-commutator which we need to compute quite carefully, ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{\\mu\\nu}\\right\\}=8\\,(x^{\\nu}p^{\\mu}-x^{\\mu}p^{\\nu})+2i\\lambda\\gamma^{\\mu\\nu}\\lambda.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "This computation is sensitive to the size of the $\\gamma$ matrix. We obtain precisely the right ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from six dimensions. We would not obtain the right ratio had we considered a theory reduced from ten dimensions with a $S p i n(9)$ symmetry. Again, we can use (4.2) to generate the terms in the $S p i n(5)$ currents which act on vector multiplets. ", "page_idx": 8}, {"type": "text", "text": "For the hypermultiplet, we take the following choice: ", "page_idx": 8}, {"type": "equation", "text": "$$\n(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Again $v_{2}$ anti-commutes with $Q^{v}$ because $\\lambda$ anti-commutes with $\\psi$ . In much the same way as before, we can argue that the anti-commutator of $v_{2}$ with the interaction term $I$ in (2.6) must vanish. We see that, ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "but $I\\gamma^{\\mu\\nu}s^{i}$ again does not contain a singlet under $S p i n(5)$ so the trace vanishes. ", "page_idx": 8}, {"type": "text", "text": "The remaining anti-commutator involves the kinetic term in the hypermultiplet charge, ", "page_idx": 8}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "Again we conclude that for appropriately chosen constants $\\alpha_{1}$ and $\\alpha_{2}$ , the choice ", "page_idx": 8}, {"type": "equation", "text": "$$\nv_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}\n$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "satisfies (4.1). A straightforward repeat of the argument given in section ${\\it3.2}$ then implies that the $S p i n(5)$ symmetry acts trivially on all normalizable ground states. ", "page_idx": 8}, {"type": "text", "text": "4.2. Theories with sixteen supercharges ", "text_level": 1, "page_idx": 8}, {"type": "text", "text": "For theories obtained by reduction from ten dimensions, the previous argument does not apply directly to the $S p i n(9)$ symmetry for reasons mentioned earlier. These theories contain scalars $y^{i}$ where $i\\,=\\,1,...\\,,9$ transforming in the adjoint representation of the gauge group. The superpartners to these scalars are real fermions $\\eta_{\\alpha}$ where $\\alpha=1,\\ldots,16$ also in the adjoint representation. ", "page_idx": 8}]
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{"preproc_blocks": [{"type": "text", "bbox": [69, 70, 540, 101], "lines": [{"bbox": [94, 73, 541, 88], "spans": [{"bbox": [94, 73, 541, 88], "score": 1.0, "content": "We are left with the following anti-commutator which we need to compute quite", "type": "text"}], "index": 0}, {"bbox": [71, 93, 118, 105], "spans": [{"bbox": [71, 93, 118, 105], "score": 1.0, "content": "carefully,", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "interline_equation", "bbox": [165, 103, 445, 133], "lines": [{"bbox": [165, 103, 445, 133], "spans": [{"bbox": [165, 103, 445, 133], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{\\mu\\nu}\\right\\}=8\\,(x^{\\nu}p^{\\mu}-x^{\\mu}p^{\\nu})+2i\\lambda\\gamma^{\\mu\\nu}\\lambda.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [69, 135, 542, 227], "lines": [{"bbox": [70, 137, 540, 153], "spans": [{"bbox": [70, 137, 326, 153], "score": 1.0, "content": "This computation is sensitive to the size of the ", "type": "text"}, {"bbox": [327, 143, 334, 151], "score": 0.9, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [334, 137, 540, 153], "score": 1.0, "content": " matrix. We obtain precisely the right", "type": "text"}], "index": 3}, {"bbox": [70, 156, 541, 171], "spans": [{"bbox": [70, 156, 541, 171], "score": 1.0, "content": "ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from", "type": "text"}], "index": 4}, {"bbox": [70, 176, 541, 190], "spans": [{"bbox": [70, 176, 541, 190], "score": 1.0, "content": "six dimensions. We would not obtain the right ratio had we considered a theory reduced", "type": "text"}], "index": 5}, {"bbox": [70, 195, 541, 209], "spans": [{"bbox": [70, 195, 221, 209], "score": 1.0, "content": "from ten dimensions with a ", "type": "text"}, {"bbox": [222, 196, 262, 208], "score": 0.92, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [263, 195, 541, 209], "score": 1.0, "content": " symmetry. Again, we can use (4.2) to generate the", "type": "text"}], "index": 6}, {"bbox": [71, 214, 392, 228], "spans": [{"bbox": [71, 214, 139, 228], "score": 1.0, "content": "terms in the ", "type": "text"}, {"bbox": [139, 215, 180, 227], "score": 0.93, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [180, 214, 392, 228], "score": 1.0, "content": " currents which act on vector multiplets.", "type": "text"}], "index": 7}], "index": 5}, {"type": "text", "bbox": [93, 229, 373, 245], "lines": [{"bbox": [95, 233, 371, 247], "spans": [{"bbox": [95, 233, 371, 247], "score": 1.0, "content": "For the hypermultiplet, we take the following choice:", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [247, 262, 365, 279], "lines": [{"bbox": [247, 262, 365, 279], "spans": [{"bbox": [247, 262, 365, 279], "score": 0.94, "content": "(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [69, 292, 541, 345], "lines": [{"bbox": [72, 295, 540, 311], "spans": [{"bbox": [72, 295, 105, 311], "score": 1.0, "content": "Again ", "type": "text"}, {"bbox": [106, 300, 117, 307], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [117, 295, 227, 311], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [227, 297, 242, 308], "score": 0.92, "content": "Q^{v}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [242, 295, 289, 311], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [289, 297, 297, 306], "score": 0.9, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [297, 295, 406, 311], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [407, 297, 415, 308], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [416, 295, 540, 311], "score": 1.0, "content": ". In much the same way", "type": "text"}], "index": 10}, {"bbox": [70, 314, 540, 328], "spans": [{"bbox": [70, 314, 343, 328], "score": 1.0, "content": "as before, we can argue that the anti-commutator of ", "type": "text"}, {"bbox": [343, 319, 354, 326], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [355, 314, 491, 328], "score": 1.0, "content": " with the interaction term ", "type": "text"}, {"bbox": [492, 316, 498, 325], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [499, 314, 540, 328], "score": 1.0, "content": " in (2.6)", "type": "text"}], "index": 11}, {"bbox": [71, 333, 209, 347], "spans": [{"bbox": [71, 333, 209, 347], "score": 1.0, "content": "must vanish. We see that,", "type": "text"}], "index": 12}], "index": 11}, {"type": "interline_equation", "bbox": [212, 356, 398, 386], "lines": [{"bbox": [212, 356, 398, 386], "spans": [{"bbox": [212, 356, 398, 386], "score": 0.92, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [70, 396, 492, 411], "lines": [{"bbox": [70, 397, 493, 415], "spans": [{"bbox": [70, 397, 93, 415], "score": 1.0, "content": "but ", "type": "text"}, {"bbox": [93, 399, 127, 412], "score": 0.94, "content": "I\\gamma^{\\mu\\nu}s^{i}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [128, 397, 335, 415], "score": 1.0, "content": " again does not contain a singlet under ", "type": "text"}, {"bbox": [336, 400, 376, 412], "score": 0.9, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [377, 397, 493, 415], "score": 1.0, "content": " so the trace vanishes.", "type": "text"}], "index": 14}], "index": 14}, {"type": "text", "bbox": [70, 415, 540, 447], "lines": [{"bbox": [93, 416, 541, 434], "spans": [{"bbox": [93, 416, 541, 434], "score": 1.0, "content": "The remaining anti-commutator involves the kinetic term in the hypermultiplet", "type": "text"}], "index": 15}, {"bbox": [70, 435, 110, 451], "spans": [{"bbox": [70, 435, 110, 451], "score": 1.0, "content": "charge,", "type": "text"}], "index": 16}], "index": 15.5}, {"type": "interline_equation", "bbox": [213, 449, 398, 480], "lines": [{"bbox": [213, 449, 398, 480], "spans": [{"bbox": [213, 449, 398, 480], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [70, 484, 498, 501], "lines": [{"bbox": [71, 486, 497, 504], "spans": [{"bbox": [71, 486, 382, 504], "score": 1.0, "content": "Again we conclude that for appropriately chosen constants ", "type": "text"}, {"bbox": [382, 492, 396, 500], "score": 0.84, "content": "\\alpha_{1}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [396, 486, 422, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [423, 492, 436, 500], "score": 0.87, "content": "\\alpha_{2}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [436, 486, 497, 504], "score": 1.0, "content": ", the choice", "type": "text"}], "index": 18}], "index": 18}, {"type": "interline_equation", "bbox": [235, 517, 375, 532], "lines": [{"bbox": [235, 517, 375, 532], "spans": [{"bbox": [235, 517, 375, 532], "score": 0.93, "content": "v_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [70, 547, 541, 582], "lines": [{"bbox": [70, 550, 541, 565], "spans": [{"bbox": [70, 550, 455, 565], "score": 1.0, "content": "satisfies (4.1). A straightforward repeat of the argument given in section ", "type": "text"}, {"bbox": [456, 550, 473, 561], "score": 0.26, "content": "{\\it3.2}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [473, 550, 541, 565], "score": 1.0, "content": " then implies", "type": "text"}], "index": 20}, {"bbox": [72, 570, 466, 584], "spans": [{"bbox": [72, 570, 117, 584], "score": 1.0, "content": "that the ", "type": "text"}, {"bbox": [118, 570, 158, 583], "score": 0.91, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [159, 570, 466, 584], "score": 1.0, "content": " symmetry acts trivially on all normalizable ground states.", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "title", "bbox": [70, 598, 278, 614], "lines": [{"bbox": [71, 601, 278, 615], "spans": [{"bbox": [71, 601, 278, 615], "score": 1.0, "content": "4.2. Theories with sixteen supercharges", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [70, 624, 542, 716], "lines": [{"bbox": [94, 627, 540, 641], "spans": [{"bbox": [94, 627, 540, 641], "score": 1.0, "content": "For theories obtained by reduction from ten dimensions, the previous argument does", "type": "text"}], "index": 23}, {"bbox": [71, 646, 541, 660], "spans": [{"bbox": [71, 646, 204, 660], "score": 1.0, "content": "not apply directly to the ", "type": "text"}, {"bbox": [204, 647, 245, 659], "score": 0.91, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 646, 541, 660], "score": 1.0, "content": " symmetry for reasons mentioned earlier. These theories", "type": "text"}], "index": 24}, {"bbox": [71, 664, 541, 680], "spans": [{"bbox": [71, 664, 155, 680], "score": 1.0, "content": "contain scalars ", "type": "text"}, {"bbox": [155, 665, 165, 678], "score": 0.91, "content": "y^{i}", "type": "inline_equation", "height": 13, "width": 10}, {"bbox": [165, 664, 207, 680], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [207, 667, 271, 678], "score": 0.93, "content": "i\\,=\\,1,...\\,,9", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [272, 664, 541, 680], "score": 1.0, "content": " transforming in the adjoint representation of the", "type": "text"}], "index": 25}, {"bbox": [70, 684, 540, 699], "spans": [{"bbox": [70, 684, 419, 699], "score": 1.0, "content": "gauge group. The superpartners to these scalars are real fermions ", "type": "text"}, {"bbox": [420, 689, 432, 696], "score": 0.91, "content": "\\eta_{\\alpha}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 684, 470, 699], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [471, 686, 540, 696], "score": 0.92, "content": "\\alpha=1,\\ldots,16", "type": "inline_equation", "height": 10, "width": 69}], "index": 26}, {"bbox": [71, 702, 248, 717], "spans": [{"bbox": [71, 702, 248, 717], "score": 1.0, "content": "also in the adjoint representation.", "type": "text"}], "index": 27}], "index": 25}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [165, 103, 445, 133], "lines": [{"bbox": [165, 103, 445, 133], "spans": [{"bbox": [165, 103, 445, 133], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{\\mu\\nu}\\right\\}=8\\,(x^{\\nu}p^{\\mu}-x^{\\mu}p^{\\nu})+2i\\lambda\\gamma^{\\mu\\nu}\\lambda.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "interline_equation", "bbox": [247, 262, 365, 279], "lines": [{"bbox": [247, 262, 365, 279], "spans": [{"bbox": [247, 262, 365, 279], "score": 0.94, "content": "(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [212, 356, 398, 386], "lines": [{"bbox": [212, 356, 398, 386], "spans": [{"bbox": [212, 356, 398, 386], "score": 0.92, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [213, 449, 398, 480], "lines": [{"bbox": [213, 449, 398, 480], "spans": [{"bbox": [213, 449, 398, 480], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "interline_equation", "bbox": [235, 517, 375, 532], "lines": [{"bbox": [235, 517, 375, 532], "spans": [{"bbox": [235, 517, 375, 532], "score": 0.93, "content": "v_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}", "type": "interline_equation"}], "index": 19}], "index": 19}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 70, 540, 101], "lines": [{"bbox": [94, 73, 541, 88], "spans": [{"bbox": [94, 73, 541, 88], "score": 1.0, "content": "We are left with the following anti-commutator which we need to compute quite", "type": "text"}], "index": 0}, {"bbox": [71, 93, 118, 105], "spans": [{"bbox": [71, 93, 118, 105], "score": 1.0, "content": "carefully,", "type": "text"}], "index": 1}], "index": 0.5, "bbox_fs": [71, 73, 541, 105]}, {"type": "interline_equation", "bbox": [165, 103, 445, 133], "lines": [{"bbox": [165, 103, 445, 133], "spans": [{"bbox": [165, 103, 445, 133], "score": 0.92, "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{\\mu\\nu}\\right\\}=8\\,(x^{\\nu}p^{\\mu}-x^{\\mu}p^{\\nu})+2i\\lambda\\gamma^{\\mu\\nu}\\lambda.", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [69, 135, 542, 227], "lines": [{"bbox": [70, 137, 540, 153], "spans": [{"bbox": [70, 137, 326, 153], "score": 1.0, "content": "This computation is sensitive to the size of the ", "type": "text"}, {"bbox": [327, 143, 334, 151], "score": 0.9, "content": "\\gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [334, 137, 540, 153], "score": 1.0, "content": " matrix. We obtain precisely the right", "type": "text"}], "index": 3}, {"bbox": [70, 156, 541, 171], "spans": [{"bbox": [70, 156, 541, 171], "score": 1.0, "content": "ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from", "type": "text"}], "index": 4}, {"bbox": [70, 176, 541, 190], "spans": [{"bbox": [70, 176, 541, 190], "score": 1.0, "content": "six dimensions. We would not obtain the right ratio had we considered a theory reduced", "type": "text"}], "index": 5}, {"bbox": [70, 195, 541, 209], "spans": [{"bbox": [70, 195, 221, 209], "score": 1.0, "content": "from ten dimensions with a ", "type": "text"}, {"bbox": [222, 196, 262, 208], "score": 0.92, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [263, 195, 541, 209], "score": 1.0, "content": " symmetry. Again, we can use (4.2) to generate the", "type": "text"}], "index": 6}, {"bbox": [71, 214, 392, 228], "spans": [{"bbox": [71, 214, 139, 228], "score": 1.0, "content": "terms in the ", "type": "text"}, {"bbox": [139, 215, 180, 227], "score": 0.93, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [180, 214, 392, 228], "score": 1.0, "content": " currents which act on vector multiplets.", "type": "text"}], "index": 7}], "index": 5, "bbox_fs": [70, 137, 541, 228]}, {"type": "text", "bbox": [93, 229, 373, 245], "lines": [{"bbox": [95, 233, 371, 247], "spans": [{"bbox": [95, 233, 371, 247], "score": 1.0, "content": "For the hypermultiplet, we take the following choice:", "type": "text"}], "index": 8}], "index": 8, "bbox_fs": [95, 233, 371, 247]}, {"type": "interline_equation", "bbox": [247, 262, 365, 279], "lines": [{"bbox": [247, 262, 365, 279], "spans": [{"bbox": [247, 262, 365, 279], "score": 0.94, "content": "(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [69, 292, 541, 345], "lines": [{"bbox": [72, 295, 540, 311], "spans": [{"bbox": [72, 295, 105, 311], "score": 1.0, "content": "Again ", "type": "text"}, {"bbox": [106, 300, 117, 307], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [117, 295, 227, 311], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [227, 297, 242, 308], "score": 0.92, "content": "Q^{v}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [242, 295, 289, 311], "score": 1.0, "content": " because ", "type": "text"}, {"bbox": [289, 297, 297, 306], "score": 0.9, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [297, 295, 406, 311], "score": 1.0, "content": " anti-commutes with ", "type": "text"}, {"bbox": [407, 297, 415, 308], "score": 0.91, "content": "\\psi", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [416, 295, 540, 311], "score": 1.0, "content": ". In much the same way", "type": "text"}], "index": 10}, {"bbox": [70, 314, 540, 328], "spans": [{"bbox": [70, 314, 343, 328], "score": 1.0, "content": "as before, we can argue that the anti-commutator of ", "type": "text"}, {"bbox": [343, 319, 354, 326], "score": 0.9, "content": "v_{2}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [355, 314, 491, 328], "score": 1.0, "content": " with the interaction term ", "type": "text"}, {"bbox": [492, 316, 498, 325], "score": 0.9, "content": "I", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [499, 314, 540, 328], "score": 1.0, "content": " in (2.6)", "type": "text"}], "index": 11}, {"bbox": [71, 333, 209, 347], "spans": [{"bbox": [71, 333, 209, 347], "score": 1.0, "content": "must vanish. We see that,", "type": "text"}], "index": 12}], "index": 11, "bbox_fs": [70, 295, 540, 347]}, {"type": "interline_equation", "bbox": [212, 356, 398, 386], "lines": [{"bbox": [212, 356, 398, 386], "spans": [{"bbox": [212, 356, 398, 386], "score": 0.92, "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),", "type": "interline_equation"}], "index": 13}], "index": 13}, {"type": "text", "bbox": [70, 396, 492, 411], "lines": [{"bbox": [70, 397, 493, 415], "spans": [{"bbox": [70, 397, 93, 415], "score": 1.0, "content": "but ", "type": "text"}, {"bbox": [93, 399, 127, 412], "score": 0.94, "content": "I\\gamma^{\\mu\\nu}s^{i}", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [128, 397, 335, 415], "score": 1.0, "content": " again does not contain a singlet under ", "type": "text"}, {"bbox": [336, 400, 376, 412], "score": 0.9, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 40}, {"bbox": [377, 397, 493, 415], "score": 1.0, "content": " so the trace vanishes.", "type": "text"}], "index": 14}], "index": 14, "bbox_fs": [70, 397, 493, 415]}, {"type": "text", "bbox": [70, 415, 540, 447], "lines": [{"bbox": [93, 416, 541, 434], "spans": [{"bbox": [93, 416, 541, 434], "score": 1.0, "content": "The remaining anti-commutator involves the kinetic term in the hypermultiplet", "type": "text"}], "index": 15}, {"bbox": [70, 435, 110, 451], "spans": [{"bbox": [70, 435, 110, 451], "score": 1.0, "content": "charge,", "type": "text"}], "index": 16}], "index": 15.5, "bbox_fs": [70, 416, 541, 451]}, {"type": "interline_equation", "bbox": [213, 449, 398, 480], "lines": [{"bbox": [213, 449, 398, 480], "spans": [{"bbox": [213, 449, 398, 480], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.", "type": "interline_equation"}], "index": 17}], "index": 17}, {"type": "text", "bbox": [70, 484, 498, 501], "lines": [{"bbox": [71, 486, 497, 504], "spans": [{"bbox": [71, 486, 382, 504], "score": 1.0, "content": "Again we conclude that for appropriately chosen constants ", "type": "text"}, {"bbox": [382, 492, 396, 500], "score": 0.84, "content": "\\alpha_{1}", "type": "inline_equation", "height": 8, "width": 14}, {"bbox": [396, 486, 422, 504], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [423, 492, 436, 500], "score": 0.87, "content": "\\alpha_{2}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [436, 486, 497, 504], "score": 1.0, "content": ", the choice", "type": "text"}], "index": 18}], "index": 18, "bbox_fs": [71, 486, 497, 504]}, {"type": "interline_equation", "bbox": [235, 517, 375, 532], "lines": [{"bbox": [235, 517, 375, 532], "spans": [{"bbox": [235, 517, 375, 532], "score": 0.93, "content": "v_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}", "type": "interline_equation"}], "index": 19}], "index": 19}, {"type": "text", "bbox": [70, 547, 541, 582], "lines": [{"bbox": [70, 550, 541, 565], "spans": [{"bbox": [70, 550, 455, 565], "score": 1.0, "content": "satisfies (4.1). A straightforward repeat of the argument given in section ", "type": "text"}, {"bbox": [456, 550, 473, 561], "score": 0.26, "content": "{\\it3.2}", "type": "inline_equation", "height": 11, "width": 17}, {"bbox": [473, 550, 541, 565], "score": 1.0, "content": " then implies", "type": "text"}], "index": 20}, {"bbox": [72, 570, 466, 584], "spans": [{"bbox": [72, 570, 117, 584], "score": 1.0, "content": "that the ", "type": "text"}, {"bbox": [118, 570, 158, 583], "score": 0.91, "content": "S p i n(5)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [159, 570, 466, 584], "score": 1.0, "content": " symmetry acts trivially on all normalizable ground states.", "type": "text"}], "index": 21}], "index": 20.5, "bbox_fs": [70, 550, 541, 584]}, {"type": "title", "bbox": [70, 598, 278, 614], "lines": [{"bbox": [71, 601, 278, 615], "spans": [{"bbox": [71, 601, 278, 615], "score": 1.0, "content": "4.2. Theories with sixteen supercharges", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [70, 624, 542, 716], "lines": [{"bbox": [94, 627, 540, 641], "spans": [{"bbox": [94, 627, 540, 641], "score": 1.0, "content": "For theories obtained by reduction from ten dimensions, the previous argument does", "type": "text"}], "index": 23}, {"bbox": [71, 646, 541, 660], "spans": [{"bbox": [71, 646, 204, 660], "score": 1.0, "content": "not apply directly to the ", "type": "text"}, {"bbox": [204, 647, 245, 659], "score": 0.91, "content": "S p i n(9)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [245, 646, 541, 660], "score": 1.0, "content": " symmetry for reasons mentioned earlier. These theories", "type": "text"}], "index": 24}, {"bbox": [71, 664, 541, 680], "spans": [{"bbox": [71, 664, 155, 680], "score": 1.0, "content": "contain scalars ", "type": "text"}, {"bbox": [155, 665, 165, 678], "score": 0.91, "content": "y^{i}", "type": "inline_equation", "height": 13, "width": 10}, {"bbox": [165, 664, 207, 680], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [207, 667, 271, 678], "score": 0.93, "content": "i\\,=\\,1,...\\,,9", "type": "inline_equation", "height": 11, "width": 64}, {"bbox": [272, 664, 541, 680], "score": 1.0, "content": " transforming in the adjoint representation of the", "type": "text"}], "index": 25}, {"bbox": [70, 684, 540, 699], "spans": [{"bbox": [70, 684, 419, 699], "score": 1.0, "content": "gauge group. The superpartners to these scalars are real fermions ", "type": "text"}, {"bbox": [420, 689, 432, 696], "score": 0.91, "content": "\\eta_{\\alpha}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [433, 684, 470, 699], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [471, 686, 540, 696], "score": 0.92, "content": "\\alpha=1,\\ldots,16", "type": "inline_equation", "height": 10, "width": 69}], "index": 26}, {"bbox": [71, 702, 248, 717], "spans": [{"bbox": [71, 702, 248, 717], "score": 1.0, "content": "also in the adjoint representation.", "type": "text"}], "index": 27}], "index": 25, "bbox_fs": [70, 627, 541, 717]}]}
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[{"type": "text", "bbox": [69, 70, 540, 101], "content": "We are left with the following anti-commutator which we need to compute quite carefully,", "index": 0}, {"type": "interline_equation", "bbox": [165, 103, 445, 133], "content": "", "index": 1}, {"type": "text", "bbox": [69, 135, 542, 227], "content": "This computation is sensitive to the size of the matrix. We obtain precisely the right ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from six dimensions. We would not obtain the right ratio had we considered a theory reduced from ten dimensions with a symmetry. Again, we can use (4.2) to generate the terms in the currents which act on vector multiplets.", "index": 2}, {"type": "text", "bbox": [93, 229, 373, 245], "content": "For the hypermultiplet, we take the following choice:", "index": 3}, {"type": "interline_equation", "bbox": [247, 262, 365, 279], "content": "", "index": 4}, {"type": "text", "bbox": [69, 292, 541, 345], "content": "Again anti-commutes with because anti-commutes with . In much the same way as before, we can argue that the anti-commutator of with the interaction term in (2.6) must vanish. We see that,", "index": 5}, {"type": "interline_equation", "bbox": [212, 356, 398, 386], "content": "", "index": 6}, {"type": "text", "bbox": [70, 396, 492, 411], "content": "but again does not contain a singlet under so the trace vanishes.", "index": 7}, {"type": "text", "bbox": [70, 415, 540, 447], "content": "The remaining anti-commutator involves the kinetic term in the hypermultiplet charge,", "index": 8}, {"type": "interline_equation", "bbox": [213, 449, 398, 480], "content": "", "index": 9}, {"type": "text", "bbox": [70, 484, 498, 501], "content": "Again we conclude that for appropriately chosen constants and , the choice", "index": 10}, {"type": "interline_equation", "bbox": [235, 517, 375, 532], "content": "", "index": 11}, {"type": "text", "bbox": [70, 547, 541, 582], "content": "satisfies (4.1). A straightforward repeat of the argument given in section then implies that the symmetry acts trivially on all normalizable ground states.", "index": 12}, {"type": "title", "bbox": [70, 598, 278, 614], "content": "4.2. Theories with sixteen supercharges", "index": 13}, {"type": "text", "bbox": [70, 624, 542, 716], "content": "For theories obtained by reduction from ten dimensions, the previous argument does not apply directly to the symmetry for reasons mentioned earlier. These theories contain scalars where transforming in the adjoint representation of the gauge group. The superpartners to these scalars are real fermions where also in the adjoint representation.", "index": 14}]
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[{"bbox": [94, 73, 541, 88], "content": "We are left with the following anti-commutator which we need to compute quite", "parent_index": 0, "line_index": 0}, {"bbox": [71, 93, 118, 105], "content": "carefully,", "parent_index": 0, "line_index": 1}, {"bbox": [70, 137, 540, 153], "content": "This computation is sensitive to the size of the matrix. We obtain precisely the right", "parent_index": 2, "line_index": 0}, {"bbox": [70, 156, 541, 171], "content": "ratio between the bosonic and fermion terms in (4.4) because the theory is reduced from", "parent_index": 2, "line_index": 1}, {"bbox": [70, 176, 541, 190], "content": "six dimensions. We would not obtain the right ratio had we considered a theory reduced", "parent_index": 2, "line_index": 2}, {"bbox": [70, 195, 541, 209], "content": "from ten dimensions with a symmetry. Again, we can use (4.2) to generate the", "parent_index": 2, "line_index": 3}, {"bbox": [71, 214, 392, 228], "content": "terms in the currents which act on vector multiplets.", "parent_index": 2, "line_index": 4}, {"bbox": [95, 233, 371, 247], "content": "For the hypermultiplet, we take the following choice:", "parent_index": 3, "line_index": 0}, {"bbox": [72, 295, 540, 311], "content": "Again anti-commutes with because anti-commutes with . In much the same way", "parent_index": 5, "line_index": 0}, {"bbox": [70, 314, 540, 328], "content": "as before, we can argue that the anti-commutator of with the interaction term in (2.6)", "parent_index": 5, "line_index": 1}, {"bbox": [71, 333, 209, 347], "content": "must vanish. We see that,", "parent_index": 5, "line_index": 2}, {"bbox": [70, 397, 493, 415], "content": "but again does not contain a singlet under so the trace vanishes.", "parent_index": 7, "line_index": 0}, {"bbox": [93, 416, 541, 434], "content": "The remaining anti-commutator involves the kinetic term in the hypermultiplet", "parent_index": 8, "line_index": 0}, {"bbox": [70, 435, 110, 451], "content": "charge,", "parent_index": 8, "line_index": 1}, {"bbox": [71, 486, 497, 504], "content": "Again we conclude that for appropriately chosen constants and , the choice", "parent_index": 10, "line_index": 0}, {"bbox": [70, 550, 541, 565], "content": "satisfies (4.1). A straightforward repeat of the argument given in section then implies", "parent_index": 12, "line_index": 0}, {"bbox": [72, 570, 466, 584], "content": "that the symmetry acts trivially on all normalizable ground states.", "parent_index": 12, "line_index": 1}, {"bbox": [71, 601, 278, 615], "content": "4.2. Theories with sixteen supercharges", "parent_index": 13, "line_index": 0}, {"bbox": [94, 627, 540, 641], "content": "For theories obtained by reduction from ten dimensions, the previous argument does", "parent_index": 14, "line_index": 0}, {"bbox": [71, 646, 541, 660], "content": "not apply directly to the symmetry for reasons mentioned earlier. These theories", "parent_index": 14, "line_index": 1}, {"bbox": [71, 664, 541, 680], "content": "contain scalars where transforming in the adjoint representation of the", "parent_index": 14, "line_index": 2}, {"bbox": [70, 684, 540, 699], "content": "gauge group. The superpartners to these scalars are real fermions where", "parent_index": 14, "line_index": 3}, {"bbox": [71, 702, 248, 717], "content": "also in the adjoint representation.", "parent_index": 14, "line_index": 4}]
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[]
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[{"bbox": [165, 103, 445, 133], "content": "\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{\\mu\\nu}\\right\\}=8\\,(x^{\\nu}p^{\\mu}-x^{\\mu}p^{\\nu})+2i\\lambda\\gamma^{\\mu\\nu}\\lambda.", "parent_index": 1, "subtype": "interline"}, {"bbox": [327, 143, 334, 151], "content": "\\gamma", "parent_index": 2, "subtype": "inline"}, {"bbox": [222, 196, 262, 208], "content": "S p i n(9)", "parent_index": 2, "subtype": "inline"}, {"bbox": [139, 215, 180, 227], "content": "S p i n(5)", "parent_index": 2, "subtype": "inline"}, {"bbox": [247, 262, 365, 279], "content": "(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.", "parent_index": 4, "subtype": "interline"}, {"bbox": [106, 300, 117, 307], "content": "v_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [227, 297, 242, 308], "content": "Q^{v}", "parent_index": 5, "subtype": "inline"}, {"bbox": [289, 297, 297, 306], "content": "\\lambda", "parent_index": 5, "subtype": "inline"}, {"bbox": [407, 297, 415, 308], "content": "\\psi", "parent_index": 5, "subtype": "inline"}, {"bbox": [343, 319, 354, 326], "content": "v_{2}", "parent_index": 5, "subtype": "inline"}, {"bbox": [492, 316, 498, 325], "content": "I", "parent_index": 5, "subtype": "inline"}, {"bbox": [212, 356, 398, 386], "content": "\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{\\mu\\nu}\\right\\}\\sim q^{i}\\mathrm{tr}\\left(I\\gamma^{\\mu\\nu}s^{i}\\right),", "parent_index": 6, "subtype": "interline"}, {"bbox": [93, 399, 127, 412], "content": "I\\gamma^{\\mu\\nu}s^{i}", "parent_index": 7, "subtype": "inline"}, {"bbox": [336, 400, 376, 412], "content": "S p i n(5)", "parent_index": 7, "subtype": "inline"}, {"bbox": [213, 449, 398, 480], "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{\\mu\\nu}\\right\\}=-i\\psi\\gamma^{\\mu\\nu}\\psi.", "parent_index": 9, "subtype": "interline"}, {"bbox": [382, 492, 396, 500], "content": "\\alpha_{1}", "parent_index": 10, "subtype": "inline"}, {"bbox": [423, 492, 436, 500], "content": "\\alpha_{2}", "parent_index": 10, "subtype": "inline"}, {"bbox": [235, 517, 375, 532], "content": "v_{a}^{\\mu\\nu}=\\alpha_{1}(v_{1})_{a}^{\\mu\\nu}+\\alpha_{2}(v_{2})_{a}^{\\mu\\nu}", "parent_index": 11, "subtype": "interline"}, {"bbox": [456, 550, 473, 561], "content": "{\\it3.2}", "parent_index": 12, "subtype": "inline"}, {"bbox": [118, 570, 158, 583], "content": "S p i n(5)", "parent_index": 12, "subtype": "inline"}, {"bbox": [204, 647, 245, 659], "content": "S p i n(9)", "parent_index": 14, "subtype": "inline"}, {"bbox": [155, 665, 165, 678], "content": "y^{i}", "parent_index": 14, "subtype": "inline"}, {"bbox": [207, 667, 271, 678], "content": "i\\,=\\,1,...\\,,9", "parent_index": 14, "subtype": "inline"}, {"bbox": [420, 689, 432, 696], "content": "\\eta_{\\alpha}", "parent_index": 14, "subtype": "inline"}, {"bbox": [471, 686, 540, 696], "content": "\\alpha=1,\\ldots,16", "parent_index": 14, "subtype": "inline"}]
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[]
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However, we can always view these theories as special cases of theories with eight supercharges. We choose any 5 of the 9 scalars $y^{i}$ to be the vector multiplet, and the remaining 4 scalars comprise an adjoint hypermultiplet. Of the original $S p i n(9)$ symmetry, only a $S p i n(5)\times S U(2)_{L}\times S U(2)_{R}$ subgroup is manifest. The scalars decompose in the following way,
$$
{\bf9}\quad\rightarrow\quad({\bf5},{\bf1},{\bf1})\oplus({\bf1},{\bf2},{\bf2}).
$$
The fermions decompose according to,
$$
{\bf16}\quad\rightarrow\quad({\bf4},{\bf1},{\bf2})\oplus({\bf4},{\bf2},{\bf1}).
$$
Our invariance argument implies that all normalizable ground states are invariant under the $S p i n(5)\times S U(2)_{R}$ symmetry. However, this is true regardless of how we embed $S p i n(5)\times$ $S U(2)_{R}$ into $S p i n(9)$ . This is only possible if the full $S p i n(9)$ symmetry acts trivially on all normalizable ground states.
# Acknowledgements
The work of S.S. is supported by the William Keck Foundation and by NSF grant PHY–9513835; that of M.S. by NSF grant DMS–9870161.
|
<html><body>
<p data-bbox="70 70 542 163">However, we can always view these theories as special cases of theories with eight supercharges. We choose any 5 of the 9 scalars $y^{i}$ to be the vector multiplet, and the remaining 4 scalars comprise an adjoint hypermultiplet. Of the original $S p i n(9)$ symmetry, only a $S p i n(5)\times S U(2)_{L}\times S U(2)_{R}$ subgroup is manifest. The scalars decompose in the following way, </p>
<div class="equation" data-bbox="233 168 378 183">$$
{\bf9}\quad\rightarrow\quad({\bf5},{\bf1},{\bf1})\oplus({\bf1},{\bf2},{\bf2}).
$$</div>
<p data-bbox="70 194 274 209">The fermions decompose according to, </p>
<div class="equation" data-bbox="230 229 381 244">$$
{\bf16}\quad\rightarrow\quad({\bf4},{\bf1},{\bf2})\oplus({\bf4},{\bf2},{\bf1}).
$$</div>
<p data-bbox="70 260 542 335">Our invariance argument implies that all normalizable ground states are invariant under the $S p i n(5)\times S U(2)_{R}$ symmetry. However, this is true regardless of how we embed $S p i n(5)\times$ $S U(2)_{R}$ into $S p i n(9)$ . This is only possible if the full $S p i n(9)$ symmetry acts trivially on all normalizable ground states. </p>
<h1 data-bbox="248 380 364 396">Acknowledgements </h1>
<p data-bbox="69 399 541 435">The work of S.S. is supported by the William Keck Foundation and by NSF grant PHY–9513835; that of M.S. by NSF grant DMS–9870161. </p>
</body></html>
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0001189v2
| 9 | 612 | 792 | 1,275 | 1,650 |
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The scalars decompose in the", "type": "text"}], "index": 3}, {"bbox": [70, 149, 146, 166], "spans": [{"bbox": [70, 149, 146, 166], "score": 1.0, "content": "following way,", "type": "text"}], "index": 4}], "index": 2, "bbox_fs": [70, 73, 541, 166]}, {"type": "interline_equation", "bbox": [233, 168, 378, 183], "lines": [{"bbox": [233, 168, 378, 183], "spans": [{"bbox": [233, 168, 378, 183], "score": 0.89, "content": "{\\bf9}\\quad\\rightarrow\\quad({\\bf5},{\\bf1},{\\bf1})\\oplus({\\bf1},{\\bf2},{\\bf2}).", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [70, 194, 274, 209], "lines": [{"bbox": [71, 196, 272, 211], "spans": [{"bbox": [71, 196, 272, 211], "score": 1.0, "content": "The fermions decompose according to,", "type": "text"}], "index": 6}], "index": 6, "bbox_fs": [71, 196, 272, 211]}, {"type": "interline_equation", "bbox": [230, 229, 381, 244], "lines": [{"bbox": [230, 229, 381, 244], "spans": [{"bbox": [230, 229, 381, 244], "score": 0.89, "content": "{\\bf16}\\quad\\rightarrow\\quad({\\bf4},{\\bf1},{\\bf2})\\oplus({\\bf4},{\\bf2},{\\bf1}).", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 260, 542, 335], "lines": [{"bbox": [72, 264, 541, 278], "spans": [{"bbox": [72, 264, 541, 278], "score": 1.0, "content": "Our invariance argument implies that all normalizable ground states are invariant under the", "type": "text"}], "index": 8}, {"bbox": [71, 283, 541, 298], "spans": [{"bbox": [71, 284, 166, 297], "score": 0.93, "content": "S p i n(5)\\times S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [166, 283, 487, 298], "score": 1.0, "content": " symmetry. However, this is true regardless of how we embed ", "type": "text"}, {"bbox": [487, 284, 541, 297], "score": 0.93, "content": "S p i n(5)\\times", "type": "inline_equation", "height": 13, "width": 54}], "index": 9}, {"bbox": [71, 301, 541, 317], "spans": [{"bbox": [71, 303, 112, 316], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [112, 301, 140, 317], "score": 1.0, "content": " into ", "type": "text"}, {"bbox": [141, 303, 181, 316], "score": 0.92, "content": "S p i n(9)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [182, 301, 355, 317], "score": 1.0, "content": ". This is only possible if the full ", "type": "text"}, {"bbox": [355, 303, 396, 316], "score": 0.9, "content": "S p i n(9)", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [397, 301, 541, 317], "score": 1.0, "content": " symmetry acts trivially on", "type": "text"}], "index": 10}, {"bbox": [71, 322, 232, 336], "spans": [{"bbox": [71, 322, 232, 336], "score": 1.0, "content": "all normalizable ground states.", "type": "text"}], "index": 11}], "index": 9.5, "bbox_fs": [71, 264, 541, 336]}, {"type": "title", "bbox": [248, 380, 364, 396], "lines": [{"bbox": [249, 384, 363, 397], "spans": [{"bbox": [249, 384, 363, 397], "score": 1.0, "content": "Acknowledgements", "type": "text"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 399, 541, 435], "lines": [{"bbox": [94, 401, 540, 419], "spans": [{"bbox": [94, 401, 540, 419], "score": 1.0, "content": "The work of S.S. is supported by the William Keck Foundation and by NSF grant", "type": "text"}], "index": 13}, {"bbox": [70, 421, 375, 438], "spans": [{"bbox": [70, 421, 375, 438], "score": 1.0, "content": "PHY–9513835; that of M.S. by NSF grant DMS–9870161.", "type": "text"}], "index": 14}], "index": 13.5, "bbox_fs": [70, 401, 540, 438]}]}
|
[{"type": "text", "bbox": [70, 70, 542, 163], "content": "However, we can always view these theories as special cases of theories with eight supercharges. We choose any 5 of the 9 scalars to be the vector multiplet, and the remaining 4 scalars comprise an adjoint hypermultiplet. Of the original symmetry, only a subgroup is manifest. The scalars decompose in the following way,", "index": 0}, {"type": "interline_equation", "bbox": [233, 168, 378, 183], "content": "", "index": 1}, {"type": "text", "bbox": [70, 194, 274, 209], "content": "The fermions decompose according to,", "index": 2}, {"type": "interline_equation", "bbox": [230, 229, 381, 244], "content": "", "index": 3}, {"type": "text", "bbox": [70, 260, 542, 335], "content": "Our invariance argument implies that all normalizable ground states are invariant under the symmetry. However, this is true regardless of how we embed into . This is only possible if the full symmetry acts trivially on all normalizable ground states.", "index": 4}, {"type": "title", "bbox": [248, 380, 364, 396], "content": "Acknowledgements", "index": 5}, {"type": "text", "bbox": [69, 399, 541, 435], "content": "The work of S.S. is supported by the William Keck Foundation and by NSF grant PHY–9513835; that of M.S. by NSF grant DMS–9870161.", "index": 6}]
|
[{"bbox": [95, 73, 540, 88], "content": "However, we can always view these theories as special cases of theories with eight", "parent_index": 0, "line_index": 0}, {"bbox": [70, 92, 541, 108], "content": "supercharges. We choose any 5 of the 9 scalars to be the vector multiplet, and the", "parent_index": 0, "line_index": 1}, {"bbox": [71, 111, 541, 128], "content": "remaining 4 scalars comprise an adjoint hypermultiplet. Of the original symmetry,", "parent_index": 0, "line_index": 2}, {"bbox": [71, 131, 541, 146], "content": "only a subgroup is manifest. The scalars decompose in the", "parent_index": 0, "line_index": 3}, {"bbox": [70, 149, 146, 166], "content": "following way,", "parent_index": 0, "line_index": 4}, {"bbox": [71, 196, 272, 211], "content": "The fermions decompose according to,", "parent_index": 2, "line_index": 0}, {"bbox": [72, 264, 541, 278], "content": "Our invariance argument implies that all normalizable ground states are invariant under the", "parent_index": 4, "line_index": 0}, {"bbox": [71, 283, 541, 298], "content": "symmetry. However, this is true regardless of how we embed", "parent_index": 4, "line_index": 1}, {"bbox": [71, 301, 541, 317], "content": "into . This is only possible if the full symmetry acts trivially on", "parent_index": 4, "line_index": 2}, {"bbox": [71, 322, 232, 336], "content": "all normalizable ground states.", "parent_index": 4, "line_index": 3}, {"bbox": [249, 384, 363, 397], "content": "Acknowledgements", "parent_index": 5, "line_index": 0}, {"bbox": [94, 401, 540, 419], "content": "The work of S.S. is supported by the William Keck Foundation and by NSF grant", "parent_index": 6, "line_index": 0}, {"bbox": [70, 421, 375, 438], "content": "PHY–9513835; that of M.S. by NSF grant DMS–9870161.", "parent_index": 6, "line_index": 1}]
|
[]
|
[{"bbox": [334, 93, 344, 106], "content": "y^{i}", "parent_index": 0, "subtype": "inline"}, {"bbox": [442, 113, 482, 126], "content": "S p i n(9)", "parent_index": 0, "subtype": "inline"}, {"bbox": [108, 132, 260, 145], "content": "S p i n(5)\\times S U(2)_{L}\\times S U(2)_{R}", "parent_index": 0, "subtype": "inline"}, {"bbox": [233, 168, 378, 183], "content": "{\\bf9}\\quad\\rightarrow\\quad({\\bf5},{\\bf1},{\\bf1})\\oplus({\\bf1},{\\bf2},{\\bf2}).", "parent_index": 1, "subtype": "interline"}, {"bbox": [230, 229, 381, 244], "content": "{\\bf16}\\quad\\rightarrow\\quad({\\bf4},{\\bf1},{\\bf2})\\oplus({\\bf4},{\\bf2},{\\bf1}).", "parent_index": 3, "subtype": "interline"}, {"bbox": [71, 284, 166, 297], "content": "S p i n(5)\\times S U(2)_{R}", "parent_index": 4, "subtype": "inline"}, {"bbox": [487, 284, 541, 297], "content": "S p i n(5)\\times", "parent_index": 4, "subtype": "inline"}, {"bbox": [71, 303, 112, 316], "content": "S U(2)_{R}", "parent_index": 4, "subtype": "inline"}, {"bbox": [141, 303, 181, 316], "content": "S p i n(9)", "parent_index": 4, "subtype": "inline"}, {"bbox": [355, 303, 396, 316], "content": "S p i n(9)", "parent_index": 4, "subtype": "inline"}]
|
[]
|
|
# Appendix A. Quaternions and Symplectic Groups
We will summarize some useful relations between quaternions and symplectic groups. Let us label a basis for our quaternions by $\{\mathbf{1},I,J,K\}$ where,
$$
I^{2}=J^{2}=K^{2}=-{\bf1},\qquad I J K=-{\bf1}.
$$
A quaternion $q$ can then be expanded in components
$$
q=q^{1}+I q^{2}+J q^{3}+K q^{4}.
$$
The conjugate quaternion $q$ has an expansion
$$
q=q^{1}-I q^{2}-J q^{3}-K q^{4}.
$$
The symmetry group $S p(1)_{R}\sim S U(2)_{R}$ is the group of unit quaternions. Let $\Lambda$ be a field transforming in the 2 of $S p(1)_{R}$ . If we view $S p(1)_{R}$ acting on $\Lambda$ as right multiplication by a unit quaternion $g$ then,
$$
\Lambda\to\Lambda g.
$$
In this formalism, $\Lambda$ is valued in the quaternions. Equivalently, we can expand $\Lambda$ in components and express the action of $g$ in the following way,
$$
\Lambda_{a}\rightarrow g_{a b}\Lambda_{b},
$$
where $g_{a b}$ implements right multiplication by the unit quaternion $g$ . For example, right multiplication by $I$ on $q$ gives
$$
\begin{array}{l}{{q\to q I}}\\ {{\to q^{1}I-q^{2}-q^{3}K+q^{4}J,}}\end{array}
$$
which can be realized by the matrix
$$
I^{R}=\left(\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\ {{1}}&{{0}}&{{0}}&{{0}}\\ {{0}}&{{0}}&{{0}}&{{1}}\\ {{0}}&{{0}}&{{-1}}&{{0}}\end{array}\right)
$$
acting on $q$ in the usual way $q_{a}\rightarrow I_{a b}^{R}\,q_{b}$ . The matrices $J^{R}$ and $K^{R}$ realize right multiplication by $J,K$ while ${\bf1}^{R}$ is the identity matrix:
$$
\begin{array}{c c}{{J^{R}=\left(\!\!\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\ {{0}}&{{0}}&{{0}}&{{-1}}\\ {{1}}&{{0}}&{{0}}&{{0}}\\ {{0}}&{{1}}&{{0}}&{{0}}\end{array}\!\!\right),~~~}}&{{K^{R}=\left(\!\!\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\ {{0}}&{{0}}&{{1}}&{{0}}\\ {{0}}&{{-1}}&{{0}}&{{0}}\\ {{1}}&{{0}}&{{0}}&{{0}}\end{array}\!\!\right).}}\end{array}
$$
|
<html><body>
<h1 data-bbox="71 70 374 86">Appendix A. Quaternions and Symplectic Groups </h1>
<p data-bbox="69 95 540 130">We will summarize some useful relations between quaternions and symplectic groups. Let us label a basis for our quaternions by $\{\mathbf{1},I,J,K\}$ where, </p>
<div class="equation" data-bbox="207 144 403 158">$$
I^{2}=J^{2}=K^{2}=-{\bf1},\qquad I J K=-{\bf1}.
$$</div>
<p data-bbox="71 171 351 186">A quaternion $q$ can then be expanded in components </p>
<div class="equation" data-bbox="236 200 375 214">$$
q=q^{1}+I q^{2}+J q^{3}+K q^{4}.
$$</div>
<p data-bbox="70 227 312 242">The conjugate quaternion $q$ has an expansion </p>
<div class="equation" data-bbox="235 257 375 271">$$
q=q^{1}-I q^{2}-J q^{3}-K q^{4}.
$$</div>
<p data-bbox="70 282 542 336">The symmetry group $S p(1)_{R}\sim S U(2)_{R}$ is the group of unit quaternions. Let $\Lambda$ be a field transforming in the 2 of $S p(1)_{R}$ . If we view $S p(1)_{R}$ acting on $\Lambda$ as right multiplication by a unit quaternion $g$ then, </p>
<div class="equation" data-bbox="284 342 327 355">$$
\Lambda\to\Lambda g.
$$</div>
<p data-bbox="70 364 541 398">In this formalism, $\Lambda$ is valued in the quaternions. Equivalently, we can expand $\Lambda$ in components and express the action of $g$ in the following way, </p>
<div class="equation" data-bbox="274 414 336 426">$$
\Lambda_{a}\rightarrow g_{a b}\Lambda_{b},
$$</div>
<p data-bbox="70 438 540 473">where $g_{a b}$ implements right multiplication by the unit quaternion $g$ . For example, right multiplication by $I$ on $q$ gives </p>
<div class="equation" data-bbox="236 483 375 522">$$
\begin{array}{l}{{q\to q I}}\\ {{\to q^{1}I-q^{2}-q^{3}K+q^{4}J,}}\end{array}
$$</div>
<p data-bbox="70 530 262 544">which can be realized by the matrix </p>
<div class="equation" data-bbox="238 554 372 610">$$
I^{R}=\left(\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\ {{1}}&{{0}}&{{0}}&{{0}}\\ {{0}}&{{0}}&{{0}}&{{1}}\\ {{0}}&{{0}}&{{-1}}&{{0}}\end{array}\right)
$$</div>
<p data-bbox="69 617 540 652">acting on $q$ in the usual way $q_{a}\rightarrow I_{a b}^{R}\,q_{b}$ . The matrices $J^{R}$ and $K^{R}$ realize right multiplication by $J,K$ while ${\bf1}^{R}$ is the identity matrix: </p>
<div class="equation" data-bbox="150 661 462 717">$$
\begin{array}{c c}{{J^{R}=\left(\!\!\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\ {{0}}&{{0}}&{{0}}&{{-1}}\\ {{1}}&{{0}}&{{0}}&{{0}}\\ {{0}}&{{1}}&{{0}}&{{0}}\end{array}\!\!\right),~~~}}&{{K^{R}=\left(\!\!\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\ {{0}}&{{0}}&{{1}}&{{0}}\\ {{0}}&{{-1}}&{{0}}&{{0}}\\ {{1}}&{{0}}&{{0}}&{{0}}\end{array}\!\!\right).}}\end{array}
$$</div>
</body></html>
|
0001189v2
| 10 | 612 | 792 | 1,275 | 1,650 |
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{"preproc_blocks": [{"type": "title", "bbox": [71, 70, 374, 86], "lines": [{"bbox": [72, 74, 372, 87], "spans": [{"bbox": [72, 74, 372, 87], "score": 1.0, "content": "Appendix A. Quaternions and Symplectic Groups", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [69, 95, 540, 130], "lines": [{"bbox": [95, 98, 539, 114], "spans": [{"bbox": [95, 98, 539, 114], "score": 1.0, "content": "We will summarize some useful relations between quaternions and symplectic groups.", "type": "text"}], "index": 1}, {"bbox": [69, 116, 394, 133], "spans": [{"bbox": [69, 116, 297, 133], "score": 1.0, "content": "Let us label a basis for our quaternions by ", "type": "text"}, {"bbox": [297, 118, 356, 131], "score": 0.95, "content": "\\{\\mathbf{1},I,J,K\\}", "type": "inline_equation", "height": 13, "width": 59}, {"bbox": [357, 116, 394, 133], "score": 1.0, "content": " where,", "type": "text"}], "index": 2}], "index": 1.5}, {"type": "interline_equation", "bbox": [207, 144, 403, 158], "lines": [{"bbox": [207, 144, 403, 158], "spans": [{"bbox": [207, 144, 403, 158], "score": 0.87, "content": "I^{2}=J^{2}=K^{2}=-{\\bf1},\\qquad I J K=-{\\bf1}.", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [71, 171, 351, 186], "lines": [{"bbox": [72, 174, 350, 188], "spans": [{"bbox": [72, 174, 144, 188], "score": 1.0, "content": "A quaternion ", "type": "text"}, {"bbox": [144, 178, 150, 186], "score": 0.88, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [151, 174, 350, 188], "score": 1.0, "content": " can then be expanded in components", "type": "text"}], "index": 4}], "index": 4}, {"type": "interline_equation", "bbox": [236, 200, 375, 214], "lines": [{"bbox": [236, 200, 375, 214], "spans": [{"bbox": [236, 200, 375, 214], "score": 0.9, "content": "q=q^{1}+I q^{2}+J q^{3}+K q^{4}.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [70, 227, 312, 242], "lines": [{"bbox": [72, 230, 310, 244], "spans": [{"bbox": [72, 230, 210, 244], "score": 1.0, "content": "The conjugate quaternion ", "type": "text"}, {"bbox": [210, 233, 216, 243], "score": 0.91, "content": "q", "type": "inline_equation", "height": 10, "width": 6}, {"bbox": [217, 230, 310, 244], "score": 1.0, "content": " has an expansion", "type": "text"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [235, 257, 375, 271], "lines": [{"bbox": [235, 257, 375, 271], "spans": [{"bbox": [235, 257, 375, 271], "score": 0.91, "content": "q=q^{1}-I q^{2}-J q^{3}-K q^{4}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [70, 282, 542, 336], "lines": [{"bbox": [70, 285, 542, 303], "spans": [{"bbox": [70, 285, 185, 303], "score": 1.0, "content": "The symmetry group ", "type": "text"}, {"bbox": [185, 287, 280, 300], "score": 0.93, "content": "S p(1)_{R}\\sim S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [280, 285, 479, 303], "score": 1.0, "content": " is the group of unit quaternions. Let ", "type": "text"}, {"bbox": [479, 288, 488, 297], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 285, 542, 303], "score": 1.0, "content": " be a field", "type": "text"}], "index": 8}, {"bbox": [70, 304, 541, 321], "spans": [{"bbox": [70, 304, 200, 321], "score": 1.0, "content": "transforming in the 2 of ", "type": "text"}, {"bbox": [200, 306, 237, 318], "score": 0.95, "content": "S p(1)_{R}", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [238, 304, 302, 321], "score": 1.0, "content": ". If we view ", "type": "text"}, {"bbox": [302, 305, 339, 318], "score": 0.93, "content": "S p(1)_{R}", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [340, 304, 395, 321], "score": 1.0, "content": " acting on ", "type": "text"}, {"bbox": [395, 307, 404, 315], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [404, 304, 541, 321], "score": 1.0, "content": " as right multiplication by", "type": "text"}], "index": 9}, {"bbox": [70, 323, 204, 338], "spans": [{"bbox": [70, 323, 166, 338], "score": 1.0, "content": "a unit quaternion ", "type": "text"}, {"bbox": [167, 328, 173, 336], "score": 0.91, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [173, 323, 204, 338], "score": 1.0, "content": " then,", "type": "text"}], "index": 10}], "index": 9}, {"type": "interline_equation", "bbox": [284, 342, 327, 355], "lines": [{"bbox": [284, 342, 327, 355], "spans": [{"bbox": [284, 342, 327, 355], "score": 0.89, "content": "\\Lambda\\to\\Lambda g.", "type": "interline_equation"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [70, 364, 541, 398], "lines": [{"bbox": [69, 366, 542, 383], "spans": [{"bbox": [69, 366, 174, 383], "score": 1.0, "content": "In this formalism, ", "type": "text"}, {"bbox": [174, 369, 183, 378], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [183, 366, 515, 383], "score": 1.0, "content": " is valued in the quaternions. 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The matrices ", "type": "text"}, {"bbox": [362, 621, 377, 632], "score": 0.92, "content": "J^{R}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [378, 619, 405, 637], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [405, 621, 424, 632], "score": 0.95, "content": "K^{R}", "type": "inline_equation", "height": 11, "width": 19}, {"bbox": [424, 619, 539, 637], "score": 1.0, "content": " realize right multipli-", "type": "text"}], "index": 20}, {"bbox": [72, 639, 317, 654], "spans": [{"bbox": [72, 639, 123, 654], "score": 1.0, "content": "cation by ", "type": "text"}, {"bbox": [124, 641, 147, 652], "score": 0.93, "content": "J,K", "type": "inline_equation", "height": 11, "width": 23}, {"bbox": [147, 639, 181, 654], "score": 1.0, "content": " while ", "type": "text"}, {"bbox": [182, 639, 196, 650], "score": 0.92, "content": "{\\bf1}^{R}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [197, 639, 317, 654], "score": 1.0, "content": " is the identity matrix:", "type": "text"}], "index": 21}], "index": 20.5, "bbox_fs": [70, 619, 539, 654]}, {"type": "interline_equation", "bbox": [150, 661, 462, 717], "lines": [{"bbox": [150, 661, 462, 717], "spans": [{"bbox": [150, 661, 462, 717], "score": 0.95, "content": "\\begin{array}{c c}{{J^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{1}}&{{0}}&{{0}}\\end{array}\\!\\!\\right),~~~}}&{{K^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{0}}&{{0}}&{{1}}&{{0}}\\\\ {{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\end{array}\\!\\!\\right).}}\\end{array}", "type": "interline_equation"}], "index": 22}], "index": 22}]}
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[{"type": "title", "bbox": [71, 70, 374, 86], "content": "Appendix A. Quaternions and Symplectic Groups", "index": 0}, {"type": "text", "bbox": [69, 95, 540, 130], "content": "We will summarize some useful relations between quaternions and symplectic groups. Let us label a basis for our quaternions by where,", "index": 1}, {"type": "interline_equation", "bbox": [207, 144, 403, 158], "content": "", "index": 2}, {"type": "text", "bbox": [71, 171, 351, 186], "content": "A quaternion can then be expanded in components", "index": 3}, {"type": "interline_equation", "bbox": [236, 200, 375, 214], "content": "", "index": 4}, {"type": "text", "bbox": [70, 227, 312, 242], "content": "The conjugate quaternion has an expansion", "index": 5}, {"type": "interline_equation", "bbox": [235, 257, 375, 271], "content": "", "index": 6}, {"type": "text", "bbox": [70, 282, 542, 336], "content": "The symmetry group is the group of unit quaternions. Let be a field transforming in the 2 of . If we view acting on as right multiplication by a unit quaternion then,", "index": 7}, {"type": "interline_equation", "bbox": [284, 342, 327, 355], "content": "", "index": 8}, {"type": "text", "bbox": [70, 364, 541, 398], "content": "In this formalism, is valued in the quaternions. Equivalently, we can expand in components and express the action of in the following way,", "index": 9}, {"type": "interline_equation", "bbox": [274, 414, 336, 426], "content": "", "index": 10}, {"type": "text", "bbox": [70, 438, 540, 473], "content": "where implements right multiplication by the unit quaternion . For example, right multiplication by on gives", "index": 11}, {"type": "interline_equation", "bbox": [236, 483, 375, 522], "content": "", "index": 12}, {"type": "text", "bbox": [70, 530, 262, 544], "content": "which can be realized by the matrix", "index": 13}, {"type": "interline_equation", "bbox": [238, 554, 372, 610], "content": "", "index": 14}, {"type": "text", "bbox": [69, 617, 540, 652], "content": "acting on in the usual way . The matrices and realize right multipli- cation by while is the identity matrix:", "index": 15}, {"type": "interline_equation", "bbox": [150, 661, 462, 717], "content": "", "index": 16}]
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[{"bbox": [72, 74, 372, 87], "content": "Appendix A. Quaternions and Symplectic Groups", "parent_index": 0, "line_index": 0}, {"bbox": [95, 98, 539, 114], "content": "We will summarize some useful relations between quaternions and symplectic groups.", "parent_index": 1, "line_index": 0}, {"bbox": [69, 116, 394, 133], "content": "Let us label a basis for our quaternions by where,", "parent_index": 1, "line_index": 1}, {"bbox": [72, 174, 350, 188], "content": "A quaternion can then be expanded in components", "parent_index": 3, "line_index": 0}, {"bbox": [72, 230, 310, 244], "content": "The conjugate quaternion has an expansion", "parent_index": 5, "line_index": 0}, {"bbox": [70, 285, 542, 303], "content": "The symmetry group is the group of unit quaternions. Let be a field", "parent_index": 7, "line_index": 0}, {"bbox": [70, 304, 541, 321], "content": "transforming in the 2 of . If we view acting on as right multiplication by", "parent_index": 7, "line_index": 1}, {"bbox": [70, 323, 204, 338], "content": "a unit quaternion then,", "parent_index": 7, "line_index": 2}, {"bbox": [69, 366, 542, 383], "content": "In this formalism, is valued in the quaternions. Equivalently, we can expand in", "parent_index": 9, "line_index": 0}, {"bbox": [70, 385, 391, 402], "content": "components and express the action of in the following way,", "parent_index": 9, "line_index": 1}, {"bbox": [70, 441, 540, 457], "content": "where implements right multiplication by the unit quaternion . For example, right", "parent_index": 11, "line_index": 0}, {"bbox": [71, 460, 228, 476], "content": "multiplication by on gives", "parent_index": 11, "line_index": 1}, {"bbox": [71, 531, 261, 546], "content": "which can be realized by the matrix", "parent_index": 13, "line_index": 0}, {"bbox": [70, 619, 539, 637], "content": "acting on in the usual way . The matrices and realize right multipli-", "parent_index": 15, "line_index": 0}, {"bbox": [72, 639, 317, 654], "content": "cation by while is the identity matrix:", "parent_index": 15, "line_index": 1}]
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[]
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[{"bbox": [297, 118, 356, 131], "content": "\\{\\mathbf{1},I,J,K\\}", "parent_index": 1, "subtype": "inline"}, {"bbox": [207, 144, 403, 158], "content": "I^{2}=J^{2}=K^{2}=-{\\bf1},\\qquad I J K=-{\\bf1}.", "parent_index": 2, "subtype": "interline"}, {"bbox": [144, 178, 150, 186], "content": "q", "parent_index": 3, "subtype": "inline"}, {"bbox": [236, 200, 375, 214], "content": "q=q^{1}+I q^{2}+J q^{3}+K q^{4}.", "parent_index": 4, "subtype": "interline"}, {"bbox": [210, 233, 216, 243], "content": "q", "parent_index": 5, "subtype": "inline"}, {"bbox": [235, 257, 375, 271], "content": "q=q^{1}-I q^{2}-J q^{3}-K q^{4}.", "parent_index": 6, "subtype": "interline"}, {"bbox": [185, 287, 280, 300], "content": "S p(1)_{R}\\sim S U(2)_{R}", "parent_index": 7, "subtype": "inline"}, {"bbox": [479, 288, 488, 297], "content": "\\Lambda", "parent_index": 7, "subtype": "inline"}, {"bbox": [200, 306, 237, 318], "content": "S p(1)_{R}", "parent_index": 7, "subtype": "inline"}, {"bbox": [302, 305, 339, 318], "content": "S p(1)_{R}", "parent_index": 7, "subtype": "inline"}, {"bbox": [395, 307, 404, 315], "content": "\\Lambda", "parent_index": 7, "subtype": "inline"}, {"bbox": [167, 328, 173, 336], "content": "g", "parent_index": 7, "subtype": "inline"}, {"bbox": [284, 342, 327, 355], "content": "\\Lambda\\to\\Lambda g.", "parent_index": 8, "subtype": "interline"}, {"bbox": [174, 369, 183, 378], "content": "\\Lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [515, 369, 524, 378], "content": "\\Lambda", "parent_index": 9, "subtype": "inline"}, {"bbox": [272, 390, 279, 398], "content": "g", "parent_index": 9, "subtype": "inline"}, {"bbox": [274, 414, 336, 426], "content": "\\Lambda_{a}\\rightarrow g_{a b}\\Lambda_{b},", "parent_index": 10, "subtype": "interline"}, {"bbox": [106, 447, 122, 455], "content": "g_{a b}", "parent_index": 11, "subtype": "inline"}, {"bbox": [424, 447, 430, 455], "content": "g", "parent_index": 11, "subtype": "inline"}, {"bbox": [164, 462, 171, 471], "content": "I", "parent_index": 11, "subtype": "inline"}, {"bbox": [191, 465, 197, 473], "content": "q", "parent_index": 11, "subtype": "inline"}, {"bbox": [236, 483, 375, 522], "content": "\\begin{array}{l}{{q\\to q I}}\\\\ {{\\to q^{1}I-q^{2}-q^{3}K+q^{4}J,}}\\end{array}", "parent_index": 12, "subtype": "interline"}, {"bbox": [238, 554, 372, 610], "content": "I^{R}=\\left(\\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{1}}\\\\ {{0}}&{{0}}&{{-1}}&{{0}}\\end{array}\\right)", "parent_index": 14, "subtype": "interline"}, {"bbox": [124, 626, 130, 634], "content": "q", "parent_index": 15, "subtype": "inline"}, {"bbox": [225, 621, 280, 635], "content": "q_{a}\\rightarrow I_{a b}^{R}\\,q_{b}", "parent_index": 15, "subtype": "inline"}, {"bbox": [362, 621, 377, 632], "content": "J^{R}", "parent_index": 15, "subtype": "inline"}, {"bbox": [405, 621, 424, 632], "content": "K^{R}", "parent_index": 15, "subtype": "inline"}, {"bbox": [124, 641, 147, 652], "content": "J,K", "parent_index": 15, "subtype": "inline"}, {"bbox": [182, 639, 196, 650], "content": "{\\bf1}^{R}", "parent_index": 15, "subtype": "inline"}, {"bbox": [150, 661, 462, 717], "content": "\\begin{array}{c c}{{J^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{1}}&{{0}}&{{0}}\\end{array}\\!\\!\\right),~~~}}&{{K^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{0}}&{{0}}&{{1}}&{{0}}\\\\ {{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\end{array}\\!\\!\\right).}}\\end{array}", "parent_index": 16, "subtype": "interline"}]
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[]
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