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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]",
"",
"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.",
"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.",
"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)",
"",
"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",
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[{"type": "text", "text": "The Automorphisms of Affine Fusion Rings ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Terry Gannon ", "page_idx": 0}, {"type": "text", "text": "Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected] ", "page_idx": 0}, {"type": "text", "text": "1. Introduction ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Verlinde\u2019s formula [33] ", "page_idx": 0}, {"type": "equation", "text": "$$\nV_{a^{1}\\ldots a^{t}}^{(g)}=\\sum_{b\\in\\Phi}(S_{0b})^{2(1-g)}\\frac{S_{a^{1}b}}{S_{0b}}\\cdot\\cdot\\cdot\\frac{S_{a^{t}b}}{S_{0b}}\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "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 \u2018primary fields\u2019. 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. ", "page_idx": 0}, {"type": "text", "text": "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 \u201cConnes\u2019 fusion\u201d; in quantum cohomology; and in Lusztig\u2019s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein. ", "page_idx": 0}, {"type": "text", "text": "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) ", "page_idx": 0}, {"type": "equation", "text": "$$\nN_{a b}^{c}\\,{\\overset{\\mathrm{def}}{=}}\\,V_{a,b,C c}^{(0)}=\\sum_{d\\in\\Phi}{\\frac{S_{a d}S_{b d}S_{c d}^{*}}{S_{0d}}}\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "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 ", "page_idx": 0}, {"type": "equation", "text": "$$\n{\\cal N}_{\\pi a,\\pi b}^{\\pi c}={\\cal N}_{a b}^{c}\\ ,\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "are precisely the symmetries of all numbers of the form (1.1a). ", "page_idx": 0}]
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The matrix ", "type": "text"}, {"bbox": [285, 353, 293, 362], "score": 0.9, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [293, 352, 461, 366], "score": 1.0, "content": " comes from a representation of ", "type": "text"}, {"bbox": [461, 352, 498, 364], "score": 0.92, "content": "\\mathrm{SL_{2}}(\\mathbb{Z})", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [499, 352, 541, 366], "score": 1.0, "content": " defined", "type": "text"}], "index": 10}, {"bbox": [68, 361, 545, 386], "spans": [{"bbox": [68, 361, 490, 386], "score": 1.0, "content": "by the chiral characters of the theory. 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See the excellent bibliography in [6] for references to the", "type": "text"}], "index": 12}, {"bbox": [70, 397, 165, 410], "spans": [{"bbox": [70, 397, 165, 410], "score": 1.0, "content": "physics literature.", "type": "text"}], "index": 13}], "index": 10.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [68, 322, 545, 410]}, {"type": "text", "bbox": [70, 408, 541, 523], "lines": [{"bbox": [94, 410, 540, 425], "spans": [{"bbox": [94, 410, 540, 425], "score": 1.0, "content": "These numbers are remarkable for also arising in several other contexts: for example,", "type": "text"}], "index": 14}, {"bbox": [71, 426, 541, 439], "spans": [{"bbox": [71, 426, 541, 439], "score": 1.0, "content": "as dimensions of spaces of generalised theta functions; as certain tensor product coefficients", "type": "text"}], "index": 15}, {"bbox": [69, 439, 542, 455], "spans": [{"bbox": [69, 439, 505, 455], "score": 1.0, "content": "in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for ", "type": "text"}, {"bbox": [506, 441, 519, 453], "score": 0.91, "content": "\\mathbb{F}_{p}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [519, 439, 542, 455], "score": 1.0, "content": "; as", "type": "text"}], "index": 16}, {"bbox": [70, 454, 540, 467], "spans": [{"bbox": [70, 454, 540, 467], "score": 1.0, "content": "certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in", "type": "text"}], "index": 17}, {"bbox": [72, 469, 540, 482], "spans": [{"bbox": [72, 469, 540, 482], "score": 1.0, "content": "algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator", "type": "text"}], "index": 18}, {"bbox": [70, 482, 541, 497], "spans": [{"bbox": [70, 482, 541, 497], "score": 1.0, "content": "algebras (VOAs); in von Neumann algebras as \u201cConnes\u2019 fusion\u201d; in quantum cohomology;", "type": "text"}], "index": 19}, {"bbox": [70, 496, 541, 511], "spans": [{"bbox": [70, 496, 541, 511], "score": 1.0, "content": "and in Lusztig\u2019s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and", "type": "text"}], "index": 20}, {"bbox": [71, 512, 167, 525], "spans": [{"bbox": [71, 512, 167, 525], "score": 1.0, "content": "references therein.", "type": "text"}], "index": 21}], "index": 17.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [69, 410, 542, 525]}, {"type": "text", "bbox": [70, 523, 541, 552], "lines": [{"bbox": [94, 525, 541, 540], "spans": [{"bbox": [94, 525, 541, 540], "score": 1.0, "content": "The more fundamental of these numbers are those corresponding to a sphere with three", "type": "text"}], "index": 22}, {"bbox": [70, 540, 520, 555], "spans": [{"bbox": [70, 540, 520, 555], "score": 1.0, "content": "punctures. It is more convenient to write these in the form (called fusion coefficients)", "type": "text"}], "index": 23}], "index": 22.5, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [70, 525, 541, 555]}, {"type": "interline_equation", "bbox": [223, 567, 387, 602], "lines": [{"bbox": [223, 567, 387, 602], "spans": [{"bbox": [223, 567, 387, 602], "score": 0.95, "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}}}", "type": "interline_equation"}], "index": 24}], "index": 24, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 613, 541, 657], "lines": [{"bbox": [71, 616, 541, 631], "spans": [{"bbox": [71, 616, 106, 631], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 618, 116, 627], "score": 0.91, "content": "C", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [116, 616, 225, 631], "score": 1.0, "content": " is a permutation of ", "type": "text"}, {"bbox": [226, 618, 234, 627], "score": 0.88, "content": "\\Phi", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [235, 616, 541, 631], "score": 1.0, "content": " called charge-conjugation and will be defined below. The", "type": "text"}], "index": 25}, {"bbox": [70, 630, 542, 645], "spans": [{"bbox": [70, 630, 542, 645], "score": 1.0, "content": "fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries", "type": "text"}], "index": 26}, {"bbox": [70, 643, 385, 660], "spans": [{"bbox": [70, 643, 305, 660], "score": 1.0, "content": "of the numbers (1.1b), i.e. the permutations ", "type": "text"}, {"bbox": [305, 649, 313, 655], "score": 0.88, "content": "\\pi", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [313, 643, 330, 660], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [330, 646, 339, 655], "score": 0.9, "content": "\\Phi", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [339, 643, 385, 660], "score": 1.0, "content": " obeying", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [70, 616, 542, 660]}, {"type": "interline_equation", "bbox": [266, 673, 344, 689], "lines": [{"bbox": [266, 673, 344, 689], "spans": [{"bbox": [266, 673, 344, 689], "score": 0.93, "content": "{\\cal N}_{\\pi a,\\pi b}^{\\pi c}={\\cal N}_{a b}^{c}\\ ,", "type": "interline_equation"}], "index": 28}], "index": 28, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 700, 399, 715], "lines": [{"bbox": [70, 703, 398, 717], "spans": [{"bbox": [70, 703, 398, 717], "score": 1.0, "content": "are precisely the symmetries of all numbers of the form (1.1a).", "type": "text"}], "index": 29}], "index": 29, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [70, 703, 398, 717]}]}
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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]
arXiv:math/0002044v1 [math.QA] 7 Feb 2000
# 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 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.
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.
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 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
$$
{\cal N}_{\pi a,\pi b}^{\pi c}={\cal N}_{a b}^{c}\ ,
$$
are precisely the symmetries of all numbers of the form (1.1a).
|
<div class="pdf-page">
<h1>The Automorphisms of Affine Fusion Rings</h1>
<p>Terry Gannon</p>
<p>Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected]</p>
<h1>1. Introduction</h1>
<p>Verlinde’s formula [33]</p>
<p>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.</p>
<p>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.</p>
<p>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>
<p>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</p>
<p>are precisely the symmetries of all numbers of the form (1.1a).</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="234" data-y="85" data-width="554" data-height="27">The Automorphisms of Affine Fusion Rings</h1>
<p class="pdf-text" data-x="435" data-y="148" data-width="152" data-height="18">Terry Gannon</p>
<p class="pdf-text" data-x="245" data-y="177" data-width="533" data-height="62">Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected]</p>
<div class="pdf-discarded" data-x="21" data-y="215" data-width="42" data-height="506" style="opacity: 0.5;">arXiv:math/0002044v1 [math.QA] 7 Feb 2000</div>
<h1 class="pdf-title" data-x="426" data-y="276" data-width="167" data-height="18">1. Introduction</h1>
<p class="pdf-text" data-x="155" data-y="314" data-width="206" data-height="19">Verlinde’s formula [33]</p>
<p class="pdf-text" data-x="117" data-y="411" data-width="788" data-height="116">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.</p>
<p class="pdf-text" data-x="117" data-y="527" data-width="788" data-height="149">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.</p>
<p class="pdf-text" data-x="117" data-y="676" data-width="788" data-height="37">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>
<p class="pdf-text" data-x="117" data-y="792" data-width="788" data-height="57">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</p>
<p class="pdf-text" data-x="117" data-y="905" data-width="550" data-height="19">are precisely the symmetries of all numbers of the form (1.1a).</p>
</div>
|
# 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
$$
{\cal N}_{\pi a,\pi b}^{\pi c}={\cal N}_{a b}^{c}\ ,
$$
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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}\\ ,",
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"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.",
"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].",
"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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[{"type": "text", "text": "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 $N_{a b}^{c}$ : ", "page_idx": 1}, {"type": "equation", "text": "$$\n\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}\n$$", "text_format": "latex", "page_idx": 1}, {"type": "text", "text": "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. ", "page_idx": 1}, {"type": "text", "text": "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 \u2014 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]. ", "page_idx": 1}, {"type": "text", "text": "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. ", "page_idx": 1}, {"type": "text", "text": "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. ", "page_idx": 1}, {"type": "text", "text": "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]. ", "page_idx": 1}]
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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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_1", "page_size": [612.0, 792.0], "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 \u2014 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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [70, 556, 541, 600]}]}
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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 :
$$
\chi_{a}\chi_{b}=\sum_{c\in\Phi}N_{a b}^{c}\chi_{c}
$$
As an abstract ring, it is not so interesting (the fusion ring over 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.
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].
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].
|
<div class="pdf-page">
<p>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 :</p>
<p>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.</p>
<p>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].</p>
<p>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>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>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>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="90" data-width="788" data-height="58">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 :</p>
<p class="pdf-text" data-x="117" data-y="223" data-width="786" data-height="93">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.</p>
<p class="pdf-text" data-x="118" data-y="318" data-width="787" data-height="296">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].</p>
<p class="pdf-text" data-x="117" data-y="615" data-width="788" data-height="54">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 class="pdf-text" data-x="118" data-y="669" data-width="787" data-height="38">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 class="pdf-text" data-x="117" data-y="716" data-width="786" data-height="57">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>
</div>
|
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.
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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 \u00a73). 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 \u03ba d=ef k + i ai\u2228 . This is the matrix S appearing in (1.1); \u03a6 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 \u2018 $0^{\\circ}$ . 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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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": "\u03ba d=ef k + i ai\u2228 . This is the matrix S appearing in (1.1); \u03a6 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 \u2018", "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). A useful fact", "type": "text"}], "index": 16}, {"bbox": [70, 484, 107, 496], "spans": [{"bbox": [70, 484, 107, 496], "score": 1.0, "content": "is that", "type": "text"}], "index": 17}], "index": 16}, {"type": "interline_equation", "bbox": [233, 497, 378, 511], "lines": [{"bbox": [233, 497, 378, 511], "spans": [{"bbox": [233, 497, 378, 511], "score": 0.89, "content": "S_{\\lambda0}>0\\qquad\\mathrm{for~all~}\\lambda\\in P_{+}\\ .", "type": "interline_equation"}], "index": 18}], "index": 18}, {"type": "text", "bbox": [93, 519, 300, 534], "lines": [{"bbox": [95, 521, 300, 536], "spans": [{"bbox": [95, 521, 300, 536], "score": 1.0, "content": "Equation (2.1a) gives us the important", "type": "text"}], "index": 19}], "index": 19}, {"type": "interline_equation", "bbox": [218, 549, 393, 580], "lines": [{"bbox": [218, 549, 393, 580], "spans": [{"bbox": [218, 549, 393, 580], "score": 0.94, "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})~,", "type": "interline_equation"}], "index": 20}], "index": 20}, {"type": "text", "bbox": [69, 592, 540, 622], "lines": [{"bbox": [70, 595, 540, 611], "spans": [{"bbox": [70, 595, 105, 611], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 597, 125, 610], "score": 0.74, "content": "\\mathrm{ch}_{{\\overline{{\\lambda}}}}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [125, 595, 275, 611], "score": 1.0, "content": " is the Weyl character of the ", "type": "text"}, {"bbox": [275, 597, 291, 608], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [291, 595, 336, 611], "score": 1.0, "content": "-module ", "type": "text"}, {"bbox": [336, 595, 361, 609], "score": 0.94, "content": "L(\\overline{{\\lambda}})", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [362, 595, 540, 611], "score": 1.0, "content": ". 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Consider the (finite) set", "type": "text"}], "index": 2}, {"bbox": [71, 156, 347, 174], "spans": [{"bbox": [71, 156, 152, 173], "score": 0.94, "content": "P_{+}=P_{+}^{k}(X_{r}^{(1)})", "type": "inline_equation", "height": 17, "width": 81}, {"bbox": [152, 156, 196, 174], "score": 1.0, "content": " of level ", "type": "text"}, {"bbox": [196, 160, 203, 169], "score": 0.9, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [203, 156, 347, 174], "score": 1.0, "content": " integrable highest weights:", "type": "text"}], "index": 3}], "index": 2, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [69, 125, 542, 174]}, {"type": "interline_equation", "bbox": [174, 187, 436, 226], "lines": [{"bbox": [174, 187, 436, 226], "spans": [{"bbox": [174, 187, 436, 226], "score": 0.93, "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\\}\\ ,", "type": "interline_equation"}], "index": 4}], "index": 4, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 239, 541, 328], "lines": [{"bbox": [69, 241, 542, 260], "spans": [{"bbox": [69, 241, 105, 260], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [105, 246, 117, 257], "score": 0.92, "content": "\\Lambda_{i}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [118, 241, 315, 260], "score": 1.0, "content": " denote the fundamental weights, and ", "type": "text"}, {"bbox": [316, 245, 329, 259], "score": 0.9, "content": "a_{j}^{\\vee}", "type": "inline_equation", "height": 14, "width": 13}, {"bbox": [329, 241, 436, 260], "score": 1.0, "content": " are the co-labels, of ", "type": "text"}, {"bbox": [436, 241, 460, 256], "score": 0.91, "content": "X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [461, 241, 488, 260], "score": 1.0, "content": "(the ", "type": "text"}, {"bbox": [488, 245, 501, 259], "score": 0.88, "content": "a_{j}^{\\vee}", "type": "inline_equation", "height": 14, "width": 13}, {"bbox": [502, 241, 542, 260], "score": 1.0, "content": " will be", "type": "text"}], "index": 5}, {"bbox": [70, 258, 540, 274], "spans": [{"bbox": [70, 258, 480, 274], "score": 1.0, "content": "given for each algebra in \u00a73). 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, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [69, 241, 542, 330]}, {"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, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"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": "\u03ba d=ef k + i ai\u2228 . This is the matrix S appearing in (1.1); \u03a6 there is P+ here.", "type": "text"}], "index": 14}], "index": 12.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [68, 393, 541, 457]}, {"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 \u2018", "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). A useful fact", "type": "text"}], "index": 16}, {"bbox": [70, 484, 107, 496], "spans": [{"bbox": [70, 484, 107, 496], "score": 1.0, "content": "is that", "type": "text"}], "index": 17}], "index": 16, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [70, 453, 541, 496]}, {"type": "interline_equation", "bbox": [233, 497, 378, 511], "lines": [{"bbox": [233, 497, 378, 511], "spans": [{"bbox": [233, 497, 378, 511], "score": 0.89, "content": "S_{\\lambda0}>0\\qquad\\mathrm{for~all~}\\lambda\\in P_{+}\\ .", "type": "interline_equation"}], "index": 18}], "index": 18, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [93, 519, 300, 534], "lines": [{"bbox": [95, 521, 300, 536], "spans": [{"bbox": [95, 521, 300, 536], "score": 1.0, "content": "Equation (2.1a) gives us the important", "type": "text"}], "index": 19}], "index": 19, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [95, 521, 300, 536]}, {"type": "interline_equation", "bbox": [218, 549, 393, 580], "lines": [{"bbox": [218, 549, 393, 580], "spans": [{"bbox": [218, 549, 393, 580], "score": 0.94, "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})~,", "type": "interline_equation"}], "index": 20}], "index": 20, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 592, 540, 622], "lines": [{"bbox": [70, 595, 540, 611], "spans": [{"bbox": [70, 595, 105, 611], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 597, 125, 610], "score": 0.74, "content": "\\mathrm{ch}_{{\\overline{{\\lambda}}}}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [125, 595, 275, 611], "score": 1.0, "content": " is the Weyl character of the ", "type": "text"}, {"bbox": [275, 597, 291, 608], "score": 0.92, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [291, 595, 336, 611], "score": 1.0, "content": "-module ", "type": "text"}, {"bbox": [336, 595, 361, 609], "score": 0.94, "content": "L(\\overline{{\\lambda}})", "type": "inline_equation", "height": 14, "width": 25}, {"bbox": [362, 595, 540, 611], "score": 1.0, "content": ". Together with the Weyl denomi-", "type": "text"}], "index": 21}, {"bbox": [70, 610, 424, 625], "spans": [{"bbox": [70, 610, 351, 625], "score": 1.0, "content": "nator formula, it provides a useful expression for the ", "type": "text"}, {"bbox": [351, 615, 357, 623], "score": 0.86, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [357, 610, 424, 625], "score": 1.0, "content": "-dimensions:", "type": "text"}], "index": 22}], "index": 21.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [70, 595, 540, 625]}, {"type": "interline_equation", "bbox": [200, 636, 410, 672], "lines": [{"bbox": [200, 636, 410, 672], "spans": [{"bbox": [200, 636, 410, 672], "score": 0.94, "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)}~,", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 685, 540, 715], "lines": [{"bbox": [71, 688, 540, 703], "spans": [{"bbox": [71, 688, 300, 703], "score": 1.0, "content": "where the product is over the positive roots ", "type": "text"}, {"bbox": [301, 688, 341, 702], "score": 0.94, "content": "\\alpha\\in\\overline{{\\Delta}}_{+}", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [341, 688, 357, 703], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [357, 690, 373, 701], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [373, 688, 540, 703], "score": 1.0, "content": ". 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, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [69, 688, 540, 718]}]}
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|
2. Generalities
# 2.1. The affine fusion ring
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:
$$
P_{+}\stackrel{\mathrm{def}}{=}\{\sum_{j=0}^{r}\lambda_{j}\Lambda_{j}\mid\lambda_{j}\in\mathbb{Z},\ \lambda_{j}\geq0,\ \sum_{j=0}^{r}a_{j}^{\vee}\lambda_{j}=k\}\ ,
$$
where 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
$$
S_{\mu\nu}=c\,\sum_{w\in\overline{{{W}}}}{\operatorname*{det}(w)}\,\exp[-2\pi\mathrm{i}\,\frac{(w(\mu+\rho)|\nu+\rho)}{\kappa}]\ .
$$
An explicit expression for the normalisation constant 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.
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
$$
S_{\lambda0}>0\qquad\mathrm{for~all~}\lambda\in P_{+}\ .
$$
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 is the Weyl character of the -module . Together with the Weyl denomi- nator formula, it provides a useful expression for the -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 of . Another consequence of (2.1b) is the Kac-Walton formula (2.4).
3
|
<div class="pdf-page">
<h1>2.1. The affine fusion ring</h1>
<p>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:</p>
<p>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</p>
<p>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.</p>
<p>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</p>
<p>Equation (2.1a) gives us the important</p>
<p>where is the Weyl character of the -module . Together with the Weyl denomi- nator formula, it provides a useful expression for the -dimensions:</p>
<p>where the product is over the positive roots of . Another consequence of (2.1b) is the Kac-Walton formula (2.4).</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="428" data-y="91" data-width="164" data-height="18" style="opacity: 0.5;">2. Generalities</div>
<h1 class="pdf-title" data-x="117" data-y="129" data-width="241" data-height="20">2.1. The affine fusion ring</h1>
<p class="pdf-text" data-x="117" data-y="159" data-width="788" data-height="62">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:</p>
<p class="pdf-text" data-x="115" data-y="309" data-width="790" data-height="115">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</p>
<p class="pdf-text" data-x="117" data-y="502" data-width="788" data-height="82">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.</p>
<p class="pdf-text" data-x="117" data-y="584" data-width="788" data-height="54">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</p>
<p class="pdf-text" data-x="155" data-y="671" data-width="346" data-height="19">Equation (2.1a) gives us the important</p>
<p class="pdf-text" data-x="115" data-y="765" data-width="788" data-height="39">where is the Weyl character of the -module . Together with the Weyl denomi- nator formula, it provides a useful expression for the -dimensions:</p>
<p class="pdf-text" data-x="115" data-y="885" data-width="788" data-height="39">where the product is over the positive roots of . Another consequence of (2.1b) is the Kac-Walton formula (2.4).</p>
<div class="pdf-discarded" data-x="503" data-y="945" data-width="17" data-height="13" style="opacity: 0.5;">3</div>
</div>
|
# 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:
$$
{\mathcal D}(\lambda)\,\overset{\mathrm{def}}{=}\frac{S_{\lambda0}}{S_{00}}=\prod_{\alpha>0}\frac{\sin(\pi\left(\lambda+\rho\left|\alpha\right)/\kappa\right)}{\sin(\pi\left(\rho\left|\alpha\right)/\kappa\right)}~,
$$
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"The source of some of the most interesting fusion data are the affine nontwisted",
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"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 denomi-",
"nator 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)."
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"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\\}\\ ,",
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"{\\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)}~,"
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"Charge-conjugation is the order 2 permutation of given by , the weight contragredient to . For instance . It has the basic property that",
"",
"and . corresponds to a symmetry of the (unextended) Dynkin diagram of , as we will see next section.",
"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 .",
"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",
"",
"The simple-currents form an abelian group, given by composition of the permutations .",
"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 .",
"Evaluating in two ways gives the useful",
"",
"and hence the reciprocity .",
"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",
"",
"This is simply the statement that the affine Weyl orbit of intersects the set at precisely one point (namely . Write . Then [16]",
"",
"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",
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[{"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\u2013Dynkin 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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We", "type": "text"}], "index": 5}, {"bbox": [68, 186, 542, 209], "spans": [{"bbox": [68, 186, 256, 209], "score": 1.0, "content": "call these conjugations. 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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\u2013Dynkin", "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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\u2013Dynkin", "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}]}
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|
Charge-conjugation is the order 2 permutation of given by , the weight contragredient to . For instance . It has the basic property that
$$
S_{C\lambda,\mu}=S_{\lambda,C\mu}=S_{\lambda\mu}^{*}
$$
and . corresponds to a symmetry of the (unextended) Dynkin diagram of , as we will see next section.
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 .
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
$$
S_{J\lambda,\mu}=\exp[2\pi\mathrm{i}\,Q_{j}(\mu)]\,S_{\lambda\mu}
$$
The simple-currents form an abelian group, given by composition of the permutations .
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 .
Evaluating 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 .
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
$$
\ell\left(\lambda+\rho\right)=\omega(\lambda^{\left(\ell\right)}+\rho)+\kappa\alpha\;.
$$
This is simply the statement that the affine Weyl orbit of intersects the set at precisely one point (namely . Write . 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 by all entries lies in the cyclotomic field where denotes the root of unity ; for any , there will be a function such that
$$
\sigma_{\ell}\bigl(S_{\lambda\mu}\bigr)=\epsilon_{\ell}\bigl(\lambda\bigr)\,S_{\lambda^{(\ell)},\mu}=\epsilon_{\ell}\bigl(\mu\bigr)\,S_{\lambda,\mu^{(\ell)}}\ .
$$
|
<div class="pdf-page">
<p>Charge-conjugation is the order 2 permutation of given by , the weight contragredient to . For instance . It has the basic property that</p>
<p>and . corresponds to a symmetry of the (unextended) Dynkin diagram of , as we will see next section.</p>
<p>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 .</p>
<p>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</p>
<p>The simple-currents form an abelian group, given by composition of the permutations .</p>
<p>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 .</p>
<p>Evaluating in two ways gives the useful</p>
<p>and hence the reciprocity .</p>
<p>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</p>
<p>This is simply the statement that the affine Weyl orbit of intersects the set at precisely one point (namely . Write . Then [16]</p>
<p>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</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="90" data-width="786" data-height="39">Charge-conjugation is the order 2 permutation of given by , the weight contragredient to . For instance . It has the basic property that</p>
<p class="pdf-text" data-x="117" data-y="184" data-width="786" data-height="38">and . corresponds to a symmetry of the (unextended) Dynkin diagram of , as we will see next section.</p>
<p class="pdf-text" data-x="117" data-y="223" data-width="788" data-height="60">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 .</p>
<p class="pdf-text" data-x="117" data-y="283" data-width="788" data-height="57">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</p>
<p class="pdf-text" data-x="117" data-y="395" data-width="783" data-height="18">The simple-currents form an abelian group, given by composition of the permutations .</p>
<p class="pdf-text" data-x="117" data-y="415" data-width="788" data-height="98">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 .</p>
<p class="pdf-text" data-x="157" data-y="513" data-width="408" data-height="19">Evaluating in two ways gives the useful</p>
<p class="pdf-text" data-x="117" data-y="588" data-width="374" data-height="18">and hence the reciprocity .</p>
<p class="pdf-text" data-x="117" data-y="607" data-width="788" data-height="74">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</p>
<p class="pdf-text" data-x="117" data-y="736" data-width="788" data-height="39">This is simply the statement that the affine Weyl orbit of intersects the set at precisely one point (namely . Write . Then [16]</p>
<p class="pdf-text" data-x="113" data-y="830" data-width="792" data-height="73">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</p>
</div>
|
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]
$$
\epsilon_{\ell}^{\prime}(\lambda)\,S_{\lambda^{(\ell)},\mu}=\epsilon_{\ell}^{\prime}(\mu)\,S_{\lambda,\mu^{(\ell)}}
$$
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"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 .",
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"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."
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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(\u03c0 (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 \u201csub-quotient\u201d $\\widetilde{\\mathcal{O}}_{k}$ of Kazhdan-Lusztig\u2019s category of level $k$ integrable highest weight $X_{r}^{(1)}$ -modules, and t o Gelfand-Kazhdan\u2019s 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(\u03c0 (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 \u201csub-quotient\u201d", "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\u2019s 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\u2019s 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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_4", "page_size": [612.0, 792.0], "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(\u03c0 (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": ". 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|
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 .
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 :
$$
N_{\lambda\mu}^{\nu}=\sum_{w\in W}\operatorname*{det}(w)\,T_{\lambda\mu}^{w.\nu}~,
$$
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.
A handy consequence of (2.4) that whenever is large enough that (i.e. that , then .
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 .
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 .
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
$$
N_{a b}^{c}=\left\{\begin{array}{c c}{{1}}&{{\mathrm{if~}c\equiv a\!+\!b\;(\mathrm{mod~}2)\;\mathrm{and~}|a\!-\!b|\leq c\leq\operatorname*{min}\{a\!+\!b,2k\!-\!a\!-\!b\}}}\\ {{0}}&{{\mathrm{otherwise}}}\end{array}\right..
$$
Equation (2.4) tells us the affine fusion rules are the structure constants for the ring 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.
|
<div class="pdf-page">
<p>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 .</p>
<p>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 :</p>
<p>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.</p>
<p>A handy consequence of (2.4) that whenever is large enough that (i.e. that , then .</p>
<p>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 .</p>
<p>The importance of (charge-)conjugation and simple-currents for us is that they respect fusions:</p>
<p>for any simple-currents .</p>
<p>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</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="90" data-width="788" data-height="57">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 .</p>
<p class="pdf-text" data-x="115" data-y="148" data-width="790" data-height="64">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 :</p>
<p class="pdf-text" data-x="117" data-y="285" data-width="788" data-height="78">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.</p>
<p class="pdf-text" data-x="117" data-y="364" data-width="788" data-height="40">A handy consequence of (2.4) that whenever is large enough that (i.e. that , then .</p>
<p class="pdf-text" data-x="117" data-y="404" data-width="788" data-height="53">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 .</p>
<p class="pdf-text" data-x="117" data-y="458" data-width="786" data-height="37">The importance of (charge-)conjugation and simple-currents for us is that they respect fusions:</p>
<p class="pdf-text" data-x="117" data-y="598" data-width="276" data-height="18">for any simple-currents .</p>
<p class="pdf-text" data-x="117" data-y="618" data-width="788" data-height="121">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</p>
<p class="pdf-text" data-x="117" data-y="806" data-width="788" data-height="119">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.</p>
</div>
|
$\epsilon_{\ell}(\lambda)/\epsilon_{\ell}^{\prime}(\lambda)=\sigma_{\ell}(c)/c$ 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
$$
N_{a b}^{c}=\left\{\begin{array}{c c}{{1}}&{{\mathrm{if~}c\equiv a\!+\!b\;(\mathrm{mod~}2)\;\mathrm{and~}|a\!-\!b|\leq c\leq\operatorname*{min}\{a\!+\!b,2k\!-\!a\!-\!b\}}}\\ {{0}}&{{\mathrm{otherwise}}}\end{array}\right..
$$
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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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"N_{\\lambda\\mu}^{\\nu}=\\sum_{w\\in W}\\operatorname*{det}(w)\\,T_{\\lambda\\mu}^{w.\\nu}~,",
"\\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_{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.."
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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 : 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).",
"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.",
"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.",
"2.2. Symmetries of fusion coefficients",
"Definition 2.1. By an isomorphism between fusion rings and (with fusion coefficients and respectively) we mean a bijection such that",
"",
"When we call an automorphism or fusion-symmetry. Call the pair of permutations an -symmetry if",
"",
"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",
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[{"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 \u00a73.3. ", "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\u2019s 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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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 \u00a73.3.", "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\u2019s 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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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 \u00a73.3.", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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\u2019s 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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}]}
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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 : 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).
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.
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.
# 2.2. Symmetries of fusion coefficients
Definition 2.1. By an isomorphism between fusion rings and (with fusion coefficients and respectively) we mean a bijection such that
$$
N_{\lambda,\mu}^{\nu}=M_{\pi\lambda,\pi\mu}^{\pi\nu}\qquad\forall\lambda,\mu,\nu\in P_{+}(X_{r,k})\ .
$$
When we call an automorphism or fusion-symmetry. Call the pair of permutations an -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 -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
$$
\pi(J\mu)=\pi(j)\,\pi(\mu)~.
$$
|
<div class="pdf-page">
<p>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).</p>
<p>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.</p>
<p>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.</p>
<h1>2.2. Symmetries of fusion coefficients</h1>
<p>Definition 2.1. By an isomorphism between fusion rings and (with fusion coefficients and respectively) we mean a bijection such that</p>
<p>When we call an automorphism or fusion-symmetry. Call the pair of permutations an -symmetry if</p>
<p>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</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="118" data-y="148" data-width="793" data-height="110">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).</p>
<p class="pdf-text" data-x="117" data-y="259" data-width="789" data-height="212">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.</p>
<p class="pdf-text" data-x="118" data-y="473" data-width="787" data-height="62">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.</p>
<h1 class="pdf-title" data-x="118" data-y="550" data-width="333" data-height="20">2.2. Symmetries of fusion coefficients</h1>
<p class="pdf-text" data-x="118" data-y="576" data-width="787" data-height="61">Definition 2.1. By an isomorphism between fusion rings and (with fusion coefficients and respectively) we mean a bijection such that</p>
<p class="pdf-text" data-x="118" data-y="667" data-width="788" data-height="37">When we call an automorphism or fusion-symmetry. Call the pair of permutations an -symmetry if</p>
<p class="pdf-text" data-x="117" data-y="746" data-width="788" data-height="159">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</p>
</div>
|
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
$$
S_{\pi\lambda,\pi^{\prime}\mu}=S_{\lambda\mu}\qquad\forall\lambda,\mu\in{\cal P}_{+}\ .
$$
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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 excep-",
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"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 rank-",
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"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",
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"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)~."
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"N_{\\lambda,\\mu}^{\\nu}=M_{\\pi\\lambda,\\pi\\mu}^{\\pi\\nu}\\qquad\\forall\\lambda,\\mu,\\nu\\in P_{+}(X_{r,k})\\ .",
"S_{\\pi\\lambda,\\pi^{\\prime}\\mu}=S_{\\lambda\\mu}\\qquad\\forall\\lambda,\\mu\\in{\\cal P}_{+}\\ .",
"\\pi(J\\mu)=\\pi(j)\\,\\pi(\\mu)~."
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"For instance must send -fixed-points to -fixed-points.",
"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.",
"The key to finding fusion-symmetries is the following Lemma.",
"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 .",
"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 . ■",
"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",
"",
"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",
"",
"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",
"",
"Let be an -symmetry, and suppose we know that for all in the fusion- generator . Then for any ,",
"",
"for all , so .",
"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."
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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\u2019s 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 $\\alpha(\\mu)=1$ . \u25a0 ", "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 \u2014 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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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": ". 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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\u2019s 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": "\u25a0", "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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\u2019s 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": "\u25a0", "type": "text"}], "index": 16}], "index": 13.5, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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 \u2014 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, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [72, 689, 541, 717]}]}
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|
For instance must send -fixed-points to -fixed-points.
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.
The key to finding fusion-symmetries is the following Lemma.
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 .
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 . ■
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
$$
Q_{j}(\lambda)\equiv\widetilde{Q}_{\pi^{\prime}j}(\pi\lambda)\equiv\widetilde{Q}_{\pi j}(\pi^{\prime}\lambda)\qquad(\mathrm{mod~1})~.
$$
Another quick consequence of the Lemma is that for any Galois automorphism and isomorphism , we have and . To see this, apply the invertibility of 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 of which generates as a ring. Diagonalising, this is equivalent to requiring that there are -variable polynomials 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 be an -symmetry, and suppose we know that for all in the fusion- generator . Then for any ,
$$
{\frac{S_{\lambda\mu}}{S_{0\mu}}}={\frac{S_{\pi\lambda,\pi^{\prime}\mu}}{S_{0,\pi^{\prime}\mu}}}=P_{\pi\lambda}({\frac{S_{\gamma_{1},\pi^{\prime}\mu}}{S_{0,\pi^{\prime}\mu}}},\dots)=P_{\pi\lambda}({\frac{S_{\gamma_{1}\mu}}{S_{0\mu}}},\dots)={\frac{S_{\pi\lambda,\mu}}{S_{0\mu}}}
$$
for all , so .
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.
|
<div class="pdf-page">
<p>For instance must send -fixed-points to -fixed-points.</p>
<p>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.</p>
<p>The key to finding fusion-symmetries is the following Lemma.</p>
<p>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 .</p>
<p>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 . ■</p>
<p>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</p>
<p>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</p>
<p>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</p>
<p>Let be an -symmetry, and suppose we know that for all in the fusion- generator . Then for any ,</p>
<p>for all , so .</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="90" data-width="538" data-height="19">For instance must send -fixed-points to -fixed-points.</p>
<p class="pdf-text" data-x="117" data-y="111" data-width="789" data-height="95">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.</p>
<p class="pdf-text" data-x="158" data-y="206" data-width="544" data-height="18">The key to finding fusion-symmetries is the following Lemma.</p>
<p class="pdf-text" data-x="117" data-y="232" data-width="791" data-height="86">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 .</p>
<p class="pdf-text" data-x="117" data-y="323" data-width="816" data-height="121">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 . ■</p>
<p class="pdf-text" data-x="117" data-y="451" data-width="788" data-height="54">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</p>
<p class="pdf-text" data-x="117" data-y="535" data-width="788" data-height="57">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</p>
<p class="pdf-text" data-x="115" data-y="641" data-width="791" data-height="57">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</p>
<p class="pdf-text" data-x="115" data-y="765" data-width="790" data-height="37">Let be an -symmetry, and suppose we know that for all in the fusion- generator . Then for any ,</p>
<p class="pdf-text" data-x="115" data-y="868" data-width="231" data-height="18">for all , so .</p>
<p class="pdf-text" data-x="117" data-y="888" data-width="788" data-height="36">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.</p>
</div>
|
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 $\pi\lambda=\lambda$ .
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"content": [
"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 simple-",
"currents 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 fusion-",
"generator \\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",
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"2.3. Standard constructions of fusion-symmetries",
"Simple-currents are a large source of fusion-symmetries. Let be any simple-current of order . Choose any number such that",
"",
"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.",
"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.",
"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 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 , 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",
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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 \u2014 see \u00a72.2); the \u2018gcd\u2019 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 $j$ \u2014 equivalently, that $\\sigma_{\\ell}(S_{00}^{2})=S_{00}^{2}$ \u2014 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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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": " \u2014 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": " \u2014 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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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 \u2014 see", "type": "text"}], "index": 12}, {"bbox": [69, 308, 539, 325], "spans": [{"bbox": [69, 308, 235, 325], "score": 1.0, "content": "\u00a72.2); the \u2018gcd\u2019 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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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": " \u2014 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": " \u2014 then the permutation", "type": "text"}], "index": 25}], "index": 24, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 687, 542, 719]}]}
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|
2.3. Standard constructions of fusion-symmetries
Simple-currents are a large source of fusion-symmetries. Let be any simple-current of order . Choose any number such that
$$
\operatorname*{gcd}(n a Q_{j}(j)+1,n)=1\ .
$$
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.
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.
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 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 , 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
$$
\pi\{\ell\}:\lambda\mapsto J(\lambda^{(\ell)})
$$
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
$$
\epsilon_{\ell}(\lambda)\,S_{\lambda^{(\ell)},0}=\sigma_{\ell}S_{\lambda0}=\epsilon_{\ell}(0)\,e^{2\pi\mathrm{i}\,Q_{j}(\lambda)}S_{\lambda0}
$$
that . 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 is an -symmetry. Incidentally, will always be order 1 or 2 because for all .
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.
|
<div class="pdf-page">
<p>2.3. Standard constructions of fusion-symmetries</p>
<p>Simple-currents are a large source of fusion-symmetries. Let be any simple-current of order . Choose any number such that</p>
<p>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.</p>
<p>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.</p>
<p>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>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.</p>
<p>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</p>
<p>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</p>
<p>that . Hence</p>
<p>and so is an -symmetry. Incidentally, will always be order 1 or 2 because for all .</p>
<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.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="120" data-y="223" data-width="433" data-height="20">2.3. Standard constructions of fusion-symmetries</p>
<p class="pdf-text" data-x="117" data-y="250" data-width="786" data-height="40">Simple-currents are a large source of fusion-symmetries. Let be any simple-current of order . Choose any number such that</p>
<p class="pdf-text" data-x="117" data-y="337" data-width="788" data-height="97">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.</p>
<p class="pdf-text" data-x="118" data-y="434" data-width="787" data-height="54">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.</p>
<p class="pdf-text" data-x="117" data-y="491" data-width="788" data-height="36">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 class="pdf-text" data-x="117" data-y="528" data-width="788" data-height="57">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.</p>
<p class="pdf-text" data-x="117" data-y="585" data-width="788" data-height="57">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</p>
<p class="pdf-text" data-x="115" data-y="690" data-width="788" data-height="39">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</p>
<p class="pdf-text" data-x="117" data-y="779" data-width="307" data-height="21">that . Hence</p>
<p class="pdf-text" data-x="117" data-y="849" data-width="789" data-height="37">and so is an -symmetry. Incidentally, will always be order 1 or 2 because for all .</p>
<p class="pdf-text" data-x="117" data-y="888" data-width="789" data-height="36">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>
</div>
|
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 $\lambda\in P_{+}$ .
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"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 fusion-",
"symmetry. 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."
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"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.",
"3.1. The algebra ,",
"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 .",
"The Kac-Peterson relation (2.1b) for takes the form",
"",
"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 .",
"The fusion (derived from the Pieri rule and (2.4))",
"",
"valid for and , will be useful.",
"There are no exceptional fusion-symmetries for :",
"Theorem 3.A. The fusion-symmetries for level are , for and any integer for which is coprime to .",
"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 .",
"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 .",
"3. Data for the Affine Algebras."
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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 \u00a72.3, 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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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. Recall the simple-current automorphism", "type": "text"}], "index": 2}, {"bbox": [71, 146, 541, 162], "spans": [{"bbox": [71, 148, 92, 161], "score": 0.92, "content": "\\pi[a]", "type": "inline_equation", "height": 13, "width": 21}, {"bbox": [92, 146, 231, 162], "score": 1.0, "content": " and Galois automorphism ", "type": "text"}, {"bbox": [231, 148, 256, 161], "score": 0.91, "content": "\\pi\\{\\ell\\}", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [257, 146, 429, 162], "score": 1.0, "content": " defined in \u00a72.3, and the notation ", "type": "text"}, {"bbox": [429, 148, 484, 159], "score": 0.94, "content": "\\kappa=k\\!+\\!h^{\\vee}", "type": "inline_equation", "height": 11, "width": 55}, {"bbox": [485, 146, 541, 162], "score": 1.0, "content": ". It will be", "type": "text"}], "index": 3}, {"bbox": [68, 161, 541, 178], "spans": [{"bbox": [68, 163, 178, 178], "score": 1.0, "content": "convenient to write ", "type": "text"}, {"bbox": [179, 164, 206, 178], "score": 0.86, "content": "X{_{r,k}}^{,}", "type": "inline_equation", "height": 14, "width": 27}, {"bbox": [206, 163, 230, 178], "score": 1.0, "content": " for ", "type": "text"}, {"bbox": [231, 161, 255, 176], "score": 0.9, "content": "\\cdot X_{r}^{(1)}", "type": "inline_equation", "height": 15, "width": 24}, {"bbox": [257, 163, 310, 178], "score": 1.0, "content": "and level ", "type": "text"}, {"bbox": [310, 165, 320, 174], "score": 0.62, "content": "k'", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [321, 163, 380, 178], "score": 1.0, "content": ". 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, "page_num": "page_8", "page_size": [612.0, 792.0], "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. 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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [69, 631, 541, 689]}]}
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|
3. Data for the Affine Algebras.
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.
# 3.1. The algebra ,
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 .
The Kac-Peterson relation (2.1b) for 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 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 .
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 and , will be useful.
There are no exceptional fusion-symmetries for :
Theorem 3.A. The fusion-symmetries for level are , for and any integer for which is coprime to .
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 .
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 .
|
<div class="pdf-page">
<p>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.</p>
<h1>3.1. The algebra ,</h1>
<p>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 .</p>
<p>The Kac-Peterson relation (2.1b) for takes the form</p>
<p>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 .</p>
<p>The fusion (derived from the Pieri rule and (2.4))</p>
<p>valid for and , will be useful.</p>
<p>There are no exceptional fusion-symmetries for :</p>
<p>Theorem 3.A. The fusion-symmetries for level are , for and any integer for which is coprime to .</p>
<p>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 .</p>
<p>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 .</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="334" data-y="91" data-width="352" data-height="18" style="opacity: 0.5;">3. Data for the Affine Algebras.</div>
<p class="pdf-text" data-x="117" data-y="128" data-width="789" data-height="118">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.</p>
<h1 class="pdf-title" data-x="118" data-y="262" data-width="246" data-height="23">3.1. The algebra ,</h1>
<p class="pdf-text" data-x="115" data-y="294" data-width="790" data-height="123">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 .</p>
<p class="pdf-text" data-x="158" data-y="418" data-width="501" data-height="20">The Kac-Peterson relation (2.1b) for takes the form</p>
<p class="pdf-text" data-x="117" data-y="510" data-width="786" data-height="57">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 .</p>
<p class="pdf-text" data-x="157" data-y="567" data-width="440" data-height="18">The fusion (derived from the Pieri rule and (2.4))</p>
<p class="pdf-text" data-x="118" data-y="641" data-width="390" data-height="18">valid for and , will be useful.</p>
<p class="pdf-text" data-x="158" data-y="661" data-width="462" data-height="20">There are no exceptional fusion-symmetries for :</p>
<p class="pdf-text" data-x="117" data-y="689" data-width="789" data-height="41">Theorem 3.A. The fusion-symmetries for level are , for and any integer for which is coprime to .</p>
<p class="pdf-text" data-x="117" data-y="738" data-width="788" data-height="73">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 .</p>
<p class="pdf-text" data-x="117" data-y="814" data-width="788" data-height="72">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 .</p>
</div>
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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 parti-",
"tion (\\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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"\\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}]\\,\\ \\",
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"3.2. The algebra ,",
"A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current: . It has .",
"The only fusion products we need are",
"for all , , and , where we drop if . We will also use the character formula (2.1b)",
"",
"where and",
"",
"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 ).",
"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.",
"Theorem 3.B. The fusion-symmetries of level for are where . For a fusion-symmetry will equal for and , .",
"When , is trivial. We have",
"3.3. The algebra ,",
"A weight of satisfies and . Charge-conjugation again is trivial, and there is a simple-current defined by , with .",
"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",
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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 \u2014 one for each Galois automorphism, since S020 = 41\u03ba 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] \u2014 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\u2019s 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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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 \u2014 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 = 41\u03ba ", "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] \u2014 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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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 \u2014 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 = 41\u03ba ", "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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] \u2014 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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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\u2019s 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, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [71, 683, 541, 719]}]}
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3.2. The algebra ,
A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current: . It has .
The only fusion products we need are
for all , , and , where we drop if . 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 and
$$
\lambda(\ell)=\sum_{i=\ell}^{r-1}\lambda_{i}+\frac{1}{2}\lambda_{r}\ .
$$
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 ).
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.
Theorem 3.B. The fusion-symmetries of level for are where . For a fusion-symmetry will equal for and , .
When , is trivial. We have
3.3. The algebra ,
A weight of satisfies and . Charge-conjugation again is trivial, and there is a simple-current defined by , with .
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
10
|
<div class="pdf-page">
<p>3.2. The algebra ,</p>
<p>A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current: . It has .</p>
<p>The only fusion products we need are</p>
<p>for all , , and , where we drop if . We will also use the character formula (2.1b)</p>
<p>where and</p>
<p>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 ).</p>
<p>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.</p>
<p>Theorem 3.B. The fusion-symmetries of level for are where . For a fusion-symmetry will equal for and , .</p>
<p>When , is trivial. We have</p>
<p>3.3. The algebra ,</p>
<p>A weight of satisfies and . Charge-conjugation again is trivial, and there is a simple-current defined by , with .</p>
<p>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</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="89" data-width="252" data-height="22">3.2. The algebra ,</p>
<p class="pdf-text" data-x="117" data-y="118" data-width="789" data-height="57">A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current: . It has .</p>
<p class="pdf-text" data-x="157" data-y="177" data-width="334" data-height="18">The only fusion products we need are</p>
<p class="pdf-text" data-x="115" data-y="265" data-width="790" data-height="38">for all , , and , where we drop if . We will also use the character formula (2.1b)</p>
<p class="pdf-text" data-x="117" data-y="381" data-width="262" data-height="19">where and</p>
<p class="pdf-text" data-x="117" data-y="456" data-width="788" data-height="94">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 ).</p>
<p class="pdf-text" data-x="117" data-y="550" data-width="788" data-height="135">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.</p>
<p class="pdf-text" data-x="117" data-y="691" data-width="789" data-height="60">Theorem 3.B. The fusion-symmetries of level for are where . For a fusion-symmetry will equal for and , .</p>
<p class="pdf-text" data-x="155" data-y="757" data-width="599" data-height="23">When , is trivial. We have</p>
<p class="pdf-text" data-x="118" data-y="796" data-width="250" data-height="22">3.3. The algebra ,</p>
<p class="pdf-text" data-x="118" data-y="826" data-width="787" data-height="57">A weight of satisfies and . Charge-conjugation again is trivial, and there is a simple-current defined by , with .</p>
<p class="pdf-text" data-x="117" data-y="883" data-width="786" data-height="42">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</p>
<div class="pdf-discarded" data-x="500" data-y="943" data-width="23" data-height="16" style="opacity: 0.5;">10</div>
</div>
|
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 $\begin{array}{r}{Q(\lambda)=(\sum_{j=1}^{r}j\lambda_{j})/2}\end{array}$ .
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"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",
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"\\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~,",
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"\\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",
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"",
"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: .",
"The only fusion product we need is",
"",
"valid for and . The following character formula (2.1b) will also be used:",
"",
"where as before.",
"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 .",
"When is even, .",
"3.4. The algebra",
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[{"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 \u00a75). 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\u2019s 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 \u00a75). 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": ". 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}, {"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": ". 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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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [69, 639, 541, 672]}]}
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|
$$
\tilde{S}_{\tau\lambda,\tau\mu}=S_{\lambda\mu}\ .
$$
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: .
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 and . 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 as before.
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 .
When is even, .
3.4. The algebra
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 .
There are three non-trivial simple-currents, , and . Explicitly, we have with , 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 . From this we compute . The fusion products we need are
11
|
<div class="pdf-page">
<p>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: .</p>
<p>The only fusion product we need is</p>
<p>valid for and . The following character formula (2.1b) will also be used:</p>
<p>where as before.</p>
<p>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 .</p>
<p>When is even, .</p>
<p>3.4. The algebra</p>
<p>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 .</p>
<p>There are three non-trivial simple-currents, , and . Explicitly, we have with , and</p>
<p>with . From this we compute . The fusion products we need are</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="204" data-width="788" data-height="61">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: .</p>
<p class="pdf-text" data-x="158" data-y="267" data-width="313" data-height="18">The only fusion product we need is</p>
<p class="pdf-text" data-x="118" data-y="338" data-width="726" data-height="20">valid for and . The following character formula (2.1b) will also be used:</p>
<p class="pdf-text" data-x="117" data-y="438" data-width="314" data-height="20">where as before.</p>
<p class="pdf-text" data-x="117" data-y="466" data-width="789" data-height="78">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 .</p>
<p class="pdf-text" data-x="157" data-y="552" data-width="355" data-height="20">When is even, .</p>
<p class="pdf-text" data-x="118" data-y="589" data-width="251" data-height="23">3.4. The algebra</p>
<p class="pdf-text" data-x="117" data-y="619" data-width="788" data-height="97">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 .</p>
<p class="pdf-text" data-x="117" data-y="716" data-width="788" data-height="39">There are three non-trivial simple-currents, , and . Explicitly, we have with , and</p>
<p class="pdf-text" data-x="115" data-y="826" data-width="790" data-height="38">with . From this we compute . The fusion products we need are</p>
<div class="pdf-discarded" data-x="500" data-y="943" data-width="22" data-height="16" style="opacity: 0.5;">11</div>
</div>
|
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
$$
J_{s}\lambda={\left\{\begin{array}{l l}{(\lambda_{r},\lambda_{r-1},\lambda_{r-2},\ldots,\lambda_{1},\lambda_{0})}&{{\mathrm{if~}}r{\mathrm{~is~even}},}\\ {(\lambda_{r-1},\lambda_{r},\lambda_{r-2},\ldots,\lambda_{1},\lambda_{0})}&{{\mathrm{if~}}r{\mathrm{~is~odd}},}\end{array}\right.}
$$
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"\\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 iso-",
"morphism \\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"
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"valid for all and . We also will use the character formula (2.1b)",
"",
"where and the orthonormal components are defined by .",
"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 .",
"When is even, the simple-currents are generated by both and . If in addition is even, we have 16 simple-current automorphisms:",
"",
"for any , forming a group isomorphic to . This notation means",
"",
"When is odd, we will have six simple-current automorphisms:",
"",
"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).",
"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.)",
"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 , .",
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[{"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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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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [72, 684, 541, 718]}]}
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valid for all and . 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 and the orthonormal components are defined by .
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 .
When is even, the simple-currents are generated by both and . If in addition 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 , forming a group isomorphic to . 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 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 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).
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.)
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 , .
12
|
<div class="pdf-page">
<p>valid for all and . We also will use the character formula (2.1b)</p>
<p>where and the orthonormal components are defined by .</p>
<p>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 .</p>
<p>When is even, the simple-currents are generated by both and . If in addition is even, we have 16 simple-current automorphisms:</p>
<p>for any , forming a group isomorphic to . This notation means</p>
<p>When is odd, we will have six simple-current automorphisms:</p>
<p>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).</p>
<p>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.)</p>
<p>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 , .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="90" data-width="727" data-height="21">valid for all and . We also will use the character formula (2.1b)</p>
<p class="pdf-text" data-x="117" data-y="190" data-width="788" data-height="40">where and the orthonormal components are defined by .</p>
<p class="pdf-text" data-x="117" data-y="230" data-width="789" data-height="111">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 .</p>
<p class="pdf-text" data-x="117" data-y="341" data-width="788" data-height="37">When is even, the simple-currents are generated by both and . If in addition is even, we have 16 simple-current automorphisms:</p>
<p class="pdf-text" data-x="115" data-y="447" data-width="713" data-height="21">for any , forming a group isomorphic to . This notation means</p>
<p class="pdf-text" data-x="117" data-y="536" data-width="563" data-height="19">When is odd, we will have six simple-current automorphisms:</p>
<p class="pdf-text" data-x="115" data-y="667" data-width="790" data-height="75">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).</p>
<p class="pdf-text" data-x="117" data-y="742" data-width="789" data-height="134">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.)</p>
<p class="pdf-text" data-x="117" data-y="884" data-width="788" data-height="41">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 , .</p>
<div class="pdf-discarded" data-x="500" data-y="943" data-width="22" data-height="16" style="opacity: 0.5;">12</div>
</div>
|
valid for $k\geq2$ and $1\leq\ell<r$ , will be useful.
There are no exceptional fusion-symmetries for $A_{r}^{(1)}$ :
|
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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 ."
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"\\chi_{\\Lambda_{1}}[\\lambda]={\\frac{S_{\\Lambda_{1}\\lambda}}{S_{0\\lambda}}}=2\\sum_{\\ell=1}^{r}\\cos(2\\pi{\\frac{\\lambda^{+}(\\ell)}{\\kappa}})~,",
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"{\\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}},"
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"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",
"3.5. The algebra",
"A weight of satisfies and . The charge-conjugation acts as . The order 3 simple-current is given by with .",
"The fusion products we need can be derived from [29] using (2.4):",
"",
"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).",
"Theorem 3.E6. The fusion-symmetries of are , for any and any for which (mod 3).",
"3.6. The algebra E7(1)",
"A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current given by . It has .",
"The only fusion products we need can be obtained from [29] and (2.4):",
"× + + + × + + × + + + + × + + + × + + + + + × + + + × + + + + + + +",
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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": "$\\Lambda_{6}$ \u00d7 $\\Lambda_{6}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{5})_{2}$ + $(2\\Lambda_{6})_{2}$ $\\Lambda_{1}$ \u00d7 $\\Lambda_{6}=(\\Lambda_{6})_{2}$ + $(\\Lambda_{7})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ $\\Lambda_{5}$ \u00d7 $\\Lambda_{6}=(\\Lambda_{4})_{3}$ + $(\\Lambda_{6})_{2}$ + $(\\Lambda_{7})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ + $(\\Lambda_{5}+\\Lambda_{6})_{3}$ $\\Lambda_{6}$ \u00d7 $(2\\Lambda_{6})=(\\Lambda_{6})_{2}$ + $(\\Lambda_{1}+\\Lambda_{6})_{3}$ + $(3\\Lambda_{6})_{3}$ + $(\\Lambda_{5}+\\Lambda_{6})_{3}$ $\\Lambda_{4}$ \u00d7 $\\Lambda_{6}=(\\Lambda_{2})_{3}$ + $(\\Lambda_{3})_{4}$ + $(\\Lambda_{5})_{3}$ + $(\\Lambda_{1}+\\Lambda_{5})_{4}$ + $(\\Lambda_{4}+\\Lambda_{6})_{4}$ + $(\\Lambda_{6}+\\Lambda_{7})_{3}$ $\\Lambda_{6}$ \u00d7 $\\Lambda_{7}=(\\Lambda_{1})_{2}$ + $(\\Lambda_{2})_{3}$ + $(\\Lambda_{5})_{2}$ + $(\\Lambda_{6}+\\Lambda_{7})_{3}$ $\\Lambda_{6}$ \u00d7 $(\\Lambda_{5}+\\Lambda_{6})=(\\Lambda_{5})_{3}$ + $(2\\Lambda_{5})_{4}$ + $(2\\Lambda_{6})_{3}$ + $(\\Lambda_{6}+\\Lambda_{7})_{3}$ + $(\\Lambda_{1}+\\Lambda_{5})_{4}$ + $(\\Lambda_{4}+\\Lambda_{6})_{4}$ + $(\\Lambda_{1}+2\\Lambda_{6})_{4}$ + $(\\Lambda_{5}+2\\Lambda_{6})_{4}$ ", "page_idx": 12}]
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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": " \u00d7", "type": "text"}, {"bbox": [152, 575, 202, 591], "score": 0.93, "content": "\\Lambda_{6}=(0)_{1}", "type": "inline_equation", 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"index": 3}, {"bbox": [71, 147, 540, 166], "spans": [{"bbox": [71, 148, 200, 163], "score": 0.92, "content": "A(D_{o d d,1})\\:=\\:\\langle{C_{1}}\\rangle\\:\\cong\\:\\mathbb{Z}_{2}", "type": "inline_equation", "height": 15, "width": 129}, {"bbox": [201, 148, 248, 166], "score": 1.0, "content": ". When ", "type": "text"}, {"bbox": [248, 149, 282, 161], "score": 0.85, "content": "r\\,>\\,4", "type": "inline_equation", "height": 12, "width": 34}, {"bbox": [282, 148, 290, 166], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [290, 147, 473, 163], "score": 0.91, "content": "{\\cal A}(D_{r,2})\\,\\cong\\,(\\mathbb{Z}_{2r}^{\\times}/\\{\\pm1\\})\\,\\times\\,\\mathbb{Z}_{2}\\,\\times\\,\\mathbb{Z}_{2}", "type": "inline_equation", "height": 16, "width": 183}, {"bbox": [473, 148, 492, 166], "score": 1.0, "content": " or ", "type": "text"}, {"bbox": [493, 148, 540, 162], "score": 0.91, "content": "\\mathbb{Z}_{r}^{\\times}\\times\\mathbb{Z}_{2}", "type": "inline_equation", "height": 14, "width": 47}], "index": 4}, {"bbox": [70, 163, 541, 178], "spans": [{"bbox": [70, 163, 90, 178], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [90, 166, 97, 174], "score": 0.65, "content": "r", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [97, 163, 162, 178], "score": 1.0, "content": " even/odd. ", "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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], 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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
# 3.5. The algebra
A weight of satisfies and . The charge-conjugation acts as . The order 3 simple-current is given by with .
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 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).
Theorem 3.E6. The fusion-symmetries of are , for any and any for which (mod 3).
# 3.6. The algebra E7(1)
A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current given by . It has .
The only fusion products we need can be obtained from [29] and (2.4):
× + + + × + + × + + + + × + + + × + + + + + × + + + × + + + + + + +
13
|
<div class="pdf-page">
<p>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</p>
<h1>3.5. The algebra</h1>
<p>A weight of satisfies and . The charge-conjugation acts as . The order 3 simple-current is given by with .</p>
<p>The fusion products we need can be derived from [29] using (2.4):</p>
<p>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).</p>
<p>Theorem 3.E6. The fusion-symmetries of are , for any and any for which (mod 3).</p>
<h1>3.6. The algebra E7(1)</h1>
<p>A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current given by . It has .</p>
<p>The only fusion products we need can be obtained from [29] and (2.4):</p>
<p>× + + + × + + × + + + + × + + + × + + + + + × + + + × + + + + + + +</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="137" data-width="788" data-height="126">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</p>
<h1 class="pdf-title" data-x="117" data-y="279" data-width="189" data-height="20">3.5. The algebra</h1>
<p class="pdf-text" data-x="117" data-y="306" data-width="788" data-height="56">A weight of satisfies and . The charge-conjugation acts as . The order 3 simple-current is given by with .</p>
<p class="pdf-text" data-x="160" data-y="363" data-width="581" data-height="18">The fusion products we need can be derived from [29] using (2.4):</p>
<p class="pdf-text" data-x="117" data-y="480" data-width="788" data-height="95">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).</p>
<p class="pdf-text" data-x="118" data-y="581" data-width="788" data-height="42">Theorem 3.E6. The fusion-symmetries of are , for any and any for which (mod 3).</p>
<h1 class="pdf-title" data-x="118" data-y="634" data-width="188" data-height="21">3.6. The algebra E7(1)</h1>
<p class="pdf-text" data-x="117" data-y="661" data-width="789" data-height="57">A weight in satisfies , and . The charge-conjugation is trivial, but there is a simple-current given by . It has .</p>
<p class="pdf-text" data-x="157" data-y="718" data-width="626" data-height="18">The only fusion products we need can be obtained from [29] and (2.4):</p>
<p class="pdf-text" data-x="137" data-y="740" data-width="749" data-height="185">× + + + × + + × + + + + × + + + × + + + + + × + + + × + + + + + + +</p>
<div class="pdf-discarded" data-x="500" data-y="945" data-width="22" data-height="13" style="opacity: 0.5;">13</div>
</div>
|
$\pi_{v}$ 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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"content": [
"At there is an order 3 Galois fusion-symmetry , which sends and fixes the other six weights.",
"Theorem 3.E7. The only nontrivial fusion-symmetries for are at even , as well as and its inverse at .",
"3.7. The algebra E8(1)",
"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).",
"The only fusion products we need can be derived from [28] and (2.4):",
"+ + + + + + × + + + + + + × + + + + × + + + + +",
"",
"",
"",
"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 .",
"Theorem 3.E8. The only nontrivial fusion-symmetries for are and , oc- curring at and 5 respectively.",
"3.8. The algebra F 4(1)",
"A weight in satisfies , and . Again, the conjugations and simple-currents are trivial.",
"14"
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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": "$(2\\Lambda_{1})\\vert\\mathrm{\\bf\\sfXI}(2\\Lambda_{1})=(0)_{4}$ + $(\\Lambda_{1})_{5}$ + $(\\Lambda_{2})_{5}$ + $(\\Lambda_{3})_{4}$ + $(\\Lambda_{7})_{4}$ + $2\\,\\mathbf{E}\\left(2\\Lambda_{1}\\right)_{46}$ + $(2\\Lambda_{2})_{6}$ $\\Lambda_{1}$ \u00d7 $\\Lambda_{4}=(\\Lambda_{3})_{5}$ + $(\\Lambda_{4})_{6}$ + $(\\Lambda_{5})_{6}$ + $(\\Lambda_{6})_{5}$ + $(\\Lambda_{1}+\\Lambda_{3})_{6}$ + $(\\Lambda_{1}+\\Lambda_{4})_{7}$ + $(\\Lambda_{1}+\\Lambda_{6})_{6}$ $\\Lambda_{1}$ \u00d7 $(\\Lambda_{1}\\!+\\!\\Lambda_{3})=(\\Lambda_{3})_{6}$ + $(\\Lambda_{4})_{6}$ + $(\\Lambda_{1}+\\Lambda_{2})_{6}$ + $2\\,\\mathtt{H}\\,(\\Lambda_{1}+\\Lambda_{3})_{67}$ + $(\\Lambda_{1}+\\Lambda_{4})_{7}$ $\\Lambda_{1}$ \u00d7 $(2\\Lambda_{7})=\\!(\\Lambda_{6})_{4}$ + $(\\Lambda_{1}+\\Lambda_{7})_{4}$ + $(2\\Lambda_{7})_{5}$ + $(\\Lambda_{2}+\\Lambda_{7})_{5}$ + $(\\Lambda_{7}+\\Lambda_{8})_{5}$ + $(\\Lambda_{1}+2\\Lambda_{7})_{6}$ $(3.7g)$ ", "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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"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, "page_num": "page_13", "page_size": [612.0, 792.0], "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. 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, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [72, 108, 540, 139]}, {"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. The algebra E8(1)", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 173, 542, 230], "lines": [{"bbox": [95, 176, 541, 190], "spans": [{"bbox": [95, 176, 144, 190], "score": 1.0, "content": "A weight ", "type": "text"}, {"bbox": [144, 177, 153, 186], "score": 0.81, "content": "\\lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [153, 176, 168, 190], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [168, 176, 185, 189], "score": 0.9, "content": "P_{+}", "type": "inline_equation", "height": 13, "width": 17}, {"bbox": [185, 176, 231, 190], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [231, 176, 514, 188], "score": 0.85, "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}", "type": "inline_equation", "height": 12, "width": 283}, {"bbox": [514, 176, 541, 190], "score": 1.0, "content": ", and", "type": "text"}], "index": 5}, {"bbox": [71, 190, 541, 205], "spans": [{"bbox": [71, 190, 128, 202], "score": 0.89, "content": "\\kappa=k+30", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [128, 190, 541, 205], "score": 1.0, "content": ". 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": 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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, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [69, 556, 541, 615]}, {"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, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [72, 619, 540, 653]}, {"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, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [72, 688, 541, 717]}]}
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At there is an order 3 Galois fusion-symmetry , which sends and fixes the other six weights.
Theorem 3.E7. The only nontrivial fusion-symmetries for are at even , as well as and its inverse at .
# 3.7. The algebra E8(1)
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).
The only fusion products we need can be derived from [28] and (2.4):
+ + + + + + × + + + + + + × + + + + × + + + + +
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 .
Theorem 3.E8. The only nontrivial fusion-symmetries for are and , oc- curring at and 5 respectively.
# 3.8. The algebra F 4(1)
A weight in satisfies , and . Again, the conjugations and simple-currents are trivial.
14
|
<div class="pdf-page">
<p>At there is an order 3 Galois fusion-symmetry , which sends and fixes the other six weights.</p>
<p>Theorem 3.E7. The only nontrivial fusion-symmetries for are at even , as well as and its inverse at .</p>
<h1>3.7. The algebra E8(1)</h1>
<p>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).</p>
<p>The only fusion products we need can be derived from [28] and (2.4):</p>
<p>+ + + + + + × + + + + + + × + + + + × + + + + +</p>
<p>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 .</p>
<p>Theorem 3.E8. The only nontrivial fusion-symmetries for are and , oc- curring at and 5 respectively.</p>
<h1>3.8. The algebra F 4(1)</h1>
<p>A weight in satisfies , and . Again, the conjugations and simple-currents are trivial.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="118" data-y="90" data-width="787" data-height="39">At there is an order 3 Galois fusion-symmetry , which sends and fixes the other six weights.</p>
<p class="pdf-text" data-x="118" data-y="138" data-width="787" data-height="39">Theorem 3.E7. The only nontrivial fusion-symmetries for are at even , as well as and its inverse at .</p>
<h1 class="pdf-title" data-x="117" data-y="193" data-width="189" data-height="22">3.7. The algebra E8(1)</h1>
<p class="pdf-text" data-x="117" data-y="223" data-width="789" data-height="74">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).</p>
<p class="pdf-text" data-x="157" data-y="297" data-width="612" data-height="19">The only fusion products we need can be derived from [28] and (2.4):</p>
<p class="pdf-text" data-x="78" data-y="470" data-width="706" data-height="22">+ + + + + + × + + + + + + × + + + + × + + + + +</p>
<p class="pdf-text" data-x="117" data-y="716" data-width="789" data-height="76">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 .</p>
<p class="pdf-text" data-x="117" data-y="799" data-width="788" data-height="41">Theorem 3.E8. The only nontrivial fusion-symmetries for are and , oc- curring at and 5 respectively.</p>
<h1 class="pdf-title" data-x="118" data-y="857" data-width="188" data-height="22">3.8. The algebra F 4(1)</h1>
<p class="pdf-text" data-x="117" data-y="886" data-width="788" data-height="38">A weight in satisfies , and . Again, the conjugations and simple-currents are trivial.</p>
<div class="pdf-discarded" data-x="500" data-y="945" data-width="22" data-height="14" style="opacity: 0.5;">14</div>
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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",
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"The only fusion products we need can be derived from [28] and (2.4):",
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"\\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}",
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"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",
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"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} , oc-",
"curring at k=4 and 5 respectively.",
"3.8. The algebra F 4(1)",
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"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 .",
"The only fusion products we need can be obtained from [29] and (2.4):",
"× + + + + × + + × + + + + + + × + + + + +",
"Theorem 3.F4. The only nontrivial fusion-symmetries of are at level 3, and for , which occur at level 4.",
"3.9. The algebra",
"A weight in satisfies , and . The conjugations and simple-currents are all trivial.",
"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 .",
"The only fusion products we will need can be obtained from [29] and (2.4):",
"× + + + × × + + + + +",
"Theorem 3.G2. The only nontrivial fusion-symmetries for are at , and at .",
"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 preserves q-dimensions: . In this subsection we use that to find a weight for each algebra which must be essentially fixed by .",
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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": "$\\Lambda_{4}$ \u00d7 $\\Lambda_{4}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{1}$ + $(2\\Lambda_{4})_{2}$ $\\Lambda_{1}$ \u00d7 $\\Lambda_{4}=(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ $\\Lambda_{3}$ \u00d7 $\\Lambda_{4}=(\\Lambda_{1})_{2}$ + $(\\Lambda_{2})_{3}$ + $(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ + $(\\Lambda_{3}+\\Lambda_{4})_{3}$ + $(2\\Lambda_{4})_{2}$ $(2\\Lambda_{4})$ \u00d7 $\\Lambda_{4}=(\\Lambda_{3})_{2}$ + $(\\Lambda_{4})_{2}$ + $(2\\Lambda_{4})_{2}$ + $(3\\Lambda_{4})_{3}$ + $(\\Lambda_{1}+\\Lambda_{4})_{3}$ + $(\\Lambda_{3}+\\Lambda_{4})_{3}$ ", "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": "$\\Lambda_{2}$ \u00d7 $\\Lambda_{2}=(0)_{1}$ + $(\\Lambda_{1})_{2}$ + $(\\Lambda_{2})_{1}$ + $(2\\Lambda_{2})_{2}$ $\\Lambda_{2}$ \u00d7 $\\Lambda_{2}$ \u00d7 $\\Lambda_{2}=(0)_{1}$ + $2\\,\\pmb{\\nabla}\\,(\\Lambda_{1})_{22}$ + $4\\,\\pmb{\\mathrm{{E}}}\\left(\\Lambda_{2}\\right)_{1122}$ + $3\\,\\pm\\,(2\\Lambda_{2})_{222}$ + $2\\,\\Xi\\,(\\Lambda_{1}+\\Lambda_{2})_{33}$ + $(3\\Lambda_{2})_{3}$ ", "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 \u2014 see e.g. [22,28] \u2014 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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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": " \u00d7", "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, 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\u00d7", "type": "text"}, {"bbox": [128, 254, 187, 270], "score": 0.93, "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 59}, {"bbox": [187, 254, 204, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [204, 254, 234, 270], "score": 0.91, "content": "(\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 30}, {"bbox": [234, 254, 251, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [251, 254, 286, 270], "score": 0.91, "content": "(2\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [286, 254, 303, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [304, 254, 339, 270], "score": 0.91, "content": "(3\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [339, 254, 355, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [356, 254, 414, 270], "score": 0.9, "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}, {"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": " \u00d7 ", "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": " \u00d7", "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": " \u00d7", "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 \u2014 see e.g.", "type": "text"}], "index": 30}, {"bbox": [71, 656, 401, 673], "spans": [{"bbox": [71, 656, 401, 673], "score": 1.0, "content": "[22,28] \u2014 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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], 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"score": 0.88, "content": "(2\\Lambda_{4})", "type": "inline_equation", "height": 16, "width": 29}, {"bbox": [112, 254, 128, 271], "score": 1.0, "content": " \u00d7", "type": "text"}, {"bbox": [128, 254, 187, 270], "score": 0.93, "content": "\\Lambda_{4}=(\\Lambda_{3})_{2}", "type": "inline_equation", "height": 16, "width": 59}, {"bbox": [187, 254, 204, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [204, 254, 234, 270], "score": 0.91, "content": "(\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 30}, {"bbox": [234, 254, 251, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [251, 254, 286, 270], "score": 0.91, "content": "(2\\Lambda_{4})_{2}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [286, 254, 303, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [304, 254, 339, 270], "score": 0.91, "content": "(3\\Lambda_{4})_{3}", "type": "inline_equation", "height": 16, "width": 35}, {"bbox": [339, 254, 355, 271], "score": 1.0, "content": " + ", "type": "text"}, {"bbox": [356, 254, 414, 270], "score": 0.9, "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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": " \u00d7 ", "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": " \u00d7", "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": " \u00d7", "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"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 \u2014 see e.g.", "type": "text"}], "index": 30}, {"bbox": [71, 656, 401, 673], "spans": [{"bbox": [71, 656, 401, 673], "score": 1.0, "content": "[22,28] \u2014 and the resulting combinatorics is often quite pretty.", "type": "text"}], "index": 31}], "index": 29.5, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [70, 671, 540, 715]}]}
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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 .
The only fusion products we need can be obtained from [29] and (2.4):
× + + + + × + + × + + + + + + × + + + + +
Theorem 3.F4. The only nontrivial fusion-symmetries of are at level 3, and for , which occur at level 4.
# 3.9. The algebra
A weight in satisfies , and . The conjugations and simple-currents are all trivial.
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 .
The only fusion products we will need can be obtained from [29] and (2.4):
× + + + × × + + + + +
Theorem 3.G2. The only nontrivial fusion-symmetries for are at , and at .
# 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 preserves q-dimensions: . In this subsection we use that to find a weight for each algebra which must be essentially fixed by .
15
|
<div class="pdf-page">
<p>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 .</p>
<p>The only fusion products we need can be obtained from [29] and (2.4):</p>
<p>× + + + + × + + × + + + + + + × + + + + +</p>
<p>Theorem 3.F4. The only nontrivial fusion-symmetries of are at level 3, and for , which occur at level 4.</p>
<h1>3.9. The algebra</h1>
<p>A weight in satisfies , and . The conjugations and simple-currents are all trivial.</p>
<p>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 .</p>
<p>The only fusion products we will need can be obtained from [29] and (2.4):</p>
<p>× + + + × × + + + + +</p>
<p>Theorem 3.G2. The only nontrivial fusion-symmetries for are at , and at .</p>
<h1>4. The Arguments</h1>
<p>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>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 .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="90" data-width="788" data-height="132">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 .</p>
<p class="pdf-text" data-x="158" data-y="222" data-width="623" data-height="18">The only fusion products we need can be obtained from [29] and (2.4):</p>
<p class="pdf-text" data-x="143" data-y="254" data-width="745" data-height="95">× + + + + × + + × + + + + + + × + + + + +</p>
<p class="pdf-text" data-x="118" data-y="364" data-width="788" data-height="39">Theorem 3.F4. The only nontrivial fusion-symmetries of are at level 3, and for , which occur at level 4.</p>
<h1 class="pdf-title" data-x="118" data-y="421" data-width="188" data-height="22">3.9. The algebra</h1>
<p class="pdf-text" data-x="117" data-y="452" data-width="788" data-height="36">A weight in satisfies , and . The conjugations and simple-currents are all trivial.</p>
<p class="pdf-text" data-x="117" data-y="490" data-width="788" data-height="93">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 .</p>
<p class="pdf-text" data-x="158" data-y="583" data-width="661" data-height="19">The only fusion products we will need can be obtained from [29] and (2.4):</p>
<p class="pdf-text" data-x="110" data-y="616" data-width="801" data-height="47">× + + + × × + + + + +</p>
<p class="pdf-text" data-x="117" data-y="678" data-width="789" data-height="40">Theorem 3.G2. The only nontrivial fusion-symmetries for are at , and at .</p>
<h1 class="pdf-title" data-x="416" data-y="753" data-width="189" data-height="20">4. The Arguments</h1>
<p class="pdf-text" data-x="117" data-y="791" data-width="788" data-height="73">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 class="pdf-text" data-x="115" data-y="866" data-width="788" data-height="55">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 .</p>
<div class="pdf-discarded" data-x="500" data-y="943" data-width="22" data-height="16" style="opacity: 0.5;">15</div>
</div>
|
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.
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"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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"4.1. -dimensions",
"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.",
"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 :",
"",
"Choose any . Then a straightforward calculation from (2.1c) gives",
"",
"for . This means that for all ,",
"",
"Proposition 4.1 [17,18]. For the following algebras and levels , and choices of weight , implies :",
"(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 .",
"The missing cases are: where where ; where , and its rank-level dual ; where ; where , and where ; where , and where ; where , and where .",
"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.",
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[{"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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Then a straightforward calculation from (2.1c) gives", "type": "text"}], "index": 7}], "index": 7, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [71, 231, 467, 246]}, {"type": "interline_equation", "bbox": [159, 257, 453, 286], "lines": [{"bbox": [159, 257, 453, 286], "spans": [{"bbox": [159, 257, 453, 286], "score": 0.92, "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", "type": "interline_equation"}], "index": 8}], "index": 8, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 296, 324, 311], "lines": [{"bbox": [70, 298, 324, 313], "spans": [{"bbox": [70, 298, 89, 313], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [90, 301, 138, 309], "score": 0.91, "content": "0<t<1", "type": "inline_equation", "height": 8, "width": 48}, {"bbox": [138, 298, 271, 313], "score": 1.0, "content": ". 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, "page_num": "page_15", "page_size": [612.0, 792.0], "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, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"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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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, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [69, 654, 543, 718]}]}
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|
4.1. -dimensions
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.
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 :
$$
C_{+}\stackrel{\mathrm{def}}{=}\{\sum_{i=0}^{r}x_{i}\Lambda_{i}\left|\right.x_{i}\in\mathbb{R},\ x_{i}>-1,\ \sum_{i=0}^{r}x_{i}a_{i}^{\vee}=k\}\ .
$$
Choose any . 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 . This means that for all ,
$$
{\mathcal{D}}(t a+(1-t)b)>\operatorname*{min}\{{\mathcal{D}}(a),\,{\mathcal{D}}(b)\}~.
$$
Proposition 4.1 [17,18]. For the following algebras and levels , and choices of weight , implies :
(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 .
- The missing cases are: where where ; where , and its rank-level dual ; where ; where , and where ; where , and where ; where , and where .
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.
16
|
<div class="pdf-page">
<p>4.1. -dimensions</p>
<p>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.</p>
<p>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 :</p>
<p>Choose any . Then a straightforward calculation from (2.1c) gives</p>
<p>for . This means that for all ,</p>
<p>Proposition 4.1 [17,18]. For the following algebras and levels , and choices of weight , implies :</p>
<p>(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 .</p>
<ul>
<li>The missing cases are: where where ; where , and its rank-level dual ; where ; where , and where ; where , and where ; where , and where .</li>
</ul>
<p>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.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="118" data-y="91" data-width="161" data-height="20">4.1. -dimensions</p>
<p class="pdf-text" data-x="117" data-y="118" data-width="788" data-height="60">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.</p>
<p class="pdf-text" data-x="117" data-y="178" data-width="788" data-height="37">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 :</p>
<p class="pdf-text" data-x="117" data-y="294" data-width="666" data-height="21">Choose any . Then a straightforward calculation from (2.1c) gives</p>
<p class="pdf-text" data-x="117" data-y="382" data-width="425" data-height="20">for . This means that for all ,</p>
<p class="pdf-text" data-x="118" data-y="465" data-width="788" data-height="39">Proposition 4.1 [17,18]. For the following algebras and levels , and choices of weight , implies :</p>
<p class="pdf-text" data-x="117" data-y="506" data-width="684" data-height="194">(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 .</p>
<ul class="pdf-list" data-x="155" data-y="707" data-width="716" data-height="133">
<li>The missing cases are: where where ; where , and its rank-level dual ; where ; where , and where ; where , and where ; where , and where .</li>
</ul>
<p class="pdf-text" data-x="117" data-y="845" data-width="788" data-height="79">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.</p>
<div class="pdf-discarded" data-x="500" data-y="945" data-width="22" data-height="14" style="opacity: 0.5;">16</div>
</div>
|
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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 observa-",
"tion. 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."
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"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\\}\\ .",
"\\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",
"{\\mathcal{D}}(t a+(1-t)b)>\\operatorname*{min}\\{{\\mathcal{D}}(a),\\,{\\mathcal{D}}(b)\\}~."
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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 and the most difficult cases.",
"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).",
"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.",
"4.2. The -series argument",
"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 .",
"If then so . Thus we can assume .",
"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.",
"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 .",
"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 .",
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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]) \u2014 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 $\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\Lambda_{1}$ ( $k$ times). Thus the only nontrivial simple-current appearing in the fusion $\\pi\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\pi\\Lambda_{1}$ 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 $\\Lambda_{1}$ \u00d7 $\\Lambda_{1}$ 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\u2019t in the tensor (hence fusion) product of $\\Lambda_{2}$ with $k-2~\\Lambda_{1}$ \u2019s (recall that $k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}$ in the usual partial order on weights). Since $\\Lambda_{2}$ \u00d7 $\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\Lambda_{1}$ 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 \u00a72.2) we know that $\\pi$ must fix everything in $P_{+}$ . ", "page_idx": 16}]
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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]) \u2014 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": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "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": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "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": " \u00d7", "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\u2019t 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": "\u2019s (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": " \u00d7", "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": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7", "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 \u00a72.2) 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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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]) \u2014 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": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "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": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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": " \u00d7", "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\u2019t 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": "\u2019s (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": " \u00d7", "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": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7", "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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 \u00a72.2) 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, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [70, 632, 543, 707]}]}
|
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0.492}, {"category_id": 13, "poly": [378, 589, 505, 589, 505, 639, 378, 639], "score": 0.95, "latex": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})"}, {"category_id": 13, "poly": [465, 1196, 674, 1196, 674, 1232, 465, 1232], "score": 0.95, "latex": "h(\\lambda)=\\prod h(x)"}, {"category_id": 13, "poly": [825, 677, 951, 677, 951, 726, 825, 726], "score": 0.95, "latex": "\\textstyle\\mathcal{D}(\\frac{k}{a_{i}^{\\vee}}\\Lambda_{i})"}, {"category_id": 13, "poly": [1340, 1030, 1421, 1030, 1421, 1059, 1340, 1059], "score": 0.95, "latex": "k\\geq2"}, {"category_id": 13, "poly": [833, 1028, 1009, 1028, 1009, 1063, 833, 1063], "score": 0.94, "latex": "\\pi=\\pi[c-1]"}, {"category_id": 13, "poly": [679, 546, 994, 546, 994, 587, 679, 587], "score": 0.94, "latex": "a\\mathrm{~-~}b\\mathrm{~=~}a_{j}^{\\vee}\\Lambda_{i}\\mathrm{~-~}a_{i}^{\\vee}\\Lambda_{j}"}, {"category_id": 13, "poly": [801, 1102, 860, 1102, 860, 1151, 801, 1151], "score": 0.94, "latex": "\\frac{\\ell!}{h(\\lambda)}"}, {"category_id": 13, "poly": [710, 593, 797, 593, 797, 628, 710, 628], "score": 0.94, "latex": "\\mathcal{D}(\\Lambda_{i})"}, {"category_id": 13, "poly": [289, 729, 375, 729, 375, 763, 289, 763], "score": 0.94, "latex": "\\mathcal{D}(\\Lambda_{i})"}, {"category_id": 13, "poly": [1201, 413, 1431, 413, 1431, 452, 1201, 452], "score": 0.94, "latex": "a-b=\\widetilde{\\Lambda_{0}}-\\widetilde{\\Lambda_{1}}"}, {"category_id": 13, "poly": [244, 459, 784, 459, 784, 502, 244, 502], "score": 0.94, "latex": "\\mathcal{D}(\\Lambda_{\\ell})\\,=\\,\\widetilde{\\mathcal{D}}(\\ell\\widetilde{\\Lambda_{1}})\\,\\geq\\,\\widetilde{\\mathcal{D}}(\\widetilde{\\Lambda_{1}})\\,=\\,\\mathcal{D}(\\Lambda_{1})"}, {"category_id": 13, "poly": [1115, 905, 1346, 905, 1346, 939, 1115, 939], "score": 0.94, "latex": "\\pi\\Lambda_{1}\\,=\\,C^{a}J^{b}\\Lambda_{1}"}, {"category_id": 13, "poly": [1294, 334, 1500, 334, 1500, 370, 1294, 370], "score": 0.93, "latex": "\\mathcal{D}(\\lambda)\\ge\\mathcal{D}(\\Lambda_{\\ell})"}, {"category_id": 13, "poly": [199, 1107, 321, 1107, 321, 1142, 199, 1142], "score": 0.93, "latex": "t(\\lambda)=\\ell"}, {"category_id": 13, "poly": [1249, 1679, 1425, 1679, 1425, 1719, 1249, 1719], "score": 0.93, "latex": "\\pi(J0)\\,=\\,J0"}, {"category_id": 13, "poly": [199, 593, 290, 593, 290, 628, 199, 628], "score": 0.93, "latex": "\\mathcal{D}(\\Lambda_{1})"}, {"category_id": 13, "poly": [1333, 297, 1451, 297, 1451, 326, 1333, 326], "score": 0.93, "latex": "\\lambda=k\\Lambda_{\\ell}"}, {"category_id": 13, "poly": [1439, 1232, 1499, 1232, 1499, 1280, 1439, 1280], "score": 0.93, "latex": "\\frac{\\ell!}{h(\\lambda)}"}, {"category_id": 13, "poly": [327, 549, 390, 549, 390, 583, 327, 583], "score": 0.93, "latex": "E_{8,k}"}, {"category_id": 13, "poly": [1125, 1801, 1492, 1801, 1492, 1839, 1125, 1839], "score": 0.93, "latex": "\\pi\\Lambda_{\\ell+1}\\in\\{\\Lambda_{\\ell+1},\\Lambda_{1}+\\Lambda_{\\ell}\\}"}, 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"latex": "\\pi\\Lambda_{1}=\\Lambda_{1}"}, {"category_id": 13, "poly": [1058, 469, 1146, 469, 1146, 493, 1058, 493], "score": 0.92, "latex": "\\ell\\,=\\,1"}, {"category_id": 13, "poly": [199, 988, 382, 988, 382, 1023, 199, 1023], "score": 0.92, "latex": "\\pi(J0)=J^{c}0"}, {"category_id": 13, "poly": [298, 1030, 379, 1030, 379, 1054, 298, 1054], "score": 0.92, "latex": "k=1"}, {"category_id": 13, "poly": [868, 991, 911, 991, 911, 1022, 868, 1022], "score": 0.92, "latex": "P_{+}"}, {"category_id": 13, "poly": [1108, 1479, 1270, 1479, 1270, 1519, 1108, 1519], "score": 0.92, "latex": "\\pi[J0]=J0"}, {"category_id": 13, "poly": [1031, 1843, 1240, 1843, 1240, 1877, 1031, 1877], "score": 0.92, "latex": "\\pi\\Lambda_{\\ell+1}=\\Lambda_{\\ell+1}"}, {"category_id": 13, "poly": [695, 1840, 966, 1840, 966, 1878, 695, 1878], "score": 0.92, "latex": "h(\\Lambda_{\\ell+1})=(\\ell+1)!"}, {"category_id": 13, "poly": [1159, 245, 1248, 245, 1248, 291, 1159, 291], "score": 0.92, "latex": "(E_{8}^{(1)})"}, {"category_id": 13, "poly": [1247, 731, 1281, 731, 1281, 760, 1247, 760], "score": 0.92, "latex": "\\Lambda_{i}"}, {"category_id": 13, "poly": [290, 423, 350, 423, 350, 456, 290, 456], "score": 0.92, "latex": "C_{r,k}"}, {"category_id": 13, "poly": [272, 1726, 330, 1726, 330, 1755, 272, 1755], "score": 0.92, "latex": "\\pi\\Lambda_{2}"}, {"category_id": 13, "poly": [467, 297, 529, 297, 529, 331, 467, 331], "score": 0.92, "latex": "A_{r,k}"}, {"category_id": 13, "poly": [1263, 1287, 1406, 1287, 1406, 1317, 1263, 1317], "score": 0.92, "latex": "J0=k\\Lambda_{1}"}, {"category_id": 13, "poly": [198, 1805, 337, 1805, 337, 1835, 198, 1835], "score": 0.92, "latex": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}"}, {"category_id": 13, "poly": [199, 1644, 519, 1644, 519, 1678, 199, 1678], "score": 0.92, "latex": "k\\Lambda_{1}\\succ(k-2)\\Lambda_{1}+\\Lambda_{2}"}, {"category_id": 13, "poly": [982, 1281, 1081, 1281, 1081, 1320, 982, 1320], "score": 0.92, "latex": "(N_{\\Lambda_{1}})^{\\ell}"}, {"category_id": 13, "poly": [699, 1442, 754, 1442, 754, 1480, 699, 1480], "score": 0.92, "latex": "\\pi[b]"}, {"category_id": 13, "poly": [267, 1841, 618, 1841, 618, 1877, 267, 1877], "score": 0.92, "latex": "h(\\Lambda_{1}+\\Lambda_{\\ell})=(\\ell+1)!/\\ell"}, {"category_id": 13, "poly": [1290, 951, 1382, 951, 1382, 975, 1290, 975], "score": 0.91, "latex": "a\\;=\\;0"}, {"category_id": 13, "poly": [983, 1068, 1164, 1068, 1164, 1099, 983, 1099], "score": 0.91, "latex": "\\Lambda_{1}\\otimes\\cdot\\cdot\\otimes\\Lambda_{1}"}, {"category_id": 13, "poly": [1060, 1235, 1241, 1235, 1241, 1272, 1060, 1272], "score": 0.91, "latex": "t(\\lambda)=\\ell\\leq k"}, {"category_id": 13, "poly": [700, 423, 760, 423, 760, 457, 700, 457], "score": 0.91, "latex": "A_{r,k}"}, {"category_id": 13, "poly": [486, 1761, 629, 1761, 629, 1795, 486, 1795], "score": 0.91, "latex": "\\pi\\Lambda_{2}=\\Lambda_{2}"}, {"category_id": 13, "poly": [1097, 1521, 1338, 1521, 1338, 1559, 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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 and the most difficult cases.
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).
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.
# 4.2. The -series argument
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 .
If then so . Thus we can assume .
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.
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 .
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 .
17
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<div class="pdf-page">
<p>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.</p>
<p>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).</p>
<p>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.</p>
<h1>4.2. The -series argument</h1>
<p>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 .</p>
<p>If then so . Thus we can assume .</p>
<p>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.</p>
<p>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 .</p>
<p>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 .</p>
</div>
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<div class="pdf-page">
<p class="pdf-text" data-x="118" data-y="90" data-width="788" data-height="41">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.</p>
<p class="pdf-text" data-x="117" data-y="133" data-width="788" data-height="115">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).</p>
<p class="pdf-text" data-x="117" data-y="249" data-width="788" data-height="123">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.</p>
<h1 class="pdf-title" data-x="120" data-y="389" data-width="244" data-height="19">4.2. The -series argument</h1>
<p class="pdf-text" data-x="117" data-y="417" data-width="788" data-height="56">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 .</p>
<p class="pdf-text" data-x="147" data-y="471" data-width="719" data-height="20">If then so . Thus we can assume .</p>
<p class="pdf-text" data-x="115" data-y="492" data-width="790" data-height="193">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.</p>
<p class="pdf-text" data-x="117" data-y="686" data-width="788" data-height="129">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 .</p>
<p class="pdf-text" data-x="117" data-y="817" data-width="788" data-height="91">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 .</p>
<div class="pdf-discarded" data-x="500" data-y="945" data-width="22" data-height="14" style="opacity: 0.5;">17</div>
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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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"is easy: and . is automatic. will be done later in this subsection. Assume now that .",
"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 .",
"Thus and for some . Similarly, . Hitting with , we may assume that fixes .",
"Now assume fixes , for . Then the fusion × says that equals or . But from (3.2) we find",
"",
"Hence will fix if it fixes , concluding the argument.",
"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]",
"",
"From this we see (mod ), so is coprime to . Hitting it with the Galois fusion-symmetry , we see that we may assume .",
"Now use (4.2) to get for all . Then equals the identity or , depending on what does to .",
"4.4. The -series argument",
"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",
"18",
"4.3. The -series argument"
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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\u2019t get mapped to each other. Also, the fusions $\\Lambda_{1}$ \u00d7 $\\Lambda_{r}=\\Lambda_{r}$ + $(\\Lambda_{1}+\\Lambda_{r})$ 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 $\\Lambda_{1}$ \u00d7 $\\Lambda_{\\ell}$ 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 \u00a73.2), 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 $\\Lambda_{1}$ \u00d7 $\\Lambda_{\\ell}$ 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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", "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\u2019t 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": " \u00d7", "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": " \u00d7", "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 \u00a73.2), 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, "page_num": "page_17", "page_size": [612.0, 792.0], "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\u2019t 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": " \u00d7", "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, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0], "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": " \u00d7", "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, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_17", "page_size": [612.0, 792.0], "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 \u00a73.2), 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, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0]}, {"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": " \u00d7 ", "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, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [70, 659, 542, 717]}]}
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4.3. The -series argument
is easy: and . is automatic. will be done later in this subsection. Assume now that .
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 .
Thus and for some . Similarly, . Hitting with , we may assume that fixes .
Now assume fixes , for . Then the fusion × says that equals or . 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 will fix if it fixes , concluding the argument.
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]
$$
\frac{S_{\gamma^{a}\gamma^{b}}}{S_{0\gamma^{b}}}=2\cos(2\pi\frac{a b}{\kappa})\ .
$$
From this we see (mod ), so is coprime to . Hitting it with the Galois fusion-symmetry , we see that we may assume .
Now use (4.2) to get for all . Then equals the identity or , depending on what does to .
# 4.4. The -series argument
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
18
|
<div class="pdf-page">
<p>is easy: and . is automatic. will be done later in this subsection. Assume now that .</p>
<p>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 .</p>
<p>Thus and for some . Similarly, . Hitting with , we may assume that fixes .</p>
<p>Now assume fixes , for . Then the fusion × says that equals or . But from (3.2) we find</p>
<p>Hence will fix if it fixes , concluding the argument.</p>
<p>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]</p>
<p>From this we see (mod ), so is coprime to . Hitting it with the Galois fusion-symmetry , we see that we may assume .</p>
<p>Now use (4.2) to get for all . Then equals the identity or , depending on what does to .</p>
<h1>4.4. The -series argument</h1>
<p>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</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="118" data-y="91" data-width="248" data-height="18" style="opacity: 0.5;">4.3. The -series argument</div>
<p class="pdf-text" data-x="117" data-y="120" data-width="788" data-height="37">is easy: and . is automatic. will be done later in this subsection. Assume now that .</p>
<p class="pdf-text" data-x="117" data-y="159" data-width="788" data-height="184">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 .</p>
<p class="pdf-text" data-x="117" data-y="345" data-width="788" data-height="36">Thus and for some . Similarly, . Hitting with , we may assume that fixes .</p>
<p class="pdf-text" data-x="117" data-y="382" data-width="786" data-height="39">Now assume fixes , for . Then the fusion × says that equals or . But from (3.2) we find</p>
<p class="pdf-text" data-x="117" data-y="528" data-width="537" data-height="21">Hence will fix if it fixes , concluding the argument.</p>
<p class="pdf-text" data-x="117" data-y="553" data-width="788" data-height="94">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]</p>
<p class="pdf-text" data-x="117" data-y="724" data-width="788" data-height="37">From this we see (mod ), so is coprime to . Hitting it with the Galois fusion-symmetry , we see that we may assume .</p>
<p class="pdf-text" data-x="118" data-y="762" data-width="787" data-height="38">Now use (4.2) to get for all . Then equals the identity or , depending on what does to .</p>
<h1 class="pdf-title" data-x="118" data-y="819" data-width="246" data-height="20">4.4. The -series argument</h1>
<p class="pdf-text" data-x="117" data-y="849" data-width="788" data-height="75">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</p>
<div class="pdf-discarded" data-x="500" data-y="945" data-width="22" data-height="13" style="opacity: 0.5;">18</div>
</div>
|
$k=1$ 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
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"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",
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"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",
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"",
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"as in . When , that inequality only holds for , but we can force by hitting if necessary with .",
"The remaining case follows because : by (2.7b) , and by (2.7a) ( is a -fixed-point).",
"4.5. The -series argument",
"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.",
"Consider first even , and even . Now, forces ; hence hitting with the simple-current automorphism , we may assume .",
"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 .",
"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",
"2)",
"for , which can be shown e.g. using Taylor series.",
"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.",
"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.",
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[{"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 $\\pi\\Lambda_{r-1}=$ $C_{1}^{i}J_{v}^{j+1}J_{s}\\Lambda_{r-1}$ \u2014 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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{"preproc_blocks": [{"type": "text", "bbox": [69, 70, 542, 100], "lines": [{"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}, {"bbox": [126, 72, 178, 90], "score": 1.0, "content": " holds for ", "type": "text"}, {"bbox": [178, 75, 191, 84], "score": 0.39, "content": "k r", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [191, 72, 312, 90], "score": 1.0, "content": " even. 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": " \u00d7 ", "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": " \u2014 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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0], "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": " \u2014 a contradiction.", "type": "text"}], "index": 32}], "index": 28.5, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [70, 658, 542, 717]}]}
|
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"\\pi^{\\prime}\\Lambda_{1}=J_{v}^{a^{\\prime}}J_{s}^{b^{\\prime}}\\Lambda_{1}"}, {"category_id": 13, "poly": [904, 1204, 1157, 1204, 1157, 1235, 904, 1235], "score": 0.94, "latex": "\\pi\\Lambda_{1}=\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}"}, {"category_id": 13, "poly": [1305, 763, 1415, 763, 1415, 799, 1305, 799], "score": 0.93, "latex": "C^{i}J_{v}^{j}\\Lambda_{r}"}, {"category_id": 13, "poly": [1142, 1873, 1364, 1873, 1364, 1911, 1142, 1911], "score": 0.93, "latex": "\\pi\\Lambda_{r}\\,=\\,C_{1}^{i}J_{v}^{j}\\Lambda_{r}"}, {"category_id": 13, "poly": [766, 1047, 886, 1047, 886, 1080, 766, 1080], "score": 0.93, "latex": "S_{J_{v}\\Lambda_{1},J0}"}, {"category_id": 13, "poly": [1204, 1679, 1307, 1679, 1307, 1712, 1204, 1712], "score": 0.93, "latex": "S_{\\Lambda_{1},J_{v}0}"}, {"category_id": 13, "poly": [312, 721, 527, 721, 527, 759, 312, 759], "score": 0.93, "latex": "\\pi\\Lambda_{1}=J_{v}^{a}J_{s}^{b}\\Lambda_{1}"}, {"category_id": 13, "poly": [581, 1082, 693, 1082, 693, 1119, 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"r<k"}, {"category_id": 13, "poly": [306, 763, 482, 763, 482, 800, 306, 800], "score": 0.92, "latex": "\\chi_{\\Lambda_{1}}[\\Lambda_{1}]>0"}, {"category_id": 13, "poly": [1048, 845, 1400, 845, 1400, 880, 1048, 880], "score": 0.92, "latex": "0\\;\\neq\\;S_{\\Lambda_{1}\\Lambda_{1}}\\;=\\;S_{\\pi\\Lambda_{1},\\pi^{\\prime}\\Lambda_{1}}"}, {"category_id": 13, "poly": [1049, 1598, 1151, 1598, 1151, 1633, 1049, 1633], "score": 0.92, "latex": "S_{\\Lambda_{1},J_{s}0}"}, {"category_id": 13, "poly": [874, 507, 1033, 507, 1033, 535, 874, 535], "score": 0.92, "latex": "\\pi^{\\prime}J0\\,=\\,J0"}, {"category_id": 13, "poly": [1040, 1006, 1221, 1006, 1221, 1037, 1040, 1037], "score": 0.92, "latex": "\\pi\\Lambda_{1}=J_{v}\\Lambda_{1}"}, {"category_id": 13, "poly": [1275, 1002, 1464, 1002, 1464, 1037, 1275, 1037], "score": 0.92, "latex": "\\pi^{\\prime}\\Lambda_{1}\\,=\\,J_{v}\\Lambda_{1}"}, {"category_id": 13, "poly": [249, 1475, 445, 1475, 445, 1513, 249, 1513], "score": 0.92, "latex": "0<x\\le2\\pi/9"}, {"category_id": 13, "poly": [759, 1716, 1086, 1716, 1086, 1752, 759, 1752], "score": 0.92, "latex": "\\chi_{\\Lambda_{1}}[\\Lambda_{r}]\\;=\\;\\chi_{J_{v}\\Lambda_{1}}[\\pi\\Lambda_{r}]"}, {"category_id": 13, "poly": [199, 251, 336, 251, 336, 280, 199, 280], "score": 0.92, "latex": "\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}"}, {"category_id": 13, "poly": [738, 1597, 829, 1597, 829, 1623, 738, 1623], "score": 0.92, "latex": "a\\prime=1"}, {"category_id": 13, "poly": [1406, 1558, 1491, 1558, 1491, 1584, 1406, 1584], "score": 0.92, "latex": "b^{\\prime}=0"}, {"category_id": 13, "poly": [475, 1672, 664, 1672, 664, 1708, 475, 1708], "score": 0.92, "latex": "\\pi^{\\prime}\\Lambda_{1}=J_{v}\\Lambda_{1}"}, {"category_id": 13, "poly": [272, 1283, 429, 1283, 429, 1316, 272, 1316], "score": 0.92, "latex": "\\pi^{\\prime}\\Lambda_{1}=\\Lambda_{1}"}, {"category_id": 13, "poly": [683, 1754, 1000, 1754, 1000, 1791, 683, 1791], "score": 0.92, "latex": 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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 . When , that inequality only holds for , but we can force by hitting if necessary with .
The remaining case follows because : by (2.7b) , and by (2.7a) ( is a -fixed-point).
# 4.5. The -series argument
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.
Consider first even , and even . Now, forces ; hence hitting with the simple-current automorphism , we may assume .
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 .
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
2)
for , which can be shown e.g. using Taylor series.
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.
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.
19
|
<div class="pdf-page">
<p>as in . When , that inequality only holds for , but we can force by hitting if necessary with .</p>
<p>The remaining case follows because : by (2.7b) , and by (2.7a) ( is a -fixed-point).</p>
<h1>4.5. The -series argument</h1>
<p>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.</p>
<p>Consider first even , and even . Now, forces ; hence hitting with the simple-current automorphism , we may assume .</p>
<p>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 .</p>
<p>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</p>
<p>2)</p>
<p>for , which can be shown e.g. using Taylor series.</p>
<p>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.</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="192" data-width="788" data-height="38">as in . When , that inequality only holds for , but we can force by hitting if necessary with .</p>
<p class="pdf-text" data-x="117" data-y="231" data-width="788" data-height="37">The remaining case follows because : by (2.7b) , and by (2.7a) ( is a -fixed-point).</p>
<h1 class="pdf-title" data-x="118" data-y="285" data-width="248" data-height="20">4.5. The -series argument</h1>
<p class="pdf-text" data-x="117" data-y="314" data-width="788" data-height="73">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.</p>
<p class="pdf-text" data-x="115" data-y="389" data-width="790" data-height="73">Consider first even , and even . Now, forces ; hence hitting with the simple-current automorphism , we may assume .</p>
<p class="pdf-text" data-x="117" data-y="462" data-width="789" data-height="112">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 .</p>
<p class="pdf-text" data-x="117" data-y="574" data-width="788" data-height="56">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</p>
<p class="pdf-text" data-x="163" data-y="646" data-width="697" data-height="21">2)</p>
<p class="pdf-text" data-x="115" data-y="682" data-width="550" data-height="18">for , which can be shown e.g. using Taylor series.</p>
<p class="pdf-text" data-x="117" data-y="702" data-width="788" data-height="147">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.</p>
<p class="pdf-text" data-x="117" data-y="850" data-width="788" data-height="74">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.</p>
<div class="pdf-discarded" data-x="500" data-y="945" data-width="22" data-height="13" style="opacity: 0.5;">19</div>
</div>
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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."
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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})\\}>"
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"Next, note that we know from × that is or . As in , the fusion × × · · · × ( times) contains the simple-current , but × × · · · × ( times) doesn’t, and thus .",
"Assume . Using the fusions × (for ), and noting that",
"",
"equals 0 only when , we see that except possibly for (hence ). For that , use q-dimensions:",
"",
"which is valid for these . So we also know for all , and we are done.",
"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.",
"For , we can force , and then eliminate or by . 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 sym- metries) given in §§3.5-3.9.",
"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 .",
"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.",
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[{"type": "text", "text": "Next, note that we know from $\\Lambda_{1}$ \u00d7 $\\Lambda_{1}$ that $\\pi\\Lambda_{2}$ is $\\Lambda_{2}$ or $2\\Lambda_{1}$ . As in $\\S4.2$ , the fusion $(2\\Lambda_{1})$ \u00d7 $\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\Lambda_{1}$ ( $k{-}2$ times) contains the simple-current $J_{v}0$ , but $\\Lambda_{2}$ \u00d7 $\\Lambda_{1}$ \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 $\\Lambda_{1}$ ( $k-2$ times) doesn\u2019t, and thus $\\pi\\Lambda_{2}=\\Lambda_{2}$ . ", "page_idx": 19}, {"type": "text", "text": "Assume $\\pi\\Lambda_{\\ell}=\\Lambda_{\\ell}$ . Using the fusions $\\Lambda_{1}$ \u00d7 $\\Lambda_{\\ell}$ (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 \u00a7\u00a73.5-3.9. ", "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)\u2013(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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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 \u00a7\u00a73.5-3.9.", "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)\u2013(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": " \u00d7 ", "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": " \u00d7 ", "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": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 ", "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": " \u00d7", "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": " \u00d7 \u00b7 \u00b7 \u00b7 \u00d7", "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\u2019t, 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, "page_num": "page_19", "page_size": [612.0, 792.0], "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": " \u00d7", "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, "page_num": "page_19", "page_size": [612.0, 792.0], "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, "page_num": "page_19", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_19", "page_size": [612.0, 792.0], "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, "page_num": "page_19", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_19", "page_size": [612.0, 792.0], "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, "page_num": "page_19", "page_size": [612.0, 792.0], "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, "page_num": "page_19", "page_size": [612.0, 792.0], "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, "page_num": "page_19", "page_size": [612.0, 792.0], "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 \u00a7\u00a73.5-3.9.", "type": "text"}], "index": 17}], "index": 16.5, "page_num": "page_19", "page_size": [612.0, 792.0], "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, "page_num": "page_19", "page_size": [612.0, 792.0], "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)\u2013(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, "page_num": "page_19", "page_size": [612.0, 792.0], "bbox_fs": [66, 564, 545, 691]}]}
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{"layout_dets": [{"category_id": 1, "poly": [195, 1215, 1504, 1215, 1504, 1576, 195, 1576], "score": 0.984}, {"category_id": 1, "poly": [195, 1581, 1506, 1581, 1506, 1911, 195, 1911], "score": 0.983}, {"category_id": 1, "poly": [197, 791, 1507, 791, 1507, 952, 197, 952], "score": 0.97}, {"category_id": 1, "poly": [194, 506, 1504, 506, 1504, 588, 194, 588], "score": 0.965}, {"category_id": 8, "poly": [568, 619, 1130, 619, 1130, 714, 568, 714], "score": 0.949}, {"category_id": 1, "poly": [197, 953, 1504, 953, 1504, 1036, 197, 1036], "score": 0.946}, {"category_id": 1, "poly": [197, 1130, 1500, 1130, 1500, 1212, 197, 1212], "score": 0.938}, {"category_id": 8, "poly": [342, 391, 1358, 391, 1358, 477, 342, 477], "score": 0.935}, {"category_id": 1, "poly": [198, 197, 1583, 197, 1583, 318, 198, 318], "score": 0.927}, {"category_id": 1, "poly": [198, 742, 1475, 742, 1475, 785, 198, 785], "score": 0.911}, {"category_id": 2, "poly": [831, 2031, 869, 2031, 869, 2062, 831, 2062], "score": 0.87}, {"category_id": 1, "poly": [260, 319, 1466, 319, 1466, 361, 260, 361], "score": 0.862}, {"category_id": 1, "poly": [201, 1071, 894, 1071, 894, 1111, 201, 1111], "score": 0.525}, {"category_id": 0, "poly": [201, 1071, 894, 1071, 894, 1111, 201, 1111], "score": 0.485}, {"category_id": 13, "poly": [428, 1305, 586, 1305, 586, 1341, 428, 1341], "score": 0.94, "latex": "\\pi[-b]^{-1}C^{a}"}, {"category_id": 13, "poly": [326, 1583, 388, 1583, 388, 1627, 326, 1627], "score": 0.94, "latex": "{E}_{8}^{(1)}"}, {"category_id": 13, "poly": [600, 964, 742, 964, 742, 993, 600, 993], "score": 0.94, "latex": "\\pi\\Lambda_{1}=\\Lambda_{1}"}, {"category_id": 13, "poly": [1151, 1223, 1368, 1223, 1368, 1258, 1151, 1258], "score": 0.94, "latex": "\\pi\\Lambda_{1}=C^{a}J^{b}\\Lambda_{1}"}, {"category_id": 13, "poly": [589, 1217, 651, 1217, 651, 1262, 589, 1262], "score": 0.94, "latex": "E_{6}^{(1)}"}, {"category_id": 13, "poly": [452, 1468, 564, 1468, 564, 1498, 452, 1498], "score": 0.93, "latex": "\\Lambda_{1}+\\Lambda_{5}"}, {"category_id": 13, "poly": [199, 842, 352, 842, 352, 876, 199, 876], "score": 0.93, "latex": "\\mathcal{D}(\\lambda^{a})=2"}, {"category_id": 13, "poly": [474, 1266, 645, 1266, 645, 1295, 474, 1295], "score": 0.93, "latex": "\\pi^{\\prime}J0\\,=\\,J^{c}0"}, {"category_id": 13, "poly": [947, 1308, 1096, 1308, 1096, 1338, 947, 1338], "score": 0.93, "latex": "\\pi\\Lambda_{1}=\\Lambda_{1}"}, {"category_id": 13, "poly": [1347, 1828, 1411, 1828, 1411, 1874, 1347, 1874], "score": 0.93, "latex": "{E}_{8}^{(1)}"}, {"category_id": 13, "poly": [1217, 1506, 1501, 1506, 1501, 1541, 1217, 1541], "score": 0.93, "latex": "\\{\\Lambda_{1},\\Lambda_{2},\\Lambda_{4},\\Lambda_{5},\\Lambda_{6}\\}"}, {"category_id": 14, "poly": [570, 626, 1125, 626, 1125, 714, 570, 714], "score": 0.92, "latex": "\\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\\ ,"}, {"category_id": 14, "poly": [345, 397, 1356, 397, 1356, 477, 345, 477], "score": 0.92, "latex": "\\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})\\}"}, {"category_id": 13, "poly": [937, 1669, 1284, 1669, 1284, 1705, 937, 1705], "score": 0.92, "latex": "\\Lambda_{3},\\Lambda_{6},\\Lambda_{8},\\Lambda_{1}+\\Lambda_{7},2\\Lambda_{7}"}, {"category_id": 13, "poly": [662, 285, 804, 285, 804, 320, 662, 320], "score": 0.92, "latex": "\\pi\\Lambda_{2}=\\Lambda_{2}"}, {"category_id": 13, "poly": [960, 1467, 1101, 1467, 1101, 1498, 960, 1498], "score": 0.92, "latex": "\\pi\\Lambda_{6}=\\Lambda_{6}"}, {"category_id": 13, "poly": [341, 1468, 380, 1468, 380, 1497, 341, 1497], "score": 0.92, "latex": "\\Lambda_{6}"}, {"category_id": 13, "poly": [265, 1839, 438, 1839, 438, 1871, 265, 1871], "score": 0.92, "latex": "\\Lambda_{5},\\Lambda_{7}+\\Lambda_{8}"}, 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|
Next, note that we know from × that is or . As in , the fusion × × · · · × ( times) contains the simple-current , but × × · · · × ( times) doesn’t, and thus .
Assume . Using the fusions × (for ), 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 , we see that except possibly for (hence ). For that , 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 . So we also know for all , and we are done.
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.
For , we can force , and then eliminate or by . 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 sym- metries) given in §§3.5-3.9.
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 .
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.
20
|
<div class="pdf-page">
<p>Next, note that we know from × that is or . As in , the fusion × × · · · × ( times) contains the simple-current , but × × · · · × ( times) doesn’t, and thus .</p>
<p>Assume . Using the fusions × (for ), and noting that</p>
<p>equals 0 only when , we see that except possibly for (hence ). For that , use q-dimensions:</p>
<p>which is valid for these . So we also know for all , and we are done.</p>
<p>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.</p>
<p>For , we can force , and then eliminate or by . The rest of the argument is as before.</p>
<p>4.6. The arguments for the exceptional algebras</p>
<p>The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym- metries) given in §§3.5-3.9.</p>
<p>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 .</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="118" data-y="90" data-width="834" data-height="57">Next, note that we know from × that is or . As in , the fusion × × · · · × ( times) contains the simple-current , but × × · · · × ( times) doesn’t, and thus .</p>
<p class="pdf-text" data-x="155" data-y="147" data-width="726" data-height="19">Assume . Using the fusions × (for ), and noting that</p>
<p class="pdf-text" data-x="115" data-y="235" data-width="790" data-height="37">equals 0 only when , we see that except possibly for (hence ). For that , use q-dimensions:</p>
<p class="pdf-text" data-x="118" data-y="345" data-width="770" data-height="19">which is valid for these . So we also know for all , and we are done.</p>
<p class="pdf-text" data-x="117" data-y="367" data-width="789" data-height="75">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.</p>
<p class="pdf-text" data-x="117" data-y="443" data-width="788" data-height="37">For , we can force , and then eliminate or by . The rest of the argument is as before.</p>
<p class="pdf-text" data-x="120" data-y="497" data-width="417" data-height="18">4.6. The arguments for the exceptional algebras</p>
<p class="pdf-text" data-x="117" data-y="524" data-width="786" data-height="39">The exceptional algebras follow quickly from the fusions (and Dynkin diagram sym- metries) given in §§3.5-3.9.</p>
<p class="pdf-text" data-x="117" data-y="565" data-width="788" data-height="168">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 .</p>
<p class="pdf-text" data-x="117" data-y="735" data-width="789" data-height="153">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.</p>
<div class="pdf-discarded" data-x="500" data-y="945" data-width="22" data-height="14" style="opacity: 0.5;">20</div>
</div>
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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 sym-",
"metries) 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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"\\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})\\}",
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"5. Affine fusion ring isomorphisms",
"We conclude the paper with the determination of all isomorphisms among the affine fusion rings . Recall Definition 2.1 and the discussion in .",
"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 .",
"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 .",
"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 .",
"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 .",
"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.",
"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,",
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[{"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\u2019t 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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"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\u2019t 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, "page_num": "page_20", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_20", "page_size": [612.0, 792.0], "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, "page_num": "page_20", "page_size": [612.0, 792.0], "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, "page_num": "page_20", "page_size": [612.0, 792.0], "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, "page_num": "page_20", "page_size": [612.0, 792.0], "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, "page_num": "page_20", "page_size": [612.0, 792.0], "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\u2019t 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, "page_num": "page_20", "page_size": [612.0, 792.0], "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, "page_num": "page_20", "page_size": [612.0, 792.0], "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, "page_num": "page_20", "page_size": [612.0, 792.0]}]}
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# 5. Affine fusion ring isomorphisms
We conclude the paper with the determination of all isomorphisms among the affine fusion rings . Recall Definition 2.1 and the discussion in .
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 .
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 .
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 .
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 .
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.
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,
$$
\begin{array}{r l}&{\displaystyle\prod_{a=0}^{r-1}\frac{\sin(\pi\ell\,(2a+1)/2\kappa)}{\sin(\pi\,(2a+1)/2\kappa)}\,\prod_{b=1}^{2r-2}\frac{\sin(\pi\ell b/\kappa)^{\left[\frac{2r-b}{2}\right]}}{\sin(\pi b/\kappa)^{\left[\frac{2r-b}{2}\right]}}\,\,,}\\ &{\displaystyle\prod_{a=1}^{r-1}\frac{\sin(\pi\ell a/\kappa)^{r-a}\,\sin(\pi\ell\,(2a-1)/2\kappa)^{r-a}}{\sin(\pi a/\kappa)^{r-a}\,\sin(\pi\,(2a-1)/2\kappa)^{r-a}}\,\prod_{b=r}^{2r-1}\frac{\sin(\pi\ell b/2\kappa)}{\sin(\pi b/2\kappa)}\,\,,}\\ &{\displaystyle\prod_{a=1}^{r-1}\frac{\sin(\pi\ell a/\kappa)^{\left[\frac{2r-a+1}{2}\right]}}{\sin(\pi a/\kappa)^{\left[\frac{2r-a+1}{2}\right]}}\,\prod_{b=r}^{2r-3}\frac{\sin(\pi\ell b/\kappa)^{\left[\frac{2r-b-1}{2}\right]}}{\sin(\pi b/\kappa)^{\left[\frac{2r-b-1}{2}\right]}}\,\,,}\end{array}
$$
21
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<div class="pdf-page">
<h1>5. Affine fusion ring isomorphisms</h1>
<p>We conclude the paper with the determination of all isomorphisms among the affine fusion rings . Recall Definition 2.1 and the discussion in .</p>
<p>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 .</p>
<p>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 .</p>
<p>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 .</p>
<p>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 .</p>
<p>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.</p>
<p>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,</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="334" data-y="91" data-width="352" data-height="20">5. Affine fusion ring isomorphisms</h1>
<p class="pdf-text" data-x="117" data-y="131" data-width="789" data-height="39">We conclude the paper with the determination of all isomorphisms among the affine fusion rings . Recall Definition 2.1 and the discussion in .</p>
<p class="pdf-text" data-x="117" data-y="178" data-width="788" data-height="156">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 .</p>
<p class="pdf-text" data-x="117" data-y="343" data-width="788" data-height="100">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 .</p>
<p class="pdf-text" data-x="117" data-y="444" data-width="788" data-height="93">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 .</p>
<p class="pdf-text" data-x="117" data-y="537" data-width="788" data-height="96">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 .</p>
<p class="pdf-text" data-x="118" data-y="633" data-width="787" data-height="56">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.</p>
<p class="pdf-text" data-x="117" data-y="690" data-width="788" data-height="58">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,</p>
<div class="pdf-discarded" data-x="498" data-y="945" data-width="22" data-height="14" style="opacity: 0.5;">21</div>
</div>
|
# 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.
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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,",
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"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 .",
"If , then that Galois argument implies , so compare numbers of highest-weights: .",
"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.",
"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.",
"For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively and for and , except for ).",
"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.",
"References",
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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.",
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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 $\\Lambda_{\\star}$ \u00d7 $\\Lambda_{\\star}$ (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 \u00a73 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\u2013166; J. Fuchs and C. Schweigert, Branes: From free fields to general backgrounds, Nucl. Phys. B530 (1998), 99\u2013136. \n2. D. Bernard, String characters from Kac\u2013Moody automorphisms, Nucl. Phys. B288 (1987), 628\u2013648. \n3. J. B\u00a8ockenhauer 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\u2013321. \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\u00b4en\u00b4echal, \u201cConformal Field Theory\u201d, Springer-Verlag, New York, 1997. \n7. G. Faltings, A proof for the Verlinde formula, J. Alg. Geom. 3 (1994), 347\u2013374. 8. M. Finkelberg, An equivalence of fusion categories, Geom. Func. Anal. 6 (1996), 249\u2013267. 9. I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteg-de Vries type equations, in: \u201cLie algebras and related topics\u201d, Lecture Notes in Math, Vol. 933, Springer-Verlag, New York, 1982. \n10. I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, \u201cOn axiomatic approaches to vertex operator algebras and modules\u201d, Memoirs Amer. Math. Soc. 104 (1993). \n11. J. Fr\u00a8ohlich and T. Kerler, \u201cQuantum groups, quantum categories and quantum field theory\u201d, Lecture Notes in Mathematics, Vol. 1542, Springer-Verlag, Berlin, 1993. \n12. J. Fuchs, Simple WZW currents, Commun. Math. Phys. 136 (1991), 345\u2013356. \n13. J. Fuchs, B. Gato-Rivera, A. N. Schellekens and C. Schweigert, Modular invariants and fusion automorphisms from Galois theory, Phys. Lett. B334 (1994), 113\u2013120. \n14. J. Fuchs and P. van Driel, WZW fusion rules, quantum groups, and the modular matrix $S$ , Nucl. Phys. B346 (1990), 632\u2013648. \n15. J. Fuchs and P. van Driel, Fusion rule engineering, Lett. Math. Phys. 23 (1991), 11\u201318. \n16. T. Gannon, WZW commutants, lattices, and level-one partition functions, Nucl. Phys. B396 (1993), 708\u2013736; P. Ruelle, E. Thiran and J. Weyers, Implications of an arithmetical symmetry of the commutant for modular invariants, Nucl. Phys. B402 (1993), 693\u2013708. \n17. T. Gannon, Symmetries of the Kac-Peterson modular matrices of affine algebras, Invent. math. 122 (1995), 341\u2013357. \n18. T. Gannon, Ph. Ruelle and M. A. Walton, Automorphism modular invariants of current algebras, Commun. Math. Phys. 179 (1996), 121\u2013156. \n19. G. Georgiev and O. Mathieu, Cat\u00b4egorie de fusion pour les groupes de Chevalley, C. R. Acad. Sci. Paris 315 (1992), 659\u2013662. \n20. F. M. Goodman and H. Wenzl, Littlewood-Richardson coefficients for Hecke algebras at roots of unity, Adv. Math. 82 (1990), 244\u2013265. \n21. Y.-Z. Huang and J. Lepowsky, Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke Math. J. 99 (1999), 113\u2013134. \n22. V. G. Kac, Simple Lie groups and the Legendre symbol, in: \u201cLie algebras, group theory, and partially ordered algebraic structures\u201d, Lecture Notes in Math, Vol. 848, Springer-Verlag, Berlin, 1981. \n23. V. G. Kac, \u201cInfinite Dimensional Lie algebras\u201d, 3rd edition, Cambridge University Press, Cambridge, 1990. \n24. V. G. Kac and D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984), 125\u2013264. \n25. V. G. Kac and M. Wakimoto, Modular and conformal constraints in representation theory of affine algebras, Adv. Math. 70 (1988), 156\u2013236. \n26. G. Lusztig, Exotic Fourier transform, Duke Math. J. 73 (1994), 227\u2013241. \n27. I. G. Macdonald, \u201cSymmetric functions and Hall polynomials\u201d, 2nd edition, Oxford University Press, New York, 1995. \n28. W. G. McKay, R. V. Moody and J. Patera, Decomposition of tensor products of $E_{8}$ representations, Alg., Groups Geom. 3 (1986), 286\u2013328. \n29. W. G. McKay, J. Patera and D. W. Rand, \u201cTables of representations of simple Lie algebras\u201d, Vol. 1, Centre de Recherches Math\u00b4ematiques, Univ\u00b4ersit\u00b4e de Montr\u00b4eal, 1990. \n30. W. Nahm, Conformal field theories, dilogarithms, and 3-dimensional manifolds, in: \u201cInterface between physics and mathematics\u201d, World-Scientific, 1994, (W. Nahm and J.-M. Shen, Eds.), World-Scientific, Singapore, 1994. \n31. A. N. Schellekens, Fusion rule automorphisms from integer spin simple currents, Phys. Lett. B244 (1990), 255\u2013260. \n32. V. G. Turaev, \u201cQuantum invariants of knots and 3-manifolds\u201d, Walter de Gruyter, Berlin, 1994. \n33. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. 300 (1988), 360\u2013376. \n34. D. Verstegen, New exceptional modular invariant partition functions for simple Kac\u2013Moody algebras, Nucl. Phys. B346 (1990), 349\u2013386. \n35. M. A. Walton, Algorithm for WZW fusion rules: a proof, Phys. Lett. B241 (1990), 365\u2013368. \n36. A. J. Wassermann, Operator algebras and conformal field theory, in: \u201cProc. ICM, Zurich\", Birkhauser, Basel, 1995. \n37. E. Witten, The Verlinde formula and the cohomology of the Grassmannian, in: \u201cGeometry, Topology and Physics\u201d, International Press, Cambridge, MA, 1995. ", "page_idx": 21}]
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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": "\u00d7 ", "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 \u00a73 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\u2013374.", "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, "page_num": "page_21", "page_size": [612.0, 792.0], "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, "page_num": "page_21", "page_size": [612.0, 792.0], "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, "page_num": "page_21", "page_size": [612.0, 792.0], "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": "\u00d7 ", "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 \u00a73 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, "page_num": "page_21", "page_size": [612.0, 792.0], "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, "page_num": "page_21", "page_size": [612.0, 792.0], "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, "page_num": "page_21", "page_size": [612.0, 792.0], "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, "page_num": "page_21", "page_size": [612.0, 792.0]}, {"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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1.0, "text": ""}, {"category_id": 15, "poly": [335.0, 205.0, 1060.0, 205.0, 1060.0, 249.0, 335.0, 249.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1081.0, 205.0, 1504.0, 205.0, 1504.0, 249.0, 1081.0, 249.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [196.0, 245.0, 991.0, 245.0, 991.0, 289.0, 196.0, 289.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1109.0, 245.0, 1214.0, 245.0, 1214.0, 289.0, 1109.0, 289.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1253.0, 245.0, 1505.0, 245.0, 1505.0, 289.0, 1253.0, 289.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [198.0, 285.0, 591.0, 285.0, 591.0, 329.0, 198.0, 329.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [685.0, 285.0, 972.0, 285.0, 972.0, 329.0, 685.0, 329.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1285.0, 285.0, 1504.0, 285.0, 1504.0, 329.0, 1285.0, 329.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [198.0, 326.0, 494.0, 326.0, 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"text": ""}, {"category_id": 15, "poly": [828.0, 2033.0, 873.0, 2033.0, 873.0, 2069.0, 828.0, 2069.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [752.0, 1270.0, 951.0, 1270.0, 951.0, 1305.0, 752.0, 1305.0], "score": 1.0, "text": ""}], "page_info": {"page_no": 21, "height": 2200, "width": 1700}}
|
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 .
If , then that Galois argument implies , so compare numbers of highest-weights: .
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.
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.
For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively and for and , except for ).
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.
# References
- 1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the clas- sification 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 hep- th/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 , 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 alge- bras 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 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.
22
|
<div class="pdf-page">
<p>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 .</p>
<p>If , then that Galois argument implies , so compare numbers of highest-weights: .</p>
<p>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.</p>
<p>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.</p>
<p>For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively and for and , except for ).</p>
<p>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.</p>
<h1>References</h1>
<ul>
<li>1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the clas- sification 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 hep- th/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 , 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 alge- bras 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 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.</li>
</ul>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="118" data-y="90" data-width="787" data-height="94">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 .</p>
<p class="pdf-text" data-x="117" data-y="186" data-width="788" data-height="40">If , then that Galois argument implies , so compare numbers of highest-weights: .</p>
<p class="pdf-text" data-x="117" data-y="227" data-width="788" data-height="93">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.</p>
<p class="pdf-text" data-x="118" data-y="321" data-width="787" data-height="91">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.</p>
<p class="pdf-text" data-x="117" data-y="415" data-width="788" data-height="36">For the orthogonal algebras at level 2, useful is the number of weights with second smallest q-dimension (respectively and for and , except for ).</p>
<p class="pdf-text" data-x="117" data-y="453" data-width="788" data-height="110">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.</p>
<h1 class="pdf-title" data-x="451" data-y="588" data-width="121" data-height="17">References</h1>
<ul class="pdf-list" data-x="130" data-y="618" data-width="775" data-height="307">
<li>1. R. E. Behrend, P. A. Pearce, V. B. Petkova and J.-B. Zuber, On the clas- sification 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 hep- th/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 , 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 alge- bras 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 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.</li>
</ul>
<div class="pdf-discarded" data-x="500" data-y="945" data-width="22" data-height="14" style="opacity: 0.5;">22</div>
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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
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"content": [
"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 clas-",
"sification 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 hep-",
"th/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 alge-",
"bras 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.",
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"Invariance Theorems for Supersymmetric Yang-Mills Theories",
"Savdeep Sethi 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",
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"We consider quantum mechanical Yang-Mills theories with eight supercharges and a 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 flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique.",
"1 [email protected] 2 [email protected]",
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hep-th/0001189 DUK-CGTP-00-03, IASSNS–HEP–00/118
# Invariance Theorems for Supersymmetric Yang-Mills Theories
arXiv:hep-th/0001189v2 18 Sep 2000
Savdeep Sethi 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 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 flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique.
1/00
1 [email protected] 2 [email protected]
|
<div class="pdf-page">
<h1>Invariance Theorems for Supersymmetric Yang-Mills Theories</h1>
<p>Savdeep Sethi and Mark Stern†2</p>
<p>∗ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA † Department of Mathematics, Duke University, Durham, NC 27706, USA</p>
<p>We consider quantum mechanical Yang-Mills theories with eight supercharges and a 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 flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique.</p>
</div>
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<div class="pdf-page">
<div class="pdf-discarded" data-x="530" data-y="89" data-width="375" data-height="41" style="opacity: 0.5;">hep-th/0001189 DUK-CGTP-00-03, IASSNS–HEP–00/118</div>
<h1 class="pdf-title" data-x="197" data-y="239" data-width="627" data-height="67">Invariance Theorems for Supersymmetric Yang-Mills Theories</h1>
<div class="pdf-discarded" data-x="23" data-y="266" data-width="37" data-height="406" style="opacity: 0.5;">arXiv:hep-th/0001189v2 18 Sep 2000</div>
<p class="pdf-text" data-x="354" data-y="363" data-width="305" data-height="22">Savdeep Sethi and Mark Stern†2</p>
<p class="pdf-text" data-x="130" data-y="399" data-width="760" data-height="19">∗ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA † Department of Mathematics, Duke University, Durham, NC 27706, USA</p>
<p class="pdf-text" data-x="118" data-y="514" data-width="788" data-height="144">We consider quantum mechanical Yang-Mills theories with eight supercharges and a 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 flavor symmetry. When combined with index results, this implies that the bound state of two D0-branes is unique.</p>
<div class="pdf-discarded" data-x="120" data-y="846" data-width="42" data-height="17" style="opacity: 0.5;">1/00</div>
<div class="pdf-discarded" data-x="137" data-y="883" data-width="199" data-height="40" style="opacity: 0.5;">1 [email protected] 2 [email protected]</div>
</div>
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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
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"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 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].",
"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.",
"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."
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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. 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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, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [69, 606, 542, 719]}]}
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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 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].
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.
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.
|
<div class="pdf-page">
<h1>1. Introduction</h1>
<p>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>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].</p>
<p>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.</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="118" data-y="91" data-width="158" data-height="20">1. Introduction</h1>
<p class="pdf-text" data-x="117" data-y="126" data-width="789" data-height="145">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 class="pdf-text" data-x="117" data-y="276" data-width="789" data-height="197">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].</p>
<p class="pdf-text" data-x="117" data-y="477" data-width="789" data-height="297">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.</p>
<p class="pdf-text" data-x="117" data-y="779" data-width="789" data-height="146">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.</p>
</div>
|
# 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.
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"We can couple these invariance theorems with results from index theory [8,9]. The index for the non-Fredholm theory of two D0-branes is proven to be one [8]. We also know that the 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 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 symmetry which commutes with the Hamil- tonian . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars with transforming in the of the symmetry group. These scalars transform in the adjoint representation of the gauge group . Let be the associated canonical momenta obeying,",
"",
"where the subscript is a group index.",
"Associated to these bosons are eight real fermions where transforming in the representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the representation. These fermions obey the usual quantization relation,",
"",
"By non-Fredholm, we mean a theory without a gap."
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[{"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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We can couple these invariance theorems with results from index theory [8,9]. The index for the non-Fredholm theory of two D0-branes is proven to be one [8]. We also know that the 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 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 symmetry which commutes with the Hamil- tonian . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars with transforming in the of the symmetry group. These scalars transform in the adjoint representation of the gauge group . Let be the associated canonical momenta obeying,
$$
[x_{A}^{\mu},p_{B}^{\nu}]=i\delta^{\mu\nu}\delta_{A B},
$$
where the subscript is a group index.
Associated to these bosons are eight real fermions where transforming in the representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the representation. These fermions obey the usual quantization relation,
$$
\left\{\lambda_{a A},\lambda_{b B}\right\}=\delta_{a b}\delta_{A B}.
$$
By non-Fredholm, we mean a theory without a gap.
|
<div class="pdf-page">
<p>We can couple these invariance theorems with results from index theory [8,9]. The index for the non-Fredholm theory of two D0-branes is proven to be one [8]. We also know that the 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 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>2. The Field Content and Symmetries</h1>
<h1>2.1. The vector multiplet supercharge</h1>
<p>The argument we wish to make requires reasonably little explicit knowledge of the gauge theory. There is a symmetry which commutes with the Hamil- tonian . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars with transforming in the of the symmetry group. These scalars transform in the adjoint representation of the gauge group . Let be the associated canonical momenta obeying,</p>
<p>where the subscript is a group index.</p>
<p>Associated to these bosons are eight real fermions where transforming in the representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the representation. These fermions obey the usual quantization relation,</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="142" data-width="789" data-height="248">We can couple these invariance theorems with results from index theory [8,9]. The index for the non-Fredholm theory of two D0-branes is proven to be one [8]. We also know that the 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 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 class="pdf-title" data-x="118" data-y="433" data-width="388" data-height="20">2. The Field Content and Symmetries</h1>
<h1 class="pdf-title" data-x="118" data-y="469" data-width="328" data-height="19">2.1. The vector multiplet supercharge</h1>
<p class="pdf-text" data-x="117" data-y="504" data-width="789" data-height="147">The argument we wish to make requires reasonably little explicit knowledge of the gauge theory. There is a symmetry which commutes with the Hamil- tonian . Since we are considering a gauge theory, we must have at least one vector multiplet. It contains five scalars with transforming in the of the symmetry group. These scalars transform in the adjoint representation of the gauge group . Let be the associated canonical momenta obeying,</p>
<p class="pdf-text" data-x="117" data-y="720" data-width="351" data-height="19">where the subscript is a group index.</p>
<p class="pdf-text" data-x="117" data-y="746" data-width="789" data-height="95">Associated to these bosons are eight real fermions where transforming in the representation of the symmetry group. These fermions are also in the adjoint representation of the gauge group. The eight supercharges also transform in the representation. These fermions obey the usual quantization relation,</p>
<div class="pdf-discarded" data-x="137" data-y="905" data-width="445" data-height="18" style="opacity: 0.5;">By non-Fredholm, we mean a theory without a gap.</div>
</div>
|
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.
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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 Hamil-",
"tonian 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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"",
"[x_{A}^{\\mu},p_{B}^{\\nu}]=i\\delta^{\\mu\\nu}\\delta_{A B},",
"\\left\\{\\lambda_{a A},\\lambda_{b B}\\right\\}=\\delta_{a b}\\delta_{A B}."
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"Let be hermitian real gamma matrices which obey,",
"",
"Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action.",
"The supercharge takes the form,",
"",
"where are the structure constants and . The real anti- symmetric matrix does not involve momenta. The -term transforms in the representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of 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 with components where . This field transforms as under the symmetry group, and in some representation of the gauge group. We again introduce canonical momenta satisfying the usual commutation relations. Now is the group of unit quaternions. We choose to act on a hypermultiplet by right multiplication by a unit quaternion. The gauge symmetry commutes with the symmetry and acts by left multiplication on . See Appendix A for a more detailed discussion.",
"The superpartner to is a real fermion with satisfying,",
"",
"These fermions transform in the representation, and the subscripts index the representation of . For hypermultiplets, the gauge group acts via a subgroup of the symmetry. In terms of the operators given in Appendix A, the hypermultiplet charge takes the form",
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[{"type": "text", "text": "Let $\\gamma^{\\mu}$ be hermitian real gamma matrices which obey, ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\{\\gamma^{\\mu},\\gamma^{\\nu}\\}=2\\delta^{\\mu\\nu}.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action. ", "page_idx": 3}, {"type": "text", "text": "The supercharge takes the form, ", "page_idx": 3}, {"type": "equation", "text": "$$\nQ_{a}^{v}=(\\gamma^{\\mu}p_{A}^{\\mu}\\lambda_{A})_{a}+\\frac{1}{2}f_{A B C}\\left(\\gamma^{\\mu\\nu}\\lambda_{A}x_{B}^{\\mu}x_{C}^{\\nu}\\right)_{a}+D_{a b A}\\lambda_{b A},\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "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. ", "page_idx": 3}, {"type": "text", "text": "2.2. The hypermultiplet supercharge ", "text_level": 1, "page_idx": 3}, {"type": "text", "text": "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. ", "page_idx": 3}, {"type": "text", "text": "The superpartner to $q$ is a real fermion $\\psi_{a}$ with $a=1,\\dotsc,8$ satisfying, ", "page_idx": 3}, {"type": "equation", "text": "$$\n\\left\\{\\psi_{a}^{R},\\psi_{b S}\\right\\}=\\delta_{a b}\\delta_{S}^{R}.\n$$", "text_format": "latex", "page_idx": 3}, {"type": "text", "text": "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 ", "page_idx": 3}, {"type": "equation", "text": "$$\nQ_{a}^{h}=s_{a b}^{j}\\psi_{b}\\,p_{j}+I_{a b}\\psi_{b}.\n$$", "text_format": "latex", "page_idx": 3}]
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The ", "type": "text"}, {"bbox": [367, 268, 378, 277], "score": 0.88, "content": "D", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [379, 266, 511, 281], "score": 1.0, "content": "-term transforms in the ", "type": "text"}, {"bbox": [511, 267, 540, 280], "score": 0.89, "content": "\\left(\\mathbf{1},\\mathbf{3}\\right)", "type": "inline_equation", "height": 13, "width": 29}], "index": 7}, {"bbox": [70, 286, 541, 301], "spans": [{"bbox": [70, 286, 541, 301], "score": 1.0, "content": "representation of the symmetry group, and in the adjoint representation of the gauge", "type": "text"}], "index": 8}, {"bbox": [70, 305, 541, 320], "spans": [{"bbox": [70, 305, 214, 320], "score": 1.0, "content": "group. The precise form of ", "type": "text"}, {"bbox": [215, 307, 225, 316], "score": 0.9, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [226, 305, 541, 320], "score": 1.0, "content": " is not important for our argument. 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The hypermultiplet supercharge", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 401, 542, 551], "lines": [{"bbox": [95, 403, 540, 420], "spans": [{"bbox": [95, 403, 533, 420], "score": 1.0, "content": "A hypermultiplet contains four real scalars which we can package into a quaternion ", "type": "text"}, {"bbox": [533, 409, 540, 417], "score": 0.86, "content": "q", "type": "inline_equation", "height": 8, "width": 7}], "index": 13}, {"bbox": [72, 423, 540, 438], "spans": [{"bbox": [72, 423, 163, 438], "score": 1.0, "content": "with components ", "type": "text"}, {"bbox": [163, 424, 172, 436], "score": 0.92, "content": "q^{i}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [173, 423, 209, 438], "score": 1.0, "content": " where ", "type": "text"}, {"bbox": [210, 425, 270, 436], "score": 0.93, "content": "i=1,2,3,4", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [271, 423, 402, 438], "score": 1.0, "content": ". 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. We again introduce canonical", "type": "text"}], "index": 15}, {"bbox": [70, 462, 541, 477], "spans": [{"bbox": [70, 462, 124, 477], "score": 1.0, "content": "momenta ", "type": "text"}, {"bbox": [124, 467, 135, 475], "score": 0.89, "content": "p_{i}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [135, 462, 406, 477], "score": 1.0, "content": " satisfying the usual commutation relations. Now ", "type": "text"}, {"bbox": [407, 463, 505, 475], "score": 0.93, "content": "S U(2)_{R}\\,\\sim\\,S p(1)_{R}", "type": "inline_equation", "height": 12, "width": 98}, {"bbox": [505, 462, 541, 477], "score": 1.0, "content": " is the", "type": "text"}], "index": 16}, {"bbox": [70, 481, 541, 496], "spans": [{"bbox": [70, 481, 286, 496], "score": 1.0, "content": "group of unit quaternions. 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. 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|
Let 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 are the structure constants and . The real anti- symmetric matrix does not involve momenta. The -term transforms in the representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of 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 with components where . This field transforms as under the symmetry group, and in some representation of the gauge group. We again introduce canonical momenta satisfying the usual commutation relations. Now is the group of unit quaternions. We choose to act on a hypermultiplet by right multiplication by a unit quaternion. The gauge symmetry commutes with the symmetry and acts by left multiplication on . See Appendix A for a more detailed discussion.
The superpartner to is a real fermion with satisfying,
$$
\left\{\psi_{a}^{R},\psi_{b S}\right\}=\delta_{a b}\delta_{S}^{R}.
$$
These fermions transform in the representation, and the subscripts index the representation of . For hypermultiplets, the gauge group acts via a subgroup of the symmetry. In terms of the operators given in Appendix A, the hypermultiplet charge takes the form
$$
Q_{a}^{h}=s_{a b}^{j}\psi_{b}\,p_{j}+I_{a b}\psi_{b}.
$$
|
<div class="pdf-page">
<p>Let be hermitian real gamma matrices which obey,</p>
<p>Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action.</p>
<p>The supercharge takes the form,</p>
<p>where are the structure constants and . The real anti- symmetric matrix does not involve momenta. The -term transforms in the representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of 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>2.2. The hypermultiplet supercharge</h1>
<p>A hypermultiplet contains four real scalars which we can package into a quaternion with components where . This field transforms as under the symmetry group, and in some representation of the gauge group. We again introduce canonical momenta satisfying the usual commutation relations. Now is the group of unit quaternions. We choose to act on a hypermultiplet by right multiplication by a unit quaternion. The gauge symmetry commutes with the symmetry and acts by left multiplication on . See Appendix A for a more detailed discussion.</p>
<p>The superpartner to is a real fermion with satisfying,</p>
<p>These fermions transform in the representation, and the subscripts index the representation of . For hypermultiplets, the gauge group acts via a subgroup of the symmetry. In terms of the operators given in Appendix A, the hypermultiplet charge takes the form</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="115" data-y="90" data-width="480" data-height="21">Let be hermitian real gamma matrices which obey,</p>
<p class="pdf-text" data-x="117" data-y="178" data-width="789" data-height="45">Appendix A includes an explicit basis for these gamma matrices along with a discussion of the symmetry group action.</p>
<p class="pdf-text" data-x="155" data-y="228" data-width="290" data-height="20">The supercharge takes the form,</p>
<p class="pdf-text" data-x="115" data-y="314" data-width="791" data-height="146">where are the structure constants and . The real anti- symmetric matrix does not involve momenta. The -term transforms in the representation of the symmetry group, and in the adjoint representation of the gauge group. The precise form of 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 class="pdf-title" data-x="118" data-y="484" data-width="318" data-height="20">2.2. The hypermultiplet supercharge</h1>
<p class="pdf-text" data-x="115" data-y="518" data-width="791" data-height="194">A hypermultiplet contains four real scalars which we can package into a quaternion with components where . This field transforms as under the symmetry group, and in some representation of the gauge group. We again introduce canonical momenta satisfying the usual commutation relations. Now is the group of unit quaternions. We choose to act on a hypermultiplet by right multiplication by a unit quaternion. The gauge symmetry commutes with the symmetry and acts by left multiplication on . See Appendix A for a more detailed discussion.</p>
<p class="pdf-text" data-x="157" data-y="717" data-width="627" data-height="21">The superpartner to is a real fermion with satisfying,</p>
<p class="pdf-text" data-x="117" data-y="804" data-width="788" data-height="95">These fermions transform in the representation, and the subscripts index the representation of . For hypermultiplets, the gauge group acts via a subgroup of the symmetry. In terms of the operators given in Appendix A, the hypermultiplet charge takes the form</p>
</div>
|
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,
$$
\left\{\psi_{a}^{R},\psi_{b S}\right\}=\delta_{a b}\delta_{S}^{R}.
$$
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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 anti-",
"symmetric 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}."
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"\\{\\gamma^{\\mu},\\gamma^{\\nu}\\}=2\\delta^{\\mu\\nu}.",
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"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,",
"",
"2.3. The currents",
"The three generators of correspond to right multiplication by and are given in terms of the gauge invariant rotation generators,",
"",
"Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three generators by :",
"",
"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 act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by:",
"",
"Adding either more vector multiplets or more hypermultiplets is again straightforward."
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[{"type": "text", "text": "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, ", "page_idx": 4}, {"type": "equation", "text": "$$\nQ_{a}=Q_{a}^{v}+Q_{a}^{h}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "2.3. The $S U(2)_{R}$ currents ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "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, ", "page_idx": 4}, {"type": "equation", "text": "$$\nW_{i j}=q_{i}p_{j}-q_{j}p_{i}.\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "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 ${\\tilde{s}}^{i}$ : ", "page_idx": 4}, {"type": "equation", "text": "$$\n\\begin{array}{l}{{\\displaystyle\\tilde{s}^{2}=W_{12}-W_{34}+\\frac{i}{2}\\,\\lambda s^{2}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{3}=W_{13}+W_{24}+\\frac{i}{2}\\,\\lambda s^{3}\\lambda}}\\\\ {{\\displaystyle\\tilde{s}^{4}=W_{14}-W_{23}+\\frac{i}{2}\\,\\lambda s^{4}\\lambda.}}\\end{array}\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "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. ", "page_idx": 4}, {"type": "text", "text": "2.4. The Spin(5) currents ", "text_level": 1, "page_idx": 4}, {"type": "text", "text": "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: ", "page_idx": 4}, {"type": "equation", "text": "$$\nT^{\\mu\\nu}=x^{\\mu}p^{\\nu}-x^{\\nu}p^{\\mu}-\\frac{i}{4}\\gamma_{a b}^{\\mu\\nu}\\left(\\lambda_{a}\\lambda_{b}+\\psi_{a}\\psi_{b}\\right).\n$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Adding either more vector multiplets or more hypermultiplets is again straightforward. ", "page_idx": 4}]
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We also need to note that ", "type": "text"}, {"bbox": [324, 95, 331, 104], "score": 0.91, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [331, 92, 431, 108], "score": 1.0, "content": " is proportional to ", "type": "text"}, {"bbox": [432, 95, 458, 106], "score": 0.94, "content": "x^{\\mu}\\gamma^{\\mu}", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [458, 92, 540, 108], "score": 1.0, "content": " with a propor-", "type": "text"}], "index": 1}, {"bbox": [70, 112, 541, 127], "spans": [{"bbox": [70, 112, 300, 127], "score": 1.0, "content": "tionality constant that commutes with the ", "type": "text"}, {"bbox": [300, 113, 341, 125], "score": 0.93, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [342, 112, 541, 127], "score": 1.0, "content": " generators. We have also suppressed", "type": "text"}], "index": 2}, {"bbox": [70, 130, 540, 146], "spans": [{"bbox": [70, 130, 265, 146], "score": 1.0, "content": "gauge indices. Note that since the ", "type": "text"}, {"bbox": [266, 131, 276, 141], "score": 0.91, "content": "s^{j}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [276, 130, 540, 146], "score": 1.0, "content": " implement right multiplication by a quaternion,", "type": "text"}], "index": 3}, {"bbox": [71, 149, 539, 166], "spans": [{"bbox": [71, 149, 178, 166], "score": 1.0, "content": "they commute with ", "type": "text"}, {"bbox": [178, 152, 192, 163], "score": 0.93, "content": "\\gamma^{\\mu}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [192, 149, 539, 166], "score": 1.0, "content": ". Again, there can be many hypermultiplets in different represen-", "type": "text"}], "index": 4}, {"bbox": [70, 168, 541, 184], "spans": [{"bbox": [70, 168, 541, 184], "score": 1.0, "content": "tations of the gauge group. In that case, the hypermultiplet supercharge (2.6) generalizes", "type": "text"}], "index": 5}, {"bbox": [70, 187, 541, 202], "spans": [{"bbox": [70, 187, 541, 202], "score": 1.0, "content": "in a straightforward way. The full Hermitian supercharge is the sum of the vector and", "type": "text"}], "index": 6}, {"bbox": [70, 204, 223, 223], "spans": [{"bbox": [70, 204, 223, 223], "score": 1.0, "content": "hypermultiplet supercharges,", "type": "text"}], "index": 7}], "index": 3.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [70, 73, 541, 223]}, {"type": "interline_equation", "bbox": [265, 224, 346, 239], "lines": [{"bbox": [265, 224, 346, 239], "spans": [{"bbox": [265, 224, 346, 239], "score": 0.92, "content": "Q_{a}=Q_{a}^{v}+Q_{a}^{h}.", "type": "interline_equation"}], "index": 8}], "index": 8, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "title", "bbox": [71, 254, 211, 270], "lines": [{"bbox": [71, 255, 210, 273], "spans": [{"bbox": [71, 255, 120, 273], "score": 1.0, "content": "2.3. The ", "type": "text"}, {"bbox": [121, 259, 162, 271], "score": 0.92, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [162, 255, 210, 273], "score": 1.0, "content": " currents", "type": "text"}], "index": 9}], "index": 9, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 280, 541, 315], "lines": [{"bbox": [93, 282, 542, 300], "spans": [{"bbox": [93, 282, 221, 300], "score": 1.0, "content": "The three generators of ", "type": "text"}, {"bbox": [221, 285, 262, 297], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [262, 282, 462, 300], "score": 1.0, "content": " correspond to right multiplication by ", "type": "text"}, {"bbox": [462, 285, 497, 297], "score": 0.95, "content": "I,J,K", "type": "inline_equation", "height": 12, "width": 35}, {"bbox": [497, 282, 542, 300], "score": 1.0, "content": " and are", "type": "text"}], "index": 10}, {"bbox": [71, 304, 372, 317], "spans": [{"bbox": [71, 304, 372, 317], "score": 1.0, "content": "given in terms of the gauge invariant rotation generators,", "type": "text"}], "index": 11}], "index": 10.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [71, 282, 542, 317]}, {"type": "interline_equation", "bbox": [258, 333, 354, 348], "lines": [{"bbox": [258, 333, 354, 348], "spans": [{"bbox": [258, 333, 354, 348], "score": 0.93, "content": "W_{i j}=q_{i}p_{j}-q_{j}p_{i}.", "type": "interline_equation"}], "index": 12}], "index": 12, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 363, 541, 397], "lines": [{"bbox": [72, 366, 541, 381], "spans": [{"bbox": [72, 366, 541, 381], "score": 1.0, "content": "Again here and in the subsequent discussion, we generally suppress gauge indices. In", "type": "text"}], "index": 13}, {"bbox": [70, 383, 459, 400], "spans": [{"bbox": [70, 383, 324, 400], "score": 1.0, "content": "accord with prior notation, we denote the three ", "type": "text"}, {"bbox": [324, 385, 365, 398], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [365, 383, 443, 400], "score": 1.0, "content": " generators by ", "type": "text"}, {"bbox": [443, 385, 453, 395], "score": 0.91, "content": "{\\tilde{s}}^{i}", "type": "inline_equation", "height": 10, "width": 10}, {"bbox": [453, 383, 459, 400], "score": 1.0, "content": ":", "type": "text"}], "index": 14}], "index": 13.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [70, 366, 541, 400]}, {"type": "interline_equation", "bbox": [235, 409, 375, 497], "lines": [{"bbox": [235, 409, 375, 497], "spans": [{"bbox": [235, 409, 375, 497], "score": 0.94, "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}", "type": "interline_equation"}], "index": 15}], "index": 15, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 502, 542, 575], "lines": [{"bbox": [70, 505, 541, 521], "spans": [{"bbox": [70, 505, 541, 521], "score": 1.0, "content": "As they should, these generators act on the bosons of the hypermultiplet and the fermions", "type": "text"}], "index": 16}, {"bbox": [70, 524, 541, 540], "spans": [{"bbox": [70, 524, 541, 540], "score": 1.0, "content": "of the vector multiplet. Adding either more vector multiplets or more hypermultiplets is", "type": "text"}], "index": 17}, {"bbox": [71, 544, 540, 558], "spans": [{"bbox": [71, 544, 540, 558], "score": 1.0, "content": "straightforward: we simply need to sum the contributions to the three currents (2.8) from", "type": "text"}], "index": 18}, {"bbox": [72, 563, 149, 578], "spans": [{"bbox": [72, 563, 149, 578], "score": 1.0, "content": "each multiplet.", "type": "text"}], "index": 19}], "index": 17.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [70, 505, 541, 578]}, {"type": "title", "bbox": [71, 592, 211, 608], "lines": [{"bbox": [72, 595, 210, 608], "spans": [{"bbox": [72, 595, 210, 608], "score": 1.0, "content": "2.4. The Spin(5) currents", "type": "text"}], "index": 20}], "index": 20, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 618, 541, 653], "lines": [{"bbox": [94, 619, 541, 637], "spans": [{"bbox": [94, 619, 207, 637], "score": 1.0, "content": "The ten generators of ", "type": "text"}, {"bbox": [207, 622, 248, 634], "score": 0.84, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [248, 619, 541, 637], "score": 1.0, "content": " act on the bosons of the vector multiplet and all fermions", "type": "text"}], "index": 21}, {"bbox": [71, 641, 307, 654], "spans": [{"bbox": [71, 641, 307, 654], "score": 1.0, "content": "in the problem. The generators are given by:", "type": "text"}], "index": 22}], "index": 21.5, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [71, 619, 541, 654]}, {"type": "interline_equation", "bbox": [192, 663, 419, 690], "lines": [{"bbox": [192, 663, 419, 690], "spans": [{"bbox": [192, 663, 419, 690], "score": 0.94, "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).", "type": "interline_equation"}], "index": 23}], "index": 23, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 700, 528, 716], "lines": [{"bbox": [70, 702, 528, 718], "spans": [{"bbox": [70, 702, 528, 718], "score": 1.0, "content": "Adding either more vector multiplets or more hypermultiplets is again straightforward.", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [70, 702, 528, 718]}]}
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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,
$$
Q_{a}=Q_{a}^{v}+Q_{a}^{h}.
$$
# 2.3. The currents
The three generators of correspond to right multiplication by 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 generators by :
$$
\begin{array}{l}{{\displaystyle\tilde{s}^{2}=W_{12}-W_{34}+\frac{i}{2}\,\lambda s^{2}\lambda}}\\ {{\displaystyle\tilde{s}^{3}=W_{13}+W_{24}+\frac{i}{2}\,\lambda s^{3}\lambda}}\\ {{\displaystyle\tilde{s}^{4}=W_{14}-W_{23}+\frac{i}{2}\,\lambda s^{4}\lambda.}}\end{array}
$$
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 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.
|
<div class="pdf-page">
<p>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,</p>
<h1>2.3. The currents</h1>
<p>The three generators of correspond to right multiplication by and are given in terms of the gauge invariant rotation generators,</p>
<p>Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three generators by :</p>
<p>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>2.4. The Spin(5) currents</h1>
<p>The ten generators of act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by:</p>
<p>Adding either more vector multiplets or more hypermultiplets is again straightforward.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="115" data-y="89" data-width="791" data-height="192">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,</p>
<h1 class="pdf-title" data-x="118" data-y="328" data-width="235" data-height="21">2.3. The currents</h1>
<p class="pdf-text" data-x="117" data-y="362" data-width="788" data-height="45">The three generators of correspond to right multiplication by and are given in terms of the gauge invariant rotation generators,</p>
<p class="pdf-text" data-x="117" data-y="469" data-width="788" data-height="44">Again here and in the subsequent discussion, we generally suppress gauge indices. In accord with prior notation, we denote the three generators by :</p>
<p class="pdf-text" data-x="115" data-y="649" data-width="791" data-height="94">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 class="pdf-title" data-x="118" data-y="765" data-width="235" data-height="21">2.4. The Spin(5) currents</h1>
<p class="pdf-text" data-x="115" data-y="799" data-width="790" data-height="45">The ten generators of act on the bosons of the vector multiplet and all fermions in the problem. The generators are given by:</p>
<p class="pdf-text" data-x="117" data-y="905" data-width="766" data-height="20">Adding either more vector multiplets or more hypermultiplets is again straightforward.</p>
</div>
|
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:
$$
T^{\mu\nu}=x^{\mu}p^{\nu}-x^{\nu}p^{\mu}-\frac{i}{4}\gamma_{a b}^{\mu\nu}\left(\lambda_{a}\lambda_{b}+\psi_{a}\psi_{b}\right).
$$
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"Again here and in the subsequent discussion, we generally suppress gauge indices. In",
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"3. An Invariance Argument for the Symmetry",
"3.1. Relating the currents to the supercharge",
"A key point in the argument is a relation between the supercharge and the currents. For some choice of , we want to show that:",
"",
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"",
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"",
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"What remains is the following anti-commutator which is not hard to compute,",
"",
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"For the hypermultiplet, we take the following candidate gauge singlet:",
"",
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[{"type": "text", "text": "3. An Invariance Argument for the $S U(2)_{R}$ Symmetry ", "text_level": 1, "page_idx": 5}, {"type": "text", "text": "3.1. Relating the $S U(2)_{R}$ currents to the supercharge ", "page_idx": 5}, {"type": "text", "text": "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: ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\tilde{s}^{i}=\\sum_{a}\\;\\{Q_{a},v_{a}^{i}\\}.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "Let us start with the vector multiplet. We take a candidate gauge singlet, ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\left(v_{1}\\right)_{a}^{i}=\\left(s^{i}\\gamma^{\\nu}\\lambda\\right)_{a}x^{\\nu}.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "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: ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{D_{a b}\\lambda_{b},(v_{1})_{a}^{i}\\right\\}=x_{A}^{\\nu}\\mathrm{tr}\\left(s^{i}\\gamma^{\\nu}D_{A}^{T}\\right),\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "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 $S p i n(5)$ . ", "page_idx": 5}, {"type": "text", "text": "What remains is the following anti-commutator which is not hard to compute, ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{(\\gamma^{\\mu}p^{\\mu}\\lambda)_{a}\\,,(v_{1})_{a}^{i}\\right\\}\\sim i\\,\\lambda s^{i}\\lambda.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "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. ", "page_idx": 5}, {"type": "text", "text": "For the hypermultiplet, we take the following candidate gauge singlet: ", "page_idx": 5}, {"type": "equation", "text": "$$\n(v_{2})_{a}^{i}=\\left(s^{i}s^{l}\\psi\\right)_{a}q^{l}.\n$$", "text_format": "latex", "page_idx": 5}, {"type": "text", "text": "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, ", "page_idx": 5}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{I_{a b}\\psi_{b},(v_{2})_{a}^{i}\\right\\}\\sim q^{l}\\mathrm{tr}\\left(s^{i}s^{l}I\\right),\n$$", "text_format": "latex", "page_idx": 5}]
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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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}]}
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# 3. An Invariance Argument for the Symmetry
3.1. Relating the currents to the supercharge
A key point in the argument is a relation between the supercharge and the currents. For some choice of , 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 because anti-commutes with . It also anti-commutes with the -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 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 .
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 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 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,
$$
\sum_{a}\left\{I_{a b}\psi_{b},(v_{2})_{a}^{i}\right\}\sim q^{l}\mathrm{tr}\left(s^{i}s^{l}I\right),
$$
5
|
<div class="pdf-page">
<h1>3. An Invariance Argument for the Symmetry</h1>
<p>3.1. Relating the currents to the supercharge</p>
<p>A key point in the argument is a relation between the supercharge and the currents. For some choice of , we want to show that:</p>
<p>Let us start with the vector multiplet. We take a candidate gauge singlet,</p>
<p>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:</p>
<p>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 .</p>
<p>What remains is the following anti-commutator which is not hard to compute,</p>
<p>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.</p>
<p>For the hypermultiplet, we take the following candidate gauge singlet:</p>
<p>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,</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="117" data-y="90" data-width="545" data-height="21">3. An Invariance Argument for the Symmetry</h1>
<p class="pdf-text" data-x="118" data-y="125" data-width="470" data-height="21">3.1. Relating the currents to the supercharge</p>
<p class="pdf-text" data-x="115" data-y="159" data-width="790" data-height="45">A key point in the argument is a relation between the supercharge and the currents. For some choice of , we want to show that:</p>
<p class="pdf-text" data-x="117" data-y="275" data-width="652" data-height="22">Let us start with the vector multiplet. We take a candidate gauge singlet,</p>
<p class="pdf-text" data-x="115" data-y="360" data-width="788" data-height="47">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:</p>
<p class="pdf-text" data-x="115" data-y="477" data-width="791" data-height="95">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 .</p>
<p class="pdf-text" data-x="155" data-y="576" data-width="693" data-height="22">What remains is the following anti-commutator which is not hard to compute,</p>
<p class="pdf-text" data-x="115" data-y="661" data-width="790" data-height="72">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.</p>
<p class="pdf-text" data-x="155" data-y="735" data-width="623" data-height="22">For the hypermultiplet, we take the following candidate gauge singlet:</p>
<p class="pdf-text" data-x="117" data-y="822" data-width="788" data-height="67">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,</p>
<div class="pdf-discarded" data-x="501" data-y="943" data-width="19" data-height="16" style="opacity: 0.5;">5</div>
</div>
|
# 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,
$$
\left(v_{1}\right)_{a}^{i}=\left(s^{i}\gamma^{\nu}\lambda\right)_{a}x^{\nu}.
$$
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[{"type": "text", "text": "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. ", "page_idx": 6}, {"type": "text", "text": "Again what remains is the anti-commutator, ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "It is easy to check that the $\\psi\\psi$ terms in the anti-commutator vanish because, ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\sum_{k}\\,\\psi\\{s^{k}\\}^{T}s^{i}s^{k}\\psi=0.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "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 ", "page_idx": 6}, {"type": "equation", "text": "$$\nv_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "satisfies (3.1). ", "page_idx": 6}, {"type": "text", "text": "3.2. Rotating a ground state ", "page_idx": 6}, {"type": "text", "text": "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, ", "page_idx": 6}, {"type": "equation", "text": "$$\n<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty.\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "For some $\\tilde{s}^{i}$ , the state ${\\tilde{s}}^{i}\\Psi$ is a non-trivial ground state. It satisfies the relation, ", "page_idx": 6}, {"type": "equation", "text": "$$\nQ_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,\n$$", "text_format": "latex", "page_idx": 6}, {"type": "text", "text": "for each $a$ by definition of a ground state. Using (3.1), we find that ", "page_idx": 6}, {"type": "equation", "text": "$$\n\\begin{array}{r c l}{{}}&{{}}&{{\\tilde{s}^{i}\\Psi=\\displaystyle\\sum_{a}\\,\\left\\{Q_{a},v_{a}^{i}\\right\\}\\Psi,}}\\\\ {{}}&{{}}&{{=\\displaystyle\\sum_{a}Q_{a}\\left(v_{a}^{i}\\Psi\\right).}}\\end{array}\n$$", "text_format": "latex", "page_idx": 6}]
|
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We therefore conclude that for appropriately", "type": "text"}], "index": 7}, {"bbox": [71, 301, 277, 315], "spans": [{"bbox": [71, 301, 162, 315], "score": 1.0, "content": "chosen constants ", "type": "text"}, {"bbox": [163, 306, 175, 313], "score": 0.9, "content": "\\alpha_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [176, 301, 202, 315], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [203, 306, 216, 313], "score": 0.87, "content": "\\alpha_{2}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [216, 301, 277, 315], "score": 1.0, "content": ", the choice", "type": "text"}], "index": 8}], "index": 7}, {"type": "interline_equation", "bbox": [245, 332, 366, 348], "lines": [{"bbox": [245, 332, 366, 348], "spans": [{"bbox": [245, 332, 366, 348], "score": 0.93, "content": "v_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "text", "bbox": [70, 366, 145, 381], "lines": [{"bbox": [70, 368, 144, 384], "spans": [{"bbox": [70, 368, 144, 384], "score": 1.0, "content": "satisfies (3.1).", "type": "text"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [71, 401, 222, 416], "lines": [{"bbox": [72, 404, 221, 417], "spans": [{"bbox": [72, 404, 221, 417], "score": 1.0, "content": "3.2. Rotating a ground state", "type": "text"}], "index": 11}], "index": 11}, {"type": "text", "bbox": [70, 427, 542, 502], "lines": [{"bbox": [95, 430, 540, 445], "spans": [{"bbox": [95, 430, 376, 445], "score": 1.0, "content": "We assume there exists a normalizable ground state ", "type": "text"}, {"bbox": [376, 433, 386, 441], "score": 0.89, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [387, 430, 540, 445], "score": 1.0, "content": " which is not a singlet under", "type": "text"}], "index": 12}, {"bbox": [71, 449, 540, 465], "spans": [{"bbox": [71, 451, 112, 464], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [112, 449, 185, 465], "score": 1.0, "content": ". Under some ", "type": "text"}, {"bbox": [185, 451, 225, 464], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [226, 449, 431, 465], "score": 1.0, "content": " rotation, we obtain another non-trivial ", "type": "text"}, {"bbox": [431, 450, 445, 461], "score": 0.9, "content": "L^{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [445, 449, 540, 465], "score": 1.0, "content": " zero-energy state.", "type": "text"}], "index": 13}, {"bbox": [70, 468, 539, 485], "spans": [{"bbox": [70, 468, 131, 485], "score": 1.0, "content": "What does ", "type": "text"}, {"bbox": [131, 470, 145, 480], "score": 0.91, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [145, 468, 468, 485], "score": 1.0, "content": " imply? Let us collectively denote all the bosonic coordinates ", "type": "text"}, {"bbox": [469, 473, 476, 480], "score": 0.69, "content": "x", "type": "inline_equation", "height": 7, "width": 7}, {"bbox": [477, 468, 502, 485], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [503, 475, 509, 483], "score": 0.87, "content": "q", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [509, 468, 529, 485], "score": 1.0, "content": " by ", "type": "text"}, {"bbox": [529, 470, 539, 483], "score": 0.9, "content": "y^{i}", "type": "inline_equation", "height": 13, "width": 10}], "index": 14}, {"bbox": [70, 489, 333, 505], "spans": [{"bbox": [70, 489, 105, 505], "score": 1.0, "content": "where ", "type": "text"}, {"bbox": [106, 492, 169, 502], "score": 0.94, "content": "i=1,\\dots,D", "type": "inline_equation", "height": 10, "width": 63}, {"bbox": [169, 489, 333, 505], "score": 1.0, "content": ". Normalizability requires that,", "type": "text"}], "index": 15}], "index": 13.5}, {"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}, {"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}, {"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}], "layout_bboxes": [], "page_idx": 6, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [249, 141, 362, 172], "lines": [{"bbox": [249, 141, 362, 172], "spans": [{"bbox": [249, 141, 362, 172], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}.", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "interline_equation", "bbox": [246, 217, 364, 247], "lines": [{"bbox": [246, 217, 364, 247], "spans": [{"bbox": [246, 217, 364, 247], "score": 0.92, "content": "\\sum_{k}\\,\\psi\\{s^{k}\\}^{T}s^{i}s^{k}\\psi=0.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [245, 332, 366, 348], "lines": [{"bbox": [245, 332, 366, 348], "spans": [{"bbox": [245, 332, 366, 348], "score": 0.93, "content": "v_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}", "type": "interline_equation"}], "index": 9}], "index": 9}, {"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": "interline_equation", "bbox": [249, 592, 362, 609], "lines": [{"bbox": [249, 592, 362, 609], "spans": [{"bbox": [249, 592, 362, 609], 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"but ", "type": "text"}, {"bbox": [94, 74, 120, 84], "score": 0.93, "content": "s^{i}s^{l}I", "type": "inline_equation", "height": 10, "width": 26}, {"bbox": [120, 71, 325, 90], "score": 1.0, "content": " does not contain a singlet under the ", "type": "text"}, {"bbox": [325, 75, 366, 87], "score": 0.86, "content": "S p i n(5)", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [366, 71, 519, 90], "score": 1.0, "content": " action on fermions because ", "type": "text"}, {"bbox": [520, 75, 527, 84], "score": 0.88, "content": "I", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [527, 71, 542, 90], "score": 1.0, "content": " is", "type": "text"}], "index": 0}, {"bbox": [72, 94, 284, 108], "spans": [{"bbox": [72, 94, 154, 108], "score": 1.0, "content": "proportional to ", "type": "text"}, {"bbox": [154, 95, 168, 106], "score": 0.92, "content": "\\gamma^{\\mu}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [168, 94, 284, 108], "score": 1.0, "content": " so the trace vanishes.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [69, 71, 542, 108]}, {"type": "text", "bbox": [93, 110, 330, 125], "lines": [{"bbox": [96, 113, 329, 127], "spans": [{"bbox": [96, 113, 329, 127], "score": 1.0, "content": "Again what remains is the anti-commutator,", "type": "text"}], "index": 2}], "index": 2, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [96, 113, 329, 127]}, {"type": "interline_equation", "bbox": [249, 141, 362, 172], "lines": [{"bbox": [249, 141, 362, 172], "spans": [{"bbox": [249, 141, 362, 172], "score": 0.93, "content": "\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}.", "type": "interline_equation"}], "index": 3}], "index": 3, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 183, 477, 199], "lines": [{"bbox": [69, 185, 477, 202], "spans": [{"bbox": [69, 185, 215, 202], "score": 1.0, "content": "It is easy to check that the ", "type": "text"}, {"bbox": [216, 188, 232, 199], "score": 0.93, "content": "\\psi\\psi", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [233, 185, 477, 202], "score": 1.0, "content": " terms in the anti-commutator vanish because,", "type": "text"}], "index": 4}], "index": 4, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [69, 185, 477, 202]}, {"type": "interline_equation", "bbox": [246, 217, 364, 247], "lines": [{"bbox": [246, 217, 364, 247], "spans": [{"bbox": [246, 217, 364, 247], "score": 0.92, "content": "\\sum_{k}\\,\\psi\\{s^{k}\\}^{T}s^{i}s^{k}\\psi=0.", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 258, 541, 313], "lines": [{"bbox": [70, 261, 540, 277], "spans": [{"bbox": [70, 261, 540, 277], "score": 1.0, "content": "With a little additional work, we find that (3.7) gives precisely the bosonic terms in (2.8)", "type": "text"}], "index": 6}, {"bbox": [69, 280, 540, 297], "spans": [{"bbox": [69, 280, 540, 297], "score": 1.0, "content": "up to an overall non-vanishing constant. We therefore conclude that for appropriately", "type": "text"}], "index": 7}, {"bbox": [71, 301, 277, 315], "spans": [{"bbox": [71, 301, 162, 315], "score": 1.0, "content": "chosen constants ", "type": "text"}, {"bbox": [163, 306, 175, 313], "score": 0.9, "content": "\\alpha_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [176, 301, 202, 315], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [203, 306, 216, 313], "score": 0.87, "content": "\\alpha_{2}", "type": "inline_equation", "height": 7, "width": 13}, {"bbox": [216, 301, 277, 315], "score": 1.0, "content": ", the choice", "type": "text"}], "index": 8}], "index": 7, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [69, 261, 540, 315]}, {"type": "interline_equation", "bbox": [245, 332, 366, 348], "lines": [{"bbox": [245, 332, 366, 348], "spans": [{"bbox": [245, 332, 366, 348], "score": 0.93, "content": "v_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 366, 145, 381], "lines": [{"bbox": [70, 368, 144, 384], "spans": [{"bbox": [70, 368, 144, 384], "score": 1.0, "content": "satisfies (3.1).", "type": "text"}], "index": 10}], "index": 10, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [70, 368, 144, 384]}, {"type": "text", "bbox": [71, 401, 222, 416], "lines": [{"bbox": [72, 404, 221, 417], "spans": [{"bbox": [72, 404, 221, 417], "score": 1.0, "content": "3.2. Rotating a ground state", "type": "text"}], "index": 11}], "index": 11, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [72, 404, 221, 417]}, {"type": "text", "bbox": [70, 427, 542, 502], "lines": [{"bbox": [95, 430, 540, 445], "spans": [{"bbox": [95, 430, 376, 445], "score": 1.0, "content": "We assume there exists a normalizable ground state ", "type": "text"}, {"bbox": [376, 433, 386, 441], "score": 0.89, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [387, 430, 540, 445], "score": 1.0, "content": " which is not a singlet under", "type": "text"}], "index": 12}, {"bbox": [71, 449, 540, 465], "spans": [{"bbox": [71, 451, 112, 464], "score": 0.94, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [112, 449, 185, 465], "score": 1.0, "content": ". Under some ", "type": "text"}, {"bbox": [185, 451, 225, 464], "score": 0.95, "content": "S U(2)_{R}", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [226, 449, 431, 465], "score": 1.0, "content": " rotation, we obtain another non-trivial ", "type": "text"}, {"bbox": [431, 450, 445, 461], "score": 0.9, "content": "L^{2}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [445, 449, 540, 465], "score": 1.0, "content": " zero-energy state.", "type": "text"}], "index": 13}, {"bbox": [70, 468, 539, 485], "spans": [{"bbox": [70, 468, 131, 485], "score": 1.0, "content": "What does ", "type": "text"}, {"bbox": [131, 470, 145, 480], "score": 0.91, "content": "L^{2}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [145, 468, 468, 485], "score": 1.0, "content": " imply? 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Normalizability requires that,", "type": "text"}], "index": 15}], "index": 13.5, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}]}
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|
but does not contain a singlet under the action on fermions because is proportional to 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 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 and , 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 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,
$$
<\Psi,\Psi>=\int d^{D}y\,\Psi^{\dag}(y^{i})\,\Psi(y^{i})<\infty.
$$
For some , the state is a non-trivial ground state. It satisfies the relation,
$$
Q_{a}\left(\tilde{s}^{i}\Psi\right)=Q_{a}\Psi=0,
$$
for each by definition of a ground state. Using (3.1), we find that
$$
\begin{array}{r c l}{{}}&{{}}&{{\tilde{s}^{i}\Psi=\displaystyle\sum_{a}\,\left\{Q_{a},v_{a}^{i}\right\}\Psi,}}\\ {{}}&{{}}&{{=\displaystyle\sum_{a}Q_{a}\left(v_{a}^{i}\Psi\right).}}\end{array}
$$
6
|
<div class="pdf-page">
<p>but does not contain a singlet under the action on fermions because is proportional to so the trace vanishes.</p>
<p>Again what remains is the anti-commutator,</p>
<p>It is easy to check that the terms in the anti-commutator vanish because,</p>
<p>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</p>
<p>satisfies (3.1).</p>
<p>3.2. Rotating a ground state</p>
<p>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,</p>
<p>For some , the state is a non-trivial ground state. It satisfies the relation,</p>
<p>for each by definition of a ground state. Using (3.1), we find that</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="115" data-y="90" data-width="790" data-height="45">but does not contain a singlet under the action on fermions because is proportional to so the trace vanishes.</p>
<p class="pdf-text" data-x="155" data-y="142" data-width="397" data-height="19">Again what remains is the anti-commutator,</p>
<p class="pdf-text" data-x="117" data-y="236" data-width="681" data-height="21">It is easy to check that the terms in the anti-commutator vanish because,</p>
<p class="pdf-text" data-x="117" data-y="333" data-width="788" data-height="71">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</p>
<p class="pdf-text" data-x="117" data-y="473" data-width="125" data-height="19">satisfies (3.1).</p>
<p class="pdf-text" data-x="118" data-y="518" data-width="253" data-height="19">3.2. Rotating a ground state</p>
<p class="pdf-text" data-x="117" data-y="552" data-width="789" data-height="97">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,</p>
<p class="pdf-text" data-x="115" data-y="720" data-width="704" data-height="20">For some , the state is a non-trivial ground state. It satisfies the relation,</p>
<p class="pdf-text" data-x="115" data-y="809" data-width="597" data-height="19">for each by definition of a ground state. Using (3.1), we find that</p>
<div class="pdf-discarded" data-x="503" data-y="945" data-width="17" data-height="13" style="opacity: 0.5;">6</div>
</div>
|
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,
$$
Q_{a}\left(\tilde{s}^{i}\Psi\right)=Q_{a}\Psi=0,
$$
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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}"
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"",
"",
"",
"",
"",
"",
"\\sum_{a}\\left\\{s_{a b}^{j}\\psi_{b}\\,p_{j},(v_{2})_{a}^{i}\\right\\}.",
"\\sum_{k}\\,\\psi\\{s^{k}\\}^{T}s^{i}s^{k}\\psi=0.",
"v_{a}^{i}=\\alpha_{1}(v_{1})_{a}^{i}+\\alpha_{2}(v_{2})_{a}^{i}",
"<\\Psi,\\Psi>=\\int d^{D}y\\,\\Psi^{\\dag}(y^{i})\\,\\Psi(y^{i})<\\infty.",
"Q_{a}\\left(\\tilde{s}^{i}\\Psi\\right)=Q_{a}\\Psi=0,",
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"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 ,",
"",
"Using (3.9) and (3.10), we see that",
"",
"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.",
"4. Invariance Under the Symmetry",
"4.1. Relating the Spin(5) currents to the supercharge",
"We want to use essentially the same argument as in the case. For some choice of , we want to show that:",
"",
"Let us start with the vector multiplet. We take a candidate gauge singlet,",
"",
"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:",
"",
"However, this combination does not contain a singlet under so (4.3) vanishes."
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[{"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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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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [70, 701, 521, 719]}]}
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|
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 ,
$$
<\tilde{s}^{i}\Psi,\tilde{s}^{i}\Psi>=\operatorname*{lim}_{R\rightarrow\infty}<\rho_{R}(y)\tilde{s}^{i}\Psi,\tilde{s}^{i}\Psi>.
$$
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 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.
# 4. Invariance Under the Symmetry
4.1. Relating the Spin(5) currents to the supercharge
We want to use essentially the same argument as in the case. For some choice of , 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 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:
$$
\sum_{a}\left\{D_{a b}\lambda_{b},(v_{1})_{a}^{\mu\nu}\right\}=D_{a b}\left\{\gamma^{\mu}x^{\nu}-\gamma^{\nu}x^{\mu}\right\}_{a b}.
$$
However, this combination does not contain a singlet under so (4.3) vanishes.
|
<div class="pdf-page">
<p>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 ,</p>
<p>Using (3.9) and (3.10), we see that</p>
<p>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.</p>
<h1>4. Invariance Under the Symmetry</h1>
<p>4.1. Relating the Spin(5) currents to the supercharge</p>
<p>We want to use essentially the same argument as in the case. For some choice of , we want to show that:</p>
<p>Let us start with the vector multiplet. We take a candidate gauge singlet,</p>
<p>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:</p>
<p>However, this combination does not contain a singlet under so (4.3) vanishes.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="115" data-y="89" data-width="791" data-height="121">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 ,</p>
<p class="pdf-text" data-x="117" data-y="270" data-width="311" data-height="20">Using (3.9) and (3.10), we see that</p>
<p class="pdf-text" data-x="117" data-y="398" data-width="788" data-height="70">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.</p>
<h1 class="pdf-title" data-x="120" data-y="506" data-width="432" data-height="21">4. Invariance Under the Symmetry</h1>
<p class="pdf-text" data-x="120" data-y="541" data-width="468" data-height="20">4.1. Relating the Spin(5) currents to the supercharge</p>
<p class="pdf-text" data-x="115" data-y="574" data-width="791" data-height="44">We want to use essentially the same argument as in the case. For some choice of , we want to show that:</p>
<p class="pdf-text" data-x="117" data-y="687" data-width="652" data-height="21">Let us start with the vector multiplet. We take a candidate gauge singlet,</p>
<p class="pdf-text" data-x="115" data-y="768" data-width="791" data-height="71">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:</p>
<p class="pdf-text" data-x="117" data-y="905" data-width="754" data-height="20">However, this combination does not contain a singlet under so (4.3) vanishes.</p>
</div>
|
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:
$$
\sum_{a}\left\{D_{a b}\lambda_{b},(v_{1})_{a}^{\mu\nu}\right\}=D_{a b}\left\{\gamma^{\mu}x^{\nu}-\gamma^{\nu}x^{\mu}\right\}_{a b}.
$$
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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 anti-",
"commutator 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."
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"<\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>=\\operatorname*{lim}_{R\\rightarrow\\infty}<\\rho_{R}(y)\\tilde{s}^{i}\\Psi,\\tilde{s}^{i}\\Psi>.",
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"T^{\\mu\\nu}=\\sum_{a}\\,\\{Q_{a},v_{a}^{\\mu\\nu}\\}.",
"(v_{1})_{a}^{\\mu\\nu}=\\{\\gamma^{\\mu}x^{\\nu}-\\gamma^{\\nu}x^{\\mu}\\}_{a b}\\,\\lambda_{b}.",
"\\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}."
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"We are left with the following anti-commutator which we need to compute quite carefully,",
"",
"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.",
"For the hypermultiplet, we take the following choice:",
"",
"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,",
"",
"but again does not contain a singlet under so the trace vanishes.",
"The remaining anti-commutator involves the kinetic term in the hypermultiplet charge,",
"",
"Again we conclude that for appropriately chosen constants and , the choice",
"",
"satisfies (4.1). A straightforward repeat of the argument given in section then implies that the 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 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."
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[{"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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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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [70, 627, 541, 717]}]}
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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 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.
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 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,
$$
\sum_{a}\left\{I_{a b}\psi_{b},(v_{2})_{a}^{\mu\nu}\right\}\sim q^{i}\mathrm{tr}\left(I\gamma^{\mu\nu}s^{i}\right),
$$
but again does not contain a singlet under 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 and , 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 then implies that the 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 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.
|
<div class="pdf-page">
<p>We are left with the following anti-commutator which we need to compute quite carefully,</p>
<p>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.</p>
<p>For the hypermultiplet, we take the following choice:</p>
<p>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,</p>
<p>but again does not contain a singlet under so the trace vanishes.</p>
<p>The remaining anti-commutator involves the kinetic term in the hypermultiplet charge,</p>
<p>Again we conclude that for appropriately chosen constants and , the choice</p>
<p>satisfies (4.1). A straightforward repeat of the argument given in section then implies that the symmetry acts trivially on all normalizable ground states.</p>
<h1>4.2. Theories with sixteen supercharges</h1>
<p>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.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="115" data-y="90" data-width="788" data-height="40">We are left with the following anti-commutator which we need to compute quite carefully,</p>
<p class="pdf-text" data-x="115" data-y="174" data-width="791" data-height="119">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.</p>
<p class="pdf-text" data-x="155" data-y="296" data-width="469" data-height="20">For the hypermultiplet, we take the following choice:</p>
<p class="pdf-text" data-x="115" data-y="377" data-width="790" data-height="69">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,</p>
<p class="pdf-text" data-x="117" data-y="512" data-width="706" data-height="19">but again does not contain a singlet under so the trace vanishes.</p>
<p class="pdf-text" data-x="117" data-y="536" data-width="786" data-height="41">The remaining anti-commutator involves the kinetic term in the hypermultiplet charge,</p>
<p class="pdf-text" data-x="117" data-y="625" data-width="716" data-height="22">Again we conclude that for appropriately chosen constants and , the choice</p>
<p class="pdf-text" data-x="117" data-y="707" data-width="788" data-height="45">satisfies (4.1). A straightforward repeat of the argument given in section then implies that the symmetry acts trivially on all normalizable ground states.</p>
<h1 class="pdf-title" data-x="117" data-y="773" data-width="348" data-height="20">4.2. Theories with sixteen supercharges</h1>
<p class="pdf-text" data-x="117" data-y="806" data-width="789" data-height="119">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.</p>
</div>
|
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
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"We are left with the following anti-commutator which we need to compute quite",
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"\\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.",
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"(v_{2})_{a}^{\\mu\\nu}=\\left(\\gamma^{\\mu\\nu}s^{i}\\psi\\right)_{a}q^{i}.",
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"4.2. Theories with sixteen supercharges",
"For theories obtained by reduction from ten dimensions, the previous argument does",
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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 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,
$$
{\bf9}\quad\rightarrow\quad({\bf5},{\bf1},{\bf1})\oplus({\bf1},{\bf2},{\bf2}).
$$
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 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.
# 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.
9
|
<div class="pdf-page">
<p>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,</p>
<p>The fermions decompose according to,</p>
<p>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.</p>
<h1>Acknowledgements</h1>
<p>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>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="117" data-y="90" data-width="789" data-height="120">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,</p>
<p class="pdf-text" data-x="117" data-y="250" data-width="341" data-height="20">The fermions decompose according to,</p>
<p class="pdf-text" data-x="117" data-y="336" data-width="789" data-height="97">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.</p>
<h1 class="pdf-title" data-x="414" data-y="491" data-width="195" data-height="21">Acknowledgements</h1>
<p class="pdf-text" data-x="115" data-y="515" data-width="790" data-height="47">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>
<div class="pdf-discarded" data-x="503" data-y="945" data-width="17" data-height="13" style="opacity: 0.5;">9</div>
</div>
|
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
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[{"type": "text", "text": "Appendix A. Quaternions and Symplectic Groups ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "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, ", "page_idx": 10}, {"type": "equation", "text": "$$\nI^{2}=J^{2}=K^{2}=-{\\bf1},\\qquad I J K=-{\\bf1}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "A quaternion $q$ can then be expanded in components ", "page_idx": 10}, {"type": "equation", "text": "$$\nq=q^{1}+I q^{2}+J q^{3}+K q^{4}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "The conjugate quaternion $q$ has an expansion ", "page_idx": 10}, {"type": "equation", "text": "$$\nq=q^{1}-I q^{2}-J q^{3}-K q^{4}.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "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, ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\Lambda\\to\\Lambda g.\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "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, ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\Lambda_{a}\\rightarrow g_{a b}\\Lambda_{b},\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "where $g_{a b}$ implements right multiplication by the unit quaternion $g$ . For example, right multiplication by $I$ on $q$ gives ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\begin{array}{l}{{q\\to q I}}\\\\ {{\\to q^{1}I-q^{2}-q^{3}K+q^{4}J,}}\\end{array}\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "which can be realized by the matrix ", "page_idx": 10}, {"type": "equation", "text": "$$\nI^{R}=\\left(\\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{1}}\\\\ {{0}}&{{0}}&{{-1}}&{{0}}\\end{array}\\right)\n$$", "text_format": "latex", "page_idx": 10}, {"type": "text", "text": "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: ", "page_idx": 10}, {"type": "equation", "text": "$$\n\\begin{array}{c c}{{J^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{1}}&{{0}}&{{0}}\\end{array}\\!\\!\\right),~~~}}&{{K^{R}=\\left(\\!\\!\\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\\\ {{0}}&{{0}}&{{1}}&{{0}}\\\\ {{0}}&{{-1}}&{{0}}&{{0}}\\\\ {{1}}&{{0}}&{{0}}&{{0}}\\end{array}\\!\\!\\right).}}\\end{array}\n$$", "text_format": "latex", "page_idx": 10}]
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# 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 where,
$$
I^{2}=J^{2}=K^{2}=-{\bf1},\qquad I J K=-{\bf1}.
$$
A quaternion can then be expanded in components
$$
q=q^{1}+I q^{2}+J q^{3}+K q^{4}.
$$
The conjugate quaternion has an expansion
$$
q=q^{1}-I q^{2}-J q^{3}-K q^{4}.
$$
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,
$$
\Lambda\to\Lambda g.
$$
In this formalism, is valued in the quaternions. Equivalently, we can expand in components and express the action of in the following way,
$$
\Lambda_{a}\rightarrow g_{a b}\Lambda_{b},
$$
where implements right multiplication by the unit quaternion . For example, right multiplication by on 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 in the usual way . The matrices and realize right multipli- cation by while is the identity matrix:
$$
\begin{array}{c c}{{J^{R}=\left(\!\!\begin{array}{c c c c}{{0}}&{{0}}&{{-1}}&{{0}}\\ {{0}}&{{0}}&{{0}}&{{-1}}\\ {{1}}&{{0}}&{{0}}&{{0}}\\ {{0}}&{{1}}&{{0}}&{{0}}\end{array}\!\!\right),~~~}}&{{K^{R}=\left(\!\!\begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{-1}}\\ {{0}}&{{0}}&{{1}}&{{0}}\\ {{0}}&{{-1}}&{{0}}&{{0}}\\ {{1}}&{{0}}&{{0}}&{{0}}\end{array}\!\!\right).}}\end{array}
$$
10
|
<div class="pdf-page">
<h1>Appendix A. Quaternions and Symplectic Groups</h1>
<p>We will summarize some useful relations between quaternions and symplectic groups. Let us label a basis for our quaternions by where,</p>
<p>A quaternion can then be expanded in components</p>
<p>The conjugate quaternion has an expansion</p>
<p>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,</p>
<p>In this formalism, is valued in the quaternions. Equivalently, we can expand in components and express the action of in the following way,</p>
<p>where implements right multiplication by the unit quaternion . For example, right multiplication by on gives</p>
<p>which can be realized by the matrix</p>
<p>acting on in the usual way . The matrices and realize right multipli- cation by while is the identity matrix:</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="118" data-y="90" data-width="507" data-height="21">Appendix A. Quaternions and Symplectic Groups</h1>
<p class="pdf-text" data-x="115" data-y="122" data-width="788" data-height="46">We will summarize some useful relations between quaternions and symplectic groups. Let us label a basis for our quaternions by where,</p>
<p class="pdf-text" data-x="118" data-y="221" data-width="469" data-height="19">A quaternion can then be expanded in components</p>
<p class="pdf-text" data-x="117" data-y="293" data-width="405" data-height="19">The conjugate quaternion has an expansion</p>
<p class="pdf-text" data-x="117" data-y="364" data-width="789" data-height="70">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,</p>
<p class="pdf-text" data-x="117" data-y="470" data-width="788" data-height="44">In this formalism, is valued in the quaternions. Equivalently, we can expand in components and express the action of in the following way,</p>
<p class="pdf-text" data-x="117" data-y="566" data-width="786" data-height="45">where implements right multiplication by the unit quaternion . For example, right multiplication by on gives</p>
<p class="pdf-text" data-x="117" data-y="685" data-width="321" data-height="18">which can be realized by the matrix</p>
<p class="pdf-text" data-x="115" data-y="797" data-width="788" data-height="45">acting on in the usual way . The matrices and realize right multipli- cation by while is the identity matrix:</p>
<div class="pdf-discarded" data-x="500" data-y="943" data-width="23" data-height="16" style="opacity: 0.5;">10</div>
</div>
|
# 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
$$
I^{R}=\left(\begin{array}{c c c c}{{0}}&{{-1}}&{{0}}&{{0}}\\ {{1}}&{{0}}&{{0}}&{{0}}\\ {{0}}&{{0}}&{{0}}&{{1}}\\ {{0}}&{{0}}&{{-1}}&{{0}}\end{array}\right)
$$
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[{"type": "text", "text": "We define operators $s^{j}$ in terms of $\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}$ ", "page_idx": 11}, {"type": "equation", "text": "$$\n{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "In a similar way, the group $S p(2)\\sim S p i n(5)$ is the group of quaternion-valued $2\\times2$ matrices with unit determinant. We will view $S p(2)$ as acting by left multiplication on a field $\\Psi$ in the defining representation. So an element $U\\in S p(2)$ acts in the following way: ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\Psi\\to U\\Psi.\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "Equivalently, in terms of components ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\Psi_{a}\\to U_{a b}\\Psi_{b}.\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions: ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,\n$$", "text_format": "latex", "page_idx": 11}, {"type": "equation", "text": "$$\n\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "In turn, $\\{I,J,K\\}$ can be expressed in terms of the Pauli matrices $\\sigma^{i}$ ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "as $4\\times4$ real anti-symmetric matrices: ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}\n$$", "text_format": "latex", "page_idx": 11}]
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{"preproc_blocks": [{"type": "text", "bbox": [69, 69, 350, 86], "lines": [{"bbox": [70, 71, 348, 88], "spans": [{"bbox": [70, 71, 179, 88], "score": 1.0, "content": "We define operators ", "type": "text"}, {"bbox": [179, 73, 190, 84], "score": 0.89, "content": "s^{j}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [190, 71, 255, 88], "score": 1.0, "content": " in terms of", "type": "text"}, {"bbox": [255, 73, 348, 88], "score": 0.9, "content": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}", "type": "inline_equation", "height": 15, "width": 93}], "index": 0}], "index": 0}, {"type": "interline_equation", "bbox": [98, 101, 500, 133], "lines": [{"bbox": [98, 101, 500, 133], "spans": [{"bbox": [98, 101, 500, 133], "score": 0.86, "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "text", "bbox": [70, 144, 542, 198], "lines": [{"bbox": [93, 147, 540, 163], "spans": [{"bbox": [93, 147, 241, 163], "score": 1.0, "content": "In a similar way, the group ", "type": "text"}, {"bbox": [242, 147, 329, 160], "score": 0.92, "content": "S p(2)\\sim S p i n(5)", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [330, 147, 512, 163], "score": 1.0, "content": " is the group of quaternion-valued ", "type": "text"}, {"bbox": [513, 149, 540, 158], "score": 0.91, "content": "2\\times2", "type": "inline_equation", "height": 9, "width": 27}], "index": 2}, {"bbox": [70, 165, 541, 182], "spans": [{"bbox": [70, 165, 316, 182], "score": 1.0, "content": "matrices with unit determinant. We will view ", "type": "text"}, {"bbox": [317, 167, 346, 180], "score": 0.93, "content": "S p(2)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [347, 165, 541, 182], "score": 1.0, "content": " as acting by left multiplication on a", "type": "text"}], "index": 3}, {"bbox": [70, 184, 541, 201], "spans": [{"bbox": [70, 184, 97, 201], "score": 1.0, "content": "field ", "type": "text"}, {"bbox": [97, 187, 106, 195], "score": 0.91, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [107, 184, 349, 201], "score": 1.0, "content": " in the defining representation. So an element ", "type": "text"}, {"bbox": [349, 186, 403, 199], "score": 0.94, "content": "U\\in S p(2)", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [403, 184, 541, 201], "score": 1.0, "content": " acts in the following way:", "type": "text"}], "index": 4}], "index": 3}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "text", "bbox": [69, 249, 267, 264], "lines": [{"bbox": [71, 251, 267, 266], "spans": [{"bbox": [71, 251, 267, 266], "score": 1.0, "content": "Equivalently, in terms of components", "type": "text"}], "index": 6}], "index": 6}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [69, 315, 541, 332], "lines": [{"bbox": [69, 318, 541, 335], "spans": [{"bbox": [69, 318, 541, 335], "score": 1.0, "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "type": "text"}], "index": 8}], "index": 8}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 426, 432, 442], "lines": [{"bbox": [70, 430, 430, 443], "spans": [{"bbox": [70, 430, 116, 443], "score": 1.0, "content": "In turn, ", "type": "text"}, {"bbox": [116, 431, 163, 443], "score": 0.94, "content": "\\{I,J,K\\}", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [163, 430, 418, 443], "score": 1.0, "content": " can be expressed in terms of the Pauli matrices ", "type": "text"}, {"bbox": [419, 430, 430, 440], "score": 0.89, "content": "\\sigma^{i}", "type": "inline_equation", "height": 10, "width": 11}], "index": 11}], "index": 11}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "text", "bbox": [69, 498, 271, 514], "lines": [{"bbox": [70, 500, 271, 514], "spans": [{"bbox": [70, 500, 86, 514], "score": 1.0, "content": "as ", "type": "text"}, {"bbox": [86, 503, 113, 512], "score": 0.93, "content": "4\\times4", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [113, 500, 271, 514], "score": 1.0, "content": " real anti-symmetric matrices:", "type": "text"}], "index": 13}], "index": 13}, {"type": "interline_equation", "bbox": [123, 527, 488, 558], "lines": [{"bbox": [123, 527, 488, 558], "spans": [{"bbox": [123, 527, 488, 558], "score": 0.93, "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [98, 101, 500, 133], "lines": [{"bbox": [98, 101, 500, 133], "spans": [{"bbox": [98, 101, 500, 133], "score": 0.86, "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": "interline_equation"}], "index": 12}], "index": 12}, {"type": "interline_equation", "bbox": [123, 527, 488, 558], "lines": [{"bbox": [123, 527, 488, 558], "spans": [{"bbox": [123, 527, 488, 558], "score": 0.93, "content": "\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}", "type": "interline_equation"}], "index": 14}], "index": 14}], "discarded_blocks": [{"type": "discarded", "bbox": [299, 730, 312, 743], "lines": [{"bbox": [298, 732, 312, 744], "spans": [{"bbox": [298, 732, 312, 744], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [69, 69, 350, 86], "lines": [{"bbox": [70, 71, 348, 88], "spans": [{"bbox": [70, 71, 179, 88], "score": 1.0, "content": "We define operators ", "type": "text"}, {"bbox": [179, 73, 190, 84], "score": 0.89, "content": "s^{j}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [190, 71, 255, 88], "score": 1.0, "content": " in terms of", "type": "text"}, {"bbox": [255, 73, 348, 88], "score": 0.9, "content": "\\left\\{\\mathbf{1}^{R},I^{R},J^{R},K^{R}\\right\\}", "type": "inline_equation", "height": 15, "width": 93}], "index": 0}], "index": 0, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 71, 348, 88]}, {"type": "interline_equation", "bbox": [98, 101, 500, 133], "lines": [{"bbox": [98, 101, 500, 133], "spans": [{"bbox": [98, 101, 500, 133], "score": 0.86, "content": "{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),", "type": "interline_equation"}], "index": 1}], "index": 1, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 144, 542, 198], "lines": [{"bbox": [93, 147, 540, 163], "spans": [{"bbox": [93, 147, 241, 163], "score": 1.0, "content": "In a similar way, the group ", "type": "text"}, {"bbox": [242, 147, 329, 160], "score": 0.92, "content": "S p(2)\\sim S p i n(5)", "type": "inline_equation", "height": 13, "width": 87}, {"bbox": [330, 147, 512, 163], "score": 1.0, "content": " is the group of quaternion-valued ", "type": "text"}, {"bbox": [513, 149, 540, 158], "score": 0.91, "content": "2\\times2", "type": "inline_equation", "height": 9, "width": 27}], "index": 2}, {"bbox": [70, 165, 541, 182], "spans": [{"bbox": [70, 165, 316, 182], "score": 1.0, "content": "matrices with unit determinant. We will view ", "type": "text"}, {"bbox": [317, 167, 346, 180], "score": 0.93, "content": "S p(2)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [347, 165, 541, 182], "score": 1.0, "content": " as acting by left multiplication on a", "type": "text"}], "index": 3}, {"bbox": [70, 184, 541, 201], "spans": [{"bbox": [70, 184, 97, 201], "score": 1.0, "content": "field ", "type": "text"}, {"bbox": [97, 187, 106, 195], "score": 0.91, "content": "\\Psi", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [107, 184, 349, 201], "score": 1.0, "content": " in the defining representation. So an element ", "type": "text"}, {"bbox": [349, 186, 403, 199], "score": 0.94, "content": "U\\in S p(2)", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [403, 184, 541, 201], "score": 1.0, "content": " acts in the following way:", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 147, 541, 201]}, {"type": "interline_equation", "bbox": [281, 219, 330, 229], "lines": [{"bbox": [281, 219, 330, 229], "spans": [{"bbox": [281, 219, 330, 229], "score": 0.88, "content": "\\Psi\\to U\\Psi.", "type": "interline_equation"}], "index": 5}], "index": 5, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 249, 267, 264], "lines": [{"bbox": [71, 251, 267, 266], "spans": [{"bbox": [71, 251, 267, 266], "score": 1.0, "content": "Equivalently, in terms of components", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [71, 251, 267, 266]}, {"type": "interline_equation", "bbox": [272, 286, 339, 298], "lines": [{"bbox": [272, 286, 339, 298], "spans": [{"bbox": [272, 286, 339, 298], "score": 0.91, "content": "\\Psi_{a}\\to U_{a b}\\Psi_{b}.", "type": "interline_equation"}], "index": 7}], "index": 7, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [69, 315, 541, 332], "lines": [{"bbox": [69, 318, 541, 335], "spans": [{"bbox": [69, 318, 541, 335], "score": 1.0, "content": "Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:", "type": "text"}], "index": 8}], "index": 8, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [69, 318, 541, 335]}, {"type": "interline_equation", "bbox": [153, 347, 457, 377], "lines": [{"bbox": [153, 347, 457, 377], "spans": [{"bbox": [153, 347, 457, 377], "score": 0.87, "content": "\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,", "type": "interline_equation"}], "index": 9}], "index": 9, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "interline_equation", "bbox": [197, 391, 414, 421], "lines": [{"bbox": [197, 391, 414, 421], "spans": [{"bbox": [197, 391, 414, 421], "score": 0.87, "content": "\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).", "type": "interline_equation"}], "index": 10}], "index": 10, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [70, 426, 432, 442], "lines": [{"bbox": [70, 430, 430, 443], "spans": [{"bbox": [70, 430, 116, 443], "score": 1.0, "content": "In turn, ", "type": "text"}, {"bbox": [116, 431, 163, 443], "score": 0.94, "content": "\\{I,J,K\\}", "type": "inline_equation", "height": 12, "width": 47}, {"bbox": [163, 430, 418, 443], "score": 1.0, "content": " can be expressed in terms of the Pauli matrices ", "type": "text"}, {"bbox": [419, 430, 430, 440], "score": 0.89, "content": "\\sigma^{i}", "type": "inline_equation", "height": 10, "width": 11}], "index": 11}], "index": 11, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [70, 430, 430, 443]}, {"type": "interline_equation", "bbox": [154, 457, 456, 488], "lines": [{"bbox": [154, 457, 456, 488], "spans": [{"bbox": [154, 457, 456, 488], "score": 0.94, "content": "\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)", "type": 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We define operators in terms of
$$
{}^{1}=\left({\begin{array}{c c}{1^{R}}&{0}\\ {0}&{1^{R}}\end{array}}\right),\quad s^{2}=\left({\begin{array}{c c}{I^{R}}&{0}\\ {0}&{I^{R}}\end{array}}\right),\quad s^{3}=\left({\begin{array}{c c}{J^{R}}&{0}\\ {0}&{J^{R}}\end{array}}\right),\quad s^{4}=\left({\begin{array}{c c}{K^{R}}&{0}\\ {0}&{K^{R}}\end{array}}\right),
$$
In a similar way, the group is the group of quaternion-valued matrices with unit determinant. We will view as acting by left multiplication on a field in the defining representation. So an element acts in the following way:
$$
\Psi\to U\Psi.
$$
Equivalently, in terms of components
$$
\Psi_{a}\to U_{a b}\Psi_{b}.
$$
Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:
$$
\gamma^{1}={\binom{1}{0}}\,\,\,\,\,\,\,\,\,\,\,\,\gamma^{2}={\binom{0}{1}}\,\,\,\,\,\,1{\binom{1}{0}}\,,\,\,\,\,\,\,\,\,\,\,\,\gamma^{3}={\binom{0}{-I}}\,\,\,\,\,\,I\,\,\,\,
$$
$$
\gamma^{4}=\left(\!\begin{array}{c c}{{0}}&{{J}}\\ {{-J}}&{{0}}\end{array}\!\right),\qquad\gamma^{5}=\left(\!\begin{array}{c c}{{0}}&{{K}}\\ {{-K}}&{{0}}\end{array}\!\right).
$$
In turn, can be expressed in terms of the Pauli matrices
$$
\sigma^{1}=\left(\!\!\begin{array}{c c}{{0}}&{{1}}\\ {{1}}&{{0}}\end{array}\!\!\right),\qquad\sigma^{2}=\left(\!\!\begin{array}{c c}{{0}}&{{-i}}\\ {{i}}&{{0}}\end{array}\!\!\right),\qquad\sigma^{3}=\left(\!\!\begin{array}{c c}{{1}}&{{0}}\\ {{0}}&{{-1}}\end{array}\!\!\right)
$$
as real anti-symmetric matrices:
$$
\begin{array}{c c c}{{I=\left(\begin{array}{c c}{{0}}&{{\sigma^{1}}}\\ {{-\sigma^{1}}}&{{0}}\end{array}\right),~~}}&{{J=\left(\begin{array}{c c}{{-i\sigma^{2}}}&{{0}}\\ {{0}}&{{-i\sigma^{2}}}\end{array}\right),~~}}&{{K=\left(\begin{array}{c c}{{0}}&{{\sigma^{3}}}\\ {{-\sigma^{3}}}&{{0}}\end{array}\right).}}\end{array}
$$
11
|
<div class="pdf-page">
<p>We define operators in terms of</p>
<p>In a similar way, the group is the group of quaternion-valued matrices with unit determinant. We will view as acting by left multiplication on a field in the defining representation. So an element acts in the following way:</p>
<p>Equivalently, in terms of components</p>
<p>Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:</p>
<p>In turn, can be expressed in terms of the Pauli matrices</p>
<p>as real anti-symmetric matrices:</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="115" data-y="89" data-width="470" data-height="22">We define operators in terms of</p>
<p class="pdf-text" data-x="117" data-y="186" data-width="789" data-height="70">In a similar way, the group is the group of quaternion-valued matrices with unit determinant. We will view as acting by left multiplication on a field in the defining representation. So an element acts in the following way:</p>
<p class="pdf-text" data-x="115" data-y="321" data-width="331" data-height="20">Equivalently, in terms of components</p>
<p class="pdf-text" data-x="115" data-y="407" data-width="790" data-height="22">Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:</p>
<p class="pdf-text" data-x="117" data-y="550" data-width="605" data-height="21">In turn, can be expressed in terms of the Pauli matrices</p>
<p class="pdf-text" data-x="115" data-y="643" data-width="338" data-height="21">as real anti-symmetric matrices:</p>
<div class="pdf-discarded" data-x="500" data-y="943" data-width="22" data-height="17" style="opacity: 0.5;">11</div>
</div>
|
We define operators $s^{j}$ in terms of $\left\{\mathbf{1}^{R},I^{R},J^{R},K^{R}\right\}$
$$
{}^{1}=\left({\begin{array}{c c}{1^{R}}&{0}\\ {0}&{1^{R}}\end{array}}\right),\quad s^{2}=\left({\begin{array}{c c}{I^{R}}&{0}\\ {0}&{I^{R}}\end{array}}\right),\quad s^{3}=\left({\begin{array}{c c}{J^{R}}&{0}\\ {0}&{J^{R}}\end{array}}\right),\quad s^{4}=\left({\begin{array}{c c}{K^{R}}&{0}\\ {0}&{K^{R}}\end{array}}\right),
$$
In a similar way, the group $S p(2)\sim S p i n(5)$ is the group of quaternion-valued $2\times2$ matrices with unit determinant. We will view $S p(2)$ as acting by left multiplication on a field $\Psi$ in the defining representation. So an element $U\in S p(2)$ acts in the following way:
$$
\Psi\to U\Psi.
$$
Equivalently, in terms of components
$$
\Psi_{a}\to U_{a b}\Psi_{b}.
$$
Lastly, we can give an explicit form for the gamma matrices (2.3) in terms of quaternions:
$$
\gamma^{1}={\binom{1}{0}}\,\,\,\,\,\,\,\,\,\,\,\,\gamma^{2}={\binom{0}{1}}\,\,\,\,\,\,1{\binom{1}{0}}\,,\,\,\,\,\,\,\,\,\,\,\,\gamma^{3}={\binom{0}{-I}}\,\,\,\,\,\,I\,\,\,\,
$$
$$
\gamma^{4}=\left(\!\begin{array}{c c}{{0}}&{{J}}\\ {{-J}}&{{0}}\end{array}\!\right),\qquad\gamma^{5}=\left(\!\begin{array}{c c}{{0}}&{{K}}\\ {{-K}}&{{0}}\end{array}\!\right).
$$
In turn, $\{I,J,K\}$ can be expressed in terms of the Pauli matrices $\sigma^{i}$
$$
\sigma^{1}=\left(\!\!\begin{array}{c c}{{0}}&{{1}}\\ {{1}}&{{0}}\end{array}\!\!\right),\qquad\sigma^{2}=\left(\!\!\begin{array}{c c}{{0}}&{{-i}}\\ {{i}}&{{0}}\end{array}\!\!\right),\qquad\sigma^{3}=\left(\!\!\begin{array}{c c}{{1}}&{{0}}\\ {{0}}&{{-1}}\end{array}\!\!\right)
$$
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"{}^{1}=\\left({\\begin{array}{c c}{1^{R}}&{0}\\\\ {0}&{1^{R}}\\end{array}}\\right),\\quad s^{2}=\\left({\\begin{array}{c c}{I^{R}}&{0}\\\\ {0}&{I^{R}}\\end{array}}\\right),\\quad s^{3}=\\left({\\begin{array}{c c}{J^{R}}&{0}\\\\ {0}&{J^{R}}\\end{array}}\\right),\\quad s^{4}=\\left({\\begin{array}{c c}{K^{R}}&{0}\\\\ {0}&{K^{R}}\\end{array}}\\right),",
"\\Psi\\to U\\Psi.",
"\\Psi_{a}\\to U_{a b}\\Psi_{b}.",
"\\gamma^{1}={\\binom{1}{0}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{2}={\\binom{0}{1}}\\,\\,\\,\\,\\,\\,1{\\binom{1}{0}}\\,,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma^{3}={\\binom{0}{-I}}\\,\\,\\,\\,\\,\\,I\\,\\,\\,\\,",
"\\gamma^{4}=\\left(\\!\\begin{array}{c c}{{0}}&{{J}}\\\\ {{-J}}&{{0}}\\end{array}\\!\\right),\\qquad\\gamma^{5}=\\left(\\!\\begin{array}{c c}{{0}}&{{K}}\\\\ {{-K}}&{{0}}\\end{array}\\!\\right).",
"\\sigma^{1}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{1}}\\\\ {{1}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{2}=\\left(\\!\\!\\begin{array}{c c}{{0}}&{{-i}}\\\\ {{i}}&{{0}}\\end{array}\\!\\!\\right),\\qquad\\sigma^{3}=\\left(\\!\\!\\begin{array}{c c}{{1}}&{{0}}\\\\ {{0}}&{{-1}}\\end{array}\\!\\!\\right)",
"\\begin{array}{c c c}{{I=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{1}}}\\\\ {{-\\sigma^{1}}}&{{0}}\\end{array}\\right),~~}}&{{J=\\left(\\begin{array}{c c}{{-i\\sigma^{2}}}&{{0}}\\\\ {{0}}&{{-i\\sigma^{2}}}\\end{array}\\right),~~}}&{{K=\\left(\\begin{array}{c c}{{0}}&{{\\sigma^{3}}}\\\\ {{-\\sigma^{3}}}&{{0}}\\end{array}\\right).}}\\end{array}"
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"[1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196. [2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85. [3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47. [4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997. [5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541. [6] N. Hitchin, math.DG/9909002. [7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689; R. Flume, Ann. Phys. 164 (1985) 189; M. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070. [8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675. [9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307. [10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184. [11] V. G. Kac and A. Smilga, hep-th/9908096. [12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007. [13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209 (2000) 77. [14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147. [15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998) 31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350. [16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367. [17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080. [18] J. Jost and K. Zuo, “Vanishing Theorems for -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,” preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998).",
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# References
- [1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196. [2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85. [3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47. [4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997. [5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541. [6] N. Hitchin, math.DG/9909002. [7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689; R. Flume, Ann. Phys. 164 (1985) 189; M. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070. [8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675. [9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307. [10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184. [11] V. G. Kac and A. Smilga, hep-th/9908096. [12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007. [13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209 (2000) 77. [14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147. [15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998) 31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350. [16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367. [17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080. [18] J. Jost and K. Zuo, “Vanishing Theorems for -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,” preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998).
12
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<div class="pdf-page">
<h1>References</h1>
<ul>
<li>[1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196. [2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85. [3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47. [4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997. [5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541. [6] N. Hitchin, math.DG/9909002. [7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689; R. Flume, Ann. Phys. 164 (1985) 189; M. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070. [8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675. [9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307. [10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184. [11] V. G. Kac and A. Smilga, hep-th/9908096. [12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007. [13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209 (2000) 77. [14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147. [15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998) 31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350. [16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367. [17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080. [18] J. Jost and K. Zuo, “Vanishing Theorems for -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,” preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998).</li>
</ul>
</div>
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<div class="pdf-page">
<h1 class="pdf-title" data-x="455" data-y="90" data-width="112" data-height="19">References</h1>
<ul class="pdf-list" data-x="110" data-y="129" data-width="798" data-height="565">
<li>[1] M. Berkooz and M. R. Douglas, hep-th/9610236, Phys. Lett. B395 (1997) 196. [2] M. R. Douglas, D. Kabat, P. Pouliot and S. Shenker, hep-th/9608024, Nucl. Phys. B485 (1997), 85. [3] S. Sethi and M. Stern, hep-th/9607145, Phys. Lett. B398 (1997), 47. [4] M. R. Douglas, hep-th/9612126, J.H.E.P. 9707:004, 1997. [5] E. Witten, hep-th/9511030, Nucl. Phys. B460 (1996) 541. [6] N. Hitchin, math.DG/9909002. [7] M. Claudson and M. Halpern, Nucl. Phys. B250 (1985) 689; R. Flume, Ann. Phys. 164 (1985) 189; M. Baake, P. Reinecke and V. Rittenberg, J. Math. Phys. 26 (1985) 1070. [8] S. Sethi and M. Stern, hep-th/9705046, Comm. Math. Phys. 194 (1998) 675. [9] P. Yi, hep-th/9704098, Nucl. Phys. B505 (1997) 307. [10] M. Porrati and A. Rozenberg, hep-th/9708119, Nucl. Phys. B515 (1998) 184. [11] V. G. Kac and A. Smilga, hep-th/9908096. [12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007. [13] G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9803265, Comm. Math. Phys. 209 (2000) 77. [14] P. Vanhove, hep-th/9903050, Class. Quant. Grav. 16 (1999) 3147. [15] W. Krauth, H. Nicolai and M. Staudacher, hep-th/9803117, Phys. Lett. B431 (1998) 31; W. Krauth and M. Staudacher, hep-th/9804199, Phys. Lett. B435 (1998) 350. [16] M. Halpern and C. Schwartz, hep-th/9712133, Int. J. Mod. Phys. A13 (1998) 4367. [17] J. Frohlich, G. M. Graf, D. Hasler, J. Hoppe and S.-T. Yau, hep-th/9904182; G. M. Graf and J. Hoppe, hep-th/9805080. [18] J. Jost and K. Zuo, “Vanishing Theorems for -cohomology on infinite coverings of compact Kahler manifolds and applications in algebraic geometry,” preprint no. 70, Max-Planck-Institut fir Mathematik in den Naturwissenschaften, Leipzig (1998).</li>
</ul>
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"[11] V. G. Kac and A. Smilga, hep-th/9908096.",
"[12] M. B. Green and M. Gutperle, hep-th/9804123, Phys. Rev. D58 (1998) 46007.",
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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 on the space of generalized connections is investigated for compact structure groups .",
"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.",
"arXiv:math-ph/0001008v1 5 Jan 2000",
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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
arXiv:math-ph/0001008v1 5 Jan 2000
January 5, 2000
# Abstract
The action of Ashtekar’s generalized gauge group on the space of generalized connections is investigated for compact structure groups .
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.
[email protected] or Christian.Fleischhack@mi
1
|
<div class="pdf-page">
<h1>Stratification of the Generalized Gauge Orbit Space</h1>
<p>Christian Fleischhack∗</p>
<p>Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany</p>
<p>Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany</p>
<p>January 5, 2000</p>
<h1>Abstract</h1>
<p>The action of Ashtekar’s generalized gauge group on the space of generalized connections is investigated for compact structure groups .</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="138" data-y="71" data-width="725" data-height="31">Stratification of the Generalized Gauge Orbit Space</h1>
<p class="pdf-text" data-x="388" data-y="125" data-width="232" data-height="21">Christian Fleischhack∗</p>
<p class="pdf-text" data-x="222" data-y="166" data-width="591" data-height="78">Mathematisches Institut Institut fir Theoretische Physik Universitat Leipzig Universitat Leipzig Augustusplatz 10/11 Augustusplatz 10/11 04109 Leipzig, Germany 04109 Leipzig, Germany</p>
<p class="pdf-text" data-x="225" data-y="262" data-width="554" data-height="58">Max-Planck-Institut fir Mathematik in den Naturwissenschaften Inselstraße 22-26 04103 Leipzig, Germany</p>
<div class="pdf-discarded" data-x="23" data-y="265" data-width="38" data-height="406" style="opacity: 0.5;">arXiv:math-ph/0001008v1 5 Jan 2000</div>
<p class="pdf-text" data-x="418" data-y="334" data-width="167" data-height="22">January 5, 2000</p>
<h1 class="pdf-title" data-x="460" data-y="411" data-width="82" data-height="18">Abstract</h1>
<p class="pdf-text" data-x="153" data-y="438" data-width="695" data-height="35">The action of Ashtekar’s generalized gauge group on the space of generalized connections is investigated for compact structure groups .</p>
<p class="pdf-text" data-x="153" data-y="474" data-width="696" data-height="138">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.</p>
<div class="pdf-discarded" data-x="214" data-y="872" data-width="480" data-height="16" style="opacity: 0.5;">[email protected] or Christian.Fleischhack@mi</div>
<div class="pdf-discarded" data-x="495" data-y="910" data-width="11" data-height="14" style="opacity: 0.5;">1</div>
</div>
|
# 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 $\mathbf{G}$ .
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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 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.",
"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 .",
"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.",
"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.",
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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 (\u201dgauge fields\u201d) 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 \u2013 at least preliminary \u2013 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 \u201dclassical\u201d 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] \u2013 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 (\u201dgauge fields\u201d) 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 \u2013 at least preliminary \u2013 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": "\u201dclassical\u201d 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, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"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 (\u201dgauge fields\u201d) 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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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 \u2013 at least preliminary \u2013 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, "page_num": "page_1", "page_size": [612.0, 792.0], "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": "\u201dclassical\u201d 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] \u2013 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, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [61, 606, 538, 680]}]}
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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 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.
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 .
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.
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.
2
|
<div class="pdf-page">
<h1>1 Introduction</h1>
<p>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.</p>
<p>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 .</p>
<p>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.</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="105" data-y="12" data-width="229" data-height="25">1 Introduction</h1>
<p class="pdf-text" data-x="105" data-y="53" data-width="795" data-height="241">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.</p>
<p class="pdf-text" data-x="105" data-y="296" data-width="795" data-height="205">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 .</p>
<p class="pdf-text" data-x="105" data-y="501" data-width="793" data-height="281">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.</p>
<p class="pdf-text" data-x="107" data-y="782" data-width="791" data-height="93">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.</p>
<div class="pdf-discarded" data-x="493" data-y="910" data-width="13" data-height="14" style="opacity: 0.5;">2</div>
</div>
|
# 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.
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"For quite a long time the geometric structure of gauge theories has been investigated. A",
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"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 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."
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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 to the space . 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 . 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].",
"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.",
"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.",
"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 , 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].",
"• 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.",
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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 \u2013 to the best of our knowlegde \u2013 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": "\u2022 Let $\\mathbf{G}$ be a compact Lie group. \u2022 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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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": ". 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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": "\u2022 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": "\u2022 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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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 \u2013 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 \u2013 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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_2", "page_size": [612.0, 792.0], "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": "\u2022 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": "\u2022 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, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [61, 612, 537, 685]}]}
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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 to the space . 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 . 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].
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.
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.
# 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 , 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].
• 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.
3
|
<div class="pdf-page">
<p>The outline of the paper is as follows:</p>
<p>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>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.</p>
<p>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].</p>
<p>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.</p>
<p>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.</p>
<h1>2 Preliminaries</h1>
<p>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].</p>
<p>• 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.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="105" data-y="200" data-width="325" data-height="18">The outline of the paper is as follows:</p>
<p class="pdf-text" data-x="105" data-y="221" data-width="793" data-height="72">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 class="pdf-text" data-x="105" data-y="294" data-width="793" data-height="55">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.</p>
<p class="pdf-text" data-x="105" data-y="350" data-width="793" data-height="111">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].</p>
<p class="pdf-text" data-x="105" data-y="462" data-width="793" data-height="55">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.</p>
<p class="pdf-text" data-x="105" data-y="518" data-width="793" data-height="56">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.</p>
<h1 class="pdf-title" data-x="103" data-y="601" data-width="241" data-height="26">2 Preliminaries</h1>
<p class="pdf-text" data-x="105" data-y="641" data-width="795" data-height="150">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].</p>
<p class="pdf-text" data-x="105" data-y="791" data-width="795" data-height="93">• 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.</p>
<div class="pdf-discarded" data-x="493" data-y="911" data-width="13" data-height="12" style="opacity: 0.5;">3</div>
</div>
|
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].
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"reader to the preceding papers [9, 10].",
"• Let \\mathbf{G} be a compact Lie group.",
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"• 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",
"",
"We have .",
"• 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",
"",
"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.",
"• 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.",
"• The type of a gauge orbit is the centralizer of the holonomy group of modulo conjugation in . (An equivalent definition uses the stabilizer itself.)",
"3 Partial Ordering of Types",
"Definition 3.1 A subgroup of is called Howe subgroup iff there is a set with .",
"Analogously to the general theory we define a partial ordering for the gauge orbit types [8].",
"Definition 3.2 Let denote the set of all Howe subgroups of . Let . Then holds iff there are and with .",
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[{"type": "text", "text": "", "page_idx": 3}, {"type": "text", "text": "\u2022 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": "\u2022 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": "\u2022 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": "\u2022 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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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": "\u2022 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": "\u2022 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": "\u2022 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": "\u2022 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, "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}, {"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}], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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": "\u2022 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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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": "\u2022 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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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": "\u2022 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, "page_num": "page_3", "page_size": [612.0, 792.0], "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": "\u2022 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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [75, 664, 409, 681]}]}
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|
• 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
$$
h_{\overline{{A}}\circ\overline{{g}}}(\gamma):=g_{\gamma(0)}^{-1}\;h_{\overline{{A}}}(\gamma)\;g_{\gamma(1)}\mathrm{~for~all~}\gamma\in\mathcal{P}.
$$
We have .
• 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
$$
\begin{array}{r l}{\tau_{\Gamma}:\;\;\overline{{\mathcal{A}}}\;\;\longrightarrow\;\;\;\;\overline{{\mathcal{A}}}_{\Gamma}\equiv\mathbf{G}^{E}\qquad\qquad\mathrm{and}\qquad\pi_{\Gamma}:\;\;\overline{{\mathcal{G}}}\;\;\longrightarrow\;\;\;\;\overline{{\mathcal{G}}}_{\Gamma}\equiv\mathbf{G}^{V}.}\\ {\overline{{\mathcal{A}}}\;\;\longmapsto\;\;\left(h_{\overline{{A}}}(e_{1}),\ldots,h_{\overline{{A}}}(e_{E})\right)\qquad\qquad\qquad\quad\overline{{g}}\;\;\longmapsto\;\;\left(g_{v_{1}},\ldots,g_{v_{V}}\right)}\end{array}
$$
The topologies on 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.
• 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.
• The type of a gauge orbit is the centralizer of the holonomy group of modulo conjugation in . (An equivalent definition uses the stabilizer itself.)
# 3 Partial Ordering of Types
Definition 3.1 A subgroup of is called Howe subgroup iff there is a set with .
Analogously to the general theory we define a partial ordering for the gauge orbit types [8].
Definition 3.2 Let denote the set of all Howe subgroups of . Let . Then holds iff there are and with .
Obviously, we have
Lemma 3.1 The maximal element in is the class of the center of , the minimal is the class of itself.
1Homomorphism means supposed is defined.
4
|
<div class="pdf-page">
<p>• 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</p>
<p>We have .</p>
<p>• 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</p>
<p>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.</p>
<p>• 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.</p>
<p>• The type of a gauge orbit is the centralizer of the holonomy group of modulo conjugation in . (An equivalent definition uses the stabilizer itself.)</p>
<h1>3 Partial Ordering of Types</h1>
<p>Definition 3.1 A subgroup of is called Howe subgroup iff there is a set with .</p>
<p>Analogously to the general theory we define a partial ordering for the gauge orbit types [8].</p>
<p>Definition 3.2 Let denote the set of all Howe subgroups of . Let . Then holds iff there are and with .</p>
<p>Obviously, we have</p>
<p>Lemma 3.1 The maximal element in is the class of the center of , the minimal is the class of itself.</p>
<p>1Homomorphism means supposed is defined.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="108" data-y="149" data-width="792" data-height="74">• 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</p>
<p class="pdf-text" data-x="128" data-y="252" data-width="305" data-height="19">We have .</p>
<p class="pdf-text" data-x="110" data-y="271" data-width="790" data-height="52">• 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</p>
<p class="pdf-text" data-x="128" data-y="371" data-width="772" data-height="55">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.</p>
<p class="pdf-text" data-x="108" data-y="426" data-width="792" data-height="92">• 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.</p>
<p class="pdf-text" data-x="108" data-y="519" data-width="790" data-height="39">• The type of a gauge orbit is the centralizer of the holonomy group of modulo conjugation in . (An equivalent definition uses the stabilizer itself.)</p>
<h1 class="pdf-title" data-x="105" data-y="584" data-width="418" data-height="26">3 Partial Ordering of Types</h1>
<p class="pdf-text" data-x="105" data-y="623" data-width="793" data-height="38">Definition 3.1 A subgroup of is called Howe subgroup iff there is a set with .</p>
<p class="pdf-text" data-x="105" data-y="677" data-width="788" data-height="21">Analogously to the general theory we define a partial ordering for the gauge orbit types [8].</p>
<p class="pdf-text" data-x="105" data-y="708" data-width="791" data-height="58">Definition 3.2 Let denote the set of all Howe subgroups of . Let . Then holds iff there are and with .</p>
<p class="pdf-text" data-x="103" data-y="779" data-width="168" data-height="18">Obviously, we have</p>
<p class="pdf-text" data-x="107" data-y="809" data-width="793" data-height="40">Lemma 3.1 The maximal element in is the class of the center of , the minimal is the class of itself.</p>
<p class="pdf-text" data-x="125" data-y="857" data-width="561" data-height="19">1Homomorphism means supposed is defined.</p>
<div class="pdf-discarded" data-x="495" data-y="911" data-width="11" data-height="13" style="opacity: 0.5;">4</div>
</div>
|
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.
# 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.
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.
• 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.
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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 cen-",
"tralizer 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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"",
"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}.",
"\\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}"
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"Definition 3.3 Let . We define the following expressions:",
"",
"All the are called strata.2",
"4 Reducing the Problem to Finite-Dimensional G- Spaces",
"4.1 Finiteness Lemma for Centralizers",
"We start with the crucial",
"Lemma 4.1 Let be a subset of a compact Lie group . Then there exist an and , such that .",
"Proof • The case is trivial.",
"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).",
"Corollary 4.2 Let .",
"Then there is a finite set , such that .3",
"Proof Due to and the just proven lemma there are an and with . On the other hand, since , there are with for all . qed",
"4.2 Reduction Mapping",
"Definition 4.1 Let . Then the map",
"",
"is called reduction mapping.",
"Lemma 4.3 Let be arbitrary.",
"Then is continuous, and for all and we have . Here acts on by the adjoint map.",
"2The justification for that notation can be found in section 8. where . To avoid cumbersome notations we denote also by . It should be clear from the context what is meant. Furthermore, is always finite.",
"5"
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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 \u2022 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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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 \u2022 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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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 \u2022 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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "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, "page_num": "page_4", "page_size": [612.0, 792.0], "bbox_fs": [137, 603, 538, 637]}]}
|
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Definition 3.3 Let . 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 are called strata.2
# 4 Reducing the Problem to Finite-Dimensional G- Spaces
# 4.1 Finiteness Lemma for Centralizers
We start with the crucial
Lemma 4.1 Let be a subset of a compact Lie group . Then there exist an and , such that .
Proof • The case is trivial.
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).
Corollary 4.2 Let .
Then there is a finite set , such that .3
Proof Due to and the just proven lemma there are an and with . On the other hand, since , there are with for all . qed
# 4.2 Reduction Mapping
Definition 4.1 Let . Then the map
$$
\varphi_{\alpha}:\;{\overline{{\mathbf{\mathcal{A}}}}}\;\;\longrightarrow\;\;{\mathbf{G}}^{\#\alpha}
$$
is called reduction mapping.
Lemma 4.3 Let be arbitrary.
Then is continuous, and for all and we have . Here acts on by the adjoint map.
2The justification for that notation can be found in section 8. where . To avoid cumbersome notations we denote also by . It should be clear from the context what is meant. Furthermore, is always finite.
5
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<div class="pdf-page">
<p>Definition 3.3 Let . We define the following expressions:</p>
<p>All the are called strata.2</p>
<h1>4 Reducing the Problem to Finite-Dimensional G- Spaces</h1>
<h1>4.1 Finiteness Lemma for Centralizers</h1>
<p>We start with the crucial</p>
<p>Lemma 4.1 Let be a subset of a compact Lie group . Then there exist an and , such that .</p>
<p>Proof • The case is trivial.</p>
<p>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).</p>
<p>Corollary 4.2 Let .</p>
<p>Then there is a finite set , such that .3</p>
<p>Proof Due to and the just proven lemma there are an and with . On the other hand, since , there are with for all . qed</p>
<h1>4.2 Reduction Mapping</h1>
<p>Definition 4.1 Let . Then the map</p>
<p>is called reduction mapping.</p>
<p>Lemma 4.3 Let be arbitrary.</p>
<p>Then is continuous, and for all and we have . Here acts on by the adjoint map.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="103" data-y="18" data-width="561" data-height="19">Definition 3.3 Let . We define the following expressions:</p>
<p class="pdf-text" data-x="256" data-y="107" data-width="267" data-height="21">All the are called strata.2</p>
<h1 class="pdf-title" data-x="107" data-y="153" data-width="789" data-height="53">4 Reducing the Problem to Finite-Dimensional G- Spaces</h1>
<h1 class="pdf-title" data-x="105" data-y="222" data-width="470" data-height="23">4.1 Finiteness Lemma for Centralizers</h1>
<p class="pdf-text" data-x="105" data-y="254" data-width="217" data-height="20">We start with the crucial</p>
<p class="pdf-text" data-x="105" data-y="281" data-width="795" data-height="40">Lemma 4.1 Let be a subset of a compact Lie group . Then there exist an and , such that .</p>
<p class="pdf-text" data-x="105" data-y="332" data-width="422" data-height="18">Proof • The case is trivial.</p>
<p class="pdf-text" data-x="179" data-y="351" data-width="719" data-height="137">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).</p>
<p class="pdf-text" data-x="103" data-y="512" data-width="246" data-height="19">Corollary 4.2 Let .</p>
<p class="pdf-text" data-x="247" data-y="532" data-width="579" data-height="21">Then there is a finite set , such that .3</p>
<p class="pdf-text" data-x="103" data-y="563" data-width="797" data-height="58">Proof Due to and the just proven lemma there are an and with . On the other hand, since , there are with for all . qed</p>
<h1 class="pdf-title" data-x="105" data-y="642" data-width="299" data-height="22">4.2 Reduction Mapping</h1>
<p class="pdf-text" data-x="105" data-y="673" data-width="398" data-height="20">Definition 4.1 Let . Then the map</p>
<p class="pdf-text" data-x="254" data-y="730" data-width="269" data-height="18">is called reduction mapping.</p>
<p class="pdf-text" data-x="105" data-y="760" data-width="350" data-height="19">Lemma 4.3 Let be arbitrary.</p>
<p class="pdf-text" data-x="229" data-y="779" data-width="669" data-height="38">Then is continuous, and for all and we have . Here acts on by the adjoint map.</p>
<div class="pdf-discarded" data-x="105" data-y="823" data-width="795" data-height="65" style="opacity: 0.5;">2The justification for that notation can be found in section 8. where . To avoid cumbersome notations we denote also by . It should be clear from the context what is meant. Furthermore, is always finite.</div>
<div class="pdf-discarded" data-x="493" data-y="910" data-width="13" data-height="14" style="opacity: 0.5;">5</div>
</div>
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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.
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.
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 $F^{-1}(\{{\overline{{A}}}\})={\overline{{S}}}$ .
# 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.
# 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
$$
\begin{array}{r c l}{\overline{{S}}_{0}}&{:=}&{\varphi^{-1}(S)\,\cap\,\psi^{-1}(\psi(\overline{{A}})),}\\ {\overline{{S}}}&{:=}&{\big(\varphi^{-1}(S)\,\cap\,\psi^{-1}(\psi(\overline{{A}}))\big)\circ{\bf B}(\overline{{A}})}&{\equiv}&{\overline{{S}}_{0}\circ{\bf B}(\overline{{A}})}\end{array}
$$
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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}",
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"4 Reducing the Problem to Finite-Dimensional G-",
"Spaces",
"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",
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"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}}}",
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"\\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}",
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"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{~=~}",
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"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",
"4.3 Adjoint Action of on",
"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]).",
"Consequently, we have for the type of the corresponding orbit",
"",
"The slice theorem reads now as follows:",
"Proposition 4.4 Let . Then there is an with , such that: • is an open neighboorhood of and • there is an equivariant retraction with .",
"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",
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"",
"Proof We have",
"5 Slice Theorem for",
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[{"type": "text", "text": "Proof \u2022 $\\varphi_{\\alpha}:{\\overline{{\\mathcal{A}}}}\\longrightarrow\\mathbf{G}^{\\#\\alpha}$ 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. \u2022 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: \u2022 $S\\circ\\mathbf{G}$ is an open neighboorhood of $\\vec{g}\\circ\\mathbf{G}$ and \u2022 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": "\u2022 $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ 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}]
|
{"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 \u2022", "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": "\u2022 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": "\u2022", "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": "\u2022 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": "\u2022", "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 \u2022", "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": "\u2022 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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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": "\u2022", "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": "\u2022 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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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": "\u2022", "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, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [148, 617, 537, 647]}]}
|
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|
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
# 4.3 Adjoint Action of on
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]).
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 . Then there is an with , such that: • is an open neighboorhood of and • there is an equivariant retraction with .
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
Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all and all we have
$$
\mathrm{Typ}\big(\varphi_{\alpha}(\overline{{A}})\big)\leq\mathrm{Typ}(\overline{{A}}).
$$
Proof We have
# 5 Slice Theorem for
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 there is an with , such that:
• is an open neighbourhood of and there is an equivariant retraction with .
6
|
<div class="pdf-page">
<p>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</p>
<h1>4.3 Adjoint Action of on</h1>
<p>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]).</p>
<p>Consequently, we have for the type of the corresponding orbit</p>
<p>The slice theorem reads now as follows:</p>
<p>Proposition 4.4 Let . Then there is an with , such that: • is an open neighboorhood of and • there is an equivariant retraction with .</p>
<p>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</p>
<p>Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all and all we have</p>
<p>Proof We have</p>
<h1>5 Slice Theorem for</h1>
<p>We state now the main theorem of the present paper.</p>
<p>Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all there is an with , such that:</p>
<p>• is an open neighbourhood of and there is an equivariant retraction with .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="102" data-y="15" data-width="799" data-height="173">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</p>
<h1 class="pdf-title" data-x="103" data-y="212" data-width="382" data-height="22">4.3 Adjoint Action of on</h1>
<p class="pdf-text" data-x="103" data-y="244" data-width="795" data-height="55">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]).</p>
<p class="pdf-text" data-x="103" data-y="338" data-width="541" data-height="17">Consequently, we have for the type of the corresponding orbit</p>
<p class="pdf-text" data-x="103" data-y="374" data-width="343" data-height="19">The slice theorem reads now as follows:</p>
<p class="pdf-text" data-x="103" data-y="404" data-width="797" data-height="75">Proposition 4.4 Let . Then there is an with , such that: • is an open neighboorhood of and • there is an equivariant retraction with .</p>
<p class="pdf-text" data-x="103" data-y="492" data-width="797" data-height="39">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</p>
<p class="pdf-text" data-x="105" data-y="543" data-width="793" data-height="37">Proposition 4.5 Any reduction mapping is type-minorifying, i.e. for all and all we have</p>
<p class="pdf-text" data-x="103" data-y="619" data-width="705" data-height="26">Proof We have</p>
<h1 class="pdf-title" data-x="103" data-y="667" data-width="342" data-height="26">5 Slice Theorem for</h1>
<p class="pdf-text" data-x="103" data-y="708" data-width="462" data-height="19">We state now the main theorem of the present paper.</p>
<p class="pdf-text" data-x="102" data-y="738" data-width="799" data-height="57">Theorem 5.1 There is a tubular neighbourhood for any gauge orbit. Equivalently we have: For all there is an with , such that:</p>
<p class="pdf-text" data-x="244" data-y="795" data-width="654" data-height="40">• is an open neighbourhood of and there is an equivariant retraction with .</p>
<div class="pdf-discarded" data-x="493" data-y="910" data-width="15" data-height="14" style="opacity: 0.5;">6</div>
</div>
|
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.
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:
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"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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"\\mathrm{Typ}(\\vec{g})=[\\mathbf{G}_{\\vec{g}}]=[Z(\\{g_{1},\\dots,g_{n}\\})]",
"\\mathrm{Typ}\\big(\\varphi_{\\alpha}(\\overline{{A}})\\big)\\leq\\mathrm{Typ}(\\overline{{A}})."
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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 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 .",
"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 .",
"We are now looking for an appropriate , such tha",
"",
"is well-defined and has the desired properties.",
"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)",
"",
"",
"all . In particular, we have for",
"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 .",
"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.",
"",
"4We have",
"7"
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[{"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": ". 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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": ". 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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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [62, 463, 463, 480]}, {"type": "text", "bbox": [62, 507, 538, 595], "lines": [], "index": 31.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [61, 509, 538, 597], "lines_deleted": true}]}
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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 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 .
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 .
We are now looking for an appropriate , 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 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)
$$
h_{\overline{{{A}}}^{\prime}}(\gamma_{m})^{-1}Z(\mathbf{H}_{\overline{{{A}}}^{\prime}})h_{\overline{{{A}}}^{\prime}}(\gamma_{x})=\pi_{x}(\mathbf{B}(\overline{{{A}}}^{\prime}))\subseteq\pi_{x}(\mathbf{B}(\overline{{{A}}}))=h_{\overline{{{A}}}}(\gamma_{m})^{-1}Z(\mathbf{H}_{\overline{{{A}}}})h_{\overline{{{A}}}}(\gamma_{x}),
$$
$$
Z({\bf H}_{\overline{{{A}}}^{\prime}})\subseteq h_{\overline{{{A}}}^{\prime}}(\gamma_{m})h_{\overline{{{A}}}}(\gamma_{m})^{-1}\,Z({\bf H}_{\overline{{{A}}}})\,h_{\overline{{{A}}}}(\gamma_{x})h_{\overline{{{A}}}^{\prime}}^{-1}(\gamma_{x})
$$
all . In particular, we have for
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 .
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.
4We have
7
|
<div class="pdf-page">
<h1>5.1 The Idea</h1>
<p>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 .</p>
<p>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 .</p>
<p>We are now looking for an appropriate , such tha</p>
<p>is well-defined and has the desired properties.</p>
<p>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)</p>
<p>all . In particular, we have for</p>
<p>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 .</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="103" data-y="15" data-width="173" data-height="22">5.1 The Idea</h1>
<p class="pdf-text" data-x="103" data-y="46" data-width="797" data-height="168">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 .</p>
<p class="pdf-text" data-x="103" data-y="214" data-width="795" data-height="113">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 .</p>
<p class="pdf-text" data-x="105" data-y="328" data-width="477" data-height="18">We are now looking for an appropriate , such tha</p>
<p class="pdf-text" data-x="103" data-y="385" data-width="397" data-height="17">is well-defined and has the desired properties.</p>
<p class="pdf-text" data-x="102" data-y="402" data-width="798" data-height="55">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)</p>
<p class="pdf-text" data-x="135" data-y="524" data-width="479" data-height="19">all . In particular, we have for</p>
<p class="pdf-text" data-x="102" data-y="543" data-width="796" data-height="55">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 .</p>
<p class="pdf-text" data-x="105" data-y="599" data-width="671" data-height="17">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.</p>
<div class="pdf-discarded" data-x="123" data-y="801" data-width="312" data-height="21" style="opacity: 0.5;">4We have</div>
<div class="pdf-discarded" data-x="493" data-y="910" data-width="13" data-height="14" style="opacity: 0.5;">7</div>
</div>
|
# 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 $\vec{g}^{\prime}\in S$ .
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"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."
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"\\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}",
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"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})"
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"content": [
"5.2 The Proof",
"Proof 1. Let . Choose for an with according to Corollary 4.2 and denote the corresponding reduction mapping shortly by .",
"2. Due to Proposition 4.4 there is an with , such that",
"is an open neighbourhood of and there exists an equivariant mapping with and .",
"3. We define the mapping",
"",
"whereas for all the (arbitrary, but fixed) path runs from to and is the trivial path.",
"4. As we motivated above we set",
"",
"and",
"",
"5. is well-defined.",
"• Let with and . Then there exist with and as well as . Due to we have , i.e. for all .",
"Furthermore, we have",
"",
"and analogously .",
"Therefore, we have , i.e. is an element of the stabilizer of , thus .",
"• Since , we have , and so for all",
"Moreover, since , we have . From for all now follows, and thus .",
"By this we have , i.e. is well-defined.",
"8"
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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": "\u2022 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": "\u2022 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": "\u2022 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": "\u2022 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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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": "\u2022 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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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": "\u2022 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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [147, 671, 425, 684]}]}
|
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"\\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)"}, {"category_id": 13, "poly": [654, 1467, 1031, 1467, 1031, 1508, 654, 1508], "score": 0.92, "latex": "f(\\varphi(\\overline{{A}}^{\\prime\\prime}\\circ\\overline{{g}}^{\\prime\\prime}))=\\varphi(\\overline{{A}})\\circ g_{m}^{\\prime\\prime}"}, {"category_id": 13, "poly": [928, 1012, 1114, 1012, 1114, 1053, 928, 1053], "score": 0.91, "latex": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}_{0}^{\\prime\\prime}\\circ\\overline{{z}}^{\\prime\\prime}"}, {"category_id": 13, "poly": [685, 1010, 856, 1010, 856, 1054, 685, 1054], "score": 0.91, "latex": "\\bar{\\vec{A}}^{\\prime}=\\overline{{A}}_{0}^{\\prime}\\circ\\overline{{z}}^{\\prime}"}, {"category_id": 13, "poly": [413, 1100, 662, 1100, 662, 1149, 413, 1149], "score": 0.91, "latex": "h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})"}, {"category_id": 13, "poly": [1108, 1507, 1251, 1507, 1251, 1548, 1108, 1548], "score": 0.91, "latex": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}"}, {"category_id": 13, "poly": [809, 1548, 1254, 1548, 1254, 1588, 809, 1588], "score": 0.91, "latex": "g_{m}^{\\prime\\prime}\\left(g_{m}^{\\prime}\\right)^{-1}\\in Z(\\varphi(\\overline{{A}}))=Z(\\mathbf{H}_{\\overline{{A}}})"}, {"category_id": 13, "poly": [527, 1054, 767, 1054, 767, 1095, 527, 1095], "score": 0.91, "latex": "{\\overline{{S}}}_{0}\\,\\subseteq\\,\\psi^{-1}(\\psi({\\overline{{A}}}))"}, {"category_id": 13, "poly": [620, 1778, 1017, 1778, 1017, 1823, 620, 1823], "score": 0.91, "latex": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\gamma_{x})=h_{\\overline{{{A}}}}(\\gamma_{x})=h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})"}, {"category_id": 13, "poly": [719, 111, 744, 111, 744, 139, 719, 139], "score": 0.91, "latex": 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{"category_id": 13, "poly": [1420, 578, 1450, 578, 1450, 593, 1420, 593], "score": 0.89, "latex": "m"}, {"category_id": 13, "poly": [549, 1641, 644, 1641, 644, 1666, 549, 1666], "score": 0.89, "latex": "x\\in M"}, {"category_id": 13, "poly": [367, 942, 393, 942, 393, 967, 367, 967], "score": 0.89, "latex": "F"}, {"category_id": 13, "poly": [467, 392, 717, 392, 717, 429, 467, 429], "score": 0.89, "latex": "f^{-1}(\\{\\varphi({\\overline{{A}}})\\})=S"}, {"category_id": 13, "poly": [1268, 1009, 1436, 1009, 1436, 1054, 1268, 1054], "score": 0.88, "latex": "\\overline{{A}}_{0}^{\\prime},\\overline{{A}}_{0}^{\\prime\\prime}\\in\\overline{{S}}_{0}"}, {"category_id": 13, "poly": [413, 1014, 601, 1014, 601, 1053, 413, 1053], "score": 0.88, "latex": "\\overline{{z}}^{\\prime},\\overline{{z}}^{\\prime\\prime}\\in\\mathbf{B}(\\overline{{A}})"}, {"category_id": 13, "poly": [420, 351, 803, 351, 803, 389, 420, 389], "score": 0.88, "latex": "\\begin{array}{r l}{-}&{{}f:S\\circ\\mathbf{G}\\longrightarrow\\varphi(\\overline{{A}})\\circ\\mathbf{G}}\\end{array}"}, {"category_id": 13, "poly": [368, 618, 386, 618, 386, 634, 368, 634], "score": 0.87, "latex": "x"}, {"category_id": 14, "poly": [730, 845, 1085, 845, 1085, 942, 730, 942], "score": 0.87, "latex": "\\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}"}, {"category_id": 14, "poly": [706, 468, 1112, 468, 1112, 567, 706, 567], "score": 0.84, "latex": "\\psi:\\ {\\overline{{\\mathcal{A}}}}\\ \\ \\longrightarrow\\ \\ \\overline{{\\mathcal{G}}},}"}, {"category_id": 13, "poly": [766, 1116, 785, 1116, 785, 1132, 766, 1132], "score": 0.83, "latex": "x"}, {"category_id": 13, "poly": [1143, 1723, 1492, 1723, 1492, 1773, 1143, 1773], "score": 0.79, "latex": "\\left(\\overline{{{z}}}^{\\prime\\prime}\\ \\overline{{{g}}}^{\\prime\\prime}\\ (\\overline{{{g}}}^{\\prime})^{-1}\\ (\\overline{{{z}}}^{\\prime})^{-1}\\right)_{m}\\ \\in"}, {"category_id": 14, "poly": [529, 689, 1329, 689, 1329, 810, 529, 810], "score": 0.79, "latex": "\\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}"}, {"category_id": 13, "poly": [1215, 1772, 1494, 1772, 1494, 1815, 1215, 1815], "score": 0.72, "latex": "\\overline{{z}}^{\\prime\\prime}\\,\\overline{{g}}^{\\prime\\prime}\\,(\\overline{{g}}^{\\prime})^{-1}\\,(\\overline{{z}}^{\\prime})^{\\overset{...}{-1}}\\in"}, {"category_id": 13, "poly": [507, 1663, 1405, 1663, 1405, 1723, 507, 1723], "score": 0.65, "latex": "h_{\\overline{{{A}}}_{0}^{\\prime}}(\\bar{\\gamma_{x}})=\\Bigl(\\overline{{{z}}}^{\\prime\\prime}\\,\\overline{{{g}}}^{\\prime\\prime}\\,(\\overline{{{g}}}^{\\prime})^{-1}\\,(\\overline{{{z}}}^{\\prime})^{-1}\\Bigr)_{m}^{-1}\\,h_{\\overline{{{A}}}_{0}^{\\prime\\prime}}(\\gamma_{x})\\,\\Bigl(\\overline{{{z}}}^{\\prime\\prime}\\,\\overline{{{g}}}^{\\prime\\prime}\\,(\\overline{{{g}}}^{\\prime})^{-1}\\,(\\overline{{{z}}}^{\\prime})^{-1}\\Bigr)_{x}."}, {"category_id": 13, "poly": [1119, 1785, 1140, 1785, 1140, 1807, 1119, 1807], "score": 0.3, "latex": "x"}, {"category_id": 15, "poly": [366.0, 562.0, 586.0, 562.0, 586.0, 606.0, 366.0, 606.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [777.0, 562.0, 1227.0, 562.0, 1227.0, 606.0, 777.0, 606.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1261.0, 562.0, 1419.0, 562.0, 1419.0, 606.0, 1261.0, 606.0], "score": 1.0, "text": ""}, {"category_id": 15, "poly": [1451.0, 562.0, 1496.0, 562.0, 1496.0, 606.0, 1451.0, 606.0], 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|
# 5.2 The Proof
Proof 1. Let . Choose for an with according to Corollary 4.2 and denote the corresponding reduction mapping shortly by .
2. Due to Proposition 4.4 there is an with , such that
is an open neighbourhood of and there exists an equivariant mapping with and .
3. We define the mapping
$$
\psi:\ {\overline{{\mathcal{A}}}}\ \ \longrightarrow\ \ \overline{{\mathcal{G}}},}
$$
whereas for all the (arbitrary, but fixed) path runs from to and 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. is well-defined.
• Let with and . Then there exist with and as well as . Due to we have , i.e. for all .
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 .
Therefore, we have , i.e. is an element of the stabilizer of , thus .
• Since , we have , and so for all
Moreover, since , we have . From for all now follows, and thus .
By this we have , i.e. is well-defined.
8
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<div class="pdf-page">
<h1>5.2 The Proof</h1>
<p>Proof 1. Let . Choose for an with according to Corollary 4.2 and denote the corresponding reduction mapping shortly by .</p>
<p>2. Due to Proposition 4.4 there is an with , such that</p>
<p>is an open neighbourhood of and there exists an equivariant mapping with and .</p>
<p>3. We define the mapping</p>
<p>whereas for all the (arbitrary, but fixed) path runs from to and is the trivial path.</p>
<p>4. As we motivated above we set</p>
<p>and</p>
<p>5. is well-defined.</p>
<p>• Let with and . Then there exist with and as well as . Due to we have , i.e. for all .</p>
<p>Furthermore, we have</p>
<p>and analogously .</p>
<p>Therefore, we have , i.e. is an element of the stabilizer of , thus .</p>
<p>• Since , we have , and so for all</p>
<p>Moreover, since , we have . From for all now follows, and thus .</p>
<p>By this we have , i.e. is well-defined.</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="102" data-y="15" data-width="192" data-height="22">5.2 The Proof</h1>
<p class="pdf-text" data-x="103" data-y="45" data-width="798" data-height="58">Proof 1. Let . Choose for an with according to Corollary 4.2 and denote the corresponding reduction mapping shortly by .</p>
<p class="pdf-text" data-x="187" data-y="103" data-width="641" data-height="19">2. Due to Proposition 4.4 there is an with , such that</p>
<p class="pdf-text" data-x="215" data-y="122" data-width="455" data-height="75">is an open neighbourhood of and there exists an equivariant mapping with and .</p>
<p class="pdf-text" data-x="185" data-y="199" data-width="239" data-height="15">3. We define the mapping</p>
<p class="pdf-text" data-x="219" data-y="259" data-width="679" data-height="37">whereas for all the (arbitrary, but fixed) path runs from to and is the trivial path.</p>
<p class="pdf-text" data-x="185" data-y="297" data-width="298" data-height="17">4. As we motivated above we set</p>
<p class="pdf-text" data-x="219" data-y="376" data-width="37" data-height="17">and</p>
<p class="pdf-text" data-x="184" data-y="433" data-width="187" data-height="16">5. is well-defined.</p>
<p class="pdf-text" data-x="214" data-y="448" data-width="686" data-height="88">• Let with and . Then there exist with and as well as . Due to we have , i.e. for all .</p>
<p class="pdf-text" data-x="219" data-y="531" data-width="221" data-height="15">Furthermore, we have</p>
<p class="pdf-text" data-x="247" data-y="678" data-width="378" data-height="22">and analogously .</p>
<p class="pdf-text" data-x="245" data-y="700" data-width="655" data-height="38">Therefore, we have , i.e. is an element of the stabilizer of , thus .</p>
<p class="pdf-text" data-x="220" data-y="739" data-width="680" data-height="35">• Since , we have , and so for all</p>
<p class="pdf-text" data-x="245" data-y="800" data-width="653" data-height="64">Moreover, since , we have . From for all now follows, and thus .</p>
<p class="pdf-text" data-x="220" data-y="864" data-width="494" data-height="20">By this we have , i.e. is well-defined.</p>
<div class="pdf-discarded" data-x="493" data-y="911" data-width="13" data-height="13" style="opacity: 0.5;">8</div>
</div>
|
# 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.
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 $\varphi(\overline{{A}})$ .
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}
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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."
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"6. is equivariant.",
"Let . Then",
"",
"7. is retracting.",
"• Let . Then .",
"8. is an open neighbourhood of .",
"Obviously, .",
"Consequently, is as a preimage of an open set again open because of the continuity of .",
"9. is continuous.",
"We consider the following diagram",
"",
"",
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"9"
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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$$", "text_format": "latex", "page_idx": 8}, {"type": "text", "text": "7. $F$ is retracting. ", "page_idx": 8}, {"type": "text", "text": "\u2022 Let $\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}$ . Then $F(\\overline{{A}}^{\\prime})=F(\\overline{{A}}\\circ\\overline{{g}})=\\overline{{A}}\\circ\\overline{{g}}=\\overline{{A}}^{\\prime}$ . ", "page_idx": 8}, {"type": "text", "text": "8. $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ is an open neighbourhood of $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}$ . ", "page_idx": 8}, {"type": "text", "text": "Obviously, $\\overline{{A}}\\circ\\overline{{\\mathcal{G}}}\\subseteq\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}$ . ", "page_idx": 8}, {"type": "text", "text": "Consequently, $\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}=\\varphi^{-1}(S\\circ\\mathbf{G})$ is as a preimage of an open set again open because of the continuity of $\\varphi$ . ", "page_idx": 8}, {"type": "text", "text": "9. $F$ is continuous. ", "page_idx": 8}, {"type": "text", "text": "We consider the following diagram ", "page_idx": 8}, {"type": "equation", "text": "$$\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"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [263, 47, 421, 127], "lines": [{"bbox": [263, 47, 421, 127], "spans": [{"bbox": [263, 47, 421, 127], "score": 0.93, "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}", "type": "interline_equation"}], "index": 2}], "index": 2}, {"type": "text", "bbox": [111, 128, 212, 141], "lines": [{"bbox": [111, 129, 211, 143], "spans": [{"bbox": [111, 129, 131, 143], "score": 1.0, "content": "7.", "type": "text"}, {"bbox": [132, 132, 141, 141], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 129, 211, 143], "score": 1.0, "content": " is retracting.", "type": "text"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [129, 141, 468, 158], "lines": [{"bbox": [135, 140, 468, 160], "spans": [{"bbox": [135, 141, 169, 160], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [169, 144, 266, 157], "score": 0.93, "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [266, 140, 304, 159], "score": 1.0, "content": ". 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}, {"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}, {"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}, {"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}, {"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}, {"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}, {"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}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [263, 47, 421, 127], "lines": [{"bbox": [263, 47, 421, 127], "spans": [{"bbox": [263, 47, 421, 127], "score": 0.93, "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}", "type": "interline_equation"}], "index": 2}], "index": 2}, {"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}], "discarded_blocks": [{"type": "discarded", "bbox": [295, 705, 303, 714], "lines": [{"bbox": [295, 706, 304, 717], "spans": [{"bbox": [295, 706, 304, 717], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 15, 218, 28], "lines": [{"bbox": [111, 17, 217, 30], "spans": [{"bbox": [111, 17, 131, 30], "score": 1.0, "content": "6.", "type": "text"}, {"bbox": [132, 19, 141, 28], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 17, 217, 30], "score": 1.0, "content": " is equivariant.", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [111, 17, 217, 30]}, {"type": "text", "bbox": [132, 28, 309, 43], "lines": [{"bbox": [146, 28, 309, 45], "spans": [{"bbox": [146, 28, 169, 45], "score": 1.0, "content": "Let ", "type": "text"}, {"bbox": [170, 30, 273, 45], "score": 0.94, "content": "\\overline{{A}}^{\\prime\\prime}=\\overline{{A}}^{\\prime}\\circ\\overline{{g}}^{\\prime}\\in\\overline{{S}}\\circ\\overline{{\\mathcal{G}}}", "type": "inline_equation", "height": 15, "width": 103}, {"bbox": [273, 28, 309, 45], "score": 1.0, "content": ". Then", "type": "text"}], "index": 1}], "index": 1, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [146, 28, 309, 45]}, {"type": "interline_equation", "bbox": [263, 47, 421, 127], "lines": [{"bbox": [263, 47, 421, 127], "spans": [{"bbox": [263, 47, 421, 127], "score": 0.93, "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}", "type": "interline_equation"}], "index": 2}], "index": 2, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [111, 128, 212, 141], "lines": [{"bbox": [111, 129, 211, 143], "spans": [{"bbox": [111, 129, 131, 143], "score": 1.0, "content": "7.", "type": "text"}, {"bbox": [132, 132, 141, 141], "score": 0.9, "content": "F", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [141, 129, 211, 143], "score": 1.0, "content": " is retracting.", "type": "text"}], "index": 3}], "index": 3, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [111, 129, 211, 143]}, {"type": "text", "bbox": [129, 141, 468, 158], "lines": [{"bbox": [135, 140, 468, 160], "spans": [{"bbox": [135, 141, 169, 160], "score": 1.0, "content": "\u2022 Let ", "type": "text"}, {"bbox": [169, 144, 266, 157], "score": 0.93, "content": "\\overline{{A}}^{\\prime}=\\overline{{A}}\\circ\\overline{{g}}\\in\\overline{{A}}\\circ\\overline{{g}}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [266, 140, 304, 159], "score": 1.0, "content": ". 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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [146, 635, 538, 678]}]}
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6. is equivariant.
Let . 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}
$$
7. is retracting.
• Let . Then .
8. is an open neighbourhood of .
Obviously, .
Consequently, is as a preimage of an open set again open because of the continuity of .
9. is continuous.
We consider the following diagram
$$
\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 , and the definition of . is the canonical homeomorphism between the orbit of and the quotient of the acting group by the stabilizer of .
9
|
<div class="pdf-page">
<p>6. is equivariant.</p>
<p>Let . Then</p>
<p>7. is retracting.</p>
<p>• Let . Then .</p>
<p>8. is an open neighbourhood of .</p>
<p>Obviously, .</p>
<p>Consequently, is as a preimage of an open set again open because of the continuity of .</p>
<p>9. is continuous.</p>
<p>We consider the following diagram</p>
<p>It is commutative due to , and the definition of . is the canonical homeomorphism between the orbit of and the quotient of the acting group by the stabilizer of .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="19" data-width="180" data-height="17">6. is equivariant.</p>
<p class="pdf-text" data-x="220" data-y="36" data-width="297" data-height="19">Let . Then</p>
<p class="pdf-text" data-x="185" data-y="165" data-width="169" data-height="17">7. is retracting.</p>
<p class="pdf-text" data-x="215" data-y="182" data-width="568" data-height="22">• Let . Then .</p>
<p class="pdf-text" data-x="185" data-y="204" data-width="390" data-height="17">8. is an open neighbourhood of .</p>
<p class="pdf-text" data-x="220" data-y="222" data-width="248" data-height="17">Obviously, .</p>
<p class="pdf-text" data-x="220" data-y="513" data-width="678" data-height="37">Consequently, is as a preimage of an open set again open because of the continuity of .</p>
<p class="pdf-text" data-x="184" data-y="552" data-width="177" data-height="15">9. is continuous.</p>
<p class="pdf-text" data-x="220" data-y="570" data-width="327" data-height="16">We consider the following diagram</p>
<p class="pdf-text" data-x="245" data-y="817" data-width="656" data-height="58">It is commutative due to , and the definition of . is the canonical homeomorphism between the orbit of and the quotient of the acting group by the stabilizer of .</p>
<div class="pdf-discarded" data-x="493" data-y="911" data-width="13" data-height="12" style="opacity: 0.5;">9</div>
</div>
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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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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}", "type": "interline_equation"}], "index": 4}], "index": 4}, {"type": "text", "bbox": [149, 123, 279, 136], "lines": [{"bbox": [149, 124, 279, 138], "spans": [{"bbox": [149, 126, 163, 135], "score": 0.91, "content": "F^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [164, 124, 279, 138], "score": 1.0, "content": " is continuous because", "type": "text"}], "index": 5}], "index": 5}, {"type": "interline_equation", "bbox": [153, 136, 527, 170], "lines": [{"bbox": [153, 136, 527, 170], "spans": [{"bbox": [153, 136, 527, 170], "score": 0.81, "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}", "type": "interline_equation"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [147, 168, 343, 180], "lines": [{"bbox": [147, 169, 342, 181], "spans": [{"bbox": [147, 169, 304, 181], "score": 1.0, "content": "is 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}, {"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}, {"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}}}}\\\\ 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[{"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}, {"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}, {"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}, {"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}, {"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}, {"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}, {"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}, {"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}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [199, 29, 427, 66], "lines": [{"bbox": [199, 29, 427, 66], "spans": [{"bbox": [199, 29, 427, 66], "score": 0.88, "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}", "type": "interline_equation"}], "index": 1}], "index": 1}, {"type": "interline_equation", "bbox": [190, 90, 477, 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Then", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [149, 661, 536, 676]}]}
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|
Since , and 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}
$$
is continuous because
$$
\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}
$$
is obviously continuous for all .
induces a map via the following commutative diagram
$$
\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}
$$
i.e.,
is well-defined.
Let with . Then
$$
\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}
$$
because for .
is continuous, because is open and surjective and and are continuous.
For there is an and a with . Thus, we have and
$$
\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}
$$
where we used .
Now, is the concatenation of the following continuous maps:
$$
\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 is the canonical homeomorphism between the orbit and the
10
|
<div class="pdf-page">
<p>Since , and are continuous, the map</p>
<p>is continuous.</p>
<p>Now, we consider the map</p>
<p>is continuous because</p>
<p>is obviously continuous for all .</p>
<p>induces a map via the following commutative diagram</p>
<p>i.e.,</p>
<p>is well-defined.</p>
<p>Let with . Then</p>
<p>because for .</p>
<p>is continuous, because is open and surjective and and are continuous.</p>
<p>For there is an and a with . Thus, we have and</p>
<p>where we used .</p>
<p>Now, is the concatenation of the following continuous maps:</p>
<p>where is the canonical homeomorphism between the orbit and the</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="245" data-y="19" data-width="375" data-height="18">Since , and are continuous, the map</p>
<p class="pdf-text" data-x="245" data-y="78" data-width="121" data-height="17">is continuous.</p>
<p class="pdf-text" data-x="220" data-y="96" data-width="258" data-height="17">Now, we consider the map</p>
<p class="pdf-text" data-x="249" data-y="159" data-width="217" data-height="16">is continuous because</p>
<p class="pdf-text" data-x="245" data-y="217" data-width="328" data-height="15">is obviously continuous for all .</p>
<p class="pdf-text" data-x="219" data-y="234" data-width="564" data-height="18">induces a map via the following commutative diagram</p>
<p class="pdf-text" data-x="244" data-y="354" data-width="505" data-height="27">i.e.,</p>
<p class="pdf-text" data-x="247" data-y="380" data-width="196" data-height="18">is well-defined.</p>
<p class="pdf-text" data-x="279" data-y="399" data-width="350" data-height="19">Let with . Then</p>
<p class="pdf-text" data-x="279" data-y="531" data-width="581" data-height="22">because for .</p>
<p class="pdf-text" data-x="249" data-y="554" data-width="651" data-height="34">is continuous, because is open and surjective and and are continuous.</p>
<p class="pdf-text" data-x="220" data-y="589" data-width="680" data-height="39">For there is an and a with . Thus, we have and</p>
<p class="pdf-text" data-x="242" data-y="766" data-width="228" data-height="20">where we used .</p>
<p class="pdf-text" data-x="219" data-y="786" data-width="569" data-height="18">Now, is the concatenation of the following continuous maps:</p>
<p class="pdf-text" data-x="247" data-y="853" data-width="654" data-height="18">where is the canonical homeomorphism between the orbit and the</p>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="23" data-height="14" style="opacity: 0.5;">10</div>
</div>
|
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.
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}
$$
It is commutative due to $\varphi(\overline{{S}}\circ\overline{{\mathcal{G}}})\subseteq S\circ\mathbf{G}$ , $\varphi(\overline{{A}}\circ\overline{{\mathcal{G}}})\subseteq\varphi(\overline{{A}})\circ\mathbf{G}$ and the definition of $F$ . $\tau_{\mathbf G}$ is the canonical homeomorphism between the orbit of $\varphi(\overline{{A}})$ and the quotient of the acting group $\mathbf{G}$ by the stabilizer of $\varphi(\overline{{A}})$ .
Since $\varphi$ , $f$ and $\tau_{\mathbf G}$ are continuous, the map
$$
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}
$$
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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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"\\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}",
"\\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}",
"\\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}",
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"acting group modulo the stabilizer of . Hence, is continuous.",
"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. .",
"6 Openness of the Strata",
"Proposition 6.1 is open for all .",
"Corollary 6.2 is open in for all .",
"Proof Since , is open w.r.t. to the relative topology on . qed Corollary 6.3 is compact for all .",
"Proof is open because is open for all . Thus, 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 , 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 has a neighbourhood that again is contained in . So, let .",
"• Variant 1 Due to the slice theorem there is an open neighbourhood of , and so of , too, and an equivariant retraction . Since every equivariant mapping reduces types, we have for all , thus . • Variant 2 Choose again for an with . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood of in and an equivariant retraction . Since and are type-reducing, we have for all , i.e. . Obviously, contains and is open as a preimage of an open set. qed",
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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}}}$ . \u201d\u2286\u201d 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}}$ . \u201d\u2287\u201d 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": "\u2022 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\u2022 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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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": "\u201d\u2287\u201d 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": ". 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"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"}, 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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": "\u2022 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": "\u2022 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, "page_num": "page_10", "page_size": [612.0, 792.0], "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": "\u201d\u2286\u201d 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": "\u201d\u2287\u201d 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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_10", "page_size": [612.0, 792.0], "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": "\u2022 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": "\u2022 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, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [106, 480, 539, 687]}]}
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- acting group modulo the stabilizer of . Hence, is continuous.
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. .
# 6 Openness of the Strata
Proposition 6.1 is open for all .
Corollary 6.2 is open in for all .
Proof Since , is open w.r.t. to the relative topology on . qed Corollary 6.3 is compact for all .
Proof is open because is open for all . Thus, 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 , 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 has a neighbourhood that again is contained in . So, let .
- • Variant 1 Due to the slice theorem there is an open neighbourhood of , and so of , too, and an equivariant retraction . Since every equivariant mapping reduces types, we have for all , thus . • Variant 2 Choose again for an with . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood of in and an equivariant retraction . Since and are type-reducing, we have for all , i.e. . Obviously, contains and is open as a preimage of an open set. qed
11
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<div class="pdf-page">
<ul>
<li>acting group modulo the stabilizer of . Hence, is continuous.</li>
</ul>
<p>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. .</p>
<h1>6 Openness of the Strata</h1>
<p>Proposition 6.1 is open for all .</p>
<p>Corollary 6.2 is open in for all .</p>
<p>Proof Since , is open w.r.t. to the relative topology on . qed Corollary 6.3 is compact for all .</p>
<p>Proof is open because is open for all . Thus, is closed and the refore compact. qed</p>
<p>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.</p>
<h1>Proof Proposition 6.1</h1>
<p>We have to show that any has a neighbourhood that again is contained in . So, let .</p>
<ul>
<li>• Variant 1 Due to the slice theorem there is an open neighbourhood of , and so of , too, and an equivariant retraction . Since every equivariant mapping reduces types, we have for all , thus . • Variant 2 Choose again for an with . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood of in and an equivariant retraction . Since and are type-reducing, we have for all , i.e. . Obviously, contains and is open as a preimage of an open set. qed</li>
</ul>
</div>
|
<div class="pdf-page">
<ul class="pdf-list" data-x="240" data-y="16" data-width="427" data-height="38">
<li>acting group modulo the stabilizer of . Hence, is continuous.</li>
</ul>
<p class="pdf-text" data-x="197" data-y="56" data-width="704" data-height="220">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. .</p>
<h1 class="pdf-title" data-x="103" data-y="311" data-width="380" data-height="26">6 Openness of the Strata</h1>
<p class="pdf-text" data-x="105" data-y="350" data-width="386" data-height="21">Proposition 6.1 is open for all .</p>
<p class="pdf-text" data-x="105" data-y="376" data-width="428" data-height="20">Corollary 6.2 is open in for all .</p>
<p class="pdf-text" data-x="105" data-y="400" data-width="796" data-height="47">Proof Since , is open w.r.t. to the relative topology on . qed Corollary 6.3 is compact for all .</p>
<p class="pdf-text" data-x="105" data-y="451" data-width="795" data-height="41">Proof is open because is open for all . Thus, is closed and the refore compact. qed</p>
<p class="pdf-text" data-x="105" data-y="499" data-width="793" data-height="55">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.</p>
<h1 class="pdf-title" data-x="103" data-y="561" data-width="226" data-height="16">Proof Proposition 6.1</h1>
<p class="pdf-text" data-x="174" data-y="579" data-width="724" data-height="37">We have to show that any has a neighbourhood that again is contained in . So, let .</p>
<ul class="pdf-list" data-x="175" data-y="618" data-width="728" data-height="267">
<li>• Variant 1 Due to the slice theorem there is an open neighbourhood of , and so of , too, and an equivariant retraction . Since every equivariant mapping reduces types, we have for all , thus . • Variant 2 Choose again for an with . Due to the slice theorem for general transformation groups there is an open, invariant neighbourhood of in and an equivariant retraction . Since and are type-reducing, we have for all , i.e. . Obviously, contains and is open as a preimage of an open set. qed</li>
</ul>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="20" data-height="14" style="opacity: 0.5;">11</div>
</div>
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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.
# 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.
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.
• 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.
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.
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}}}$ .
# 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.
# 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.
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}
$$
It is commutative due to $\varphi(\overline{{S}}\circ\overline{{\mathcal{G}}})\subseteq S\circ\mathbf{G}$ , $\varphi(\overline{{A}}\circ\overline{{\mathcal{G}}})\subseteq\varphi(\overline{{A}})\circ\mathbf{G}$ and the definition of $F$ . $\tau_{\mathbf G}$ is the canonical homeomorphism between the orbit of $\varphi(\overline{{A}})$ and the quotient of the acting group $\mathbf{G}$ by the stabilizer of $\varphi(\overline{{A}})$ .
Since $\varphi$ , $f$ and $\tau_{\mathbf G}$ are continuous, the map
$$
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
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}$ .
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"content": [
"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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"7 Denseness of the Strata",
"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 .",
"As we will see in a moment, the next proposition will be very helpful.",
"Proposition 7.1 Let and be finitely many graphs. Then there is for any an with and for all .",
"Namely, we have",
"Corollary 7.2 is dense in for all .",
"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, .",
"Along with the proposition about the openness of the strata we get",
"Corollary 7.3 For all the closure of w.r.t. is equal to .",
"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",
"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 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 .",
"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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[{"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 \u2013 in contrast to the slice theorem and the openness of the strata \u2013 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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This assertion does \u2013 in contrast to the slice theorem and the openness of the strata \u2013", "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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 \u2013 in contrast to the slice theorem and the openness of the strata \u2013", "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [63, 649, 536, 678]}]}
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|
# 7 Denseness of the Strata
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 .
As we will see in a moment, the next proposition will be very helpful.
Proposition 7.1 Let and be finitely many graphs. Then there is for any an with and for all .
Namely, we have
Corollary 7.2 is dense in for all .
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, .
Along with the proposition about the openness of the strata we get
Corollary 7.3 For all the closure of w.r.t. is equal to .
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
# 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 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 .
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.
12
|
<div class="pdf-page">
<h1>7 Denseness of the Strata</h1>
<p>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 .</p>
<p>As we will see in a moment, the next proposition will be very helpful.</p>
<p>Proposition 7.1 Let and be finitely many graphs. Then there is for any an with and for all .</p>
<p>Namely, we have</p>
<p>Corollary 7.2 is dense in for all .</p>
<p>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, .</p>
<p>Along with the proposition about the openness of the strata we get</p>
<p>Corollary 7.3 For all the closure of w.r.t. is equal to .</p>
<p>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</p>
<h1>7.1 How to Prove Proposition 7.1?</h1>
<p>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 .</p>
<p>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>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="105" data-y="12" data-width="386" data-height="25">7 Denseness of the Strata</h1>
<p class="pdf-text" data-x="103" data-y="51" data-width="797" data-height="75">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 .</p>
<p class="pdf-text" data-x="105" data-y="126" data-width="599" data-height="21">As we will see in a moment, the next proposition will be very helpful.</p>
<p class="pdf-text" data-x="105" data-y="156" data-width="795" data-height="61">Proposition 7.1 Let and be finitely many graphs. Then there is for any an with and for all .</p>
<p class="pdf-text" data-x="103" data-y="227" data-width="147" data-height="19">Namely, we have</p>
<p class="pdf-text" data-x="103" data-y="257" data-width="435" data-height="22">Corollary 7.2 is dense in for all .</p>
<p class="pdf-text" data-x="103" data-y="292" data-width="797" data-height="142">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, .</p>
<p class="pdf-text" data-x="105" data-y="464" data-width="582" data-height="19">Along with the proposition about the openness of the strata we get</p>
<p class="pdf-text" data-x="103" data-y="493" data-width="639" data-height="22">Corollary 7.3 For all the closure of w.r.t. is equal to .</p>
<p class="pdf-text" data-x="105" data-y="530" data-width="795" data-height="104">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</p>
<h1 class="pdf-title" data-x="105" data-y="655" data-width="430" data-height="22">7.1 How to Prove Proposition 7.1?</h1>
<p class="pdf-text" data-x="103" data-y="686" data-width="797" data-height="150">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 .</p>
<p class="pdf-text" data-x="105" data-y="837" data-width="793" data-height="37">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>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="22" data-height="14" style="opacity: 0.5;">12</div>
</div>
|
# 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$ .
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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",
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"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",
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"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 .",
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"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",
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"we use the induction on the number of generators of t . In conclusion, we have to construct",
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"Definition 7.1 Let .",
"We say that and have the same initial segment (shortly: ↑↑ ) iff there exist such that and coincide up to the parametrization. We say analogously that the final segment of coincides with the initial segment of (shortly: ) iff there exist such that and coincide up to the parametrization. Iff the corresponding relations are not fulfilled, we write ↑↑ and , respectively.",
"Finally, we recall the decomposition lemma.",
"Lemma 7.4 Let be a point. Any can be written (up to parametrization) as a product with , such that • int or • int .",
"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 from as given in [10].",
"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 • .",
"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",
"",
"Obviously, there exists such an because is supposed to be at least two- dimensional. Set and . Finally, define a connection for , and as follows:",
"2. Construction of",
"• 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 .",
"13"
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[{"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}$ \u2191\u2191 $\\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}$ \u2191\u2191 $\\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 \u2022 int $\\gamma_{i}\\cap\\{x\\}=\\emptyset$ or \u2022 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: \u2022 $h_{\\overline{{{A}}}^{\\prime}}(\\pmb{\\alpha})=h_{\\overline{{{A}}}}(\\pmb{\\alpha})$ , \u2022 $\\pi_{\\Gamma_{i}}(\\overline{{{A}}}^{\\prime})=\\pi_{\\Gamma_{i}}(\\overline{{{A}}})$ for all $i$ , \u2022 $h_{\\overline{{{A}}}^{\\prime}}(e)=g$ for an $e\\in{\\mathcal{H}}{\\mathcal{G}}$ and \u2022 $\\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": "\u2022 Let $\\delta\\in\\mathcal{P}$ be for the moment a \u201dgenuine\u201d 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 t\uf8f3rivial path $\\delta$ set $h_{\\overline{{A}}^{\\prime}}(\\delta)=e_{\\mathbf{G}}$ . ", "page_idx": 12}]
|
{"preproc_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}, {"type": "text", "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": " \u2191\u2191", "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}, {"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": " \u2191\u2191", "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": "\u2022 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": "\u2022 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": "\u2022", "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": "\u2022", "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": "\u2022", "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": "\u2022", "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": "\u2022 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 \u201dgenuine\u201d 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 t\uf8f3rivial 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, "page_num": "page_12", "page_size": [612.0, 792.0], "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": " \u2191\u2191", "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": " \u2191\u2191", "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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": "\u2022 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": "\u2022 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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_12", "page_size": [612.0, 792.0], "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": "\u2022", "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": "\u2022", "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": "\u2022", "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": "\u2022", "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0], "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": "\u2022 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 \u201dgenuine\u201d 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 t\uf8f3rivial 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, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [127, 561, 536, 690]}]}
|
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Definition 7.1 Let .
- We say that and have the same initial segment (shortly: ↑↑ ) iff there exist such that and coincide up to the parametrization. We say analogously that the final segment of coincides with the initial segment of (shortly: ) iff there exist such that and coincide up to the parametrization. Iff the corresponding relations are not fulfilled, we write ↑↑ and , respectively.
Finally, we recall the decomposition lemma.
Lemma 7.4 Let be a point. Any can be written (up to parametrization) as a product with , such that • int or • int .
# 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 from as given in [10].
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 • .
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
$$
\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 because is supposed to be at least two- dimensional. Set and . Finally, define a connection for , and as follows:
2. Construction of
• 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 .
13
|
<div class="pdf-page">
<p>Definition 7.1 Let .</p>
<ul>
<li>We say that and have the same initial segment (shortly: ↑↑ ) iff there exist such that and coincide up to the parametrization. We say analogously that the final segment of coincides with the initial segment of (shortly: ) iff there exist such that and coincide up to the parametrization. Iff the corresponding relations are not fulfilled, we write ↑↑ and , respectively.</li>
</ul>
<p>Finally, we recall the decomposition lemma.</p>
<p>Lemma 7.4 Let be a point. Any can be written (up to parametrization) as a product with , such that • int or • int .</p>
<h1>7.2 Successive Magnifying of the Types</h1>
<ul>
<li>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 from as given in [10].</li>
</ul>
<p>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 • .</p>
<p>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</p>
<p>Obviously, there exists such an because is supposed to be at least two- dimensional. Set and . Finally, define a connection for , and as follows:</p>
<p>2. Construction of</p>
<p>• 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 .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="103" data-y="18" data-width="281" data-height="19">Definition 7.1 Let .</p>
<ul class="pdf-list" data-x="100" data-y="21" data-width="801" data-height="167">
<li>We say that and have the same initial segment (shortly: ↑↑ ) iff there exist such that and coincide up to the parametrization. We say analogously that the final segment of coincides with the initial segment of (shortly: ) iff there exist such that and coincide up to the parametrization. Iff the corresponding relations are not fulfilled, we write ↑↑ and , respectively.</li>
</ul>
<p class="pdf-text" data-x="103" data-y="199" data-width="380" data-height="19">Finally, we recall the decomposition lemma.</p>
<p class="pdf-text" data-x="105" data-y="230" data-width="793" data-height="77">Lemma 7.4 Let be a point. Any can be written (up to parametrization) as a product with , such that • int or • int .</p>
<h1 class="pdf-title" data-x="105" data-y="328" data-width="485" data-height="23">7.2 Successive Magnifying of the Types</h1>
<ul class="pdf-list" data-x="105" data-y="360" data-width="793" data-height="39">
<li>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 from as given in [10].</li>
</ul>
<p class="pdf-text" data-x="102" data-y="409" data-width="796" data-height="140">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 • .</p>
<p class="pdf-text" data-x="103" data-y="562" data-width="795" data-height="56">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</p>
<p class="pdf-text" data-x="199" data-y="646" data-width="699" data-height="57">Obviously, there exists such an because is supposed to be at least two- dimensional. Set and . Finally, define a connection for , and as follows:</p>
<p class="pdf-text" data-x="175" data-y="704" data-width="196" data-height="17">2. Construction of</p>
<p class="pdf-text" data-x="205" data-y="722" data-width="693" data-height="167">• 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 .</p>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="22" data-height="14" style="opacity: 0.5;">13</div>
</div>
|
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}}$ .
• 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}
$$
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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 two-",
"dimensional. 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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"• 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 .",
"We know from [10] that is indeed a connectio n.",
"3. The assertion for all is an immediate consequence of the construction because im . As well, we get .",
"4. Moreover, from (4), the fact that has no self-intersections and the definition of we get and so",
"",
"5. We have .",
"Let , i.e. for all . • From follows , i.e. . From im e′ ∩ im follows , i.e. for all .",
"Let be a path from to , such that int or int . Set . Then by construction we have",
"",
"There are four cases:",
"↑↑ and :",
"",
"↑↑ and :",
"",
"↑↑ and :",
"",
"↑↑ and :",
"",
"Thus, in each case we get .",
"Now, let be arbitrary and .",
"By the Decomposition Lemma 7.4 there is a decomposition with int or int for all . Thus, . Using the result just proven we get .",
"14"
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[{"type": "text", "text": "\u2022 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$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "5. We have $Z(\\mathbf{H}_{\\overline{{{A}}}^{\\prime}})=Z(\\{g\\}\\cup\\mathbf{H}_{\\overline{{{A}}}})$ . ", "page_idx": 13}, {"type": "text", "text": "Let $f\\in\\cal Z(\\mathbf{H}_{\\overline{{\\cal A}}^{\\prime}})$ , i.e. $f\\;h_{\\overline{{{A}}}^{\\prime}}(\\alpha)=h_{\\overline{{{A}}}^{\\prime}}(\\alpha)\\;f$ for all $\\alpha\\in{\\mathcal{H}}{\\mathcal{G}}$ . \n\u2022 From $h_{\\overline{{{A}}}^{\\prime}}(e)=g$ follows $f g=g f$ , i.e. $f\\in Z(\\{g\\})$ . From im e\u2032 \u2229 im $({\\pmb{\\alpha}})=\\emptyset$ follows $h_{\\overline{{A}}}(\\alpha_{i})=h_{\\overline{{A}}^{\\prime}}(\\alpha_{i})$ , i.e. $f\\,\\in\\,Z(h_{\\overline{{A}}}(\\alpha_{i}))$ for all $i$ . ", "page_idx": 13}, {"type": "text", "text": "Let $\\alpha^{\\prime}$ be a path from $m^{\\prime}$ to $m^{\\prime}$ , such that int $\\alpha^{\\prime}\\cap\\{m^{\\prime}\\}=\\emptyset$ or int $\\alpha^{\\prime}=$ $\\{m^{\\prime}\\}$ . Set $\\alpha:=\\gamma\\,\\alpha^{\\prime}\\,\\gamma^{-1}$ . Then by construction we have ", "page_idx": 13}, {"type": "equation", "text": "$$\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$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "There are four cases: ", "page_idx": 13}, {"type": "text", "text": "$-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}$ \u2191\u2191 $e^{\\prime}$ and $\\alpha^{\\prime}\\,\\#\\,e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\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$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$-\\mathrm{~\\boldmath~\\alpha~}\\alpha^{\\prime}$ \u2191\u2191 $e^{\\prime}$ and $\\alpha^{\\prime}\\,\\#\\,e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\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$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$\\alpha^{\\prime}$ \u2191\u2191 $e^{\\prime}$ and $\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\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$$", "text_format": "latex", "page_idx": 13}, {"type": "text", "text": "$\\alpha^{\\prime}$ \u2191\u2191 $e^{\\prime}$ and $\\alpha^{\\prime}\\downarrow\\uparrow e^{\\prime}$ : ", "page_idx": 13}, {"type": "equation", "text": "$$\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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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": "\u2022 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\u2032 \u2229 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": " \u2191\u2191", "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": " \u2191\u2191", "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": " \u2191\u2191", "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": " \u2191\u2191", "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": "\u2022 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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_13", "page_size": [612.0, 792.0], "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": "\u2022 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\u2032 \u2229 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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_13", "page_size": [612.0, 792.0], "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": " \u2191\u2191", "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"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": " \u2191\u2191", "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"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": " \u2191\u2191", "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"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": " \u2191\u2191", "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [174, 620, 538, 686]}]}
|
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• 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 .
We know from [10] that is indeed a connectio n.
3. The assertion for all is an immediate consequence of the construction because im . As well, we get .
4. Moreover, from (4), the fact that has no self-intersections and the definition of we get 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.
$$
5. We have .
- Let , i.e. for all . • From follows , i.e. . From im e′ ∩ im follows , i.e. for all .
Let be a path from to , such that int or int . Set . Then by construction we have
$$
\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}
$$
There are four cases:
↑↑ and :
$$
\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}
$$
↑↑ and :
$$
\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}
$$
↑↑ and :
$$
\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}
$$
↑↑ and :
$$
\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 .
Now, let be arbitrary and .
By the Decomposition Lemma 7.4 there is a decomposition with int or int for all . Thus, . Using the result just proven we get .
14
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<div class="pdf-page">
<p>• 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 .</p>
<p>We know from [10] that is indeed a connectio n.</p>
<p>3. The assertion for all is an immediate consequence of the construction because im . As well, we get .</p>
<p>4. Moreover, from (4), the fact that has no self-intersections and the definition of we get and so</p>
<p>5. We have .</p>
<ul>
<li>Let , i.e. for all . • From follows , i.e. . From im e′ ∩ im follows , i.e. for all .</li>
</ul>
<p>Let be a path from to , such that int or int . Set . Then by construction we have</p>
<p>There are four cases:</p>
<p>↑↑ and :</p>
<p>↑↑ and :</p>
<p>↑↑ and :</p>
<p>↑↑ and :</p>
<p>Thus, in each case we get .</p>
<p>Now, let be arbitrary and .</p>
<p>By the Decomposition Lemma 7.4 there is a decomposition with int or int for all . Thus, . Using the result just proven we get .</p>
</div>
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<div class="pdf-page">
<p class="pdf-text" data-x="209" data-y="18" data-width="689" data-height="58">• 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 .</p>
<p class="pdf-text" data-x="210" data-y="76" data-width="432" data-height="19">We know from [10] that is indeed a connectio n.</p>
<p class="pdf-text" data-x="177" data-y="96" data-width="723" data-height="38">3. The assertion for all is an immediate consequence of the construction because im . As well, we get .</p>
<p class="pdf-text" data-x="175" data-y="135" data-width="726" data-height="35">4. Moreover, from (4), the fact that has no self-intersections and the definition of we get and so</p>
<p class="pdf-text" data-x="175" data-y="190" data-width="320" data-height="19">5. We have .</p>
<ul class="pdf-list" data-x="245" data-y="209" data-width="651" data-height="74">
<li>Let , i.e. for all . • From follows , i.e. . From im e′ ∩ im follows , i.e. for all .</li>
</ul>
<p class="pdf-text" data-x="262" data-y="321" data-width="638" data-height="37">Let be a path from to , such that int or int . Set . Then by construction we have</p>
<p class="pdf-text" data-x="291" data-y="408" data-width="180" data-height="17">There are four cases:</p>
<p class="pdf-text" data-x="291" data-y="426" data-width="212" data-height="18">↑↑ and :</p>
<p class="pdf-text" data-x="289" data-y="495" data-width="216" data-height="19">↑↑ and :</p>
<p class="pdf-text" data-x="289" data-y="584" data-width="216" data-height="19">↑↑ and :</p>
<p class="pdf-text" data-x="322" data-y="671" data-width="183" data-height="19">↑↑ and :</p>
<p class="pdf-text" data-x="281" data-y="761" data-width="389" data-height="19">Thus, in each case we get .</p>
<p class="pdf-text" data-x="264" data-y="782" data-width="438" data-height="17">Now, let be arbitrary and .</p>
<p class="pdf-text" data-x="286" data-y="800" data-width="612" data-height="86">By the Decomposition Lemma 7.4 there is a decomposition with int or int for all . Thus, . Using the result just proven we get .</p>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="23" data-height="14" style="opacity: 0.5;">14</div>
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• 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.
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.
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}}}$ .
# 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.
# 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.
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}
$$
It is commutative due to $\varphi(\overline{{S}}\circ\overline{{\mathcal{G}}})\subseteq S\circ\mathbf{G}$ , $\varphi(\overline{{A}}\circ\overline{{\mathcal{G}}})\subseteq\varphi(\overline{{A}})\circ\mathbf{G}$ and the definition of $F$ . $\tau_{\mathbf G}$ is the canonical homeomorphism between the orbit of $\varphi(\overline{{A}})$ and the quotient of the acting group $\mathbf{G}$ by the stabilizer of $\varphi(\overline{{A}})$ .
Since $\varphi$ , $f$ and $\tau_{\mathbf G}$ are continuous, the map
$$
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
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
# 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.
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}}$ .
• 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$ .
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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)\\}) ."
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"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.",
"\\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}",
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"\\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}",
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"Thus, . Due to the definition of we have .",
"7.3 Construction of Arbitrary Types",
"Finally, we can now prove the desired proposition.",
"Proof Proposition 7.1",
"• Let and . Then there exist a Howe subgroup with and a , such that . Since is a Howe subgroup, we have and so by Lemma 4.1 there exist certain , such that . • Now let with an appropriate as in Corollary 4.2. Because of we have . • We now use inductively Lemma 7.5. Let and . Construct for all a connection and an from and by that lemma, such that for all , , and . Setting we get Finally, we define . Now, we get for all , and . Thus, i.e., . qed",
"The proposition just proven has a further immediate consequence.",
"Corollary 7.6 is non-empty for all .",
"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",
"This corollary solves the problem which gauge orbit types exist for generalized connections.",
"Theorem 7.7 The set of all gauge orbit types on is the set of all conjugacy classes of Howe subgroups of .",
"Furthermore we have",
"Corollary 7.8 Let be some graph. Then . In other words: is surjective even on the generic connections.",
"Proof is surjective on as proven in [10]. By Proposition 7.1 there is now an with and . qed",
"15",
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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": "\u2022 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\u2022 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\u2022 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}]
|
{"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": "\u2022 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": "\u2022 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": "\u2022 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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"type": "list", "bbox": [106, 124, 538, 417], "lines": [{"bbox": [106, 126, 538, 140], "spans": [{"bbox": [106, 126, 144, 140], "score": 1.0, "content": "\u2022 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": "\u2022 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": "\u2022 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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [61, 656, 539, 687]}]}
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|
Thus, . Due to the definition of we have .
qed
# 7.3 Construction of Arbitrary Types
Finally, we can now prove the desired proposition.
# Proof Proposition 7.1
- • Let and . Then there exist a Howe subgroup with and a , such that . Since is a Howe subgroup, we have and so by Lemma 4.1 there exist certain , such that . • Now let with an appropriate as in Corollary 4.2. Because of we have . • We now use inductively Lemma 7.5. Let and . Construct for all a connection and an from and by that lemma, such that for all , , and . Setting we get Finally, we define . Now, we get for all , and . Thus, i.e., . qed
The proposition just proven has a further immediate consequence.
Corollary 7.6 is non-empty for all .
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
This corollary solves the problem which gauge orbit types exist for generalized connections.
Theorem 7.7 The set of all gauge orbit types on is the set of all conjugacy classes of Howe subgroups of .
Furthermore we have
Corollary 7.8 Let be some graph. Then . In other words: is surjective even on the generic connections.
Proof is surjective on as proven in [10]. By Proposition 7.1 there is now an with and . qed
15
|
<div class="pdf-page">
<p>Thus, . Due to the definition of we have .</p>
<h1>7.3 Construction of Arbitrary Types</h1>
<p>Finally, we can now prove the desired proposition.</p>
<h1>Proof Proposition 7.1</h1>
<ul>
<li>• Let and . Then there exist a Howe subgroup with and a , such that . Since is a Howe subgroup, we have and so by Lemma 4.1 there exist certain , such that . • Now let with an appropriate as in Corollary 4.2. Because of we have . • We now use inductively Lemma 7.5. Let and . Construct for all a connection and an from and by that lemma, such that for all , , and . Setting we get Finally, we define . Now, we get for all , and . Thus, i.e., . qed</li>
</ul>
<p>The proposition just proven has a further immediate consequence.</p>
<p>Corollary 7.6 is non-empty for all .</p>
<p>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</p>
<p>This corollary solves the problem which gauge orbit types exist for generalized connections.</p>
<p>Theorem 7.7 The set of all gauge orbit types on is the set of all conjugacy classes of Howe subgroups of .</p>
<p>Furthermore we have</p>
<p>Corollary 7.8 Let be some graph. Then . In other words: is surjective even on the generic connections.</p>
<p>Proof is surjective on as proven in [10]. By Proposition 7.1 there is now an with and . qed</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="207" data-y="18" data-width="542" data-height="40">Thus, . Due to the definition of we have .</p>
<div class="pdf-discarded" data-x="860" data-y="38" data-width="38" data-height="17" style="opacity: 0.5;">qed</div>
<h1 class="pdf-title" data-x="103" data-y="78" data-width="454" data-height="22">7.3 Construction of Arbitrary Types</h1>
<p class="pdf-text" data-x="103" data-y="109" data-width="435" data-height="20">Finally, we can now prove the desired proposition.</p>
<h1 class="pdf-title" data-x="103" data-y="140" data-width="228" data-height="19">Proof Proposition 7.1</h1>
<ul class="pdf-list" data-x="177" data-y="160" data-width="723" data-height="379">
<li>• Let and . Then there exist a Howe subgroup with and a , such that . Since is a Howe subgroup, we have and so by Lemma 4.1 there exist certain , such that . • Now let with an appropriate as in Corollary 4.2. Because of we have . • We now use inductively Lemma 7.5. Let and . Construct for all a connection and an from and by that lemma, such that for all , , and . Setting we get Finally, we define . Now, we get for all , and . Thus, i.e., . qed</li>
</ul>
<p class="pdf-text" data-x="107" data-y="550" data-width="568" data-height="20">The proposition just proven has a further immediate consequence.</p>
<p class="pdf-text" data-x="105" data-y="577" data-width="415" data-height="22">Corollary 7.6 is non-empty for all .</p>
<p class="pdf-text" data-x="105" data-y="612" data-width="793" data-height="60">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</p>
<p class="pdf-text" data-x="105" data-y="683" data-width="790" data-height="21">This corollary solves the problem which gauge orbit types exist for generalized connections.</p>
<p class="pdf-text" data-x="105" data-y="713" data-width="795" data-height="40">Theorem 7.7 The set of all gauge orbit types on is the set of all conjugacy classes of Howe subgroups of .</p>
<p class="pdf-text" data-x="103" data-y="764" data-width="184" data-height="18">Furthermore we have</p>
<p class="pdf-text" data-x="105" data-y="792" data-width="795" data-height="40">Corollary 7.8 Let be some graph. Then . In other words: is surjective even on the generic connections.</p>
<p class="pdf-text" data-x="105" data-y="844" data-width="795" data-height="42">Proof is surjective on as proven in [10]. By Proposition 7.1 there is now an with and . qed</p>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="22" data-height="14" style="opacity: 0.5;">15</div>
</div>
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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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"8 Stratification of",
"First we recall the general definition of a stratification [12].",
"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 .",
"Theorem 8.1 is a topologically regular stratification of . Analogously, is a topologically regular stratification of .",
"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",
"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 be a connection. 1. is called complete . 2. is called almost complete . 3. is called non-complete .",
"Obviously, we have",
"denotes again the closure of , here w.r.t. .",
"16"
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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 \u2022 $U\\cap V\\neq\\emptyset\\Longrightarrow U=V$ , \u2022 $\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{U}}\\supseteq V$ and \u2022 $\\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 \u2022 Obviously, $_S$ is a covering of $\\overline{{\\mathcal{A}}}$ . \u2022 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]). \u2022 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}}$ . \u2022 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}}$ . \u2022 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}}$ . \u2022 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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"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 \u2022 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": "\u2022 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": "\u2022 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": "\u2022 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": "\u2022 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": "\u2022 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. ", "type": "text"}, {"bbox": [298, 665, 306, 674], "score": 0.84, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [307, 663, 309, 678], "score": 1.0, "content": ".", "type": "text"}]}]}, {"type": "discarded", "bbox": [293, 704, 306, 715], "lines": [{"bbox": [292, 705, 307, 718], "spans": [{"bbox": [292, 705, 307, 718], "score": 1.0, "content": "16", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [61, 12, 248, 33], "lines": [{"bbox": [64, 16, 245, 32], "spans": [{"bbox": [64, 20, 73, 30], "score": 1.0, "content": "8", "type": "text"}, {"bbox": [91, 17, 231, 32], "score": 1.0, "content": "Stratification of ", "type": "text"}, {"bbox": [231, 16, 245, 32], "score": 0.31, "content": "\\overline{{\\mathcal{A}}}", "type": "inline_equation", "height": 16, "width": 14}], "index": 0}], "index": 0, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [62, 43, 371, 59], "lines": [{"bbox": [62, 46, 366, 61], "spans": [{"bbox": [62, 46, 366, 61], "score": 1.0, "content": "First we recall the general definition of a stratification [12].", "type": "text"}], "index": 1}], "index": 1, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [62, 46, 366, 61]}, {"type": "text", "bbox": [61, 68, 537, 185], "lines": [{"bbox": [61, 70, 537, 87], "spans": [{"bbox": [61, 70, 252, 87], "score": 1.0, "content": "Definition 8.1 A countable family ", "type": "text"}, {"bbox": [252, 72, 261, 82], "score": 0.83, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [261, 70, 482, 87], "score": 1.0, "content": " of non-empty subsets of a topological space ", "type": "text"}, {"bbox": [482, 73, 493, 82], "score": 0.91, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [493, 70, 537, 87], "score": 1.0, "content": " is called", "type": "text"}], "index": 2}, {"bbox": [152, 85, 534, 100], "spans": [{"bbox": [152, 85, 245, 100], "score": 1.0, "content": "stratification of ", "type": "text"}, {"bbox": [245, 87, 257, 96], "score": 0.86, "content": "X", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [257, 85, 273, 100], "score": 1.0, "content": " iff", "type": "text"}, {"bbox": [273, 86, 282, 96], "score": 0.82, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 10, "width": 9}, {"bbox": [283, 85, 371, 100], "score": 1.0, "content": " is a covering for ", "type": "text"}, {"bbox": [371, 87, 382, 96], "score": 0.88, "content": "X", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [383, 85, 443, 100], "score": 1.0, "content": " and for all ", "type": "text"}, {"bbox": [443, 87, 488, 99], "score": 0.93, "content": "U,V\\in S", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [489, 85, 534, 100], "score": 1.0, "content": " we have", "type": "text"}], "index": 3}, {"bbox": [151, 100, 291, 114], "spans": [{"bbox": [151, 100, 170, 114], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [171, 101, 286, 113], "score": 0.89, "content": "U\\cap V\\neq\\emptyset\\Longrightarrow U=V", "type": "inline_equation", "height": 12, "width": 115}, {"bbox": [287, 100, 291, 114], "score": 1.0, "content": ",", "type": "text"}], "index": 4}, {"bbox": [152, 111, 311, 129], "spans": [{"bbox": [152, 112, 170, 129], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [170, 114, 286, 127], "score": 0.89, "content": "\\overline{{U}}\\cap V\\neq\\emptyset\\Longrightarrow\\overline{{U}}\\supseteq V", "type": "inline_equation", "height": 13, "width": 116}, {"bbox": [286, 111, 311, 128], "score": 1.0, "content": " and", "type": "text"}], "index": 5}, {"bbox": [152, 127, 345, 143], "spans": [{"bbox": [152, 127, 169, 143], "score": 1.0, "content": "\u2022", "type": "text"}, {"bbox": [170, 128, 341, 142], "score": 0.88, "content": "\\overline{{{U}}}\\cap V\\neq\\varnothing\\Longrightarrow\\overline{{{V}}}\\cap(U\\cup V)=V", "type": "inline_equation", "height": 14, "width": 171}, {"bbox": [341, 127, 345, 143], "score": 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, "page_num": "page_15", "page_size": [612.0, 792.0], "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, "page_num": "page_15", "page_size": [612.0, 792.0], "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 \u2022 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": "\u2022 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": "\u2022 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": "\u2022 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": "\u2022 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": "\u2022 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, "page_num": "page_15", "page_size": [612.0, 792.0], "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, "page_num": "page_15", "page_size": [612.0, 792.0], "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, "page_num": "page_15", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_15", "page_size": [612.0, 792.0], "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, "page_num": "page_15", "page_size": [612.0, 792.0], "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, "page_num": "page_15", "page_size": [612.0, 792.0], "bbox_fs": [63, 644, 162, 657]}]}
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# 8 Stratification of
First we recall the general definition of a stratification [12].
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 .
Theorem 8.1 is a topologically regular stratification of . Analogously, is a topologically regular stratification of .
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
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 be a connection. 1. is called complete . 2. is called almost complete . 3. is called non-complete .
Obviously, we have
denotes again the closure of , here w.r.t. .
16
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<div class="pdf-page">
<h1>8 Stratification of</h1>
<p>First we recall the general definition of a stratification [12].</p>
<p>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 .</p>
<p>Theorem 8.1 is a topologically regular stratification of . Analogously, is a topologically regular stratification of .</p>
<p>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</p>
<p>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>9 Non-complete Connections</h1>
<p>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>Definition 9.1 Let be a connection. 1. is called complete . 2. is called almost complete . 3. is called non-complete .</p>
<p>Obviously, we have</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="102" data-y="15" data-width="312" data-height="27">8 Stratification of</h1>
<p class="pdf-text" data-x="103" data-y="55" data-width="517" data-height="21">First we recall the general definition of a stratification [12].</p>
<p class="pdf-text" data-x="102" data-y="87" data-width="796" data-height="152">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 .</p>
<p class="pdf-text" data-x="103" data-y="252" data-width="797" data-height="59">Theorem 8.1 is a topologically regular stratification of . Analogously, is a topologically regular stratification of .</p>
<p class="pdf-text" data-x="135" data-y="325" data-width="765" data-height="189">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</p>
<p class="pdf-text" data-x="103" data-y="530" data-width="797" data-height="58">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 class="pdf-title" data-x="103" data-y="614" data-width="435" data-height="26">9 Non-complete Connections</h1>
<p class="pdf-text" data-x="103" data-y="652" data-width="797" data-height="77">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 class="pdf-text" data-x="102" data-y="739" data-width="575" data-height="78">Definition 9.1 Let be a connection. 1. is called complete . 2. is called almost complete . 3. is called non-complete .</p>
<p class="pdf-text" data-x="103" data-y="830" data-width="169" data-height="18">Obviously, we have</p>
<div class="pdf-discarded" data-x="127" data-y="855" data-width="390" data-height="19" style="opacity: 0.5;">denotes again the closure of , here w.r.t. .</div>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="22" data-height="14" style="opacity: 0.5;">16</div>
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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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"Lemma 9.1 If is complete (almost complete, non-complete), so is complete (almost complete, non-complete) for all .",
"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 non-complete}. Then is contained in a set of -measure zero whereas is the induced Haar measure on . [2, 6, 10]",
"Since is gauge invariant, we have",
"Corollary 9.3 Let non-complete}. Then is contained in a set of -measure zero.",
"For the proof of the proposition we still need the follow",
"Lemma 9.4 Let be measurable with and . Then is contained in a set of -measure zero.",
"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",
"",
"• From for all follows . But, for all , i.e. , because 1. qed",
"Proof Proposition 9.2",
"• 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",
"17",
"6Such a graph does indeed exist for . For instance, take circles with centers in and radii . By means of an appropriate chart mapping around these circles define a graph with the desired properties."
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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 \u2022 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}$ \u2022 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}$ . \u2022 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": "\u2022 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": "\u2022 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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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 \u2022 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": "\u2022 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": "\u2022 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": "\u2022 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": "\u2022 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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0], "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 \u2022 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": "\u2022 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": "\u2022 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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [106, 411, 537, 455], "lines": [{"bbox": [106, 413, 538, 429], "spans": [{"bbox": [106, 413, 154, 429], "score": 1.0, "content": "\u2022 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, "page_num": "page_16", "page_size": [612.0, 792.0], "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, "page_num": "page_16", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [106, 478, 539, 646], "lines": [{"bbox": [107, 479, 538, 494], "spans": [{"bbox": [107, 479, 144, 494], "score": 1.0, "content": "\u2022 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, "page_num": "page_16", "page_size": [612.0, 792.0], "bbox_fs": [107, 479, 539, 648]}]}
|
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{"category_id": 1, "poly": [295, 1329, 1498, 1329, 1498, 1796, 295, 1796], "score": 0.645}, {"category_id": 1, "poly": [297, 1144, 1494, 1144, 1494, 1265, 297, 1265], "score": 0.586}, {"category_id": 1, "poly": [299, 1144, 1494, 1144, 1494, 1265, 299, 1265], "score": 0.441}, {"category_id": 1, "poly": [181, 689, 1495, 689, 1495, 975, 181, 975], "score": 0.355}, {"category_id": 1, "poly": [298, 898, 1493, 898, 1493, 977, 298, 977], "score": 0.303}, {"category_id": 13, "poly": [1263, 1467, 1481, 1467, 1481, 1502, 1263, 1502], "score": 0.96, "latex": "\\mu_{\\mathrm{Haar}}(U_{k,i})\\,>\\,0"}, {"category_id": 13, "poly": [511, 1414, 780, 1414, 780, 1465, 511, 1465], "score": 0.95, "latex": "N^{\\prime}:=\\cup_{k}\\mathopen{}\\left(\\cup_{i}\\,N_{U_{k,i}}\\right)"}, {"category_id": 13, "poly": [783, 1557, 945, 1557, 945, 1593, 783, 1593], "score": 0.94, "latex": "\\mu_{0}(N^{\\ast})=0"}, {"category_id": 13, "poly": [421, 49, 519, 49, 519, 79, 421, 79], "score": 0.94, "latex": "{\\overline{{A}}}\\in{\\overline{{A}}}"}, {"category_id": 13, "poly": [712, 1764, 833, 1764, 833, 1797, 712, 1797], "score": 0.94, "latex": "U_{k,i}\\subseteq U"}, {"category_id": 13, "poly": [1189, 868, 1345, 868, 1345, 901, 1189, 901], "score": 0.94, "latex": "N_{U}\\subseteq N_{k,U}"}, {"category_id": 13, "poly": [1240, 49, 1317, 49, 1317, 85, 1240, 85], "score": 0.94, "latex": "\\overline{{A}}\\circ\\overline{{g}}"}, {"category_id": 13, "poly": [957, 89, 1038, 89, 1038, 125, 957, 125], "score": 0.94, "latex": "{\\overline{{g}}}\\in{\\overline{{g}}}"}, {"category_id": 13, "poly": [1046, 1507, 1247, 1507, 1247, 1548, 1046, 1548], "score": 0.94, "latex": "\\mu_{0}(N_{U_{k,i}}^{*})\\,=\\,0"}, {"category_id": 13, "poly": [1091, 1337, 1185, 1337, 1185, 1372, 1091, 1372], "score": 0.94, "latex": "\\{U_{k,i}\\}_{i}"}, {"category_id": 13, "poly": [343, 1550, 699, 1550, 699, 1601, 343, 1601], "score": 0.94, "latex": "N^{\\prime}\\subseteq N^{*}:=\\cup_{k}\\bigl(\\cup_{i}N_{U_{k,i}}^{*}\\bigr)"}, {"category_id": 13, "poly": [470, 1725, 521, 1725, 521, 1757, 470, 1757], "score": 0.94, "latex": "U_{k,i}"}, {"category_id": 13, "poly": [973, 1762, 1169, 1762, 1169, 1798, 973, 1798], "score": 0.94, "latex": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U_{k,i}"}, {"category_id": 13, "poly": [1027, 1642, 1207, 1642, 1207, 1677, 1027, 1677], "score": 0.93, "latex": "\\mathbf{H}_{\\overline{{A}}}\\subseteq\\mathbf{G}\\setminus U"}, {"category_id": 13, "poly": [761, 1509, 955, 1509, 955, 1548, 761, 1548], "score": 0.93, "latex": "N_{U_{k,i}}\\ \\subseteq\\ N_{U_{k,i}}^{*}"}, {"category_id": 13, "poly": [562, 823, 874, 823, 874, 860, 562, 860], "score": 0.93, "latex": "N_{k,U}:=\\pi_{k}^{-1}((\\mathbf G\\backslash U)^{k})"}, {"category_id": 13, "poly": [600, 1724, 721, 1724, 721, 1757, 600, 1757], "score": 0.93, "latex": "m\\in U_{k,i}"}, {"category_id": 13, "poly": [866, 560, 1059, 560, 1059, 594, 866, 594], 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Lemma 9.1 If is complete (almost complete, non-complete), so is complete (almost complete, non-complete) for all .
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 non-complete}. Then is contained in a set of -measure zero whereas is the induced Haar measure on . [2, 6, 10]
Since is gauge invariant, we have
Corollary 9.3 Let non-complete}. Then is contained in a set of -measure zero.
For the proof of the proposition we still need the follow
Lemma 9.4 Let be measurable with and . Then is contained in a set of -measure zero.
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
$$
\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 for all follows . But, for all , i.e. , because 1. qed
# Proof Proposition 9.2
• 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
6Such a graph does indeed exist for . For instance, take circles with centers in and radii . By means of an appropriate chart mapping around these circles define a graph with the desired properties.
17
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<div class="pdf-page">
<p>Lemma 9.1 If is complete (almost complete, non-complete), so is complete (almost complete, non-complete) for all .</p>
<p>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>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]</p>
<p>Since is gauge invariant, we have</p>
<p>Corollary 9.3 Let non-complete}. Then is contained in a set of -measure zero.</p>
<p>For the proof of the proposition we still need the follow</p>
<p>Lemma 9.4 Let be measurable with and . Then is contained in a set of -measure zero.</p>
<p>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</p>
<p>• From for all follows . But, for all , i.e. , because 1. qed</p>
<h1>Proof Proposition 9.2</h1>
<p>• 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</p>
</div>
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<div class="pdf-page">
<p class="pdf-text" data-x="105" data-y="15" data-width="795" data-height="41">Lemma 9.1 If is complete (almost complete, non-complete), so is complete (almost complete, non-complete) for all .</p>
<p class="pdf-text" data-x="103" data-y="64" data-width="797" data-height="38">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 class="pdf-text" data-x="105" data-y="107" data-width="795" data-height="41">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]</p>
<p class="pdf-text" data-x="103" data-y="155" data-width="311" data-height="20">Since is gauge invariant, we have</p>
<p class="pdf-text" data-x="105" data-y="179" data-width="795" data-height="42">Corollary 9.3 Let non-complete}. Then is contained in a set of -measure zero.</p>
<p class="pdf-text" data-x="103" data-y="227" data-width="479" data-height="19">For the proof of the proposition we still need the follow</p>
<p class="pdf-text" data-x="105" data-y="253" data-width="795" data-height="58">Lemma 9.4 Let be measurable with and . Then is contained in a set of -measure zero.</p>
<p class="pdf-text" data-x="108" data-y="320" data-width="792" data-height="133">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</p>
<p class="pdf-text" data-x="177" data-y="531" data-width="721" data-height="57">• From for all follows . But, for all , i.e. , because 1. qed</p>
<h1 class="pdf-title" data-x="102" data-y="598" data-width="229" data-height="18">Proof Proposition 9.2</h1>
<p class="pdf-text" data-x="177" data-y="618" data-width="724" data-height="217">• 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</p>
<div class="pdf-discarded" data-x="103" data-y="840" data-width="795" data-height="48" style="opacity: 0.5;">6Such a graph does indeed exist for . For instance, take circles with centers in and radii . By means of an appropriate chart mapping around these circles define a graph with the desired properties.</div>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="22" data-height="14" style="opacity: 0.5;">17</div>
</div>
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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.
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.
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}}}$ .
# 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.
# 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.
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}
$$
It is commutative due to $\varphi(\overline{{S}}\circ\overline{{\mathcal{G}}})\subseteq S\circ\mathbf{G}$ , $\varphi(\overline{{A}}\circ\overline{{\mathcal{G}}})\subseteq\varphi(\overline{{A}})\circ\mathbf{G}$ and the definition of $F$ . $\tau_{\mathbf G}$ is the canonical homeomorphism between the orbit of $\varphi(\overline{{A}})$ and the quotient of the acting group $\mathbf{G}$ by the stabilizer of $\varphi(\overline{{A}})$ .
Since $\varphi$ , $f$ and $\tau_{\mathbf G}$ are continuous, the map
$$
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
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
# 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.
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}}$ .
• 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)\})$ .
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
# 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
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
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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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"Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1.",
"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",
"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.",
"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.",
"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 -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.",
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[{"type": "text", "text": "Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has \u00b50-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 \u201dsize\u201d 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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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 \u201dsize\u201d 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": "\u00b50-measure 1.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0], "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 \u201dsize\u201d 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, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_17", "page_size": [612.0, 792.0], "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, "page_num": "page_17", "page_size": [612.0, 792.0], "bbox_fs": [61, 631, 538, 689]}]}
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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 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
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.
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.
# 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 -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.
18
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<div class="pdf-page">
<p>Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1.</p>
<p>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</p>
<p>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.</p>
<p>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.</p>
<h1>10 Summary and Discussion</h1>
<p>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>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.</p>
</div>
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<div class="pdf-page">
<p class="pdf-text" data-x="105" data-y="18" data-width="793" data-height="37">Corollary 9.5 The set of all generic connections (i.e. connections of maximal type) has µ0-measure 1.</p>
<p class="pdf-text" data-x="105" data-y="71" data-width="795" data-height="76">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</p>
<p class="pdf-text" data-x="105" data-y="162" data-width="793" data-height="340">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.</p>
<p class="pdf-text" data-x="105" data-y="502" data-width="795" data-height="75">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.</p>
<h1 class="pdf-title" data-x="105" data-y="605" data-width="423" data-height="25">10 Summary and Discussion</h1>
<p class="pdf-text" data-x="105" data-y="643" data-width="795" data-height="168">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 class="pdf-text" data-x="107" data-y="814" data-width="793" data-height="74">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.</p>
<div class="pdf-discarded" data-x="491" data-y="910" data-width="21" data-height="14" style="opacity: 0.5;">18</div>
</div>
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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.
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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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"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.",
"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.",
"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.",
"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 connected itself (at least for connected )? Is globally trivial over , at least for the generic stratum with ? What sections do exist in these bundles, i.e. what gauge fixings do exist in ? 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, non- commutative 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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[{"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 \u2013 to the best of our knowlegde \u2013 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 \u201dtransfer\u201d 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 \u2013 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 \u2013 and in this paper completely ignored \u2013 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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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 \u201dtransfer\u201d 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 \u2013 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 \u2013 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": "\u2013 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, "page_num": "page_18", "page_size": [612.0, 792.0], "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, "page_num": "page_18", "page_size": [612.0, 792.0], "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 \u2013 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": "\u2013 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, "page_num": "page_18", "page_size": [612.0, 792.0], "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 \u201dtransfer\u201d 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 \u2013 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, "page_num": "page_18", "page_size": [612.0, 792.0], "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 \u2013 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": "\u2013 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, "page_num": "page_18", "page_size": [612.0, 792.0], "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": ")? 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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, "page_num": "page_18", "page_size": [612.0, 792.0], "bbox_fs": [78, 518, 539, 649]}]}
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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.
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.
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.
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 connected itself (at least for connected )? Is globally trivial over , at least for the generic stratum with ? What sections do exist in these bundles, i.e. what gauge fixings do exist in ? 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, non- commutative 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.
19
|
<div class="pdf-page">
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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>
<ul>
<li>Topology What is the topological structure of the strata? Are they connected or is connected itself (at least for connected )? Is globally trivial over , at least for the generic stratum with ? What sections do exist in these bundles, i.e. what gauge fixings do exist in ? 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, non- commutative 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.</li>
</ul>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="105" data-y="56" data-width="795" data-height="185">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.</p>
<p class="pdf-text" data-x="105" data-y="244" data-width="795" data-height="297">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.</p>
<p class="pdf-text" data-x="107" data-y="541" data-width="791" data-height="56">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.</p>
<p class="pdf-text" data-x="105" data-y="611" data-width="793" data-height="56">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>
<ul class="pdf-list" data-x="107" data-y="669" data-width="793" data-height="167">
<li>Topology What is the topological structure of the strata? Are they connected or is connected itself (at least for connected )? Is globally trivial over , at least for the generic stratum with ? What sections do exist in these bundles, i.e. what gauge fixings do exist in ? 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, non- commutative 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.</li>
</ul>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="22" data-height="14" style="opacity: 0.5;">19</div>
</div>
|
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:
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"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?",
"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.",
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"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.",
"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 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 connec- tions via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995. [4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra- tion for gauge theories. J. Math. Phys., 36:2170–2191, 1995. [5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thie- mann. 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 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 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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[{"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\u2019s 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\u20131468, 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\u2013230, 1995. \n[4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integration for gauge theories. J. Math. Phys., 36:2170\u20132191, 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\u20135482, 1997. \n[6] John C. Baez and Stephen Sawin. Functional integration on spaces of connections. J. Funct. Anal., 150:1\u201326, 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). Izdatel\u044cstvo \u00abMir\u00bb, 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\u2013102, 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\u2013434, 1986. \n[14] Jerzy Lewandowski. Group of loops, holonomy maps, path bundle and path connection. Class. Quant. Grav., 10:879\u2013904, 1993. \n[15] Donald Marolf and Jose M. Mourao. On the support of the Ashtekar-Lewandowski measure. Commun. Math. Phys., 170:583\u2013606, 1995. \n[16] P. K. Mitter. Geometry of the space of gauge orbits and the Yang-Mills dynamical system. Lectures given at Carg\u00b4ese 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\u2013608, 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\u201312, 1978. \n[20] T. Thiemann. Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories. Class. Quant. Grav., 15:1487\u20131512, 1998. ", "page_idx": 19}]
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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, "page_num": "page_19", "page_size": [612.0, 792.0], "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, "page_num": "page_19", "page_size": [612.0, 792.0]}, {"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": ". 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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?
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.
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 . 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 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 connec- tions via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995. [4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra- tion for gauge theories. J. Math. Phys., 36:2170–2191, 1995. [5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thie- mann. 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 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 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.
7First abstract attempts can be found, e.g., in [4, 3].
20
|
<div class="pdf-page">
<p>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?</p>
<p>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.</p>
<p>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>Acknowledgements</h1>
<p>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.</p>
<h1>References</h1>
<ul>
<li>[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 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 connec- tions via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995. [4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra- tion for gauge theories. J. Math. Phys., 36:2170–2191, 1995. [5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thie- mann. 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 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 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.</li>
</ul>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="112" data-y="19" data-width="784" data-height="54">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?</p>
<p class="pdf-text" data-x="123" data-y="91" data-width="775" data-height="131">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.</p>
<p class="pdf-text" data-x="103" data-y="224" data-width="797" data-height="74">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 class="pdf-title" data-x="105" data-y="325" data-width="273" data-height="26">Acknowledgements</h1>
<p class="pdf-text" data-x="105" data-y="364" data-width="795" data-height="94">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.</p>
<h1 class="pdf-title" data-x="105" data-y="486" data-width="154" data-height="24">References</h1>
<ul class="pdf-list" data-x="110" data-y="523" data-width="790" data-height="341">
<li>[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 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 connec- tions via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995. [4] Abhay Ashtekar and Jerzy Lewandowski. Projective techniques and functional integra- tion for gauge theories. J. Math. Phys., 36:2170–2191, 1995. [5] Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, and Thomas Thie- mann. 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 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 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.</li>
</ul>
<div class="pdf-discarded" data-x="125" data-y="861" data-width="387" data-height="15" style="opacity: 0.5;">7First abstract attempts can be found, e.g., in [4, 3].</div>
<div class="pdf-discarded" data-x="490" data-y="910" data-width="22" data-height="14" style="opacity: 0.5;">20</div>
</div>
|
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
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"content": [
"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 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 \\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",
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"[3] Abhay Ashtekar and Jerzy Lewandowski. Differential geometry on the space of connec-",
"tions via graphs and projective limits. J. Geom. Phys., 17:191–230, 1995.",
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"3/2000, math-ph/0001007.",
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"connections. J. Geom. Phys., 3:421–434, 1986.",
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"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",
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"10:605–608, 1993.",
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"1978.",
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],
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"Genus one 1-bridge knots and Dunwoody manifolds∗",
"Luigi Grasselli Michele Mulazzani",
"November 1, 2018",
"Abstract",
"In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.",
"2000 Mathematics Subject Classification: Primary 57M12, 57M25; Secondary 20F05, 57M05.",
"Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli- cally presented groups, geometric presentations of groups, Heegaard diagrams.",
"1 Introduction and preliminaries",
"The problem of determining if a balanced presentation of a group is geomet- ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec- tions between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con- ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented man- ifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide sub- class are proved to be cyclic coverings of , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the funda- mental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.",
"∗Work performed under the auspices of G.N.S.A.G.A. of C.N.R. of Italy and supported by the University of Bologna, funds for selected research topics.",
"arXiv:math/0003042v1 [math.GT] 7 Mar 2000"
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[{"type": "text", "text": "Genus one 1-bridge knots and Dunwoody manifolds\u2217 ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "Luigi Grasselli Michele Mulazzani ", "page_idx": 0}, {"type": "text", "text": "November 1, 2018 ", "page_idx": 0}, {"type": "text", "text": "Abstract ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\\mathbf{S^{3}}$ ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\\mathbf{S^{3}}$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. ", "page_idx": 0}, {"type": "text", "text": "2000 Mathematics Subject Classification: Primary 57M12, 57M25; \nSecondary 20F05, 57M05. ", "page_idx": 0}, {"type": "text", "text": "Keywords: Genus one 1-bridge knots, branched cyclic coverings, cyclically presented groups, geometric presentations of groups, Heegaard diagrams. ", "page_idx": 0}, {"type": "text", "text": "1 Introduction and preliminaries ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "The problem of determining if a balanced presentation of a group is geometric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connections between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable conditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented manifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide subclass are proved to be cyclic coverings of $\\mathrm{{S^{3}}}$ , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation. ", "page_idx": 0}]
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Dunwoody introduces in [6]", "type": "text", "cross_page": true}], "index": 4}, {"bbox": [109, 200, 500, 214], "spans": [{"bbox": [109, 200, 500, 214], "score": 1.0, "content": "a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [110, 215, 500, 228], "spans": [{"bbox": [110, 215, 500, 228], "score": 1.0, "content": "graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con-", "type": "text", "cross_page": true}], "index": 6}, {"bbox": [110, 230, 500, 243], "spans": [{"bbox": [110, 230, 500, 243], "score": 1.0, "content": "ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram", "type": "text", "cross_page": true}], "index": 7}, {"bbox": [110, 244, 500, 257], "spans": [{"bbox": [110, 244, 500, 257], "score": 1.0, "content": "such that the presentation of the fundamental group of the represented man-", "type": "text", "cross_page": true}], "index": 8}, {"bbox": [109, 258, 500, 271], "spans": [{"bbox": [109, 258, 500, 271], "score": 1.0, "content": "ifold is cyclic. This construction gives rise to a wide class of closed orientable", "type": "text", "cross_page": true}], "index": 9}, {"bbox": [109, 272, 501, 286], "spans": [{"bbox": [109, 272, 501, 286], "score": 1.0, "content": "3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and", "type": "text", "cross_page": true}], "index": 10}, {"bbox": [110, 288, 500, 301], "spans": [{"bbox": [110, 288, 500, 301], "score": 1.0, "content": "admitting geometric cyclic presentations for their fundamental groups. Our", "type": "text", "cross_page": true}], "index": 11}, {"bbox": [109, 301, 501, 316], "spans": [{"bbox": [109, 301, 501, 316], "score": 1.0, "content": "main result is that each Dunwoody manifold is a cyclic covering of a lens", "type": "text", "cross_page": true}], "index": 12}, {"bbox": [109, 315, 500, 330], "spans": [{"bbox": [109, 315, 500, 330], "score": 1.0, "content": "space (eventually the 3-sphere), branched over a genus one 1-bridge knot.", "type": "text", "cross_page": true}], "index": 13}, {"bbox": [109, 329, 500, 344], "spans": [{"bbox": [109, 329, 500, 344], "score": 1.0, "content": "As a direct consequence, the Dunwoody manifolds belonging to a wide sub-", "type": "text", "cross_page": true}], "index": 14}, {"bbox": [109, 344, 500, 359], "spans": [{"bbox": [109, 344, 326, 359], "score": 1.0, "content": "class are proved to be cyclic coverings of ", "type": "text", "cross_page": true}, {"bbox": [326, 345, 339, 355], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13, "cross_page": true}, {"bbox": [339, 344, 500, 359], "score": 1.0, "content": ", branched over suitable knots,", "type": "text", "cross_page": true}], "index": 15}, {"bbox": [110, 360, 500, 373], "spans": [{"bbox": [110, 360, 500, 373], "score": 1.0, "content": "thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover,", "type": "text", "cross_page": true}], "index": 16}, {"bbox": [110, 373, 500, 388], "spans": [{"bbox": [110, 373, 500, 388], "score": 1.0, "content": "we show that all branched cyclic coverings of knots with classical (i.e. genus", "type": "text", "cross_page": true}], "index": 17}, {"bbox": [110, 388, 499, 401], "spans": [{"bbox": [110, 388, 499, 401], "score": 1.0, "content": "zero) bridge number two belong to this subclass; as a corollary, the funda-", "type": "text", "cross_page": true}], "index": 18}, {"bbox": [110, 402, 500, 416], "spans": [{"bbox": [110, 402, 500, 416], "score": 1.0, "content": "mental group of each branched cyclic covering of a 2-bridge knot admits a", "type": "text", "cross_page": true}], "index": 19}, {"bbox": [110, 417, 262, 431], "spans": [{"bbox": [110, 417, 262, 431], "score": 1.0, "content": "geometric cyclic presentation.", "type": "text", "cross_page": true}], "index": 20}], "index": 21, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [110, 597, 500, 641]}]}
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# Genus one 1-bridge knots and Dunwoody manifolds∗
arXiv:math/0003042v1 [math.GT] 7 Mar 2000
Luigi Grasselli Michele Mulazzani
November 1, 2018
# Abstract
In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.
- 2000 Mathematics Subject Classification: Primary 57M12, 57M25; Secondary 20F05, 57M05.
Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli- cally presented groups, geometric presentations of groups, Heegaard diagrams.
# 1 Introduction and preliminaries
The problem of determining if a balanced presentation of a group is geomet- ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec- tions between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con- ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented man- ifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide sub- class are proved to be cyclic coverings of , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the funda- mental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.
∗Work performed under the auspices of G.N.S.A.G.A. of C.N.R. of Italy and supported by the University of Bologna, funds for selected research topics.
|
<div class="pdf-page">
<h1>Genus one 1-bridge knots and Dunwoody manifolds∗</h1>
<p>Luigi Grasselli Michele Mulazzani</p>
<p>November 1, 2018</p>
<h1>Abstract</h1>
<p>In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.</p>
<ul>
<li>2000 Mathematics Subject Classification: Primary 57M12, 57M25; Secondary 20F05, 57M05.</li>
</ul>
<p>Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli- cally presented groups, geometric presentations of groups, Heegaard diagrams.</p>
<h1>1 Introduction and preliminaries</h1>
<p>The problem of determining if a balanced presentation of a group is geomet- ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec- tions between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con- ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented man- ifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide sub- class are proved to be cyclic coverings of , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the funda- mental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="220" data-y="214" data-width="578" data-height="61">Genus one 1-bridge knots and Dunwoody manifolds∗</h1>
<div class="pdf-discarded" data-x="23" data-y="214" data-width="40" data-height="507" style="opacity: 0.5;">arXiv:math/0003042v1 [math.GT] 7 Mar 2000</div>
<p class="pdf-text" data-x="306" data-y="298" data-width="406" data-height="22">Luigi Grasselli Michele Mulazzani</p>
<p class="pdf-text" data-x="416" data-y="337" data-width="188" data-height="19">November 1, 2018</p>
<h1 class="pdf-title" data-x="468" data-y="413" data-width="82" data-height="17">Abstract</h1>
<p class="pdf-text" data-x="230" data-y="439" data-width="558" data-height="157">In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.</p>
<ul class="pdf-list" data-x="232" data-y="614" data-width="552" data-height="35">
<li>2000 Mathematics Subject Classification: Primary 57M12, 57M25; Secondary 20F05, 57M05.</li>
</ul>
<p class="pdf-text" data-x="230" data-y="651" data-width="556" data-height="51">Keywords: Genus one 1-bridge knots, branched cyclic coverings, cycli- cally presented groups, geometric presentations of groups, Heegaard diagrams.</p>
<h1 class="pdf-title" data-x="185" data-y="730" data-width="482" data-height="25">1 Introduction and preliminaries</h1>
<p class="pdf-text" data-x="184" data-y="770" data-width="652" data-height="56">The problem of determining if a balanced presentation of a group is geomet- ric (i.e. induced by a Heegaard diagram of a closed orientable 3-manifold) is quite important within geometric topology and has been deeply investigated by many authors (see [9], [22], [25], [26], [27], [28], [33]); further, the connec- tions between branched cyclic coverings of links and cyclic presentations of groups induced by suitable Heegaard diagrams have been recently pointed out in several papers (see [1], [3], [4], [6], [11], [12], [16], [18], [17], [20], [34]). In order to investigate these connections, M.J. Dunwoody introduces in [6] a class of planar, 3-regular graphs endowed with a cyclic symmetry. Each graph is defined by a 6-tuple of integers; if this 6-tuple satisfies suitable con- ditions (admissible 6-tuple), the graph uniquely defines a Heegaard diagram such that the presentation of the fundamental group of the represented man- ifold is cyclic. This construction gives rise to a wide class of closed orientable 3-manifolds (Dunwoody manifolds), depending on 6-tuples of integers and admitting geometric cyclic presentations for their fundamental groups. Our main result is that each Dunwoody manifold is a cyclic covering of a lens space (eventually the 3-sphere), branched over a genus one 1-bridge knot. As a direct consequence, the Dunwoody manifolds belonging to a wide sub- class are proved to be cyclic coverings of , branched over suitable knots, thus giving a positive answer to a conjecture of Dunwoody [6]. Moreover, we show that all branched cyclic coverings of knots with classical (i.e. genus zero) bridge number two belong to this subclass; as a corollary, the funda- mental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.</p>
<div class="pdf-discarded" data-x="184" data-y="833" data-width="654" data-height="31" style="opacity: 0.5;">∗Work performed under the auspices of G.N.S.A.G.A. of C.N.R. of Italy and supported by the University of Bologna, funds for selected research topics.</div>
</div>
|
# Genus one 1-bridge knots and Dunwoody manifolds∗
Luigi Grasselli Michele Mulazzani
November 1, 2018
# Abstract
In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\mathbf{S^{3}}$ ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\mathbf{S^{3}}$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.
2000 Mathematics Subject Classification: Primary 57M12, 57M25;
Secondary 20F05, 57M05.
Keywords: Genus one 1-bridge knots, branched cyclic coverings, cyclically presented groups, geometric presentations of groups, Heegaard diagrams.
# 1 Introduction and preliminaries
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For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15].
We recall that a finite balanced presentation of a group is said to be a cyclic presentation if there exists a word in the free group generated by such that the relators of the presentation are , , where denotes the automorphism defined by (mod ), . Let us denote this cyclic presentation (and the related group) by the symbol , so that:
$$
G_{n}(w)=<x_{1},x_{2},\ldots,x_{n}|w,\theta_{n}(w),\ldots,\theta_{n}^{n-1}(w)>.
$$
A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word is the integer given by the sum of the exponents of its letters; in other terms, where is the homomorphism defined by for each .
2
|
<div class="pdf-page">
<p>For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15].</p>
<p>We recall that a finite balanced presentation of a group is said to be a cyclic presentation if there exists a word in the free group generated by such that the relators of the presentation are , , where denotes the automorphism defined by (mod ), . Let us denote this cyclic presentation (and the related group) by the symbol , so that:</p>
<p>A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word is the integer given by the sum of the exponents of its letters; in other terms, where is the homomorphism defined by for each .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="182" data-y="554" data-width="656" data-height="75">For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15].</p>
<p class="pdf-text" data-x="182" data-y="629" data-width="654" data-height="129">We recall that a finite balanced presentation of a group is said to be a cyclic presentation if there exists a word in the free group generated by such that the relators of the presentation are , , where denotes the automorphism defined by (mod ), . Let us denote this cyclic presentation (and the related group) by the symbol , so that:</p>
<p class="pdf-text" data-x="184" data-y="787" data-width="652" data-height="75">A group is said to be cyclically presented if it admits a cyclic presentation. We recall that the exponent-sum of a word is the integer given by the sum of the exponents of its letters; in other terms, where is the homomorphism defined by for each .</p>
<div class="pdf-discarded" data-x="501" data-y="893" data-width="14" data-height="14" style="opacity: 0.5;">2</div>
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|
For the theory of Heegaard splittings of 3-manifolds, and in particular for Singer moves on Heegaard diagrams realizing the homeomorphism of the represented manifolds, we refer to [13] and [31]. For the theory of cyclically presented groups, we refer to [15].
We recall that a finite balanced presentation of a group $<$ $x_{1},\dots\,,x_{n}|r_{1},...\ ,r_{n}\,>$ is said to be a cyclic presentation if there exists a word $w$ in the free group $F_{n}$ generated by $x_{1},\ldots,x_{n}$ such that the relators of the presentation are $r_{k}=\theta_{n}^{k-1}(w)$ , $k=1,\dotsc,n$ , where $\theta_{n}:F_{n}\to F_{n}$ denotes the automorphism defined by $\theta_{n}(x_{i})\,=\,x_{i+1}$ (mod $n$ ), $i=1,\dots,n$ . Let us denote this cyclic presentation (and the related group) by the symbol $G_{n}(w)$ , so that:
$$
G_{n}(w)=<x_{1},x_{2},\ldots,x_{n}|w,\theta_{n}(w),\ldots,\theta_{n}^{n-1}(w)>.
$$
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"Following [10], we recall the definition of genus bridge number of a link, which is a generalization of the classical concept of bridge number for links in (see [5]).",
"A set of mutually disjoint arcs properly embedded in a handlebody is trivial if there is a set of mutually disjoint discs such that , and for and . Let and be the two handlebodies of a Hee- gaard splitting of the closed orientable 3-manifold and let be their common surface: a link in is in -bridge position with respect to if intersects transversally and if the set of arcs has components and is trivial both in and in . A link in 1-bridge position is obviously a knot.",
"The genus bridge number of a link in , , is the smallest integer for which is in -bridge position with respect to some genus Heegaard surface in . If the genus bridge number of a link is , we say that is a genus -bridge link or simply a -link. Of course, the genus bridge number of a link in a manifold of Heegaard genus is defined only for and the genus 0 bridge number of a link in is the classical bridge number. Moreover, a -link is a knot, for each .",
"In what follows, we shall deal with -knots, i.e. knots in or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of -knots in contains all torus knots (trivially) and all 2-bridge knots (i.e. -knots) [23].",
"2 Dunwoody manifolds",
"Let us sketch now the construction of Dunwoody manifolds given in [6]. Let be integers such that , and . Let be the planar regular trivalent graph drawn in Figure 1.",
"It contains upper cycles and lower cycles , each having vertices. For each , the cycle (resp. ) is connected to the cycle (resp. ) by parallel arcs, to the cycle by parallel arcs and to the cycle by parallel arcs (assume ). We set and . Moreover, denote by (resp. ) the set of the arcs of belonging to a cycle of (resp. ) and by",
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[{"type": "text", "text": "Following [10], we recall the definition of genus $g$ bridge number of a link, which is a generalization of the classical concept of bridge number for links in $\\mathrm{{S^{3}}}$ (see [5]). ", "page_idx": 2}, {"type": "text", "text": "A set of mutually disjoint arcs $\\{t_{1},\\ldots,t_{n}\\}$ properly embedded in a handlebody $U$ is trivial if there is a set of mutually disjoint discs $D\\,=$ $\\{D_{1},...\\,,D_{n}\\}$ such that $t_{i}\\cap D_{i}=t_{i}\\cap\\partial D_{i}=t_{i}$ , $t_{i}\\cap D_{j}=\\emptyset$ and $\\partial D_{i}-t_{i}\\subset\\partial U$ for $1\\leq i,j\\leq n$ and $i\\neq j$ . Let $U_{1}$ and $U_{2}$ be the two handlebodies of a Heegaard splitting of the closed orientable 3-manifold $M$ and let $T$ be their common surface: a link $L$ in $M$ is in $n$ -bridge position with respect to $T$ if $L$ intersects $T$ transversally and if the set of arcs $L\\cap U_{i}$ has $n$ components and is trivial both in $U_{1}$ and in $U_{2}$ . A link in 1-bridge position is obviously a knot. ", "page_idx": 2}, {"type": "text", "text": "The genus $g$ bridge number of a link $L$ in $M$ , $b_{g}(L)$ , is the smallest integer $n$ for which $L$ is in $n$ -bridge position with respect to some genus $g$ Heegaard surface in $M$ . If the genus $g$ bridge number of a link $L$ is $b$ , we say that $L$ is a genus $g$ $b$ -bridge link or simply a $(g,b)$ -link. Of course, the genus $g$ bridge number of a link in a manifold of Heegaard genus $g^{\\prime}$ is defined only for $g\\geq g^{\\prime}$ and the genus 0 bridge number of a link in $\\mathbf{S^{3}}$ is the classical bridge number. Moreover, a $(g,1)$ -link is a knot, for each $g\\geq0$ . ", "page_idx": 2}, {"type": "text", "text": "In what follows, we shall deal with $(1,1)$ -knots, i.e. knots in $\\mathrm{{S^{3}}}$ or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of $(1,1)$ -knots in $\\mathbf{S^{3}}$ contains all torus knots (trivially) and all 2-bridge knots (i.e. $(0,2)$ -knots) [23]. ", "page_idx": 2}, {"type": "text", "text": "2 Dunwoody manifolds ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $a,b,c,n$ be integers such that $n\\ >\\ 0$ , $a,b,c\\,\\geq\\,0$ and $a+b+c>0$ . Let $\\Gamma=\\Gamma(a,b,c,n)$ be the planar regular trivalent graph drawn in Figure 1. ", "page_idx": 2}, {"type": "text", "text": "It contains $n$ upper cycles $C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}$ and $n$ lower cycles $C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}$ , each having $d=2a+b+c$ vertices. For each $i=1,\\dots,n$ , the cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) is connected to the cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ) by $a$ parallel arcs, to the cycle $C_{i}^{\\prime\\prime}$ by $c$ parallel arcs and to the cycle $C_{i+1}^{\\prime\\prime}$ by $b$ parallel arcs (assume $n+1=1$ ). We set $\\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\}$ and $\\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\}$ . Moreover, denote by $A^{\\prime}$ (resp. $A^{\\prime\\prime}$ ) the set of the arcs of $\\Gamma$ belonging to a cycle of $\\mathcal{C}^{\\prime}$ (resp. $\\mathcal{C^{\\prime\\prime}}$ ) and by ", "page_idx": 2}]
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{"preproc_blocks": [{"type": "text", "bbox": [109, 125, 500, 168], "lines": [{"bbox": [127, 127, 498, 142], "spans": [{"bbox": [127, 127, 367, 142], "score": 1.0, "content": "Following [10], we recall the definition of genus ", "type": "text"}, {"bbox": [368, 133, 374, 141], "score": 0.9, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [374, 127, 498, 142], "score": 1.0, "content": " bridge number of a link,", "type": "text"}], "index": 0}, {"bbox": [109, 142, 500, 156], "spans": [{"bbox": [109, 142, 500, 156], "score": 1.0, "content": "which is a generalization of the classical concept of bridge number for links", "type": "text"}], "index": 1}, {"bbox": [109, 154, 186, 171], "spans": [{"bbox": [109, 154, 123, 171], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [124, 157, 137, 167], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [137, 154, 186, 171], "score": 1.0, "content": " (see [5]).", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [109, 169, 500, 298], "lines": [{"bbox": [126, 170, 501, 187], "spans": [{"bbox": [126, 170, 303, 187], "score": 1.0, "content": "A set of mutually disjoint arcs ", "type": "text"}, {"bbox": [303, 172, 363, 185], "score": 0.94, "content": "\\{t_{1},\\ldots,t_{n}\\}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [363, 170, 501, 187], "score": 1.0, "content": " properly embedded in a", "type": "text"}], "index": 3}, {"bbox": [109, 185, 500, 199], "spans": [{"bbox": [109, 185, 175, 199], "score": 1.0, "content": "handlebody ", "type": "text"}, {"bbox": [176, 187, 185, 196], "score": 0.9, "content": "U", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [185, 185, 472, 199], "score": 1.0, "content": " is trivial if there is a set of mutually disjoint discs ", "type": "text"}, {"bbox": [473, 186, 500, 198], "score": 0.84, "content": "D\\,=", "type": "inline_equation", "height": 12, "width": 27}], "index": 4}, {"bbox": [110, 200, 499, 214], "spans": [{"bbox": [110, 201, 180, 213], "score": 0.94, "content": "\\{D_{1},...\\,,D_{n}\\}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [181, 200, 233, 214], "score": 1.0, "content": " such that ", "type": "text"}, {"bbox": [234, 201, 342, 212], "score": 0.9, "content": "t_{i}\\cap D_{i}=t_{i}\\cap\\partial D_{i}=t_{i}", "type": "inline_equation", "height": 11, "width": 108}, {"bbox": [343, 200, 348, 214], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [348, 201, 402, 214], "score": 0.94, "content": "t_{i}\\cap D_{j}=\\emptyset", "type": "inline_equation", "height": 13, "width": 54}, {"bbox": [403, 200, 427, 214], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [428, 201, 499, 212], "score": 0.92, "content": "\\partial D_{i}-t_{i}\\subset\\partial U", "type": "inline_equation", "height": 11, "width": 71}], "index": 5}, {"bbox": [109, 214, 500, 228], "spans": [{"bbox": [109, 214, 128, 228], "score": 1.0, "content": "for ", "type": "text"}, {"bbox": [128, 216, 188, 227], "score": 0.94, "content": "1\\leq i,j\\leq n", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [188, 214, 213, 228], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [214, 216, 239, 227], "score": 0.93, "content": "i\\neq j", "type": "inline_equation", "height": 11, "width": 25}, {"bbox": [240, 214, 268, 228], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [268, 216, 281, 226], "score": 0.92, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [282, 214, 307, 228], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [308, 216, 321, 226], "score": 0.92, "content": "U_{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [321, 214, 500, 228], "score": 1.0, "content": " be the two handlebodies of a Hee-", "type": "text"}], "index": 6}, {"bbox": [109, 230, 500, 242], "spans": [{"bbox": [109, 230, 381, 242], "score": 1.0, "content": "gaard splitting of the closed orientable 3-manifold ", "type": "text"}, {"bbox": [381, 231, 394, 240], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [395, 230, 442, 242], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [443, 231, 452, 239], "score": 0.91, "content": "T", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [452, 230, 500, 242], "score": 1.0, "content": " be their", "type": "text"}], "index": 7}, {"bbox": [109, 244, 501, 257], "spans": [{"bbox": [109, 244, 235, 257], "score": 1.0, "content": "common surface: a link ", "type": "text"}, {"bbox": [236, 245, 244, 254], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [244, 244, 262, 257], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [262, 245, 275, 254], "score": 0.91, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [275, 244, 304, 257], "score": 1.0, "content": " is in ", "type": "text"}, {"bbox": [305, 248, 312, 254], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [313, 244, 478, 257], "score": 1.0, "content": "-bridge position with respect to ", "type": "text"}, {"bbox": [479, 245, 488, 254], "score": 0.91, "content": "T", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [488, 244, 501, 257], "score": 1.0, "content": " if", "type": "text"}], "index": 8}, {"bbox": [110, 257, 500, 271], "spans": [{"bbox": [110, 259, 118, 268], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [119, 257, 174, 271], "score": 1.0, "content": " intersects ", "type": "text"}, {"bbox": [174, 259, 183, 268], "score": 0.92, "content": "T", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [184, 257, 368, 271], "score": 1.0, "content": " transversally and if the set of arcs ", "type": "text"}, {"bbox": [368, 259, 402, 270], "score": 0.94, "content": "L\\cap U_{i}", "type": "inline_equation", "height": 11, "width": 34}, {"bbox": [402, 257, 426, 271], "score": 1.0, "content": " has ", "type": "text"}, {"bbox": [426, 263, 434, 268], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [434, 257, 500, 271], "score": 1.0, "content": " components", "type": "text"}], "index": 9}, {"bbox": [109, 271, 499, 286], "spans": [{"bbox": [109, 271, 222, 286], "score": 1.0, "content": "and is trivial both in ", "type": "text"}, {"bbox": [222, 274, 235, 284], "score": 0.92, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [235, 271, 276, 286], "score": 1.0, "content": " and in ", "type": "text"}, {"bbox": [276, 274, 289, 284], "score": 0.92, "content": "U_{2}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [289, 271, 499, 286], "score": 1.0, "content": ". A link in 1-bridge position is obviously", "type": "text"}], "index": 10}, {"bbox": [109, 287, 147, 299], "spans": [{"bbox": [109, 287, 147, 299], "score": 1.0, "content": "a knot.", "type": "text"}], "index": 11}], "index": 7}, {"type": "text", "bbox": [109, 299, 500, 399], "lines": [{"bbox": [127, 300, 500, 315], "spans": [{"bbox": [127, 300, 182, 315], "score": 1.0, "content": "The genus ", "type": "text"}, {"bbox": [182, 306, 189, 314], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [189, 300, 312, 315], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [313, 303, 321, 312], "score": 0.9, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [321, 300, 336, 315], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [337, 303, 349, 312], "score": 0.89, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [350, 300, 356, 315], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [356, 302, 383, 315], "score": 0.95, "content": "b_{g}(L)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [383, 300, 500, 315], "score": 1.0, "content": ", is the smallest integer", "type": "text"}], "index": 12}, {"bbox": [110, 315, 501, 330], "spans": [{"bbox": [110, 320, 117, 326], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [118, 315, 172, 330], "score": 1.0, "content": " for which ", "type": "text"}, {"bbox": [172, 317, 180, 326], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [180, 315, 208, 330], "score": 1.0, "content": " is in ", "type": "text"}, {"bbox": [208, 320, 216, 326], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [216, 315, 441, 330], "score": 1.0, "content": "-bridge position with respect to some genus ", "type": "text"}, {"bbox": [441, 320, 447, 328], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [448, 315, 501, 330], "score": 1.0, "content": " Heegaard", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 163, 344], "score": 1.0, "content": "surface in ", "type": "text"}, {"bbox": [163, 331, 176, 340], "score": 0.92, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [176, 330, 247, 344], "score": 1.0, "content": ". If the genus ", "type": "text"}, {"bbox": [248, 335, 254, 343], "score": 0.9, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [254, 330, 381, 344], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [390, 330, 404, 344], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [405, 331, 410, 340], "score": 0.88, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [410, 330, 479, 344], "score": 1.0, "content": ", we say that ", "type": "text"}, {"bbox": [479, 331, 487, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [487, 330, 500, 344], "score": 1.0, "content": " is", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 345, 152, 358], "score": 1.0, "content": "a genus ", "type": "text"}, {"bbox": [152, 349, 158, 357], "score": 0.85, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [159, 345, 162, 358], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [162, 346, 167, 355], "score": 0.84, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [168, 345, 291, 358], "score": 1.0, "content": "-bridge link or simply a ", "type": "text"}, {"bbox": [292, 345, 317, 358], "score": 0.94, "content": "(g,b)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [317, 345, 457, 358], "score": 1.0, "content": "-link. Of course, the genus ", "type": "text"}, {"bbox": [457, 349, 463, 357], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [463, 345, 500, 358], "score": 1.0, "content": " bridge", "type": "text"}], "index": 15}, {"bbox": [109, 358, 499, 374], "spans": [{"bbox": [109, 358, 362, 374], "score": 1.0, "content": "number of a link in a manifold of Heegaard genus ", "type": "text"}, {"bbox": [362, 360, 371, 372], "score": 0.92, "content": "g^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [371, 358, 468, 374], "score": 1.0, "content": " is defined only for ", "type": "text"}, {"bbox": [468, 360, 499, 372], "score": 0.94, "content": "g\\geq g^{\\prime}", "type": "inline_equation", "height": 12, "width": 31}], "index": 16}, {"bbox": [110, 373, 499, 387], "spans": [{"bbox": [110, 373, 329, 387], "score": 1.0, "content": "and the genus 0 bridge number of a link in ", "type": "text"}, {"bbox": [329, 374, 342, 384], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [342, 373, 499, 387], "score": 1.0, "content": " is the classical bridge number.", "type": "text"}], "index": 17}, {"bbox": [110, 387, 356, 401], "spans": [{"bbox": [110, 387, 174, 401], "score": 1.0, "content": "Moreover, a ", "type": "text"}, {"bbox": [174, 389, 201, 401], "score": 0.94, "content": "(g,1)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [201, 387, 324, 401], "score": 1.0, "content": "-link is a knot, for each ", "type": "text"}, {"bbox": [324, 390, 352, 401], "score": 0.93, "content": "g\\geq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [352, 387, 356, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 15}, {"type": "text", "bbox": [109, 400, 500, 473], "lines": [{"bbox": [126, 402, 501, 416], "spans": [{"bbox": [126, 402, 306, 416], "score": 1.0, "content": "In what follows, we shall deal with ", "type": "text"}, {"bbox": [307, 403, 333, 416], "score": 0.93, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [333, 402, 435, 416], "score": 1.0, "content": "-knots, i.e. knots in ", "type": "text"}, {"bbox": [435, 403, 448, 413], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 402, 501, 416], "score": 1.0, "content": " or in lens", "type": "text"}], "index": 19}, {"bbox": [109, 416, 501, 430], "spans": [{"bbox": [109, 416, 501, 430], "score": 1.0, "content": "spaces. This class of knots is very important in the light of some results and", "type": "text"}], "index": 20}, {"bbox": [110, 431, 499, 446], "spans": [{"bbox": [110, 431, 499, 446], "score": 1.0, "content": "conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).", "type": "text"}], "index": 21}, {"bbox": [109, 444, 499, 460], "spans": [{"bbox": [109, 444, 233, 460], "score": 1.0, "content": "Notice that the class of ", "type": "text"}, {"bbox": [234, 446, 259, 459], "score": 0.94, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [260, 444, 308, 460], "score": 1.0, "content": "-knots in ", "type": "text"}, {"bbox": [309, 446, 321, 456], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [322, 444, 499, 460], "score": 1.0, "content": " contains all torus knots (trivially)", "type": "text"}], "index": 22}, {"bbox": [110, 459, 340, 475], "spans": [{"bbox": [110, 459, 251, 475], "score": 1.0, "content": "and all 2-bridge knots (i.e. ", "type": "text"}, {"bbox": [251, 461, 277, 474], "score": 0.93, "content": "(0,2)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [278, 459, 340, 475], "score": 1.0, "content": "-knots) [23].", "type": "text"}], "index": 23}], "index": 21}, {"type": "title", "bbox": [110, 493, 318, 513], "lines": [{"bbox": [110, 496, 318, 514], "spans": [{"bbox": [110, 498, 122, 511], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [137, 496, 318, 514], "score": 1.0, "content": "Dunwoody manifolds", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [109, 524, 500, 567], "lines": [{"bbox": [109, 525, 499, 541], "spans": [{"bbox": [109, 525, 499, 541], "score": 1.0, "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let", "type": "text"}], "index": 25}, {"bbox": [110, 541, 500, 554], "spans": [{"bbox": [110, 542, 149, 554], "score": 0.93, "content": "a,b,c,n", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [150, 541, 270, 554], "score": 1.0, "content": " be integers such that ", "type": "text"}, {"bbox": [270, 543, 304, 551], "score": 0.9, "content": "n\\ >\\ 0", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 541, 312, 554], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [312, 542, 366, 554], "score": 0.92, "content": "a,b,c\\,\\geq\\,0", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [366, 541, 394, 554], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 542, 469, 552], "score": 0.93, "content": "a+b+c>0", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [470, 541, 500, 554], "score": 1.0, "content": ". Let", "type": "text"}], "index": 26}, {"bbox": [110, 555, 481, 569], "spans": [{"bbox": [110, 556, 189, 568], "score": 0.93, "content": "\\Gamma=\\Gamma(a,b,c,n)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [189, 555, 481, 569], "score": 1.0, "content": " be the planar regular trivalent graph drawn in Figure 1.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [110, 579, 500, 667], "lines": [{"bbox": [123, 578, 503, 599], "spans": [{"bbox": [123, 578, 184, 599], "score": 1.0, "content": "It contains ", "type": "text"}, {"bbox": [185, 586, 192, 592], "score": 0.86, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [192, 578, 260, 599], "score": 1.0, "content": " upper cycles ", "type": "text"}, {"bbox": [260, 583, 316, 595], "score": 0.93, "content": "C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [316, 578, 340, 599], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 586, 347, 592], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 578, 412, 599], "score": 1.0, "content": " lower cycles ", "type": "text"}, {"bbox": [413, 583, 470, 595], "score": 0.93, "content": "C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [470, 578, 503, 599], "score": 1.0, "content": ", each", "type": "text"}], "index": 28}, {"bbox": [109, 595, 500, 611], "spans": [{"bbox": [109, 595, 147, 611], "score": 1.0, "content": "having ", "type": "text"}, {"bbox": [147, 597, 218, 607], "score": 0.94, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [218, 595, 314, 611], "score": 1.0, "content": " vertices. For each ", "type": "text"}, {"bbox": [315, 598, 375, 609], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [376, 595, 430, 611], "score": 1.0, "content": ", the cycle ", "type": "text"}, {"bbox": [430, 597, 443, 609], "score": 0.91, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [443, 595, 479, 611], "score": 1.0, "content": " (resp. 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If the genus ", "type": "text"}, {"bbox": [248, 335, 254, 343], "score": 0.9, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [254, 330, 381, 344], "score": 1.0, "content": " bridge number of a link ", "type": "text"}, {"bbox": [381, 331, 389, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [390, 330, 404, 344], "score": 1.0, "content": " is ", "type": "text"}, {"bbox": [405, 331, 410, 340], "score": 0.88, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [410, 330, 479, 344], "score": 1.0, "content": ", we say that ", "type": "text"}, {"bbox": [479, 331, 487, 340], "score": 0.91, "content": "L", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [487, 330, 500, 344], "score": 1.0, "content": " is", "type": "text"}], "index": 14}, {"bbox": [110, 345, 500, 358], "spans": [{"bbox": [110, 345, 152, 358], "score": 1.0, "content": "a genus ", "type": "text"}, {"bbox": [152, 349, 158, 357], "score": 0.85, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [159, 345, 162, 358], "score": 1.0, "content": " ", "type": "text"}, {"bbox": [162, 346, 167, 355], "score": 0.84, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [168, 345, 291, 358], "score": 1.0, "content": "-bridge link or simply a ", "type": "text"}, {"bbox": [292, 345, 317, 358], "score": 0.94, "content": "(g,b)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [317, 345, 457, 358], "score": 1.0, "content": "-link. Of course, the genus ", "type": "text"}, {"bbox": [457, 349, 463, 357], "score": 0.89, "content": "g", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [463, 345, 500, 358], "score": 1.0, "content": " bridge", "type": "text"}], "index": 15}, {"bbox": [109, 358, 499, 374], "spans": [{"bbox": [109, 358, 362, 374], "score": 1.0, "content": "number of a link in a manifold of Heegaard genus ", "type": "text"}, {"bbox": [362, 360, 371, 372], "score": 0.92, "content": "g^{\\prime}", "type": "inline_equation", "height": 12, "width": 9}, {"bbox": [371, 358, 468, 374], "score": 1.0, "content": " is defined only for ", "type": "text"}, {"bbox": [468, 360, 499, 372], "score": 0.94, "content": "g\\geq g^{\\prime}", "type": "inline_equation", "height": 12, "width": 31}], "index": 16}, {"bbox": [110, 373, 499, 387], "spans": [{"bbox": [110, 373, 329, 387], "score": 1.0, "content": "and the genus 0 bridge number of a link in ", "type": "text"}, {"bbox": [329, 374, 342, 384], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [342, 373, 499, 387], "score": 1.0, "content": " is the classical bridge number.", "type": "text"}], "index": 17}, {"bbox": [110, 387, 356, 401], "spans": [{"bbox": [110, 387, 174, 401], "score": 1.0, "content": "Moreover, a ", "type": "text"}, {"bbox": [174, 389, 201, 401], "score": 0.94, "content": "(g,1)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [201, 387, 324, 401], "score": 1.0, "content": "-link is a knot, for each ", "type": "text"}, {"bbox": [324, 390, 352, 401], "score": 0.93, "content": "g\\geq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [352, 387, 356, 401], "score": 1.0, "content": ".", "type": "text"}], "index": 18}], "index": 15, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 300, 501, 401]}, {"type": "text", "bbox": [109, 400, 500, 473], "lines": [{"bbox": [126, 402, 501, 416], "spans": [{"bbox": [126, 402, 306, 416], "score": 1.0, "content": "In what follows, we shall deal with ", "type": "text"}, {"bbox": [307, 403, 333, 416], "score": 0.93, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [333, 402, 435, 416], "score": 1.0, "content": "-knots, i.e. knots in ", "type": "text"}, {"bbox": [435, 403, 448, 413], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 402, 501, 416], "score": 1.0, "content": " or in lens", "type": "text"}], "index": 19}, {"bbox": [109, 416, 501, 430], "spans": [{"bbox": [109, 416, 501, 430], "score": 1.0, "content": "spaces. This class of knots is very important in the light of some results and", "type": "text"}], "index": 20}, {"bbox": [110, 431, 499, 446], "spans": [{"bbox": [110, 431, 499, 446], "score": 1.0, "content": "conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).", "type": "text"}], "index": 21}, {"bbox": [109, 444, 499, 460], "spans": [{"bbox": [109, 444, 233, 460], "score": 1.0, "content": "Notice that the class of ", "type": "text"}, {"bbox": [234, 446, 259, 459], "score": 0.94, "content": "(1,1)", "type": "inline_equation", "height": 13, "width": 25}, {"bbox": [260, 444, 308, 460], "score": 1.0, "content": "-knots in ", "type": "text"}, {"bbox": [309, 446, 321, 456], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [322, 444, 499, 460], "score": 1.0, "content": " contains all torus knots (trivially)", "type": "text"}], "index": 22}, {"bbox": [110, 459, 340, 475], "spans": [{"bbox": [110, 459, 251, 475], "score": 1.0, "content": "and all 2-bridge knots (i.e. ", "type": "text"}, {"bbox": [251, 461, 277, 474], "score": 0.93, "content": "(0,2)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [278, 459, 340, 475], "score": 1.0, "content": "-knots) [23].", "type": "text"}], "index": 23}], "index": 21, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 402, 501, 475]}, {"type": "title", "bbox": [110, 493, 318, 513], "lines": [{"bbox": [110, 496, 318, 514], "spans": [{"bbox": [110, 498, 122, 511], "score": 1.0, "content": "2", "type": "text"}, {"bbox": [137, 496, 318, 514], "score": 1.0, "content": "Dunwoody manifolds", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 524, 500, 567], "lines": [{"bbox": [109, 525, 499, 541], "spans": [{"bbox": [109, 525, 499, 541], "score": 1.0, "content": "Let us sketch now the construction of Dunwoody manifolds given in [6]. Let", "type": "text"}], "index": 25}, {"bbox": [110, 541, 500, 554], "spans": [{"bbox": [110, 542, 149, 554], "score": 0.93, "content": "a,b,c,n", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [150, 541, 270, 554], "score": 1.0, "content": " be integers such that ", "type": "text"}, {"bbox": [270, 543, 304, 551], "score": 0.9, "content": "n\\ >\\ 0", "type": "inline_equation", "height": 8, "width": 34}, {"bbox": [304, 541, 312, 554], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [312, 542, 366, 554], "score": 0.92, "content": "a,b,c\\,\\geq\\,0", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [366, 541, 394, 554], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [394, 542, 469, 552], "score": 0.93, "content": "a+b+c>0", "type": "inline_equation", "height": 10, "width": 75}, {"bbox": [470, 541, 500, 554], "score": 1.0, "content": ". Let", "type": "text"}], "index": 26}, {"bbox": [110, 555, 481, 569], "spans": [{"bbox": [110, 556, 189, 568], "score": 0.93, "content": "\\Gamma=\\Gamma(a,b,c,n)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [189, 555, 481, 569], "score": 1.0, "content": " be the planar regular trivalent graph drawn in Figure 1.", "type": "text"}], "index": 27}], "index": 26, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [109, 525, 500, 569]}, {"type": "text", "bbox": [110, 579, 500, 667], "lines": [{"bbox": [123, 578, 503, 599], "spans": [{"bbox": [123, 578, 184, 599], "score": 1.0, "content": "It contains ", "type": "text"}, {"bbox": [185, 586, 192, 592], "score": 0.86, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [192, 578, 260, 599], "score": 1.0, "content": " upper cycles ", "type": "text"}, {"bbox": [260, 583, 316, 595], "score": 0.93, "content": "C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 56}, {"bbox": [316, 578, 340, 599], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 586, 347, 592], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [348, 578, 412, 599], "score": 1.0, "content": " lower cycles ", "type": "text"}, {"bbox": [413, 583, 470, 595], "score": 0.93, "content": "C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 57}, {"bbox": [470, 578, 503, 599], "score": 1.0, "content": ", each", "type": "text"}], "index": 28}, {"bbox": [109, 595, 500, 611], "spans": [{"bbox": [109, 595, 147, 611], "score": 1.0, "content": "having ", "type": "text"}, {"bbox": [147, 597, 218, 607], "score": 0.94, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 71}, {"bbox": [218, 595, 314, 611], "score": 1.0, "content": " vertices. For each ", "type": "text"}, {"bbox": [315, 598, 375, 609], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 11, "width": 60}, {"bbox": [376, 595, 430, 611], "score": 1.0, "content": ", the cycle ", "type": "text"}, {"bbox": [430, 597, 443, 609], "score": 0.91, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [443, 595, 479, 611], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [480, 597, 494, 609], "score": 0.87, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [494, 595, 500, 611], "score": 1.0, "content": ")", "type": "text"}], "index": 29}, {"bbox": [108, 609, 499, 625], "spans": [{"bbox": [108, 609, 237, 625], "score": 1.0, "content": "is connected to the cycle ", "type": "text"}, {"bbox": [238, 612, 261, 624], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [261, 609, 298, 625], "score": 1.0, "content": " (resp. 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Moreover, denote by ", "type": "text"}, {"bbox": [487, 641, 499, 650], "score": 0.89, "content": "A^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}], "index": 32}, {"bbox": [110, 653, 499, 668], "spans": [{"bbox": [110, 653, 144, 668], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [144, 655, 158, 664], "score": 0.84, "content": "A^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [158, 653, 271, 668], "score": 1.0, "content": ") the set of the arcs of ", "type": "text"}, {"bbox": [271, 656, 278, 664], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [279, 653, 397, 668], "score": 1.0, "content": " belonging to a cycle of ", "type": "text"}, {"bbox": [397, 655, 407, 664], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [408, 653, 444, 668], "score": 1.0, "content": " (resp. 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Following [10], we recall the definition of genus bridge number of a link, which is a generalization of the classical concept of bridge number for links in (see [5]).
A set of mutually disjoint arcs properly embedded in a handlebody is trivial if there is a set of mutually disjoint discs such that , and for and . Let and be the two handlebodies of a Hee- gaard splitting of the closed orientable 3-manifold and let be their common surface: a link in is in -bridge position with respect to if intersects transversally and if the set of arcs has components and is trivial both in and in . A link in 1-bridge position is obviously a knot.
The genus bridge number of a link in , , is the smallest integer for which is in -bridge position with respect to some genus Heegaard surface in . If the genus bridge number of a link is , we say that is a genus -bridge link or simply a -link. Of course, the genus bridge number of a link in a manifold of Heegaard genus is defined only for and the genus 0 bridge number of a link in is the classical bridge number. Moreover, a -link is a knot, for each .
In what follows, we shall deal with -knots, i.e. knots in or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of -knots in contains all torus knots (trivially) and all 2-bridge knots (i.e. -knots) [23].
# 2 Dunwoody manifolds
Let us sketch now the construction of Dunwoody manifolds given in [6]. Let be integers such that , and . Let be the planar regular trivalent graph drawn in Figure 1.
It contains upper cycles and lower cycles , each having vertices. For each , the cycle (resp. ) is connected to the cycle (resp. ) by parallel arcs, to the cycle by parallel arcs and to the cycle by parallel arcs (assume ). We set and . Moreover, denote by (resp. ) the set of the arcs of belonging to a cycle of (resp. ) and by
3
|
<div class="pdf-page">
<p>Following [10], we recall the definition of genus bridge number of a link, which is a generalization of the classical concept of bridge number for links in (see [5]).</p>
<p>A set of mutually disjoint arcs properly embedded in a handlebody is trivial if there is a set of mutually disjoint discs such that , and for and . Let and be the two handlebodies of a Hee- gaard splitting of the closed orientable 3-manifold and let be their common surface: a link in is in -bridge position with respect to if intersects transversally and if the set of arcs has components and is trivial both in and in . A link in 1-bridge position is obviously a knot.</p>
<p>The genus bridge number of a link in , , is the smallest integer for which is in -bridge position with respect to some genus Heegaard surface in . If the genus bridge number of a link is , we say that is a genus -bridge link or simply a -link. Of course, the genus bridge number of a link in a manifold of Heegaard genus is defined only for and the genus 0 bridge number of a link in is the classical bridge number. Moreover, a -link is a knot, for each .</p>
<p>In what follows, we shall deal with -knots, i.e. knots in or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of -knots in contains all torus knots (trivially) and all 2-bridge knots (i.e. -knots) [23].</p>
<h1>2 Dunwoody manifolds</h1>
<p>Let us sketch now the construction of Dunwoody manifolds given in [6]. Let be integers such that , and . Let be the planar regular trivalent graph drawn in Figure 1.</p>
<p>It contains upper cycles and lower cycles , each having vertices. For each , the cycle (resp. ) is connected to the cycle (resp. ) by parallel arcs, to the cycle by parallel arcs and to the cycle by parallel arcs (assume ). We set and . Moreover, denote by (resp. ) the set of the arcs of belonging to a cycle of (resp. ) and by</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="182" data-y="161" data-width="654" data-height="56">Following [10], we recall the definition of genus bridge number of a link, which is a generalization of the classical concept of bridge number for links in (see [5]).</p>
<p class="pdf-text" data-x="182" data-y="218" data-width="654" data-height="167">A set of mutually disjoint arcs properly embedded in a handlebody is trivial if there is a set of mutually disjoint discs such that , and for and . Let and be the two handlebodies of a Hee- gaard splitting of the closed orientable 3-manifold and let be their common surface: a link in is in -bridge position with respect to if intersects transversally and if the set of arcs has components and is trivial both in and in . A link in 1-bridge position is obviously a knot.</p>
<p class="pdf-text" data-x="182" data-y="386" data-width="654" data-height="129">The genus bridge number of a link in , , is the smallest integer for which is in -bridge position with respect to some genus Heegaard surface in . If the genus bridge number of a link is , we say that is a genus -bridge link or simply a -link. Of course, the genus bridge number of a link in a manifold of Heegaard genus is defined only for and the genus 0 bridge number of a link in is the classical bridge number. Moreover, a -link is a knot, for each .</p>
<p class="pdf-text" data-x="182" data-y="517" data-width="654" data-height="94">In what follows, we shall deal with -knots, i.e. knots in or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of -knots in contains all torus knots (trivially) and all 2-bridge knots (i.e. -knots) [23].</p>
<h1 class="pdf-title" data-x="184" data-y="637" data-width="348" data-height="26">2 Dunwoody manifolds</h1>
<p class="pdf-text" data-x="182" data-y="677" data-width="654" data-height="56">Let us sketch now the construction of Dunwoody manifolds given in [6]. Let be integers such that , and . Let be the planar regular trivalent graph drawn in Figure 1.</p>
<p class="pdf-text" data-x="184" data-y="748" data-width="652" data-height="114">It contains upper cycles and lower cycles , each having vertices. For each , the cycle (resp. ) is connected to the cycle (resp. ) by parallel arcs, to the cycle by parallel arcs and to the cycle by parallel arcs (assume ). We set and . Moreover, denote by (resp. ) the set of the arcs of belonging to a cycle of (resp. ) and by</p>
<div class="pdf-discarded" data-x="501" data-y="893" data-width="14" data-height="14" style="opacity: 0.5;">3</div>
</div>
|
Following [10], we recall the definition of genus $g$ bridge number of a link, which is a generalization of the classical concept of bridge number for links in $\mathrm{{S^{3}}}$ (see [5]).
A set of mutually disjoint arcs $\{t_{1},\ldots,t_{n}\}$ properly embedded in a handlebody $U$ is trivial if there is a set of mutually disjoint discs $D\,=$ $\{D_{1},...\,,D_{n}\}$ such that $t_{i}\cap D_{i}=t_{i}\cap\partial D_{i}=t_{i}$ , $t_{i}\cap D_{j}=\emptyset$ and $\partial D_{i}-t_{i}\subset\partial U$ for $1\leq i,j\leq n$ and $i\neq j$ . Let $U_{1}$ and $U_{2}$ be the two handlebodies of a Heegaard splitting of the closed orientable 3-manifold $M$ and let $T$ be their common surface: a link $L$ in $M$ is in $n$ -bridge position with respect to $T$ if $L$ intersects $T$ transversally and if the set of arcs $L\cap U_{i}$ has $n$ components and is trivial both in $U_{1}$ and in $U_{2}$ . A link in 1-bridge position is obviously a knot.
The genus $g$ bridge number of a link $L$ in $M$ , $b_{g}(L)$ , is the smallest integer $n$ for which $L$ is in $n$ -bridge position with respect to some genus $g$ Heegaard surface in $M$ . If the genus $g$ bridge number of a link $L$ is $b$ , we say that $L$ is a genus $g$ $b$ -bridge link or simply a $(g,b)$ -link. Of course, the genus $g$ bridge number of a link in a manifold of Heegaard genus $g^{\prime}$ is defined only for $g\geq g^{\prime}$ and the genus 0 bridge number of a link in $\mathbf{S^{3}}$ is the classical bridge number. Moreover, a $(g,1)$ -link is a knot, for each $g\geq0$ .
In what follows, we shall deal with $(1,1)$ -knots, i.e. knots in $\mathrm{{S^{3}}}$ or in lens spaces. This class of knots is very important in the light of some results and conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]). Notice that the class of $(1,1)$ -knots in $\mathbf{S^{3}}$ contains all torus knots (trivially) and all 2-bridge knots (i.e. $(0,2)$ -knots) [23].
# 2 Dunwoody manifolds
Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $a,b,c,n$ be integers such that $n\ >\ 0$ , $a,b,c\,\geq\,0$ and $a+b+c>0$ . Let $\Gamma=\Gamma(a,b,c,n)$ be the planar regular trivalent graph drawn in Figure 1.
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"Following [10], we recall the definition of genus g bridge number of a link,",
"which is a generalization of the classical concept of bridge number for links",
"in \\mathrm{{S^{3}}} (see [5]).",
"A set of mutually disjoint arcs \\{t_{1},\\ldots,t_{n}\\} properly embedded in a",
"handlebody U is trivial if there is a set of mutually disjoint discs D\\,=",
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"for 1\\leq i,j\\leq n and i\\neq j . Let U_{1} and U_{2} be the two handlebodies of a Hee-",
"gaard splitting of the closed orientable 3-manifold M and let T be their",
"common surface: a link L in M is in n -bridge position with respect to T if",
"L intersects T transversally and if the set of arcs L\\cap U_{i} has n components",
"and is trivial both in U_{1} and in U_{2} . A link in 1-bridge position is obviously",
"a knot.",
"The genus g bridge number of a link L in M , b_{g}(L) , is the smallest integer",
"n for which L is in n -bridge position with respect to some genus g Heegaard",
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"Moreover, a (g,1) -link is a knot, for each g\\geq0 .",
"In what follows, we shall deal with (1,1) -knots, i.e. knots in \\mathrm{{S^{3}}} or in lens",
"spaces. This class of knots is very important in the light of some results and",
"conjectures involving Dehn surgery on knots (see [2], [7], [8], [35], [36], [37]).",
"Notice that the class of (1,1) -knots in \\mathbf{S^{3}} contains all torus knots (trivially)",
"and all 2-bridge knots (i.e. (0,2) -knots) [23].",
"2 Dunwoody manifolds",
"Let us sketch now the construction of Dunwoody manifolds given in [6]. Let",
"a,b,c,n be integers such that n\\ >\\ 0 , a,b,c\\,\\geq\\,0 and a+b+c>0 . Let",
"\\Gamma=\\Gamma(a,b,c,n) be the planar regular trivalent graph drawn in Figure 1.",
"It contains n upper cycles C_{1}^{\\prime},\\ldots\\,,C_{n}^{\\prime} and n lower cycles C_{1}^{\\prime\\prime},\\ldots\\,,C_{n}^{\\prime\\prime} , each",
"having d=2a+b+c vertices. For each i=1,\\dots,n , the cycle C_{i}^{\\prime} (resp. C_{i}^{\\prime\\prime} )",
"is connected to the cycle C_{i+1}^{\\prime} (resp. C_{i+1}^{\\prime\\prime} ) by a parallel arcs, to the cycle C_{i}^{\\prime\\prime}",
"by c parallel arcs and to the cycle C_{i+1}^{\\prime\\prime} by b parallel arcs (assume n+1=1 ).",
"We set \\mathcal{C}^{\\prime}=\\{C_{1}^{\\prime},\\ldots,C_{n}^{\\prime}\\} and \\mathcal{C}^{\\prime\\prime}=\\{C_{1}^{\\prime\\prime},\\ldots,C_{n}^{\\prime\\prime}\\} . Moreover, denote by A^{\\prime}",
"(resp. A^{\\prime\\prime} ) the set of the arcs of \\Gamma belonging to a cycle of \\mathcal{C}^{\\prime} (resp. \\mathcal{C^{\\prime\\prime}} ) and by"
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"Figure 1: The graph .",
"the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph in ; it is evident that the graph is invariant with respect to a rotation of the sphere by radians along a suitable axis intersecting in two points not belonging to the graph. Obviously, sends to and to (mod ), for each .",
"By cutting the sphere along all and and by removing the interior of the corresponding discs, we obtain a sphere with holes. Let now and be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of (resp. of ) and label their vertices from 1 to , in accordance with these orientations (see Figure 2) so that:",
"- the vertex 1 of each is the endpoint of the first arc of connecting Ci′ with Ci′+1; - the vertex (mod ) of each is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1.",
"Then glue the cycle with the cycle (mod ) so that equally labelled vertices are identified together."
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[{"type": "image", "img_path": "images/75b9851aa6f587d46bce7b58819bdbd6615243f7ae4ed18bcd8b379ea5f40807.jpg", "img_caption": ["Figure 1: The graph $\\Gamma$ . "], "img_footnote": [], "page_idx": 3}, {"type": "text", "text": "$A$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $\\Gamma$ in $\\mathbf{S^{2}}$ ; it is evident that the graph is invariant with respect to a rotation $\\rho_{n}$ of the sphere by $2\\pi/n$ radians along a suitable axis intersecting $\\mathbf{S^{2}}$ in two points not belonging to the graph. Obviously, $\\rho_{n}$ sends $C_{i}^{\\prime}$ to $C_{i+1}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ to $C_{i+1}^{\\prime\\prime}$ (mod $n$ ), for each $i=1,\\dots,n$ . ", "page_idx": 3}, {"type": "text", "text": "By cutting the sphere along all $C_{i}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ and by removing the interior of the corresponding discs, we obtain a sphere with $2n$ holes. Let now $r$ and $s$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $\\mathcal{C}^{\\prime}$ (resp. of $\\mathcal{C^{\\prime\\prime}}$ ) and label their vertices from 1 to $d$ , in accordance with these orientations (see Figure 2) so that: ", "page_idx": 3}, {"type": "text", "text": "- the vertex 1 of each $C_{i}^{\\prime}$ is the endpoint of the first arc of $A$ connecting Ci\u2032 with Ci\u2032+1; \n- the vertex $1-r$ (mod $d$ ) of each $C_{i}^{\\prime\\prime}$ is the endpoint of the first arc of A connecting Ci\u2032\u2032 with Ci\u2032\u2032+1. ", "page_idx": 3}, {"type": "text", "text": "Then glue the cycle $C_{i}^{\\prime}$ with the cycle $C_{i-s}^{\\prime\\prime}$ (mod $n$ ) so that equally labelled vertices are identified together. ", "page_idx": 3}]
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The one-point compactification of", "type": "text"}], "index": 15}, {"bbox": [110, 400, 501, 415], "spans": [{"bbox": [110, 400, 369, 415], "score": 1.0, "content": "the plane leads to a 2-cell embedding of the graph ", "type": "text"}, {"bbox": [369, 402, 377, 411], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [377, 400, 393, 415], "score": 1.0, "content": " in ", "type": "text"}, {"bbox": [394, 401, 407, 411], "score": 0.9, "content": "\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [407, 400, 501, 415], "score": 1.0, "content": "; it is evident that", "type": "text"}], "index": 16}, {"bbox": [109, 414, 499, 430], "spans": [{"bbox": [109, 414, 366, 430], "score": 1.0, "content": "the graph is invariant with respect to a rotation ", "type": "text"}, {"bbox": [367, 420, 379, 428], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [379, 414, 473, 430], "score": 1.0, "content": " of the sphere by ", "type": "text"}, {"bbox": [473, 416, 499, 428], "score": 0.94, "content": "2\\pi/n", "type": "inline_equation", "height": 12, "width": 26}], "index": 17}, {"bbox": [109, 429, 501, 443], "spans": [{"bbox": [109, 429, 324, 443], "score": 1.0, "content": "radians along a suitable axis intersecting ", "type": "text"}, {"bbox": [324, 430, 338, 440], "score": 0.9, "content": "\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [338, 429, 501, 443], "score": 1.0, "content": " in two points not belonging to", "type": "text"}], "index": 18}, {"bbox": [108, 442, 501, 461], "spans": [{"bbox": [108, 442, 225, 461], "score": 1.0, "content": "the graph. Obviously, ", "type": "text"}, {"bbox": [225, 448, 237, 456], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [237, 442, 271, 461], "score": 1.0, "content": " sends ", "type": "text"}, {"bbox": [272, 445, 284, 457], "score": 0.93, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [284, 442, 301, 461], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [302, 445, 324, 457], "score": 0.94, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [325, 442, 350, 461], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [351, 445, 365, 457], "score": 0.93, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [365, 442, 383, 461], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [383, 445, 405, 457], "score": 0.93, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 22}, {"bbox": [406, 442, 439, 461], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [440, 448, 447, 454], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [447, 442, 501, 461], "score": 1.0, "content": "), for each", "type": "text"}], "index": 19}, {"bbox": [110, 458, 175, 473], "spans": [{"bbox": [110, 460, 171, 471], "score": 0.93, "content": "i=1,\\dots,n", "type": "inline_equation", "height": 11, "width": 61}, {"bbox": [172, 458, 175, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 20}], "index": 17.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [108, 385, 501, 473]}, {"type": "text", "bbox": [110, 470, 501, 543], "lines": [{"bbox": [127, 472, 502, 487], "spans": [{"bbox": [127, 472, 286, 487], "score": 1.0, "content": "By cutting the sphere along all ", "type": "text"}, {"bbox": [286, 474, 299, 486], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [299, 472, 324, 487], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [324, 474, 338, 486], "score": 0.93, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [339, 472, 502, 487], "score": 1.0, "content": "and by removing the interior of", "type": "text"}], "index": 21}, {"bbox": [109, 487, 499, 501], "spans": [{"bbox": [109, 487, 364, 501], "score": 1.0, "content": "the corresponding discs, we obtain a sphere with ", "type": "text"}, {"bbox": [364, 489, 377, 497], "score": 0.87, "content": "2n", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [378, 487, 460, 501], "score": 1.0, "content": " holes. Let now ", "type": "text"}, {"bbox": [461, 491, 466, 497], "score": 0.87, "content": "r", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [467, 487, 493, 501], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [493, 492, 499, 497], "score": 0.87, "content": "s", "type": "inline_equation", "height": 5, "width": 6}], "index": 22}, {"bbox": [109, 501, 501, 516], "spans": [{"bbox": [109, 501, 501, 516], "score": 1.0, "content": "be two new integers; give a clockwise (resp. counterclockwise) orientation to", "type": "text"}], "index": 23}, {"bbox": [110, 516, 500, 529], "spans": [{"bbox": [110, 516, 174, 529], "score": 1.0, "content": "the cycles of ", "type": "text"}, {"bbox": [174, 517, 184, 526], "score": 0.88, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [184, 516, 232, 529], "score": 1.0, "content": " (resp. of ", "type": "text"}, {"bbox": [232, 517, 245, 527], "score": 0.89, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [245, 516, 417, 529], "score": 1.0, "content": ") and label their vertices from 1 to", "type": "text"}, {"bbox": [418, 518, 424, 526], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [425, 516, 500, 529], "score": 1.0, "content": ", in accordance", "type": "text"}], "index": 24}, {"bbox": [110, 530, 345, 544], "spans": [{"bbox": [110, 530, 345, 544], "score": 1.0, "content": "with these orientations (see Figure 2) so that:", "type": "text"}], "index": 25}], "index": 23, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [109, 472, 502, 544]}, {"type": "list", "bbox": [126, 551, 502, 622], "lines": [{"bbox": [129, 554, 500, 569], "spans": [{"bbox": [129, 554, 244, 569], "score": 1.0, "content": "- the vertex 1 of each ", "type": "text"}, {"bbox": [244, 556, 257, 568], "score": 0.93, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [257, 554, 431, 569], "score": 1.0, "content": " is the endpoint of the first arc of ", "type": "text"}, {"bbox": [432, 556, 440, 565], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [441, 554, 500, 569], "score": 1.0, "content": " connecting", "type": "text"}], "index": 26, "is_list_start_line": true}, {"bbox": [139, 568, 210, 586], "spans": [{"bbox": [139, 568, 210, 586], "score": 1.0, "content": "Ci\u2032 with Ci\u2032+1;", "type": "text"}], "index": 27, "is_list_end_line": true}, {"bbox": [128, 591, 502, 610], "spans": [{"bbox": [128, 591, 195, 610], "score": 1.0, "content": "- the vertex ", "type": "text"}, {"bbox": [195, 595, 222, 605], "score": 0.91, "content": "1-r", "type": "inline_equation", "height": 10, "width": 27}, {"bbox": [222, 591, 257, 610], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [257, 595, 263, 604], "score": 0.8, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [264, 591, 311, 610], "score": 1.0, "content": ") of each ", "type": "text"}, {"bbox": [312, 595, 326, 607], "score": 0.92, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [326, 591, 502, 610], "score": 1.0, "content": " is the endpoint of the first arc of", "type": "text"}], "index": 28, "is_list_start_line": true}, {"bbox": [137, 602, 285, 625], "spans": [{"bbox": [137, 602, 285, 625], "score": 1.0, "content": "A connecting Ci\u2032\u2032 with Ci\u2032\u2032+1.", "type": "text"}], "index": 29, "is_list_end_line": true}], "index": 27.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [128, 554, 502, 625]}, {"type": "text", "bbox": [109, 630, 500, 659], "lines": [{"bbox": [110, 632, 500, 647], "spans": [{"bbox": [110, 632, 214, 647], "score": 1.0, "content": "Then glue the cycle ", "type": "text"}, {"bbox": [215, 633, 227, 646], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 13, "width": 12}, {"bbox": [227, 632, 306, 647], "score": 1.0, "content": " with the cycle ", "type": "text"}, {"bbox": [307, 633, 329, 646], "score": 0.91, "content": "C_{i-s}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [329, 632, 363, 647], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [363, 637, 371, 642], "score": 0.78, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [371, 632, 500, 647], "score": 1.0, "content": ") so that equally labelled", "type": "text"}], "index": 30}, {"bbox": [111, 647, 269, 660], "spans": [{"bbox": [111, 647, 269, 660], "score": 1.0, "content": "vertices are identified together.", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [110, 632, 500, 660]}]}
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![Image]()
![Image]()
Figure 1: The graph .
![Image]() *Figure 1: The graph .*
the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph in ; it is evident that the graph is invariant with respect to a rotation of the sphere by radians along a suitable axis intersecting in two points not belonging to the graph. Obviously, sends to and to (mod ), for each .
By cutting the sphere along all and and by removing the interior of the corresponding discs, we obtain a sphere with holes. Let now and be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of (resp. of ) and label their vertices from 1 to , in accordance with these orientations (see Figure 2) so that:
- - the vertex 1 of each is the endpoint of the first arc of connecting Ci′ with Ci′+1; - the vertex (mod ) of each is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1.
Then glue the cycle with the cycle (mod ) so that equally labelled vertices are identified together.
|
<div class="pdf-page">
<em>Figure 1: The graph .</em>
<p>the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph in ; it is evident that the graph is invariant with respect to a rotation of the sphere by radians along a suitable axis intersecting in two points not belonging to the graph. Obviously, sends to and to (mod ), for each .</p>
<p>By cutting the sphere along all and and by removing the interior of the corresponding discs, we obtain a sphere with holes. Let now and be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of (resp. of ) and label their vertices from 1 to , in accordance with these orientations (see Figure 2) so that:</p>
<ul>
<li>- the vertex 1 of each is the endpoint of the first arc of connecting Ci′ with Ci′+1; - the vertex (mod ) of each is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1.</li>
</ul>
<p>Then glue the cycle with the cycle (mod ) so that equally labelled vertices are identified together.</p>
</div>
|
<div class="pdf-page">
<figcaption class="pdf-image-caption" data-x="408" data-y="451" data-width="201" data-height="19">Figure 1: The graph .</figcaption>
<p class="pdf-text" data-x="182" data-y="495" data-width="654" data-height="112">the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph in ; it is evident that the graph is invariant with respect to a rotation of the sphere by radians along a suitable axis intersecting in two points not belonging to the graph. Obviously, sends to and to (mod ), for each .</p>
<p class="pdf-text" data-x="184" data-y="607" data-width="654" data-height="95">By cutting the sphere along all and and by removing the interior of the corresponding discs, we obtain a sphere with holes. Let now and be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of (resp. of ) and label their vertices from 1 to , in accordance with these orientations (see Figure 2) so that:</p>
<ul class="pdf-list" data-x="210" data-y="712" data-width="629" data-height="92">
<li>- the vertex 1 of each is the endpoint of the first arc of connecting Ci′ with Ci′+1; - the vertex (mod ) of each is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1.</li>
</ul>
<p class="pdf-text" data-x="182" data-y="814" data-width="654" data-height="38">Then glue the cycle with the cycle (mod ) so that equally labelled vertices are identified together.</p>
</div>
|
Figure 1: The graph $\Gamma$ .
$A$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $\Gamma$ in $\mathbf{S^{2}}$ ; it is evident that the graph is invariant with respect to a rotation $\rho_{n}$ of the sphere by $2\pi/n$ radians along a suitable axis intersecting $\mathbf{S^{2}}$ in two points not belonging to the graph. Obviously, $\rho_{n}$ sends $C_{i}^{\prime}$ to $C_{i+1}^{\prime}$ and $C_{i}^{\prime\prime}$ to $C_{i+1}^{\prime\prime}$ (mod $n$ ), for each $i=1,\dots,n$ .
By cutting the sphere along all $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ and by removing the interior of the corresponding discs, we obtain a sphere with $2n$ holes. Let now $r$ and $s$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $\mathcal{C}^{\prime}$ (resp. of $\mathcal{C^{\prime\prime}}$ ) and label their vertices from 1 to $d$ , in accordance with these orientations (see Figure 2) so that:
- the vertex 1 of each $C_{i}^{\prime}$ is the endpoint of the first arc of $A$ connecting Ci′ with Ci′+1;
- the vertex $1-r$ (mod $d$ ) of each $C_{i}^{\prime\prime}$ is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1.
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![Image]()
![Image]()
Figure 2:
![Image]() *Figure 2:*
It is evident by construction that the integers and can be taken mod and mod respectively. Denote by the set of all the 6-tuples such that , and .
The described gluing gives rise to an orientable surface of genus and the arcs belonging to are pairwise connected through their endpoints, realizing cycles on . It is straightforward that the cut of along the cycles does not disconnect the surface. Set and .
If and if the cut along the cycles of does not disconnect , then the two systems of meridian curves and in represent a genus Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold.
Thus, we define to be admissible the 6-tuples of satisfying the following conditions:
(1) the set contains exactly cycles;
(2) the surface is not disconnected by the cut along the cycles of .
5
|
<div class="pdf-page">
<em>Figure 2:</em>
<p>It is evident by construction that the integers and can be taken mod and mod respectively. Denote by the set of all the 6-tuples such that , and .</p>
<p>The described gluing gives rise to an orientable surface of genus and the arcs belonging to are pairwise connected through their endpoints, realizing cycles on . It is straightforward that the cut of along the cycles does not disconnect the surface. Set and .</p>
<p>If and if the cut along the cycles of does not disconnect , then the two systems of meridian curves and in represent a genus Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold.</p>
<p>Thus, we define to be admissible the 6-tuples of satisfying the following conditions:</p>
<p>(1) the set contains exactly cycles;</p>
<p>(2) the surface is not disconnected by the cut along the cycles of .</p>
</div>
|
<div class="pdf-page">
<figcaption class="pdf-image-caption" data-x="463" data-y="487" data-width="84" data-height="19">Figure 2:</figcaption>
<p class="pdf-text" data-x="182" data-y="530" data-width="654" data-height="55">It is evident by construction that the integers and can be taken mod and mod respectively. Denote by the set of all the 6-tuples such that , and .</p>
<p class="pdf-text" data-x="182" data-y="586" data-width="654" data-height="94">The described gluing gives rise to an orientable surface of genus and the arcs belonging to are pairwise connected through their endpoints, realizing cycles on . It is straightforward that the cut of along the cycles does not disconnect the surface. Set and .</p>
<p class="pdf-text" data-x="182" data-y="680" data-width="654" data-height="91">If and if the cut along the cycles of does not disconnect , then the two systems of meridian curves and in represent a genus Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold.</p>
<p class="pdf-text" data-x="182" data-y="773" data-width="656" data-height="37">Thus, we define to be admissible the 6-tuples of satisfying the following conditions:</p>
<p class="pdf-text" data-x="197" data-y="822" data-width="343" data-height="19">(1) the set contains exactly cycles;</p>
<p class="pdf-text" data-x="197" data-y="853" data-width="616" data-height="19">(2) the surface is not disconnected by the cut along the cycles of .</p>
<div class="pdf-discarded" data-x="501" data-y="893" data-width="16" data-height="14" style="opacity: 0.5;">5</div>
</div>
|
Figure 2:
It is evident by construction that the integers $r$ and $s$ can be taken mod $d$ and mod $n$ respectively. Denote by $\boldsymbol{S}$ the set of all the 6-tuples $(a,b,c,n,r,s)\in\mathbf{Z}^{6}$ such that $n>0$ , $a,b,c\geq0$ and $a+b+c>0$ .
The described gluing gives rise to an orientable surface $T_{n}$ of genus $n$ and the $n d$ arcs belonging to $A$ are pairwise connected through their endpoints, realizing ${\mathit{m}}$ cycles $D_{1},\ldots,D_{m}$ on $T_{n}$ . It is straightforward that the cut of $T_{n}$ along the $n$ cycles $C_{i}=C_{i}^{\prime}=C_{i-s}^{\prime\prime}$ does not disconnect the surface. Set $\mathcal{C}=\{C_{1},\ldots,C_{n}\}$ and $\mathcal{D}=\{D_{1},\dotsc,D_{m}\}$ .
If $m\,=\,n$ and if the cut along the cycles of $\mathcal{D}$ does not disconnect $T_{n}^{'}$ , then the two systems of meridian curves $\scriptscriptstyle\mathcal{C}$ and $\mathcal{D}$ in $T_{n}$ represent a genus $n$ Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold.
Thus, we define to be admissible the 6-tuples $(a,b,c,n,r,s)$ of $\boldsymbol{S}$ satisfying the following conditions:
(1) the set $\mathcal{D}$ contains exactly $n$ cycles;
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"The “open” Heegaard diagram and the Dunwoody manifold associated to the admissible 6-tuple will be denoted by and respectively.",
"Remark 1. It is easy to see that not all the 6-tuples in are admissible. For example, the 6-tuples , with , give rise to exactly cycles in ; thus, they are not admissible if . The 6-tuples are not admissible if is even, since, in this case, we obtain exactly one cycle , but the cut along it disconnects the torus .",
"Consider now a 6-tuple . The graph becomes, via the gluing quotient map, a regular 4-valent graph denoted by embedded in . Its vertices are the intersection points of the spaces and ; hence they inherit the labelling of the corresponding glued vertices of . Since the gluing of the cycles of and is invariant with respect to the rotation , the group naturally induces a cyclic action of order on such that the quotient is homeomorphic to a torus. The labelling of the vertices of is invariant under the rotation and (mod ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of .",
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"b) If is admissible, then also is admissible and the Heegaard diagram is the quotient of the Heegaard diagram respect to .",
"Proof. a) First of all, note that ; thus the group also acts on the spaces and (and hence on the set ). If the 6-tuple is admissible, then is connected, and hence the quotient must be connected too. This implies that has a unique connected component. Since has exactly connected components, the cyclic group of order defines a simply transitive cyclic action on the cycles of .",
"b) Let the two curves and . Then, the two systems of curves and on define a Heegaard diagram of genus one. The graph corresponding to is the quotient of the graph corresponding to , respect to . Moreover, the gluings on are invariant respect to . Therefore, the gluings on give rise to the Heegaard diagram above. This show that the 6-tuple is admissible and obviously is the quotient of respect to .",
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[{"type": "text", "text": "The \u201copen\u201d Heegaard diagram $\\Gamma$ and the Dunwoody manifold associated to the admissible 6-tuple $\\sigma$ will be denoted by $H(\\sigma)$ and $M(\\sigma)$ respectively. ", "page_idx": 5}, {"type": "text", "text": "Remark 1. It is easy to see that not all the 6-tuples in $\\boldsymbol{S}$ are admissible. For example, the 6-tuples $(a,0,a,1,a,0)$ , with $a\\geq1$ , give rise to exactly $a$ cycles in $\\mathcal{D}$ ; thus, they are not admissible if $a>1$ . The 6-tuples $(1,0,c,1,2,0)$ are not admissible if $c$ is even, since, in this case, we obtain exactly one cycle $D_{1}$ , but the cut along it disconnects the torus $T_{1}$ . ", "page_idx": 5}, {"type": "text", "text": "Consider now a 6-tuple $\\sigma\\,\\in\\,S$ . The graph $\\Gamma$ becomes, via the gluing quotient map, a regular 4-valent graph denoted by $\\Gamma^{\\prime}$ embedded in $T_{n}^{'}$ . Its vertices are the intersection points of the spaces $\\Omega=\\cup_{i=1}^{n}C_{i}$ and $\\Lambda=\\cup_{j=1}^{m}D_{j}$ ; hence they inherit the labelling of the corresponding glued vertices of $\\Gamma$ . Since the gluing of the cycles of $\\mathcal{C}^{\\prime}$ and $\\mathcal{C^{\\prime\\prime}}$ is invariant with respect to the rotation $\\rho_{n}$ , the group $\\mathcal{G}_{n}=<\\rho_{n}>$ naturally induces a cyclic action of order $n$ on $T_{n}^{'}$ such that the quotient $T_{1}=T_{n}/\\mathcal{G}_{n}$ is homeomorphic to a torus. The labelling of the vertices of $\\Gamma^{\\prime}$ is invariant under the rotation $\\rho_{n}$ and $\\rho_{n}(C_{i})=C_{i+1}$ (mod $n$ ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of $\\mathcal{D}$ . ", "page_idx": 5}, {"type": "text", "text": "Lemma 1 a) Let $\\sigma\\;=\\;(a,b,c,n,r,s)$ be an admissible 6-tuple. Then $\\rho_{n}$ induces a cyclic permutation on the curves of $\\mathcal{D}$ . Thus, if $D$ is a cycle of $\\mathcal{D}$ , then ${\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)|k=1,\\ldots,n\\}$ . ", "page_idx": 5}, {"type": "text", "text": "b) If $(a,b,c,n,r,s)$ is admissible, then also $(a,b,c,1,r,0)$ is admissible and the Heegaard diagram $H(a,b,c,1,r,0)$ is the quotient of the Heegaard diagram $H(a,b,c,n,r,s)$ respect to $\\mathcal{G}_{n}$ . ", "page_idx": 5}, {"type": "text", "text": "Proof. a) First of all, note that $\\rho_{n}(\\Lambda)=\\Lambda$ ; thus the group $\\mathcal{G}_{n}$ also acts on the spaces $T_{n}\\mathrm{~-~}\\Lambda$ and $\\Lambda$ (and hence on the set $\\mathcal{D}$ ). If the 6-tuple $\\sigma$ is admissible, then $T_{n}-\\Lambda$ is connected, and hence the quotient $(T_{n}-\\Lambda)/\\mathcal{G}_{n}=$ $T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}$ must be connected too. This implies that $\\Lambda/\\mathcal{G}_{n}$ has a unique connected component. Since $\\Lambda$ has exactly $n$ connected components, the cyclic group $\\mathcal{G}_{n}$ of order $n$ defines a simply transitive cyclic action on the cycles of $\\mathcal{D}$ . ", "page_idx": 5}, {"type": "text", "text": "b) Let $C,D\\ \\subset\\ T_{1}$ the two curves $C\\;=\\;\\Omega/\\mathcal{G}_{n}$ and $D\\,=\\,\\Lambda/\\mathcal{G}_{n}$ . Then, the two systems of curves ${\\mathcal{C}}=\\{C\\}$ and $\\mathcal{D}=\\{D\\}$ on $T_{1}$ define a Heegaard diagram of genus one. The graph $\\Gamma_{1}$ corresponding to $\\sigma_{1}\\,=\\,(a,b,c,1,r,0)$ is the quotient of the graph $\\Gamma_{n}$ corresponding to $\\sigma=(a,b,c,n,r,s)$ , respect to $\\mathcal{G}_{n}$ . Moreover, the gluings on $\\Gamma_{n}$ are invariant respect to $\\rho_{n}$ . Therefore, the gluings on $\\Gamma_{1}$ give rise to the Heegaard diagram above. This show that the 6-tuple $\\sigma_{1}$ is admissible and obviously $H(a,b,c,1,r,0)$ is the quotient of $H(a,b,c,n,r,s)$ respect to $\\mathcal{G}_{n}$ . ", "page_idx": 5}]
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{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [127, 128, 500, 142], "spans": [{"bbox": [127, 128, 287, 142], "score": 1.0, "content": "The \u201copen\u201d Heegaard diagram ", "type": "text"}, {"bbox": [288, 129, 295, 138], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [295, 128, 500, 142], "score": 1.0, "content": " and the Dunwoody manifold associated", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 156], "spans": [{"bbox": [109, 142, 241, 156], "score": 1.0, "content": "to the admissible 6-tuple ", "type": "text"}, {"bbox": [241, 147, 248, 153], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [248, 142, 350, 156], "score": 1.0, "content": " will be denoted by ", "type": "text"}, {"bbox": [351, 143, 378, 156], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [378, 142, 403, 156], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [404, 143, 433, 156], "score": 0.94, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [433, 142, 498, 156], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [109, 160, 501, 232], "lines": [{"bbox": [110, 163, 500, 176], "spans": [{"bbox": [110, 163, 390, 176], "score": 1.0, "content": "Remark 1. It is easy to see that not all the 6-tuples in ", "type": "text"}, {"bbox": [391, 164, 399, 173], "score": 0.88, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [399, 163, 500, 176], "score": 1.0, "content": " are admissible. For", "type": "text"}], "index": 2}, {"bbox": [110, 178, 500, 191], "spans": [{"bbox": [110, 178, 222, 191], "score": 1.0, "content": "example, the 6-tuples ", "type": "text"}, {"bbox": [223, 178, 294, 190], "score": 0.92, "content": "(a,0,a,1,a,0)", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [294, 178, 326, 191], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [326, 179, 355, 189], "score": 0.92, "content": "a\\geq1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [355, 178, 459, 191], "score": 1.0, "content": ", give rise to exactly ", "type": "text"}, {"bbox": [460, 182, 466, 187], "score": 0.86, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [466, 178, 500, 191], "score": 1.0, "content": " cycles", "type": "text"}], "index": 3}, {"bbox": [109, 191, 500, 206], "spans": [{"bbox": [109, 191, 123, 206], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [124, 194, 134, 202], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [134, 191, 304, 206], "score": 1.0, "content": "; thus, they are not admissible if ", "type": "text"}, {"bbox": [304, 194, 333, 202], "score": 0.92, "content": "a>1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [333, 191, 409, 206], "score": 1.0, "content": ". The 6-tuples ", "type": "text"}, {"bbox": [410, 192, 479, 205], "score": 0.92, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [480, 191, 500, 206], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [110, 206, 499, 219], "spans": [{"bbox": [110, 206, 201, 219], "score": 1.0, "content": "not admissible if ", "type": "text"}, {"bbox": [201, 211, 207, 216], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [207, 206, 499, 219], "score": 1.0, "content": " is even, since, in this case, we obtain exactly one cycle", "type": "text"}], "index": 5}, {"bbox": [110, 220, 365, 234], "spans": [{"bbox": [110, 222, 125, 232], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [125, 220, 348, 234], "score": 1.0, "content": ", but the cut along it disconnects the torus ", "type": "text"}, {"bbox": [348, 222, 360, 232], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [361, 220, 365, 234], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4}, {"type": "text", "bbox": [109, 238, 500, 383], "lines": [{"bbox": [127, 239, 500, 256], "spans": [{"bbox": [127, 239, 255, 256], "score": 1.0, "content": "Consider now a 6-tuple ", "type": "text"}, {"bbox": [255, 243, 290, 252], "score": 0.92, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [290, 239, 361, 256], "score": 1.0, "content": ". The graph ", "type": "text"}, {"bbox": [362, 243, 369, 251], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [370, 239, 500, 256], "score": 1.0, "content": " becomes, via the gluing", "type": "text"}], "index": 7}, {"bbox": [110, 256, 501, 269], "spans": [{"bbox": [110, 256, 378, 269], "score": 1.0, "content": "quotient map, a regular 4-valent graph denoted by ", "type": "text"}, {"bbox": [378, 257, 388, 266], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [388, 256, 462, 269], "score": 1.0, "content": " embedded in ", "type": "text"}, {"bbox": [462, 257, 475, 267], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [476, 256, 501, 269], "score": 1.0, "content": ". Its", "type": "text"}], "index": 8}, {"bbox": [108, 267, 501, 285], "spans": [{"bbox": [108, 267, 351, 285], "score": 1.0, "content": "vertices are the intersection points of the spaces ", "type": "text"}, {"bbox": [352, 271, 410, 283], "score": 0.94, "content": "\\Omega=\\cup_{i=1}^{n}C_{i}", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [410, 267, 434, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [434, 271, 496, 285], "score": 0.94, "content": "\\Lambda=\\cup_{j=1}^{m}D_{j}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [496, 267, 501, 285], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [109, 284, 501, 298], "spans": [{"bbox": [109, 284, 456, 298], "score": 1.0, "content": "hence they inherit the labelling of the corresponding glued vertices of ", "type": "text"}, {"bbox": [457, 286, 464, 294], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [465, 284, 501, 298], "score": 1.0, "content": ". Since", "type": "text"}], "index": 10}, {"bbox": [110, 299, 500, 312], "spans": [{"bbox": [110, 299, 244, 312], "score": 1.0, "content": "the gluing of the cycles of ", "type": "text"}, {"bbox": [245, 300, 255, 309], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 299, 280, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [281, 300, 293, 309], "score": 0.92, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [294, 299, 500, 312], "score": 1.0, "content": " is invariant with respect to the rotation", "type": "text"}], "index": 11}, {"bbox": [110, 313, 499, 328], "spans": [{"bbox": [110, 318, 122, 326], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 313, 181, 328], "score": 1.0, "content": ", the group ", "type": "text"}, {"bbox": [182, 315, 244, 326], "score": 0.93, "content": "\\mathcal{G}_{n}=<\\rho_{n}>", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [244, 313, 459, 328], "score": 1.0, "content": " naturally induces a cyclic action of order ", "type": "text"}, {"bbox": [459, 318, 466, 323], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [467, 313, 486, 328], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [486, 315, 499, 325], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}], "index": 12}, {"bbox": [110, 328, 500, 342], "spans": [{"bbox": [110, 328, 225, 342], "score": 1.0, "content": "such that the quotient ", "type": "text"}, {"bbox": [225, 329, 284, 341], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [284, 328, 500, 342], "score": 1.0, "content": " is homeomorphic to a torus. The labelling", "type": "text"}], "index": 13}, {"bbox": [109, 341, 501, 357], "spans": [{"bbox": [109, 341, 195, 357], "score": 1.0, "content": "of the vertices of ", "type": "text"}, {"bbox": [195, 343, 205, 352], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 341, 361, 357], "score": 1.0, "content": " is invariant under the rotation", "type": "text"}, {"bbox": [362, 347, 374, 355], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [374, 341, 398, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 343, 469, 355], "score": 0.94, "content": "\\rho_{n}(C_{i})=C_{i+1}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [469, 341, 501, 357], "score": 1.0, "content": " (mod", "type": "text"}], "index": 14}, {"bbox": [110, 357, 499, 371], "spans": [{"bbox": [110, 361, 118, 367], "score": 0.69, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [118, 357, 499, 371], "score": 1.0, "content": "). We are going to show that, if the 6-tuple is admissible, this last property", "type": "text"}], "index": 15}, {"bbox": [110, 371, 263, 384], "spans": [{"bbox": [110, 371, 249, 384], "score": 1.0, "content": "also holds for the cycles of ", "type": "text"}, {"bbox": [249, 372, 259, 381], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 371, 263, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11.5}, {"type": "text", "bbox": [109, 396, 500, 439], "lines": [{"bbox": [108, 395, 498, 414], "spans": [{"bbox": [108, 395, 208, 414], "score": 1.0, "content": "Lemma 1 a) Let ", "type": "text"}, {"bbox": [209, 399, 308, 411], "score": 0.92, "content": "\\sigma\\;=\\;(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 99}, {"bbox": [308, 395, 487, 414], "score": 1.0, "content": " be an admissible 6-tuple. Then ", "type": "text"}, {"bbox": [487, 403, 498, 411], "score": 0.86, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 11}], "index": 17}, {"bbox": [111, 412, 499, 426], "spans": [{"bbox": [111, 412, 344, 426], "score": 1.0, "content": "induces a cyclic permutation on the curves of ", "type": "text"}, {"bbox": [344, 414, 354, 423], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [355, 412, 407, 426], "score": 1.0, "content": ". Thus, if ", "type": "text"}, {"bbox": [407, 414, 417, 423], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [417, 412, 485, 426], "score": 1.0, "content": " is a cycle of ", "type": "text"}, {"bbox": [486, 415, 496, 423], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [496, 412, 499, 426], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 426, 285, 441], "spans": [{"bbox": [110, 426, 136, 441], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [136, 427, 281, 441], "score": 0.93, "content": "{\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)|k=1,\\ldots,n\\}", "type": "inline_equation", "height": 14, "width": 145}, {"bbox": [282, 426, 285, 441], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [110, 440, 501, 483], "lines": [{"bbox": [127, 441, 501, 455], "spans": [{"bbox": [127, 441, 157, 455], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [157, 442, 226, 455], "score": 0.94, "content": "(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [226, 441, 359, 455], "score": 1.0, "content": " is admissible, then also ", "type": "text"}, {"bbox": [359, 442, 428, 455], "score": 0.92, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [428, 441, 501, 455], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 20}, {"bbox": [111, 456, 501, 470], "spans": [{"bbox": [111, 456, 252, 470], "score": 1.0, "content": "and the Heegaard diagram ", "type": "text"}, {"bbox": [252, 457, 331, 469], "score": 0.9, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [332, 456, 501, 470], "score": 1.0, "content": " is the quotient of the Heegaard", "type": "text"}], "index": 21}, {"bbox": [111, 469, 309, 486], "spans": [{"bbox": [111, 469, 155, 486], "score": 1.0, "content": "diagram ", "type": "text"}, {"bbox": [155, 471, 235, 484], "score": 0.92, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [235, 469, 290, 486], "score": 1.0, "content": " respect to ", "type": "text"}, {"bbox": [291, 472, 304, 483], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [304, 469, 309, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21}, {"type": "text", "bbox": [109, 496, 501, 596], "lines": [{"bbox": [126, 497, 501, 512], "spans": [{"bbox": [126, 497, 297, 512], "score": 1.0, "content": "Proof. a) First of all, note that ", "type": "text"}, {"bbox": [298, 499, 352, 511], "score": 0.94, "content": "\\rho_{n}(\\Lambda)=\\Lambda", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [352, 497, 437, 512], "score": 1.0, "content": "; thus the group ", "type": "text"}, {"bbox": [438, 500, 451, 510], "score": 0.91, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [451, 497, 501, 512], "score": 1.0, "content": " also acts", "type": "text"}], "index": 23}, {"bbox": [110, 513, 500, 526], "spans": [{"bbox": [110, 513, 184, 526], "score": 1.0, "content": "on the spaces ", "type": "text"}, {"bbox": [185, 514, 222, 524], "score": 0.93, "content": "T_{n}\\mathrm{~-~}\\Lambda", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [222, 513, 249, 526], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [249, 514, 258, 523], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [258, 513, 380, 526], "score": 1.0, "content": " (and hence on the set ", "type": "text"}, {"bbox": [380, 514, 390, 523], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [390, 513, 479, 526], "score": 1.0, "content": "). If the 6-tuple ", "type": "text"}, {"bbox": [479, 517, 486, 523], "score": 0.84, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [487, 513, 500, 526], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [109, 526, 501, 541], "spans": [{"bbox": [109, 526, 196, 541], "score": 1.0, "content": "admissible, then ", "type": "text"}, {"bbox": [196, 528, 231, 539], "score": 0.93, "content": "T_{n}-\\Lambda", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [231, 526, 424, 541], "score": 1.0, "content": " is connected, and hence the quotient ", "type": "text"}, {"bbox": [424, 528, 501, 540], "score": 0.93, "content": "(T_{n}-\\Lambda)/\\mathcal{G}_{n}=", "type": "inline_equation", "height": 12, "width": 77}], "index": 25}, {"bbox": [110, 542, 499, 555], "spans": [{"bbox": [110, 542, 183, 555], "score": 0.94, "content": "T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [183, 542, 403, 555], "score": 1.0, "content": " must be connected too. This implies that ", "type": "text"}, {"bbox": [403, 542, 430, 555], "score": 0.94, "content": "\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [430, 542, 499, 555], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 26}, {"bbox": [110, 556, 500, 569], "spans": [{"bbox": [110, 556, 266, 569], "score": 1.0, "content": "connected component. Since ", "type": "text"}, {"bbox": [267, 557, 275, 566], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [276, 556, 344, 569], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [344, 560, 352, 566], "score": 0.85, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [352, 556, 500, 569], "score": 1.0, "content": " connected components, the", "type": "text"}], "index": 27}, {"bbox": [110, 570, 500, 583], "spans": [{"bbox": [110, 570, 177, 583], "score": 1.0, "content": "cyclic group ", "type": "text"}, {"bbox": [178, 572, 190, 582], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [191, 570, 241, 583], "score": 1.0, "content": " of order ", "type": "text"}, {"bbox": [241, 575, 249, 580], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [249, 570, 500, 583], "score": 1.0, "content": " defines a simply transitive cyclic action on the", "type": "text"}], "index": 28}, {"bbox": [110, 585, 172, 599], "spans": [{"bbox": [110, 585, 156, 599], "score": 1.0, "content": "cycles of ", "type": "text"}, {"bbox": [157, 587, 167, 595], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [167, 585, 172, 599], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26}, {"type": "text", "bbox": [109, 597, 500, 669], "lines": [{"bbox": [127, 599, 499, 613], "spans": [{"bbox": [127, 599, 167, 613], "score": 1.0, "content": "b) Let ", "type": "text"}, {"bbox": [167, 601, 223, 612], "score": 0.93, "content": "C,D\\ \\subset\\ T_{1}", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [223, 599, 311, 613], "score": 1.0, "content": " the two curves ", "type": "text"}, {"bbox": [311, 600, 369, 613], "score": 0.95, "content": "C\\;=\\;\\Omega/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [369, 599, 398, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 600, 456, 613], "score": 0.95, "content": "D\\,=\\,\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [456, 599, 499, 613], "score": 1.0, "content": ". Then,", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 247, 627], "score": 1.0, "content": "the two systems of curves ", "type": "text"}, {"bbox": [247, 614, 293, 627], "score": 0.95, "content": "{\\mathcal{C}}=\\{C\\}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [293, 613, 320, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [320, 614, 370, 627], "score": 0.94, "content": "\\mathcal{D}=\\{D\\}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [370, 613, 390, 627], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [390, 615, 402, 626], "score": 0.94, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [402, 613, 500, 627], "score": 1.0, "content": " define a Heegaard", "type": "text"}], "index": 31}, {"bbox": [110, 627, 499, 642], "spans": [{"bbox": [110, 627, 291, 642], "score": 1.0, "content": "diagram of genus one. The graph ", "type": "text"}, {"bbox": [292, 630, 304, 640], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [304, 627, 399, 642], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [400, 629, 499, 641], "score": 0.92, "content": "\\sigma_{1}\\,=\\,(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 99}], "index": 32}, {"bbox": [109, 642, 500, 657], "spans": [{"bbox": [109, 642, 255, 657], "score": 1.0, "content": "is the quotient of the graph ", "type": "text"}, {"bbox": [256, 644, 269, 654], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [270, 642, 362, 657], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [363, 643, 455, 655], "score": 0.92, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [456, 642, 500, 657], "score": 1.0, "content": ", respect", "type": "text"}], "index": 33}, {"bbox": [109, 657, 500, 671], "spans": [{"bbox": [109, 657, 124, 671], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [125, 658, 138, 669], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [138, 657, 281, 671], "score": 1.0, "content": ". Moreover, the gluings on ", "type": "text"}, {"bbox": [281, 658, 294, 669], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [295, 657, 424, 671], "score": 1.0, "content": " are invariant respect to ", "type": "text"}, {"bbox": [424, 662, 436, 669], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [437, 657, 500, 671], "score": 1.0, "content": ". Therefore,", "type": "text"}], "index": 34}], "index": 32}], "layout_bboxes": [], "page_idx": 5, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "6", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [127, 128, 500, 142], "spans": [{"bbox": [127, 128, 287, 142], "score": 1.0, "content": "The \u201copen\u201d Heegaard diagram ", "type": "text"}, {"bbox": [288, 129, 295, 138], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [295, 128, 500, 142], "score": 1.0, "content": " and the Dunwoody manifold associated", "type": "text"}], "index": 0}, {"bbox": [109, 142, 498, 156], "spans": [{"bbox": [109, 142, 241, 156], "score": 1.0, "content": "to the admissible 6-tuple ", "type": "text"}, {"bbox": [241, 147, 248, 153], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [248, 142, 350, 156], "score": 1.0, "content": " will be denoted by ", "type": "text"}, {"bbox": [351, 143, 378, 156], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [378, 142, 403, 156], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [404, 143, 433, 156], "score": 0.94, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [433, 142, 498, 156], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 500, 156]}, {"type": "text", "bbox": [109, 160, 501, 232], "lines": [{"bbox": [110, 163, 500, 176], "spans": [{"bbox": [110, 163, 390, 176], "score": 1.0, "content": "Remark 1. It is easy to see that not all the 6-tuples in ", "type": "text"}, {"bbox": [391, 164, 399, 173], "score": 0.88, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [399, 163, 500, 176], "score": 1.0, "content": " are admissible. For", "type": "text"}], "index": 2}, {"bbox": [110, 178, 500, 191], "spans": [{"bbox": [110, 178, 222, 191], "score": 1.0, "content": "example, the 6-tuples ", "type": "text"}, {"bbox": [223, 178, 294, 190], "score": 0.92, "content": "(a,0,a,1,a,0)", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [294, 178, 326, 191], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [326, 179, 355, 189], "score": 0.92, "content": "a\\geq1", "type": "inline_equation", "height": 10, "width": 29}, {"bbox": [355, 178, 459, 191], "score": 1.0, "content": ", give rise to exactly ", "type": "text"}, {"bbox": [460, 182, 466, 187], "score": 0.86, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [466, 178, 500, 191], "score": 1.0, "content": " cycles", "type": "text"}], "index": 3}, {"bbox": [109, 191, 500, 206], "spans": [{"bbox": [109, 191, 123, 206], "score": 1.0, "content": "in ", "type": "text"}, {"bbox": [124, 194, 134, 202], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [134, 191, 304, 206], "score": 1.0, "content": "; thus, they are not admissible if ", "type": "text"}, {"bbox": [304, 194, 333, 202], "score": 0.92, "content": "a>1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [333, 191, 409, 206], "score": 1.0, "content": ". The 6-tuples ", "type": "text"}, {"bbox": [410, 192, 479, 205], "score": 0.92, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [480, 191, 500, 206], "score": 1.0, "content": " are", "type": "text"}], "index": 4}, {"bbox": [110, 206, 499, 219], "spans": [{"bbox": [110, 206, 201, 219], "score": 1.0, "content": "not admissible if ", "type": "text"}, {"bbox": [201, 211, 207, 216], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [207, 206, 499, 219], "score": 1.0, "content": " is even, since, in this case, we obtain exactly one cycle", "type": "text"}], "index": 5}, {"bbox": [110, 220, 365, 234], "spans": [{"bbox": [110, 222, 125, 232], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [125, 220, 348, 234], "score": 1.0, "content": ", but the cut along it disconnects the torus ", "type": "text"}, {"bbox": [348, 222, 360, 232], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [361, 220, 365, 234], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 163, 500, 234]}, {"type": "text", "bbox": [109, 238, 500, 383], "lines": [{"bbox": [127, 239, 500, 256], "spans": [{"bbox": [127, 239, 255, 256], "score": 1.0, "content": "Consider now a 6-tuple ", "type": "text"}, {"bbox": [255, 243, 290, 252], "score": 0.92, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 35}, {"bbox": [290, 239, 361, 256], "score": 1.0, "content": ". The graph ", "type": "text"}, {"bbox": [362, 243, 369, 251], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [370, 239, 500, 256], "score": 1.0, "content": " becomes, via the gluing", "type": "text"}], "index": 7}, {"bbox": [110, 256, 501, 269], "spans": [{"bbox": [110, 256, 378, 269], "score": 1.0, "content": "quotient map, a regular 4-valent graph denoted by ", "type": "text"}, {"bbox": [378, 257, 388, 266], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [388, 256, 462, 269], "score": 1.0, "content": " embedded in ", "type": "text"}, {"bbox": [462, 257, 475, 267], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [476, 256, 501, 269], "score": 1.0, "content": ". Its", "type": "text"}], "index": 8}, {"bbox": [108, 267, 501, 285], "spans": [{"bbox": [108, 267, 351, 285], "score": 1.0, "content": "vertices are the intersection points of the spaces ", "type": "text"}, {"bbox": [352, 271, 410, 283], "score": 0.94, "content": "\\Omega=\\cup_{i=1}^{n}C_{i}", "type": "inline_equation", "height": 12, "width": 58}, {"bbox": [410, 267, 434, 285], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [434, 271, 496, 285], "score": 0.94, "content": "\\Lambda=\\cup_{j=1}^{m}D_{j}", "type": "inline_equation", "height": 14, "width": 62}, {"bbox": [496, 267, 501, 285], "score": 1.0, "content": ";", "type": "text"}], "index": 9}, {"bbox": [109, 284, 501, 298], "spans": [{"bbox": [109, 284, 456, 298], "score": 1.0, "content": "hence they inherit the labelling of the corresponding glued vertices of ", "type": "text"}, {"bbox": [457, 286, 464, 294], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [465, 284, 501, 298], "score": 1.0, "content": ". Since", "type": "text"}], "index": 10}, {"bbox": [110, 299, 500, 312], "spans": [{"bbox": [110, 299, 244, 312], "score": 1.0, "content": "the gluing of the cycles of ", "type": "text"}, {"bbox": [245, 300, 255, 309], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [255, 299, 280, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [281, 300, 293, 309], "score": 0.92, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [294, 299, 500, 312], "score": 1.0, "content": " is invariant with respect to the rotation", "type": "text"}], "index": 11}, {"bbox": [110, 313, 499, 328], "spans": [{"bbox": [110, 318, 122, 326], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 313, 181, 328], "score": 1.0, "content": ", the group ", "type": "text"}, {"bbox": [182, 315, 244, 326], "score": 0.93, "content": "\\mathcal{G}_{n}=<\\rho_{n}>", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [244, 313, 459, 328], "score": 1.0, "content": " naturally induces a cyclic action of order ", "type": "text"}, {"bbox": [459, 318, 466, 323], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [467, 313, 486, 328], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [486, 315, 499, 325], "score": 0.92, "content": "T_{n}^{'}", "type": "inline_equation", "height": 10, "width": 13}], "index": 12}, {"bbox": [110, 328, 500, 342], "spans": [{"bbox": [110, 328, 225, 342], "score": 1.0, "content": "such that the quotient ", "type": "text"}, {"bbox": [225, 329, 284, 341], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 59}, {"bbox": [284, 328, 500, 342], "score": 1.0, "content": " is homeomorphic to a torus. The labelling", "type": "text"}], "index": 13}, {"bbox": [109, 341, 501, 357], "spans": [{"bbox": [109, 341, 195, 357], "score": 1.0, "content": "of the vertices of ", "type": "text"}, {"bbox": [195, 343, 205, 352], "score": 0.91, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [205, 341, 361, 357], "score": 1.0, "content": " is invariant under the rotation", "type": "text"}, {"bbox": [362, 347, 374, 355], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [374, 341, 398, 357], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 343, 469, 355], "score": 0.94, "content": "\\rho_{n}(C_{i})=C_{i+1}", "type": "inline_equation", "height": 12, "width": 71}, {"bbox": [469, 341, 501, 357], "score": 1.0, "content": " (mod", "type": "text"}], "index": 14}, {"bbox": [110, 357, 499, 371], "spans": [{"bbox": [110, 361, 118, 367], "score": 0.69, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [118, 357, 499, 371], "score": 1.0, "content": "). We are going to show that, if the 6-tuple is admissible, this last property", "type": "text"}], "index": 15}, {"bbox": [110, 371, 263, 384], "spans": [{"bbox": [110, 371, 249, 384], "score": 1.0, "content": "also holds for the cycles of ", "type": "text"}, {"bbox": [249, 372, 259, 381], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [259, 371, 263, 384], "score": 1.0, "content": ".", "type": "text"}], "index": 16}], "index": 11.5, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [108, 239, 501, 384]}, {"type": "text", "bbox": [109, 396, 500, 439], "lines": [{"bbox": [108, 395, 498, 414], "spans": [{"bbox": [108, 395, 208, 414], "score": 1.0, "content": "Lemma 1 a) Let ", "type": "text"}, {"bbox": [209, 399, 308, 411], "score": 0.92, "content": "\\sigma\\;=\\;(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 99}, {"bbox": [308, 395, 487, 414], "score": 1.0, "content": " be an admissible 6-tuple. Then ", "type": "text"}, {"bbox": [487, 403, 498, 411], "score": 0.86, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 11}], "index": 17}, {"bbox": [111, 412, 499, 426], "spans": [{"bbox": [111, 412, 344, 426], "score": 1.0, "content": "induces a cyclic permutation on the curves of ", "type": "text"}, {"bbox": [344, 414, 354, 423], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [355, 412, 407, 426], "score": 1.0, "content": ". Thus, if ", "type": "text"}, {"bbox": [407, 414, 417, 423], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [417, 412, 485, 426], "score": 1.0, "content": " is a cycle of ", "type": "text"}, {"bbox": [486, 415, 496, 423], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [496, 412, 499, 426], "score": 1.0, "content": ",", "type": "text"}], "index": 18}, {"bbox": [110, 426, 285, 441], "spans": [{"bbox": [110, 426, 136, 441], "score": 1.0, "content": "then ", "type": "text"}, {"bbox": [136, 427, 281, 441], "score": 0.93, "content": "{\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)|k=1,\\ldots,n\\}", "type": "inline_equation", "height": 14, "width": 145}, {"bbox": [282, 426, 285, 441], "score": 1.0, "content": ".", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [108, 395, 499, 441]}, {"type": "text", "bbox": [110, 440, 501, 483], "lines": [{"bbox": [127, 441, 501, 455], "spans": [{"bbox": [127, 441, 157, 455], "score": 1.0, "content": "b) If ", "type": "text"}, {"bbox": [157, 442, 226, 455], "score": 0.94, "content": "(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [226, 441, 359, 455], "score": 1.0, "content": " is admissible, then also ", "type": "text"}, {"bbox": [359, 442, 428, 455], "score": 0.92, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [428, 441, 501, 455], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 20}, {"bbox": [111, 456, 501, 470], "spans": [{"bbox": [111, 456, 252, 470], "score": 1.0, "content": "and the Heegaard diagram ", "type": "text"}, {"bbox": [252, 457, 331, 469], "score": 0.9, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 79}, {"bbox": [332, 456, 501, 470], "score": 1.0, "content": " is the quotient of the Heegaard", "type": "text"}], "index": 21}, {"bbox": [111, 469, 309, 486], "spans": [{"bbox": [111, 469, 155, 486], "score": 1.0, "content": "diagram ", "type": "text"}, {"bbox": [155, 471, 235, 484], "score": 0.92, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [235, 469, 290, 486], "score": 1.0, "content": " respect to ", "type": "text"}, {"bbox": [291, 472, 304, 483], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [304, 469, 309, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 22}], "index": 21, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [111, 441, 501, 486]}, {"type": "text", "bbox": [109, 496, 501, 596], "lines": [{"bbox": [126, 497, 501, 512], "spans": [{"bbox": [126, 497, 297, 512], "score": 1.0, "content": "Proof. a) First of all, note that ", "type": "text"}, {"bbox": [298, 499, 352, 511], "score": 0.94, "content": "\\rho_{n}(\\Lambda)=\\Lambda", "type": "inline_equation", "height": 12, "width": 54}, {"bbox": [352, 497, 437, 512], "score": 1.0, "content": "; thus the group ", "type": "text"}, {"bbox": [438, 500, 451, 510], "score": 0.91, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [451, 497, 501, 512], "score": 1.0, "content": " also acts", "type": "text"}], "index": 23}, {"bbox": [110, 513, 500, 526], "spans": [{"bbox": [110, 513, 184, 526], "score": 1.0, "content": "on the spaces ", "type": "text"}, {"bbox": [185, 514, 222, 524], "score": 0.93, "content": "T_{n}\\mathrm{~-~}\\Lambda", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [222, 513, 249, 526], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [249, 514, 258, 523], "score": 0.87, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [258, 513, 380, 526], "score": 1.0, "content": " (and hence on the set ", "type": "text"}, {"bbox": [380, 514, 390, 523], "score": 0.86, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [390, 513, 479, 526], "score": 1.0, "content": "). If the 6-tuple ", "type": "text"}, {"bbox": [479, 517, 486, 523], "score": 0.84, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [487, 513, 500, 526], "score": 1.0, "content": " is", "type": "text"}], "index": 24}, {"bbox": [109, 526, 501, 541], "spans": [{"bbox": [109, 526, 196, 541], "score": 1.0, "content": "admissible, then ", "type": "text"}, {"bbox": [196, 528, 231, 539], "score": 0.93, "content": "T_{n}-\\Lambda", "type": "inline_equation", "height": 11, "width": 35}, {"bbox": [231, 526, 424, 541], "score": 1.0, "content": " is connected, and hence the quotient ", "type": "text"}, {"bbox": [424, 528, 501, 540], "score": 0.93, "content": "(T_{n}-\\Lambda)/\\mathcal{G}_{n}=", "type": "inline_equation", "height": 12, "width": 77}], "index": 25}, {"bbox": [110, 542, 499, 555], "spans": [{"bbox": [110, 542, 183, 555], "score": 0.94, "content": "T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 73}, {"bbox": [183, 542, 403, 555], "score": 1.0, "content": " must be connected too. This implies that ", "type": "text"}, {"bbox": [403, 542, 430, 555], "score": 0.94, "content": "\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [430, 542, 499, 555], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 26}, {"bbox": [110, 556, 500, 569], "spans": [{"bbox": [110, 556, 266, 569], "score": 1.0, "content": "connected component. Since ", "type": "text"}, {"bbox": [267, 557, 275, 566], "score": 0.89, "content": "\\Lambda", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [276, 556, 344, 569], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [344, 560, 352, 566], "score": 0.85, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [352, 556, 500, 569], "score": 1.0, "content": " connected components, the", "type": "text"}], "index": 27}, {"bbox": [110, 570, 500, 583], "spans": [{"bbox": [110, 570, 177, 583], "score": 1.0, "content": "cyclic group ", "type": "text"}, {"bbox": [178, 572, 190, 582], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [191, 570, 241, 583], "score": 1.0, "content": " of order ", "type": "text"}, {"bbox": [241, 575, 249, 580], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [249, 570, 500, 583], "score": 1.0, "content": " defines a simply transitive cyclic action on the", "type": "text"}], "index": 28}, {"bbox": [110, 585, 172, 599], "spans": [{"bbox": [110, 585, 156, 599], "score": 1.0, "content": "cycles of ", "type": "text"}, {"bbox": [157, 587, 167, 595], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [167, 585, 172, 599], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 497, 501, 599]}, {"type": "text", "bbox": [109, 597, 500, 669], "lines": [{"bbox": [127, 599, 499, 613], "spans": [{"bbox": [127, 599, 167, 613], "score": 1.0, "content": "b) Let ", "type": "text"}, {"bbox": [167, 601, 223, 612], "score": 0.93, "content": "C,D\\ \\subset\\ T_{1}", "type": "inline_equation", "height": 11, "width": 56}, {"bbox": [223, 599, 311, 613], "score": 1.0, "content": " the two curves ", "type": "text"}, {"bbox": [311, 600, 369, 613], "score": 0.95, "content": "C\\;=\\;\\Omega/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [369, 599, 398, 613], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [398, 600, 456, 613], "score": 0.95, "content": "D\\,=\\,\\Lambda/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 58}, {"bbox": [456, 599, 499, 613], "score": 1.0, "content": ". Then,", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 247, 627], "score": 1.0, "content": "the two systems of curves ", "type": "text"}, {"bbox": [247, 614, 293, 627], "score": 0.95, "content": "{\\mathcal{C}}=\\{C\\}", "type": "inline_equation", "height": 13, "width": 46}, {"bbox": [293, 613, 320, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [320, 614, 370, 627], "score": 0.94, "content": "\\mathcal{D}=\\{D\\}", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [370, 613, 390, 627], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [390, 615, 402, 626], "score": 0.94, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [402, 613, 500, 627], "score": 1.0, "content": " define a Heegaard", "type": "text"}], "index": 31}, {"bbox": [110, 627, 499, 642], "spans": [{"bbox": [110, 627, 291, 642], "score": 1.0, "content": "diagram of genus one. The graph ", "type": "text"}, {"bbox": [292, 630, 304, 640], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [304, 627, 399, 642], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [400, 629, 499, 641], "score": 0.92, "content": "\\sigma_{1}\\,=\\,(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 99}], "index": 32}, {"bbox": [109, 642, 500, 657], "spans": [{"bbox": [109, 642, 255, 657], "score": 1.0, "content": "is the quotient of the graph ", "type": "text"}, {"bbox": [256, 644, 269, 654], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [270, 642, 362, 657], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [363, 643, 455, 655], "score": 0.92, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [456, 642, 500, 657], "score": 1.0, "content": ", respect", "type": "text"}], "index": 33}, {"bbox": [109, 657, 500, 671], "spans": [{"bbox": [109, 657, 124, 671], "score": 1.0, "content": "to ", "type": "text"}, {"bbox": [125, 658, 138, 669], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [138, 657, 281, 671], "score": 1.0, "content": ". Moreover, the gluings on ", "type": "text"}, {"bbox": [281, 658, 294, 669], "score": 0.92, "content": "\\Gamma_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [295, 657, 424, 671], "score": 1.0, "content": " are invariant respect to ", "type": "text"}, {"bbox": [424, 662, 436, 669], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [437, 657, 500, 671], "score": 1.0, "content": ". Therefore,", "type": "text"}], "index": 34}, {"bbox": [110, 128, 500, 142], "spans": [{"bbox": [110, 128, 187, 142], "score": 1.0, "content": "the gluings on ", "type": "text", "cross_page": true}, {"bbox": [187, 130, 199, 140], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12, "cross_page": true}, {"bbox": [199, 128, 500, 142], "score": 1.0, "content": " give rise to the Heegaard diagram above. This show that", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 142, 501, 156], "spans": [{"bbox": [110, 142, 170, 156], "score": 1.0, "content": "the 6-tuple ", "type": "text", "cross_page": true}, {"bbox": [170, 147, 181, 154], "score": 0.88, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 11, "cross_page": true}, {"bbox": [182, 142, 328, 156], "score": 1.0, "content": " is admissible and obviously ", "type": "text", "cross_page": true}, {"bbox": [329, 143, 407, 155], "score": 0.94, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 78, "cross_page": true}, {"bbox": [408, 142, 501, 156], "score": 1.0, "content": " is the quotient of", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 155, 281, 172], "spans": [{"bbox": [110, 158, 190, 170], "score": 0.93, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 80, "cross_page": true}, {"bbox": [190, 155, 248, 172], "score": 1.0, "content": " respect to ", "type": "text", "cross_page": true}, {"bbox": [248, 158, 261, 169], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13, "cross_page": true}, {"bbox": [261, 155, 281, 172], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 2}], "index": 32, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [109, 599, 500, 671]}]}
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The “open” Heegaard diagram and the Dunwoody manifold associated to the admissible 6-tuple will be denoted by and respectively.
Remark 1. It is easy to see that not all the 6-tuples in are admissible. For example, the 6-tuples , with , give rise to exactly cycles in ; thus, they are not admissible if . The 6-tuples are not admissible if is even, since, in this case, we obtain exactly one cycle , but the cut along it disconnects the torus .
Consider now a 6-tuple . The graph becomes, via the gluing quotient map, a regular 4-valent graph denoted by embedded in . Its vertices are the intersection points of the spaces and ; hence they inherit the labelling of the corresponding glued vertices of . Since the gluing of the cycles of and is invariant with respect to the rotation , the group naturally induces a cyclic action of order on such that the quotient is homeomorphic to a torus. The labelling of the vertices of is invariant under the rotation and (mod ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of .
Lemma 1 a) Let be an admissible 6-tuple. Then induces a cyclic permutation on the curves of . Thus, if is a cycle of , then .
b) If is admissible, then also is admissible and the Heegaard diagram is the quotient of the Heegaard diagram respect to .
Proof. a) First of all, note that ; thus the group also acts on the spaces and (and hence on the set ). If the 6-tuple is admissible, then is connected, and hence the quotient must be connected too. This implies that has a unique connected component. Since has exactly connected components, the cyclic group of order defines a simply transitive cyclic action on the cycles of .
b) Let the two curves and . Then, the two systems of curves and on define a Heegaard diagram of genus one. The graph corresponding to is the quotient of the graph corresponding to , respect to . Moreover, the gluings on are invariant respect to . Therefore, the gluings on give rise to the Heegaard diagram above. This show that the 6-tuple is admissible and obviously is the quotient of respect to .
6
|
<div class="pdf-page">
<p>The “open” Heegaard diagram and the Dunwoody manifold associated to the admissible 6-tuple will be denoted by and respectively.</p>
<p>Remark 1. It is easy to see that not all the 6-tuples in are admissible. For example, the 6-tuples , with , give rise to exactly cycles in ; thus, they are not admissible if . The 6-tuples are not admissible if is even, since, in this case, we obtain exactly one cycle , but the cut along it disconnects the torus .</p>
<p>Consider now a 6-tuple . The graph becomes, via the gluing quotient map, a regular 4-valent graph denoted by embedded in . Its vertices are the intersection points of the spaces and ; hence they inherit the labelling of the corresponding glued vertices of . Since the gluing of the cycles of and is invariant with respect to the rotation , the group naturally induces a cyclic action of order on such that the quotient is homeomorphic to a torus. The labelling of the vertices of is invariant under the rotation and (mod ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of .</p>
<p>Lemma 1 a) Let be an admissible 6-tuple. Then induces a cyclic permutation on the curves of . Thus, if is a cycle of , then .</p>
<p>b) If is admissible, then also is admissible and the Heegaard diagram is the quotient of the Heegaard diagram respect to .</p>
<p>Proof. a) First of all, note that ; thus the group also acts on the spaces and (and hence on the set ). If the 6-tuple is admissible, then is connected, and hence the quotient must be connected too. This implies that has a unique connected component. Since has exactly connected components, the cyclic group of order defines a simply transitive cyclic action on the cycles of .</p>
<p>b) Let the two curves and . Then, the two systems of curves and on define a Heegaard diagram of genus one. The graph corresponding to is the quotient of the graph corresponding to , respect to . Moreover, the gluings on are invariant respect to . Therefore, the gluings on give rise to the Heegaard diagram above. This show that the 6-tuple is admissible and obviously is the quotient of respect to .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="161" data-width="654" data-height="38">The “open” Heegaard diagram and the Dunwoody manifold associated to the admissible 6-tuple will be denoted by and respectively.</p>
<p class="pdf-text" data-x="182" data-y="206" data-width="656" data-height="93">Remark 1. It is easy to see that not all the 6-tuples in are admissible. For example, the 6-tuples , with , give rise to exactly cycles in ; thus, they are not admissible if . The 6-tuples are not admissible if is even, since, in this case, we obtain exactly one cycle , but the cut along it disconnects the torus .</p>
<p class="pdf-text" data-x="182" data-y="307" data-width="654" data-height="188">Consider now a 6-tuple . The graph becomes, via the gluing quotient map, a regular 4-valent graph denoted by embedded in . Its vertices are the intersection points of the spaces and ; hence they inherit the labelling of the corresponding glued vertices of . Since the gluing of the cycles of and is invariant with respect to the rotation , the group naturally induces a cyclic action of order on such that the quotient is homeomorphic to a torus. The labelling of the vertices of is invariant under the rotation and (mod ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of .</p>
<p class="pdf-text" data-x="182" data-y="512" data-width="654" data-height="55">Lemma 1 a) Let be an admissible 6-tuple. Then induces a cyclic permutation on the curves of . Thus, if is a cycle of , then .</p>
<p class="pdf-text" data-x="184" data-y="568" data-width="654" data-height="56">b) If is admissible, then also is admissible and the Heegaard diagram is the quotient of the Heegaard diagram respect to .</p>
<p class="pdf-text" data-x="182" data-y="641" data-width="656" data-height="129">Proof. a) First of all, note that ; thus the group also acts on the spaces and (and hence on the set ). If the 6-tuple is admissible, then is connected, and hence the quotient must be connected too. This implies that has a unique connected component. Since has exactly connected components, the cyclic group of order defines a simply transitive cyclic action on the cycles of .</p>
<p class="pdf-text" data-x="182" data-y="771" data-width="654" data-height="93">b) Let the two curves and . Then, the two systems of curves and on define a Heegaard diagram of genus one. The graph corresponding to is the quotient of the graph corresponding to , respect to . Moreover, the gluings on are invariant respect to . Therefore, the gluings on give rise to the Heegaard diagram above. This show that the 6-tuple is admissible and obviously is the quotient of respect to .</p>
<div class="pdf-discarded" data-x="501" data-y="893" data-width="14" data-height="14" style="opacity: 0.5;">6</div>
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"The “open” Heegaard diagram \\Gamma and the Dunwoody manifold associated",
"to the admissible 6-tuple \\sigma will be denoted by H(\\sigma) and M(\\sigma) respectively.",
"Remark 1. It is easy to see that not all the 6-tuples in \\boldsymbol{S} are admissible. For",
"example, the 6-tuples (a,0,a,1,a,0) , with a\\geq1 , give rise to exactly a cycles",
"in \\mathcal{D} ; thus, they are not admissible if a>1 . The 6-tuples (1,0,c,1,2,0) are",
"not admissible if c is even, since, in this case, we obtain exactly one cycle",
"D_{1} , but the cut along it disconnects the torus T_{1} .",
"Consider now a 6-tuple \\sigma\\,\\in\\,S . The graph \\Gamma becomes, via the gluing",
"quotient map, a regular 4-valent graph denoted by \\Gamma^{\\prime} embedded in T_{n}^{'} . Its",
"vertices are the intersection points of the spaces \\Omega=\\cup_{i=1}^{n}C_{i} and \\Lambda=\\cup_{j=1}^{m}D_{j} ;",
"hence they inherit the labelling of the corresponding glued vertices of \\Gamma . Since",
"the gluing of the cycles of \\mathcal{C}^{\\prime} and \\mathcal{C^{\\prime\\prime}} is invariant with respect to the rotation",
"\\rho_{n} , the group \\mathcal{G}_{n}=<\\rho_{n}> naturally induces a cyclic action of order n on T_{n}^{'}",
"such that the quotient T_{1}=T_{n}/\\mathcal{G}_{n} is homeomorphic to a torus. The labelling",
"of the vertices of \\Gamma^{\\prime} is invariant under the rotation \\rho_{n} and \\rho_{n}(C_{i})=C_{i+1} (mod",
"n ). We are going to show that, if the 6-tuple is admissible, this last property",
"also holds for the cycles of \\mathcal{D} .",
"Lemma 1 a) Let \\sigma\\;=\\;(a,b,c,n,r,s) be an admissible 6-tuple. Then \\rho_{n}",
"induces a cyclic permutation on the curves of \\mathcal{D} . Thus, if D is a cycle of \\mathcal{D} ,",
"then {\\mathcal{D}}=\\{\\rho_{n}^{k-1}(D)|k=1,\\ldots,n\\} .",
"b) If (a,b,c,n,r,s) is admissible, then also (a,b,c,1,r,0) is admissible",
"and the Heegaard diagram H(a,b,c,1,r,0) is the quotient of the Heegaard",
"diagram H(a,b,c,n,r,s) respect to \\mathcal{G}_{n} .",
"Proof. a) First of all, note that \\rho_{n}(\\Lambda)=\\Lambda ; thus the group \\mathcal{G}_{n} also acts",
"on the spaces T_{n}\\mathrm{~-~}\\Lambda and \\Lambda (and hence on the set \\mathcal{D} ). If the 6-tuple \\sigma is",
"admissible, then T_{n}-\\Lambda is connected, and hence the quotient (T_{n}-\\Lambda)/\\mathcal{G}_{n}=",
"T_{n}/\\mathcal{G}_{n}-\\Lambda/\\mathcal{G}_{n} must be connected too. This implies that \\Lambda/\\mathcal{G}_{n} has a unique",
"connected component. Since \\Lambda has exactly n connected components, the",
"cyclic group \\mathcal{G}_{n} of order n defines a simply transitive cyclic action on the",
"cycles of \\mathcal{D} .",
"b) Let C,D\\ \\subset\\ T_{1} the two curves C\\;=\\;\\Omega/\\mathcal{G}_{n} and D\\,=\\,\\Lambda/\\mathcal{G}_{n} . Then,",
"the two systems of curves {\\mathcal{C}}=\\{C\\} and \\mathcal{D}=\\{D\\} on T_{1} define a Heegaard",
"diagram of genus one. The graph \\Gamma_{1} corresponding to \\sigma_{1}\\,=\\,(a,b,c,1,r,0)",
"is the quotient of the graph \\Gamma_{n} corresponding to \\sigma=(a,b,c,n,r,s) , respect",
"to \\mathcal{G}_{n} . Moreover, the gluings on \\Gamma_{n} are invariant respect to \\rho_{n} . Therefore,",
"the gluings on \\Gamma_{1} give rise to the Heegaard diagram above. This show that",
"the 6-tuple \\sigma_{1} is admissible and obviously H(a,b,c,1,r,0) is the quotient of",
"H(a,b,c,n,r,s) respect to \\mathcal{G}_{n} ."
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"Remark 2. More generally, given two positive integer and such that divides , if is admissible, then is admissible too. Moreover, the Heegaard diagram is the quotient of respect to the action of a cyclic group of order .",
"It is easy to see that, for admissible 6-tuples, each cycle in contains vertices with different labels and is composed by exactly arcs of (in fact, horizontal arcs, oblique arcs and vertical arcs).",
"An important consequence of point a) of Lemma is that, if is an ad- missible 6-tuple, the presentation of the fundamental group of induced by the Heegaard diagram is cyclic.",
"To see this, let be the vertex belonging to the cycle and labelled by ; denote by the curve of containing and by the vertex of corresponding to . Orient the arc of the graph containing so that is its first endpoint and orient the curve in accordance with the orientation of this arc. Now, set , for each ; the orientation on induces, via , an orientation also on these curves. Moreover, these orientation on the cycles of induce an orientation on the arcs of the graph belonging to . By orienting the arcs of and in accordance with the fixed orientations of the cycles and , the graph becomes an oriented graph, whose orientation is invariant under the action of the group . Let us define to be canonical this orientation of .",
"Let now be the word obtained by reading the oriented arcs of corresponding to the oriented cycle , starting from the vertex . The letters of are in one-to-one correspondence with the oriented arcs ; more precisely, the letter of corresponding to is if comes out from the cycle and is if comes out from the cycle . Note that the word in the cyclic presentation is obtained by reading the cycle along the given orientation, for (roughly speaking, the automorphism is “geometrically” realized by ).",
"This proves that each admissible 6-tuple uniquely defines, via the asso- ciated Heegaard diagram , a word and a cyclic presentation for the fundamental group of the Dunwoody manifold . Note that the sequence of the exponents in the word , and hence its exponent- sum , only depends on the integers .",
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[{"type": "text", "text": "", "page_idx": 6}, {"type": "text", "text": "Remark 2. More generally, given two positive integer $n$ and $n^{\\prime}$ such that $n^{\\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then $(a,b,c,r,n^{\\prime},b)$ is admissible too. Moreover, the Heegaard diagram ${\\cal H}(a,b,c,r,n^{\\prime},b)$ is the quotient of $H(a,b,c,r,n,b)$ respect to the action of a cyclic group of order $n/n^{\\prime}$ . ", "page_idx": 6}, {"type": "text", "text": "It is easy to see that, for admissible 6-tuples, each cycle in $\\mathcal{D}$ contains $d$ vertices with different labels and is composed by exactly $d$ arcs of $\\Gamma$ (in fact, $\\mathit{2a}$ horizontal arcs, $b$ oblique arcs and $c$ vertical arcs). ", "page_idx": 6}, {"type": "text", "text": "An important consequence of point a) of Lemma $1$ is that, if $\\sigma$ is an admissible 6-tuple, the presentation of the fundamental group of $M(\\sigma)$ induced by the Heegaard diagram $H(\\sigma)$ is cyclic. ", "page_idx": 6}, {"type": "text", "text": "To see this, let $\\upsilon$ be the vertex belonging to the cycle $C_{1}$ and labelled by $a+b+1$ ; denote by $D_{1}$ the curve of $\\mathcal{D}$ containing $v$ and by $v^{\\prime}$ the vertex of $C_{1}^{\\prime}$ corresponding to $v$ . Orient the arc $e^{\\prime}\\in A$ of the graph $\\Gamma$ containing $v^{\\prime}$ so that $v^{\\prime}$ is its first endpoint and orient the curve $D_{1}$ in accordance with the orientation of this arc. Now, set $D_{k}=\\rho_{n}^{k-1}(D_{1})$ , for each $k=1,\\dotsc,n$ ; the orientation on $D_{1}$ induces, via $\\rho_{n}$ , an orientation also on these curves. Moreover, these orientation on the cycles of $\\mathcal{D}$ induce an orientation on the arcs of the graph $\\Gamma$ belonging to $A$ . By orienting the arcs of $C^{\\prime}$ and $C^{\\prime\\prime}$ in accordance with the fixed orientations of the cycles $C_{i}^{\\prime}$ and $C_{i}^{\\prime\\prime}$ , the graph $\\Gamma$ becomes an oriented graph, whose orientation is invariant under the action of the group $\\mathcal{G}_{n}$ . Let us define to be canonical this orientation of $\\Gamma$ . ", "page_idx": 6}, {"type": "text", "text": "Let now $w\\in F_{n}$ be the word obtained by reading the oriented arcs $e_{1}=$ $e^{\\prime},e_{2},\\ldots,e_{d}$ of $\\Gamma$ corresponding to the oriented cycle $D_{1}$ , starting from the vertex $v^{\\prime}$ . The letters of $w$ are in one-to-one correspondence with the oriented arcs $e_{h}$ ; more precisely, the letter of $w$ corresponding to $e_{h}$ is $x_{i}$ if $e_{h}$ comes out from the cycle $C_{i}^{\\prime}$ and is $x_{i}^{-1}$ if $e_{h}$ comes out from the cycle $C_{i-s}^{\\prime\\prime}$ . Note that the word $\\theta_{n}^{k-1}(w)$ in the cyclic presentation $G_{n}(w)$ is obtained by reading the cycle $D_{k}$ along the given orientation, for $1\\leq k\\leq n$ (roughly speaking, the automorphism $\\theta_{n}$ is \u201cgeometrically\u201d realized by $\\rho_{n}$ ). ", "page_idx": 6}, {"type": "text", "text": "This proves that each admissible 6-tuple $\\sigma$ uniquely defines, via the associated Heegaard diagram $H(\\sigma)$ , a word $w=w(\\sigma)$ and a cyclic presentation $G_{n}(w)$ for the fundamental group of the Dunwoody manifold $M(\\sigma)$ . Note that the sequence of the exponents in the word $w(\\sigma)$ , and hence its exponentsum $\\varepsilon_{w(\\sigma)}$ , only depends on the integers $a,b,c,r$ . ", "page_idx": 6}]
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{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 500, 169], "lines": [{"bbox": [110, 128, 500, 142], "spans": [{"bbox": [110, 128, 187, 142], "score": 1.0, "content": "the gluings on ", "type": "text"}, {"bbox": [187, 130, 199, 140], "score": 0.91, "content": "\\Gamma_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [199, 128, 500, 142], "score": 1.0, "content": " give rise to the Heegaard diagram above. This show that", "type": "text"}], "index": 0}, {"bbox": [110, 142, 501, 156], "spans": [{"bbox": [110, 142, 170, 156], "score": 1.0, "content": "the 6-tuple ", "type": "text"}, {"bbox": [170, 147, 181, 154], "score": 0.88, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [182, 142, 328, 156], "score": 1.0, "content": " is admissible and obviously ", "type": "text"}, {"bbox": [329, 143, 407, 155], "score": 0.94, "content": "H(a,b,c,1,r,0)", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [408, 142, 501, 156], "score": 1.0, "content": " is the quotient of", "type": "text"}], "index": 1}, {"bbox": [110, 155, 281, 172], "spans": [{"bbox": [110, 158, 190, 170], "score": 0.93, "content": "H(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 80}, {"bbox": [190, 155, 248, 172], "score": 1.0, "content": " respect to ", "type": "text"}, {"bbox": [248, 158, 261, 169], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [261, 155, 281, 172], "score": 1.0, "content": ".", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [110, 174, 501, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 401, 191], "score": 1.0, "content": "Remark 2. More generally, given two positive integer ", "type": "text"}, {"bbox": [401, 182, 408, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [408, 176, 435, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [436, 178, 446, 187], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 176, 500, 191], "score": 1.0, "content": " such that", "type": "text"}], "index": 3}, {"bbox": [110, 191, 500, 205], "spans": [{"bbox": [110, 193, 120, 202], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [120, 191, 163, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [163, 196, 171, 202], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [171, 191, 188, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [188, 192, 257, 205], "score": 0.94, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [257, 191, 359, 205], "score": 1.0, "content": " is admissible, then ", "type": "text"}, {"bbox": [359, 192, 430, 205], "score": 0.95, "content": "(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [430, 191, 500, 205], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 206, 502, 220], "spans": [{"bbox": [109, 206, 318, 220], "score": 1.0, "content": "too. Moreover, the Heegaard diagram ", "type": "text"}, {"bbox": [319, 207, 401, 219], "score": 0.93, "content": "{\\cal H}(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [401, 206, 502, 220], "score": 1.0, "content": " is the quotient of", "type": "text"}], "index": 5}, {"bbox": [110, 220, 462, 235], "spans": [{"bbox": [110, 221, 189, 234], "score": 0.93, "content": "H(a,b,c,r,n,b)", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [190, 220, 435, 235], "score": 1.0, "content": " respect to the action of a cyclic group of order ", "type": "text"}, {"bbox": [435, 221, 458, 234], "score": 0.94, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [458, 220, 462, 235], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5}, {"type": "text", "bbox": [110, 239, 500, 281], "lines": [{"bbox": [127, 241, 499, 254], "spans": [{"bbox": [127, 241, 433, 254], "score": 1.0, "content": "It is easy to see that, for admissible 6-tuples, each cycle in ", "type": "text"}, {"bbox": [433, 243, 443, 251], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [443, 241, 492, 254], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [493, 243, 499, 251], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 6}], "index": 7}, {"bbox": [110, 255, 500, 269], "spans": [{"bbox": [110, 255, 400, 269], "score": 1.0, "content": "vertices with different labels and is composed by exactly ", "type": "text"}, {"bbox": [401, 257, 407, 266], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [407, 255, 447, 269], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [447, 257, 455, 266], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [455, 255, 500, 269], "score": 1.0, "content": " (in fact,", "type": "text"}], "index": 8}, {"bbox": [110, 270, 384, 284], "spans": [{"bbox": [110, 272, 123, 280], "score": 0.85, "content": "\\mathit{2a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [123, 270, 208, 284], "score": 1.0, "content": " horizontal arcs, ", "type": "text"}, {"bbox": [208, 271, 213, 280], "score": 0.89, "content": "b", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [214, 270, 304, 284], "score": 1.0, "content": " oblique arcs and ", "type": "text"}, {"bbox": [305, 275, 310, 280], "score": 0.89, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [311, 270, 384, 284], "score": 1.0, "content": " vertical arcs).", "type": "text"}], "index": 9}], "index": 8}, {"type": "text", "bbox": [110, 282, 500, 325], "lines": [{"bbox": [127, 285, 499, 297], "spans": [{"bbox": [127, 285, 382, 297], "score": 1.0, "content": "An important consequence of point a) of Lemma ", "type": "text"}, {"bbox": [382, 286, 388, 294], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [388, 285, 443, 297], "score": 1.0, "content": " is that, if ", "type": "text"}, {"bbox": [443, 289, 450, 294], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [451, 285, 499, 297], "score": 1.0, "content": " is an ad-", "type": "text"}], "index": 10}, {"bbox": [110, 298, 500, 312], "spans": [{"bbox": [110, 298, 426, 312], "score": 1.0, "content": "missible 6-tuple, the presentation of the fundamental group of ", "type": "text"}, {"bbox": [427, 299, 456, 312], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [456, 298, 500, 312], "score": 1.0, "content": " induced", "type": "text"}], "index": 11}, {"bbox": [110, 313, 317, 326], "spans": [{"bbox": [110, 313, 243, 326], "score": 1.0, "content": "by the Heegaard diagram ", "type": "text"}, {"bbox": [244, 314, 271, 326], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [271, 313, 317, 326], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [109, 326, 500, 484], "lines": [{"bbox": [127, 327, 499, 341], "spans": [{"bbox": [127, 327, 206, 341], "score": 1.0, "content": "To see this, let ", "type": "text"}, {"bbox": [206, 333, 213, 338], "score": 0.88, "content": "\\upsilon", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 327, 403, 341], "score": 1.0, "content": " be the vertex belonging to the cycle ", "type": "text"}, {"bbox": [403, 329, 417, 340], "score": 0.92, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [417, 327, 499, 341], "score": 1.0, "content": " and labelled by", "type": "text"}], "index": 13}, {"bbox": [110, 342, 500, 356], "spans": [{"bbox": [110, 344, 159, 353], "score": 0.93, "content": "a+b+1", "type": "inline_equation", "height": 9, "width": 49}, {"bbox": [159, 342, 222, 356], "score": 1.0, "content": "; denote by ", "type": "text"}, {"bbox": [222, 344, 237, 354], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [237, 342, 308, 356], "score": 1.0, "content": " the curve of ", "type": "text"}, {"bbox": [309, 344, 318, 352], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [319, 342, 380, 356], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [381, 347, 387, 352], "score": 0.89, "content": "v", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [387, 342, 432, 356], "score": 1.0, "content": " and by ", "type": "text"}, {"bbox": [433, 343, 442, 352], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [442, 342, 500, 356], "score": 1.0, "content": " the vertex", "type": "text"}], "index": 14}, {"bbox": [110, 356, 500, 370], "spans": [{"bbox": [110, 356, 124, 370], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [124, 358, 137, 370], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [137, 356, 232, 370], "score": 1.0, "content": " corresponding to ", "type": "text"}, {"bbox": [232, 361, 239, 367], "score": 0.89, "content": "v", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [239, 356, 327, 370], "score": 1.0, "content": ". Orient the arc ", "type": "text"}, {"bbox": [327, 358, 361, 367], "score": 0.95, "content": "e^{\\prime}\\in A", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [361, 356, 433, 370], "score": 1.0, "content": " of the graph ", "type": "text"}, {"bbox": [434, 358, 441, 367], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [442, 356, 500, 370], "score": 1.0, "content": " containing", "type": "text"}], "index": 15}, {"bbox": [110, 370, 500, 385], "spans": [{"bbox": [110, 372, 119, 381], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 370, 162, 385], "score": 1.0, "content": " so that ", "type": "text"}, {"bbox": [162, 372, 172, 381], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [172, 370, 384, 385], "score": 1.0, "content": " is its first endpoint and orient the curve ", "type": "text"}, {"bbox": [384, 372, 398, 383], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [399, 370, 500, 385], "score": 1.0, "content": " in accordance with", "type": "text"}], "index": 16}, {"bbox": [108, 385, 501, 401], "spans": [{"bbox": [108, 385, 301, 401], "score": 1.0, "content": "the orientation of this arc. Now, set ", "type": "text"}, {"bbox": [301, 385, 379, 398], "score": 0.94, "content": "D_{k}=\\rho_{n}^{k-1}(D_{1})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [379, 385, 430, 401], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [431, 387, 496, 398], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [497, 385, 501, 401], "score": 1.0, "content": ";", "type": "text"}], "index": 17}, {"bbox": [109, 399, 501, 414], "spans": [{"bbox": [109, 399, 208, 414], "score": 1.0, "content": "the orientation on ", "type": "text"}, {"bbox": [208, 402, 223, 412], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [223, 399, 294, 414], "score": 1.0, "content": " induces, via ", "type": "text"}, {"bbox": [294, 405, 306, 412], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [306, 399, 501, 414], "score": 1.0, "content": ", an orientation also on these curves.", "type": "text"}], "index": 18}, {"bbox": [110, 415, 499, 427], "spans": [{"bbox": [110, 415, 338, 427], "score": 1.0, "content": "Moreover, these orientation on the cycles of ", "type": "text"}, {"bbox": [339, 416, 349, 424], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [349, 415, 499, 427], "score": 1.0, "content": " induce an orientation on the", "type": "text"}], "index": 19}, {"bbox": [109, 428, 501, 442], "spans": [{"bbox": [109, 428, 203, 442], "score": 1.0, "content": "arcs of the graph ", "type": "text"}, {"bbox": [203, 430, 211, 439], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [211, 428, 283, 442], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [284, 430, 293, 439], "score": 0.88, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [293, 428, 430, 442], "score": 1.0, "content": ". By orienting the arcs of ", "type": "text"}, {"bbox": [430, 430, 442, 439], "score": 0.91, "content": "C^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [443, 428, 470, 442], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [470, 430, 485, 439], "score": 0.92, "content": "C^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [485, 428, 501, 442], "score": 1.0, "content": " in", "type": "text"}], "index": 20}, {"bbox": [109, 443, 499, 458], "spans": [{"bbox": [109, 443, 377, 458], "score": 1.0, "content": "accordance with the fixed orientations of the cycles ", "type": "text"}, {"bbox": [377, 444, 389, 456], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [390, 443, 416, 458], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 444, 431, 456], "score": 0.92, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [431, 443, 491, 458], "score": 1.0, "content": ", the graph ", "type": "text"}, {"bbox": [491, 445, 499, 454], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "becomes an oriented graph, whose orientation is invariant under the action", "type": "text"}], "index": 22}, {"bbox": [110, 471, 459, 486], "spans": [{"bbox": [110, 471, 176, 486], "score": 1.0, "content": "of the group ", "type": "text"}, {"bbox": [177, 474, 190, 484], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [190, 471, 446, 486], "score": 1.0, "content": ". Let us define to be canonical this orientation of ", "type": "text"}, {"bbox": [446, 474, 453, 482], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [454, 471, 459, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 18}, {"type": "text", "bbox": [109, 485, 501, 599], "lines": [{"bbox": [126, 486, 500, 500], "spans": [{"bbox": [126, 486, 173, 500], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [173, 488, 210, 498], "score": 0.94, "content": "w\\in F_{n}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [210, 486, 475, 500], "score": 1.0, "content": " be the word obtained by reading the oriented arcs ", "type": "text"}, {"bbox": [476, 488, 500, 499], "score": 0.82, "content": "e_{1}=", "type": "inline_equation", "height": 11, "width": 24}], "index": 24}, {"bbox": [110, 500, 501, 516], "spans": [{"bbox": [110, 502, 172, 514], "score": 0.89, "content": "e^{\\prime},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 12, "width": 62}, {"bbox": [173, 500, 190, 516], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [190, 502, 198, 511], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [198, 500, 388, 516], "score": 1.0, "content": " corresponding to the oriented cycle ", "type": "text"}, {"bbox": [388, 502, 403, 513], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [403, 500, 501, 516], "score": 1.0, "content": ", starting from the", "type": "text"}], "index": 25}, {"bbox": [110, 515, 500, 529], "spans": [{"bbox": [110, 515, 144, 529], "score": 1.0, "content": "vertex ", "type": "text"}, {"bbox": [145, 516, 154, 525], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [154, 515, 232, 529], "score": 1.0, "content": ". 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Note", "type": "text"}], "index": 28}, {"bbox": [109, 558, 500, 573], "spans": [{"bbox": [109, 558, 181, 573], "score": 1.0, "content": "that the word ", "type": "text"}, {"bbox": [181, 559, 221, 572], "score": 0.94, "content": "\\theta_{n}^{k-1}(w)", "type": "inline_equation", "height": 13, "width": 40}, {"bbox": [221, 558, 352, 573], "score": 1.0, "content": " in the cyclic presentation ", "type": "text"}, {"bbox": [352, 560, 385, 572], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [386, 558, 500, 573], "score": 1.0, "content": " is obtained by reading", "type": "text"}], "index": 29}, {"bbox": [109, 572, 499, 588], "spans": [{"bbox": [109, 572, 160, 588], "score": 1.0, "content": "the cycle ", "type": "text"}, {"bbox": [160, 575, 175, 586], "score": 0.92, "content": "D_{k}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [176, 572, 344, 588], "score": 1.0, "content": " along the given orientation, for ", "type": "text"}, {"bbox": [345, 575, 399, 585], "score": 0.92, "content": "1\\leq k\\leq n", "type": "inline_equation", "height": 10, "width": 54}, {"bbox": [400, 572, 499, 588], "score": 1.0, "content": " (roughly speaking,", "type": "text"}], "index": 30}, {"bbox": [109, 587, 398, 602], "spans": [{"bbox": [109, 587, 207, 602], "score": 1.0, "content": "the automorphism ", "type": "text"}, {"bbox": [208, 589, 219, 600], "score": 0.92, "content": "\\theta_{n}", "type": "inline_equation", "height": 11, "width": 11}, {"bbox": [219, 587, 377, 602], "score": 1.0, "content": " is \u201cgeometrically\u201d realized by ", "type": "text"}, {"bbox": [377, 592, 389, 600], "score": 0.89, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [389, 587, 398, 602], "score": 1.0, "content": ").", "type": "text"}], "index": 31}], "index": 27.5}, {"type": "text", "bbox": [109, 601, 500, 672], "lines": [{"bbox": [127, 601, 499, 616], "spans": [{"bbox": [127, 601, 336, 616], "score": 1.0, "content": "This proves that each admissible 6-tuple ", "type": "text"}, {"bbox": [336, 607, 343, 612], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [344, 601, 499, 616], "score": 1.0, "content": " uniquely defines, via the asso-", "type": "text"}], "index": 32}, {"bbox": [110, 617, 499, 630], "spans": [{"bbox": [110, 617, 242, 630], "score": 1.0, "content": "ciated Heegaard diagram ", "type": "text"}, {"bbox": [243, 617, 270, 630], "score": 0.94, "content": "H(\\sigma)", "type": "inline_equation", "height": 13, "width": 27}, {"bbox": [270, 617, 316, 630], "score": 1.0, "content": ", a word ", "type": "text"}, {"bbox": [316, 617, 366, 630], "score": 0.95, "content": "w=w(\\sigma)", "type": "inline_equation", "height": 13, "width": 50}, {"bbox": [367, 617, 499, 630], "score": 1.0, "content": " and a cyclic presentation", "type": "text"}], "index": 33}, {"bbox": [110, 631, 500, 644], "spans": [{"bbox": [110, 632, 143, 644], "score": 0.95, "content": "G_{n}(w)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [144, 631, 434, 644], "score": 1.0, "content": " for the fundamental group of the Dunwoody manifold ", "type": "text"}, {"bbox": [434, 632, 463, 644], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [464, 631, 500, 644], "score": 1.0, "content": ". Note", "type": "text"}], "index": 34}, {"bbox": [110, 645, 499, 659], "spans": [{"bbox": [110, 645, 348, 659], "score": 1.0, "content": "that the sequence of the exponents in the word ", "type": "text"}, {"bbox": [348, 646, 373, 658], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [373, 645, 499, 659], "score": 1.0, "content": ", and hence its exponent-", "type": "text"}], "index": 35}, {"bbox": [109, 659, 359, 675], "spans": [{"bbox": [109, 659, 134, 675], "score": 1.0, "content": "sum ", "type": "text"}, {"bbox": [135, 664, 159, 674], "score": 0.92, "content": "\\varepsilon_{w(\\sigma)}", "type": "inline_equation", "height": 10, "width": 24}, {"bbox": [160, 659, 316, 675], "score": 1.0, "content": ", only depends on the integers ", "type": "text"}, {"bbox": [317, 661, 354, 672], "score": 0.94, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 37}, {"bbox": [355, 659, 359, 675], "score": 1.0, "content": ".", "type": "text"}], "index": 36}], "index": 34}], "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": "text", "bbox": [110, 125, 500, 169], "lines": [], "index": 1, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [110, 128, 501, 172], "lines_deleted": true}, {"type": "text", "bbox": [110, 174, 501, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 401, 191], "score": 1.0, "content": "Remark 2. More generally, given two positive integer ", "type": "text"}, {"bbox": [401, 182, 408, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [408, 176, 435, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [436, 178, 446, 187], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [446, 176, 500, 191], "score": 1.0, "content": " such that", "type": "text"}], "index": 3}, {"bbox": [110, 191, 500, 205], "spans": [{"bbox": [110, 193, 120, 202], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [120, 191, 163, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [163, 196, 171, 202], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [171, 191, 188, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [188, 192, 257, 205], "score": 0.94, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 69}, {"bbox": [257, 191, 359, 205], "score": 1.0, "content": " is admissible, then ", "type": "text"}, {"bbox": [359, 192, 430, 205], "score": 0.95, "content": "(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [430, 191, 500, 205], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 206, 502, 220], "spans": [{"bbox": [109, 206, 318, 220], "score": 1.0, "content": "too. Moreover, the Heegaard diagram ", "type": "text"}, {"bbox": [319, 207, 401, 219], "score": 0.93, "content": "{\\cal H}(a,b,c,r,n^{\\prime},b)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [401, 206, 502, 220], "score": 1.0, "content": " is the quotient of", "type": "text"}], "index": 5}, {"bbox": [110, 220, 462, 235], "spans": [{"bbox": [110, 221, 189, 234], "score": 0.93, "content": "H(a,b,c,r,n,b)", "type": "inline_equation", "height": 13, "width": 79}, {"bbox": [190, 220, 435, 235], "score": 1.0, "content": " respect to the action of a cyclic group of order ", "type": "text"}, {"bbox": [435, 221, 458, 234], "score": 0.94, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [458, 220, 462, 235], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [109, 176, 502, 235]}, {"type": "text", "bbox": [110, 239, 500, 281], "lines": [{"bbox": [127, 241, 499, 254], "spans": [{"bbox": [127, 241, 433, 254], "score": 1.0, "content": "It is easy to see that, for admissible 6-tuples, each cycle in ", "type": "text"}, {"bbox": [433, 243, 443, 251], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [443, 241, 492, 254], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [493, 243, 499, 251], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 6}], "index": 7}, {"bbox": [110, 255, 500, 269], "spans": [{"bbox": [110, 255, 400, 269], "score": 1.0, "content": "vertices with different labels and is composed by exactly ", "type": "text"}, {"bbox": [401, 257, 407, 266], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [407, 255, 447, 269], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [447, 257, 455, 266], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 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[{"bbox": [127, 285, 382, 297], "score": 1.0, "content": "An important consequence of point a) of Lemma ", "type": "text"}, {"bbox": [382, 286, 388, 294], "score": 0.44, "content": "1", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [388, 285, 443, 297], "score": 1.0, "content": " is that, if ", "type": "text"}, {"bbox": [443, 289, 450, 294], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [451, 285, 499, 297], "score": 1.0, "content": " is an ad-", "type": "text"}], "index": 10}, {"bbox": [110, 298, 500, 312], "spans": [{"bbox": [110, 298, 426, 312], "score": 1.0, "content": "missible 6-tuple, the presentation of the fundamental group of ", "type": "text"}, {"bbox": [427, 299, 456, 312], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 13, "width": 29}, {"bbox": [456, 298, 500, 312], "score": 1.0, "content": " induced", "type": "text"}], "index": 11}, {"bbox": [110, 313, 317, 326], "spans": [{"bbox": [110, 313, 243, 326], "score": 1.0, "content": "by the Heegaard diagram ", "type": "text"}, {"bbox": [244, 314, 271, 326], "score": 0.95, "content": "H(\\sigma)", "type": "inline_equation", "height": 12, "width": 27}, {"bbox": [271, 313, 317, 326], "score": 1.0, "content": " is cyclic.", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [110, 285, 500, 326]}, {"type": "text", "bbox": [109, 326, 500, 484], "lines": [{"bbox": [127, 327, 499, 341], "spans": [{"bbox": [127, 327, 206, 341], "score": 1.0, "content": "To see this, let ", "type": "text"}, {"bbox": [206, 333, 213, 338], "score": 0.88, "content": "\\upsilon", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [213, 327, 403, 341], "score": 1.0, "content": " be the vertex belonging to the cycle ", "type": "text"}, {"bbox": [403, 329, 417, 340], "score": 0.92, "content": "C_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [417, 327, 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Orient the arc ", "type": "text"}, {"bbox": [327, 358, 361, 367], "score": 0.95, "content": "e^{\\prime}\\in A", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [361, 356, 433, 370], "score": 1.0, "content": " of the graph ", "type": "text"}, {"bbox": [434, 358, 441, 367], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [442, 356, 500, 370], "score": 1.0, "content": " containing", "type": "text"}], "index": 15}, {"bbox": [110, 370, 500, 385], "spans": [{"bbox": [110, 372, 119, 381], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [119, 370, 162, 385], "score": 1.0, "content": " so that ", "type": "text"}, {"bbox": [162, 372, 172, 381], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [172, 370, 384, 385], "score": 1.0, "content": " is its first endpoint and orient the curve ", "type": "text"}, {"bbox": [384, 372, 398, 383], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [399, 370, 500, 385], "score": 1.0, "content": " in accordance with", "type": "text"}], "index": 16}, {"bbox": [108, 385, 501, 401], "spans": [{"bbox": [108, 385, 301, 401], "score": 1.0, "content": "the orientation of this arc. Now, set ", "type": "text"}, {"bbox": [301, 385, 379, 398], "score": 0.94, "content": "D_{k}=\\rho_{n}^{k-1}(D_{1})", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [379, 385, 430, 401], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [431, 387, 496, 398], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 11, "width": 65}, {"bbox": [497, 385, 501, 401], "score": 1.0, "content": ";", "type": "text"}], "index": 17}, {"bbox": [109, 399, 501, 414], "spans": [{"bbox": [109, 399, 208, 414], "score": 1.0, "content": "the orientation on ", "type": "text"}, {"bbox": [208, 402, 223, 412], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [223, 399, 294, 414], "score": 1.0, "content": " induces, via ", "type": "text"}, {"bbox": [294, 405, 306, 412], "score": 0.91, "content": "\\rho_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [306, 399, 501, 414], "score": 1.0, "content": ", an orientation also on these curves.", "type": "text"}], "index": 18}, {"bbox": [110, 415, 499, 427], "spans": [{"bbox": [110, 415, 338, 427], "score": 1.0, "content": "Moreover, these orientation on the cycles of ", "type": "text"}, {"bbox": [339, 416, 349, 424], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [349, 415, 499, 427], "score": 1.0, "content": " induce an orientation on the", "type": "text"}], "index": 19}, {"bbox": [109, 428, 501, 442], "spans": [{"bbox": [109, 428, 203, 442], "score": 1.0, "content": "arcs of the graph ", "type": "text"}, {"bbox": [203, 430, 211, 439], "score": 0.88, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [211, 428, 283, 442], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [284, 430, 293, 439], "score": 0.88, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [293, 428, 430, 442], "score": 1.0, "content": ". By orienting the arcs of ", "type": "text"}, {"bbox": [430, 430, 442, 439], "score": 0.91, "content": "C^{\\prime}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [443, 428, 470, 442], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [470, 430, 485, 439], "score": 0.92, "content": "C^{\\prime\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [485, 428, 501, 442], "score": 1.0, "content": " in", "type": "text"}], "index": 20}, {"bbox": [109, 443, 499, 458], "spans": [{"bbox": [109, 443, 377, 458], "score": 1.0, "content": "accordance with the fixed orientations of the cycles ", "type": "text"}, {"bbox": [377, 444, 389, 456], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [390, 443, 416, 458], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [416, 444, 431, 456], "score": 0.92, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [431, 443, 491, 458], "score": 1.0, "content": ", the graph ", "type": "text"}, {"bbox": [491, 445, 499, 454], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}], "index": 21}, {"bbox": [109, 457, 500, 471], "spans": [{"bbox": [109, 457, 500, 471], "score": 1.0, "content": "becomes an oriented graph, whose orientation is invariant under the action", "type": "text"}], "index": 22}, {"bbox": [110, 471, 459, 486], "spans": [{"bbox": [110, 471, 176, 486], "score": 1.0, "content": "of the group ", "type": "text"}, {"bbox": [177, 474, 190, 484], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [190, 471, 446, 486], "score": 1.0, "content": ". Let us define to be canonical this orientation of ", "type": "text"}, {"bbox": [446, 474, 453, 482], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [454, 471, 459, 486], "score": 1.0, "content": ".", "type": "text"}], "index": 23}], "index": 18, "page_num": "page_6", "page_size": [612.0, 792.0], "bbox_fs": [108, 327, 501, 486]}, {"type": "text", "bbox": [109, 485, 501, 599], "lines": [{"bbox": [126, 486, 500, 500], "spans": [{"bbox": [126, 486, 173, 500], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [173, 488, 210, 498], "score": 0.94, "content": "w\\in F_{n}", "type": "inline_equation", "height": 10, "width": 37}, {"bbox": [210, 486, 475, 500], "score": 1.0, "content": " be the word obtained by reading the oriented arcs ", "type": "text"}, {"bbox": [476, 488, 500, 499], "score": 0.82, "content": "e_{1}=", "type": "inline_equation", "height": 11, "width": 24}], "index": 24}, {"bbox": [110, 500, 501, 516], "spans": [{"bbox": [110, 502, 172, 514], "score": 0.89, "content": "e^{\\prime},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 12, "width": 62}, {"bbox": [173, 500, 190, 516], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [190, 502, 198, 511], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [198, 500, 388, 516], "score": 1.0, "content": " corresponding to the oriented cycle ", "type": "text"}, {"bbox": [388, 502, 403, 513], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [403, 500, 501, 516], "score": 1.0, "content": ", starting from the", "type": "text"}], "index": 25}, {"bbox": [110, 515, 500, 529], "spans": [{"bbox": [110, 515, 144, 529], "score": 1.0, "content": "vertex ", "type": "text"}, {"bbox": [145, 516, 154, 525], "score": 0.9, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [154, 515, 232, 529], "score": 1.0, "content": ". 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Remark 2. More generally, given two positive integer and such that divides , if is admissible, then is admissible too. Moreover, the Heegaard diagram is the quotient of respect to the action of a cyclic group of order .
It is easy to see that, for admissible 6-tuples, each cycle in contains vertices with different labels and is composed by exactly arcs of (in fact, horizontal arcs, oblique arcs and vertical arcs).
An important consequence of point a) of Lemma is that, if is an ad- missible 6-tuple, the presentation of the fundamental group of induced by the Heegaard diagram is cyclic.
To see this, let be the vertex belonging to the cycle and labelled by ; denote by the curve of containing and by the vertex of corresponding to . Orient the arc of the graph containing so that is its first endpoint and orient the curve in accordance with the orientation of this arc. Now, set , for each ; the orientation on induces, via , an orientation also on these curves. Moreover, these orientation on the cycles of induce an orientation on the arcs of the graph belonging to . By orienting the arcs of and in accordance with the fixed orientations of the cycles and , the graph becomes an oriented graph, whose orientation is invariant under the action of the group . Let us define to be canonical this orientation of .
Let now be the word obtained by reading the oriented arcs of corresponding to the oriented cycle , starting from the vertex . The letters of are in one-to-one correspondence with the oriented arcs ; more precisely, the letter of corresponding to is if comes out from the cycle and is if comes out from the cycle . Note that the word in the cyclic presentation is obtained by reading the cycle along the given orientation, for (roughly speaking, the automorphism is “geometrically” realized by ).
This proves that each admissible 6-tuple uniquely defines, via the asso- ciated Heegaard diagram , a word and a cyclic presentation for the fundamental group of the Dunwoody manifold . Note that the sequence of the exponents in the word , and hence its exponent- sum , only depends on the integers .
7
|
<div class="pdf-page">
<p>Remark 2. More generally, given two positive integer and such that divides , if is admissible, then is admissible too. Moreover, the Heegaard diagram is the quotient of respect to the action of a cyclic group of order .</p>
<p>It is easy to see that, for admissible 6-tuples, each cycle in contains vertices with different labels and is composed by exactly arcs of (in fact, horizontal arcs, oblique arcs and vertical arcs).</p>
<p>An important consequence of point a) of Lemma is that, if is an ad- missible 6-tuple, the presentation of the fundamental group of induced by the Heegaard diagram is cyclic.</p>
<p>To see this, let be the vertex belonging to the cycle and labelled by ; denote by the curve of containing and by the vertex of corresponding to . Orient the arc of the graph containing so that is its first endpoint and orient the curve in accordance with the orientation of this arc. Now, set , for each ; the orientation on induces, via , an orientation also on these curves. Moreover, these orientation on the cycles of induce an orientation on the arcs of the graph belonging to . By orienting the arcs of and in accordance with the fixed orientations of the cycles and , the graph becomes an oriented graph, whose orientation is invariant under the action of the group . Let us define to be canonical this orientation of .</p>
<p>Let now be the word obtained by reading the oriented arcs of corresponding to the oriented cycle , starting from the vertex . The letters of are in one-to-one correspondence with the oriented arcs ; more precisely, the letter of corresponding to is if comes out from the cycle and is if comes out from the cycle . Note that the word in the cyclic presentation is obtained by reading the cycle along the given orientation, for (roughly speaking, the automorphism is “geometrically” realized by ).</p>
<p>This proves that each admissible 6-tuple uniquely defines, via the asso- ciated Heegaard diagram , a word and a cyclic presentation for the fundamental group of the Dunwoody manifold . Note that the sequence of the exponents in the word , and hence its exponent- sum , only depends on the integers .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="224" data-width="654" data-height="75">Remark 2. More generally, given two positive integer and such that divides , if is admissible, then is admissible too. Moreover, the Heegaard diagram is the quotient of respect to the action of a cyclic group of order .</p>
<p class="pdf-text" data-x="184" data-y="309" data-width="652" data-height="54">It is easy to see that, for admissible 6-tuples, each cycle in contains vertices with different labels and is composed by exactly arcs of (in fact, horizontal arcs, oblique arcs and vertical arcs).</p>
<p class="pdf-text" data-x="184" data-y="364" data-width="652" data-height="56">An important consequence of point a) of Lemma is that, if is an ad- missible 6-tuple, the presentation of the fundamental group of induced by the Heegaard diagram is cyclic.</p>
<p class="pdf-text" data-x="182" data-y="421" data-width="654" data-height="204">To see this, let be the vertex belonging to the cycle and labelled by ; denote by the curve of containing and by the vertex of corresponding to . Orient the arc of the graph containing so that is its first endpoint and orient the curve in accordance with the orientation of this arc. Now, set , for each ; the orientation on induces, via , an orientation also on these curves. Moreover, these orientation on the cycles of induce an orientation on the arcs of the graph belonging to . By orienting the arcs of and in accordance with the fixed orientations of the cycles and , the graph becomes an oriented graph, whose orientation is invariant under the action of the group . Let us define to be canonical this orientation of .</p>
<p class="pdf-text" data-x="182" data-y="627" data-width="656" data-height="147">Let now be the word obtained by reading the oriented arcs of corresponding to the oriented cycle , starting from the vertex . The letters of are in one-to-one correspondence with the oriented arcs ; more precisely, the letter of corresponding to is if comes out from the cycle and is if comes out from the cycle . Note that the word in the cyclic presentation is obtained by reading the cycle along the given orientation, for (roughly speaking, the automorphism is “geometrically” realized by ).</p>
<p class="pdf-text" data-x="182" data-y="777" data-width="654" data-height="91">This proves that each admissible 6-tuple uniquely defines, via the asso- ciated Heegaard diagram , a word and a cyclic presentation for the fundamental group of the Dunwoody manifold . Note that the sequence of the exponents in the word , and hence its exponent- sum , only depends on the integers .</p>
<div class="pdf-discarded" data-x="501" data-y="893" data-width="16" data-height="14" style="opacity: 0.5;">7</div>
</div>
|
Remark 2. More generally, given two positive integer $n$ and $n^{\prime}$ such that $n^{\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then $(a,b,c,r,n^{\prime},b)$ is admissible too. Moreover, the Heegaard diagram ${\cal H}(a,b,c,r,n^{\prime},b)$ is the quotient of $H(a,b,c,r,n,b)$ respect to the action of a cyclic group of order $n/n^{\prime}$ .
It is easy to see that, for admissible 6-tuples, each cycle in $\mathcal{D}$ contains $d$ vertices with different labels and is composed by exactly $d$ arcs of $\Gamma$ (in fact, $\mathit{2a}$ horizontal arcs, $b$ oblique arcs and $c$ vertical arcs).
An important consequence of point a) of Lemma $1$ is that, if $\sigma$ is an admissible 6-tuple, the presentation of the fundamental group of $M(\sigma)$ induced by the Heegaard diagram $H(\sigma)$ is cyclic.
To see this, let $\upsilon$ be the vertex belonging to the cycle $C_{1}$ and labelled by $a+b+1$ ; denote by $D_{1}$ the curve of $\mathcal{D}$ containing $v$ and by $v^{\prime}$ the vertex of $C_{1}^{\prime}$ corresponding to $v$ . Orient the arc $e^{\prime}\in A$ of the graph $\Gamma$ containing $v^{\prime}$ so that $v^{\prime}$ is its first endpoint and orient the curve $D_{1}$ in accordance with the orientation of this arc. Now, set $D_{k}=\rho_{n}^{k-1}(D_{1})$ , for each $k=1,\dotsc,n$ ; the orientation on $D_{1}$ induces, via $\rho_{n}$ , an orientation also on these curves. Moreover, these orientation on the cycles of $\mathcal{D}$ induce an orientation on the arcs of the graph $\Gamma$ belonging to $A$ . By orienting the arcs of $C^{\prime}$ and $C^{\prime\prime}$ in accordance with the fixed orientations of the cycles $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ , the graph $\Gamma$ becomes an oriented graph, whose orientation is invariant under the action of the group $\mathcal{G}_{n}$ . Let us define to be canonical this orientation of $\Gamma$ .
Let now $w\in F_{n}$ be the word obtained by reading the oriented arcs $e_{1}=$ $e^{\prime},e_{2},\ldots,e_{d}$ of $\Gamma$ corresponding to the oriented cycle $D_{1}$ , starting from the vertex $v^{\prime}$ . The letters of $w$ are in one-to-one correspondence with the oriented arcs $e_{h}$ ; more precisely, the letter of $w$ corresponding to $e_{h}$ is $x_{i}$ if $e_{h}$ comes out from the cycle $C_{i}^{\prime}$ and is $x_{i}^{-1}$ if $e_{h}$ comes out from the cycle $C_{i-s}^{\prime\prime}$ . Note that the word $\theta_{n}^{k-1}(w)$ in the cyclic presentation $G_{n}(w)$ is obtained by reading the cycle $D_{k}$ along the given orientation, for $1\leq k\leq n$ (roughly speaking, the automorphism $\theta_{n}$ is “geometrically” realized by $\rho_{n}$ ).
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"Remark 2. More generally, given two positive integer n and n^{\\prime} such that",
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"Let now w\\in F_{n} be the word obtained by reading the oriented arcs e_{1}=",
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"Let us consider now the Dunwoody manifolds with (and hence ), which arises from a genus one Heegaard diagram.",
"Proposition 2 Let be an admissible 6-tuple and let be the associated word. Then the Dunwoody manifold is homeomorphic to:",
"i) , if ; ii) , if ; iii) a lens space with , if .",
"Proof. From we obtain . Thus, .",
"Example 1. The Dunwoody manifolds , and , with coprime, are homeomorphic to , and to the lens space , respectively. Moreover, all lens spaces also arise with ; in fact, for each , is homeomorphic with the lens space , if and are coprime, since it is easy to see that can be transformed into the canonical genus one Heegaard diagram of by Singer moves of type IB.",
"Let us see now how the admissibility conditions for the 6-tuples of can be given in terms of labelling of the vertices of , belonging to the curve . With this aim, consider the following properties for a 6-tuple :",
"(i’) the set of the labels of the vertices belonging to the cycle is the set of all integers from 1 to ;",
"(ii’) the vertices of the cycle have different labels.",
"It is easy to see that, if a 6-tuple is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple satisfies (i’) and (ii’), then the curves , with , which are all different from each other, are precisely the curves of . Thus, has exactly curves and they are cyclically permutated by . However, this does not imply that is admissible; for example, the 6-tuple satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for , property (ii’) always holds, while condition (i’) holds if and only if has a unique cycle.",
"If a 6-tuple satisfies property (i’), then acts transitively (not necessarily simply) on , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of and on the graph , by extending, via , the orientation of to the other cycles of .",
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[{"type": "text", "text": "Let us consider now the Dunwoody manifolds $M(a,b,c,n,r,s)$ with $n=1$ (and hence $s=0$ ), which arises from a genus one Heegaard diagram. ", "page_idx": 7}, {"type": "text", "text": "Proposition 2 Let $(a,b,c,1,r,0)$ be an admissible 6-tuple and let $w\\ =$ $w(a,b,c,1,r,0)$ be the associated word. Then the Dunwoody manifold $M(a,b,c,1,r,0)$ is homeomorphic to: ", "page_idx": 7}, {"type": "text", "text": "i) $\\mathbf{S^{3}}$ , if $\\varepsilon_{w}=\\pm1$ ; \nii) $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ , if $\\varepsilon_{w}=0$ ; \niii) a lens space $L(\\alpha,\\beta)$ with $\\alpha=|\\varepsilon_{w}|$ , if $\\left|\\varepsilon_{w}\\right|>1$ . ", "page_idx": 7}, {"type": "text", "text": "Proof. From $n=1$ we obtain $w\\in F_{1}\\cong\\mathbf{Z}\\cong<x|\\emptyset>$ . Thus, $\\pi_{1}(M)\\cong$ $G_{1}(w)\\cong<x|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{|\\varepsilon_{w}|}$ . ", "page_idx": 7}, {"type": "text", "text": "Example 1. The Dunwoody manifolds $M(0,0,1,1,0,0)$ , $M(1,0,0,1,1,0)$ and $M(0,0,c,1,r,0)$ , with $c,r$ coprime, are homeomorphic to $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}\\times S^{2}}$ and to the lens space $L(c,r)$ , respectively. Moreover, all lens spaces also arise with $a\\neq0$ ; in fact, for each $a>0$ , $M(a,0,c,1,a,0)$ is homeomorphic with the lens space $L(c,a)$ , if $a$ and $c$ are coprime, since it is easy to see that $H(a,0,c,1,a,0)$ can be transformed into the canonical genus one Heegaard diagram of $L(c,a)$ by Singer moves of type IB. ", "page_idx": 7}, {"type": "text", "text": "Let us see now how the admissibility conditions for the 6-tuples of $\\boldsymbol{S}$ can be given in terms of labelling of the vertices of $\\Gamma^{\\prime}$ , belonging to the curve $D_{1}\\in\\mathcal{D}$ . With this aim, consider the following properties for a 6-tuple $\\sigma\\in S$ : ", "page_idx": 7}, {"type": "text", "text": "(i\u2019) the set of the labels of the vertices belonging to the cycle $D_{1}$ is the set of all integers from 1 to $d$ ; ", "page_idx": 7}, {"type": "text", "text": "(ii\u2019) the vertices of the cycle $D_{1}$ have different labels. ", "page_idx": 7}, {"type": "text", "text": "It is easy to see that, if a 6-tuple $\\sigma\\in S$ is admissible, then it satisfies (i\u2019) and (ii\u2019). On the other side, if a 6-tuple $\\sigma\\,\\in\\,S$ satisfies (i\u2019) and (ii\u2019), then the curves $\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}$ , with $k\\,=\\,1,\\ldots,n$ , which are all different from each other, are precisely the curves of $\\mathcal{D}$ . Thus, $\\mathcal{D}$ has exactly $n$ curves and they are cyclically permutated by $\\rho_{n}$ . However, this does not imply that $\\sigma$ is admissible; for example, the 6-tuple $(1,0,2,1,2,0)$ satisfies (i\u2019) and (ii\u2019), but it is not admissible (see Remark 1). Note that, for $n=1$ , property (ii\u2019) always holds, while condition (i\u2019) holds if and only if $\\mathcal{D}$ has a unique cycle. ", "page_idx": 7}, {"type": "text", "text": "If a 6-tuple satisfies property (i\u2019), then $\\mathcal{G}_{n}$ acts transitively (not necessarily simply) on $\\mathcal{D}$ , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of $\\mathcal{D}$ and on the graph $\\Gamma$ , by extending, via $\\rho_{n}$ , the orientation of $D_{1}$ to the other cycles of $\\mathcal{D}$ . ", "page_idx": 7}]
|
{"preproc_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [126, 127, 499, 142], "spans": [{"bbox": [126, 127, 359, 142], "score": 1.0, "content": "Let us consider now the Dunwoody manifolds ", "type": "text"}, {"bbox": [360, 129, 441, 141], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [442, 127, 469, 142], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [470, 130, 499, 138], "score": 0.9, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}], "index": 0}, {"bbox": [111, 142, 462, 156], "spans": [{"bbox": [111, 142, 169, 156], "score": 1.0, "content": "(and hence ", "type": "text"}, {"bbox": [170, 144, 198, 153], "score": 0.87, "content": "s=0", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [198, 142, 462, 156], "score": 1.0, "content": "), which arises from a genus one Heegaard diagram.", "type": "text"}], "index": 1}], "index": 0.5}, {"type": "text", "bbox": [109, 166, 501, 210], "lines": [{"bbox": [110, 169, 500, 183], "spans": [{"bbox": [110, 169, 218, 183], "score": 1.0, "content": "Proposition 2 Let ", "type": "text"}, {"bbox": [218, 169, 286, 183], "score": 0.91, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [287, 169, 472, 183], "score": 1.0, "content": " be an admissible 6-tuple and let ", "type": "text"}, {"bbox": [472, 170, 500, 182], "score": 0.79, "content": "w\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 2}, {"bbox": [110, 183, 500, 198], "spans": [{"bbox": [110, 183, 188, 197], "score": 0.9, "content": "w(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [188, 184, 500, 198], "score": 1.0, "content": " be the associated word. Then the Dunwoody manifold", "type": "text"}], "index": 3}, {"bbox": [110, 198, 300, 212], "spans": [{"bbox": [110, 198, 191, 212], "score": 0.91, "content": "M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [191, 198, 300, 212], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 4}], "index": 3}, {"type": "text", "bbox": [126, 211, 385, 255], "lines": [{"bbox": [127, 212, 221, 227], "spans": [{"bbox": [127, 212, 139, 227], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 212, 154, 225], "score": 0.63, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [154, 212, 171, 227], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [171, 213, 216, 225], "score": 0.9, "content": "\\varepsilon_{w}=\\pm1", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [217, 212, 221, 227], "score": 1.0, "content": ";", "type": "text"}], "index": 5}, {"bbox": [127, 226, 243, 241], "spans": [{"bbox": [127, 226, 143, 241], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 226, 185, 239], "score": 0.88, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [185, 226, 202, 241], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 227, 238, 240], "score": 0.9, "content": "\\varepsilon_{w}=0", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [238, 226, 243, 241], "score": 1.0, "content": ";", "type": "text"}], "index": 6}, {"bbox": [127, 241, 384, 255], "spans": [{"bbox": [127, 242, 211, 255], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 241, 249, 255], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 242, 278, 255], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 241, 321, 255], "score": 0.9, "content": "\\alpha=|\\varepsilon_{w}|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [322, 242, 339, 255], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [340, 243, 381, 255], "score": 0.85, "content": "\\left|\\varepsilon_{w}\\right|>1", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [381, 242, 384, 255], "score": 1.0, "content": ".", "type": "text"}], "index": 7}], "index": 6}, {"type": "text", "bbox": [109, 266, 500, 296], "lines": [{"bbox": [126, 269, 501, 284], "spans": [{"bbox": [126, 269, 201, 284], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [201, 270, 232, 280], "score": 0.88, "content": "n=1", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [232, 269, 290, 284], "score": 1.0, "content": " we obtain ", "type": "text"}, {"bbox": [290, 270, 409, 282], "score": 0.92, "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x|\\emptyset>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [409, 269, 452, 284], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [452, 270, 501, 283], "score": 0.9, "content": "\\pi_{1}(M)\\cong", "type": "inline_equation", "height": 13, "width": 49}], "index": 8}, {"bbox": [110, 281, 261, 300], "spans": [{"bbox": [110, 282, 243, 298], "score": 0.89, "content": "G_{1}(w)\\cong<x|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{|\\varepsilon_{w}|}", "type": "inline_equation", "height": 16, "width": 133}, {"bbox": [243, 281, 261, 300], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8.5}, {"type": "text", "bbox": [109, 300, 500, 403], "lines": [{"bbox": [110, 304, 499, 318], "spans": [{"bbox": [110, 304, 324, 318], "score": 1.0, "content": "Example 1. The Dunwoody manifolds ", "type": "text"}, {"bbox": [324, 304, 408, 317], "score": 0.89, "content": "M(0,0,1,1,0,0)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [408, 304, 415, 318], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [416, 305, 499, 317], "score": 0.91, "content": "M(1,0,0,1,1,0)", "type": "inline_equation", "height": 12, "width": 83}], "index": 10}, {"bbox": [109, 318, 499, 333], "spans": [{"bbox": [109, 318, 133, 333], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 318, 216, 332], "score": 0.9, "content": "M(0,0,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [216, 318, 250, 333], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [251, 323, 267, 331], "score": 0.78, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [268, 318, 435, 333], "score": 1.0, "content": " coprime, are homeomorphic to ", "type": "text"}, {"bbox": [436, 319, 449, 329], "score": 0.85, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 318, 456, 333], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [457, 319, 499, 330], "score": 0.89, "content": "\\mathbf{S^{1}\\times S^{2}}", "type": "inline_equation", "height": 11, "width": 42}], "index": 11}, {"bbox": [110, 333, 500, 347], "spans": [{"bbox": [110, 333, 218, 347], "score": 1.0, "content": "and to the lens space ", "type": "text"}, {"bbox": [218, 333, 252, 347], "score": 0.92, "content": "L(c,r)", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [252, 333, 500, 347], "score": 1.0, "content": ", respectively. Moreover, all lens spaces also arise", "type": "text"}], "index": 12}, {"bbox": [110, 347, 499, 362], "spans": [{"bbox": [110, 347, 137, 362], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [137, 349, 165, 360], "score": 0.93, "content": "a\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [166, 347, 262, 362], "score": 1.0, "content": " ; in fact, for each ", "type": "text"}, {"bbox": [262, 349, 291, 358], "score": 0.89, "content": "a>0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [291, 347, 298, 362], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [298, 348, 381, 361], "score": 0.93, "content": "M(a,0,c,1,a,0)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [381, 347, 499, 362], "score": 1.0, "content": " is homeomorphic with", "type": "text"}], "index": 13}, {"bbox": [110, 362, 501, 376], "spans": [{"bbox": [110, 362, 188, 376], "score": 1.0, "content": "the lens space ", "type": "text"}, {"bbox": [189, 363, 222, 375], "score": 0.94, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [223, 362, 243, 376], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [243, 367, 249, 372], "score": 0.88, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [250, 362, 278, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 367, 284, 372], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [284, 362, 501, 376], "score": 1.0, "content": " are coprime, since it is easy to see that", "type": "text"}], "index": 14}, {"bbox": [110, 375, 501, 390], "spans": [{"bbox": [110, 377, 191, 389], "score": 0.92, "content": "H(a,0,c,1,a,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [191, 375, 501, 390], "score": 1.0, "content": " can be transformed into the canonical genus one Heegaard", "type": "text"}], "index": 15}, {"bbox": [109, 390, 349, 405], "spans": [{"bbox": [109, 390, 169, 405], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [169, 392, 202, 404], "score": 0.96, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [203, 390, 349, 405], "score": 1.0, "content": " by Singer moves of type IB.", "type": "text"}], "index": 16}], "index": 13}, {"type": "text", "bbox": [110, 409, 500, 452], "lines": [{"bbox": [126, 411, 499, 425], "spans": [{"bbox": [126, 411, 490, 425], "score": 1.0, "content": "Let us see now how the admissibility conditions for the 6-tuples of ", "type": "text"}, {"bbox": [490, 413, 499, 421], "score": 0.87, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}], "index": 17}, {"bbox": [109, 426, 501, 440], "spans": [{"bbox": [109, 426, 368, 440], "score": 1.0, "content": "can be given in terms of labelling of the vertices of ", "type": "text"}, {"bbox": [369, 427, 379, 436], "score": 0.89, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [379, 426, 501, 440], "score": 1.0, "content": ", belonging to the curve", "type": "text"}], "index": 18}, {"bbox": [110, 440, 500, 454], "spans": [{"bbox": [110, 442, 149, 452], "score": 0.93, "content": "D_{1}\\in\\mathcal{D}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [149, 440, 465, 454], "score": 1.0, "content": ". With this aim, consider the following properties for a 6-tuple ", "type": "text"}, {"bbox": [466, 442, 496, 451], "score": 0.92, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [496, 440, 500, 454], "score": 1.0, "content": ":", "type": "text"}], "index": 19}], "index": 18}, {"type": "text", "bbox": [116, 461, 499, 491], "lines": [{"bbox": [118, 464, 501, 479], "spans": [{"bbox": [118, 464, 434, 479], "score": 1.0, "content": "(i\u2019) the set of the labels of the vertices belonging to the cycle ", "type": "text"}, {"bbox": [434, 466, 449, 477], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [450, 464, 501, 479], "score": 1.0, "content": " is the set", "type": "text"}], "index": 20}, {"bbox": [139, 478, 275, 492], "spans": [{"bbox": [139, 478, 264, 492], "score": 1.0, "content": "of all integers from 1 to ", "type": "text"}, {"bbox": [264, 480, 271, 489], "score": 0.87, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [271, 478, 275, 492], "score": 1.0, "content": ";", "type": "text"}], "index": 21}], "index": 20.5}, {"type": "text", "bbox": [114, 501, 388, 515], "lines": [{"bbox": [115, 503, 388, 516], "spans": [{"bbox": [115, 503, 264, 516], "score": 1.0, "content": "(ii\u2019) the vertices of the cycle ", "type": "text"}, {"bbox": [264, 505, 279, 515], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [279, 503, 388, 516], "score": 1.0, "content": " have different labels.", "type": "text"}], "index": 22}], "index": 22}, {"type": "text", "bbox": [109, 525, 500, 641], "lines": [{"bbox": [126, 527, 499, 542], "spans": [{"bbox": [126, 527, 297, 542], "score": 1.0, "content": "It is easy to see that, if a 6-tuple ", "type": "text"}, {"bbox": [298, 529, 327, 538], "score": 0.93, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [328, 527, 499, 542], "score": 1.0, "content": " is admissible, then it satisfies (i\u2019)", "type": "text"}], "index": 23}, {"bbox": [110, 542, 500, 556], "spans": [{"bbox": [110, 542, 325, 556], "score": 1.0, "content": "and (ii\u2019). On the other side, if a 6-tuple ", "type": "text"}, {"bbox": [325, 544, 357, 553], "score": 0.94, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [357, 542, 500, 556], "score": 1.0, "content": " satisfies (i\u2019) and (ii\u2019), then", "type": "text"}], "index": 24}, {"bbox": [109, 556, 501, 572], "spans": [{"bbox": [109, 556, 168, 572], "score": 1.0, "content": "the curves ", "type": "text"}, {"bbox": [168, 557, 242, 570], "score": 0.94, "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [243, 556, 278, 572], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [279, 558, 347, 569], "score": 0.93, "content": "k\\,=\\,1,\\ldots,n", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [347, 556, 501, 572], "score": 1.0, "content": ", which are all different from", "type": "text"}], "index": 25}, {"bbox": [109, 571, 501, 586], "spans": [{"bbox": [109, 571, 307, 586], "score": 1.0, "content": "each other, are precisely the curves of ", "type": "text"}, {"bbox": [307, 573, 317, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [317, 571, 358, 586], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [358, 573, 368, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [369, 571, 433, 586], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [433, 576, 440, 581], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [441, 571, 501, 586], "score": 1.0, "content": " curves and", "type": "text"}], "index": 26}, {"bbox": [109, 585, 499, 600], "spans": [{"bbox": [109, 585, 288, 600], "score": 1.0, "content": "they are cyclically permutated by ", "type": "text"}, {"bbox": [288, 590, 300, 598], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [300, 585, 491, 600], "score": 1.0, "content": ". However, this does not imply that ", "type": "text"}, {"bbox": [492, 590, 499, 596], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}], "index": 27}, {"bbox": [109, 600, 499, 614], "spans": [{"bbox": [109, 600, 313, 614], "score": 1.0, "content": "is admissible; for example, the 6-tuple ", "type": "text"}, {"bbox": [314, 600, 385, 613], "score": 0.92, "content": "(1,0,2,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [385, 600, 499, 614], "score": 1.0, "content": " satisfies (i\u2019) and (ii\u2019),", "type": "text"}], "index": 28}, {"bbox": [108, 614, 499, 629], "spans": [{"bbox": [108, 614, 395, 629], "score": 1.0, "content": "but it is not admissible (see Remark 1). Note that, for ", "type": "text"}, {"bbox": [396, 617, 425, 625], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [425, 614, 499, 629], "score": 1.0, "content": ", property (ii\u2019)", "type": "text"}], "index": 29}, {"bbox": [110, 629, 494, 644], "spans": [{"bbox": [110, 629, 383, 644], "score": 1.0, "content": "always holds, while condition (i\u2019) holds if and only if ", "type": "text"}, {"bbox": [384, 631, 393, 639], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [394, 629, 494, 644], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 30}], "index": 26.5}, {"type": "text", "bbox": [109, 642, 501, 670], "lines": [{"bbox": [126, 642, 499, 659], "spans": [{"bbox": [126, 642, 321, 659], "score": 1.0, "content": "If a 6-tuple satisfies property (i\u2019), then ", "type": "text"}, {"bbox": [321, 645, 334, 655], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [334, 642, 499, 659], "score": 1.0, "content": " acts transitively (not necessarily", "type": "text"}], "index": 31}, {"bbox": [110, 657, 500, 673], "spans": [{"bbox": [110, 657, 167, 673], "score": 1.0, "content": "simply) on ", "type": "text"}, {"bbox": [168, 660, 177, 668], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [178, 657, 500, 673], "score": 1.0, "content": ", and hence it is possible to induce an orientation (which is still", "type": "text"}], "index": 32}], "index": 31.5}], "layout_bboxes": [], "page_idx": 7, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "8", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 501, 154], "lines": [{"bbox": [126, 127, 499, 142], "spans": [{"bbox": [126, 127, 359, 142], "score": 1.0, "content": "Let us consider now the Dunwoody manifolds ", "type": "text"}, {"bbox": [360, 129, 441, 141], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [442, 127, 469, 142], "score": 1.0, "content": " with", "type": "text"}, {"bbox": [470, 130, 499, 138], "score": 0.9, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}], "index": 0}, {"bbox": [111, 142, 462, 156], "spans": [{"bbox": [111, 142, 169, 156], "score": 1.0, "content": "(and hence ", "type": "text"}, {"bbox": [170, 144, 198, 153], "score": 0.87, "content": "s=0", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [198, 142, 462, 156], "score": 1.0, "content": "), which arises from a genus one Heegaard diagram.", "type": "text"}], "index": 1}], "index": 0.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [111, 127, 499, 156]}, {"type": "text", "bbox": [109, 166, 501, 210], "lines": [{"bbox": [110, 169, 500, 183], "spans": [{"bbox": [110, 169, 218, 183], "score": 1.0, "content": "Proposition 2 Let ", "type": "text"}, {"bbox": [218, 169, 286, 183], "score": 0.91, "content": "(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 68}, {"bbox": [287, 169, 472, 183], "score": 1.0, "content": " be an admissible 6-tuple and let ", "type": "text"}, {"bbox": [472, 170, 500, 182], "score": 0.79, "content": "w\\ =", "type": "inline_equation", "height": 12, "width": 28}], "index": 2}, {"bbox": [110, 183, 500, 198], "spans": [{"bbox": [110, 183, 188, 197], "score": 0.9, "content": "w(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 78}, {"bbox": [188, 184, 500, 198], "score": 1.0, "content": " be the associated word. Then the Dunwoody manifold", "type": "text"}], "index": 3}, {"bbox": [110, 198, 300, 212], "spans": [{"bbox": [110, 198, 191, 212], "score": 0.91, "content": "M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 81}, {"bbox": [191, 198, 300, 212], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 4}], "index": 3, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [110, 169, 500, 212]}, {"type": "list", "bbox": [126, 211, 385, 255], "lines": [{"bbox": [127, 212, 221, 227], "spans": [{"bbox": [127, 212, 139, 227], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 212, 154, 225], "score": 0.63, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [154, 212, 171, 227], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [171, 213, 216, 225], "score": 0.9, "content": "\\varepsilon_{w}=\\pm1", "type": "inline_equation", "height": 12, "width": 45}, {"bbox": [217, 212, 221, 227], "score": 1.0, "content": ";", "type": "text"}], "index": 5, "is_list_end_line": true}, {"bbox": [127, 226, 243, 241], "spans": [{"bbox": [127, 226, 143, 241], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 226, 185, 239], "score": 0.88, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 13, "width": 41}, {"bbox": [185, 226, 202, 241], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 227, 238, 240], "score": 0.9, "content": "\\varepsilon_{w}=0", "type": "inline_equation", "height": 13, "width": 35}, {"bbox": [238, 226, 243, 241], "score": 1.0, "content": ";", "type": "text"}], "index": 6, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [127, 241, 384, 255], "spans": [{"bbox": [127, 242, 211, 255], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 241, 249, 255], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 242, 278, 255], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 241, 321, 255], "score": 0.9, "content": "\\alpha=|\\varepsilon_{w}|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [322, 242, 339, 255], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [340, 243, 381, 255], "score": 0.85, "content": "\\left|\\varepsilon_{w}\\right|>1", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [381, 242, 384, 255], "score": 1.0, "content": ".", "type": "text"}], "index": 7, "is_list_start_line": true, "is_list_end_line": true}], "index": 6, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [127, 212, 384, 255]}, {"type": "text", "bbox": [109, 266, 500, 296], "lines": [{"bbox": [126, 269, 501, 284], "spans": [{"bbox": [126, 269, 201, 284], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [201, 270, 232, 280], "score": 0.88, "content": "n=1", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [232, 269, 290, 284], "score": 1.0, "content": " we obtain ", "type": "text"}, {"bbox": [290, 270, 409, 282], "score": 0.92, "content": "w\\in F_{1}\\cong\\mathbf{Z}\\cong<x|\\emptyset>", "type": "inline_equation", "height": 12, "width": 119}, {"bbox": [409, 269, 452, 284], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [452, 270, 501, 283], "score": 0.9, "content": "\\pi_{1}(M)\\cong", "type": "inline_equation", "height": 13, "width": 49}], "index": 8}, {"bbox": [110, 281, 261, 300], "spans": [{"bbox": [110, 282, 243, 298], "score": 0.89, "content": "G_{1}(w)\\cong<x|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{|\\varepsilon_{w}|}", "type": "inline_equation", "height": 16, "width": 133}, {"bbox": [243, 281, 261, 300], "score": 1.0, "content": ".", "type": "text"}], "index": 9}], "index": 8.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [110, 269, 501, 300]}, {"type": "text", "bbox": [109, 300, 500, 403], "lines": [{"bbox": [110, 304, 499, 318], "spans": [{"bbox": [110, 304, 324, 318], "score": 1.0, "content": "Example 1. The Dunwoody manifolds ", "type": "text"}, {"bbox": [324, 304, 408, 317], "score": 0.89, "content": "M(0,0,1,1,0,0)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [408, 304, 415, 318], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [416, 305, 499, 317], "score": 0.91, "content": "M(1,0,0,1,1,0)", "type": "inline_equation", "height": 12, "width": 83}], "index": 10}, {"bbox": [109, 318, 499, 333], "spans": [{"bbox": [109, 318, 133, 333], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 318, 216, 332], "score": 0.9, "content": "M(0,0,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 82}, {"bbox": [216, 318, 250, 333], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [251, 323, 267, 331], "score": 0.78, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [268, 318, 435, 333], "score": 1.0, "content": " coprime, are homeomorphic to ", "type": "text"}, {"bbox": [436, 319, 449, 329], "score": 0.85, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [449, 318, 456, 333], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [457, 319, 499, 330], "score": 0.89, "content": "\\mathbf{S^{1}\\times S^{2}}", "type": "inline_equation", "height": 11, "width": 42}], "index": 11}, {"bbox": [110, 333, 500, 347], "spans": [{"bbox": [110, 333, 218, 347], "score": 1.0, "content": "and to the lens space ", "type": "text"}, {"bbox": [218, 333, 252, 347], "score": 0.92, "content": "L(c,r)", "type": "inline_equation", "height": 14, "width": 34}, {"bbox": [252, 333, 500, 347], "score": 1.0, "content": ", respectively. Moreover, all lens spaces also arise", "type": "text"}], "index": 12}, {"bbox": [110, 347, 499, 362], "spans": [{"bbox": [110, 347, 137, 362], "score": 1.0, "content": "with ", "type": "text"}, {"bbox": [137, 349, 165, 360], "score": 0.93, "content": "a\\neq0", "type": "inline_equation", "height": 11, "width": 28}, {"bbox": [166, 347, 262, 362], "score": 1.0, "content": " ; in fact, for each ", "type": "text"}, {"bbox": [262, 349, 291, 358], "score": 0.89, "content": "a>0", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [291, 347, 298, 362], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [298, 348, 381, 361], "score": 0.93, "content": "M(a,0,c,1,a,0)", "type": "inline_equation", "height": 13, "width": 83}, {"bbox": [381, 347, 499, 362], "score": 1.0, "content": " is homeomorphic with", "type": "text"}], "index": 13}, {"bbox": [110, 362, 501, 376], "spans": [{"bbox": [110, 362, 188, 376], "score": 1.0, "content": "the lens space ", "type": "text"}, {"bbox": [189, 363, 222, 375], "score": 0.94, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [223, 362, 243, 376], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [243, 367, 249, 372], "score": 0.88, "content": "a", "type": "inline_equation", "height": 5, "width": 6}, {"bbox": [250, 362, 278, 376], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 367, 284, 372], "score": 0.88, "content": "c", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [284, 362, 501, 376], "score": 1.0, "content": " are coprime, since it is easy to see that", "type": "text"}], "index": 14}, {"bbox": [110, 375, 501, 390], "spans": [{"bbox": [110, 377, 191, 389], "score": 0.92, "content": "H(a,0,c,1,a,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [191, 375, 501, 390], "score": 1.0, "content": " can be transformed into the canonical genus one Heegaard", "type": "text"}], "index": 15}, {"bbox": [109, 390, 349, 405], "spans": [{"bbox": [109, 390, 169, 405], "score": 1.0, "content": "diagram of ", "type": "text"}, {"bbox": [169, 392, 202, 404], "score": 0.96, "content": "L(c,a)", "type": "inline_equation", "height": 12, "width": 33}, {"bbox": [203, 390, 349, 405], "score": 1.0, "content": " by Singer moves of type IB.", "type": "text"}], "index": 16}], "index": 13, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [109, 304, 501, 405]}, {"type": "text", "bbox": [110, 409, 500, 452], "lines": [{"bbox": [126, 411, 499, 425], "spans": [{"bbox": [126, 411, 490, 425], "score": 1.0, "content": "Let us see now how the admissibility conditions for the 6-tuples of ", "type": "text"}, {"bbox": [490, 413, 499, 421], "score": 0.87, "content": "\\boldsymbol{S}", "type": "inline_equation", "height": 8, "width": 9}], "index": 17}, {"bbox": [109, 426, 501, 440], "spans": [{"bbox": [109, 426, 368, 440], "score": 1.0, "content": "can be given in terms of labelling of the vertices of ", "type": "text"}, {"bbox": [369, 427, 379, 436], "score": 0.89, "content": "\\Gamma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [379, 426, 501, 440], "score": 1.0, "content": ", belonging to the curve", "type": "text"}], "index": 18}, {"bbox": [110, 440, 500, 454], "spans": [{"bbox": [110, 442, 149, 452], "score": 0.93, "content": "D_{1}\\in\\mathcal{D}", "type": "inline_equation", "height": 10, "width": 39}, {"bbox": [149, 440, 465, 454], "score": 1.0, "content": ". With this aim, consider the following properties for a 6-tuple ", "type": "text"}, {"bbox": [466, 442, 496, 451], "score": 0.92, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [496, 440, 500, 454], "score": 1.0, "content": ":", "type": "text"}], "index": 19}], "index": 18, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [109, 411, 501, 454]}, {"type": "text", "bbox": [116, 461, 499, 491], "lines": [{"bbox": [118, 464, 501, 479], "spans": [{"bbox": [118, 464, 434, 479], "score": 1.0, "content": "(i\u2019) the set of the labels of the vertices belonging to the cycle ", "type": "text"}, {"bbox": [434, 466, 449, 477], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [450, 464, 501, 479], "score": 1.0, "content": " is the set", "type": "text"}], "index": 20}, {"bbox": [139, 478, 275, 492], "spans": [{"bbox": [139, 478, 264, 492], "score": 1.0, "content": "of all integers from 1 to ", "type": "text"}, {"bbox": [264, 480, 271, 489], "score": 0.87, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [271, 478, 275, 492], "score": 1.0, "content": ";", "type": "text"}], "index": 21}], "index": 20.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [118, 464, 501, 492]}, {"type": "text", "bbox": [114, 501, 388, 515], "lines": [{"bbox": [115, 503, 388, 516], "spans": [{"bbox": [115, 503, 264, 516], "score": 1.0, "content": "(ii\u2019) the vertices of the cycle ", "type": "text"}, {"bbox": [264, 505, 279, 515], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [279, 503, 388, 516], "score": 1.0, "content": " have different labels.", "type": "text"}], "index": 22}], "index": 22, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [115, 503, 388, 516]}, {"type": "text", "bbox": [109, 525, 500, 641], "lines": [{"bbox": [126, 527, 499, 542], "spans": [{"bbox": [126, 527, 297, 542], "score": 1.0, "content": "It is easy to see that, if a 6-tuple ", "type": "text"}, {"bbox": [298, 529, 327, 538], "score": 0.93, "content": "\\sigma\\in S", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [328, 527, 499, 542], "score": 1.0, "content": " is admissible, then it satisfies (i\u2019)", "type": "text"}], "index": 23}, {"bbox": [110, 542, 500, 556], "spans": [{"bbox": [110, 542, 325, 556], "score": 1.0, "content": "and (ii\u2019). On the other side, if a 6-tuple ", "type": "text"}, {"bbox": [325, 544, 357, 553], "score": 0.94, "content": "\\sigma\\,\\in\\,S", "type": "inline_equation", "height": 9, "width": 32}, {"bbox": [357, 542, 500, 556], "score": 1.0, "content": " satisfies (i\u2019) and (ii\u2019), then", "type": "text"}], "index": 24}, {"bbox": [109, 556, 501, 572], "spans": [{"bbox": [109, 556, 168, 572], "score": 1.0, "content": "the curves ", "type": "text"}, {"bbox": [168, 557, 242, 570], "score": 0.94, "content": "\\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D}", "type": "inline_equation", "height": 13, "width": 74}, {"bbox": [243, 556, 278, 572], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [279, 558, 347, 569], "score": 0.93, "content": "k\\,=\\,1,\\ldots,n", "type": "inline_equation", "height": 11, "width": 68}, {"bbox": [347, 556, 501, 572], "score": 1.0, "content": ", which are all different from", "type": "text"}], "index": 25}, {"bbox": [109, 571, 501, 586], "spans": [{"bbox": [109, 571, 307, 586], "score": 1.0, "content": "each other, are precisely the curves of ", "type": "text"}, {"bbox": [307, 573, 317, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [317, 571, 358, 586], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [358, 573, 368, 581], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [369, 571, 433, 586], "score": 1.0, "content": " has exactly ", "type": "text"}, {"bbox": [433, 576, 440, 581], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [441, 571, 501, 586], "score": 1.0, "content": " curves and", "type": "text"}], "index": 26}, {"bbox": [109, 585, 499, 600], "spans": [{"bbox": [109, 585, 288, 600], "score": 1.0, "content": "they are cyclically permutated by ", "type": "text"}, {"bbox": [288, 590, 300, 598], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [300, 585, 491, 600], "score": 1.0, "content": ". However, this does not imply that ", "type": "text"}, {"bbox": [492, 590, 499, 596], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}], "index": 27}, {"bbox": [109, 600, 499, 614], "spans": [{"bbox": [109, 600, 313, 614], "score": 1.0, "content": "is admissible; for example, the 6-tuple ", "type": "text"}, {"bbox": [314, 600, 385, 613], "score": 0.92, "content": "(1,0,2,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [385, 600, 499, 614], "score": 1.0, "content": " satisfies (i\u2019) and (ii\u2019),", "type": "text"}], "index": 28}, {"bbox": [108, 614, 499, 629], "spans": [{"bbox": [108, 614, 395, 629], "score": 1.0, "content": "but it is not admissible (see Remark 1). Note that, for ", "type": "text"}, {"bbox": [396, 617, 425, 625], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [425, 614, 499, 629], "score": 1.0, "content": ", property (ii\u2019)", "type": "text"}], "index": 29}, {"bbox": [110, 629, 494, 644], "spans": [{"bbox": [110, 629, 383, 644], "score": 1.0, "content": "always holds, while condition (i\u2019) holds if and only if ", "type": "text"}, {"bbox": [384, 631, 393, 639], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [394, 629, 494, 644], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 30}], "index": 26.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [108, 527, 501, 644]}, {"type": "text", "bbox": [109, 642, 501, 670], "lines": [{"bbox": [126, 642, 499, 659], "spans": [{"bbox": [126, 642, 321, 659], "score": 1.0, "content": "If a 6-tuple satisfies property (i\u2019), then ", "type": "text"}, {"bbox": [321, 645, 334, 655], "score": 0.93, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [334, 642, 499, 659], "score": 1.0, "content": " acts transitively (not necessarily", "type": "text"}], "index": 31}, {"bbox": [110, 657, 500, 673], "spans": [{"bbox": [110, 657, 167, 673], "score": 1.0, "content": "simply) on ", "type": "text"}, {"bbox": [168, 660, 177, 668], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [178, 657, 500, 673], "score": 1.0, "content": ", and hence it is possible to induce an orientation (which is still", "type": "text"}], "index": 32}, {"bbox": [109, 127, 499, 142], "spans": [{"bbox": [109, 127, 306, 142], "score": 1.0, "content": "said to be canonical) on the cycles of ", "type": "text", "cross_page": true}, {"bbox": [307, 130, 316, 138], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 9, "cross_page": true}, {"bbox": [317, 127, 414, 142], "score": 1.0, "content": " and on the graph ", "type": "text", "cross_page": true}, {"bbox": [414, 129, 422, 138], "score": 0.87, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 8, "cross_page": true}, {"bbox": [422, 127, 499, 142], "score": 1.0, "content": ", by extending,", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [110, 143, 385, 155], "spans": [{"bbox": [110, 143, 129, 155], "score": 1.0, "content": "via ", "type": "text", "cross_page": true}, {"bbox": [129, 147, 141, 155], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12, "cross_page": true}, {"bbox": [141, 143, 241, 155], "score": 1.0, "content": ", the orientation of ", "type": "text", "cross_page": true}, {"bbox": [241, 144, 255, 154], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 14, "cross_page": true}, {"bbox": [256, 143, 371, 155], "score": 1.0, "content": " to the other cycles of ", "type": "text", "cross_page": true}, {"bbox": [371, 144, 381, 153], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10, "cross_page": true}, {"bbox": [381, 143, 385, 155], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 1}], "index": 31.5, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [110, 642, 500, 673]}]}
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Let us consider now the Dunwoody manifolds with (and hence ), which arises from a genus one Heegaard diagram.
Proposition 2 Let be an admissible 6-tuple and let be the associated word. Then the Dunwoody manifold is homeomorphic to:
- i) , if ; ii) , if ; iii) a lens space with , if .
Proof. From we obtain . Thus, .
Example 1. The Dunwoody manifolds , and , with coprime, are homeomorphic to , and to the lens space , respectively. Moreover, all lens spaces also arise with ; in fact, for each , is homeomorphic with the lens space , if and are coprime, since it is easy to see that can be transformed into the canonical genus one Heegaard diagram of by Singer moves of type IB.
Let us see now how the admissibility conditions for the 6-tuples of can be given in terms of labelling of the vertices of , belonging to the curve . With this aim, consider the following properties for a 6-tuple :
(i’) the set of the labels of the vertices belonging to the cycle is the set of all integers from 1 to ;
(ii’) the vertices of the cycle have different labels.
It is easy to see that, if a 6-tuple is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple satisfies (i’) and (ii’), then the curves , with , which are all different from each other, are precisely the curves of . Thus, has exactly curves and they are cyclically permutated by . However, this does not imply that is admissible; for example, the 6-tuple satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for , property (ii’) always holds, while condition (i’) holds if and only if has a unique cycle.
If a 6-tuple satisfies property (i’), then acts transitively (not necessarily simply) on , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of and on the graph , by extending, via , the orientation of to the other cycles of .
8
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<div class="pdf-page">
<p>Let us consider now the Dunwoody manifolds with (and hence ), which arises from a genus one Heegaard diagram.</p>
<p>Proposition 2 Let be an admissible 6-tuple and let be the associated word. Then the Dunwoody manifold is homeomorphic to:</p>
<ul>
<li>i) , if ; ii) , if ; iii) a lens space with , if .</li>
</ul>
<p>Proof. From we obtain . Thus, .</p>
<p>Example 1. The Dunwoody manifolds , and , with coprime, are homeomorphic to , and to the lens space , respectively. Moreover, all lens spaces also arise with ; in fact, for each , is homeomorphic with the lens space , if and are coprime, since it is easy to see that can be transformed into the canonical genus one Heegaard diagram of by Singer moves of type IB.</p>
<p>Let us see now how the admissibility conditions for the 6-tuples of can be given in terms of labelling of the vertices of , belonging to the curve . With this aim, consider the following properties for a 6-tuple :</p>
<p>(i’) the set of the labels of the vertices belonging to the cycle is the set of all integers from 1 to ;</p>
<p>(ii’) the vertices of the cycle have different labels.</p>
<p>It is easy to see that, if a 6-tuple is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple satisfies (i’) and (ii’), then the curves , with , which are all different from each other, are precisely the curves of . Thus, has exactly curves and they are cyclically permutated by . However, this does not imply that is admissible; for example, the 6-tuple satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for , property (ii’) always holds, while condition (i’) holds if and only if has a unique cycle.</p>
<p>If a 6-tuple satisfies property (i’), then acts transitively (not necessarily simply) on , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of and on the graph , by extending, via , the orientation of to the other cycles of .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="161" data-width="654" data-height="38">Let us consider now the Dunwoody manifolds with (and hence ), which arises from a genus one Heegaard diagram.</p>
<p class="pdf-text" data-x="182" data-y="214" data-width="656" data-height="57">Proposition 2 Let be an admissible 6-tuple and let be the associated word. Then the Dunwoody manifold is homeomorphic to:</p>
<ul class="pdf-list" data-x="210" data-y="272" data-width="434" data-height="57">
<li>i) , if ; ii) , if ; iii) a lens space with , if .</li>
</ul>
<p class="pdf-text" data-x="182" data-y="343" data-width="654" data-height="39">Proof. From we obtain . Thus, .</p>
<p class="pdf-text" data-x="182" data-y="387" data-width="654" data-height="134">Example 1. The Dunwoody manifolds , and , with coprime, are homeomorphic to , and to the lens space , respectively. Moreover, all lens spaces also arise with ; in fact, for each , is homeomorphic with the lens space , if and are coprime, since it is easy to see that can be transformed into the canonical genus one Heegaard diagram of by Singer moves of type IB.</p>
<p class="pdf-text" data-x="184" data-y="528" data-width="652" data-height="56">Let us see now how the admissibility conditions for the 6-tuples of can be given in terms of labelling of the vertices of , belonging to the curve . With this aim, consider the following properties for a 6-tuple :</p>
<p class="pdf-text" data-x="194" data-y="596" data-width="640" data-height="38">(i’) the set of the labels of the vertices belonging to the cycle is the set of all integers from 1 to ;</p>
<p class="pdf-text" data-x="190" data-y="647" data-width="459" data-height="18">(ii’) the vertices of the cycle have different labels.</p>
<p class="pdf-text" data-x="182" data-y="678" data-width="654" data-height="150">It is easy to see that, if a 6-tuple is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple satisfies (i’) and (ii’), then the curves , with , which are all different from each other, are precisely the curves of . Thus, has exactly curves and they are cyclically permutated by . However, this does not imply that is admissible; for example, the 6-tuple satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for , property (ii’) always holds, while condition (i’) holds if and only if has a unique cycle.</p>
<p class="pdf-text" data-x="182" data-y="830" data-width="656" data-height="36">If a 6-tuple satisfies property (i’), then acts transitively (not necessarily simply) on , and hence it is possible to induce an orientation (which is still said to be canonical) on the cycles of and on the graph , by extending, via , the orientation of to the other cycles of .</p>
<div class="pdf-discarded" data-x="501" data-y="893" data-width="14" data-height="14" style="opacity: 0.5;">8</div>
</div>
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# 2 Dunwoody manifolds
Let us sketch now the construction of Dunwoody manifolds given in [6]. Let $a,b,c,n$ be integers such that $n\ >\ 0$ , $a,b,c\,\geq\,0$ and $a+b+c>0$ . Let $\Gamma=\Gamma(a,b,c,n)$ be the planar regular trivalent graph drawn in Figure 1.
It contains $n$ upper cycles $C_{1}^{\prime},\ldots\,,C_{n}^{\prime}$ and $n$ lower cycles $C_{1}^{\prime\prime},\ldots\,,C_{n}^{\prime\prime}$ , each having $d=2a+b+c$ vertices. For each $i=1,\dots,n$ , the cycle $C_{i}^{\prime}$ (resp. $C_{i}^{\prime\prime}$ ) is connected to the cycle $C_{i+1}^{\prime}$ (resp. $C_{i+1}^{\prime\prime}$ ) by $a$ parallel arcs, to the cycle $C_{i}^{\prime\prime}$ by $c$ parallel arcs and to the cycle $C_{i+1}^{\prime\prime}$ by $b$ parallel arcs (assume $n+1=1$ ). We set $\mathcal{C}^{\prime}=\{C_{1}^{\prime},\ldots,C_{n}^{\prime}\}$ and $\mathcal{C}^{\prime\prime}=\{C_{1}^{\prime\prime},\ldots,C_{n}^{\prime\prime}\}$ . Moreover, denote by $A^{\prime}$ (resp. $A^{\prime\prime}$ ) the set of the arcs of $\Gamma$ belonging to a cycle of $\mathcal{C}^{\prime}$ (resp. $\mathcal{C^{\prime\prime}}$ ) and by
Figure 1: The graph $\Gamma$ .
$A$ the set of the other arcs of the graph. The one-point compactification of the plane leads to a 2-cell embedding of the graph $\Gamma$ in $\mathbf{S^{2}}$ ; it is evident that the graph is invariant with respect to a rotation $\rho_{n}$ of the sphere by $2\pi/n$ radians along a suitable axis intersecting $\mathbf{S^{2}}$ in two points not belonging to the graph. Obviously, $\rho_{n}$ sends $C_{i}^{\prime}$ to $C_{i+1}^{\prime}$ and $C_{i}^{\prime\prime}$ to $C_{i+1}^{\prime\prime}$ (mod $n$ ), for each $i=1,\dots,n$ .
By cutting the sphere along all $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ and by removing the interior of the corresponding discs, we obtain a sphere with $2n$ holes. Let now $r$ and $s$ be two new integers; give a clockwise (resp. counterclockwise) orientation to the cycles of $\mathcal{C}^{\prime}$ (resp. of $\mathcal{C^{\prime\prime}}$ ) and label their vertices from 1 to $d$ , in accordance with these orientations (see Figure 2) so that:
- the vertex 1 of each $C_{i}^{\prime}$ is the endpoint of the first arc of $A$ connecting Ci′ with Ci′+1;
- the vertex $1-r$ (mod $d$ ) of each $C_{i}^{\prime\prime}$ is the endpoint of the first arc of A connecting Ci′′ with Ci′′+1.
Then glue the cycle $C_{i}^{\prime}$ with the cycle $C_{i-s}^{\prime\prime}$ (mod $n$ ) so that equally labelled vertices are identified together.
Figure 2:
It is evident by construction that the integers $r$ and $s$ can be taken mod $d$ and mod $n$ respectively. Denote by $\boldsymbol{S}$ the set of all the 6-tuples $(a,b,c,n,r,s)\in\mathbf{Z}^{6}$ such that $n>0$ , $a,b,c\geq0$ and $a+b+c>0$ .
The described gluing gives rise to an orientable surface $T_{n}$ of genus $n$ and the $n d$ arcs belonging to $A$ are pairwise connected through their endpoints, realizing ${\mathit{m}}$ cycles $D_{1},\ldots,D_{m}$ on $T_{n}$ . It is straightforward that the cut of $T_{n}$ along the $n$ cycles $C_{i}=C_{i}^{\prime}=C_{i-s}^{\prime\prime}$ does not disconnect the surface. Set $\mathcal{C}=\{C_{1},\ldots,C_{n}\}$ and $\mathcal{D}=\{D_{1},\dotsc,D_{m}\}$ .
If $m\,=\,n$ and if the cut along the cycles of $\mathcal{D}$ does not disconnect $T_{n}^{'}$ , then the two systems of meridian curves $\scriptscriptstyle\mathcal{C}$ and $\mathcal{D}$ in $T_{n}$ represent a genus $n$ Heegaard diagram of a closed orientable 3-manifold, which is completely determined by the 6-tuple. Each manifold arising in this way is called a Dunwoody manifold.
Thus, we define to be admissible the 6-tuples $(a,b,c,n,r,s)$ of $\boldsymbol{S}$ satisfying the following conditions:
(1) the set $\mathcal{D}$ contains exactly $n$ cycles;
(2) the surface $T_{n}^{'}$ is not disconnected by the cut along the cycles of $\mathcal{D}$ .
The “open” Heegaard diagram $\Gamma$ and the Dunwoody manifold associated to the admissible 6-tuple $\sigma$ will be denoted by $H(\sigma)$ and $M(\sigma)$ respectively.
Remark 1. It is easy to see that not all the 6-tuples in $\boldsymbol{S}$ are admissible. For example, the 6-tuples $(a,0,a,1,a,0)$ , with $a\geq1$ , give rise to exactly $a$ cycles in $\mathcal{D}$ ; thus, they are not admissible if $a>1$ . The 6-tuples $(1,0,c,1,2,0)$ are not admissible if $c$ is even, since, in this case, we obtain exactly one cycle $D_{1}$ , but the cut along it disconnects the torus $T_{1}$ .
Consider now a 6-tuple $\sigma\,\in\,S$ . The graph $\Gamma$ becomes, via the gluing quotient map, a regular 4-valent graph denoted by $\Gamma^{\prime}$ embedded in $T_{n}^{'}$ . Its vertices are the intersection points of the spaces $\Omega=\cup_{i=1}^{n}C_{i}$ and $\Lambda=\cup_{j=1}^{m}D_{j}$ ; hence they inherit the labelling of the corresponding glued vertices of $\Gamma$ . Since the gluing of the cycles of $\mathcal{C}^{\prime}$ and $\mathcal{C^{\prime\prime}}$ is invariant with respect to the rotation $\rho_{n}$ , the group $\mathcal{G}_{n}=<\rho_{n}>$ naturally induces a cyclic action of order $n$ on $T_{n}^{'}$ such that the quotient $T_{1}=T_{n}/\mathcal{G}_{n}$ is homeomorphic to a torus. The labelling of the vertices of $\Gamma^{\prime}$ is invariant under the rotation $\rho_{n}$ and $\rho_{n}(C_{i})=C_{i+1}$ (mod $n$ ). We are going to show that, if the 6-tuple is admissible, this last property also holds for the cycles of $\mathcal{D}$ .
Lemma 1 a) Let $\sigma\;=\;(a,b,c,n,r,s)$ be an admissible 6-tuple. Then $\rho_{n}$ induces a cyclic permutation on the curves of $\mathcal{D}$ . Thus, if $D$ is a cycle of $\mathcal{D}$ , then ${\mathcal{D}}=\{\rho_{n}^{k-1}(D)|k=1,\ldots,n\}$ .
b) If $(a,b,c,n,r,s)$ is admissible, then also $(a,b,c,1,r,0)$ is admissible and the Heegaard diagram $H(a,b,c,1,r,0)$ is the quotient of the Heegaard diagram $H(a,b,c,n,r,s)$ respect to $\mathcal{G}_{n}$ .
Proof. a) First of all, note that $\rho_{n}(\Lambda)=\Lambda$ ; thus the group $\mathcal{G}_{n}$ also acts on the spaces $T_{n}\mathrm{~-~}\Lambda$ and $\Lambda$ (and hence on the set $\mathcal{D}$ ). If the 6-tuple $\sigma$ is admissible, then $T_{n}-\Lambda$ is connected, and hence the quotient $(T_{n}-\Lambda)/\mathcal{G}_{n}=$ $T_{n}/\mathcal{G}_{n}-\Lambda/\mathcal{G}_{n}$ must be connected too. This implies that $\Lambda/\mathcal{G}_{n}$ has a unique connected component. Since $\Lambda$ has exactly $n$ connected components, the cyclic group $\mathcal{G}_{n}$ of order $n$ defines a simply transitive cyclic action on the cycles of $\mathcal{D}$ .
b) Let $C,D\ \subset\ T_{1}$ the two curves $C\;=\;\Omega/\mathcal{G}_{n}$ and $D\,=\,\Lambda/\mathcal{G}_{n}$ . Then, the two systems of curves ${\mathcal{C}}=\{C\}$ and $\mathcal{D}=\{D\}$ on $T_{1}$ define a Heegaard diagram of genus one. The graph $\Gamma_{1}$ corresponding to $\sigma_{1}\,=\,(a,b,c,1,r,0)$ is the quotient of the graph $\Gamma_{n}$ corresponding to $\sigma=(a,b,c,n,r,s)$ , respect to $\mathcal{G}_{n}$ . Moreover, the gluings on $\Gamma_{n}$ are invariant respect to $\rho_{n}$ . Therefore, the gluings on $\Gamma_{1}$ give rise to the Heegaard diagram above. This show that the 6-tuple $\sigma_{1}$ is admissible and obviously $H(a,b,c,1,r,0)$ is the quotient of $H(a,b,c,n,r,s)$ respect to $\mathcal{G}_{n}$ .
Remark 2. More generally, given two positive integer $n$ and $n^{\prime}$ such that $n^{\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then $(a,b,c,r,n^{\prime},b)$ is admissible too. Moreover, the Heegaard diagram ${\cal H}(a,b,c,r,n^{\prime},b)$ is the quotient of $H(a,b,c,r,n,b)$ respect to the action of a cyclic group of order $n/n^{\prime}$ .
It is easy to see that, for admissible 6-tuples, each cycle in $\mathcal{D}$ contains $d$ vertices with different labels and is composed by exactly $d$ arcs of $\Gamma$ (in fact, $\mathit{2a}$ horizontal arcs, $b$ oblique arcs and $c$ vertical arcs).
An important consequence of point a) of Lemma $1$ is that, if $\sigma$ is an admissible 6-tuple, the presentation of the fundamental group of $M(\sigma)$ induced by the Heegaard diagram $H(\sigma)$ is cyclic.
To see this, let $\upsilon$ be the vertex belonging to the cycle $C_{1}$ and labelled by $a+b+1$ ; denote by $D_{1}$ the curve of $\mathcal{D}$ containing $v$ and by $v^{\prime}$ the vertex of $C_{1}^{\prime}$ corresponding to $v$ . Orient the arc $e^{\prime}\in A$ of the graph $\Gamma$ containing $v^{\prime}$ so that $v^{\prime}$ is its first endpoint and orient the curve $D_{1}$ in accordance with the orientation of this arc. Now, set $D_{k}=\rho_{n}^{k-1}(D_{1})$ , for each $k=1,\dotsc,n$ ; the orientation on $D_{1}$ induces, via $\rho_{n}$ , an orientation also on these curves. Moreover, these orientation on the cycles of $\mathcal{D}$ induce an orientation on the arcs of the graph $\Gamma$ belonging to $A$ . By orienting the arcs of $C^{\prime}$ and $C^{\prime\prime}$ in accordance with the fixed orientations of the cycles $C_{i}^{\prime}$ and $C_{i}^{\prime\prime}$ , the graph $\Gamma$ becomes an oriented graph, whose orientation is invariant under the action of the group $\mathcal{G}_{n}$ . Let us define to be canonical this orientation of $\Gamma$ .
Let now $w\in F_{n}$ be the word obtained by reading the oriented arcs $e_{1}=$ $e^{\prime},e_{2},\ldots,e_{d}$ of $\Gamma$ corresponding to the oriented cycle $D_{1}$ , starting from the vertex $v^{\prime}$ . The letters of $w$ are in one-to-one correspondence with the oriented arcs $e_{h}$ ; more precisely, the letter of $w$ corresponding to $e_{h}$ is $x_{i}$ if $e_{h}$ comes out from the cycle $C_{i}^{\prime}$ and is $x_{i}^{-1}$ if $e_{h}$ comes out from the cycle $C_{i-s}^{\prime\prime}$ . Note that the word $\theta_{n}^{k-1}(w)$ in the cyclic presentation $G_{n}(w)$ is obtained by reading the cycle $D_{k}$ along the given orientation, for $1\leq k\leq n$ (roughly speaking, the automorphism $\theta_{n}$ is “geometrically” realized by $\rho_{n}$ ).
This proves that each admissible 6-tuple $\sigma$ uniquely defines, via the associated Heegaard diagram $H(\sigma)$ , a word $w=w(\sigma)$ and a cyclic presentation $G_{n}(w)$ for the fundamental group of the Dunwoody manifold $M(\sigma)$ . Note that the sequence of the exponents in the word $w(\sigma)$ , and hence its exponentsum $\varepsilon_{w(\sigma)}$ , only depends on the integers $a,b,c,r$ .
Let us consider now the Dunwoody manifolds $M(a,b,c,n,r,s)$ with $n=1$ (and hence $s=0$ ), which arises from a genus one Heegaard diagram.
Proposition 2 Let $(a,b,c,1,r,0)$ be an admissible 6-tuple and let $w\ =$ $w(a,b,c,1,r,0)$ be the associated word. Then the Dunwoody manifold $M(a,b,c,1,r,0)$ is homeomorphic to:
i) $\mathbf{S^{3}}$ , if $\varepsilon_{w}=\pm1$ ;
ii) $\mathbf{S^{1}}\times\mathbf{S^{2}}$ , if $\varepsilon_{w}=0$ ;
iii) a lens space $L(\alpha,\beta)$ with $\alpha=|\varepsilon_{w}|$ , if $\left|\varepsilon_{w}\right|>1$ .
Proof. From $n=1$ we obtain $w\in F_{1}\cong\mathbf{Z}\cong<x|\emptyset>$ . Thus, $\pi_{1}(M)\cong$ $G_{1}(w)\cong<x|x^{\varepsilon_{w}}>\cong{\mathbf{Z}}_{|\varepsilon_{w}|}$ .
Example 1. The Dunwoody manifolds $M(0,0,1,1,0,0)$ , $M(1,0,0,1,1,0)$ and $M(0,0,c,1,r,0)$ , with $c,r$ coprime, are homeomorphic to $\mathbf{S^{3}}$ , $\mathbf{S^{1}\times S^{2}}$ and to the lens space $L(c,r)$ , respectively. Moreover, all lens spaces also arise with $a\neq0$ ; in fact, for each $a>0$ , $M(a,0,c,1,a,0)$ is homeomorphic with the lens space $L(c,a)$ , if $a$ and $c$ are coprime, since it is easy to see that $H(a,0,c,1,a,0)$ can be transformed into the canonical genus one Heegaard diagram of $L(c,a)$ by Singer moves of type IB.
Let us see now how the admissibility conditions for the 6-tuples of $\boldsymbol{S}$ can be given in terms of labelling of the vertices of $\Gamma^{\prime}$ , belonging to the curve $D_{1}\in\mathcal{D}$ . With this aim, consider the following properties for a 6-tuple $\sigma\in S$ :
(i’) the set of the labels of the vertices belonging to the cycle $D_{1}$ is the set of all integers from 1 to $d$ ;
(ii’) the vertices of the cycle $D_{1}$ have different labels.
It is easy to see that, if a 6-tuple $\sigma\in S$ is admissible, then it satisfies (i’) and (ii’). On the other side, if a 6-tuple $\sigma\,\in\,S$ satisfies (i’) and (ii’), then the curves $\rho_{n}^{k-1}(D_{1})\,\in\,\mathcal{D}$ , with $k\,=\,1,\ldots,n$ , which are all different from each other, are precisely the curves of $\mathcal{D}$ . Thus, $\mathcal{D}$ has exactly $n$ curves and they are cyclically permutated by $\rho_{n}$ . However, this does not imply that $\sigma$ is admissible; for example, the 6-tuple $(1,0,2,1,2,0)$ satisfies (i’) and (ii’), but it is not admissible (see Remark 1). Note that, for $n=1$ , property (ii’) always holds, while condition (i’) holds if and only if $\mathcal{D}$ has a unique cycle.
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"Let us consider now the Dunwoody manifolds M(a,b,c,n,r,s) with n=1",
"(and hence s=0 ), which arises from a genus one Heegaard diagram.",
"Proposition 2 Let (a,b,c,1,r,0) be an admissible 6-tuple and let w\\ =",
"w(a,b,c,1,r,0) be the associated word. Then the Dunwoody manifold",
"M(a,b,c,1,r,0) is homeomorphic to:",
"i) \\mathbf{S^{3}} , if \\varepsilon_{w}=\\pm1 ;",
"ii) \\mathbf{S^{1}}\\times\\mathbf{S^{2}} , if \\varepsilon_{w}=0 ;",
"iii) a lens space L(\\alpha,\\beta) with \\alpha=|\\varepsilon_{w}| , if \\left|\\varepsilon_{w}\\right|>1 .",
"Proof. From n=1 we obtain w\\in F_{1}\\cong\\mathbf{Z}\\cong<x|\\emptyset> . Thus, \\pi_{1}(M)\\cong",
"G_{1}(w)\\cong<x|x^{\\varepsilon_{w}}>\\cong{\\mathbf{Z}}_{|\\varepsilon_{w}|} .",
"Example 1. The Dunwoody manifolds M(0,0,1,1,0,0) , M(1,0,0,1,1,0)",
"and M(0,0,c,1,r,0) , with c,r coprime, are homeomorphic to \\mathbf{S^{3}} , \\mathbf{S^{1}\\times S^{2}}",
"and to the lens space L(c,r) , respectively. Moreover, all lens spaces also arise",
"with a\\neq0 ; in fact, for each a>0 , M(a,0,c,1,a,0) is homeomorphic with",
"the lens space L(c,a) , if a and c are coprime, since it is easy to see that",
"H(a,0,c,1,a,0) can be transformed into the canonical genus one Heegaard",
"diagram of L(c,a) by Singer moves of type IB.",
"Let us see now how the admissibility conditions for the 6-tuples of \\boldsymbol{S}",
"can be given in terms of labelling of the vertices of \\Gamma^{\\prime} , belonging to the curve",
"D_{1}\\in\\mathcal{D} . With this aim, consider the following properties for a 6-tuple \\sigma\\in S :",
"(i’) the set of the labels of the vertices belonging to the cycle D_{1} is the set",
"of all integers from 1 to d ;",
"(ii’) the vertices of the cycle D_{1} have different labels.",
"It is easy to see that, if a 6-tuple \\sigma\\in S is admissible, then it satisfies (i’)",
"and (ii’). On the other side, if a 6-tuple \\sigma\\,\\in\\,S satisfies (i’) and (ii’), then",
"the curves \\rho_{n}^{k-1}(D_{1})\\,\\in\\,\\mathcal{D} , with k\\,=\\,1,\\ldots,n , which are all different from",
"each other, are precisely the curves of \\mathcal{D} . Thus, \\mathcal{D} has exactly n curves and",
"they are cyclically permutated by \\rho_{n} . However, this does not imply that \\sigma",
"is admissible; for example, the 6-tuple (1,0,2,1,2,0) satisfies (i’) and (ii’),",
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"If a 6-tuple satisfies property (i’), then \\mathcal{G}_{n} acts transitively (not necessarily",
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"Property (i’) implies that the cycles of naturally induce a cyclic per- mutation on the set of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex labelled , one sequentially meets vertices (whose labels are different from each other), and then a new vertex labelled which can be different from . The sequence of the labellings of these consecutive vertices defines the cyclic permutation on . Further, each cycle of precisely contains arcs, with , and if and only if the 6-tuple satisfies (ii’) too. More- over, property (i’) is independent from the integers and ; hence, given two 6-tuples and , then satisfies (i’) if and only if satisfies (i’).",
"Let now be a 6-tuple satisfying (i’) and suppose that is canonically oriented. An arc of belonging to is said to be of type I if it is oriented from a cycle of to a cycle of , of type II if it is oriented from a cycle of to a cycle of and of type III otherwise (it joins cycles of or cycles of . Moreover, the arc is said to be of type I’ if it is oriented from a cycle (resp. ) to a cycle (resp. ), of type II’ if it is oriented from a cycle (resp. ) to a cycle (resp. ) and of type III’ otherwise (it joins with ). Let be the set of the first arcs of , following the canonical orientation, starting from the arc coming out from the vertex of labelled . Obviously, the set contains all the arcs of if and only if the 6-tuple also satisfies (ii’).",
"Now, denote by (resp. ) the number of the arcs of type I (resp. of type II) of and set . Similarly, denote by (resp. ) the number of the arcs of type (resp. of type II’) of and set . Note that has the same parity of and has the same parity of and hence of . It is evident that and only depend on the integers .",
"The integers and give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle of , starting from a vertex and let be the cycle of containing . If is the first vertex with the same label of and if is the cycle of containing , we have . Thus, the cycle contains arcs if and only if (mod ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a",
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[{"type": "text", "text": "", "page_idx": 8}, {"type": "text", "text": "Property (i\u2019) implies that the cycles of $\\mathcal{D}$ naturally induce a cyclic permutation on the set $\\mathcal{N}=\\{1,\\dotsc,d\\}$ of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex $v$ labelled $j$ , one sequentially meets $d$ vertices (whose labels are different from each other), and then a new vertex $\\bar{v}^{\\prime}$ labelled $j$ which can be different from $v$ . The sequence of the labellings of these $d$ consecutive vertices defines the cyclic permutation on $\\mathcal{N}$ . Further, each cycle of $\\mathcal{D}$ precisely contains $d^{\\prime}=l d$ arcs, with $l\\geq1$ , and $l=1$ if and only if the 6-tuple satisfies (ii\u2019) too. Moreover, property (i\u2019) is independent from the integers $n$ and $s$ ; hence, given two 6-tuples $\\sigma\\,=\\,(a,b,c,n,r,s)$ and ${\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)$ , then $\\sigma$ satisfies (i\u2019) if and only if $\\sigma^{\\prime}$ satisfies (i\u2019). ", "page_idx": 8}, {"type": "text", "text": "Let now $\\sigma$ be a 6-tuple satisfying (i\u2019) and suppose that $\\Gamma$ is canonically oriented. An arc of $\\Gamma$ belonging to $A$ is said to be of type I if it is oriented from a cycle of $\\mathcal{C}^{\\prime}$ to a cycle of $\\mathcal{C^{\\prime\\prime}}$ , of type II if it is oriented from a cycle of $\\mathcal{C^{\\prime\\prime}}$ to a cycle of $\\mathcal{C}^{\\prime}$ and of type III otherwise (it joins cycles of $\\mathcal{C}^{\\prime}$ or cycles of $\\mathcal{C}^{\\prime\\prime})$ . Moreover, the arc is said to be of type I\u2019 if it is oriented from a cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) to a cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ), of type II\u2019 if it is oriented from a cycle $C_{i+1}^{\\prime}$ (resp. $C_{i+1}^{\\prime\\prime}$ ) to a cycle $C_{i}^{\\prime}$ (resp. $C_{i}^{\\prime\\prime}$ ) and of type III\u2019 otherwise (it joins $C_{i}^{\\prime}$ with $C_{i}^{\\prime\\prime}$ ). Let $\\Delta$ be the set of the first $d$ arcs of $D_{1}$ , following the canonical orientation, starting from the arc coming out from the vertex $v^{\\prime}$ of $C_{1}^{\\prime}$ labelled $a+b+1$ . Obviously, the set $\\Delta$ contains all the arcs of $D_{1}$ if and only if the 6-tuple $\\sigma$ also satisfies (ii\u2019). ", "page_idx": 8}, {"type": "text", "text": "Now, denote by $p_{\\sigma}^{\\prime}$ (resp. $p_{\\sigma}^{\\prime\\prime}$ ) the number of the arcs of type I (resp. of type II) of $\\Delta$ and set $p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}$ . Similarly, denote by $q_{\\sigma}^{\\prime}$ (resp. $q_{\\sigma}^{\\prime\\prime}$ ) the number of the arcs of type $\\Gamma$ (resp. of type II\u2019) of $\\Delta$ and set $q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}$ . Note that $p_{\\sigma}$ has the same parity of $b\\!+\\!c$ and $q_{\\sigma}$ has the same parity of $2a+b$ and hence of $b$ . It is evident that $p_{\\sigma}$ and $q_{\\sigma}$ only depend on the integers $a,b,c,r$ . ", "page_idx": 8}, {"type": "text", "text": "The integers $p_{\\sigma}$ and $q_{\\sigma}$ give an useful tool for verifying condition (ii\u2019). In fact, suppose to walk along the canonically oriented cycle $D_{j}$ of $\\mathcal{D}$ , starting from a vertex $v$ and let $C_{i}$ be the cycle of $\\mathcal{C}$ containing $v$ . If $\\bar{v}^{\\prime}$ is the first vertex with the same label of $v$ and if $C_{i^{\\prime}}$ is the cycle of $\\mathcal{C}$ containing $\\bar{v}^{\\prime}$ , we have $\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}$ . Thus, the cycle $D_{j}$ contains $d$ arcs if and only if $q_{\\sigma}+s p_{\\sigma}\\equiv0$ (mod $n$ ). This proves that the 6-tuple satisfies (ii\u2019). Thus, (i\u2019) and (ii\u2019) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a ", "page_idx": 8}]
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In fact, by walking", "type": "text"}], "index": 3}, {"bbox": [109, 185, 499, 199], "spans": [{"bbox": [109, 185, 492, 199], "score": 1.0, "content": "along these canonically oriented cycles, starting from an arbitrary vertex ", "type": "text"}, {"bbox": [493, 189, 499, 196], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}], "index": 4}, {"bbox": [109, 200, 501, 215], "spans": [{"bbox": [109, 200, 153, 215], "score": 1.0, "content": "labelled ", "type": "text"}, {"bbox": [153, 202, 159, 213], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [159, 200, 285, 215], "score": 1.0, "content": ", one sequentially meets ", "type": "text"}, {"bbox": [285, 201, 291, 210], "score": 0.88, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [291, 200, 501, 215], "score": 1.0, "content": " vertices (whose labels are different from", "type": "text"}], "index": 5}, {"bbox": [110, 214, 501, 229], "spans": [{"bbox": [110, 214, 293, 229], "score": 1.0, "content": "each other), and then a new vertex ", "type": "text"}, {"bbox": [293, 216, 302, 225], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 214, 348, 229], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [349, 216, 354, 227], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [355, 214, 501, 229], "score": 1.0, "content": " which can be different from", "type": "text"}], "index": 6}, {"bbox": [110, 229, 500, 243], "spans": [{"bbox": [110, 232, 117, 240], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 229, 328, 243], "score": 1.0, "content": ". The sequence of the labellings of these ", "type": "text"}, {"bbox": [329, 231, 335, 240], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [335, 229, 500, 243], "score": 1.0, "content": " consecutive vertices defines the", "type": "text"}], "index": 7}, {"bbox": [110, 244, 499, 257], "spans": [{"bbox": [110, 244, 224, 257], "score": 1.0, "content": "cyclic permutation on ", "type": "text"}, {"bbox": [225, 245, 236, 254], "score": 0.89, "content": "\\mathcal{N}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [237, 244, 357, 257], "score": 1.0, "content": ". Further, each cycle of ", "type": "text"}, {"bbox": [358, 245, 368, 254], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [368, 244, 464, 257], "score": 1.0, "content": " precisely contains ", "type": "text"}, {"bbox": [464, 245, 499, 254], "score": 0.93, "content": "d^{\\prime}=l d", "type": "inline_equation", "height": 9, "width": 35}], "index": 8}, {"bbox": [110, 258, 500, 272], "spans": [{"bbox": [110, 258, 164, 272], "score": 1.0, "content": "arcs, with ", "type": "text"}, {"bbox": [164, 259, 190, 270], "score": 0.92, "content": "l\\geq1", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [190, 258, 219, 272], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [219, 259, 245, 268], "score": 0.93, "content": "l=1", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [245, 258, 500, 272], "score": 1.0, "content": " if and only if the 6-tuple satisfies (ii\u2019) too. More-", "type": "text"}], "index": 9}, {"bbox": [109, 272, 500, 286], "spans": [{"bbox": [109, 272, 370, 286], "score": 1.0, "content": "over, property (i\u2019) is independent from the integers ", "type": "text"}, {"bbox": [370, 277, 378, 282], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [378, 272, 403, 286], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [403, 277, 408, 282], "score": 0.86, "content": "s", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [409, 272, 500, 286], "score": 1.0, "content": "; hence, given two", "type": "text"}], "index": 10}, {"bbox": [109, 286, 502, 301], "spans": [{"bbox": [109, 286, 155, 301], "score": 1.0, "content": "6-tuples ", "type": "text"}, {"bbox": [155, 288, 250, 300], "score": 0.93, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [251, 286, 279, 301], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 288, 380, 300], "score": 0.93, "content": "{\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [380, 286, 415, 301], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [415, 291, 423, 297], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [423, 286, 502, 301], "score": 1.0, "content": " satisfies (i\u2019) if", "type": "text"}], "index": 11}, {"bbox": [110, 301, 246, 314], "spans": [{"bbox": [110, 301, 169, 314], "score": 1.0, "content": "and only if ", "type": "text"}, {"bbox": [169, 302, 179, 311], "score": 0.91, "content": "\\sigma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [180, 301, 246, 314], "score": 1.0, "content": " satisfies (i\u2019).", "type": "text"}], "index": 12}], "index": 7}, {"type": "text", "bbox": [109, 313, 501, 471], "lines": [{"bbox": [125, 314, 498, 330], "spans": [{"bbox": [125, 314, 173, 330], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [174, 320, 181, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [181, 314, 417, 330], "score": 1.0, "content": " be a 6-tuple satisfying (i\u2019) and suppose that ", "type": "text"}, {"bbox": [418, 317, 425, 326], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 314, 498, 330], "score": 1.0, "content": " is canonically", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 213, 344], "score": 1.0, "content": "oriented. An arc of ", "type": "text"}, {"bbox": [214, 331, 221, 340], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [222, 330, 293, 344], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [293, 331, 302, 340], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [303, 330, 500, 344], "score": 1.0, "content": " is said to be of type I if it is oriented", "type": "text"}], "index": 14}, {"bbox": [110, 345, 501, 358], "spans": [{"bbox": [110, 345, 190, 358], "score": 1.0, "content": "from a cycle of ", "type": "text"}, {"bbox": [190, 346, 200, 355], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [200, 345, 270, 358], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [271, 346, 283, 355], "score": 0.9, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [284, 345, 501, 358], "score": 1.0, "content": ", of type II if it is oriented from a cycle of", "type": "text"}], "index": 15}, {"bbox": [110, 358, 502, 374], "spans": [{"bbox": [110, 360, 122, 370], "score": 0.91, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [123, 358, 191, 374], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [192, 360, 202, 369], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [202, 358, 427, 374], "score": 1.0, "content": " and of type III otherwise (it joins cycles of ", "type": "text"}, {"bbox": [428, 360, 438, 369], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [438, 358, 502, 374], "score": 1.0, "content": " or cycles of", "type": "text"}], "index": 16}, {"bbox": [110, 372, 500, 387], "spans": [{"bbox": [110, 374, 127, 387], "score": 0.66, "content": "\\mathcal{C}^{\\prime\\prime})", "type": "inline_equation", "height": 13, "width": 17}, {"bbox": [127, 372, 500, 387], "score": 1.0, "content": ". Moreover, the arc is said to be of type I\u2019 if it is oriented from a cycle", "type": "text"}], "index": 17}, {"bbox": [110, 387, 502, 406], "spans": [{"bbox": [110, 389, 122, 401], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [123, 387, 160, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [160, 389, 174, 401], "score": 0.88, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [174, 387, 235, 406], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [236, 389, 258, 402], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [259, 387, 296, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [296, 389, 319, 402], "score": 0.9, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [320, 387, 502, 406], "score": 1.0, "content": "), of type II\u2019 if it is oriented from a", "type": "text"}], "index": 18}, {"bbox": [108, 401, 501, 419], "spans": [{"bbox": [108, 401, 138, 419], "score": 1.0, "content": "cycle ", "type": "text"}, {"bbox": [139, 403, 162, 416], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [162, 401, 198, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [199, 403, 222, 416], "score": 0.92, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [222, 401, 281, 419], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [281, 403, 293, 415], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [294, 401, 330, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [331, 403, 345, 415], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [345, 401, 501, 419], "score": 1.0, "content": ") and of type III\u2019 otherwise (it", "type": "text"}], "index": 19}, {"bbox": [108, 417, 500, 430], "spans": [{"bbox": [108, 417, 138, 430], "score": 1.0, "content": "joins ", "type": "text"}, {"bbox": [138, 418, 150, 430], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [151, 417, 182, 430], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [182, 418, 196, 430], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [196, 417, 232, 430], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [232, 418, 243, 427], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 417, 361, 430], "score": 1.0, "content": "be the set of the first ", "type": "text"}, {"bbox": [361, 418, 368, 427], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [368, 417, 410, 430], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [410, 418, 425, 429], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [425, 417, 500, 430], "score": 1.0, "content": ", following the", "type": "text"}], "index": 20}, {"bbox": [108, 432, 502, 444], "spans": [{"bbox": [108, 432, 477, 444], "score": 1.0, "content": "canonical orientation, starting from the arc coming out from the vertex ", "type": "text"}, {"bbox": [477, 432, 486, 441], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [486, 432, 502, 444], "score": 1.0, "content": " of", "type": "text"}], "index": 21}, {"bbox": [110, 446, 500, 459], "spans": [{"bbox": [110, 447, 123, 459], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [123, 446, 170, 458], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [170, 447, 215, 457], "score": 0.94, "content": "a+b+1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [216, 446, 318, 458], "score": 1.0, "content": ". Obviously, the set ", "type": "text"}, {"bbox": [318, 447, 329, 456], "score": 0.89, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [329, 446, 451, 458], "score": 1.0, "content": "contains all the arcs of ", "type": "text"}, {"bbox": [451, 447, 466, 457], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [466, 446, 500, 458], "score": 1.0, "content": " if and", "type": "text"}], "index": 22}, {"bbox": [110, 461, 307, 473], "spans": [{"bbox": [110, 461, 206, 473], "score": 1.0, "content": "only if the 6-tuple ", "type": "text"}, {"bbox": [207, 465, 214, 470], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 461, 307, 473], "score": 1.0, "content": " also satisfies (ii\u2019).", "type": "text"}], "index": 23}], "index": 18}, {"type": "text", "bbox": [110, 473, 500, 559], "lines": [{"bbox": [126, 473, 502, 490], "spans": [{"bbox": [126, 473, 212, 490], "score": 1.0, "content": "Now, denote by ", "type": "text"}, {"bbox": [213, 476, 224, 488], "score": 0.92, "content": "p_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [225, 473, 263, 490], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [263, 476, 275, 488], "score": 0.88, "content": "p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [275, 473, 502, 490], "score": 1.0, "content": ") the number of the arcs of type I (resp. of", "type": "text"}], "index": 24}, {"bbox": [109, 489, 501, 503], "spans": [{"bbox": [109, 489, 168, 503], "score": 1.0, "content": "type II) of ", "type": "text"}, {"bbox": [169, 491, 179, 499], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [179, 489, 225, 503], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [225, 490, 293, 502], "score": 0.94, "content": "p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [294, 489, 411, 503], "score": 1.0, "content": ". Similarly, denote by ", "type": "text"}, {"bbox": [411, 490, 422, 502], "score": 0.92, "content": "q_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [423, 489, 462, 503], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [462, 490, 474, 502], "score": 0.88, "content": "q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [474, 489, 501, 503], "score": 1.0, "content": ") the", "type": "text"}], "index": 25}, {"bbox": [109, 502, 500, 518], "spans": [{"bbox": [109, 502, 252, 518], "score": 1.0, "content": "number of the arcs of type ", "type": "text"}, {"bbox": [252, 505, 261, 514], "score": 0.47, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [261, 502, 374, 518], "score": 1.0, "content": " (resp. of type II\u2019) of ", "type": "text"}, {"bbox": [375, 505, 385, 514], "score": 0.91, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [385, 502, 430, 518], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [430, 504, 496, 516], "score": 0.93, "content": "q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [496, 502, 500, 518], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [109, 517, 499, 532], "spans": [{"bbox": [109, 517, 162, 532], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [162, 523, 174, 531], "score": 0.9, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [174, 517, 292, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [293, 519, 315, 529], "score": 0.93, "content": "b\\!+\\!c", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [315, 517, 340, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 523, 351, 531], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [352, 517, 469, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [470, 520, 499, 529], "score": 0.92, "content": "2a+b", "type": "inline_equation", "height": 9, "width": 29}], "index": 27}, {"bbox": [109, 532, 500, 546], "spans": [{"bbox": [109, 532, 182, 546], "score": 1.0, "content": "and hence of ", "type": "text"}, {"bbox": [182, 534, 188, 542], "score": 0.88, "content": "b", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [188, 532, 295, 546], "score": 1.0, "content": ". It is evident that ", "type": "text"}, {"bbox": [295, 537, 307, 545], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [307, 532, 335, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [336, 537, 347, 545], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [347, 532, 500, 546], "score": 1.0, "content": " only depend on the integers", "type": "text"}], "index": 28}, {"bbox": [110, 547, 154, 561], "spans": [{"bbox": [110, 548, 148, 559], "score": 0.93, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [148, 547, 154, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26.5}, {"type": "text", "bbox": [109, 559, 500, 675], "lines": [{"bbox": [127, 560, 500, 576], "spans": [{"bbox": [127, 560, 194, 576], "score": 1.0, "content": "The integers ", "type": "text"}, {"bbox": [194, 565, 206, 574], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [207, 560, 232, 576], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 566, 243, 574], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [244, 560, 500, 576], "score": 1.0, "content": " give an useful tool for verifying condition (ii\u2019). In", "type": "text"}], "index": 30}, {"bbox": [109, 575, 500, 590], "spans": [{"bbox": [109, 575, 410, 590], "score": 1.0, "content": "fact, suppose to walk along the canonically oriented cycle ", "type": "text"}, {"bbox": [410, 577, 424, 590], "score": 0.92, "content": "D_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [425, 575, 441, 590], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [442, 577, 452, 586], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [452, 575, 500, 590], "score": 1.0, "content": ", starting", "type": "text"}], "index": 31}, {"bbox": [110, 590, 500, 604], "spans": [{"bbox": [110, 590, 185, 604], "score": 1.0, "content": "from a vertex ", "type": "text"}, {"bbox": [185, 593, 191, 600], "score": 0.88, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [191, 590, 236, 604], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [236, 592, 248, 602], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [249, 590, 333, 604], "score": 1.0, "content": " be the cycle of ", "type": "text"}, {"bbox": [334, 592, 341, 601], "score": 0.91, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [342, 590, 403, 604], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [403, 593, 409, 601], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [410, 590, 432, 604], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [432, 591, 441, 600], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [441, 590, 500, 604], "score": 1.0, "content": " is the first", "type": "text"}], "index": 32}, {"bbox": [109, 604, 500, 618], "spans": [{"bbox": [109, 604, 270, 618], "score": 1.0, "content": "vertex with the same label of ", "type": "text"}, {"bbox": [271, 608, 277, 615], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [277, 604, 317, 618], "score": 1.0, "content": " and if ", "type": "text"}, {"bbox": [318, 606, 332, 617], "score": 0.92, "content": "C_{i^{\\prime}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [333, 604, 416, 618], "score": 1.0, "content": " is the cycle of ", "type": "text"}, {"bbox": [416, 606, 424, 615], "score": 0.9, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [424, 604, 486, 618], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [486, 606, 496, 615], "score": 0.89, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [496, 604, 500, 618], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [109, 617, 501, 635], "spans": [{"bbox": [109, 617, 155, 635], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [156, 620, 244, 632], "score": 0.92, "content": "\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [245, 617, 340, 635], "score": 1.0, "content": ". Thus, the cycle ", "type": "text"}, {"bbox": [340, 621, 354, 633], "score": 0.93, "content": "D_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [355, 617, 406, 635], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [406, 621, 412, 629], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [413, 617, 501, 635], "score": 1.0, "content": " arcs if and only", "type": "text"}], "index": 34}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 121, 648], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 635, 188, 646], "score": 0.92, "content": "q_{\\sigma}+s p_{\\sigma}\\equiv0", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [189, 633, 223, 648], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [224, 638, 231, 644], "score": 0.78, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [232, 633, 500, 648], "score": 1.0, "content": "). This proves that the 6-tuple satisfies (ii\u2019). Thus,", "type": "text"}], "index": 35}, {"bbox": [110, 648, 499, 662], "spans": [{"bbox": [110, 648, 499, 662], "score": 1.0, "content": "(i\u2019) and (ii\u2019) are respectively, in a different language, conditions (i) and (ii)", "type": "text"}], "index": 36}, {"bbox": [110, 662, 500, 676], "spans": [{"bbox": [110, 662, 500, 676], "score": 1.0, "content": "of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "type": "text"}], "index": 37}], "index": 33.5}], "layout_bboxes": [], "page_idx": 8, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [300, 691, 308, 702], "lines": [{"bbox": [301, 693, 309, 704], "spans": [{"bbox": [301, 693, 309, 704], "score": 1.0, "content": "9", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 499, 153], "lines": [], "index": 0.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [109, 127, 499, 155], "lines_deleted": true}, {"type": "text", "bbox": [109, 155, 500, 313], "lines": [{"bbox": [126, 155, 499, 171], "spans": [{"bbox": [126, 155, 332, 171], "score": 1.0, "content": "Property (i\u2019) implies that the cycles of ", "type": "text"}, {"bbox": [332, 159, 342, 167], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [343, 155, 499, 171], "score": 1.0, "content": " naturally induce a cyclic per-", "type": "text"}], "index": 2}, {"bbox": [109, 171, 501, 186], "spans": [{"bbox": [109, 171, 214, 186], "score": 1.0, "content": "mutation on the set ", "type": "text"}, {"bbox": [214, 172, 294, 185], "score": 0.95, "content": "\\mathcal{N}=\\{1,\\dotsc,d\\}", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [294, 171, 501, 186], "score": 1.0, "content": " of the vertex labels. In fact, by walking", "type": "text"}], "index": 3}, {"bbox": [109, 185, 499, 199], "spans": [{"bbox": [109, 185, 492, 199], "score": 1.0, "content": "along these canonically oriented cycles, starting from an arbitrary vertex ", "type": "text"}, {"bbox": [493, 189, 499, 196], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}], "index": 4}, {"bbox": [109, 200, 501, 215], "spans": [{"bbox": [109, 200, 153, 215], "score": 1.0, "content": "labelled ", "type": "text"}, {"bbox": [153, 202, 159, 213], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 6}, {"bbox": [159, 200, 285, 215], "score": 1.0, "content": ", one sequentially meets ", "type": "text"}, {"bbox": [285, 201, 291, 210], "score": 0.88, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [291, 200, 501, 215], "score": 1.0, "content": " vertices (whose labels are different from", "type": "text"}], "index": 5}, {"bbox": [110, 214, 501, 229], "spans": [{"bbox": [110, 214, 293, 229], "score": 1.0, "content": "each other), and then a new vertex ", "type": "text"}, {"bbox": [293, 216, 302, 225], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [302, 214, 348, 229], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [349, 216, 354, 227], "score": 0.89, "content": "j", "type": "inline_equation", "height": 11, "width": 5}, {"bbox": [355, 214, 501, 229], "score": 1.0, "content": " which can be different from", "type": "text"}], "index": 6}, {"bbox": [110, 229, 500, 243], "spans": [{"bbox": [110, 232, 117, 240], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [117, 229, 328, 243], "score": 1.0, "content": ". The sequence of the labellings of these ", "type": "text"}, {"bbox": [329, 231, 335, 240], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [335, 229, 500, 243], "score": 1.0, "content": " consecutive vertices defines the", "type": "text"}], "index": 7}, {"bbox": [110, 244, 499, 257], "spans": [{"bbox": [110, 244, 224, 257], "score": 1.0, "content": "cyclic permutation on ", "type": "text"}, {"bbox": [225, 245, 236, 254], "score": 0.89, "content": "\\mathcal{N}", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [237, 244, 357, 257], "score": 1.0, "content": ". Further, each cycle of ", "type": "text"}, {"bbox": [358, 245, 368, 254], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [368, 244, 464, 257], "score": 1.0, "content": " precisely contains ", "type": "text"}, {"bbox": [464, 245, 499, 254], "score": 0.93, "content": "d^{\\prime}=l d", "type": "inline_equation", "height": 9, "width": 35}], "index": 8}, {"bbox": [110, 258, 500, 272], "spans": [{"bbox": [110, 258, 164, 272], "score": 1.0, "content": "arcs, with ", "type": "text"}, {"bbox": [164, 259, 190, 270], "score": 0.92, "content": "l\\geq1", "type": "inline_equation", "height": 11, "width": 26}, {"bbox": [190, 258, 219, 272], "score": 1.0, "content": ", and ", "type": "text"}, {"bbox": [219, 259, 245, 268], "score": 0.93, "content": "l=1", "type": "inline_equation", "height": 9, "width": 26}, {"bbox": [245, 258, 500, 272], "score": 1.0, "content": " if and only if the 6-tuple satisfies (ii\u2019) too. More-", "type": "text"}], "index": 9}, {"bbox": [109, 272, 500, 286], "spans": [{"bbox": [109, 272, 370, 286], "score": 1.0, "content": "over, property (i\u2019) is independent from the integers ", "type": "text"}, {"bbox": [370, 277, 378, 282], "score": 0.89, "content": "n", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [378, 272, 403, 286], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [403, 277, 408, 282], "score": 0.86, "content": "s", "type": "inline_equation", "height": 5, "width": 5}, {"bbox": [409, 272, 500, 286], "score": 1.0, "content": "; hence, given two", "type": "text"}], "index": 10}, {"bbox": [109, 286, 502, 301], "spans": [{"bbox": [109, 286, 155, 301], "score": 1.0, "content": "6-tuples ", "type": "text"}, {"bbox": [155, 288, 250, 300], "score": 0.93, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 95}, {"bbox": [251, 286, 279, 301], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [279, 288, 380, 300], "score": 0.93, "content": "{\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 101}, {"bbox": [380, 286, 415, 301], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [415, 291, 423, 297], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [423, 286, 502, 301], "score": 1.0, "content": " satisfies (i\u2019) if", "type": "text"}], "index": 11}, {"bbox": [110, 301, 246, 314], "spans": [{"bbox": [110, 301, 169, 314], "score": 1.0, "content": "and only if ", "type": "text"}, {"bbox": [169, 302, 179, 311], "score": 0.91, "content": "\\sigma^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [180, 301, 246, 314], "score": 1.0, "content": " satisfies (i\u2019).", "type": "text"}], "index": 12}], "index": 7, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [109, 155, 502, 314]}, {"type": "text", "bbox": [109, 313, 501, 471], "lines": [{"bbox": [125, 314, 498, 330], "spans": [{"bbox": [125, 314, 173, 330], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [174, 320, 181, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [181, 314, 417, 330], "score": 1.0, "content": " be a 6-tuple satisfying (i\u2019) and suppose that ", "type": "text"}, {"bbox": [418, 317, 425, 326], "score": 0.89, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [426, 314, 498, 330], "score": 1.0, "content": " is canonically", "type": "text"}], "index": 13}, {"bbox": [110, 330, 500, 344], "spans": [{"bbox": [110, 330, 213, 344], "score": 1.0, "content": "oriented. An arc of ", "type": "text"}, {"bbox": [214, 331, 221, 340], "score": 0.9, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [222, 330, 293, 344], "score": 1.0, "content": " belonging to ", "type": "text"}, {"bbox": [293, 331, 302, 340], "score": 0.89, "content": "A", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [303, 330, 500, 344], "score": 1.0, "content": " is said to be of type I if it is oriented", "type": "text"}], "index": 14}, {"bbox": [110, 345, 501, 358], "spans": [{"bbox": [110, 345, 190, 358], "score": 1.0, "content": "from a cycle of ", "type": "text"}, {"bbox": [190, 346, 200, 355], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [200, 345, 270, 358], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [271, 346, 283, 355], "score": 0.9, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [284, 345, 501, 358], "score": 1.0, "content": ", of type II if it is oriented from a cycle of", "type": "text"}], "index": 15}, {"bbox": [110, 358, 502, 374], "spans": [{"bbox": [110, 360, 122, 370], "score": 0.91, "content": "\\mathcal{C^{\\prime\\prime}}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [123, 358, 191, 374], "score": 1.0, "content": " to a cycle of ", "type": "text"}, {"bbox": [192, 360, 202, 369], "score": 0.9, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [202, 358, 427, 374], "score": 1.0, "content": " and of type III otherwise (it joins cycles of ", "type": "text"}, {"bbox": [428, 360, 438, 369], "score": 0.91, "content": "\\mathcal{C}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [438, 358, 502, 374], "score": 1.0, "content": " or cycles of", "type": "text"}], "index": 16}, {"bbox": [110, 372, 500, 387], "spans": [{"bbox": [110, 374, 127, 387], "score": 0.66, "content": "\\mathcal{C}^{\\prime\\prime})", "type": "inline_equation", "height": 13, "width": 17}, {"bbox": [127, 372, 500, 387], "score": 1.0, "content": ". Moreover, the arc is said to be of type I\u2019 if it is oriented from a cycle", "type": "text"}], "index": 17}, {"bbox": [110, 387, 502, 406], "spans": [{"bbox": [110, 389, 122, 401], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [123, 387, 160, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [160, 389, 174, 401], "score": 0.88, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [174, 387, 235, 406], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [236, 389, 258, 402], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 22}, {"bbox": [259, 387, 296, 406], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [296, 389, 319, 402], "score": 0.9, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [320, 387, 502, 406], "score": 1.0, "content": "), of type II\u2019 if it is oriented from a", "type": "text"}], "index": 18}, {"bbox": [108, 401, 501, 419], "spans": [{"bbox": [108, 401, 138, 419], "score": 1.0, "content": "cycle ", "type": "text"}, {"bbox": [139, 403, 162, 416], "score": 0.93, "content": "C_{i+1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [162, 401, 198, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [199, 403, 222, 416], "score": 0.92, "content": "C_{i+1}^{\\prime\\prime}", "type": "inline_equation", "height": 13, "width": 23}, {"bbox": [222, 401, 281, 419], "score": 1.0, "content": ") to a cycle ", "type": "text"}, {"bbox": [281, 403, 293, 415], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [294, 401, 330, 419], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [331, 403, 345, 415], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [345, 401, 501, 419], "score": 1.0, "content": ") and of type III\u2019 otherwise (it", "type": "text"}], "index": 19}, {"bbox": [108, 417, 500, 430], "spans": [{"bbox": [108, 417, 138, 430], "score": 1.0, "content": "joins ", "type": "text"}, {"bbox": [138, 418, 150, 430], "score": 0.92, "content": "C_{i}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [151, 417, 182, 430], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [182, 418, 196, 430], "score": 0.9, "content": "C_{i}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [196, 417, 232, 430], "score": 1.0, "content": "). Let ", "type": "text"}, {"bbox": [232, 418, 243, 427], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [243, 417, 361, 430], "score": 1.0, "content": "be the set of the first ", "type": "text"}, {"bbox": [361, 418, 368, 427], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [368, 417, 410, 430], "score": 1.0, "content": " arcs of ", "type": "text"}, {"bbox": [410, 418, 425, 429], "score": 0.92, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [425, 417, 500, 430], "score": 1.0, "content": ", following the", "type": "text"}], "index": 20}, {"bbox": [108, 432, 502, 444], "spans": [{"bbox": [108, 432, 477, 444], "score": 1.0, "content": "canonical orientation, starting from the arc coming out from the vertex ", "type": "text"}, {"bbox": [477, 432, 486, 441], "score": 0.91, "content": "v^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [486, 432, 502, 444], "score": 1.0, "content": " of", "type": "text"}], "index": 21}, {"bbox": [110, 446, 500, 459], "spans": [{"bbox": [110, 447, 123, 459], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [123, 446, 170, 458], "score": 1.0, "content": " labelled ", "type": "text"}, {"bbox": [170, 447, 215, 457], "score": 0.94, "content": "a+b+1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [216, 446, 318, 458], "score": 1.0, "content": ". Obviously, the set ", "type": "text"}, {"bbox": [318, 447, 329, 456], "score": 0.89, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [329, 446, 451, 458], "score": 1.0, "content": "contains all the arcs of ", "type": "text"}, {"bbox": [451, 447, 466, 457], "score": 0.93, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [466, 446, 500, 458], "score": 1.0, "content": " if and", "type": "text"}], "index": 22}, {"bbox": [110, 461, 307, 473], "spans": [{"bbox": [110, 461, 206, 473], "score": 1.0, "content": "only if the 6-tuple ", "type": "text"}, {"bbox": [207, 465, 214, 470], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 461, 307, 473], "score": 1.0, "content": " also satisfies (ii\u2019).", "type": "text"}], "index": 23}], "index": 18, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [108, 314, 502, 473]}, {"type": "text", "bbox": [110, 473, 500, 559], "lines": [{"bbox": [126, 473, 502, 490], "spans": [{"bbox": [126, 473, 212, 490], "score": 1.0, "content": "Now, denote by ", "type": "text"}, {"bbox": [213, 476, 224, 488], "score": 0.92, "content": "p_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [225, 473, 263, 490], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [263, 476, 275, 488], "score": 0.88, "content": "p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [275, 473, 502, 490], "score": 1.0, "content": ") the number of the arcs of type I (resp. of", "type": "text"}], "index": 24}, {"bbox": [109, 489, 501, 503], "spans": [{"bbox": [109, 489, 168, 503], "score": 1.0, "content": "type II) of ", "type": "text"}, {"bbox": [169, 491, 179, 499], "score": 0.9, "content": "\\Delta", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [179, 489, 225, 503], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [225, 490, 293, 502], "score": 0.94, "content": "p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [294, 489, 411, 503], "score": 1.0, "content": ". Similarly, denote by ", "type": "text"}, {"bbox": [411, 490, 422, 502], "score": 0.92, "content": "q_{\\sigma}^{\\prime}", "type": "inline_equation", "height": 12, "width": 11}, {"bbox": [423, 489, 462, 503], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [462, 490, 474, 502], "score": 0.88, "content": "q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 12}, {"bbox": [474, 489, 501, 503], "score": 1.0, "content": ") the", "type": "text"}], "index": 25}, {"bbox": [109, 502, 500, 518], "spans": [{"bbox": [109, 502, 252, 518], "score": 1.0, "content": "number of the arcs of type ", "type": "text"}, {"bbox": [252, 505, 261, 514], "score": 0.47, "content": "\\Gamma", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [261, 502, 374, 518], "score": 1.0, "content": " (resp. of type II\u2019) of ", "type": "text"}, {"bbox": [375, 505, 385, 514], "score": 0.91, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [385, 502, 430, 518], "score": 1.0, "content": "and set ", "type": "text"}, {"bbox": [430, 504, 496, 516], "score": 0.93, "content": "q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 66}, {"bbox": [496, 502, 500, 518], "score": 1.0, "content": ".", "type": "text"}], "index": 26}, {"bbox": [109, 517, 499, 532], "spans": [{"bbox": [109, 517, 162, 532], "score": 1.0, "content": "Note that ", "type": "text"}, {"bbox": [162, 523, 174, 531], "score": 0.9, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [174, 517, 292, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [293, 519, 315, 529], "score": 0.93, "content": "b\\!+\\!c", "type": "inline_equation", "height": 10, "width": 22}, {"bbox": [315, 517, 340, 532], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [340, 523, 351, 531], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [352, 517, 469, 532], "score": 1.0, "content": " has the same parity of ", "type": "text"}, {"bbox": [470, 520, 499, 529], "score": 0.92, "content": "2a+b", "type": "inline_equation", "height": 9, "width": 29}], "index": 27}, {"bbox": [109, 532, 500, 546], "spans": [{"bbox": [109, 532, 182, 546], "score": 1.0, "content": "and hence of ", "type": "text"}, {"bbox": [182, 534, 188, 542], "score": 0.88, "content": "b", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [188, 532, 295, 546], "score": 1.0, "content": ". It is evident that ", "type": "text"}, {"bbox": [295, 537, 307, 545], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [307, 532, 335, 546], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [336, 537, 347, 545], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [347, 532, 500, 546], "score": 1.0, "content": " only depend on the integers", "type": "text"}], "index": 28}, {"bbox": [110, 547, 154, 561], "spans": [{"bbox": [110, 548, 148, 559], "score": 0.93, "content": "a,b,c,r", "type": "inline_equation", "height": 11, "width": 38}, {"bbox": [148, 547, 154, 561], "score": 1.0, "content": ".", "type": "text"}], "index": 29}], "index": 26.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [109, 473, 502, 561]}, {"type": "text", "bbox": [109, 559, 500, 675], "lines": [{"bbox": [127, 560, 500, 576], "spans": [{"bbox": [127, 560, 194, 576], "score": 1.0, "content": "The integers ", "type": "text"}, {"bbox": [194, 565, 206, 574], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [207, 560, 232, 576], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [232, 566, 243, 574], "score": 0.91, "content": "q_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [244, 560, 500, 576], "score": 1.0, "content": " give an useful tool for verifying condition (ii\u2019). In", "type": "text"}], "index": 30}, {"bbox": [109, 575, 500, 590], "spans": [{"bbox": [109, 575, 410, 590], "score": 1.0, "content": "fact, suppose to walk along the canonically oriented cycle ", "type": "text"}, {"bbox": [410, 577, 424, 590], "score": 0.92, "content": "D_{j}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [425, 575, 441, 590], "score": 1.0, "content": " of ", "type": "text"}, {"bbox": [442, 577, 452, 586], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [452, 575, 500, 590], "score": 1.0, "content": ", starting", "type": "text"}], "index": 31}, {"bbox": [110, 590, 500, 604], "spans": [{"bbox": [110, 590, 185, 604], "score": 1.0, "content": "from a vertex ", "type": "text"}, {"bbox": [185, 593, 191, 600], "score": 0.88, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [191, 590, 236, 604], "score": 1.0, "content": " and let ", "type": "text"}, {"bbox": [236, 592, 248, 602], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [249, 590, 333, 604], "score": 1.0, "content": " be the cycle of ", "type": "text"}, {"bbox": [334, 592, 341, 601], "score": 0.91, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [342, 590, 403, 604], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [403, 593, 409, 601], "score": 0.89, "content": "v", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [410, 590, 432, 604], "score": 1.0, "content": ". If ", "type": "text"}, {"bbox": [432, 591, 441, 600], "score": 0.91, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [441, 590, 500, 604], "score": 1.0, "content": " is the first", "type": "text"}], "index": 32}, {"bbox": [109, 604, 500, 618], "spans": [{"bbox": [109, 604, 270, 618], "score": 1.0, "content": "vertex with the same label of ", "type": "text"}, {"bbox": [271, 608, 277, 615], "score": 0.89, "content": "v", "type": "inline_equation", "height": 7, "width": 6}, {"bbox": [277, 604, 317, 618], "score": 1.0, "content": " and if ", "type": "text"}, {"bbox": [318, 606, 332, 617], "score": 0.92, "content": "C_{i^{\\prime}}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [333, 604, 416, 618], "score": 1.0, "content": " is the cycle of ", "type": "text"}, {"bbox": [416, 606, 424, 615], "score": 0.9, "content": "\\mathcal{C}", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [424, 604, 486, 618], "score": 1.0, "content": " containing ", "type": "text"}, {"bbox": [486, 606, 496, 615], "score": 0.89, "content": "\\bar{v}^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [496, 604, 500, 618], "score": 1.0, "content": ",", "type": "text"}], "index": 33}, {"bbox": [109, 617, 501, 635], "spans": [{"bbox": [109, 617, 155, 635], "score": 1.0, "content": "we have ", "type": "text"}, {"bbox": [156, 620, 244, 632], "score": 0.92, "content": "\\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma}", "type": "inline_equation", "height": 12, "width": 88}, {"bbox": [245, 617, 340, 635], "score": 1.0, "content": ". Thus, the cycle ", "type": "text"}, {"bbox": [340, 621, 354, 633], "score": 0.93, "content": "D_{j}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [355, 617, 406, 635], "score": 1.0, "content": " contains ", "type": "text"}, {"bbox": [406, 621, 412, 629], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [413, 617, 501, 635], "score": 1.0, "content": " arcs if and only", "type": "text"}], "index": 34}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 121, 648], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 635, 188, 646], "score": 0.92, "content": "q_{\\sigma}+s p_{\\sigma}\\equiv0", "type": "inline_equation", "height": 11, "width": 67}, {"bbox": [189, 633, 223, 648], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [224, 638, 231, 644], "score": 0.78, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [232, 633, 500, 648], "score": 1.0, "content": "). This proves that the 6-tuple satisfies (ii\u2019). Thus,", "type": "text"}], "index": 35}, {"bbox": [110, 648, 499, 662], "spans": [{"bbox": [110, 648, 499, 662], "score": 1.0, "content": "(i\u2019) and (ii\u2019) are respectively, in a different language, conditions (i) and (ii)", "type": "text"}], "index": 36}, {"bbox": [110, 662, 500, 676], "spans": [{"bbox": [110, 662, 500, 676], "score": 1.0, "content": "of Theorem 2 of [6], which gives a necessary and sufficient condition for a", "type": "text"}], "index": 37}], "index": 33.5, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [109, 560, 501, 676]}]}
|
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Property (i’) implies that the cycles of naturally induce a cyclic per- mutation on the set of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex labelled , one sequentially meets vertices (whose labels are different from each other), and then a new vertex labelled which can be different from . The sequence of the labellings of these consecutive vertices defines the cyclic permutation on . Further, each cycle of precisely contains arcs, with , and if and only if the 6-tuple satisfies (ii’) too. More- over, property (i’) is independent from the integers and ; hence, given two 6-tuples and , then satisfies (i’) if and only if satisfies (i’).
Let now be a 6-tuple satisfying (i’) and suppose that is canonically oriented. An arc of belonging to is said to be of type I if it is oriented from a cycle of to a cycle of , of type II if it is oriented from a cycle of to a cycle of and of type III otherwise (it joins cycles of or cycles of . Moreover, the arc is said to be of type I’ if it is oriented from a cycle (resp. ) to a cycle (resp. ), of type II’ if it is oriented from a cycle (resp. ) to a cycle (resp. ) and of type III’ otherwise (it joins with ). Let be the set of the first arcs of , following the canonical orientation, starting from the arc coming out from the vertex of labelled . Obviously, the set contains all the arcs of if and only if the 6-tuple also satisfies (ii’).
Now, denote by (resp. ) the number of the arcs of type I (resp. of type II) of and set . Similarly, denote by (resp. ) the number of the arcs of type (resp. of type II’) of and set . Note that has the same parity of and has the same parity of and hence of . It is evident that and only depend on the integers .
The integers and give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle of , starting from a vertex and let be the cycle of containing . If is the first vertex with the same label of and if is the cycle of containing , we have . Thus, the cycle contains arcs if and only if (mod ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a
9
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<div class="pdf-page">
<p>Property (i’) implies that the cycles of naturally induce a cyclic per- mutation on the set of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex labelled , one sequentially meets vertices (whose labels are different from each other), and then a new vertex labelled which can be different from . The sequence of the labellings of these consecutive vertices defines the cyclic permutation on . Further, each cycle of precisely contains arcs, with , and if and only if the 6-tuple satisfies (ii’) too. More- over, property (i’) is independent from the integers and ; hence, given two 6-tuples and , then satisfies (i’) if and only if satisfies (i’).</p>
<p>Let now be a 6-tuple satisfying (i’) and suppose that is canonically oriented. An arc of belonging to is said to be of type I if it is oriented from a cycle of to a cycle of , of type II if it is oriented from a cycle of to a cycle of and of type III otherwise (it joins cycles of or cycles of . Moreover, the arc is said to be of type I’ if it is oriented from a cycle (resp. ) to a cycle (resp. ), of type II’ if it is oriented from a cycle (resp. ) to a cycle (resp. ) and of type III’ otherwise (it joins with ). Let be the set of the first arcs of , following the canonical orientation, starting from the arc coming out from the vertex of labelled . Obviously, the set contains all the arcs of if and only if the 6-tuple also satisfies (ii’).</p>
<p>Now, denote by (resp. ) the number of the arcs of type I (resp. of type II) of and set . Similarly, denote by (resp. ) the number of the arcs of type (resp. of type II’) of and set . Note that has the same parity of and has the same parity of and hence of . It is evident that and only depend on the integers .</p>
<p>The integers and give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle of , starting from a vertex and let be the cycle of containing . If is the first vertex with the same label of and if is the cycle of containing , we have . Thus, the cycle contains arcs if and only if (mod ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="182" data-y="200" data-width="654" data-height="204">Property (i’) implies that the cycles of naturally induce a cyclic per- mutation on the set of the vertex labels. In fact, by walking along these canonically oriented cycles, starting from an arbitrary vertex labelled , one sequentially meets vertices (whose labels are different from each other), and then a new vertex labelled which can be different from . The sequence of the labellings of these consecutive vertices defines the cyclic permutation on . Further, each cycle of precisely contains arcs, with , and if and only if the 6-tuple satisfies (ii’) too. More- over, property (i’) is independent from the integers and ; hence, given two 6-tuples and , then satisfies (i’) if and only if satisfies (i’).</p>
<p class="pdf-text" data-x="182" data-y="404" data-width="656" data-height="204">Let now be a 6-tuple satisfying (i’) and suppose that is canonically oriented. An arc of belonging to is said to be of type I if it is oriented from a cycle of to a cycle of , of type II if it is oriented from a cycle of to a cycle of and of type III otherwise (it joins cycles of or cycles of . Moreover, the arc is said to be of type I’ if it is oriented from a cycle (resp. ) to a cycle (resp. ), of type II’ if it is oriented from a cycle (resp. ) to a cycle (resp. ) and of type III’ otherwise (it joins with ). Let be the set of the first arcs of , following the canonical orientation, starting from the arc coming out from the vertex of labelled . Obviously, the set contains all the arcs of if and only if the 6-tuple also satisfies (ii’).</p>
<p class="pdf-text" data-x="184" data-y="611" data-width="652" data-height="111">Now, denote by (resp. ) the number of the arcs of type I (resp. of type II) of and set . Similarly, denote by (resp. ) the number of the arcs of type (resp. of type II’) of and set . Note that has the same parity of and has the same parity of and hence of . It is evident that and only depend on the integers .</p>
<p class="pdf-text" data-x="182" data-y="722" data-width="654" data-height="150">The integers and give an useful tool for verifying condition (ii’). In fact, suppose to walk along the canonically oriented cycle of , starting from a vertex and let be the cycle of containing . If is the first vertex with the same label of and if is the cycle of containing , we have . Thus, the cycle contains arcs if and only if (mod ). This proves that the 6-tuple satisfies (ii’). Thus, (i’) and (ii’) are respectively, in a different language, conditions (i) and (ii) of Theorem 2 of [6], which gives a necessary and sufficient condition for a</p>
<div class="pdf-discarded" data-x="501" data-y="893" data-width="14" data-height="14" style="opacity: 0.5;">9</div>
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"Property (i’) implies that the cycles of \\mathcal{D} naturally induce a cyclic per-",
"mutation on the set \\mathcal{N}=\\{1,\\dotsc,d\\} of the vertex labels. In fact, by walking",
"along these canonically oriented cycles, starting from an arbitrary vertex v",
"labelled j , one sequentially meets d vertices (whose labels are different from",
"each other), and then a new vertex \\bar{v}^{\\prime} labelled j which can be different from",
"v . The sequence of the labellings of these d consecutive vertices defines the",
"cyclic permutation on \\mathcal{N} . Further, each cycle of \\mathcal{D} precisely contains d^{\\prime}=l d",
"arcs, with l\\geq1 , and l=1 if and only if the 6-tuple satisfies (ii’) too. More-",
"over, property (i’) is independent from the integers n and s ; hence, given two",
"6-tuples \\sigma\\,=\\,(a,b,c,n,r,s) and {\\boldsymbol{\\sigma}}^{\\prime}\\,=\\,(a,b,c,n^{\\prime},r,s) , then \\sigma satisfies (i’) if",
"and only if \\sigma^{\\prime} satisfies (i’).",
"Let now \\sigma be a 6-tuple satisfying (i’) and suppose that \\Gamma is canonically",
"oriented. An arc of \\Gamma belonging to A is said to be of type I if it is oriented",
"from a cycle of \\mathcal{C}^{\\prime} to a cycle of \\mathcal{C^{\\prime\\prime}} , of type II if it is oriented from a cycle of",
"\\mathcal{C^{\\prime\\prime}} to a cycle of \\mathcal{C}^{\\prime} and of type III otherwise (it joins cycles of \\mathcal{C}^{\\prime} or cycles of",
"\\mathcal{C}^{\\prime\\prime}) . Moreover, the arc is said to be of type I’ if it is oriented from a cycle",
"C_{i}^{\\prime} (resp. C_{i}^{\\prime\\prime} ) to a cycle C_{i+1}^{\\prime} (resp. C_{i+1}^{\\prime\\prime} ), of type II’ if it is oriented from a",
"cycle C_{i+1}^{\\prime} (resp. C_{i+1}^{\\prime\\prime} ) to a cycle C_{i}^{\\prime} (resp. C_{i}^{\\prime\\prime} ) and of type III’ otherwise (it",
"joins C_{i}^{\\prime} with C_{i}^{\\prime\\prime} ). Let \\Delta be the set of the first d arcs of D_{1} , following the",
"canonical orientation, starting from the arc coming out from the vertex v^{\\prime} of",
"C_{1}^{\\prime} labelled a+b+1 . Obviously, the set \\Delta contains all the arcs of D_{1} if and",
"only if the 6-tuple \\sigma also satisfies (ii’).",
"Now, denote by p_{\\sigma}^{\\prime} (resp. p_{\\sigma}^{\\prime\\prime} ) the number of the arcs of type I (resp. of",
"type II) of \\Delta and set p_{\\sigma}=p_{\\sigma}^{\\prime}-p_{\\sigma}^{\\prime\\prime} . Similarly, denote by q_{\\sigma}^{\\prime} (resp. q_{\\sigma}^{\\prime\\prime} ) the",
"number of the arcs of type \\Gamma (resp. of type II’) of \\Delta and set q_{\\sigma}=q_{\\sigma}^{\\prime}-q_{\\sigma}^{\\prime\\prime} .",
"Note that p_{\\sigma} has the same parity of b\\!+\\!c and q_{\\sigma} has the same parity of 2a+b",
"and hence of b . It is evident that p_{\\sigma} and q_{\\sigma} only depend on the integers",
"a,b,c,r .",
"The integers p_{\\sigma} and q_{\\sigma} give an useful tool for verifying condition (ii’). In",
"fact, suppose to walk along the canonically oriented cycle D_{j} of \\mathcal{D} , starting",
"from a vertex v and let C_{i} be the cycle of \\mathcal{C} containing v . If \\bar{v}^{\\prime} is the first",
"vertex with the same label of v and if C_{i^{\\prime}} is the cycle of \\mathcal{C} containing \\bar{v}^{\\prime} ,",
"we have \\d j^{\\prime}=\\d j+\\d q_{\\sigma}+\\d s p_{\\sigma} . Thus, the cycle D_{j} contains d arcs if and only",
"if q_{\\sigma}+s p_{\\sigma}\\equiv0 (mod n ). This proves that the 6-tuple satisfies (ii’). Thus,",
"(i’) and (ii’) are respectively, in a different language, conditions (i) and (ii)",
"of Theorem 2 of [6], which gives a necessary and sufficient condition for a"
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"6-tuple to be admissible when is odd. In fact, we have the following result:",
"Lemma 3 ([6], Theorem 2) Let be a 6-tuple with odd. Then is admissible if and only if it satisfies and (ii’).",
"Remark 3. This result does not hold when is even. In fact, the 6-tuples , with even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.",
"An immediate consequence of Lemma 3 is the following result:",
"Corollary 4 Let be a -tuple with odd and . Then is admissible if and only if has a unique cycle.",
"Proof. If is admissible, then it is straightforward that has a unique cycle. Vice versa, if has a unique cycle, then (i’) holds. Since implies (ii’), the result is a direct consequence of the above lemma.",
"The parameter associated to an admissible 6-tuple is strictly related to the word associated to . In fact, we have:",
"Lemma 5 Let be an admissible 6-tuple, the associated word and its exponent-sum. Then",
"",
"Proof. Since is admissible, the arcs of are precisely the arcs of orientation on . Let , and let be the sequence of these arcs, following the canonical , with . We have: ,where . Since if is of type I, if is of type II and if is of type III, the result immediately follows.",
"In [6] Dunwoody investigates a wide subclass of manifolds such that and he conjectures that all the elements of this subclass are cyclic coverings of branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.",
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[{"type": "text", "text": "6-tuple to be admissible when $d$ is odd. In fact, we have the following result: ", "page_idx": 9}, {"type": "text", "text": "Lemma 3 ([6], Theorem 2) Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be a 6-tuple with $d\\,=$ $2a+b+c$ odd. Then $\\sigma$ is admissible if and only if it satisfies $(i\\,?)$ and (ii\u2019). ", "page_idx": 9}, {"type": "text", "text": "Remark 3. This result does not hold when $d$ is even. In fact, the 6-tuples $(1,0,c,1,2,0)$ , with $c$ even, satisfy (i\u2019) and (ii\u2019), but they are not admissible, as pointed out in Remark 1. ", "page_idx": 9}, {"type": "text", "text": "An immediate consequence of Lemma 3 is the following result: ", "page_idx": 9}, {"type": "text", "text": "Corollary 4 Let $\\sigma=(a,b,c,n,r,s)$ be a $\\it6$ -tuple with $d=2a+b+c$ odd and $n=1$ . Then $\\sigma$ is admissible if and only if $\\mathcal{D}$ has a unique cycle. ", "page_idx": 9}, {"type": "text", "text": "Proof. If $\\sigma$ is admissible, then it is straightforward that $\\mathcal{D}$ has a unique cycle. Vice versa, if $\\mathcal{D}$ has a unique cycle, then (i\u2019) holds. Since $n=1$ implies (ii\u2019), the result is a direct consequence of the above lemma. ", "page_idx": 9}, {"type": "text", "text": "The parameter $p_{\\sigma}$ associated to an admissible 6-tuple $\\sigma$ is strictly related to the word $w(\\sigma)$ associated to $\\sigma$ . In fact, we have: ", "page_idx": 9}, {"type": "text", "text": "Lemma 5 Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be an admissible 6-tuple, $w\\,=\\,w(\\sigma)$ the associated word and $\\varepsilon_{w}$ its exponent-sum. Then ", "page_idx": 9}, {"type": "equation", "text": "$$\np_{\\sigma}=\\varepsilon_{w}.\n$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "Proof. Since $\\sigma$ is admissible, the arcs of $\\Delta$ are precisely the arcs of orientation on $D_{1}$ . Let $e_{1},e_{2},\\ldots,e_{d}$ $D_{1}$ , and let be the sequence of these arcs, following the canonical $\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}$ , with $u_{h}\\,\\in\\,\\{+1,-1\\}$ . We have: $\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}$ ,where $d+1=1$ . Since $u_{h}\\!+\\!u_{h+1}=+2$ if $e_{h}$ is of type I, $u_{h}+u_{h+1}=-2$ if $e_{h}$ is of type II and $u_{h}+u_{h+1}=0$ if $e_{h}$ is of type III, the result immediately follows. ", "page_idx": 9}, {"type": "text", "text": "In [6] Dunwoody investigates a wide subclass of manifolds $M(\\sigma)$ such that $p_{\\sigma}=\\pm1$ and he conjectures that all the elements of this subclass are cyclic coverings of $\\mathbf{S^{3}}$ branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem. ", "page_idx": 9}]
|
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In fact, we have the following result:", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [109, 152, 499, 182], "lines": [{"bbox": [108, 154, 501, 170], "spans": [{"bbox": [108, 154, 284, 170], "score": 1.0, "content": "Lemma 3 ([6], Theorem 2) Let ", "type": "text"}, {"bbox": [284, 156, 380, 169], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [381, 154, 478, 170], "score": 1.0, "content": " be a 6-tuple with ", "type": "text"}, {"bbox": [478, 156, 501, 167], "score": 0.77, "content": "d\\,=", "type": "inline_equation", "height": 11, "width": 23}], "index": 1}, {"bbox": [110, 168, 497, 184], "spans": [{"bbox": [110, 171, 162, 181], "score": 0.91, "content": "2a+b+c", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [162, 168, 222, 184], "score": 1.0, "content": " odd. Then ", "type": "text"}, {"bbox": [222, 174, 229, 180], "score": 0.69, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [230, 168, 429, 184], "score": 1.0, "content": " is admissible if and only if it satisfies ", "type": "text"}, {"bbox": [429, 169, 447, 183], "score": 0.4, "content": "(i\\,?)", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [447, 168, 497, 184], "score": 1.0, "content": " and (ii\u2019).", "type": "text"}], "index": 2}], "index": 1.5}, {"type": "text", "bbox": [110, 194, 501, 237], "lines": [{"bbox": [110, 197, 499, 210], "spans": [{"bbox": [110, 197, 342, 210], "score": 1.0, "content": "Remark 3. This result does not hold when ", "type": "text"}, {"bbox": [342, 199, 349, 207], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [349, 197, 499, 210], "score": 1.0, "content": " is even. In fact, the 6-tuples", "type": "text"}], "index": 3}, {"bbox": [110, 212, 498, 225], "spans": [{"bbox": [110, 212, 181, 225], "score": 0.88, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [181, 212, 213, 225], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [213, 216, 218, 222], "score": 0.66, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [219, 212, 498, 225], "score": 1.0, "content": " even, satisfy (i\u2019) and (ii\u2019), but they are not admissible,", "type": "text"}], "index": 4}, {"bbox": [110, 227, 255, 239], "spans": [{"bbox": [110, 227, 255, 239], "score": 1.0, "content": "as pointed out in Remark 1.", "type": "text"}], "index": 5}], "index": 4}, {"type": "text", "bbox": [125, 244, 450, 258], "lines": [{"bbox": [127, 246, 449, 260], "spans": [{"bbox": [127, 246, 449, 260], "score": 1.0, "content": "An immediate consequence of Lemma 3 is the following result:", "type": "text"}], "index": 6}], "index": 6}, {"type": "text", "bbox": [109, 271, 502, 300], "lines": [{"bbox": [110, 273, 501, 288], "spans": [{"bbox": [110, 273, 202, 288], "score": 1.0, "content": "Corollary 4 Let ", "type": "text"}, {"bbox": [202, 275, 294, 287], "score": 0.93, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [295, 273, 322, 288], "score": 1.0, "content": " be a ", "type": "text"}, {"bbox": [322, 276, 329, 284], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [329, 273, 385, 288], "score": 1.0, "content": "-tuple with ", "type": "text"}, {"bbox": [385, 275, 455, 285], "score": 0.93, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [456, 273, 501, 288], "score": 1.0, "content": " odd and", "type": "text"}], "index": 7}, {"bbox": [110, 288, 439, 302], "spans": [{"bbox": [110, 290, 139, 298], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [140, 288, 178, 302], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [178, 293, 185, 298], "score": 0.6, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [186, 288, 329, 302], "score": 1.0, "content": " is admissible if and only if ", "type": "text"}, {"bbox": [329, 290, 339, 298], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [339, 288, 439, 302], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 8}], "index": 7.5}, {"type": "text", "bbox": [109, 312, 500, 356], "lines": [{"bbox": [126, 315, 500, 329], "spans": [{"bbox": [126, 315, 181, 329], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [181, 320, 188, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [189, 315, 420, 329], "score": 1.0, "content": " is admissible, then it is straightforward that ", "type": "text"}, {"bbox": [420, 317, 430, 326], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [431, 315, 500, 329], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 9}, {"bbox": [109, 330, 499, 344], "spans": [{"bbox": [109, 330, 211, 344], "score": 1.0, "content": "cycle. Vice versa, if ", "type": "text"}, {"bbox": [211, 332, 221, 340], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [222, 330, 431, 344], "score": 1.0, "content": " has a unique cycle, then (i\u2019) holds. Since ", "type": "text"}, {"bbox": [431, 332, 460, 340], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [460, 330, 499, 344], "score": 1.0, "content": " implies", "type": "text"}], "index": 10}, {"bbox": [110, 344, 431, 358], "spans": [{"bbox": [110, 344, 431, 358], "score": 1.0, "content": "(ii\u2019), the result is a direct consequence of the above lemma.", "type": "text"}], "index": 11}], "index": 10}, {"type": "text", "bbox": [109, 357, 501, 385], "lines": [{"bbox": [126, 358, 500, 375], "spans": [{"bbox": [126, 358, 206, 375], "score": 1.0, "content": "The parameter ", "type": "text"}, {"bbox": [207, 364, 218, 372], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [218, 358, 402, 375], "score": 1.0, "content": " associated to an admissible 6-tuple ", "type": "text"}, {"bbox": [402, 364, 409, 369], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [410, 358, 500, 375], "score": 1.0, "content": " is strictly related", "type": "text"}], "index": 12}, {"bbox": [110, 373, 374, 388], "spans": [{"bbox": [110, 373, 173, 388], "score": 1.0, "content": "to the word ", "type": "text"}, {"bbox": [173, 374, 199, 387], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [199, 373, 272, 388], "score": 1.0, "content": " associated to ", "type": "text"}, {"bbox": [272, 378, 280, 384], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [280, 373, 374, 388], "score": 1.0, "content": ". In fact, we have:", "type": "text"}], "index": 13}], "index": 12.5}, {"type": "text", "bbox": [109, 398, 500, 427], "lines": [{"bbox": [109, 400, 501, 415], "spans": [{"bbox": [109, 400, 191, 415], "score": 1.0, "content": "Lemma 5 Let ", "type": "text"}, {"bbox": [191, 402, 287, 414], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [287, 400, 425, 415], "score": 1.0, "content": " be an admissible 6-tuple, ", "type": "text"}, {"bbox": [426, 402, 479, 414], "score": 0.95, "content": "w\\,=\\,w(\\sigma)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [479, 400, 501, 415], "score": 1.0, "content": " the", "type": "text"}], "index": 14}, {"bbox": [110, 416, 355, 428], "spans": [{"bbox": [110, 416, 215, 428], "score": 1.0, "content": "associated word and ", "type": "text"}, {"bbox": [216, 420, 228, 427], "score": 0.88, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [228, 416, 355, 428], "score": 1.0, "content": " its exponent-sum. Then", "type": "text"}], "index": 15}], "index": 14.5}, {"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16}, {"type": "text", "bbox": [109, 466, 501, 555], "lines": [{"bbox": [127, 468, 501, 481], "spans": [{"bbox": [127, 468, 205, 481], "score": 1.0, "content": "Proof. Since ", "type": "text"}, {"bbox": [206, 474, 213, 479], "score": 0.86, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 468, 356, 481], "score": 1.0, "content": " is admissible, the arcs of ", "type": "text"}, {"bbox": [356, 470, 366, 479], "score": 0.88, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 468, 501, 481], "score": 1.0, "content": "are precisely the arcs of", "type": "text"}], "index": 17}, {"bbox": [105, 482, 504, 518], "spans": [{"bbox": [105, 491, 187, 518], "score": 1.0, "content": "orientation on ", "type": "text"}, {"bbox": [110, 485, 125, 496], "score": 0.89, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [125, 482, 155, 499], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [155, 488, 220, 496], "score": 0.86, "content": "e_{1},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 8, "width": 65}, {"bbox": [188, 500, 203, 510], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [203, 491, 252, 518], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [220, 482, 501, 499], "score": 1.0, "content": " be the sequence of these arcs, following the canonical", "type": "text"}, {"bbox": [252, 496, 327, 513], "score": 0.94, "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "type": "inline_equation", "height": 17, "width": 75}, {"bbox": [327, 491, 362, 518], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [362, 499, 440, 511], "score": 0.94, "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [440, 491, 504, 518], "score": 1.0, "content": ". We have:", "type": "text"}], "index": 18}, {"bbox": [110, 508, 499, 533], "spans": [{"bbox": [110, 513, 302, 529], "score": 0.91, "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "type": "inline_equation", "height": 16, "width": 192}, {"bbox": [302, 508, 337, 533], "score": 1.0, "content": ",where ", "type": "text"}, {"bbox": [338, 516, 383, 526], "score": 0.92, "content": "d+1=1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [384, 508, 421, 533], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [421, 517, 499, 528], "score": 0.92, "content": "u_{h}\\!+\\!u_{h+1}=+2", "type": "inline_equation", "height": 11, "width": 78}], "index": 19}, {"bbox": [109, 529, 500, 543], "spans": [{"bbox": [109, 529, 120, 543], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 534, 132, 541], "score": 0.88, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [132, 529, 197, 543], "score": 1.0, "content": " is of type I, ", "type": "text"}, {"bbox": [198, 531, 277, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=-2", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [277, 529, 291, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [291, 534, 302, 541], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [302, 529, 392, 543], "score": 1.0, "content": " is of type II and ", "type": "text"}, {"bbox": [392, 532, 462, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=0", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [462, 529, 476, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [476, 534, 487, 541], "score": 0.9, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [488, 529, 500, 543], "score": 1.0, "content": " is", "type": "text"}], "index": 20}, {"bbox": [109, 543, 346, 557], "spans": [{"bbox": [109, 543, 346, 557], "score": 1.0, "content": "of type III, the result immediately follows.", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [109, 556, 501, 614], "lines": [{"bbox": [126, 558, 500, 572], "spans": [{"bbox": [126, 558, 420, 572], "score": 1.0, "content": "In [6] Dunwoody investigates a wide subclass of manifolds ", "type": "text"}, {"bbox": [420, 559, 449, 571], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [449, 558, 500, 572], "score": 1.0, "content": " such that", "type": "text"}], "index": 22}, {"bbox": [110, 573, 500, 586], "spans": [{"bbox": [110, 574, 154, 585], "score": 0.91, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [154, 573, 500, 586], "score": 1.0, "content": " and he conjectures that all the elements of this subclass are cyclic", "type": "text"}], "index": 23}, {"bbox": [109, 586, 500, 601], "spans": [{"bbox": [109, 586, 173, 601], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [173, 587, 186, 597], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [186, 586, 500, 601], "score": 1.0, "content": " branched over knots. In the next chapter this conjecture will", "type": "text"}], "index": 24}, {"bbox": [109, 600, 375, 616], "spans": [{"bbox": [109, 600, 375, 616], "score": 1.0, "content": "be proved as a corollary of a more general theorem.", "type": "text"}], "index": 25}], "index": 23.5}], "layout_bboxes": [], "page_idx": 9, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16}], "discarded_blocks": [{"type": "discarded", "bbox": [298, 691, 312, 702], "lines": [{"bbox": [297, 692, 312, 705], "spans": [{"bbox": [297, 692, 312, 705], "score": 1.0, "content": "10", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [109, 125, 499, 140], "lines": [{"bbox": [109, 127, 499, 141], "spans": [{"bbox": [109, 127, 265, 141], "score": 1.0, "content": "6-tuple to be admissible when ", "type": "text"}, {"bbox": [266, 129, 272, 138], "score": 0.89, "content": "d", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [272, 127, 499, 141], "score": 1.0, "content": " is odd. In fact, we have the following result:", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [109, 127, 499, 141]}, {"type": "text", "bbox": [109, 152, 499, 182], "lines": [{"bbox": [108, 154, 501, 170], "spans": [{"bbox": [108, 154, 284, 170], "score": 1.0, "content": "Lemma 3 ([6], Theorem 2) Let ", "type": "text"}, {"bbox": [284, 156, 380, 169], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [381, 154, 478, 170], "score": 1.0, "content": " be a 6-tuple with ", "type": "text"}, {"bbox": [478, 156, 501, 167], "score": 0.77, "content": "d\\,=", "type": "inline_equation", "height": 11, "width": 23}], "index": 1}, {"bbox": [110, 168, 497, 184], "spans": [{"bbox": [110, 171, 162, 181], "score": 0.91, "content": "2a+b+c", "type": "inline_equation", "height": 10, "width": 52}, {"bbox": [162, 168, 222, 184], "score": 1.0, "content": " odd. Then ", "type": "text"}, {"bbox": [222, 174, 229, 180], "score": 0.69, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [230, 168, 429, 184], "score": 1.0, "content": " is admissible if and only if it satisfies ", "type": "text"}, {"bbox": [429, 169, 447, 183], "score": 0.4, "content": "(i\\,?)", "type": "inline_equation", "height": 14, "width": 18}, {"bbox": [447, 168, 497, 184], "score": 1.0, "content": " and (ii\u2019).", "type": "text"}], "index": 2}], "index": 1.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [108, 154, 501, 184]}, {"type": "text", "bbox": [110, 194, 501, 237], "lines": [{"bbox": [110, 197, 499, 210], "spans": [{"bbox": [110, 197, 342, 210], "score": 1.0, "content": "Remark 3. This result does not hold when ", "type": "text"}, {"bbox": [342, 199, 349, 207], "score": 0.88, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [349, 197, 499, 210], "score": 1.0, "content": " is even. In fact, the 6-tuples", "type": "text"}], "index": 3}, {"bbox": [110, 212, 498, 225], "spans": [{"bbox": [110, 212, 181, 225], "score": 0.88, "content": "(1,0,c,1,2,0)", "type": "inline_equation", "height": 13, "width": 71}, {"bbox": [181, 212, 213, 225], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [213, 216, 218, 222], "score": 0.66, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [219, 212, 498, 225], "score": 1.0, "content": " even, satisfy (i\u2019) and (ii\u2019), but they are not admissible,", "type": "text"}], "index": 4}, {"bbox": [110, 227, 255, 239], "spans": [{"bbox": [110, 227, 255, 239], "score": 1.0, "content": "as pointed out in Remark 1.", "type": "text"}], "index": 5}], "index": 4, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [110, 197, 499, 239]}, {"type": "text", "bbox": [125, 244, 450, 258], "lines": [{"bbox": [127, 246, 449, 260], "spans": [{"bbox": [127, 246, 449, 260], "score": 1.0, "content": "An immediate consequence of Lemma 3 is the following result:", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [127, 246, 449, 260]}, {"type": "text", "bbox": [109, 271, 502, 300], "lines": [{"bbox": [110, 273, 501, 288], "spans": [{"bbox": [110, 273, 202, 288], "score": 1.0, "content": "Corollary 4 Let ", "type": "text"}, {"bbox": [202, 275, 294, 287], "score": 0.93, "content": "\\sigma=(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 92}, {"bbox": [295, 273, 322, 288], "score": 1.0, "content": " be a ", "type": "text"}, {"bbox": [322, 276, 329, 284], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [329, 273, 385, 288], "score": 1.0, "content": "-tuple with ", "type": "text"}, {"bbox": [385, 275, 455, 285], "score": 0.93, "content": "d=2a+b+c", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [456, 273, 501, 288], "score": 1.0, "content": " odd and", "type": "text"}], "index": 7}, {"bbox": [110, 288, 439, 302], "spans": [{"bbox": [110, 290, 139, 298], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [140, 288, 178, 302], "score": 1.0, "content": ". Then ", "type": "text"}, {"bbox": [178, 293, 185, 298], "score": 0.6, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [186, 288, 329, 302], "score": 1.0, "content": " is admissible if and only if ", "type": "text"}, {"bbox": [329, 290, 339, 298], "score": 0.84, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [339, 288, 439, 302], "score": 1.0, "content": " has a unique cycle.", "type": "text"}], "index": 8}], "index": 7.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [110, 273, 501, 302]}, {"type": "text", "bbox": [109, 312, 500, 356], "lines": [{"bbox": [126, 315, 500, 329], "spans": [{"bbox": [126, 315, 181, 329], "score": 1.0, "content": "Proof. If ", "type": "text"}, {"bbox": [181, 320, 188, 326], "score": 0.88, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [189, 315, 420, 329], "score": 1.0, "content": " is admissible, then it is straightforward that ", "type": "text"}, {"bbox": [420, 317, 430, 326], "score": 0.9, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [431, 315, 500, 329], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 9}, {"bbox": [109, 330, 499, 344], "spans": [{"bbox": [109, 330, 211, 344], "score": 1.0, "content": "cycle. Vice versa, if ", "type": "text"}, {"bbox": [211, 332, 221, 340], "score": 0.91, "content": "\\mathcal{D}", "type": "inline_equation", "height": 8, "width": 10}, {"bbox": [222, 330, 431, 344], "score": 1.0, "content": " has a unique cycle, then (i\u2019) holds. Since ", "type": "text"}, {"bbox": [431, 332, 460, 340], "score": 0.91, "content": "n=1", "type": "inline_equation", "height": 8, "width": 29}, {"bbox": [460, 330, 499, 344], "score": 1.0, "content": " implies", "type": "text"}], "index": 10}, {"bbox": [110, 344, 431, 358], "spans": [{"bbox": [110, 344, 431, 358], "score": 1.0, "content": "(ii\u2019), the result is a direct consequence of the above lemma.", "type": "text"}], "index": 11}], "index": 10, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [109, 315, 500, 358]}, {"type": "text", "bbox": [109, 357, 501, 385], "lines": [{"bbox": [126, 358, 500, 375], "spans": [{"bbox": [126, 358, 206, 375], "score": 1.0, "content": "The parameter ", "type": "text"}, {"bbox": [207, 364, 218, 372], "score": 0.91, "content": "p_{\\sigma}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [218, 358, 402, 375], "score": 1.0, "content": " associated to an admissible 6-tuple ", "type": "text"}, {"bbox": [402, 364, 409, 369], "score": 0.89, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [410, 358, 500, 375], "score": 1.0, "content": " is strictly related", "type": "text"}], "index": 12}, {"bbox": [110, 373, 374, 388], "spans": [{"bbox": [110, 373, 173, 388], "score": 1.0, "content": "to the word ", "type": "text"}, {"bbox": [173, 374, 199, 387], "score": 0.94, "content": "w(\\sigma)", "type": "inline_equation", "height": 13, "width": 26}, {"bbox": [199, 373, 272, 388], "score": 1.0, "content": " associated to ", "type": "text"}, {"bbox": [272, 378, 280, 384], "score": 0.87, "content": "\\sigma", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [280, 373, 374, 388], "score": 1.0, "content": ". In fact, we have:", "type": "text"}], "index": 13}], "index": 12.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [110, 358, 500, 388]}, {"type": "text", "bbox": [109, 398, 500, 427], "lines": [{"bbox": [109, 400, 501, 415], "spans": [{"bbox": [109, 400, 191, 415], "score": 1.0, "content": "Lemma 5 Let ", "type": "text"}, {"bbox": [191, 402, 287, 414], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 96}, {"bbox": [287, 400, 425, 415], "score": 1.0, "content": " be an admissible 6-tuple, ", "type": "text"}, {"bbox": [426, 402, 479, 414], "score": 0.95, "content": "w\\,=\\,w(\\sigma)", "type": "inline_equation", "height": 12, "width": 53}, {"bbox": [479, 400, 501, 415], "score": 1.0, "content": " the", "type": "text"}], "index": 14}, {"bbox": [110, 416, 355, 428], "spans": [{"bbox": [110, 416, 215, 428], "score": 1.0, "content": "associated word and ", "type": "text"}, {"bbox": [216, 420, 228, 427], "score": 0.88, "content": "\\varepsilon_{w}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [228, 416, 355, 428], "score": 1.0, "content": " its exponent-sum. Then", "type": "text"}], "index": 15}], "index": 14.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [109, 400, 501, 428]}, {"type": "interline_equation", "bbox": [282, 445, 326, 454], "lines": [{"bbox": [282, 445, 326, 454], "spans": [{"bbox": [282, 445, 326, 454], "score": 0.85, "content": "p_{\\sigma}=\\varepsilon_{w}.", "type": "interline_equation"}], "index": 16}], "index": 16, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [109, 466, 501, 555], "lines": [{"bbox": [127, 468, 501, 481], "spans": [{"bbox": [127, 468, 205, 481], "score": 1.0, "content": "Proof. Since ", "type": "text"}, {"bbox": [206, 474, 213, 479], "score": 0.86, "content": "\\sigma", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [214, 468, 356, 481], "score": 1.0, "content": " is admissible, the arcs of ", "type": "text"}, {"bbox": [356, 470, 366, 479], "score": 0.88, "content": "\\Delta", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [367, 468, 501, 481], "score": 1.0, "content": "are precisely the arcs of", "type": "text"}], "index": 17}, {"bbox": [105, 482, 504, 518], "spans": [{"bbox": [105, 491, 187, 518], "score": 1.0, "content": "orientation on ", "type": "text"}, {"bbox": [110, 485, 125, 496], "score": 0.89, "content": "D_{1}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [125, 482, 155, 499], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [155, 488, 220, 496], "score": 0.86, "content": "e_{1},e_{2},\\ldots,e_{d}", "type": "inline_equation", "height": 8, "width": 65}, {"bbox": [188, 500, 203, 510], "score": 0.91, "content": "D_{1}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [203, 491, 252, 518], "score": 1.0, "content": ", and let ", "type": "text"}, {"bbox": [220, 482, 501, 499], "score": 1.0, "content": " be the sequence of these arcs, following the canonical", "type": "text"}, {"bbox": [252, 496, 327, 513], "score": 0.94, "content": "\\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array}", "type": "inline_equation", "height": 17, "width": 75}, {"bbox": [327, 491, 362, 518], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [362, 499, 440, 511], "score": 0.94, "content": "u_{h}\\,\\in\\,\\{+1,-1\\}", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [440, 491, 504, 518], "score": 1.0, "content": ". We have:", "type": "text"}], "index": 18}, {"bbox": [110, 508, 499, 533], "spans": [{"bbox": [110, 513, 302, 529], "score": 0.91, "content": "\\begin{array}{r}{\\varepsilon_{w}=\\sum_{h=1}^{d}u_{h}=1/2\\sum_{h=1}^{d}(u_{h}+u_{h+1})}\\end{array}", "type": "inline_equation", "height": 16, "width": 192}, {"bbox": [302, 508, 337, 533], "score": 1.0, "content": ",where ", "type": "text"}, {"bbox": [338, 516, 383, 526], "score": 0.92, "content": "d+1=1", "type": "inline_equation", "height": 10, "width": 45}, {"bbox": [384, 508, 421, 533], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [421, 517, 499, 528], "score": 0.92, "content": "u_{h}\\!+\\!u_{h+1}=+2", "type": "inline_equation", "height": 11, "width": 78}], "index": 19}, {"bbox": [109, 529, 500, 543], "spans": [{"bbox": [109, 529, 120, 543], "score": 1.0, "content": "if ", "type": "text"}, {"bbox": [121, 534, 132, 541], "score": 0.88, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [132, 529, 197, 543], "score": 1.0, "content": " is of type I, ", "type": "text"}, {"bbox": [198, 531, 277, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=-2", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [277, 529, 291, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [291, 534, 302, 541], "score": 0.89, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [302, 529, 392, 543], "score": 1.0, "content": " is of type II and ", "type": "text"}, {"bbox": [392, 532, 462, 542], "score": 0.93, "content": "u_{h}+u_{h+1}=0", "type": "inline_equation", "height": 10, "width": 70}, {"bbox": [462, 529, 476, 543], "score": 1.0, "content": " if ", "type": "text"}, {"bbox": [476, 534, 487, 541], "score": 0.9, "content": "e_{h}", "type": "inline_equation", "height": 7, "width": 11}, {"bbox": [488, 529, 500, 543], "score": 1.0, "content": " is", "type": "text"}], "index": 20}, {"bbox": [109, 543, 346, 557], "spans": [{"bbox": [109, 543, 346, 557], "score": 1.0, "content": "of type III, the result immediately follows.", "type": "text"}], "index": 21}], "index": 19, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [105, 468, 504, 557]}, {"type": "text", "bbox": [109, 556, 501, 614], "lines": [{"bbox": [126, 558, 500, 572], "spans": [{"bbox": [126, 558, 420, 572], "score": 1.0, "content": "In [6] Dunwoody investigates a wide subclass of manifolds ", "type": "text"}, {"bbox": [420, 559, 449, 571], "score": 0.95, "content": "M(\\sigma)", "type": "inline_equation", "height": 12, "width": 29}, {"bbox": [449, 558, 500, 572], "score": 1.0, "content": " such that", "type": "text"}], "index": 22}, {"bbox": [110, 573, 500, 586], "spans": [{"bbox": [110, 574, 154, 585], "score": 0.91, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 11, "width": 44}, {"bbox": [154, 573, 500, 586], "score": 1.0, "content": " and he conjectures that all the elements of this subclass are cyclic", "type": "text"}], "index": 23}, {"bbox": [109, 586, 500, 601], "spans": [{"bbox": [109, 586, 173, 601], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [173, 587, 186, 597], "score": 0.91, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [186, 586, 500, 601], "score": 1.0, "content": " branched over knots. In the next chapter this conjecture will", "type": "text"}], "index": 24}, {"bbox": [109, 600, 375, 616], "spans": [{"bbox": [109, 600, 375, 616], "score": 1.0, "content": "be proved as a corollary of a more general theorem.", "type": "text"}], "index": 25}], "index": 23.5, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [109, 558, 500, 616]}]}
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6-tuple to be admissible when is odd. In fact, we have the following result:
Lemma 3 ([6], Theorem 2) Let be a 6-tuple with odd. Then is admissible if and only if it satisfies and (ii’).
Remark 3. This result does not hold when is even. In fact, the 6-tuples , with even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.
An immediate consequence of Lemma 3 is the following result:
Corollary 4 Let be a -tuple with odd and . Then is admissible if and only if has a unique cycle.
Proof. If is admissible, then it is straightforward that has a unique cycle. Vice versa, if has a unique cycle, then (i’) holds. Since implies (ii’), the result is a direct consequence of the above lemma.
The parameter associated to an admissible 6-tuple is strictly related to the word associated to . In fact, we have:
Lemma 5 Let be an admissible 6-tuple, the associated word and its exponent-sum. Then
$$
p_{\sigma}=\varepsilon_{w}.
$$
Proof. Since is admissible, the arcs of are precisely the arcs of orientation on . Let , and let be the sequence of these arcs, following the canonical , with . We have: ,where . Since if is of type I, if is of type II and if is of type III, the result immediately follows.
In [6] Dunwoody investigates a wide subclass of manifolds such that and he conjectures that all the elements of this subclass are cyclic coverings of branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.
10
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<div class="pdf-page">
<p>6-tuple to be admissible when is odd. In fact, we have the following result:</p>
<p>Lemma 3 ([6], Theorem 2) Let be a 6-tuple with odd. Then is admissible if and only if it satisfies and (ii’).</p>
<p>Remark 3. This result does not hold when is even. In fact, the 6-tuples , with even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.</p>
<p>An immediate consequence of Lemma 3 is the following result:</p>
<p>Corollary 4 Let be a -tuple with odd and . Then is admissible if and only if has a unique cycle.</p>
<p>Proof. If is admissible, then it is straightforward that has a unique cycle. Vice versa, if has a unique cycle, then (i’) holds. Since implies (ii’), the result is a direct consequence of the above lemma.</p>
<p>The parameter associated to an admissible 6-tuple is strictly related to the word associated to . In fact, we have:</p>
<p>Lemma 5 Let be an admissible 6-tuple, the associated word and its exponent-sum. Then</p>
<p>Proof. Since is admissible, the arcs of are precisely the arcs of orientation on . Let , and let be the sequence of these arcs, following the canonical , with . We have: ,where . Since if is of type I, if is of type II and if is of type III, the result immediately follows.</p>
<p>In [6] Dunwoody investigates a wide subclass of manifolds such that and he conjectures that all the elements of this subclass are cyclic coverings of branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="182" data-y="161" data-width="652" data-height="20">6-tuple to be admissible when is odd. In fact, we have the following result:</p>
<p class="pdf-text" data-x="182" data-y="196" data-width="652" data-height="39">Lemma 3 ([6], Theorem 2) Let be a 6-tuple with odd. Then is admissible if and only if it satisfies and (ii’).</p>
<p class="pdf-text" data-x="184" data-y="250" data-width="654" data-height="56">Remark 3. This result does not hold when is even. In fact, the 6-tuples , with even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.</p>
<p class="pdf-text" data-x="209" data-y="315" data-width="543" data-height="18">An immediate consequence of Lemma 3 is the following result:</p>
<p class="pdf-text" data-x="182" data-y="350" data-width="657" data-height="37">Corollary 4 Let be a -tuple with odd and . Then is admissible if and only if has a unique cycle.</p>
<p class="pdf-text" data-x="182" data-y="403" data-width="654" data-height="57">Proof. If is admissible, then it is straightforward that has a unique cycle. Vice versa, if has a unique cycle, then (i’) holds. Since implies (ii’), the result is a direct consequence of the above lemma.</p>
<p class="pdf-text" data-x="182" data-y="461" data-width="656" data-height="36">The parameter associated to an admissible 6-tuple is strictly related to the word associated to . In fact, we have:</p>
<p class="pdf-text" data-x="182" data-y="514" data-width="654" data-height="38">Lemma 5 Let be an admissible 6-tuple, the associated word and its exponent-sum. Then</p>
<p class="pdf-text" data-x="182" data-y="602" data-width="656" data-height="115">Proof. Since is admissible, the arcs of are precisely the arcs of orientation on . Let , and let be the sequence of these arcs, following the canonical , with . We have: ,where . Since if is of type I, if is of type II and if is of type III, the result immediately follows.</p>
<p class="pdf-text" data-x="182" data-y="718" data-width="656" data-height="75">In [6] Dunwoody investigates a wide subclass of manifolds such that and he conjectures that all the elements of this subclass are cyclic coverings of branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.</p>
<div class="pdf-discarded" data-x="498" data-y="893" data-width="24" data-height="14" style="opacity: 0.5;">10</div>
</div>
|
6-tuple to be admissible when $d$ is odd. In fact, we have the following result:
Lemma 3 ([6], Theorem 2) Let $\sigma\,=\,(a,b,c,n,r,s)$ be a 6-tuple with $d\,=$ $2a+b+c$ odd. Then $\sigma$ is admissible if and only if it satisfies $(i\,?)$ and (ii’).
Remark 3. This result does not hold when $d$ is even. In fact, the 6-tuples $(1,0,c,1,2,0)$ , with $c$ even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.
An immediate consequence of Lemma 3 is the following result:
Corollary 4 Let $\sigma=(a,b,c,n,r,s)$ be a $\it6$ -tuple with $d=2a+b+c$ odd and $n=1$ . Then $\sigma$ is admissible if and only if $\mathcal{D}$ has a unique cycle.
Proof. If $\sigma$ is admissible, then it is straightforward that $\mathcal{D}$ has a unique cycle. Vice versa, if $\mathcal{D}$ has a unique cycle, then (i’) holds. Since $n=1$ implies (ii’), the result is a direct consequence of the above lemma.
The parameter $p_{\sigma}$ associated to an admissible 6-tuple $\sigma$ is strictly related to the word $w(\sigma)$ associated to $\sigma$ . In fact, we have:
Lemma 5 Let $\sigma\,=\,(a,b,c,n,r,s)$ be an admissible 6-tuple, $w\,=\,w(\sigma)$ the associated word and $\varepsilon_{w}$ its exponent-sum. Then
$$
p_{\sigma}=\varepsilon_{w}.
$$
Proof. Since $\sigma$ is admissible, the arcs of $\Delta$ are precisely the arcs of orientation on $D_{1}$ . Let $e_{1},e_{2},\ldots,e_{d}$ $D_{1}$ , and let be the sequence of these arcs, following the canonical $\begin{array}{r}{w\,=\,\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\end{array}$ , with $u_{h}\,\in\,\{+1,-1\}$ . We have: $\begin{array}{r}{\varepsilon_{w}=\sum_{h=1}^{d}u_{h}=1/2\sum_{h=1}^{d}(u_{h}+u_{h+1})}\end{array}$ ,where $d+1=1$ . Since $u_{h}\!+\!u_{h+1}=+2$ if $e_{h}$ is of type I, $u_{h}+u_{h+1}=-2$ if $e_{h}$ is of type II and $u_{h}+u_{h+1}=0$ if $e_{h}$ is of type III, the result immediately follows.
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"6-tuple to be admissible when d is odd. In fact, we have the following result:",
"Lemma 3 ([6], Theorem 2) Let \\sigma\\,=\\,(a,b,c,n,r,s) be a 6-tuple with d\\,=",
"2a+b+c odd. Then \\sigma is admissible if and only if it satisfies (i\\,?) and (ii’).",
"Remark 3. This result does not hold when d is even. In fact, the 6-tuples",
"(1,0,c,1,2,0) , with c even, satisfy (i’) and (ii’), but they are not admissible,",
"as pointed out in Remark 1.",
"An immediate consequence of Lemma 3 is the following result:",
"Corollary 4 Let \\sigma=(a,b,c,n,r,s) be a \\it6 -tuple with d=2a+b+c odd and",
"n=1 . Then \\sigma is admissible if and only if \\mathcal{D} has a unique cycle.",
"Proof. If \\sigma is admissible, then it is straightforward that \\mathcal{D} has a unique",
"cycle. Vice versa, if \\mathcal{D} has a unique cycle, then (i’) holds. Since n=1 implies",
"(ii’), the result is a direct consequence of the above lemma.",
"The parameter p_{\\sigma} associated to an admissible 6-tuple \\sigma is strictly related",
"to the word w(\\sigma) associated to \\sigma . In fact, we have:",
"Lemma 5 Let \\sigma\\,=\\,(a,b,c,n,r,s) be an admissible 6-tuple, w\\,=\\,w(\\sigma) the",
"associated word and \\varepsilon_{w} its exponent-sum. Then",
"p_{\\sigma}=\\varepsilon_{w}.",
"Proof. Since \\sigma is admissible, the arcs of \\Delta are precisely the arcs of",
"orientation on D_{1} . Let e_{1},e_{2},\\ldots,e_{d} D_{1} , and let be the sequence of these arcs, following the canonical \\begin{array}{r}{w\\,=\\,\\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\\end{array} , with u_{h}\\,\\in\\,\\{+1,-1\\} . We have:",
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"of type III, the result immediately follows.",
"In [6] Dunwoody investigates a wide subclass of manifolds M(\\sigma) such that",
"p_{\\sigma}=\\pm1 and he conjectures that all the elements of this subclass are cyclic",
"coverings of \\mathbf{S^{3}} branched over knots. In the next chapter this conjecture will",
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"3 Main results",
"The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of or of lens spaces, branched over suitable knots.",
"Theorem 6 Let be an admissible 6-tuple, with . Then the Dunwoody manifold is the -fold cyclic cov- ering of the manifold , branched over a genus one 1- bridge knot only depending on the integers . Further, is homeomorphic to:",
"i) , if , ii) , if , iii) a lens space with , if .",
"Proof. Since the two systems of curves and on define a Heegaard diagram of , there exist two handle- bodies and of genus , with , such that . Let now be the cyclic group of order generated by the homeomorphism on . The action of on extends to both the handlebodies and (see [29]), and hence to the 3-manifold . Let (resp. ) be a disc properly embedded in (resp. in ) such that (resp. ). Since and (mod ), the discs (resp. , for , form a system of meridian discs for the handlebody (resp. ). By arguments contained in [38], the quotients and are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of or- der . The intersection of these orbifolds is a 2-orbifold with two singular points of order , which is topologically the torus ; the curve (resp. ), which is the image via the quotient map of the curves (resp. of the curves ), is non-homotopically trivial in . These curves, each of which is a fundamental system of curves in , define a Heegaard diagram of (induced by . The union of the orbifolds and is a 3-orbifold topologically homeomorphic to , having a genus one 1-bridge knot as singular set of order . Thus, is homeomorphic to and hence is the -fold cyclic covering of , branched over .",
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[{"type": "text", "text": "3 Main results ", "text_level": 1, "page_idx": 10}, {"type": "text", "text": "The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $\\mathrm{{S^{3}}}$ or of lens spaces, branched over suitable knots. ", "page_idx": 10}, {"type": "text", "text": "Theorem 6 Let $\\sigma\\,=\\,(a,b,c,n,r,s)$ be an admissible 6-tuple, with $n\\,>\\,1$ . Then the Dunwoody manifold $M=M(a,b,c,n,r,s)$ is the $n$ -fold cyclic covering of the manifold $M^{\\prime}=M(a,b,c,1,r,0)$ , branched over a genus one 1- bridge knot $K=K(a,b,c,r)$ only depending on the integers $a,b,c,r$ . Further, $M^{\\prime}$ is homeomorphic to: ", "page_idx": 10}, {"type": "text", "text": "i) $\\mathbf{S^{3}}$ , if $p_{\\sigma}=\\pm1$ , \nii) $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ , if $p_{\\sigma}=0$ , \niii) a lens space $L(\\alpha,\\beta)$ with $\\alpha=|p_{\\sigma}|$ , if $|p_{\\sigma}|>1$ . ", "page_idx": 10}, {"type": "text", "text": "Proof. Since the two systems of curves $\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}$ and $\\mathcal{D}\\,=$ $\\{D_{1},...\\,,D_{n}\\}$ on $T_{n}$ define a Heegaard diagram of $M$ , there exist two handlebodies $U_{n}$ and $U_{n}^{\\prime}$ of genus $n$ , with $\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}$ , such that $M=U_{n}\\cup U_{n}^{\\prime}$ . Let now $\\mathcal{G}_{n}$ be the cyclic group of order $n$ generated by the homeomorphism $\\rho_{n}$ on $T_{n}$ . The action of $\\mathcal{G}_{n}$ on $T_{n}$ extends to both the handlebodies $U_{n}$ and $U_{n}^{\\prime}$ (see [29]), and hence to the 3-manifold $M$ . Let $B_{1}$ (resp. $B_{1}^{\\prime}$ ) be a disc properly embedded in $U_{n}$ (resp. in $U_{n}^{\\prime}$ ) such that $\\partial B_{1}\\,=\\,C_{1}$ (resp. $\\partial B_{1}^{\\prime}\\;=\\;D_{1}$ ). Since $\\rho_{n}(C_{i})\\,=\\,C_{i+1}$ and $\\rho_{n}(D_{i})\\,=\\,D_{i+1}$ (mod $n$ ), the discs $B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})$ (resp. $B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))$ , for $k=1,\\dotsc,n$ , form a system of meridian discs for the handlebody $U_{n}$ (resp. $U_{n}^{\\prime}$ ). By arguments contained in [38], the quotients $U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}$ and $U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}$ are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of order $n$ . The intersection of these orbifolds is a 2-orbifold with two singular points of order $n$ , which is topologically the torus $T_{1}=T_{n}/\\mathcal{G}_{n}$ ; the curve $C$ (resp. $D$ ), which is the image via the quotient map of the curves $C_{i}$ (resp. of the curves $D_{i}$ ), is non-homotopically trivial in $T_{1}^{\\prime}$ . These curves, each of which is a fundamental system of curves in $T_{1}$ , define a Heegaard diagram of $M^{\\prime}$ (induced by $H(a,b,c,1,r,0))$ . The union of the orbifolds $U_{1}$ and $U_{1}^{\\prime}$ is a 3-orbifold topologically homeomorphic to $M^{\\prime}$ , having a genus one 1-bridge knot $K\\,\\subset\\,M^{\\prime}$ as singular set of order $n$ . Thus, $M^{\\prime}$ is homeomorphic to $M/\\mathcal{G}_{n}$ and hence $M$ is the $n$ -fold cyclic covering of $M^{\\prime}$ , branched over $K$ . ", "page_idx": 10}]
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{"preproc_blocks": [{"type": "title", "bbox": [109, 121, 247, 140], "lines": [{"bbox": [110, 123, 246, 140], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [137, 123, 246, 140], "score": 1.0, "content": "Main results", "type": "text"}], "index": 0}], "index": 0}, {"type": "text", "bbox": [110, 151, 500, 209], "lines": [{"bbox": [110, 154, 500, 168], "spans": [{"bbox": [110, 154, 500, 168], "score": 1.0, "content": "The following theorem is the main result of this paper and shows how the", "type": "text"}], "index": 1}, {"bbox": [110, 169, 500, 182], "spans": [{"bbox": [110, 169, 500, 182], "score": 1.0, "content": "cyclic action on the Heegaard diagrams naturally extends to a cyclic action", "type": "text"}], "index": 2}, {"bbox": [109, 182, 499, 197], "spans": [{"bbox": [109, 182, 499, 197], "score": 1.0, "content": "on the associated Dunwoody manifolds, which turn out to be cyclic coverings", "type": "text"}], "index": 3}, {"bbox": [109, 197, 381, 211], "spans": [{"bbox": [109, 197, 123, 211], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 198, 136, 208], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [137, 197, 381, 211], "score": 1.0, "content": " or of lens spaces, branched over suitable knots.", "type": "text"}], "index": 4}], "index": 2.5}, {"type": "text", "bbox": [110, 222, 500, 294], "lines": [{"bbox": [109, 223, 500, 239], "spans": [{"bbox": [109, 223, 200, 239], "score": 1.0, "content": "Theorem 6 Let ", "type": "text"}, {"bbox": [201, 225, 297, 238], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [297, 223, 462, 239], "score": 1.0, "content": " be an admissible 6-tuple, with ", "type": "text"}, {"bbox": [462, 227, 496, 236], "score": 0.86, "content": "n\\,>\\,1", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [496, 223, 500, 239], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [111, 239, 501, 253], "spans": [{"bbox": [111, 239, 264, 253], "score": 1.0, "content": "Then the Dunwoody manifold ", "type": "text"}, {"bbox": [265, 239, 375, 252], "score": 0.92, "content": "M=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 110}, {"bbox": [376, 239, 411, 253], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [411, 240, 419, 250], "score": 0.46, "content": "n", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [420, 239, 501, 253], "score": 1.0, "content": "-fold cyclic cov-", "type": "text"}], "index": 6}, {"bbox": [110, 253, 501, 268], "spans": [{"bbox": [110, 253, 223, 268], "score": 1.0, "content": "ering of the manifold ", "type": "text"}, {"bbox": [224, 253, 338, 267], "score": 0.9, "content": "M^{\\prime}=M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [339, 253, 501, 268], "score": 1.0, "content": ", branched over a genus one 1-", "type": "text"}], "index": 7}, {"bbox": [111, 267, 499, 282], "spans": [{"bbox": [111, 268, 168, 282], "score": 1.0, "content": "bridge knot", "type": "text"}, {"bbox": [169, 267, 253, 281], "score": 0.93, "content": "K=K(a,b,c,r)", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [254, 268, 410, 282], "score": 1.0, "content": " only depending on the integers ", "type": "text"}, {"bbox": [411, 269, 449, 281], "score": 0.9, "content": "a,b,c,r", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [449, 268, 499, 282], "score": 1.0, "content": ". Further,", "type": "text"}], "index": 8}, {"bbox": [110, 282, 235, 296], "spans": [{"bbox": [110, 284, 126, 293], "score": 0.88, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [126, 282, 235, 296], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 9}], "index": 7}, {"type": "text", "bbox": [126, 295, 384, 339], "lines": [{"bbox": [127, 296, 218, 311], "spans": [{"bbox": [127, 296, 139, 311], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 297, 154, 308], "score": 0.65, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [154, 296, 171, 311], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [172, 297, 215, 310], "score": 0.87, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [216, 296, 218, 311], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [127, 310, 241, 325], "spans": [{"bbox": [127, 310, 143, 325], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 311, 185, 323], "score": 0.85, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [185, 310, 203, 325], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 311, 237, 324], "score": 0.89, "content": "p_{\\sigma}=0", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [237, 310, 241, 325], "score": 1.0, "content": ",", "type": "text"}], "index": 11}, {"bbox": [127, 325, 382, 339], "spans": [{"bbox": [127, 326, 211, 339], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 325, 249, 339], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 326, 278, 339], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 325, 321, 339], "score": 0.91, "content": "\\alpha=|p_{\\sigma}|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [321, 326, 339, 339], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [339, 325, 379, 339], "score": 0.88, "content": "|p_{\\sigma}|>1", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [380, 326, 382, 339], "score": 1.0, "content": ".", "type": "text"}], "index": 12}], "index": 11}, {"type": "text", "bbox": [109, 350, 501, 669], "lines": [{"bbox": [126, 352, 500, 368], "spans": [{"bbox": [126, 352, 348, 368], "score": 1.0, "content": "Proof. Since the two systems of curves ", "type": "text"}, {"bbox": [348, 354, 445, 367], "score": 0.92, "content": "\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [445, 352, 473, 368], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 354, 500, 366], "score": 0.83, "content": "\\mathcal{D}\\,=", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [110, 367, 498, 382], "spans": [{"bbox": [110, 369, 180, 381], "score": 0.93, "content": "\\{D_{1},...\\,,D_{n}\\}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [181, 367, 198, 382], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [199, 369, 212, 380], "score": 0.89, "content": "T_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [212, 367, 364, 382], "score": 1.0, "content": " define a Heegaard diagram of ", "type": "text"}, {"bbox": [365, 369, 377, 378], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [378, 367, 498, 382], "score": 1.0, "content": ", there exist two handle-", "type": "text"}], "index": 14}, {"bbox": [108, 380, 498, 397], "spans": [{"bbox": [108, 380, 146, 397], "score": 1.0, "content": "bodies ", "type": "text"}, {"bbox": [146, 384, 160, 394], "score": 0.93, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [160, 380, 185, 397], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [185, 383, 199, 396], "score": 0.91, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [199, 380, 246, 397], "score": 1.0, "content": " of genus ", "type": "text"}, {"bbox": [246, 385, 254, 393], "score": 0.7, "content": "n", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [254, 380, 286, 397], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [286, 382, 372, 396], "score": 0.92, "content": "\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 380, 429, 397], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [429, 383, 496, 396], "score": 0.94, "content": "M=U_{n}\\cup U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [496, 380, 498, 397], "score": 1.0, "content": ".", "type": "text"}], "index": 15}, {"bbox": [109, 397, 500, 411], "spans": [{"bbox": [109, 397, 154, 411], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [155, 398, 168, 409], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [168, 397, 315, 411], "score": 1.0, "content": " be the cyclic group of order ", "type": "text"}, {"bbox": [315, 401, 322, 407], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [323, 397, 500, 411], "score": 1.0, "content": " generated by the homeomorphism", "type": "text"}], "index": 16}, {"bbox": [110, 410, 498, 426], "spans": [{"bbox": [110, 416, 122, 424], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 410, 145, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [145, 413, 158, 423], "score": 0.91, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [158, 410, 248, 426], "score": 1.0, "content": ". The action of ", "type": "text"}, {"bbox": [248, 413, 261, 423], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [261, 410, 284, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [284, 413, 297, 423], "score": 0.9, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [297, 410, 484, 426], "score": 1.0, "content": " extends to both the handlebodies ", "type": "text"}, {"bbox": [485, 413, 498, 423], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 13}], "index": 17}, {"bbox": [110, 426, 500, 439], "spans": [{"bbox": [110, 426, 133, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 427, 147, 439], "score": 0.92, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [148, 426, 361, 439], "score": 1.0, "content": " (see [29]), and hence to the 3-manifold ", "type": "text"}, {"bbox": [362, 427, 374, 436], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [375, 426, 407, 439], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [408, 427, 421, 438], "score": 0.92, "content": "B_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [422, 426, 463, 439], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [464, 427, 477, 439], "score": 0.9, "content": "B_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [478, 426, 500, 439], "score": 1.0, "content": ") be", "type": "text"}], "index": 18}, {"bbox": [108, 439, 500, 455], "spans": [{"bbox": [108, 439, 264, 455], "score": 1.0, "content": "a disc properly embedded in ", "type": "text"}, {"bbox": [264, 442, 278, 452], "score": 0.92, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 439, 335, 455], "score": 1.0, "content": " (resp. in ", "type": "text"}, {"bbox": [335, 441, 349, 453], "score": 0.9, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [349, 439, 412, 455], "score": 1.0, "content": ") such that ", "type": "text"}, {"bbox": [412, 442, 465, 452], "score": 0.93, "content": "\\partial B_{1}\\,=\\,C_{1}", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [465, 439, 500, 455], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 19}, {"bbox": [110, 453, 501, 470], "spans": [{"bbox": [110, 456, 165, 468], "score": 0.92, "content": "\\partial B_{1}^{\\prime}\\;=\\;D_{1}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [165, 453, 213, 470], "score": 1.0, "content": "). Since ", "type": "text"}, {"bbox": [214, 455, 289, 468], "score": 0.95, "content": "\\rho_{n}(C_{i})\\,=\\,C_{i+1}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [289, 453, 317, 470], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [318, 455, 396, 468], "score": 0.93, "content": "\\rho_{n}(D_{i})\\,=\\,D_{i+1}", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [396, 453, 432, 470], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [433, 459, 440, 465], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [440, 453, 501, 470], "score": 1.0, "content": "), the discs", "type": "text"}], "index": 20}, {"bbox": [110, 466, 503, 485], "spans": [{"bbox": [110, 469, 187, 482], "score": 0.93, "content": "B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [188, 466, 228, 485], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [228, 469, 308, 482], "score": 0.93, "content": "B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [309, 466, 336, 485], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [336, 470, 403, 482], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [403, 466, 503, 485], "score": 1.0, "content": ", form a system of", "type": "text"}], "index": 21}, {"bbox": [108, 483, 501, 498], "spans": [{"bbox": [108, 483, 291, 498], "score": 1.0, "content": "meridian discs for the handlebody ", "type": "text"}, {"bbox": [291, 485, 305, 495], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [305, 483, 344, 498], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [345, 484, 359, 496], "score": 0.89, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [359, 483, 501, 498], "score": 1.0, "content": "). By arguments contained", "type": "text"}], "index": 22}, {"bbox": [108, 497, 500, 512], "spans": [{"bbox": [108, 497, 225, 512], "score": 1.0, "content": "in [38], the quotients ", "type": "text"}, {"bbox": [225, 498, 290, 511], "score": 0.95, "content": "U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [291, 497, 319, 512], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [319, 498, 384, 511], "score": 0.94, "content": "U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [385, 497, 500, 512], "score": 1.0, "content": " are both handlebody", "type": "text"}], "index": 23}, {"bbox": [110, 513, 501, 527], "spans": [{"bbox": [110, 513, 501, 527], "score": 1.0, "content": "orbifolds topologically homeomorphic to a genus one handlebody with one", "type": "text"}], "index": 24}, {"bbox": [108, 526, 501, 542], "spans": [{"bbox": [108, 526, 501, 542], "score": 1.0, "content": "arc trivially embedded as its singular set with a cyclic isotropy group of or-", "type": "text"}], "index": 25}, {"bbox": [110, 542, 500, 554], "spans": [{"bbox": [110, 542, 131, 554], "score": 1.0, "content": "der ", "type": "text"}, {"bbox": [131, 546, 138, 551], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [138, 542, 500, 554], "score": 1.0, "content": ". The intersection of these orbifolds is a 2-orbifold with two singular", "type": "text"}], "index": 26}, {"bbox": [109, 555, 499, 569], "spans": [{"bbox": [109, 555, 190, 569], "score": 1.0, "content": "points of order ", "type": "text"}, {"bbox": [190, 560, 197, 566], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [198, 555, 370, 569], "score": 1.0, "content": ", which is topologically the torus ", "type": "text"}, {"bbox": [370, 556, 430, 569], "score": 0.95, "content": "T_{1}=T_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 60}, {"bbox": [430, 555, 489, 569], "score": 1.0, "content": "; the curve ", "type": "text"}, {"bbox": [489, 557, 499, 566], "score": 0.9, "content": "C", "type": "inline_equation", "height": 9, "width": 10}], "index": 27}, {"bbox": [110, 570, 500, 584], "spans": [{"bbox": [110, 570, 145, 584], "score": 1.0, "content": "(resp. ", "type": "text"}, {"bbox": [146, 572, 156, 581], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [156, 570, 453, 584], "score": 1.0, "content": "), which is the image via the quotient map of the curves ", "type": "text"}, {"bbox": [454, 572, 466, 582], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [466, 570, 500, 584], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 28}, {"bbox": [109, 584, 502, 598], "spans": [{"bbox": [109, 584, 181, 598], "score": 1.0, "content": "of the curves ", "type": "text"}, {"bbox": [181, 586, 194, 597], "score": 0.89, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [195, 584, 366, 598], "score": 1.0, "content": "), is non-homotopically trivial in ", "type": "text"}, {"bbox": [367, 586, 379, 596], "score": 0.92, "content": "T_{1}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [379, 584, 502, 598], "score": 1.0, "content": ". These curves, each of", "type": "text"}], "index": 29}, {"bbox": [109, 598, 500, 612], "spans": [{"bbox": [109, 598, 339, 612], "score": 1.0, "content": "which is a fundamental system of curves in ", "type": "text"}, {"bbox": [340, 600, 352, 611], "score": 0.92, "content": "T_{1}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [352, 598, 500, 612], "score": 1.0, "content": ", define a Heegaard diagram", "type": "text"}], "index": 30}, {"bbox": [110, 613, 500, 627], "spans": [{"bbox": [110, 613, 123, 627], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 614, 139, 624], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [139, 613, 206, 627], "score": 1.0, "content": " (induced by ", "type": "text"}, {"bbox": [207, 614, 289, 626], "score": 0.93, "content": "H(a,b,c,1,r,0))", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [290, 613, 435, 627], "score": 1.0, "content": ". The union of the orbifolds ", "type": "text"}, {"bbox": [435, 615, 448, 625], "score": 0.93, "content": "U_{1}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [448, 613, 474, 627], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 614, 487, 627], "score": 0.93, "content": "U_{1}^{\\prime}", "type": "inline_equation", "height": 13, "width": 13}, {"bbox": [488, 613, 500, 627], "score": 1.0, "content": " is", "type": "text"}], "index": 31}, {"bbox": [110, 628, 500, 642], "spans": [{"bbox": [110, 628, 334, 642], "score": 1.0, "content": "a 3-orbifold topologically homeomorphic to ", "type": "text"}, {"bbox": [334, 629, 349, 638], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [350, 628, 500, 642], "score": 1.0, "content": ", having a genus one 1-bridge", "type": "text"}], "index": 32}, {"bbox": [109, 642, 501, 656], "spans": [{"bbox": [109, 642, 138, 656], "score": 1.0, "content": "knot ", "type": "text"}, {"bbox": [138, 644, 186, 653], "score": 0.93, "content": "K\\,\\subset\\,M^{\\prime}", "type": "inline_equation", "height": 9, "width": 48}, {"bbox": [186, 642, 319, 656], "score": 1.0, "content": " as singular set of order ", "type": "text"}, {"bbox": [319, 647, 326, 653], "score": 0.88, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [327, 642, 374, 656], "score": 1.0, "content": ". Thus, ", "type": "text"}, {"bbox": [374, 644, 389, 653], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [390, 642, 501, 656], "score": 1.0, "content": " is homeomorphic to", "type": "text"}], "index": 33}, {"bbox": [110, 656, 501, 671], "spans": [{"bbox": [110, 658, 141, 670], "score": 0.94, "content": "M/\\mathcal{G}_{n}", "type": "inline_equation", "height": 12, "width": 31}, {"bbox": [141, 656, 203, 671], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [203, 658, 216, 667], "score": 0.92, "content": "M", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [216, 656, 254, 671], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [254, 662, 261, 667], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [262, 656, 383, 671], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [383, 658, 399, 667], "score": 0.91, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [399, 656, 484, 671], "score": 1.0, "content": ", branched over ", "type": "text"}, {"bbox": [484, 659, 496, 667], "score": 0.91, "content": "K", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [496, 656, 501, 671], "score": 1.0, "content": ".", "type": "text"}], "index": 34}], "index": 23.5}], "layout_bboxes": [], "page_idx": 10, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 310, 702], "lines": [{"bbox": [298, 692, 312, 704], "spans": [{"bbox": [298, 692, 312, 704], "score": 1.0, "content": "11", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "title", "bbox": [109, 121, 247, 140], "lines": [{"bbox": [110, 123, 246, 140], "spans": [{"bbox": [110, 126, 121, 138], "score": 1.0, "content": "3", "type": "text"}, {"bbox": [137, 123, 246, 140], "score": 1.0, "content": "Main results", "type": "text"}], "index": 0}], "index": 0, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [110, 151, 500, 209], "lines": [{"bbox": [110, 154, 500, 168], "spans": [{"bbox": [110, 154, 500, 168], "score": 1.0, "content": "The following theorem is the main result of this paper and shows how the", "type": "text"}], "index": 1}, {"bbox": [110, 169, 500, 182], "spans": [{"bbox": [110, 169, 500, 182], "score": 1.0, "content": "cyclic action on the Heegaard diagrams naturally extends to a cyclic action", "type": "text"}], "index": 2}, {"bbox": [109, 182, 499, 197], "spans": [{"bbox": [109, 182, 499, 197], "score": 1.0, "content": "on the associated Dunwoody manifolds, which turn out to be cyclic coverings", "type": "text"}], "index": 3}, {"bbox": [109, 197, 381, 211], "spans": [{"bbox": [109, 197, 123, 211], "score": 1.0, "content": "of ", "type": "text"}, {"bbox": [123, 198, 136, 208], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [137, 197, 381, 211], "score": 1.0, "content": " or of lens spaces, branched over suitable knots.", "type": "text"}], "index": 4}], "index": 2.5, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [109, 154, 500, 211]}, {"type": "text", "bbox": [110, 222, 500, 294], "lines": [{"bbox": [109, 223, 500, 239], "spans": [{"bbox": [109, 223, 200, 239], "score": 1.0, "content": "Theorem 6 Let ", "type": "text"}, {"bbox": [201, 225, 297, 238], "score": 0.92, "content": "\\sigma\\,=\\,(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [297, 223, 462, 239], "score": 1.0, "content": " be an admissible 6-tuple, with ", "type": "text"}, {"bbox": [462, 227, 496, 236], "score": 0.86, "content": "n\\,>\\,1", "type": "inline_equation", "height": 9, "width": 34}, {"bbox": [496, 223, 500, 239], "score": 1.0, "content": ".", "type": "text"}], "index": 5}, {"bbox": [111, 239, 501, 253], "spans": [{"bbox": [111, 239, 264, 253], "score": 1.0, "content": "Then the Dunwoody manifold ", "type": "text"}, {"bbox": [265, 239, 375, 252], "score": 0.92, "content": "M=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 110}, {"bbox": [376, 239, 411, 253], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [411, 240, 419, 250], "score": 0.46, "content": "n", "type": "inline_equation", "height": 10, "width": 8}, {"bbox": [420, 239, 501, 253], "score": 1.0, "content": "-fold cyclic cov-", "type": "text"}], "index": 6}, {"bbox": [110, 253, 501, 268], "spans": [{"bbox": [110, 253, 223, 268], "score": 1.0, "content": "ering of the manifold ", "type": "text"}, {"bbox": [224, 253, 338, 267], "score": 0.9, "content": "M^{\\prime}=M(a,b,c,1,r,0)", "type": "inline_equation", "height": 14, "width": 114}, {"bbox": [339, 253, 501, 268], "score": 1.0, "content": ", branched over a genus one 1-", "type": "text"}], "index": 7}, {"bbox": [111, 267, 499, 282], "spans": [{"bbox": [111, 268, 168, 282], "score": 1.0, "content": "bridge knot", "type": "text"}, {"bbox": [169, 267, 253, 281], "score": 0.93, "content": "K=K(a,b,c,r)", "type": "inline_equation", "height": 14, "width": 84}, {"bbox": [254, 268, 410, 282], "score": 1.0, "content": " only depending on the integers ", "type": "text"}, {"bbox": [411, 269, 449, 281], "score": 0.9, "content": "a,b,c,r", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [449, 268, 499, 282], "score": 1.0, "content": ". Further,", "type": "text"}], "index": 8}, {"bbox": [110, 282, 235, 296], "spans": [{"bbox": [110, 284, 126, 293], "score": 0.88, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 16}, {"bbox": [126, 282, 235, 296], "score": 1.0, "content": " is homeomorphic to:", "type": "text"}], "index": 9}], "index": 7, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [109, 223, 501, 296]}, {"type": "list", "bbox": [126, 295, 384, 339], "lines": [{"bbox": [127, 296, 218, 311], "spans": [{"bbox": [127, 296, 139, 311], "score": 1.0, "content": "i) ", "type": "text"}, {"bbox": [139, 297, 154, 308], "score": 0.65, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [154, 296, 171, 311], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [172, 297, 215, 310], "score": 0.87, "content": "p_{\\sigma}=\\pm1", "type": "inline_equation", "height": 13, "width": 43}, {"bbox": [216, 296, 218, 311], "score": 1.0, "content": ",", "type": "text"}], "index": 10, "is_list_end_line": true}, {"bbox": [127, 310, 241, 325], "spans": [{"bbox": [127, 310, 143, 325], "score": 1.0, "content": "ii) ", "type": "text"}, {"bbox": [144, 311, 185, 323], "score": 0.85, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 12, "width": 41}, {"bbox": [185, 310, 203, 325], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [203, 311, 237, 324], "score": 0.89, "content": "p_{\\sigma}=0", "type": "inline_equation", "height": 13, "width": 34}, {"bbox": [237, 310, 241, 325], "score": 1.0, "content": ",", "type": "text"}], "index": 11, "is_list_start_line": true, "is_list_end_line": true}, {"bbox": [127, 325, 382, 339], "spans": [{"bbox": [127, 326, 211, 339], "score": 1.0, "content": "iii) a lens space ", "type": "text"}, {"bbox": [212, 325, 249, 339], "score": 0.93, "content": "L(\\alpha,\\beta)", "type": "inline_equation", "height": 14, "width": 37}, {"bbox": [250, 326, 278, 339], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [278, 325, 321, 339], "score": 0.91, "content": "\\alpha=|p_{\\sigma}|", "type": "inline_equation", "height": 14, "width": 43}, {"bbox": [321, 326, 339, 339], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [339, 325, 379, 339], "score": 0.88, "content": "|p_{\\sigma}|>1", "type": "inline_equation", "height": 14, "width": 40}, {"bbox": [380, 326, 382, 339], "score": 1.0, "content": ".", "type": "text"}], "index": 12, "is_list_start_line": true}], "index": 11, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [127, 296, 382, 339]}, {"type": "text", "bbox": [109, 350, 501, 669], "lines": [{"bbox": [126, 352, 500, 368], "spans": [{"bbox": [126, 352, 348, 368], "score": 1.0, "content": "Proof. Since the two systems of curves ", "type": "text"}, {"bbox": [348, 354, 445, 367], "score": 0.92, "content": "\\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\}", "type": "inline_equation", "height": 13, "width": 97}, {"bbox": [445, 352, 473, 368], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [474, 354, 500, 366], "score": 0.83, "content": "\\mathcal{D}\\,=", "type": "inline_equation", "height": 12, "width": 26}], "index": 13}, {"bbox": [110, 367, 498, 382], "spans": [{"bbox": [110, 369, 180, 381], "score": 0.93, "content": "\\{D_{1},...\\,,D_{n}\\}", "type": "inline_equation", "height": 12, "width": 70}, {"bbox": [181, 367, 198, 382], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [199, 369, 212, 380], "score": 0.89, "content": "T_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [212, 367, 364, 382], "score": 1.0, "content": " define a Heegaard diagram of ", "type": "text"}, {"bbox": [365, 369, 377, 378], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [378, 367, 498, 382], "score": 1.0, "content": ", there exist two handle-", "type": "text"}], "index": 14}, {"bbox": [108, 380, 498, 397], "spans": [{"bbox": [108, 380, 146, 397], "score": 1.0, "content": "bodies ", "type": "text"}, {"bbox": [146, 384, 160, 394], "score": 0.93, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [160, 380, 185, 397], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [185, 383, 199, 396], "score": 0.91, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 14}, {"bbox": [199, 380, 246, 397], "score": 1.0, "content": " of genus ", "type": "text"}, {"bbox": [246, 385, 254, 393], "score": 0.7, "content": "n", "type": "inline_equation", "height": 8, "width": 8}, {"bbox": [254, 380, 286, 397], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [286, 382, 372, 396], "score": 0.92, "content": "\\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n}", "type": "inline_equation", "height": 14, "width": 86}, {"bbox": [372, 380, 429, 397], "score": 1.0, "content": ", such that ", "type": "text"}, {"bbox": [429, 383, 496, 396], "score": 0.94, "content": "M=U_{n}\\cup U_{n}^{\\prime}", "type": "inline_equation", "height": 13, "width": 67}, {"bbox": [496, 380, 498, 397], "score": 1.0, "content": ".", "type": "text"}], "index": 15}, {"bbox": [109, 397, 500, 411], "spans": [{"bbox": [109, 397, 154, 411], "score": 1.0, "content": "Let now ", "type": "text"}, {"bbox": [155, 398, 168, 409], "score": 0.92, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [168, 397, 315, 411], "score": 1.0, "content": " be the cyclic group of order ", "type": "text"}, {"bbox": [315, 401, 322, 407], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [323, 397, 500, 411], "score": 1.0, "content": " generated by the homeomorphism", "type": "text"}], "index": 16}, {"bbox": [110, 410, 498, 426], "spans": [{"bbox": [110, 416, 122, 424], "score": 0.9, "content": "\\rho_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [122, 410, 145, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [145, 413, 158, 423], "score": 0.91, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [158, 410, 248, 426], "score": 1.0, "content": ". The action of ", "type": "text"}, {"bbox": [248, 413, 261, 423], "score": 0.9, "content": "\\mathcal{G}_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [261, 410, 284, 426], "score": 1.0, "content": " on ", "type": "text"}, {"bbox": [284, 413, 297, 423], "score": 0.9, "content": "T_{n}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [297, 410, 484, 426], "score": 1.0, "content": " extends to both the handlebodies ", "type": "text"}, {"bbox": [485, 413, 498, 423], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 13}], "index": 17}, {"bbox": [110, 426, 500, 439], "spans": [{"bbox": [110, 426, 133, 439], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [134, 427, 147, 439], "score": 0.92, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [148, 426, 361, 439], "score": 1.0, "content": " (see [29]), and hence to the 3-manifold ", "type": "text"}, {"bbox": [362, 427, 374, 436], "score": 0.9, "content": "M", "type": "inline_equation", "height": 9, "width": 12}, {"bbox": [375, 426, 407, 439], "score": 1.0, "content": ". Let ", "type": "text"}, {"bbox": [408, 427, 421, 438], "score": 0.92, "content": "B_{1}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [422, 426, 463, 439], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [464, 427, 477, 439], "score": 0.9, "content": "B_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [478, 426, 500, 439], "score": 1.0, "content": ") be", "type": "text"}], "index": 18}, {"bbox": [108, 439, 500, 455], "spans": [{"bbox": [108, 439, 264, 455], "score": 1.0, "content": "a disc properly embedded in ", "type": "text"}, {"bbox": [264, 442, 278, 452], "score": 0.92, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [279, 439, 335, 455], "score": 1.0, "content": " (resp. in ", "type": "text"}, {"bbox": [335, 441, 349, 453], "score": 0.9, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [349, 439, 412, 455], "score": 1.0, "content": ") such that ", "type": "text"}, {"bbox": [412, 442, 465, 452], "score": 0.93, "content": "\\partial B_{1}\\,=\\,C_{1}", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [465, 439, 500, 455], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 19}, {"bbox": [110, 453, 501, 470], "spans": [{"bbox": [110, 456, 165, 468], "score": 0.92, "content": "\\partial B_{1}^{\\prime}\\;=\\;D_{1}", "type": "inline_equation", "height": 12, "width": 55}, {"bbox": [165, 453, 213, 470], "score": 1.0, "content": "). Since ", "type": "text"}, {"bbox": [214, 455, 289, 468], "score": 0.95, "content": "\\rho_{n}(C_{i})\\,=\\,C_{i+1}", "type": "inline_equation", "height": 13, "width": 75}, {"bbox": [289, 453, 317, 470], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [318, 455, 396, 468], "score": 0.93, "content": "\\rho_{n}(D_{i})\\,=\\,D_{i+1}", "type": "inline_equation", "height": 13, "width": 78}, {"bbox": [396, 453, 432, 470], "score": 1.0, "content": " (mod ", "type": "text"}, {"bbox": [433, 459, 440, 465], "score": 0.81, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [440, 453, 501, 470], "score": 1.0, "content": "), the discs", "type": "text"}], "index": 20}, {"bbox": [110, 466, 503, 485], "spans": [{"bbox": [110, 469, 187, 482], "score": 0.93, "content": "B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1})", "type": "inline_equation", "height": 13, "width": 77}, {"bbox": [188, 466, 228, 485], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [228, 469, 308, 482], "score": 0.93, "content": "B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime}))", "type": "inline_equation", "height": 13, "width": 80}, {"bbox": [309, 466, 336, 485], "score": 1.0, "content": ", for ", "type": "text"}, {"bbox": [336, 470, 403, 482], "score": 0.92, "content": "k=1,\\dotsc,n", "type": "inline_equation", "height": 12, "width": 67}, {"bbox": [403, 466, 503, 485], "score": 1.0, "content": ", form a system of", "type": "text"}], "index": 21}, {"bbox": [108, 483, 501, 498], "spans": [{"bbox": [108, 483, 291, 498], "score": 1.0, "content": "meridian discs for the handlebody ", "type": "text"}, {"bbox": [291, 485, 305, 495], "score": 0.91, "content": "U_{n}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [305, 483, 344, 498], "score": 1.0, "content": " (resp. ", "type": "text"}, {"bbox": [345, 484, 359, 496], "score": 0.89, "content": "U_{n}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [359, 483, 501, 498], "score": 1.0, "content": "). By arguments contained", "type": "text"}], "index": 22}, {"bbox": [108, 497, 500, 512], "spans": [{"bbox": [108, 497, 225, 512], "score": 1.0, "content": "in [38], the quotients ", "type": "text"}, {"bbox": [225, 498, 290, 511], "score": 0.95, "content": "U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [291, 497, 319, 512], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [319, 498, 384, 511], "score": 0.94, "content": "U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n}", "type": "inline_equation", "height": 13, "width": 65}, {"bbox": [385, 497, 500, 512], "score": 1.0, "content": " are both handlebody", "type": "text"}], "index": 23}, {"bbox": [110, 513, 501, 527], "spans": [{"bbox": [110, 513, 501, 527], "score": 1.0, "content": "orbifolds topologically homeomorphic to a genus one handlebody with one", "type": "text"}], "index": 24}, {"bbox": [108, 526, 501, 542], "spans": [{"bbox": [108, 526, 501, 542], "score": 1.0, "content": "arc trivially embedded as its singular set with a cyclic isotropy group of or-", "type": "text"}], "index": 25}, {"bbox": [110, 542, 500, 554], "spans": [{"bbox": [110, 542, 131, 554], "score": 1.0, "content": "der ", "type": "text"}, {"bbox": [131, 546, 138, 551], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [138, 542, 500, 554], "score": 1.0, "content": ". 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", "type": "text"}, {"bbox": [146, 572, 156, 581], "score": 0.85, "content": "D", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [156, 570, 453, 584], "score": 1.0, "content": "), which is the image via the quotient map of the curves ", "type": "text"}, {"bbox": [454, 572, 466, 582], "score": 0.92, "content": "C_{i}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [466, 570, 500, 584], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 28}, {"bbox": [109, 584, 502, 598], "spans": [{"bbox": [109, 584, 181, 598], "score": 1.0, "content": "of the curves ", "type": "text"}, {"bbox": [181, 586, 194, 597], "score": 0.89, "content": "D_{i}", "type": "inline_equation", "height": 11, "width": 13}, {"bbox": [195, 584, 366, 598], "score": 1.0, "content": "), is non-homotopically trivial in ", "type": "text"}, {"bbox": [367, 586, 379, 596], "score": 0.92, "content": "T_{1}^{\\prime}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [379, 584, 502, 598], "score": 1.0, "content": ". 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# 3 Main results
The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of or of lens spaces, branched over suitable knots.
Theorem 6 Let be an admissible 6-tuple, with . Then the Dunwoody manifold is the -fold cyclic cov- ering of the manifold , branched over a genus one 1- bridge knot only depending on the integers . Further, is homeomorphic to:
- i) , if , ii) , if , iii) a lens space with , if .
Proof. Since the two systems of curves and on define a Heegaard diagram of , there exist two handle- bodies and of genus , with , such that . Let now be the cyclic group of order generated by the homeomorphism on . The action of on extends to both the handlebodies and (see [29]), and hence to the 3-manifold . Let (resp. ) be a disc properly embedded in (resp. in ) such that (resp. ). Since and (mod ), the discs (resp. , for , form a system of meridian discs for the handlebody (resp. ). By arguments contained in [38], the quotients and are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of or- der . The intersection of these orbifolds is a 2-orbifold with two singular points of order , which is topologically the torus ; the curve (resp. ), which is the image via the quotient map of the curves (resp. of the curves ), is non-homotopically trivial in . These curves, each of which is a fundamental system of curves in , define a Heegaard diagram of (induced by . The union of the orbifolds and is a 3-orbifold topologically homeomorphic to , having a genus one 1-bridge knot as singular set of order . Thus, is homeomorphic to and hence is the -fold cyclic covering of , branched over .
11
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<div class="pdf-page">
<h1>3 Main results</h1>
<p>The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of or of lens spaces, branched over suitable knots.</p>
<p>Theorem 6 Let be an admissible 6-tuple, with . Then the Dunwoody manifold is the -fold cyclic cov- ering of the manifold , branched over a genus one 1- bridge knot only depending on the integers . Further, is homeomorphic to:</p>
<ul>
<li>i) , if , ii) , if , iii) a lens space with , if .</li>
</ul>
<p>Proof. Since the two systems of curves and on define a Heegaard diagram of , there exist two handle- bodies and of genus , with , such that . Let now be the cyclic group of order generated by the homeomorphism on . The action of on extends to both the handlebodies and (see [29]), and hence to the 3-manifold . Let (resp. ) be a disc properly embedded in (resp. in ) such that (resp. ). Since and (mod ), the discs (resp. , for , form a system of meridian discs for the handlebody (resp. ). By arguments contained in [38], the quotients and are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of or- der . The intersection of these orbifolds is a 2-orbifold with two singular points of order , which is topologically the torus ; the curve (resp. ), which is the image via the quotient map of the curves (resp. of the curves ), is non-homotopically trivial in . These curves, each of which is a fundamental system of curves in , define a Heegaard diagram of (induced by . The union of the orbifolds and is a 3-orbifold topologically homeomorphic to , having a genus one 1-bridge knot as singular set of order . Thus, is homeomorphic to and hence is the -fold cyclic covering of , branched over .</p>
</div>
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<div class="pdf-page">
<h1 class="pdf-title" data-x="182" data-y="156" data-width="231" data-height="25">3 Main results</h1>
<p class="pdf-text" data-x="184" data-y="195" data-width="652" data-height="75">The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of or of lens spaces, branched over suitable knots.</p>
<p class="pdf-text" data-x="184" data-y="287" data-width="652" data-height="93">Theorem 6 Let be an admissible 6-tuple, with . Then the Dunwoody manifold is the -fold cyclic cov- ering of the manifold , branched over a genus one 1- bridge knot only depending on the integers . Further, is homeomorphic to:</p>
<ul class="pdf-list" data-x="210" data-y="381" data-width="432" data-height="57">
<li>i) , if , ii) , if , iii) a lens space with , if .</li>
</ul>
<p class="pdf-text" data-x="182" data-y="452" data-width="656" data-height="412">Proof. Since the two systems of curves and on define a Heegaard diagram of , there exist two handle- bodies and of genus , with , such that . Let now be the cyclic group of order generated by the homeomorphism on . The action of on extends to both the handlebodies and (see [29]), and hence to the 3-manifold . Let (resp. ) be a disc properly embedded in (resp. in ) such that (resp. ). Since and (mod ), the discs (resp. , for , form a system of meridian discs for the handlebody (resp. ). By arguments contained in [38], the quotients and are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of or- der . The intersection of these orbifolds is a 2-orbifold with two singular points of order , which is topologically the torus ; the curve (resp. ), which is the image via the quotient map of the curves (resp. of the curves ), is non-homotopically trivial in . These curves, each of which is a fundamental system of curves in , define a Heegaard diagram of (induced by . The union of the orbifolds and is a 3-orbifold topologically homeomorphic to , having a genus one 1-bridge knot as singular set of order . Thus, is homeomorphic to and hence is the -fold cyclic covering of , branched over .</p>
<div class="pdf-discarded" data-x="500" data-y="893" data-width="18" data-height="14" style="opacity: 0.5;">11</div>
</div>
|
# 3 Main results
The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $\mathrm{{S^{3}}}$ or of lens spaces, branched over suitable knots.
Theorem 6 Let $\sigma\,=\,(a,b,c,n,r,s)$ be an admissible 6-tuple, with $n\,>\,1$ . Then the Dunwoody manifold $M=M(a,b,c,n,r,s)$ is the $n$ -fold cyclic covering of the manifold $M^{\prime}=M(a,b,c,1,r,0)$ , branched over a genus one 1- bridge knot $K=K(a,b,c,r)$ only depending on the integers $a,b,c,r$ . Further, $M^{\prime}$ is homeomorphic to:
i) $\mathbf{S^{3}}$ , if $p_{\sigma}=\pm1$ ,
ii) $\mathbf{S^{1}}\times\mathbf{S^{2}}$ , if $p_{\sigma}=0$ ,
iii) a lens space $L(\alpha,\beta)$ with $\alpha=|p_{\sigma}|$ , if $|p_{\sigma}|>1$ .
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"3 Main results",
"The following theorem is the main result of this paper and shows how the",
"cyclic action on the Heegaard diagrams naturally extends to a cyclic action",
"on the associated Dunwoody manifolds, which turn out to be cyclic coverings",
"of \\mathrm{{S^{3}}} or of lens spaces, branched over suitable knots.",
"Theorem 6 Let \\sigma\\,=\\,(a,b,c,n,r,s) be an admissible 6-tuple, with n\\,>\\,1 .",
"Then the Dunwoody manifold M=M(a,b,c,n,r,s) is the n -fold cyclic cov-",
"ering of the manifold M^{\\prime}=M(a,b,c,1,r,0) , branched over a genus one 1-",
"bridge knot K=K(a,b,c,r) only depending on the integers a,b,c,r . Further,",
"M^{\\prime} is homeomorphic to:",
"i) \\mathbf{S^{3}} , if p_{\\sigma}=\\pm1 ,",
"ii) \\mathbf{S^{1}}\\times\\mathbf{S^{2}} , if p_{\\sigma}=0 ,",
"iii) a lens space L(\\alpha,\\beta) with \\alpha=|p_{\\sigma}| , if |p_{\\sigma}|>1 .",
"Proof. Since the two systems of curves \\mathcal{C}\\,=\\,\\{C_{1},\\ldots\\,,C_{n}\\} and \\mathcal{D}\\,=",
"\\{D_{1},...\\,,D_{n}\\} on T_{n} define a Heegaard diagram of M , there exist two handle-",
"bodies U_{n} and U_{n}^{\\prime} of genus n , with \\partial U_{n}=\\partial U_{n}^{\\prime}=T_{n} , such that M=U_{n}\\cup U_{n}^{\\prime} .",
"Let now \\mathcal{G}_{n} be the cyclic group of order n generated by the homeomorphism",
"\\rho_{n} on T_{n} . The action of \\mathcal{G}_{n} on T_{n} extends to both the handlebodies U_{n}",
"and U_{n}^{\\prime} (see [29]), and hence to the 3-manifold M . Let B_{1} (resp. B_{1}^{\\prime} ) be",
"a disc properly embedded in U_{n} (resp. in U_{n}^{\\prime} ) such that \\partial B_{1}\\,=\\,C_{1} (resp.",
"\\partial B_{1}^{\\prime}\\;=\\;D_{1} ). Since \\rho_{n}(C_{i})\\,=\\,C_{i+1} and \\rho_{n}(D_{i})\\,=\\,D_{i+1} (mod n ), the discs",
"B_{k}\\,=\\,\\rho_{n}^{k-1}(B_{1}) (resp. B_{k}^{\\prime}\\,=\\,\\rho_{n}^{k-1}(B_{1}^{\\prime})) , for k=1,\\dotsc,n , form a system of",
"meridian discs for the handlebody U_{n} (resp. U_{n}^{\\prime} ). By arguments contained",
"in [38], the quotients U_{1}\\,=\\,U_{n}/\\mathcal{G}_{n} and U_{1}^{\\prime}\\,=\\,U_{n}^{\\prime}/\\mathcal{G}_{n} are both handlebody",
"orbifolds topologically homeomorphic to a genus one handlebody with one",
"arc trivially embedded as its singular set with a cyclic isotropy group of or-",
"der n . The intersection of these orbifolds is a 2-orbifold with two singular",
"points of order n , which is topologically the torus T_{1}=T_{n}/\\mathcal{G}_{n} ; the curve C",
"(resp. D ), which is the image via the quotient map of the curves C_{i} (resp.",
"of the curves D_{i} ), is non-homotopically trivial in T_{1}^{\\prime} . These curves, each of",
"which is a fundamental system of curves in T_{1} , define a Heegaard diagram",
"of M^{\\prime} (induced by H(a,b,c,1,r,0)) . The union of the orbifolds U_{1} and U_{1}^{\\prime} is",
"a 3-orbifold topologically homeomorphic to M^{\\prime} , having a genus one 1-bridge",
"knot K\\,\\subset\\,M^{\\prime} as singular set of order n . Thus, M^{\\prime} is homeomorphic to",
"M/\\mathcal{G}_{n} and hence M is the n -fold cyclic covering of M^{\\prime} , branched over K ."
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"Since the handlebody orbifolds and their gluing only depend on , the same holds for the branching set . The homeomorphism type of follows from Proposition 2 and Lemma 5.",
"Remark 4. More generally, given two positive integers and such that divides , if is admissible, then the Dunwoody man- ifold is the -fold cyclic covering of the manifold , branched over an -knot in .",
"Example 2. The Dunwoody manifolds , and , with coprime, are -fold cyclic coverings of the manifolds , and respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of copies of , and respectively.",
"Let us consider now the class of the Dunwoody manifolds with (and hence odd) and . Many ex- amples of these manifolds appear in Table 1 of [6], where it was conjectured that they are -fold cyclic coverings of , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.",
"Corollary 7 Let be an admissible 6-tuple with and . Then the -tuple is admissible for each and the Dunwoody manifold is a n-fold cyclic coverings of , branched over a genus one -bridge knot , which is independent on .",
"Proof. Obviously . Since is admissible, it satisfies (i’). This proves that satisfies , for each . Since and , we obtain , for each , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, is odd, since . Thus, Lemma 3 proves that is admissible. The final result is then a direct consequence of Theorem 6.",
"We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].",
"An interesting problem which naturally arises is that of characterizing the set of branching knots in involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers and , with odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type is equivalent to the 2-bridge knot of type , then can be assumed to be even.",
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[{"type": "text", "text": "Since the handlebody orbifolds and their gluing only depend on $a,b,c,r$ , the same holds for the branching set $K$ . The homeomorphism type of $M^{\\prime}$ follows from Proposition 2 and Lemma 5. ", "page_idx": 11}, {"type": "text", "text": "Remark 4. More generally, given two positive integers $n$ and $n^{\\prime}$ such that $n^{\\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then the Dunwoody manifold $M(a,b,c,n,r,s)$ is the $n/n^{\\prime}$ -fold cyclic covering of the manifold $M^{\\prime}=$ $M(a,b,c,n^{\\prime},r,s)$ , branched over an $(n^{\\prime},1)$ -knot in $M^{\\prime}$ . ", "page_idx": 11}, {"type": "text", "text": "Example 2. The Dunwoody manifolds $M(0,0,1,n,0,0)$ , $M(1,0,0,n,1,0)$ and $M(0,0,c,n,r,0)$ , with $c,r$ coprime, are $n$ -fold cyclic coverings of the manifolds $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ and $L(c,r)$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $n$ copies of $\\mathbf{S^{3}}$ , $\\mathbf{S^{1}}\\times\\mathbf{S^{2}}$ and $L(c,r)$ respectively. ", "page_idx": 11}, {"type": "text", "text": "Let us consider now the class of the Dunwoody manifolds $\\textstyle M_{n}\\ =$ $M(a,b,c,n,r,s)$ with $p=\\pm1$ (and hence $d$ odd) and $s\\,=\\,-p q$ . Many examples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $n$ -fold cyclic coverings of $\\mathbf{S^{3}}$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture. ", "page_idx": 11}, {"type": "text", "text": "Corollary 7 Let $\\sigma_{1}=(a,b,c,1,r,0)$ be an admissible 6-tuple with $p_{\\sigma_{1}}=\\pm1$ and $s=-p_{\\sigma_{1}}q_{\\sigma_{1}}$ . Then the $\\it6$ -tuple $\\sigma_{n}=(a,b,c,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,b,c,n,r,s)$ is a n-fold cyclic coverings of $\\mathrm{{S^{3}}}$ , branched over a genus one $\\mathit{1}$ -bridge knot $K\\subset{\\bf S^{3}}$ , which is independent on $n$ . ", "page_idx": 11}, {"type": "text", "text": "Proof. Obviously $(a,b,c,1,r,s)=\\sigma_{1}$ . Since $\\sigma_{1}$ is admissible, it satisfies (i\u2019). This proves that $\\sigma_{n}$ satisfies $\\left(\\mathrm{i}^{\\,\\circ}\\right)$ , for each $n>1$ . Since $s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=$ $-p_{\\sigma_{n}}q_{\\sigma_{n}}$ and $p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1$ , we obtain $q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0$ , for each $n>1$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii\u2019). Moreover, $d$ is odd, since $[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1$ . Thus, Lemma 3 proves that $\\sigma_{n}$ is admissible. The final result is then a direct consequence of Theorem 6. ", "page_idx": 11}, {"type": "text", "text": "We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32]. ", "page_idx": 11}, {"type": "text", "text": "An interesting problem which naturally arises is that of characterizing the set $\\kappa$ of branching knots in $\\mathrm{{S^{3}}}$ involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers $\\alpha$ and $\\beta$ , with $\\alpha~>~0$ odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type $(\\alpha,\\beta)$ is equivalent to the 2-bridge knot of type $(\\alpha,\\alpha-\\beta)$ , then $\\beta$ can be assumed to be even. ", "page_idx": 11}]
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The homeomorphism type of ", "type": "text"}, {"bbox": [446, 144, 461, 153], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [461, 142, 500, 156], "score": 1.0, "content": " follows", "type": "text"}], "index": 1}, {"bbox": [110, 158, 302, 169], "spans": [{"bbox": [110, 158, 302, 169], "score": 1.0, "content": "from Proposition 2 and Lemma 5.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [110, 173, 500, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 421, 191], "score": 1.0, "content": "Remark 4. More generally, given two positive integers ", "type": "text"}, {"bbox": [422, 181, 429, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [430, 176, 460, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [460, 178, 470, 187], "score": 0.89, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [470, 176, 500, 191], "score": 1.0, "content": " such", "type": "text"}], "index": 3}, {"bbox": [109, 190, 500, 205], "spans": [{"bbox": [109, 190, 136, 205], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 192, 146, 201], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [147, 190, 190, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [191, 196, 198, 201], "score": 0.85, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [198, 190, 217, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [218, 191, 286, 204], "score": 0.93, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [287, 190, 500, 205], "score": 1.0, "content": " is admissible, then the Dunwoody man-", "type": "text"}], "index": 4}, {"bbox": [109, 205, 500, 219], "spans": [{"bbox": [109, 205, 137, 219], "score": 1.0, "content": "ifold ", "type": "text"}, {"bbox": [137, 206, 218, 218], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [219, 205, 256, 219], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [256, 206, 279, 218], "score": 0.93, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [280, 205, 469, 219], "score": 1.0, "content": "-fold cyclic covering of the manifold ", "type": "text"}, {"bbox": [470, 206, 500, 218], "score": 0.83, "content": "M^{\\prime}=", "type": "inline_equation", "height": 12, "width": 30}], "index": 5}, {"bbox": [110, 219, 388, 234], "spans": [{"bbox": [110, 221, 195, 233], "score": 0.92, "content": "M(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [195, 219, 293, 233], "score": 1.0, "content": ", branched over an ", "type": "text"}, {"bbox": [293, 221, 324, 234], "score": 0.93, "content": "(n^{\\prime},1)", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [324, 219, 367, 233], "score": 1.0, "content": "-knot in ", "type": "text"}, {"bbox": [368, 221, 383, 230], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [384, 219, 388, 233], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5}, {"type": "text", "bbox": [109, 237, 500, 309], "lines": [{"bbox": [110, 239, 499, 254], "spans": [{"bbox": [110, 239, 322, 254], "score": 1.0, "content": "Example 2. The Dunwoody manifolds ", "type": "text"}, {"bbox": [322, 241, 407, 253], "score": 0.92, "content": "M(0,0,1,n,0,0)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [407, 239, 414, 254], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [414, 241, 499, 253], "score": 0.91, "content": "M(1,0,0,n,1,0)", "type": "inline_equation", "height": 12, "width": 85}], "index": 7}, {"bbox": [110, 254, 500, 268], "spans": [{"bbox": [110, 254, 134, 268], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [135, 255, 217, 267], "score": 0.92, "content": "M(0,0,c,n,r,0)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [217, 254, 254, 268], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [254, 259, 270, 267], "score": 0.9, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [271, 254, 345, 268], "score": 1.0, "content": " coprime, are ", "type": "text"}, {"bbox": [346, 259, 353, 264], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [354, 254, 500, 268], "score": 1.0, "content": "-fold cyclic coverings of the", "type": "text"}], "index": 8}, {"bbox": [109, 268, 500, 283], "spans": [{"bbox": [109, 268, 163, 283], "score": 1.0, "content": "manifolds ", "type": "text"}, {"bbox": [163, 269, 176, 279], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [177, 268, 183, 283], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [183, 269, 223, 280], "score": 0.91, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [223, 268, 249, 283], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 270, 282, 282], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [283, 268, 500, 283], "score": 1.0, "content": " respectively, branched over a trivial knot.", "type": "text"}], "index": 9}, {"bbox": [109, 282, 499, 297], "spans": [{"bbox": [109, 282, 424, 297], "score": 1.0, "content": "In fact, these Dunwoody manifolds are the connected sum of ", "type": "text"}, {"bbox": [424, 288, 432, 294], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [432, 282, 482, 297], "score": 1.0, "content": " copies of ", "type": "text"}, {"bbox": [483, 284, 496, 294], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [496, 282, 499, 297], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [110, 296, 276, 312], "spans": [{"bbox": [110, 298, 150, 308], "score": 0.93, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [151, 296, 177, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [177, 298, 210, 311], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [210, 296, 276, 312], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 11}], "index": 9}, {"type": "text", "bbox": [110, 315, 500, 387], "lines": [{"bbox": [126, 316, 499, 331], "spans": [{"bbox": [126, 316, 462, 331], "score": 1.0, "content": "Let us consider now the class of the Dunwoody manifolds ", "type": "text"}, {"bbox": [463, 318, 499, 330], "score": 0.85, "content": "\\textstyle M_{n}\\ =", "type": "inline_equation", "height": 12, "width": 36}], "index": 12}, {"bbox": [110, 331, 500, 347], "spans": [{"bbox": [110, 333, 192, 345], "score": 0.92, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [192, 331, 223, 347], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 334, 264, 345], "score": 0.92, "content": "p=\\pm1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [264, 331, 330, 347], "score": 1.0, "content": " (and hence ", "type": "text"}, {"bbox": [330, 334, 337, 342], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [337, 331, 393, 347], "score": 1.0, "content": " odd) and ", "type": "text"}, {"bbox": [393, 335, 439, 345], "score": 0.92, "content": "s\\,=\\,-p q", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [439, 331, 500, 347], "score": 1.0, "content": ". Many ex-", "type": "text"}], "index": 13}, {"bbox": [111, 347, 499, 360], "spans": [{"bbox": [111, 347, 499, 360], "score": 1.0, "content": "amples of these manifolds appear in Table 1 of [6], where it was conjectured", "type": "text"}], "index": 14}, {"bbox": [109, 361, 501, 375], "spans": [{"bbox": [109, 361, 179, 375], "score": 1.0, "content": "that they are ", "type": "text"}, {"bbox": [179, 366, 186, 371], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [187, 361, 306, 375], "score": 1.0, "content": "-fold cyclic coverings of ", "type": "text"}, {"bbox": [306, 361, 319, 371], "score": 0.9, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [320, 361, 501, 375], "score": 1.0, "content": ", branched over suitable knots. The", "type": "text"}], "index": 15}, {"bbox": [109, 375, 395, 389], "spans": [{"bbox": [109, 375, 395, 389], "score": 1.0, "content": "following corollary of Theorem 6 proves this conjecture.", "type": "text"}], "index": 16}], "index": 14}, {"type": "text", "bbox": [110, 398, 500, 470], "lines": [{"bbox": [109, 399, 499, 416], "spans": [{"bbox": [109, 399, 202, 416], "score": 1.0, "content": "Corollary 7 Let ", "type": "text"}, {"bbox": [203, 401, 298, 414], "score": 0.92, "content": "\\sigma_{1}=(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [298, 399, 452, 416], "score": 1.0, "content": " be an admissible 6-tuple with ", "type": "text"}, {"bbox": [452, 402, 499, 414], "score": 0.9, "content": "p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 12, "width": 47}], "index": 17}, {"bbox": [109, 414, 502, 431], "spans": [{"bbox": [109, 414, 132, 431], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [132, 417, 194, 428], "score": 0.89, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [194, 414, 252, 431], "score": 1.0, "content": ". Then the ", "type": "text"}, {"bbox": [252, 417, 258, 425], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [258, 414, 289, 431], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [289, 415, 387, 428], "score": 0.91, "content": "\\sigma_{n}=(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [387, 414, 502, 431], "score": 1.0, "content": " is admissible for each", "type": "text"}], "index": 18}, {"bbox": [110, 429, 501, 444], "spans": [{"bbox": [110, 431, 140, 440], "score": 0.89, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [141, 429, 291, 444], "score": 1.0, "content": " and the Dunwoody manifold", "type": "text"}, {"bbox": [291, 430, 408, 443], "score": 0.91, "content": "M_{n}=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [408, 429, 501, 444], "score": 1.0, "content": " is a n-fold cyclic", "type": "text"}], "index": 19}, {"bbox": [110, 443, 501, 456], "spans": [{"bbox": [110, 443, 175, 456], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [175, 444, 189, 454], "score": 0.88, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [189, 443, 337, 456], "score": 1.0, "content": ", branched over a genus one ", "type": "text"}, {"bbox": [338, 445, 343, 454], "score": 0.4, "content": "\\mathit{1}", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [344, 443, 408, 456], "score": 1.0, "content": "-bridge knot ", "type": "text"}, {"bbox": [408, 444, 449, 455], "score": 0.9, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [450, 443, 501, 456], "score": 1.0, "content": ", which is", "type": "text"}], "index": 20}, {"bbox": [110, 458, 204, 473], "spans": [{"bbox": [110, 458, 191, 473], "score": 1.0, "content": "independent on ", "type": "text"}, {"bbox": [191, 463, 199, 469], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [199, 458, 204, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19}, {"type": "text", "bbox": [109, 481, 500, 582], "lines": [{"bbox": [126, 483, 500, 497], "spans": [{"bbox": [126, 483, 225, 497], "score": 1.0, "content": "Proof. Obviously ", "type": "text"}, {"bbox": [225, 484, 321, 497], "score": 0.93, "content": "(a,b,c,1,r,s)=\\sigma_{1}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [321, 483, 360, 497], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [360, 488, 371, 496], "score": 0.84, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [372, 483, 500, 497], "score": 1.0, "content": " is admissible, it satisfies", "type": "text"}], "index": 22}, {"bbox": [110, 496, 500, 514], "spans": [{"bbox": [110, 496, 225, 514], "score": 1.0, "content": "(i\u2019). This proves that ", "type": "text"}, {"bbox": [225, 503, 237, 510], "score": 0.88, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [237, 496, 285, 514], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [285, 498, 300, 511], "score": 0.28, "content": "\\left(\\mathrm{i}^{\\,\\circ}\\right)", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [301, 496, 352, 514], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [353, 500, 383, 509], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [383, 496, 423, 514], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [423, 500, 500, 511], "score": 0.86, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=", "type": "inline_equation", "height": 11, "width": 77}], "index": 23}, {"bbox": [110, 513, 500, 527], "spans": [{"bbox": [110, 515, 151, 525], "score": 0.9, "content": "-p_{\\sigma_{n}}q_{\\sigma_{n}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [151, 513, 176, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [176, 514, 255, 525], "score": 0.9, "content": "p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [255, 513, 313, 527], "score": 1.0, "content": ", we obtain ", "type": "text"}, {"bbox": [314, 514, 385, 525], "score": 0.92, "content": "q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [385, 513, 433, 527], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [434, 514, 463, 523], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [463, 513, 500, 527], "score": 1.0, "content": ", which", "type": "text"}], "index": 24}, {"bbox": [110, 526, 500, 541], "spans": [{"bbox": [110, 526, 481, 541], "score": 1.0, "content": "implies condition (ii) of Theorem 2 of [6], or equivalently (ii\u2019). Moreover, ", "type": "text"}, {"bbox": [481, 528, 488, 537], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [488, 526, 500, 541], "score": 1.0, "content": " is", "type": "text"}], "index": 25}, {"bbox": [109, 541, 501, 556], "spans": [{"bbox": [109, 541, 165, 556], "score": 1.0, "content": "odd, since ", "type": "text"}, {"bbox": [165, 542, 420, 555], "score": 0.88, "content": "[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1", "type": "inline_equation", "height": 13, "width": 255}, {"bbox": [420, 541, 501, 556], "score": 1.0, "content": ". Thus, Lemma", "type": "text"}], "index": 26}, {"bbox": [110, 556, 499, 569], "spans": [{"bbox": [110, 556, 181, 569], "score": 1.0, "content": "3 proves that ", "type": "text"}, {"bbox": [182, 560, 194, 568], "score": 0.9, "content": "\\sigma_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [194, 556, 499, 569], "score": 1.0, "content": " is admissible. The final result is then a direct consequence", "type": "text"}], "index": 27}, {"bbox": [110, 569, 200, 583], "spans": [{"bbox": [110, 569, 200, 583], "score": 1.0, "content": "of Theorem 6.", "type": "text"}], "index": 28}], "index": 25}, {"type": "text", "bbox": [109, 587, 500, 617], "lines": [{"bbox": [127, 589, 499, 604], "spans": [{"bbox": [127, 589, 499, 604], "score": 1.0, "content": "We point out that the above result has been independently obtained by", "type": "text"}], "index": 29}, {"bbox": [110, 605, 281, 618], "spans": [{"bbox": [110, 605, 281, 618], "score": 1.0, "content": "H. J. Song and S. H. Kim in [32].", "type": "text"}], "index": 30}], "index": 29.5}, {"type": "text", "bbox": [109, 618, 500, 674], "lines": [{"bbox": [127, 618, 500, 633], "spans": [{"bbox": [127, 618, 500, 633], "score": 1.0, "content": "An interesting problem which naturally arises is that of characterizing the", "type": "text"}], "index": 31}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 128, 648], "score": 1.0, "content": "set ", "type": "text"}, {"bbox": [129, 635, 138, 644], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [139, 633, 258, 648], "score": 1.0, "content": " of branching knots in ", "type": "text"}, {"bbox": [258, 634, 271, 644], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [272, 633, 500, 648], "score": 1.0, "content": " involved in Corollary 7. The next theorem", "type": "text"}], "index": 32}, {"bbox": [110, 649, 500, 661], "spans": [{"bbox": [110, 649, 500, 661], "score": 1.0, "content": "shows that it contains all 2-bridge knots. We recall that a 2-bridge knot", "type": "text"}], "index": 33}, {"bbox": [109, 661, 501, 676], "spans": [{"bbox": [109, 661, 320, 676], "score": 1.0, "content": "is determined by two coprime integers ", "type": "text"}, {"bbox": [320, 667, 328, 672], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [329, 661, 358, 676], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [358, 664, 366, 675], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [366, 661, 403, 676], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [404, 664, 440, 673], "score": 0.92, "content": "\\alpha~>~0", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [440, 661, 501, 676], "score": 1.0, "content": " odd. The", "type": "text"}], "index": 34}], "index": 32.5}], "layout_bboxes": [], "page_idx": 11, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 311, 702], "lines": [{"bbox": [297, 692, 313, 705], "spans": [{"bbox": [297, 692, 313, 705], "score": 1.0, "content": "12", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [{"bbox": [109, 128, 500, 142], "spans": [{"bbox": [109, 128, 438, 142], "score": 1.0, "content": "Since the handlebody orbifolds and their gluing only depend on ", "type": "text"}, {"bbox": [438, 129, 476, 141], "score": 0.94, "content": "a,b,c,r", "type": "inline_equation", "height": 12, "width": 38}, {"bbox": [476, 128, 500, 142], "score": 1.0, "content": ", the", "type": "text"}], "index": 0}, {"bbox": [109, 142, 500, 156], "spans": [{"bbox": [109, 142, 277, 156], "score": 1.0, "content": "same holds for the branching set ", "type": "text"}, {"bbox": [277, 144, 288, 153], "score": 0.9, "content": "K", "type": "inline_equation", "height": 9, "width": 11}, {"bbox": [289, 142, 445, 156], "score": 1.0, "content": ". The homeomorphism type of ", "type": "text"}, {"bbox": [446, 144, 461, 153], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [461, 142, 500, 156], "score": 1.0, "content": " follows", "type": "text"}], "index": 1}, {"bbox": [110, 158, 302, 169], "spans": [{"bbox": [110, 158, 302, 169], "score": 1.0, "content": "from Proposition 2 and Lemma 5.", "type": "text"}], "index": 2}], "index": 1, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 128, 500, 169]}, {"type": "text", "bbox": [110, 173, 500, 232], "lines": [{"bbox": [109, 176, 500, 191], "spans": [{"bbox": [109, 176, 421, 191], "score": 1.0, "content": "Remark 4. More generally, given two positive integers ", "type": "text"}, {"bbox": [422, 181, 429, 187], "score": 0.87, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [430, 176, 460, 191], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [460, 178, 470, 187], "score": 0.89, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [470, 176, 500, 191], "score": 1.0, "content": " such", "type": "text"}], "index": 3}, {"bbox": [109, 190, 500, 205], "spans": [{"bbox": [109, 190, 136, 205], "score": 1.0, "content": "that ", "type": "text"}, {"bbox": [136, 192, 146, 201], "score": 0.9, "content": "n^{\\prime}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [147, 190, 190, 205], "score": 1.0, "content": " divides ", "type": "text"}, {"bbox": [191, 196, 198, 201], "score": 0.85, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [198, 190, 217, 205], "score": 1.0, "content": ", if ", "type": "text"}, {"bbox": [218, 191, 286, 204], "score": 0.93, "content": "(a,b,c,r,n,s)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [287, 190, 500, 205], "score": 1.0, "content": " is admissible, then the Dunwoody man-", "type": "text"}], "index": 4}, {"bbox": [109, 205, 500, 219], "spans": [{"bbox": [109, 205, 137, 219], "score": 1.0, "content": "ifold ", "type": "text"}, {"bbox": [137, 206, 218, 218], "score": 0.93, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [219, 205, 256, 219], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [256, 206, 279, 218], "score": 0.93, "content": "n/n^{\\prime}", "type": "inline_equation", "height": 12, "width": 23}, {"bbox": [280, 205, 469, 219], "score": 1.0, "content": "-fold cyclic covering of the manifold ", "type": "text"}, {"bbox": [470, 206, 500, 218], "score": 0.83, "content": "M^{\\prime}=", "type": "inline_equation", "height": 12, "width": 30}], "index": 5}, {"bbox": [110, 219, 388, 234], "spans": [{"bbox": [110, 221, 195, 233], "score": 0.92, "content": "M(a,b,c,n^{\\prime},r,s)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [195, 219, 293, 233], "score": 1.0, "content": ", branched over an ", "type": "text"}, {"bbox": [293, 221, 324, 234], "score": 0.93, "content": "(n^{\\prime},1)", "type": "inline_equation", "height": 13, "width": 31}, {"bbox": [324, 219, 367, 233], "score": 1.0, "content": "-knot in ", "type": "text"}, {"bbox": [368, 221, 383, 230], "score": 0.92, "content": "M^{\\prime}", "type": "inline_equation", "height": 9, "width": 15}, {"bbox": [384, 219, 388, 233], "score": 1.0, "content": ".", "type": "text"}], "index": 6}], "index": 4.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 176, 500, 234]}, {"type": "text", "bbox": [109, 237, 500, 309], "lines": [{"bbox": [110, 239, 499, 254], "spans": [{"bbox": [110, 239, 322, 254], "score": 1.0, "content": "Example 2. The Dunwoody manifolds ", "type": "text"}, {"bbox": [322, 241, 407, 253], "score": 0.92, "content": "M(0,0,1,n,0,0)", "type": "inline_equation", "height": 12, "width": 85}, {"bbox": [407, 239, 414, 254], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [414, 241, 499, 253], "score": 0.91, "content": "M(1,0,0,n,1,0)", "type": "inline_equation", "height": 12, "width": 85}], "index": 7}, {"bbox": [110, 254, 500, 268], "spans": [{"bbox": [110, 254, 134, 268], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [135, 255, 217, 267], "score": 0.92, "content": "M(0,0,c,n,r,0)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [217, 254, 254, 268], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [254, 259, 270, 267], "score": 0.9, "content": "c,r", "type": "inline_equation", "height": 8, "width": 16}, {"bbox": [271, 254, 345, 268], "score": 1.0, "content": " coprime, are ", "type": "text"}, {"bbox": [346, 259, 353, 264], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [354, 254, 500, 268], "score": 1.0, "content": "-fold cyclic coverings of the", "type": "text"}], "index": 8}, {"bbox": [109, 268, 500, 283], "spans": [{"bbox": [109, 268, 163, 283], "score": 1.0, "content": "manifolds ", "type": "text"}, {"bbox": [163, 269, 176, 279], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [177, 268, 183, 283], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [183, 269, 223, 280], "score": 0.91, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [223, 268, 249, 283], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [250, 270, 282, 282], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 12, "width": 32}, {"bbox": [283, 268, 500, 283], "score": 1.0, "content": " respectively, branched over a trivial knot.", "type": "text"}], "index": 9}, {"bbox": [109, 282, 499, 297], "spans": [{"bbox": [109, 282, 424, 297], "score": 1.0, "content": "In fact, these Dunwoody manifolds are the connected sum of ", "type": "text"}, {"bbox": [424, 288, 432, 294], "score": 0.89, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [432, 282, 482, 297], "score": 1.0, "content": " copies of ", "type": "text"}, {"bbox": [483, 284, 496, 294], "score": 0.89, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [496, 282, 499, 297], "score": 1.0, "content": ",", "type": "text"}], "index": 10}, {"bbox": [110, 296, 276, 312], "spans": [{"bbox": [110, 298, 150, 308], "score": 0.93, "content": "\\mathbf{S^{1}}\\times\\mathbf{S^{2}}", "type": "inline_equation", "height": 10, "width": 40}, {"bbox": [151, 296, 177, 312], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [177, 298, 210, 311], "score": 0.95, "content": "L(c,r)", "type": "inline_equation", "height": 13, "width": 33}, {"bbox": [210, 296, 276, 312], "score": 1.0, "content": " respectively.", "type": "text"}], "index": 11}], "index": 9, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 239, 500, 312]}, {"type": "text", "bbox": [110, 315, 500, 387], "lines": [{"bbox": [126, 316, 499, 331], "spans": [{"bbox": [126, 316, 462, 331], "score": 1.0, "content": "Let us consider now the class of the Dunwoody manifolds ", "type": "text"}, {"bbox": [463, 318, 499, 330], "score": 0.85, "content": "\\textstyle M_{n}\\ =", "type": "inline_equation", "height": 12, "width": 36}], "index": 12}, {"bbox": [110, 331, 500, 347], "spans": [{"bbox": [110, 333, 192, 345], "score": 0.92, "content": "M(a,b,c,n,r,s)", "type": "inline_equation", "height": 12, "width": 82}, {"bbox": [192, 331, 223, 347], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [224, 334, 264, 345], "score": 0.92, "content": "p=\\pm1", "type": "inline_equation", "height": 11, "width": 40}, {"bbox": [264, 331, 330, 347], "score": 1.0, "content": " (and hence ", "type": "text"}, {"bbox": [330, 334, 337, 342], "score": 0.89, "content": "d", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [337, 331, 393, 347], "score": 1.0, "content": " odd) and ", "type": "text"}, {"bbox": [393, 335, 439, 345], "score": 0.92, "content": "s\\,=\\,-p q", "type": "inline_equation", "height": 10, "width": 46}, {"bbox": [439, 331, 500, 347], "score": 1.0, "content": ". Many ex-", "type": "text"}], "index": 13}, {"bbox": [111, 347, 499, 360], "spans": [{"bbox": [111, 347, 499, 360], "score": 1.0, "content": "amples of these manifolds appear in Table 1 of [6], where it was conjectured", "type": "text"}], "index": 14}, {"bbox": [109, 361, 501, 375], "spans": [{"bbox": [109, 361, 179, 375], "score": 1.0, "content": "that they are ", "type": "text"}, {"bbox": [179, 366, 186, 371], "score": 0.87, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [187, 361, 306, 375], "score": 1.0, "content": "-fold cyclic coverings of ", "type": "text"}, {"bbox": [306, 361, 319, 371], "score": 0.9, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [320, 361, 501, 375], "score": 1.0, "content": ", branched over suitable knots. The", "type": "text"}], "index": 15}, {"bbox": [109, 375, 395, 389], "spans": [{"bbox": [109, 375, 395, 389], "score": 1.0, "content": "following corollary of Theorem 6 proves this conjecture.", "type": "text"}], "index": 16}], "index": 14, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 316, 501, 389]}, {"type": "text", "bbox": [110, 398, 500, 470], "lines": [{"bbox": [109, 399, 499, 416], "spans": [{"bbox": [109, 399, 202, 416], "score": 1.0, "content": "Corollary 7 Let ", "type": "text"}, {"bbox": [203, 401, 298, 414], "score": 0.92, "content": "\\sigma_{1}=(a,b,c,1,r,0)", "type": "inline_equation", "height": 13, "width": 95}, {"bbox": [298, 399, 452, 416], "score": 1.0, "content": " be an admissible 6-tuple with ", "type": "text"}, {"bbox": [452, 402, 499, 414], "score": 0.9, "content": "p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 12, "width": 47}], "index": 17}, {"bbox": [109, 414, 502, 431], "spans": [{"bbox": [109, 414, 132, 431], "score": 1.0, "content": "and ", "type": "text"}, {"bbox": [132, 417, 194, 428], "score": 0.89, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 62}, {"bbox": [194, 414, 252, 431], "score": 1.0, "content": ". Then the ", "type": "text"}, {"bbox": [252, 417, 258, 425], "score": 0.37, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [258, 414, 289, 431], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [289, 415, 387, 428], "score": 0.91, "content": "\\sigma_{n}=(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 98}, {"bbox": [387, 414, 502, 431], "score": 1.0, "content": " is admissible for each", "type": "text"}], "index": 18}, {"bbox": [110, 429, 501, 444], "spans": [{"bbox": [110, 431, 140, 440], "score": 0.89, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [141, 429, 291, 444], "score": 1.0, "content": " and the Dunwoody manifold", "type": "text"}, {"bbox": [291, 430, 408, 443], "score": 0.91, "content": "M_{n}=M(a,b,c,n,r,s)", "type": "inline_equation", "height": 13, "width": 117}, {"bbox": [408, 429, 501, 444], "score": 1.0, "content": " is a n-fold cyclic", "type": "text"}], "index": 19}, {"bbox": [110, 443, 501, 456], "spans": [{"bbox": [110, 443, 175, 456], "score": 1.0, "content": "coverings of ", "type": "text"}, {"bbox": [175, 444, 189, 454], "score": 0.88, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [189, 443, 337, 456], "score": 1.0, "content": ", branched over a genus one ", "type": "text"}, {"bbox": [338, 445, 343, 454], "score": 0.4, "content": "\\mathit{1}", "type": "inline_equation", "height": 9, "width": 5}, {"bbox": [344, 443, 408, 456], "score": 1.0, "content": "-bridge knot ", "type": "text"}, {"bbox": [408, 444, 449, 455], "score": 0.9, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 11, "width": 41}, {"bbox": [450, 443, 501, 456], "score": 1.0, "content": ", which is", "type": "text"}], "index": 20}, {"bbox": [110, 458, 204, 473], "spans": [{"bbox": [110, 458, 191, 473], "score": 1.0, "content": "independent on ", "type": "text"}, {"bbox": [191, 463, 199, 469], "score": 0.84, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [199, 458, 204, 473], "score": 1.0, "content": ".", "type": "text"}], "index": 21}], "index": 19, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 399, 502, 473]}, {"type": "text", "bbox": [109, 481, 500, 582], "lines": [{"bbox": [126, 483, 500, 497], "spans": [{"bbox": [126, 483, 225, 497], "score": 1.0, "content": "Proof. Obviously ", "type": "text"}, {"bbox": [225, 484, 321, 497], "score": 0.93, "content": "(a,b,c,1,r,s)=\\sigma_{1}", "type": "inline_equation", "height": 13, "width": 96}, {"bbox": [321, 483, 360, 497], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [360, 488, 371, 496], "score": 0.84, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 11}, {"bbox": [372, 483, 500, 497], "score": 1.0, "content": " is admissible, it satisfies", "type": "text"}], "index": 22}, {"bbox": [110, 496, 500, 514], "spans": [{"bbox": [110, 496, 225, 514], "score": 1.0, "content": "(i\u2019). This proves that ", "type": "text"}, {"bbox": [225, 503, 237, 510], "score": 0.88, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [237, 496, 285, 514], "score": 1.0, "content": " satisfies ", "type": "text"}, {"bbox": [285, 498, 300, 511], "score": 0.28, "content": "\\left(\\mathrm{i}^{\\,\\circ}\\right)", "type": "inline_equation", "height": 13, "width": 15}, {"bbox": [301, 496, 352, 514], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [353, 500, 383, 509], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 30}, {"bbox": [383, 496, 423, 514], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [423, 500, 500, 511], "score": 0.86, "content": "s=-p_{\\sigma_{1}}q_{\\sigma_{1}}=", "type": "inline_equation", "height": 11, "width": 77}], "index": 23}, {"bbox": [110, 513, 500, 527], "spans": [{"bbox": [110, 515, 151, 525], "score": 0.9, "content": "-p_{\\sigma_{n}}q_{\\sigma_{n}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [151, 513, 176, 527], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [176, 514, 255, 525], "score": 0.9, "content": "p_{\\sigma_{n}}=p_{\\sigma_{1}}=\\pm1", "type": "inline_equation", "height": 11, "width": 79}, {"bbox": [255, 513, 313, 527], "score": 1.0, "content": ", we obtain ", "type": "text"}, {"bbox": [314, 514, 385, 525], "score": 0.92, "content": "q_{\\sigma_{n}}+s p_{\\sigma_{n}}=0", "type": "inline_equation", "height": 11, "width": 71}, {"bbox": [385, 513, 433, 527], "score": 1.0, "content": ", for each ", "type": "text"}, {"bbox": [434, 514, 463, 523], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [463, 513, 500, 527], "score": 1.0, "content": ", which", "type": "text"}], "index": 24}, {"bbox": [110, 526, 500, 541], "spans": [{"bbox": [110, 526, 481, 541], "score": 1.0, "content": "implies condition (ii) of Theorem 2 of [6], or equivalently (ii\u2019). Moreover, ", "type": "text"}, {"bbox": [481, 528, 488, 537], "score": 0.9, "content": "d", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [488, 526, 500, 541], "score": 1.0, "content": " is", "type": "text"}], "index": 25}, {"bbox": [109, 541, 501, 556], "spans": [{"bbox": [109, 541, 165, 556], "score": 1.0, "content": "odd, since ", "type": "text"}, {"bbox": [165, 542, 420, 555], "score": 0.88, "content": "[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\\sigma_{n}}]_{2}=[p_{\\sigma_{1}}]_{2}=1", "type": "inline_equation", "height": 13, "width": 255}, {"bbox": [420, 541, 501, 556], "score": 1.0, "content": ". Thus, Lemma", "type": "text"}], "index": 26}, {"bbox": [110, 556, 499, 569], "spans": [{"bbox": [110, 556, 181, 569], "score": 1.0, "content": "3 proves that ", "type": "text"}, {"bbox": [182, 560, 194, 568], "score": 0.9, "content": "\\sigma_{n}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [194, 556, 499, 569], "score": 1.0, "content": " is admissible. The final result is then a direct consequence", "type": "text"}], "index": 27}, {"bbox": [110, 569, 200, 583], "spans": [{"bbox": [110, 569, 200, 583], "score": 1.0, "content": "of Theorem 6.", "type": "text"}], "index": 28}], "index": 25, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 483, 501, 583]}, {"type": "text", "bbox": [109, 587, 500, 617], "lines": [{"bbox": [127, 589, 499, 604], "spans": [{"bbox": [127, 589, 499, 604], "score": 1.0, "content": "We point out that the above result has been independently obtained by", "type": "text"}], "index": 29}, {"bbox": [110, 605, 281, 618], "spans": [{"bbox": [110, 605, 281, 618], "score": 1.0, "content": "H. J. Song and S. H. Kim in [32].", "type": "text"}], "index": 30}], "index": 29.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [110, 589, 499, 618]}, {"type": "text", "bbox": [109, 618, 500, 674], "lines": [{"bbox": [127, 618, 500, 633], "spans": [{"bbox": [127, 618, 500, 633], "score": 1.0, "content": "An interesting problem which naturally arises is that of characterizing the", "type": "text"}], "index": 31}, {"bbox": [109, 633, 500, 648], "spans": [{"bbox": [109, 633, 128, 648], "score": 1.0, "content": "set ", "type": "text"}, {"bbox": [129, 635, 138, 644], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [139, 633, 258, 648], "score": 1.0, "content": " of branching knots in ", "type": "text"}, {"bbox": [258, 634, 271, 644], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [272, 633, 500, 648], "score": 1.0, "content": " involved in Corollary 7. The next theorem", "type": "text"}], "index": 32}, {"bbox": [110, 649, 500, 661], "spans": [{"bbox": [110, 649, 500, 661], "score": 1.0, "content": "shows that it contains all 2-bridge knots. We recall that a 2-bridge knot", "type": "text"}], "index": 33}, {"bbox": [109, 661, 501, 676], "spans": [{"bbox": [109, 661, 320, 676], "score": 1.0, "content": "is determined by two coprime integers ", "type": "text"}, {"bbox": [320, 667, 328, 672], "score": 0.89, "content": "\\alpha", "type": "inline_equation", "height": 5, "width": 8}, {"bbox": [329, 661, 358, 676], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [358, 664, 366, 675], "score": 0.89, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [366, 661, 403, 676], "score": 1.0, "content": ", with ", "type": "text"}, {"bbox": [404, 664, 440, 673], "score": 0.92, "content": "\\alpha~>~0", "type": "inline_equation", "height": 9, "width": 36}, {"bbox": [440, 661, 501, 676], "score": 1.0, "content": " odd. The", "type": "text"}], "index": 34}, {"bbox": [110, 128, 500, 141], "spans": [{"bbox": [110, 128, 500, 141], "score": 1.0, "content": "classification of 2-bridge knots and links has been obtained by Schubert in", "type": "text", "cross_page": true}], "index": 0}, {"bbox": [111, 142, 500, 156], "spans": [{"bbox": [111, 142, 297, 156], "score": 1.0, "content": "[30]. Since the 2-bridge knot of type ", "type": "text", "cross_page": true}, {"bbox": [298, 143, 326, 155], "score": 0.94, "content": "(\\alpha,\\beta)", "type": "inline_equation", "height": 12, "width": 28, "cross_page": true}, {"bbox": [327, 142, 500, 156], "score": 1.0, "content": " is equivalent to the 2-bridge knot", "type": "text", "cross_page": true}], "index": 1}, {"bbox": [110, 157, 387, 170], "spans": [{"bbox": [110, 157, 149, 170], "score": 1.0, "content": "of type ", "type": "text", "cross_page": true}, {"bbox": [150, 158, 201, 170], "score": 0.92, "content": "(\\alpha,\\alpha-\\beta)", "type": "inline_equation", "height": 12, "width": 51, "cross_page": true}, {"bbox": [201, 157, 234, 170], "score": 1.0, "content": ", then ", "type": "text", "cross_page": true}, {"bbox": [235, 158, 243, 169], "score": 0.9, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8, "cross_page": true}, {"bbox": [243, 157, 387, 170], "score": 1.0, "content": " can be assumed to be even.", "type": "text", "cross_page": true}], "index": 2}], "index": 32.5, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [109, 618, 501, 676]}]}
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|
Since the handlebody orbifolds and their gluing only depend on , the same holds for the branching set . The homeomorphism type of follows from Proposition 2 and Lemma 5.
Remark 4. More generally, given two positive integers and such that divides , if is admissible, then the Dunwoody man- ifold is the -fold cyclic covering of the manifold , branched over an -knot in .
Example 2. The Dunwoody manifolds , and , with coprime, are -fold cyclic coverings of the manifolds , and respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of copies of , and respectively.
Let us consider now the class of the Dunwoody manifolds with (and hence odd) and . Many ex- amples of these manifolds appear in Table 1 of [6], where it was conjectured that they are -fold cyclic coverings of , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.
Corollary 7 Let be an admissible 6-tuple with and . Then the -tuple is admissible for each and the Dunwoody manifold is a n-fold cyclic coverings of , branched over a genus one -bridge knot , which is independent on .
Proof. Obviously . Since is admissible, it satisfies (i’). This proves that satisfies , for each . Since and , we obtain , for each , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, is odd, since . Thus, Lemma 3 proves that is admissible. The final result is then a direct consequence of Theorem 6.
We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].
An interesting problem which naturally arises is that of characterizing the set of branching knots in involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers and , with odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type is equivalent to the 2-bridge knot of type , then can be assumed to be even.
12
|
<div class="pdf-page">
<p>Since the handlebody orbifolds and their gluing only depend on , the same holds for the branching set . The homeomorphism type of follows from Proposition 2 and Lemma 5.</p>
<p>Remark 4. More generally, given two positive integers and such that divides , if is admissible, then the Dunwoody man- ifold is the -fold cyclic covering of the manifold , branched over an -knot in .</p>
<p>Example 2. The Dunwoody manifolds , and , with coprime, are -fold cyclic coverings of the manifolds , and respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of copies of , and respectively.</p>
<p>Let us consider now the class of the Dunwoody manifolds with (and hence odd) and . Many ex- amples of these manifolds appear in Table 1 of [6], where it was conjectured that they are -fold cyclic coverings of , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.</p>
<p>Corollary 7 Let be an admissible 6-tuple with and . Then the -tuple is admissible for each and the Dunwoody manifold is a n-fold cyclic coverings of , branched over a genus one -bridge knot , which is independent on .</p>
<p>Proof. Obviously . Since is admissible, it satisfies (i’). This proves that satisfies , for each . Since and , we obtain , for each , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, is odd, since . Thus, Lemma 3 proves that is admissible. The final result is then a direct consequence of Theorem 6.</p>
<p>We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].</p>
<p>An interesting problem which naturally arises is that of characterizing the set of branching knots in involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers and , with odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type is equivalent to the 2-bridge knot of type , then can be assumed to be even.</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="161" data-width="652" data-height="56">Since the handlebody orbifolds and their gluing only depend on , the same holds for the branching set . The homeomorphism type of follows from Proposition 2 and Lemma 5.</p>
<p class="pdf-text" data-x="184" data-y="223" data-width="652" data-height="76">Remark 4. More generally, given two positive integers and such that divides , if is admissible, then the Dunwoody man- ifold is the -fold cyclic covering of the manifold , branched over an -knot in .</p>
<p class="pdf-text" data-x="182" data-y="306" data-width="654" data-height="93">Example 2. The Dunwoody manifolds , and , with coprime, are -fold cyclic coverings of the manifolds , and respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of copies of , and respectively.</p>
<p class="pdf-text" data-x="184" data-y="407" data-width="652" data-height="93">Let us consider now the class of the Dunwoody manifolds with (and hence odd) and . Many ex- amples of these manifolds appear in Table 1 of [6], where it was conjectured that they are -fold cyclic coverings of , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.</p>
<p class="pdf-text" data-x="184" data-y="514" data-width="652" data-height="93">Corollary 7 Let be an admissible 6-tuple with and . Then the -tuple is admissible for each and the Dunwoody manifold is a n-fold cyclic coverings of , branched over a genus one -bridge knot , which is independent on .</p>
<p class="pdf-text" data-x="182" data-y="621" data-width="654" data-height="131">Proof. Obviously . Since is admissible, it satisfies (i’). This proves that satisfies , for each . Since and , we obtain , for each , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, is odd, since . Thus, Lemma 3 proves that is admissible. The final result is then a direct consequence of Theorem 6.</p>
<p class="pdf-text" data-x="182" data-y="758" data-width="654" data-height="39">We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].</p>
<p class="pdf-text" data-x="182" data-y="799" data-width="654" data-height="72">An interesting problem which naturally arises is that of characterizing the set of branching knots in involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers and , with odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type is equivalent to the 2-bridge knot of type , then can be assumed to be even.</p>
<div class="pdf-discarded" data-x="500" data-y="893" data-width="20" data-height="14" style="opacity: 0.5;">12</div>
</div>
|
Since the handlebody orbifolds and their gluing only depend on $a,b,c,r$ , the same holds for the branching set $K$ . The homeomorphism type of $M^{\prime}$ follows from Proposition 2 and Lemma 5.
Remark 4. More generally, given two positive integers $n$ and $n^{\prime}$ such that $n^{\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then the Dunwoody manifold $M(a,b,c,n,r,s)$ is the $n/n^{\prime}$ -fold cyclic covering of the manifold $M^{\prime}=$ $M(a,b,c,n^{\prime},r,s)$ , branched over an $(n^{\prime},1)$ -knot in $M^{\prime}$ .
Example 2. The Dunwoody manifolds $M(0,0,1,n,0,0)$ , $M(1,0,0,n,1,0)$ and $M(0,0,c,n,r,0)$ , with $c,r$ coprime, are $n$ -fold cyclic coverings of the manifolds $\mathbf{S^{3}}$ , $\mathbf{S^{1}}\times\mathbf{S^{2}}$ and $L(c,r)$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $n$ copies of $\mathbf{S^{3}}$ , $\mathbf{S^{1}}\times\mathbf{S^{2}}$ and $L(c,r)$ respectively.
Let us consider now the class of the Dunwoody manifolds $\textstyle M_{n}\ =$ $M(a,b,c,n,r,s)$ with $p=\pm1$ (and hence $d$ odd) and $s\,=\,-p q$ . Many examples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $n$ -fold cyclic coverings of $\mathbf{S^{3}}$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.
Corollary 7 Let $\sigma_{1}=(a,b,c,1,r,0)$ be an admissible 6-tuple with $p_{\sigma_{1}}=\pm1$ and $s=-p_{\sigma_{1}}q_{\sigma_{1}}$ . Then the $\it6$ -tuple $\sigma_{n}=(a,b,c,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,b,c,n,r,s)$ is a n-fold cyclic coverings of $\mathrm{{S^{3}}}$ , branched over a genus one $\mathit{1}$ -bridge knot $K\subset{\bf S^{3}}$ , which is independent on $n$ .
Proof. Obviously $(a,b,c,1,r,s)=\sigma_{1}$ . Since $\sigma_{1}$ is admissible, it satisfies (i’). This proves that $\sigma_{n}$ satisfies $\left(\mathrm{i}^{\,\circ}\right)$ , for each $n>1$ . Since $s=-p_{\sigma_{1}}q_{\sigma_{1}}=$ $-p_{\sigma_{n}}q_{\sigma_{n}}$ and $p_{\sigma_{n}}=p_{\sigma_{1}}=\pm1$ , we obtain $q_{\sigma_{n}}+s p_{\sigma_{n}}=0$ , for each $n>1$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, $d$ is odd, since $[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\sigma_{n}}]_{2}=[p_{\sigma_{1}}]_{2}=1$ . Thus, Lemma 3 proves that $\sigma_{n}$ is admissible. The final result is then a direct consequence of Theorem 6.
We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].
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"Theorem 8 The -tuple with is admissi- ble. Moreover, if , then the -tuple is admissible for each and the Dunwoody manifold is the - fold cyclic covering of , branched over the 2-bridge knot of type . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.",
"Proof. From it immediately follows that has a unique cycle in . Since is odd, Corollary 4 proves that is admissible. Since , all assumptions of Corollary 7 hold; hence is admissible for each and is an -fold cyclic covering of , branched over a knot which is independent on . In order to determine this knot, we can restrict our attention to the case . Note that and hence is always even. Thus, in the case we can suppose . Let us consider now the genus two Heegaard diagram . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space (see Figure 10). Since the representation of lens spaces (including ) as 2-fold branched coverings of is unique [14], the result immediately holds.",
"Remark 5. The Dunwoody manifold of Theorem 8 is home- omorphic to the Minkus manifold [21] and the Lins-Mandel manifold [19, 24].",
"An immediate consequence of Theorem 8 is:",
"Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric.",
"Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric.",
"About the set of knots in involved in Corollary 7, we propose the following:",
"Conjecture. The set contains all torus knots.",
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[{"type": "text", "text": "", "page_idx": 12}, {"type": "text", "text": "Theorem 8 The $\\it6$ -tuple $\\sigma_{1}=(a,0,1,1,r,0)$ with $(2a\\!+\\!1,2r)=1$ is admissible. Moreover, if $s=-q_{\\sigma_{1}}$ , then the $\\it6$ -tuple $\\sigma_{n}=(a,0,1,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,0,1,n,r,s)$ is the $n$ - fold cyclic covering of $\\mathrm{{S^{3}}}$ , branched over the 2-bridge knot of type $(2a\\!+\\!1,2r)$ . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds. ", "page_idx": 12}, {"type": "text", "text": "Proof. From $(2a+1,2r)=1$ it immediately follows that $\\sigma_{1}$ has a unique cycle in $\\mathcal{D}$ . Since $d=2a+1$ is odd, Corollary 4 proves that $\\sigma_{1}$ is admissible. Since $p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1$ , all assumptions of Corollary 7 hold; hence $\\sigma_{n}$ is admissible for each $n>1$ and $M_{n}$ is an $n$ -fold cyclic covering of $\\mathrm{{S^{3}}}$ , branched over a knot $K\\subset{\\bf S^{3}}$ which is independent on $n$ . In order to determine this knot, we can restrict our attention to the case $n\\;=\\;2$ . Note that $[s]_{2}\\,=$ $[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0$ and hence $s$ is always even. Thus, in the case $n\\,=\\,2$ we can suppose $s\\implies0$ . Let us consider now the genus two Heegaard diagram $H(a,0,1,2,r,0)$ . The sequence of Singer moves [31] on this diagram, drawn in Figures 3\u201310 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ (see Figure 10). Since the representation of lens spaces (including $\\mathbf{S^{3}}$ ) as 2-fold branched coverings of $\\mathrm{{S^{3}}}$ is unique [14], the result immediately holds. ", "page_idx": 12}, {"type": "text", "text": "Remark 5. The Dunwoody manifold $M(a,0,1,n,r,s)$ of Theorem 8 is homeomorphic to the Minkus manifold $M_{n}(2a+1,2r)$ [21] and the Lins-Mandel manifold $S(n,2a+1,2r,1)$ [19, 24]. ", "page_idx": 12}, {"type": "text", "text": "An immediate consequence of Theorem 8 is: ", "page_idx": 12}, {"type": "text", "text": "Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric. ", "page_idx": 12}, {"type": "text", "text": "Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric. ", "page_idx": 12}, {"type": "text", "text": "About the set $\\kappa$ of knots in $\\mathrm{{S^{3}}}$ involved in Corollary 7, we propose the following: ", "page_idx": 12}, {"type": "text", "text": "Conjecture. The set $\\kappa$ contains all torus knots. ", "page_idx": 12}]
|
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Since the 2-bridge knot of type ", "type": "text"}, {"bbox": [298, 143, 326, 155], "score": 0.94, "content": "(\\alpha,\\beta)", "type": "inline_equation", "height": 12, "width": 28}, {"bbox": [327, 142, 500, 156], "score": 1.0, "content": " is equivalent to the 2-bridge knot", "type": "text"}], "index": 1}, {"bbox": [110, 157, 387, 170], "spans": [{"bbox": [110, 157, 149, 170], "score": 1.0, "content": "of type ", "type": "text"}, {"bbox": [150, 158, 201, 170], "score": 0.92, "content": "(\\alpha,\\alpha-\\beta)", "type": "inline_equation", "height": 12, "width": 51}, {"bbox": [201, 157, 234, 170], "score": 1.0, "content": ", then ", "type": "text"}, {"bbox": [235, 158, 243, 169], "score": 0.9, "content": "\\beta", "type": "inline_equation", "height": 11, "width": 8}, {"bbox": [243, 157, 387, 170], "score": 1.0, "content": " can be assumed to be even.", "type": "text"}], "index": 2}], "index": 1}, {"type": "text", "bbox": [109, 180, 500, 253], "lines": [{"bbox": [109, 181, 501, 198], "spans": [{"bbox": [109, 181, 203, 198], "score": 1.0, "content": "Theorem 8 The ", "type": "text"}, {"bbox": [204, 185, 210, 194], "score": 0.32, "content": "\\it6", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [210, 181, 240, 198], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [240, 184, 338, 196], "score": 0.92, "content": "\\sigma_{1}=(a,0,1,1,r,0)", "type": "inline_equation", "height": 12, "width": 98}, {"bbox": [338, 181, 365, 198], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [366, 184, 442, 197], "score": 0.92, "content": "(2a\\!+\\!1,2r)=1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [442, 181, 501, 198], "score": 1.0, "content": " is admissi-", "type": "text"}], "index": 3}, {"bbox": [109, 196, 501, 214], "spans": [{"bbox": [109, 196, 198, 214], "score": 1.0, "content": "ble. Moreover, if ", "type": "text"}, {"bbox": [198, 200, 243, 211], "score": 0.93, "content": "s=-q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [243, 196, 295, 214], "score": 1.0, "content": ", then the ", "type": "text"}, {"bbox": [295, 200, 301, 208], "score": 0.68, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [302, 196, 332, 214], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [333, 198, 432, 211], "score": 0.91, "content": "\\sigma_{n}=(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [432, 196, 501, 214], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 212, 501, 227], "spans": [{"bbox": [109, 212, 155, 227], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [155, 214, 184, 223], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [185, 212, 334, 227], "score": 1.0, "content": " and the Dunwoody manifold ", "type": "text"}, {"bbox": [335, 213, 451, 225], "score": 0.92, "content": "M_{n}=M(a,0,1,n,r,s)", "type": "inline_equation", "height": 12, "width": 116}, {"bbox": [452, 212, 487, 227], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [487, 217, 495, 223], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [495, 212, 501, 227], "score": 1.0, "content": "-", "type": "text"}], "index": 5}, {"bbox": [109, 226, 500, 241], "spans": [{"bbox": [109, 226, 223, 241], "score": 1.0, "content": "fold cyclic covering of ", "type": "text"}, {"bbox": [223, 227, 237, 237], "score": 0.87, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [237, 226, 441, 241], "score": 1.0, "content": ", branched over the 2-bridge knot of type ", "type": "text"}, {"bbox": [441, 226, 496, 240], "score": 0.91, "content": "(2a\\!+\\!1,2r)", "type": "inline_equation", "height": 14, "width": 55}, {"bbox": [496, 226, 500, 241], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [110, 240, 499, 256], "spans": [{"bbox": [110, 240, 499, 256], "score": 1.0, "content": "Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "type": "text"}], "index": 7}], "index": 5}, {"type": "text", "bbox": [109, 266, 500, 453], "lines": [{"bbox": [126, 267, 500, 281], "spans": [{"bbox": [126, 267, 199, 281], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [199, 269, 277, 281], "score": 0.94, "content": "(2a+1,2r)=1", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [277, 267, 420, 281], "score": 1.0, "content": " it immediately follows that ", "type": "text"}, {"bbox": [420, 272, 432, 280], "score": 0.81, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [432, 267, 500, 281], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 8}, {"bbox": [110, 282, 499, 296], "spans": [{"bbox": [110, 282, 152, 296], "score": 1.0, "content": "cycle in ", "type": "text"}, {"bbox": [152, 284, 162, 293], "score": 0.89, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [162, 282, 200, 296], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [200, 284, 253, 294], "score": 0.93, "content": "d=2a+1", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [254, 282, 416, 296], "score": 1.0, "content": " is odd, Corollary 4 proves that ", "type": "text"}, {"bbox": [416, 287, 428, 294], "score": 0.86, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [428, 282, 499, 296], "score": 1.0, "content": " is admissible.", "type": "text"}], "index": 9}, {"bbox": [109, 296, 501, 312], "spans": [{"bbox": [109, 296, 141, 312], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [142, 299, 232, 310], "score": 0.91, "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "type": "inline_equation", "height": 11, "width": 90}, {"bbox": [232, 296, 473, 312], "score": 1.0, "content": ", all assumptions of Corollary 7 hold; hence ", "type": "text"}, {"bbox": [473, 302, 485, 309], "score": 0.85, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [486, 296, 501, 312], "score": 1.0, "content": " is", "type": "text"}], "index": 10}, {"bbox": [110, 311, 501, 326], "spans": [{"bbox": [110, 311, 209, 326], "score": 1.0, "content": "admissible for each ", "type": "text"}, {"bbox": [209, 313, 238, 322], "score": 0.91, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [238, 311, 263, 326], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [263, 313, 280, 323], "score": 0.93, "content": "M_{n}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [280, 311, 309, 326], "score": 1.0, "content": " is an", "type": "text"}, {"bbox": [310, 316, 317, 321], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [317, 311, 432, 326], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [433, 312, 446, 322], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [446, 311, 501, 326], "score": 1.0, "content": ", branched", "type": "text"}], "index": 11}, {"bbox": [109, 325, 500, 339], "spans": [{"bbox": [109, 325, 172, 339], "score": 1.0, "content": "over a knot ", "type": "text"}, {"bbox": [173, 326, 214, 336], "score": 0.93, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [214, 325, 347, 339], "score": 1.0, "content": " which is independent on ", "type": "text"}, {"bbox": [348, 330, 355, 336], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 325, 500, 339], "score": 1.0, "content": ". In order to determine this", "type": "text"}], "index": 12}, {"bbox": [109, 340, 500, 353], "spans": [{"bbox": [109, 340, 363, 353], "score": 1.0, "content": "knot, we can restrict our attention to the case ", "type": "text"}, {"bbox": [363, 342, 398, 350], "score": 0.92, "content": "n\\;=\\;2", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [398, 340, 466, 353], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [467, 341, 500, 353], "score": 0.91, "content": "[s]_{2}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 13}, {"bbox": [110, 354, 501, 369], "spans": [{"bbox": [110, 356, 203, 368], "score": 0.93, "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [203, 354, 263, 369], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [263, 359, 269, 365], "score": 0.88, "content": "s", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 354, 450, 369], "score": 1.0, "content": " is always even. Thus, in the case ", "type": "text"}, {"bbox": [450, 357, 481, 365], "score": 0.91, "content": "n\\,=\\,2", "type": "inline_equation", "height": 8, "width": 31}, {"bbox": [482, 354, 501, 369], "score": 1.0, "content": " we", "type": "text"}], "index": 14}, {"bbox": [109, 369, 500, 384], "spans": [{"bbox": [109, 369, 177, 384], "score": 1.0, "content": "can suppose ", "type": "text"}, {"bbox": [177, 371, 208, 380], "score": 0.92, "content": "s\\implies0", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [208, 369, 500, 384], "score": 1.0, "content": ". Let us consider now the genus two Heegaard diagram", "type": "text"}], "index": 15}, {"bbox": [110, 383, 500, 398], "spans": [{"bbox": [110, 384, 191, 397], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [191, 383, 500, 398], "score": 1.0, "content": ". The sequence of Singer moves [31] on this diagram, drawn", "type": "text"}], "index": 16}, {"bbox": [110, 398, 500, 412], "spans": [{"bbox": [110, 398, 500, 412], "score": 1.0, "content": "in Figures 3\u201310 and described in the Appendix of the paper, leads to the", "type": "text"}], "index": 17}, {"bbox": [109, 412, 500, 426], "spans": [{"bbox": [109, 412, 407, 426], "score": 1.0, "content": "canonical genus one Heegaard diagram of the lens space ", "type": "text"}, {"bbox": [407, 413, 475, 426], "score": 0.94, "content": "L(2a+1,2r)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [475, 412, 500, 426], "score": 1.0, "content": " (see", "type": "text"}], "index": 18}, {"bbox": [110, 426, 500, 441], "spans": [{"bbox": [110, 426, 433, 441], "score": 1.0, "content": "Figure 10). Since the representation of lens spaces (including ", "type": "text"}, {"bbox": [433, 428, 446, 437], "score": 0.86, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [446, 426, 500, 441], "score": 1.0, "content": ") as 2-fold", "type": "text"}], "index": 19}, {"bbox": [109, 440, 482, 455], "spans": [{"bbox": [109, 440, 224, 455], "score": 1.0, "content": "branched coverings of ", "type": "text"}, {"bbox": [224, 442, 237, 451], "score": 0.92, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [237, 440, 482, 455], "score": 1.0, "content": " is unique [14], the result immediately holds.", "type": "text"}], "index": 20}], "index": 14}, {"type": "text", "bbox": [110, 459, 500, 503], "lines": [{"bbox": [109, 461, 500, 476], "spans": [{"bbox": [109, 461, 302, 476], "score": 1.0, "content": "Remark 5. The Dunwoody manifold ", "type": "text"}, {"bbox": [302, 462, 386, 475], "score": 0.94, "content": "M(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [386, 461, 500, 476], "score": 1.0, "content": " of Theorem 8 is home-", "type": "text"}], "index": 21}, {"bbox": [110, 476, 500, 489], "spans": [{"bbox": [110, 476, 289, 489], "score": 1.0, "content": "omorphic to the Minkus manifold ", "type": "text"}, {"bbox": [289, 477, 365, 489], "score": 0.92, "content": "M_{n}(2a+1,2r)", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [365, 476, 500, 489], "score": 1.0, "content": " [21] and the Lins-Mandel", "type": "text"}], "index": 22}, {"bbox": [110, 490, 293, 505], "spans": [{"bbox": [110, 490, 158, 505], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [158, 491, 248, 504], "score": 0.92, "content": "S(n,2a+1,2r,1)", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [248, 490, 293, 505], "score": 1.0, "content": " [19, 24].", "type": "text"}], "index": 23}], "index": 22}, {"type": "text", "bbox": [127, 508, 354, 522], "lines": [{"bbox": [128, 510, 354, 524], "spans": [{"bbox": [128, 510, 354, 524], "score": 1.0, "content": "An immediate consequence of Theorem 8 is:", "type": "text"}], "index": 24}], "index": 24}, {"type": "text", "bbox": [110, 535, 500, 563], "lines": [{"bbox": [110, 537, 501, 551], "spans": [{"bbox": [110, 537, 501, 551], "score": 1.0, "content": "Corollary 9 The fundamental group of every branched cyclic covering of a", "type": "text"}], "index": 25}, {"bbox": [111, 551, 428, 565], "spans": [{"bbox": [111, 551, 428, 565], "score": 1.0, "content": "2-bridge knot admits a cyclic presentation which is geometric.", "type": "text"}], "index": 26}], "index": 25.5}, {"type": "text", "bbox": [110, 576, 500, 619], "lines": [{"bbox": [109, 577, 500, 592], "spans": [{"bbox": [109, 577, 500, 592], "score": 1.0, "content": "Remark 6. In [21] is shown that the fundamental group of every branched", "type": "text"}], "index": 27}, {"bbox": [110, 594, 500, 606], "spans": [{"bbox": [110, 594, 500, 606], "score": 1.0, "content": "cyclic covering of a 2-bridge knot admits a cyclic presentation, but without", "type": "text"}], "index": 28}, {"bbox": [110, 608, 357, 621], "spans": [{"bbox": [110, 608, 357, 621], "score": 1.0, "content": "pointing out that this presentation is geometric.", "type": "text"}], "index": 29}], "index": 28}, {"type": "text", "bbox": [109, 625, 501, 654], "lines": [{"bbox": [127, 626, 500, 641], "spans": [{"bbox": [127, 626, 203, 641], "score": 1.0, "content": "About the set ", "type": "text"}, {"bbox": [204, 629, 213, 638], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 626, 277, 641], "score": 1.0, "content": " of knots in ", "type": "text"}, {"bbox": [277, 628, 290, 638], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [291, 626, 500, 641], "score": 1.0, "content": " involved in Corollary 7, we propose the", "type": "text"}], "index": 30}, {"bbox": [109, 641, 160, 657], "spans": [{"bbox": [109, 641, 160, 657], "score": 1.0, "content": "following:", "type": "text"}], "index": 31}], "index": 30.5}, {"type": "text", "bbox": [110, 660, 362, 674], "lines": [{"bbox": [111, 662, 362, 675], "spans": [{"bbox": [111, 662, 225, 675], "score": 1.0, "content": "Conjecture. The set ", "type": "text"}, {"bbox": [226, 664, 235, 672], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [236, 662, 362, 675], "score": 1.0, "content": " contains all torus knots.", "type": "text"}], "index": 32}], "index": 32}], "layout_bboxes": [], "page_idx": 12, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [299, 691, 311, 702], "lines": [{"bbox": [297, 692, 313, 705], "spans": [{"bbox": [297, 692, 313, 705], "score": 1.0, "content": "13", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 168], "lines": [], "index": 1, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [110, 128, 500, 170], "lines_deleted": true}, {"type": "text", "bbox": [109, 180, 500, 253], "lines": [{"bbox": [109, 181, 501, 198], "spans": [{"bbox": [109, 181, 203, 198], "score": 1.0, "content": "Theorem 8 The ", "type": "text"}, {"bbox": [204, 185, 210, 194], "score": 0.32, "content": "\\it6", "type": "inline_equation", "height": 9, "width": 6}, {"bbox": [210, 181, 240, 198], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [240, 184, 338, 196], "score": 0.92, "content": "\\sigma_{1}=(a,0,1,1,r,0)", "type": "inline_equation", "height": 12, "width": 98}, {"bbox": [338, 181, 365, 198], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [366, 184, 442, 197], "score": 0.92, "content": "(2a\\!+\\!1,2r)=1", "type": "inline_equation", "height": 13, "width": 76}, {"bbox": [442, 181, 501, 198], "score": 1.0, "content": " is admissi-", "type": "text"}], "index": 3}, {"bbox": [109, 196, 501, 214], "spans": [{"bbox": [109, 196, 198, 214], "score": 1.0, "content": "ble. Moreover, if ", "type": "text"}, {"bbox": [198, 200, 243, 211], "score": 0.93, "content": "s=-q_{\\sigma_{1}}", "type": "inline_equation", "height": 11, "width": 45}, {"bbox": [243, 196, 295, 214], "score": 1.0, "content": ", then the ", "type": "text"}, {"bbox": [295, 200, 301, 208], "score": 0.68, "content": "\\it6", "type": "inline_equation", "height": 8, "width": 6}, {"bbox": [302, 196, 332, 214], "score": 1.0, "content": "-tuple ", "type": "text"}, {"bbox": [333, 198, 432, 211], "score": 0.91, "content": "\\sigma_{n}=(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 99}, {"bbox": [432, 196, 501, 214], "score": 1.0, "content": " is admissible", "type": "text"}], "index": 4}, {"bbox": [109, 212, 501, 227], "spans": [{"bbox": [109, 212, 155, 227], "score": 1.0, "content": "for each ", "type": "text"}, {"bbox": [155, 214, 184, 223], "score": 0.9, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [185, 212, 334, 227], "score": 1.0, "content": " and the Dunwoody manifold ", "type": "text"}, {"bbox": [335, 213, 451, 225], "score": 0.92, "content": "M_{n}=M(a,0,1,n,r,s)", "type": "inline_equation", "height": 12, "width": 116}, {"bbox": [452, 212, 487, 227], "score": 1.0, "content": " is the ", "type": "text"}, {"bbox": [487, 217, 495, 223], "score": 0.68, "content": "n", "type": "inline_equation", "height": 6, "width": 8}, {"bbox": [495, 212, 501, 227], "score": 1.0, "content": "-", "type": "text"}], "index": 5}, {"bbox": [109, 226, 500, 241], "spans": [{"bbox": [109, 226, 223, 241], "score": 1.0, "content": "fold cyclic covering of ", "type": "text"}, {"bbox": [223, 227, 237, 237], "score": 0.87, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 14}, {"bbox": [237, 226, 441, 241], "score": 1.0, "content": ", branched over the 2-bridge knot of type ", "type": "text"}, {"bbox": [441, 226, 496, 240], "score": 0.91, "content": "(2a\\!+\\!1,2r)", "type": "inline_equation", "height": 14, "width": 55}, {"bbox": [496, 226, 500, 241], "score": 1.0, "content": ".", "type": "text"}], "index": 6}, {"bbox": [110, 240, 499, 256], "spans": [{"bbox": [110, 240, 499, 256], "score": 1.0, "content": "Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.", "type": "text"}], "index": 7}], "index": 5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 181, 501, 256]}, {"type": "text", "bbox": [109, 266, 500, 453], "lines": [{"bbox": [126, 267, 500, 281], "spans": [{"bbox": [126, 267, 199, 281], "score": 1.0, "content": "Proof. From ", "type": "text"}, {"bbox": [199, 269, 277, 281], "score": 0.94, "content": "(2a+1,2r)=1", "type": "inline_equation", "height": 12, "width": 78}, {"bbox": [277, 267, 420, 281], "score": 1.0, "content": " it immediately follows that ", "type": "text"}, {"bbox": [420, 272, 432, 280], "score": 0.81, "content": "\\sigma_{1}", "type": "inline_equation", "height": 8, "width": 12}, {"bbox": [432, 267, 500, 281], "score": 1.0, "content": " has a unique", "type": "text"}], "index": 8}, {"bbox": [110, 282, 499, 296], "spans": [{"bbox": [110, 282, 152, 296], "score": 1.0, "content": "cycle in ", "type": "text"}, {"bbox": [152, 284, 162, 293], "score": 0.89, "content": "\\mathcal{D}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [162, 282, 200, 296], "score": 1.0, "content": ". Since ", "type": "text"}, {"bbox": [200, 284, 253, 294], "score": 0.93, "content": "d=2a+1", "type": "inline_equation", "height": 10, "width": 53}, {"bbox": [254, 282, 416, 296], "score": 1.0, "content": " is odd, Corollary 4 proves that ", "type": "text"}, {"bbox": [416, 287, 428, 294], "score": 0.86, "content": "\\sigma_{1}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [428, 282, 499, 296], "score": 1.0, "content": " is admissible.", "type": "text"}], "index": 9}, {"bbox": [109, 296, 501, 312], "spans": [{"bbox": [109, 296, 141, 312], "score": 1.0, "content": "Since ", "type": "text"}, {"bbox": [142, 299, 232, 310], "score": 0.91, "content": "p_{\\sigma_{n}}~=~p_{\\sigma_{1}}~=~+1", "type": "inline_equation", "height": 11, "width": 90}, {"bbox": [232, 296, 473, 312], "score": 1.0, "content": ", all assumptions of Corollary 7 hold; hence ", "type": "text"}, {"bbox": [473, 302, 485, 309], "score": 0.85, "content": "\\sigma_{n}", "type": "inline_equation", "height": 7, "width": 12}, {"bbox": [486, 296, 501, 312], "score": 1.0, "content": " is", "type": "text"}], "index": 10}, {"bbox": [110, 311, 501, 326], "spans": [{"bbox": [110, 311, 209, 326], "score": 1.0, "content": "admissible for each ", "type": "text"}, {"bbox": [209, 313, 238, 322], "score": 0.91, "content": "n>1", "type": "inline_equation", "height": 9, "width": 29}, {"bbox": [238, 311, 263, 326], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [263, 313, 280, 323], "score": 0.93, "content": "M_{n}", "type": "inline_equation", "height": 10, "width": 17}, {"bbox": [280, 311, 309, 326], "score": 1.0, "content": " is an", "type": "text"}, {"bbox": [310, 316, 317, 321], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [317, 311, 432, 326], "score": 1.0, "content": "-fold cyclic covering of ", "type": "text"}, {"bbox": [433, 312, 446, 322], "score": 0.9, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [446, 311, 501, 326], "score": 1.0, "content": ", branched", "type": "text"}], "index": 11}, {"bbox": [109, 325, 500, 339], "spans": [{"bbox": [109, 325, 172, 339], "score": 1.0, "content": "over a knot ", "type": "text"}, {"bbox": [173, 326, 214, 336], "score": 0.93, "content": "K\\subset{\\bf S^{3}}", "type": "inline_equation", "height": 10, "width": 41}, {"bbox": [214, 325, 347, 339], "score": 1.0, "content": " which is independent on ", "type": "text"}, {"bbox": [348, 330, 355, 336], "score": 0.9, "content": "n", "type": "inline_equation", "height": 6, "width": 7}, {"bbox": [355, 325, 500, 339], "score": 1.0, "content": ". In order to determine this", "type": "text"}], "index": 12}, {"bbox": [109, 340, 500, 353], "spans": [{"bbox": [109, 340, 363, 353], "score": 1.0, "content": "knot, we can restrict our attention to the case ", "type": "text"}, {"bbox": [363, 342, 398, 350], "score": 0.92, "content": "n\\;=\\;2", "type": "inline_equation", "height": 8, "width": 35}, {"bbox": [398, 340, 466, 353], "score": 1.0, "content": ". Note that ", "type": "text"}, {"bbox": [467, 341, 500, 353], "score": 0.91, "content": "[s]_{2}\\,=", "type": "inline_equation", "height": 12, "width": 33}], "index": 13}, {"bbox": [110, 354, 501, 369], "spans": [{"bbox": [110, 356, 203, 368], "score": 0.93, "content": "[-q_{\\sigma_{1}}]_{2}=[b]_{2}=0", "type": "inline_equation", "height": 12, "width": 93}, {"bbox": [203, 354, 263, 369], "score": 1.0, "content": " and hence ", "type": "text"}, {"bbox": [263, 359, 269, 365], "score": 0.88, "content": "s", "type": "inline_equation", "height": 6, "width": 6}, {"bbox": [270, 354, 450, 369], "score": 1.0, "content": " is always even. Thus, in the case ", "type": "text"}, {"bbox": [450, 357, 481, 365], "score": 0.91, "content": "n\\,=\\,2", "type": "inline_equation", "height": 8, "width": 31}, {"bbox": [482, 354, 501, 369], "score": 1.0, "content": " we", "type": "text"}], "index": 14}, {"bbox": [109, 369, 500, 384], "spans": [{"bbox": [109, 369, 177, 384], "score": 1.0, "content": "can suppose ", "type": "text"}, {"bbox": [177, 371, 208, 380], "score": 0.92, "content": "s\\implies0", "type": "inline_equation", "height": 9, "width": 31}, {"bbox": [208, 369, 500, 384], "score": 1.0, "content": ". Let us consider now the genus two Heegaard diagram", "type": "text"}], "index": 15}, {"bbox": [110, 383, 500, 398], "spans": [{"bbox": [110, 384, 191, 397], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [191, 383, 500, 398], "score": 1.0, "content": ". The sequence of Singer moves [31] on this diagram, drawn", "type": "text"}], "index": 16}, {"bbox": [110, 398, 500, 412], "spans": [{"bbox": [110, 398, 500, 412], "score": 1.0, "content": "in Figures 3\u201310 and described in the Appendix of the paper, leads to the", "type": "text"}], "index": 17}, {"bbox": [109, 412, 500, 426], "spans": [{"bbox": [109, 412, 407, 426], "score": 1.0, "content": "canonical genus one Heegaard diagram of the lens space ", "type": "text"}, {"bbox": [407, 413, 475, 426], "score": 0.94, "content": "L(2a+1,2r)", "type": "inline_equation", "height": 13, "width": 68}, {"bbox": [475, 412, 500, 426], "score": 1.0, "content": " (see", "type": "text"}], "index": 18}, {"bbox": [110, 426, 500, 441], "spans": [{"bbox": [110, 426, 433, 441], "score": 1.0, "content": "Figure 10). Since the representation of lens spaces (including ", "type": "text"}, {"bbox": [433, 428, 446, 437], "score": 0.86, "content": "\\mathbf{S^{3}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [446, 426, 500, 441], "score": 1.0, "content": ") as 2-fold", "type": "text"}], "index": 19}, {"bbox": [109, 440, 482, 455], "spans": [{"bbox": [109, 440, 224, 455], "score": 1.0, "content": "branched coverings of ", "type": "text"}, {"bbox": [224, 442, 237, 451], "score": 0.92, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 9, "width": 13}, {"bbox": [237, 440, 482, 455], "score": 1.0, "content": " is unique [14], the result immediately holds.", "type": "text"}], "index": 20}], "index": 14, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 267, 501, 455]}, {"type": "text", "bbox": [110, 459, 500, 503], "lines": [{"bbox": [109, 461, 500, 476], "spans": [{"bbox": [109, 461, 302, 476], "score": 1.0, "content": "Remark 5. The Dunwoody manifold ", "type": "text"}, {"bbox": [302, 462, 386, 475], "score": 0.94, "content": "M(a,0,1,n,r,s)", "type": "inline_equation", "height": 13, "width": 84}, {"bbox": [386, 461, 500, 476], "score": 1.0, "content": " of Theorem 8 is home-", "type": "text"}], "index": 21}, {"bbox": [110, 476, 500, 489], "spans": [{"bbox": [110, 476, 289, 489], "score": 1.0, "content": "omorphic to the Minkus manifold ", "type": "text"}, {"bbox": [289, 477, 365, 489], "score": 0.92, "content": "M_{n}(2a+1,2r)", "type": "inline_equation", "height": 12, "width": 76}, {"bbox": [365, 476, 500, 489], "score": 1.0, "content": " [21] and the Lins-Mandel", "type": "text"}], "index": 22}, {"bbox": [110, 490, 293, 505], "spans": [{"bbox": [110, 490, 158, 505], "score": 1.0, "content": "manifold ", "type": "text"}, {"bbox": [158, 491, 248, 504], "score": 0.92, "content": "S(n,2a+1,2r,1)", "type": "inline_equation", "height": 13, "width": 90}, {"bbox": [248, 490, 293, 505], "score": 1.0, "content": " [19, 24].", "type": "text"}], "index": 23}], "index": 22, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 461, 500, 505]}, {"type": "text", "bbox": [127, 508, 354, 522], "lines": [{"bbox": [128, 510, 354, 524], "spans": [{"bbox": [128, 510, 354, 524], "score": 1.0, "content": "An immediate consequence of Theorem 8 is:", "type": "text"}], "index": 24}], "index": 24, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [128, 510, 354, 524]}, {"type": "text", "bbox": [110, 535, 500, 563], "lines": [{"bbox": [110, 537, 501, 551], "spans": [{"bbox": [110, 537, 501, 551], "score": 1.0, "content": "Corollary 9 The fundamental group of every branched cyclic covering of a", "type": "text"}], "index": 25}, {"bbox": [111, 551, 428, 565], "spans": [{"bbox": [111, 551, 428, 565], "score": 1.0, "content": "2-bridge knot admits a cyclic presentation which is geometric.", "type": "text"}], "index": 26}], "index": 25.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [110, 537, 501, 565]}, {"type": "text", "bbox": [110, 576, 500, 619], "lines": [{"bbox": [109, 577, 500, 592], "spans": [{"bbox": [109, 577, 500, 592], "score": 1.0, "content": "Remark 6. In [21] is shown that the fundamental group of every branched", "type": "text"}], "index": 27}, {"bbox": [110, 594, 500, 606], "spans": [{"bbox": [110, 594, 500, 606], "score": 1.0, "content": "cyclic covering of a 2-bridge knot admits a cyclic presentation, but without", "type": "text"}], "index": 28}, {"bbox": [110, 608, 357, 621], "spans": [{"bbox": [110, 608, 357, 621], "score": 1.0, "content": "pointing out that this presentation is geometric.", "type": "text"}], "index": 29}], "index": 28, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 577, 500, 621]}, {"type": "text", "bbox": [109, 625, 501, 654], "lines": [{"bbox": [127, 626, 500, 641], "spans": [{"bbox": [127, 626, 203, 641], "score": 1.0, "content": "About the set ", "type": "text"}, {"bbox": [204, 629, 213, 638], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [214, 626, 277, 641], "score": 1.0, "content": " of knots in ", "type": "text"}, {"bbox": [277, 628, 290, 638], "score": 0.91, "content": "\\mathrm{{S^{3}}}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [291, 626, 500, 641], "score": 1.0, "content": " involved in Corollary 7, we propose the", "type": "text"}], "index": 30}, {"bbox": [109, 641, 160, 657], "spans": [{"bbox": [109, 641, 160, 657], "score": 1.0, "content": "following:", "type": "text"}], "index": 31}], "index": 30.5, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [109, 626, 500, 657]}, {"type": "text", "bbox": [110, 660, 362, 674], "lines": [{"bbox": [111, 662, 362, 675], "spans": [{"bbox": [111, 662, 225, 675], "score": 1.0, "content": "Conjecture. The set ", "type": "text"}, {"bbox": [226, 664, 235, 672], "score": 0.9, "content": "\\kappa", "type": "inline_equation", "height": 8, "width": 9}, {"bbox": [236, 662, 362, 675], "score": 1.0, "content": " contains all torus knots.", "type": "text"}], "index": 32}], "index": 32, "page_num": "page_12", "page_size": [612.0, 792.0], "bbox_fs": [111, 662, 362, 675]}]}
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Theorem 8 The -tuple with is admissi- ble. Moreover, if , then the -tuple is admissible for each and the Dunwoody manifold is the - fold cyclic covering of , branched over the 2-bridge knot of type . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.
Proof. From it immediately follows that has a unique cycle in . Since is odd, Corollary 4 proves that is admissible. Since , all assumptions of Corollary 7 hold; hence is admissible for each and is an -fold cyclic covering of , branched over a knot which is independent on . In order to determine this knot, we can restrict our attention to the case . Note that and hence is always even. Thus, in the case we can suppose . Let us consider now the genus two Heegaard diagram . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space (see Figure 10). Since the representation of lens spaces (including ) as 2-fold branched coverings of is unique [14], the result immediately holds.
Remark 5. The Dunwoody manifold of Theorem 8 is home- omorphic to the Minkus manifold [21] and the Lins-Mandel manifold [19, 24].
An immediate consequence of Theorem 8 is:
Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric.
Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric.
About the set of knots in involved in Corollary 7, we propose the following:
Conjecture. The set contains all torus knots.
13
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<div class="pdf-page">
<p>Theorem 8 The -tuple with is admissi- ble. Moreover, if , then the -tuple is admissible for each and the Dunwoody manifold is the - fold cyclic covering of , branched over the 2-bridge knot of type . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.</p>
<p>Proof. From it immediately follows that has a unique cycle in . Since is odd, Corollary 4 proves that is admissible. Since , all assumptions of Corollary 7 hold; hence is admissible for each and is an -fold cyclic covering of , branched over a knot which is independent on . In order to determine this knot, we can restrict our attention to the case . Note that and hence is always even. Thus, in the case we can suppose . Let us consider now the genus two Heegaard diagram . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space (see Figure 10). Since the representation of lens spaces (including ) as 2-fold branched coverings of is unique [14], the result immediately holds.</p>
<p>Remark 5. The Dunwoody manifold of Theorem 8 is home- omorphic to the Minkus manifold [21] and the Lins-Mandel manifold [19, 24].</p>
<p>An immediate consequence of Theorem 8 is:</p>
<p>Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric.</p>
<p>Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric.</p>
<p>About the set of knots in involved in Corollary 7, we propose the following:</p>
<p>Conjecture. The set contains all torus knots.</p>
</div>
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<div class="pdf-page">
<p class="pdf-text" data-x="182" data-y="232" data-width="654" data-height="95">Theorem 8 The -tuple with is admissi- ble. Moreover, if , then the -tuple is admissible for each and the Dunwoody manifold is the - fold cyclic covering of , branched over the 2-bridge knot of type . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.</p>
<p class="pdf-text" data-x="182" data-y="343" data-width="654" data-height="242">Proof. From it immediately follows that has a unique cycle in . Since is odd, Corollary 4 proves that is admissible. Since , all assumptions of Corollary 7 hold; hence is admissible for each and is an -fold cyclic covering of , branched over a knot which is independent on . In order to determine this knot, we can restrict our attention to the case . Note that and hence is always even. Thus, in the case we can suppose . Let us consider now the genus two Heegaard diagram . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space (see Figure 10). Since the representation of lens spaces (including ) as 2-fold branched coverings of is unique [14], the result immediately holds.</p>
<p class="pdf-text" data-x="184" data-y="593" data-width="652" data-height="57">Remark 5. The Dunwoody manifold of Theorem 8 is home- omorphic to the Minkus manifold [21] and the Lins-Mandel manifold [19, 24].</p>
<p class="pdf-text" data-x="212" data-y="656" data-width="380" data-height="18">An immediate consequence of Theorem 8 is:</p>
<p class="pdf-text" data-x="184" data-y="691" data-width="652" data-height="36">Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric.</p>
<p class="pdf-text" data-x="184" data-y="744" data-width="652" data-height="56">Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric.</p>
<p class="pdf-text" data-x="182" data-y="808" data-width="656" data-height="37">About the set of knots in involved in Corollary 7, we propose the following:</p>
<p class="pdf-text" data-x="184" data-y="853" data-width="421" data-height="18">Conjecture. The set contains all torus knots.</p>
<div class="pdf-discarded" data-x="500" data-y="893" data-width="20" data-height="14" style="opacity: 0.5;">13</div>
</div>
|
Lemma 3 ([6], Theorem 2) Let $\sigma\,=\,(a,b,c,n,r,s)$ be a 6-tuple with $d\,=$ $2a+b+c$ odd. Then $\sigma$ is admissible if and only if it satisfies $(i\,?)$ and (ii’).
Remark 3. This result does not hold when $d$ is even. In fact, the 6-tuples $(1,0,c,1,2,0)$ , with $c$ even, satisfy (i’) and (ii’), but they are not admissible, as pointed out in Remark 1.
An immediate consequence of Lemma 3 is the following result:
Corollary 4 Let $\sigma=(a,b,c,n,r,s)$ be a $\it6$ -tuple with $d=2a+b+c$ odd and $n=1$ . Then $\sigma$ is admissible if and only if $\mathcal{D}$ has a unique cycle.
Proof. If $\sigma$ is admissible, then it is straightforward that $\mathcal{D}$ has a unique cycle. Vice versa, if $\mathcal{D}$ has a unique cycle, then (i’) holds. Since $n=1$ implies (ii’), the result is a direct consequence of the above lemma.
The parameter $p_{\sigma}$ associated to an admissible 6-tuple $\sigma$ is strictly related to the word $w(\sigma)$ associated to $\sigma$ . In fact, we have:
Lemma 5 Let $\sigma\,=\,(a,b,c,n,r,s)$ be an admissible 6-tuple, $w\,=\,w(\sigma)$ the associated word and $\varepsilon_{w}$ its exponent-sum. Then
$$
p_{\sigma}=\varepsilon_{w}.
$$
Proof. Since $\sigma$ is admissible, the arcs of $\Delta$ are precisely the arcs of orientation on $D_{1}$ . Let $e_{1},e_{2},\ldots,e_{d}$ $D_{1}$ , and let be the sequence of these arcs, following the canonical $\begin{array}{r}{w\,=\,\prod_{h=1}^{d}x_{i_{h}}^{u_{h}}}\end{array}$ , with $u_{h}\,\in\,\{+1,-1\}$ . We have: $\begin{array}{r}{\varepsilon_{w}=\sum_{h=1}^{d}u_{h}=1/2\sum_{h=1}^{d}(u_{h}+u_{h+1})}\end{array}$ ,where $d+1=1$ . Since $u_{h}\!+\!u_{h+1}=+2$ if $e_{h}$ is of type I, $u_{h}+u_{h+1}=-2$ if $e_{h}$ is of type II and $u_{h}+u_{h+1}=0$ if $e_{h}$ is of type III, the result immediately follows.
In [6] Dunwoody investigates a wide subclass of manifolds $M(\sigma)$ such that $p_{\sigma}=\pm1$ and he conjectures that all the elements of this subclass are cyclic coverings of $\mathbf{S^{3}}$ branched over knots. In the next chapter this conjecture will be proved as a corollary of a more general theorem.
# 3 Main results
The following theorem is the main result of this paper and shows how the cyclic action on the Heegaard diagrams naturally extends to a cyclic action on the associated Dunwoody manifolds, which turn out to be cyclic coverings of $\mathrm{{S^{3}}}$ or of lens spaces, branched over suitable knots.
Theorem 6 Let $\sigma\,=\,(a,b,c,n,r,s)$ be an admissible 6-tuple, with $n\,>\,1$ . Then the Dunwoody manifold $M=M(a,b,c,n,r,s)$ is the $n$ -fold cyclic covering of the manifold $M^{\prime}=M(a,b,c,1,r,0)$ , branched over a genus one 1- bridge knot $K=K(a,b,c,r)$ only depending on the integers $a,b,c,r$ . Further, $M^{\prime}$ is homeomorphic to:
i) $\mathbf{S^{3}}$ , if $p_{\sigma}=\pm1$ ,
ii) $\mathbf{S^{1}}\times\mathbf{S^{2}}$ , if $p_{\sigma}=0$ ,
iii) a lens space $L(\alpha,\beta)$ with $\alpha=|p_{\sigma}|$ , if $|p_{\sigma}|>1$ .
Proof. Since the two systems of curves $\mathcal{C}\,=\,\{C_{1},\ldots\,,C_{n}\}$ and $\mathcal{D}\,=$ $\{D_{1},...\,,D_{n}\}$ on $T_{n}$ define a Heegaard diagram of $M$ , there exist two handlebodies $U_{n}$ and $U_{n}^{\prime}$ of genus $n$ , with $\partial U_{n}=\partial U_{n}^{\prime}=T_{n}$ , such that $M=U_{n}\cup U_{n}^{\prime}$ . Let now $\mathcal{G}_{n}$ be the cyclic group of order $n$ generated by the homeomorphism $\rho_{n}$ on $T_{n}$ . The action of $\mathcal{G}_{n}$ on $T_{n}$ extends to both the handlebodies $U_{n}$ and $U_{n}^{\prime}$ (see [29]), and hence to the 3-manifold $M$ . Let $B_{1}$ (resp. $B_{1}^{\prime}$ ) be a disc properly embedded in $U_{n}$ (resp. in $U_{n}^{\prime}$ ) such that $\partial B_{1}\,=\,C_{1}$ (resp. $\partial B_{1}^{\prime}\;=\;D_{1}$ ). Since $\rho_{n}(C_{i})\,=\,C_{i+1}$ and $\rho_{n}(D_{i})\,=\,D_{i+1}$ (mod $n$ ), the discs $B_{k}\,=\,\rho_{n}^{k-1}(B_{1})$ (resp. $B_{k}^{\prime}\,=\,\rho_{n}^{k-1}(B_{1}^{\prime}))$ , for $k=1,\dotsc,n$ , form a system of meridian discs for the handlebody $U_{n}$ (resp. $U_{n}^{\prime}$ ). By arguments contained in [38], the quotients $U_{1}\,=\,U_{n}/\mathcal{G}_{n}$ and $U_{1}^{\prime}\,=\,U_{n}^{\prime}/\mathcal{G}_{n}$ are both handlebody orbifolds topologically homeomorphic to a genus one handlebody with one arc trivially embedded as its singular set with a cyclic isotropy group of order $n$ . The intersection of these orbifolds is a 2-orbifold with two singular points of order $n$ , which is topologically the torus $T_{1}=T_{n}/\mathcal{G}_{n}$ ; the curve $C$ (resp. $D$ ), which is the image via the quotient map of the curves $C_{i}$ (resp. of the curves $D_{i}$ ), is non-homotopically trivial in $T_{1}^{\prime}$ . These curves, each of which is a fundamental system of curves in $T_{1}$ , define a Heegaard diagram of $M^{\prime}$ (induced by $H(a,b,c,1,r,0))$ . The union of the orbifolds $U_{1}$ and $U_{1}^{\prime}$ is a 3-orbifold topologically homeomorphic to $M^{\prime}$ , having a genus one 1-bridge knot $K\,\subset\,M^{\prime}$ as singular set of order $n$ . Thus, $M^{\prime}$ is homeomorphic to $M/\mathcal{G}_{n}$ and hence $M$ is the $n$ -fold cyclic covering of $M^{\prime}$ , branched over $K$ .
Since the handlebody orbifolds and their gluing only depend on $a,b,c,r$ , the same holds for the branching set $K$ . The homeomorphism type of $M^{\prime}$ follows from Proposition 2 and Lemma 5.
Remark 4. More generally, given two positive integers $n$ and $n^{\prime}$ such that $n^{\prime}$ divides $n$ , if $(a,b,c,r,n,s)$ is admissible, then the Dunwoody manifold $M(a,b,c,n,r,s)$ is the $n/n^{\prime}$ -fold cyclic covering of the manifold $M^{\prime}=$ $M(a,b,c,n^{\prime},r,s)$ , branched over an $(n^{\prime},1)$ -knot in $M^{\prime}$ .
Example 2. The Dunwoody manifolds $M(0,0,1,n,0,0)$ , $M(1,0,0,n,1,0)$ and $M(0,0,c,n,r,0)$ , with $c,r$ coprime, are $n$ -fold cyclic coverings of the manifolds $\mathbf{S^{3}}$ , $\mathbf{S^{1}}\times\mathbf{S^{2}}$ and $L(c,r)$ respectively, branched over a trivial knot. In fact, these Dunwoody manifolds are the connected sum of $n$ copies of $\mathbf{S^{3}}$ , $\mathbf{S^{1}}\times\mathbf{S^{2}}$ and $L(c,r)$ respectively.
Let us consider now the class of the Dunwoody manifolds $\textstyle M_{n}\ =$ $M(a,b,c,n,r,s)$ with $p=\pm1$ (and hence $d$ odd) and $s\,=\,-p q$ . Many examples of these manifolds appear in Table 1 of [6], where it was conjectured that they are $n$ -fold cyclic coverings of $\mathbf{S^{3}}$ , branched over suitable knots. The following corollary of Theorem 6 proves this conjecture.
Corollary 7 Let $\sigma_{1}=(a,b,c,1,r,0)$ be an admissible 6-tuple with $p_{\sigma_{1}}=\pm1$ and $s=-p_{\sigma_{1}}q_{\sigma_{1}}$ . Then the $\it6$ -tuple $\sigma_{n}=(a,b,c,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,b,c,n,r,s)$ is a n-fold cyclic coverings of $\mathrm{{S^{3}}}$ , branched over a genus one $\mathit{1}$ -bridge knot $K\subset{\bf S^{3}}$ , which is independent on $n$ .
Proof. Obviously $(a,b,c,1,r,s)=\sigma_{1}$ . Since $\sigma_{1}$ is admissible, it satisfies (i’). This proves that $\sigma_{n}$ satisfies $\left(\mathrm{i}^{\,\circ}\right)$ , for each $n>1$ . Since $s=-p_{\sigma_{1}}q_{\sigma_{1}}=$ $-p_{\sigma_{n}}q_{\sigma_{n}}$ and $p_{\sigma_{n}}=p_{\sigma_{1}}=\pm1$ , we obtain $q_{\sigma_{n}}+s p_{\sigma_{n}}=0$ , for each $n>1$ , which implies condition (ii) of Theorem 2 of [6], or equivalently (ii’). Moreover, $d$ is odd, since $[d]_{2}=[2a+b+c]_{2}=[b+c]_{2}=[p_{\sigma_{n}}]_{2}=[p_{\sigma_{1}}]_{2}=1$ . Thus, Lemma 3 proves that $\sigma_{n}$ is admissible. The final result is then a direct consequence of Theorem 6.
We point out that the above result has been independently obtained by H. J. Song and S. H. Kim in [32].
An interesting problem which naturally arises is that of characterizing the set $\kappa$ of branching knots in $\mathrm{{S^{3}}}$ involved in Corollary 7. The next theorem shows that it contains all 2-bridge knots. We recall that a 2-bridge knot is determined by two coprime integers $\alpha$ and $\beta$ , with $\alpha~>~0$ odd. The classification of 2-bridge knots and links has been obtained by Schubert in [30]. Since the 2-bridge knot of type $(\alpha,\beta)$ is equivalent to the 2-bridge knot of type $(\alpha,\alpha-\beta)$ , then $\beta$ can be assumed to be even.
Theorem 8 The $\it6$ -tuple $\sigma_{1}=(a,0,1,1,r,0)$ with $(2a\!+\!1,2r)=1$ is admissible. Moreover, if $s=-q_{\sigma_{1}}$ , then the $\it6$ -tuple $\sigma_{n}=(a,0,1,n,r,s)$ is admissible for each $n>1$ and the Dunwoody manifold $M_{n}=M(a,0,1,n,r,s)$ is the $n$ - fold cyclic covering of $\mathrm{{S^{3}}}$ , branched over the 2-bridge knot of type $(2a\!+\!1,2r)$ . Thus, all branched cyclic coverings of 2-bridge knots are Dunwoody manifolds.
Proof. From $(2a+1,2r)=1$ it immediately follows that $\sigma_{1}$ has a unique cycle in $\mathcal{D}$ . Since $d=2a+1$ is odd, Corollary 4 proves that $\sigma_{1}$ is admissible. Since $p_{\sigma_{n}}~=~p_{\sigma_{1}}~=~+1$ , all assumptions of Corollary 7 hold; hence $\sigma_{n}$ is admissible for each $n>1$ and $M_{n}$ is an $n$ -fold cyclic covering of $\mathrm{{S^{3}}}$ , branched over a knot $K\subset{\bf S^{3}}$ which is independent on $n$ . In order to determine this knot, we can restrict our attention to the case $n\;=\;2$ . Note that $[s]_{2}\,=$ $[-q_{\sigma_{1}}]_{2}=[b]_{2}=0$ and hence $s$ is always even. Thus, in the case $n\,=\,2$ we can suppose $s\implies0$ . Let us consider now the genus two Heegaard diagram $H(a,0,1,2,r,0)$ . The sequence of Singer moves [31] on this diagram, drawn in Figures 3–10 and described in the Appendix of the paper, leads to the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ (see Figure 10). Since the representation of lens spaces (including $\mathbf{S^{3}}$ ) as 2-fold branched coverings of $\mathrm{{S^{3}}}$ is unique [14], the result immediately holds.
Remark 5. The Dunwoody manifold $M(a,0,1,n,r,s)$ of Theorem 8 is homeomorphic to the Minkus manifold $M_{n}(2a+1,2r)$ [21] and the Lins-Mandel manifold $S(n,2a+1,2r,1)$ [19, 24].
An immediate consequence of Theorem 8 is:
Corollary 9 The fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation which is geometric.
Remark 6. In [21] is shown that the fundamental group of every branched cyclic covering of a 2-bridge knot admits a cyclic presentation, but without pointing out that this presentation is geometric.
About the set $\kappa$ of knots in $\mathrm{{S^{3}}}$ involved in Corollary 7, we propose the following:
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"If this conjecture is true, the set contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds (resp. are the -fold branched cyclic coverings of the 4-bridge torus knot (resp. of the 5-bridge torus knot ).",
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[{"type": "text", "text": "If this conjecture is true, the set $\\kappa$ contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds $M(1,2,3,n,4,4)$ (resp. $M(1,3,4,n,5,5))$ are the $n$ -fold branched cyclic coverings of the 4-bridge torus knot $K(4,5)$ (resp. of the 5-bridge torus knot $K(5,6)$ ). ", "page_idx": 13}, {"type": "text", "text": "4 Appendix ", "text_level": 1, "page_idx": 13}, {"type": "text", "text": "Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram $H(a,0,1,2,r,0)$ of Figure 3, the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ of Figure 10. The result will be achieved by a sequence of exactly $a+4$ Singer moves: one of type ID, $a+2$ of type IC and the final one of type III. ", "page_idx": 13}, {"type": "text", "text": "Figure 3 shows the open Heegaard diagram $H(a,0,1,2,r,0)$ . Note that, since $s\\,=\\,0$ , the cycle $C_{1}^{\\prime}$ (resp. $C_{2}^{\\prime}$ ) is glued with the cycle $C_{1}^{\\prime\\prime}$ (resp. $C_{2}^{\\prime\\prime}$ ). Let $D_{1}$ (resp. $D_{2}$ ) be the cycle of the Heegaard diagram corresponding to the arc $e^{\\prime}$ (resp. $e^{\\prime\\prime}$ ) coming out from the vertex $v^{\\prime}$ of $C_{1}^{\\prime}$ (resp. $v^{\\prime\\prime}$ of $C_{2}^{\\prime}$ ) labelled $a+1$ . Orient $D_{1}$ (resp. $D_{2}$ ) so that the arc $e^{\\prime}$ (resp. $e^{\\prime\\prime}$ ) is oriented from up to down (resp. from down to up). This orientation on $D_{2}$ is opposite to the canonical one but, in this way, all the $2a$ arcs connecting $C_{1}^{\\prime}$ with $C_{2}^{\\prime}$ are oriented from $C_{1}^{\\prime}$ to $C_{2}^{\\prime}$ and all the $2a$ arcs connecting $C_{1}^{\\prime\\prime}$ with $C_{2}^{\\prime\\prime}$ are oriented from $C_{2}^{\\prime\\prime}$ to $C_{1}^{\\prime\\prime}$ . The cycle $D_{1}$ , besides the arc $e^{\\prime}$ , has two arcs for each $k=0,\\dotsc,a-1$ , one joining the vertex of $C_{1}^{\\prime}$ labelled $a+1-(1+2k)r$ with the vertex of $C_{2}^{\\prime}$ labelled $a+1+(1+2k)r$ , and the other one joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $a+1+(1+2k)r$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $a+1-(3+2k)r$ . The cycle $D_{2}$ , besides the arc $a_{2}$ , has two arcs for each $k=0,\\dotsc,a-1$ , one joining the vertex of $C_{1}^{\\prime}$ labelled $a+1-(2+2k)r$ with the vertex of $C_{2}^{\\prime}$ labelled $a+1+(2+2k)r$ , the other joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $a+1+2k r$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $a+1-(2+2k)r$ . ", "page_idx": 13}, {"type": "text", "text": "The first Singer move consists of replacing the curve $D_{2}$ with the curve $D_{2}^{\\prime}=D_{1}\\!+\\!D_{2}$ (move of type ID of [31]) obtained by isotopically approaching the arcs $e^{\\prime}$ and $e^{\\prime\\prime}$ until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, $D_{1}$ in $D_{1}^{\\prime}$ so that it becomes disjoint from $D_{2}^{\\prime}$ . ", "page_idx": 13}, {"type": "text", "text": "The resulting Heegaard diagram is drawn in Figure 4. The new $2a+1$ pairs of vertices obtained on $C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime}$ are labelled by simply adding a prime to the old label, while the $4a+2$ pairs of fixed vertices keep their old labelling. Note that each new vertex labelled $j^{\\prime}$ is placed, in the cycles $C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ , between the old vertices labelled $j$ and $j+1$ respectively. The cycles $C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ are no longer connected by any arc, while the cycles $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ are connected by a unique arc (belonging to $D_{1}^{\\prime}$ ) joining the vertex labelled $(a+1)^{\\prime}$ of $C_{1}^{\\prime}$ with the vertex labelled $(a+1-r)^{\\prime}$ of $C_{1}^{\\prime\\prime}$ . All the $\\mathrm{3}a$ arcs connecting $C_{1}^{\\prime}$ and $C_{2}^{\\prime}$ are oriented from $C_{1}^{\\prime}$ to $C_{2}^{\\prime}$ and all the $\\mathrm{3}a$ arcs which now connect $C_{1}^{\\prime\\prime}$ with $C_{2}^{\\prime\\prime}$ are oriented from $C_{2}^{\\prime\\prime}$ to $C_{1}^{\\prime\\prime}$ . The cycle $D_{2}^{\\prime}$ contains exactly $4a+2$ arcs; more precisely, for each $i=1,\\ldots,2a+1$ , it has one arc joining the vertex labelled $i$ of $C_{1}^{\\prime}$ with the vertex labelled $2a+2-i$ of $C_{2}^{\\prime}$ and one arc joining the vertex labelled $i$ of $C_{2}^{\\prime\\prime}$ with the vertex labelled $2a+2-2r-i$ of $C_{2}^{\\prime}$ . The cycle $D_{1}^{\\prime}$ is a copy of the cycle $D_{1}$ and hence it contains $2a+1$ arcs. One of these arcs connects $C_{1}^{\\prime}$ with $C_{1}^{\\prime\\prime}$ ; moreover, for each $k=0,\\ldots,a-1$ , $D_{1}^{\\prime}$ has one arc joining the vertex of $C_{1}^{\\prime}$ labelled $(a+1-(1+2k)r)^{\\prime}$ with the vertex of $C_{2}^{\\prime}$ labelled $(a+1+(1+2k)r)^{\\prime}$ and one arc joining the vertex of $C_{2}^{\\prime\\prime}$ labelled $(a+1+(1+2k)r)^{\\prime}$ with the vertex of $C_{1}^{\\prime\\prime}$ labelled $(a+1-(3+2k)r)^{\\prime}$ . ", "page_idx": 13}]
|
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The", "type": "text"}], "index": 2}, {"bbox": [109, 171, 501, 186], "spans": [{"bbox": [109, 171, 501, 186], "score": 1.0, "content": "above conjecture is supported by several cases contained in Table 1 of [6]", "type": "text"}], "index": 3}, {"bbox": [110, 185, 499, 200], "spans": [{"bbox": [110, 185, 380, 200], "score": 1.0, "content": "(see [32]). For example, the Dunwoody manifolds ", "type": "text"}, {"bbox": [380, 186, 465, 199], "score": 0.92, "content": "M(1,2,3,n,4,4)", "type": "inline_equation", "height": 13, "width": 85}, {"bbox": [466, 185, 499, 200], "score": 1.0, "content": " (resp.", "type": "text"}], "index": 4}, {"bbox": [110, 200, 500, 214], "spans": [{"bbox": [110, 201, 197, 213], "score": 0.91, "content": "M(1,3,4,n,5,5))", "type": "inline_equation", "height": 12, "width": 87}, {"bbox": [198, 200, 246, 214], "score": 1.0, "content": " are the ", "type": "text"}, {"bbox": [247, 205, 254, 210], "score": 0.88, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [255, 200, 500, 214], "score": 1.0, "content": "-fold branched cyclic coverings of the 4-bridge", "type": "text"}], "index": 5}, {"bbox": [109, 214, 423, 228], "spans": [{"bbox": [109, 214, 167, 228], "score": 1.0, "content": "torus knot ", "type": "text"}, {"bbox": [167, 215, 204, 228], "score": 0.94, "content": "K(4,5)", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [205, 214, 378, 228], "score": 1.0, "content": " (resp. of the 5-bridge torus knot ", "type": "text"}, {"bbox": [378, 215, 415, 228], "score": 0.94, "content": "K(5,6)", "type": "inline_equation", "height": 13, "width": 37}, {"bbox": [415, 214, 423, 228], "score": 1.0, "content": ").", "type": "text"}], "index": 6}], "index": 3}, {"type": "title", "bbox": [110, 247, 223, 268], "lines": [{"bbox": [111, 250, 223, 268], "spans": [{"bbox": [111, 253, 122, 264], "score": 1.0, "content": "4", "type": "text"}, {"bbox": [137, 250, 223, 268], "score": 1.0, "content": "Appendix", "type": "text"}], "index": 7}], "index": 7}, {"type": "text", "bbox": [110, 278, 500, 350], "lines": [{"bbox": [109, 279, 500, 295], "spans": [{"bbox": [109, 279, 500, 295], "score": 1.0, "content": "Now we show how to obtain, by means of Singer moves [31] on the genus", "type": "text"}], "index": 8}, {"bbox": [109, 294, 501, 310], "spans": [{"bbox": [109, 294, 229, 310], "score": 1.0, "content": "two Heegaard diagram ", "type": "text"}, {"bbox": [230, 295, 311, 308], "score": 0.94, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 13, "width": 81}, {"bbox": [311, 294, 501, 310], "score": 1.0, "content": " of Figure 3, the canonical genus one", "type": "text"}], "index": 9}, {"bbox": [110, 310, 500, 323], "spans": [{"bbox": [110, 310, 299, 323], "score": 1.0, "content": "Heegaard diagram of the lens space ", "type": "text"}, {"bbox": [299, 311, 367, 323], "score": 0.93, "content": "L(2a+1,2r)", "type": "inline_equation", "height": 12, "width": 68}, {"bbox": [367, 310, 500, 323], "score": 1.0, "content": " of Figure 10. The result", "type": "text"}], "index": 10}, {"bbox": [110, 325, 499, 338], "spans": [{"bbox": [110, 325, 319, 338], "score": 1.0, "content": "will be achieved by a sequence of exactly ", "type": "text"}, {"bbox": [320, 326, 345, 335], "score": 0.91, "content": "a+4", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [345, 325, 499, 338], "score": 1.0, "content": " Singer moves: one of type ID,", "type": "text"}], "index": 11}, {"bbox": [110, 338, 343, 352], "spans": [{"bbox": [110, 341, 137, 350], "score": 0.93, "content": "a+2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [137, 338, 343, 352], "score": 1.0, "content": " of type IC and the final one of type III.", "type": "text"}], "index": 12}], "index": 10}, {"type": "text", "bbox": [110, 351, 500, 581], "lines": [{"bbox": [128, 353, 499, 367], "spans": [{"bbox": [128, 353, 355, 367], "score": 1.0, "content": "Figure 3 shows the open Heegaard diagram ", "type": "text"}, {"bbox": [356, 354, 437, 366], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [437, 353, 499, 367], "score": 1.0, "content": ". 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The new ", "type": "text"}, {"bbox": [466, 658, 499, 668], "score": 0.92, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 33}], "index": 34}], "index": 34}], "layout_bboxes": [], "page_idx": 13, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [], "discarded_blocks": [{"type": "discarded", "bbox": [298, 691, 311, 702], "lines": [{"bbox": [297, 691, 312, 705], "spans": [{"bbox": [297, 691, 312, 705], "score": 1.0, "content": "14", "type": "text"}]}]}], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [110, 125, 500, 226], "lines": [{"bbox": [126, 127, 500, 142], "spans": [{"bbox": [126, 127, 291, 142], "score": 1.0, "content": "If this conjecture is true, the set ", "type": "text"}, {"bbox": [291, 129, 301, 138], "score": 0.92, "content": "\\kappa", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [301, 127, 500, 142], "score": 1.0, "content": " contains knots with an arbitrarily high", "type": "text"}], "index": 0}, {"bbox": [110, 143, 499, 156], "spans": [{"bbox": [110, 143, 499, 156], "score": 1.0, "content": "number of bridges. 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The result", "type": "text"}], "index": 10}, {"bbox": [110, 325, 499, 338], "spans": [{"bbox": [110, 325, 319, 338], "score": 1.0, "content": "will be achieved by a sequence of exactly ", "type": "text"}, {"bbox": [320, 326, 345, 335], "score": 0.91, "content": "a+4", "type": "inline_equation", "height": 9, "width": 25}, {"bbox": [345, 325, 499, 338], "score": 1.0, "content": " Singer moves: one of type ID,", "type": "text"}], "index": 11}, {"bbox": [110, 338, 343, 352], "spans": [{"bbox": [110, 341, 137, 350], "score": 0.93, "content": "a+2", "type": "inline_equation", "height": 9, "width": 27}, {"bbox": [137, 338, 343, 352], "score": 1.0, "content": " of type IC and the final one of type III.", "type": "text"}], "index": 12}], "index": 10, "page_num": "page_13", "page_size": [612.0, 792.0], "bbox_fs": [109, 279, 501, 352]}, {"type": "text", "bbox": [110, 351, 500, 581], "lines": [{"bbox": [128, 353, 499, 367], "spans": [{"bbox": [128, 353, 355, 367], "score": 1.0, "content": "Figure 3 shows the open Heegaard diagram ", "type": "text"}, {"bbox": [356, 354, 437, 366], "score": 0.93, "content": "H(a,0,1,2,r,0)", "type": "inline_equation", "height": 12, "width": 81}, {"bbox": [437, 353, 499, 367], "score": 1.0, "content": ". 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If this conjecture is true, the set contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds (resp. are the -fold branched cyclic coverings of the 4-bridge torus knot (resp. of the 5-bridge torus knot ).
# 4 Appendix
Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram of Figure 3, the canonical genus one Heegaard diagram of the lens space of Figure 10. The result will be achieved by a sequence of exactly Singer moves: one of type ID, of type IC and the final one of type III.
Figure 3 shows the open Heegaard diagram . Note that, since , the cycle (resp. ) is glued with the cycle (resp. ). Let (resp. ) be the cycle of the Heegaard diagram corresponding to the arc (resp. ) coming out from the vertex of (resp. of ) labelled . Orient (resp. ) so that the arc (resp. ) is oriented from up to down (resp. from down to up). This orientation on is opposite to the canonical one but, in this way, all the arcs connecting with are oriented from to and all the arcs connecting with are oriented from to . The cycle , besides the arc , has two arcs for each , one joining the vertex of labelled with the vertex of labelled , and the other one joining the vertex of labelled with the vertex of labelled . The cycle , besides the arc , has two arcs for each , one joining the vertex of labelled with the vertex of labelled , the other joining the vertex of labelled with the vertex of labelled .
The first Singer move consists of replacing the curve with the curve (move of type ID of [31]) obtained by isotopically approaching the arcs and until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, in so that it becomes disjoint from .
The resulting Heegaard diagram is drawn in Figure 4. The new pairs of vertices obtained on are labelled by simply adding a prime to the old label, while the pairs of fixed vertices keep their old labelling. Note that each new vertex labelled is placed, in the cycles and , between the old vertices labelled and respectively. The cycles and are no longer connected by any arc, while the cycles and are connected by a unique arc (belonging to ) joining the vertex labelled of with the vertex labelled of . All the arcs connecting and are oriented from to and all the arcs which now connect with are oriented from to . The cycle contains exactly arcs; more precisely, for each , it has one arc joining the vertex labelled of with the vertex labelled of and one arc joining the vertex labelled of with the vertex labelled of . The cycle is a copy of the cycle and hence it contains arcs. One of these arcs connects with ; moreover, for each , has one arc joining the vertex of labelled with the vertex of labelled and one arc joining the vertex of labelled with the vertex of labelled .
14
|
<div class="pdf-page">
<p>If this conjecture is true, the set contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds (resp. are the -fold branched cyclic coverings of the 4-bridge torus knot (resp. of the 5-bridge torus knot ).</p>
<h1>4 Appendix</h1>
<p>Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram of Figure 3, the canonical genus one Heegaard diagram of the lens space of Figure 10. The result will be achieved by a sequence of exactly Singer moves: one of type ID, of type IC and the final one of type III.</p>
<p>Figure 3 shows the open Heegaard diagram . Note that, since , the cycle (resp. ) is glued with the cycle (resp. ). Let (resp. ) be the cycle of the Heegaard diagram corresponding to the arc (resp. ) coming out from the vertex of (resp. of ) labelled . Orient (resp. ) so that the arc (resp. ) is oriented from up to down (resp. from down to up). This orientation on is opposite to the canonical one but, in this way, all the arcs connecting with are oriented from to and all the arcs connecting with are oriented from to . The cycle , besides the arc , has two arcs for each , one joining the vertex of labelled with the vertex of labelled , and the other one joining the vertex of labelled with the vertex of labelled . The cycle , besides the arc , has two arcs for each , one joining the vertex of labelled with the vertex of labelled , the other joining the vertex of labelled with the vertex of labelled .</p>
<p>The first Singer move consists of replacing the curve with the curve (move of type ID of [31]) obtained by isotopically approaching the arcs and until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, in so that it becomes disjoint from .</p>
<p>The resulting Heegaard diagram is drawn in Figure 4. The new pairs of vertices obtained on are labelled by simply adding a prime to the old label, while the pairs of fixed vertices keep their old labelling. Note that each new vertex labelled is placed, in the cycles and , between the old vertices labelled and respectively. The cycles and are no longer connected by any arc, while the cycles and are connected by a unique arc (belonging to ) joining the vertex labelled of with the vertex labelled of . All the arcs connecting and are oriented from to and all the arcs which now connect with are oriented from to . The cycle contains exactly arcs; more precisely, for each , it has one arc joining the vertex labelled of with the vertex labelled of and one arc joining the vertex labelled of with the vertex labelled of . The cycle is a copy of the cycle and hence it contains arcs. One of these arcs connects with ; moreover, for each , has one arc joining the vertex of labelled with the vertex of labelled and one arc joining the vertex of labelled with the vertex of labelled .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="161" data-width="652" data-height="131">If this conjecture is true, the set contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds (resp. are the -fold branched cyclic coverings of the 4-bridge torus knot (resp. of the 5-bridge torus knot ).</p>
<h1 class="pdf-title" data-x="184" data-y="319" data-width="189" data-height="27">4 Appendix</h1>
<p class="pdf-text" data-x="184" data-y="359" data-width="652" data-height="93">Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram of Figure 3, the canonical genus one Heegaard diagram of the lens space of Figure 10. The result will be achieved by a sequence of exactly Singer moves: one of type ID, of type IC and the final one of type III.</p>
<p class="pdf-text" data-x="184" data-y="453" data-width="652" data-height="298">Figure 3 shows the open Heegaard diagram . Note that, since , the cycle (resp. ) is glued with the cycle (resp. ). Let (resp. ) be the cycle of the Heegaard diagram corresponding to the arc (resp. ) coming out from the vertex of (resp. of ) labelled . Orient (resp. ) so that the arc (resp. ) is oriented from up to down (resp. from down to up). This orientation on is opposite to the canonical one but, in this way, all the arcs connecting with are oriented from to and all the arcs connecting with are oriented from to . The cycle , besides the arc , has two arcs for each , one joining the vertex of labelled with the vertex of labelled , and the other one joining the vertex of labelled with the vertex of labelled . The cycle , besides the arc , has two arcs for each , one joining the vertex of labelled with the vertex of labelled , the other joining the vertex of labelled with the vertex of labelled .</p>
<p class="pdf-text" data-x="184" data-y="752" data-width="652" data-height="93">The first Singer move consists of replacing the curve with the curve (move of type ID of [31]) obtained by isotopically approaching the arcs and until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, in so that it becomes disjoint from .</p>
<p class="pdf-text" data-x="212" data-y="845" data-width="624" data-height="19">The resulting Heegaard diagram is drawn in Figure 4. The new pairs of vertices obtained on are labelled by simply adding a prime to the old label, while the pairs of fixed vertices keep their old labelling. Note that each new vertex labelled is placed, in the cycles and , between the old vertices labelled and respectively. The cycles and are no longer connected by any arc, while the cycles and are connected by a unique arc (belonging to ) joining the vertex labelled of with the vertex labelled of . All the arcs connecting and are oriented from to and all the arcs which now connect with are oriented from to . The cycle contains exactly arcs; more precisely, for each , it has one arc joining the vertex labelled of with the vertex labelled of and one arc joining the vertex labelled of with the vertex labelled of . The cycle is a copy of the cycle and hence it contains arcs. One of these arcs connects with ; moreover, for each , has one arc joining the vertex of labelled with the vertex of labelled and one arc joining the vertex of labelled with the vertex of labelled .</p>
<div class="pdf-discarded" data-x="498" data-y="893" data-width="22" data-height="14" style="opacity: 0.5;">14</div>
</div>
|
If this conjecture is true, the set $\kappa$ contains knots with an arbitrarily high number of bridges. Moreover, the conjecture implies that every branched cyclic covering of a torus knot admits a geometric cyclic presentation. The above conjecture is supported by several cases contained in Table 1 of [6] (see [32]). For example, the Dunwoody manifolds $M(1,2,3,n,4,4)$ (resp. $M(1,3,4,n,5,5))$ are the $n$ -fold branched cyclic coverings of the 4-bridge torus knot $K(4,5)$ (resp. of the 5-bridge torus knot $K(5,6)$ ).
# 4 Appendix
Now we show how to obtain, by means of Singer moves [31] on the genus two Heegaard diagram $H(a,0,1,2,r,0)$ of Figure 3, the canonical genus one Heegaard diagram of the lens space $L(2a+1,2r)$ of Figure 10. The result will be achieved by a sequence of exactly $a+4$ Singer moves: one of type ID, $a+2$ of type IC and the final one of type III.
Figure 3 shows the open Heegaard diagram $H(a,0,1,2,r,0)$ . Note that, since $s\,=\,0$ , the cycle $C_{1}^{\prime}$ (resp. $C_{2}^{\prime}$ ) is glued with the cycle $C_{1}^{\prime\prime}$ (resp. $C_{2}^{\prime\prime}$ ). Let $D_{1}$ (resp. $D_{2}$ ) be the cycle of the Heegaard diagram corresponding to the arc $e^{\prime}$ (resp. $e^{\prime\prime}$ ) coming out from the vertex $v^{\prime}$ of $C_{1}^{\prime}$ (resp. $v^{\prime\prime}$ of $C_{2}^{\prime}$ ) labelled $a+1$ . Orient $D_{1}$ (resp. $D_{2}$ ) so that the arc $e^{\prime}$ (resp. $e^{\prime\prime}$ ) is oriented from up to down (resp. from down to up). This orientation on $D_{2}$ is opposite to the canonical one but, in this way, all the $2a$ arcs connecting $C_{1}^{\prime}$ with $C_{2}^{\prime}$ are oriented from $C_{1}^{\prime}$ to $C_{2}^{\prime}$ and all the $2a$ arcs connecting $C_{1}^{\prime\prime}$ with $C_{2}^{\prime\prime}$ are oriented from $C_{2}^{\prime\prime}$ to $C_{1}^{\prime\prime}$ . The cycle $D_{1}$ , besides the arc $e^{\prime}$ , has two arcs for each $k=0,\dotsc,a-1$ , one joining the vertex of $C_{1}^{\prime}$ labelled $a+1-(1+2k)r$ with the vertex of $C_{2}^{\prime}$ labelled $a+1+(1+2k)r$ , and the other one joining the vertex of $C_{2}^{\prime\prime}$ labelled $a+1+(1+2k)r$ with the vertex of $C_{1}^{\prime\prime}$ labelled $a+1-(3+2k)r$ . The cycle $D_{2}$ , besides the arc $a_{2}$ , has two arcs for each $k=0,\dotsc,a-1$ , one joining the vertex of $C_{1}^{\prime}$ labelled $a+1-(2+2k)r$ with the vertex of $C_{2}^{\prime}$ labelled $a+1+(2+2k)r$ , the other joining the vertex of $C_{2}^{\prime\prime}$ labelled $a+1+2k r$ with the vertex of $C_{1}^{\prime\prime}$ labelled $a+1-(2+2k)r$ .
The first Singer move consists of replacing the curve $D_{2}$ with the curve $D_{2}^{\prime}=D_{1}\!+\!D_{2}$ (move of type ID of [31]) obtained by isotopically approaching the arcs $e^{\prime}$ and $e^{\prime\prime}$ until their intersection becomes a small arc and by removing the interior of this arc. The move is completed by shifting, with a small isotopy, $D_{1}$ in $D_{1}^{\prime}$ so that it becomes disjoint from $D_{2}^{\prime}$ .
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"If this conjecture is true, the set \\kappa contains knots with an arbitrarily high",
"number of bridges. Moreover, the conjecture implies that every branched",
"cyclic covering of a torus knot admits a geometric cyclic presentation. The",
"above conjecture is supported by several cases contained in Table 1 of [6]",
"(see [32]). For example, the Dunwoody manifolds M(1,2,3,n,4,4) (resp.",
"M(1,3,4,n,5,5)) are the n -fold branched cyclic coverings of the 4-bridge",
"torus knot K(4,5) (resp. of the 5-bridge torus knot K(5,6) ).",
"4 Appendix",
"Now we show how to obtain, by means of Singer moves [31] on the genus",
"two Heegaard diagram H(a,0,1,2,r,0) of Figure 3, the canonical genus one",
"Heegaard diagram of the lens space L(2a+1,2r) of Figure 10. The result",
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"a+2 of type IC and the final one of type III.",
"Figure 3 shows the open Heegaard diagram H(a,0,1,2,r,0) . Note that,",
"since s\\,=\\,0 , the cycle C_{1}^{\\prime} (resp. C_{2}^{\\prime} ) is glued with the cycle C_{1}^{\\prime\\prime} (resp. C_{2}^{\\prime\\prime} ).",
"Let D_{1} (resp. D_{2} ) be the cycle of the Heegaard diagram corresponding to",
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"labelled a+1 . Orient D_{1} (resp. D_{2} ) so that the arc e^{\\prime} (resp. e^{\\prime\\prime} ) is oriented",
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"are oriented from C_{1}^{\\prime} to C_{2}^{\\prime} and all the 2a arcs connecting C_{1}^{\\prime\\prime} with C_{2}^{\\prime\\prime} are",
"oriented from C_{2}^{\\prime\\prime} to C_{1}^{\\prime\\prime} . The cycle D_{1} , besides the arc e^{\\prime} , has two arcs for",
"each k=0,\\dotsc,a-1 , one joining the vertex of C_{1}^{\\prime} labelled a+1-(1+2k)r",
"with the vertex of C_{2}^{\\prime} labelled a+1+(1+2k)r , and the other one joining",
"the vertex of C_{2}^{\\prime\\prime} labelled a+1+(1+2k)r with the vertex of C_{1}^{\\prime\\prime} labelled",
"a+1-(3+2k)r . The cycle D_{2} , besides the arc a_{2} , has two arcs for each",
"k=0,\\dotsc,a-1 , one joining the vertex of C_{1}^{\\prime} labelled a+1-(2+2k)r with",
"the vertex of C_{2}^{\\prime} labelled a+1+(2+2k)r , the other joining the vertex of C_{2}^{\\prime\\prime}",
"labelled a+1+2k r with the vertex of C_{1}^{\\prime\\prime} labelled a+1-(2+2k)r .",
"The first Singer move consists of replacing the curve D_{2} with the curve",
"D_{2}^{\\prime}=D_{1}\\!+\\!D_{2} (move of type ID of [31]) obtained by isotopically approaching",
"the arcs e^{\\prime} and e^{\\prime\\prime} until their intersection becomes a small arc and by removing",
"the interior of this arc. The move is completed by shifting, with a small",
"isotopy, D_{1} in D_{1}^{\\prime} so that it becomes disjoint from D_{2}^{\\prime} .",
"The resulting Heegaard diagram is drawn in Figure 4. The new 2a+1",
"pairs of vertices obtained on C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime},C_{2}^{\\prime\\prime} are labelled by simply adding a",
"prime to the old label, while the 4a+2 pairs of fixed vertices keep their",
"old labelling. Note that each new vertex labelled j^{\\prime} is placed, in the cycles",
"C_{1}^{\\prime},C_{1}^{\\prime\\prime},C_{2}^{\\prime} and C_{2}^{\\prime\\prime} , between the old vertices labelled j and j+1 respectively.",
"The cycles C_{2}^{\\prime} and C_{2}^{\\prime\\prime} are no longer connected by any arc, while the cycles C_{1}^{\\prime}",
"and C_{1}^{\\prime\\prime} are connected by a unique arc (belonging to D_{1}^{\\prime} ) joining the vertex",
"labelled (a+1)^{\\prime} of C_{1}^{\\prime} with the vertex labelled (a+1-r)^{\\prime} of C_{1}^{\\prime\\prime} . All the",
"\\mathrm{3}a arcs connecting C_{1}^{\\prime} and C_{2}^{\\prime} are oriented from C_{1}^{\\prime} to C_{2}^{\\prime} and all the \\mathrm{3}a arcs",
"which now connect C_{1}^{\\prime\\prime} with C_{2}^{\\prime\\prime} are oriented from C_{2}^{\\prime\\prime} to C_{1}^{\\prime\\prime} . The cycle D_{2}^{\\prime}",
"contains exactly 4a+2 arcs; more precisely, for each i=1,\\ldots,2a+1 , it has",
"one arc joining the vertex labelled i of C_{1}^{\\prime} with the vertex labelled 2a+2-i",
"of C_{2}^{\\prime} and one arc joining the vertex labelled i of C_{2}^{\\prime\\prime} with the vertex labelled",
"2a+2-2r-i of C_{2}^{\\prime} . The cycle D_{1}^{\\prime} is a copy of the cycle D_{1} and hence",
"it contains 2a+1 arcs. One of these arcs connects C_{1}^{\\prime} with C_{1}^{\\prime\\prime} ; moreover,",
"for each k=0,\\ldots,a-1 , D_{1}^{\\prime} has one arc joining the vertex of C_{1}^{\\prime} labelled",
"(a+1-(1+2k)r)^{\\prime} with the vertex of C_{2}^{\\prime} labelled (a+1+(1+2k)r)^{\\prime} and",
"one arc joining the vertex of C_{2}^{\\prime\\prime} labelled (a+1+(1+2k)r)^{\\prime} with the vertex",
"of C_{1}^{\\prime\\prime} labelled (a+1-(3+2k)r)^{\\prime} ."
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"",
"Now, apply to the diagram a Singer move of type IC, cutting along the cycle (drawn in Figure 4) containing and and gluing the curve of the resulting disc with .",
"The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have a unique vertex ( and respectively). The cycle (resp. ) is connected with (resp. with ) by an arc joining (resp. ) with the vertex labelled (resp. , oriented as in Figure 5. The cycles and are joined by arcs, all oriented from to ; of them belong to and the other belong to . More precisely, for each , there is an arc of joining the vertex labelled of with the vertex labelled of ; while, for each , there is an arc of joining the vertex labelled of with the vertex labelled of .",
"Apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 5) containing and and gluing the curve of the resulting disc with .",
"The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have one vertex less. It is easy to see that the cycle has exactly the same arcs connecting and , all oriented from to ; if the labelling of the vertices of and is induced by the labelling of shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle instead has one arc less than in the previous step. In fact, it has arcs, connecting and , all oriented from to and joining the vertex labelled of with the vertex labelled of , for .",
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[{"type": "text", "text": "", "page_idx": 14}, {"type": "text", "text": "Now, apply to the diagram a Singer move of type IC, cutting along the cycle $E$ (drawn in Figure 4) containing $C_{1}^{\\prime\\prime}$ and $C_{2}^{\\prime\\prime}$ and gluing the curve $C_{2}^{\\prime\\prime}$ of the resulting disc with $C_{2}^{\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles $E^{\\prime}$ and $E^{\\prime\\prime}$ , which are copies of the cutting cycle $E$ . These cycles replace $C_{2}^{\\prime}$ and $C_{2}^{\\prime\\prime}$ and they both have a unique vertex ( $w^{\\prime}$ and $w^{\\prime\\prime}$ respectively). The cycle $E^{\\prime}$ (resp. $E^{\\prime\\prime}$ ) is connected with $C_{1}^{\\prime}$ (resp. with $C_{1}^{\\prime\\prime}$ ) by an arc joining $w^{\\prime}$ (resp. $w^{\\prime\\prime}$ ) with the vertex labelled $(a+1)^{\\prime}$ (resp. $(a+1-r)^{\\prime})$ , oriented as in Figure 5. The cycles $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ are joined by $3a+1$ arcs, all oriented from $C_{1}^{\\prime}$ to $C_{1}^{\\prime\\prime}$ ; $2a+1$ of them belong to $D_{2}^{\\prime}$ and the other $a$ belong to $D_{1}^{\\prime}$ . More precisely, for each $i=1,\\dots,2a+1$ , there is an arc of $D_{2}^{\\prime}$ joining the vertex labelled $i$ of $C_{1}^{\\prime}$ with the vertex labelled $i-2r$ of $C_{1}^{\\prime\\prime}$ ; while, for each $k=0,\\dotsc,a-1$ , there is an arc of $D_{1}^{\\prime}$ joining the vertex labelled $(a+1-(1+2k)r)^{\\prime}$ of $C_{1}^{\\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\\prime}$ of $C_{1}^{\\prime\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "Apply again a Singer move of type IC, cutting along the cycle $F_{1}$ (drawn in Figure 5) containing $C_{1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $C_{1}^{\\prime\\prime}$ of the resulting disc with $C_{1}^{\\prime}$ . ", "page_idx": 14}, {"type": "text", "text": "The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , which are copies of the cutting cycle $F_{1}$ . These cycles replace $C_{1}^{\\prime}$ and $C_{1}^{\\prime\\prime}$ and they both have one vertex less. It is easy to see that the cycle $D_{2}^{\\prime}$ has exactly the same $2a+1$ arcs connecting $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , all oriented from $F_{1}^{\\prime}$ to $F_{1}^{\\prime\\prime}$ ; if the labelling of the vertices of $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ is induced by the labelling of $F_{1}$ shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle $D_{1}^{\\prime}$ instead has one arc less than in the previous step. In fact, it has $a-1$ arcs, connecting $F_{1}^{\\prime}$ and $F_{1}^{\\prime\\prime}$ , all oriented from $F_{1}^{\\prime}$ to $F_{1}^{\\prime\\prime}$ and joining the vertex labelled $(a+1-(1+2k)r)^{\\prime}$ of $F_{1}^{\\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\\prime}$ of $F_{1}^{\\prime\\prime}$ , for $k=1,\\dotsc,a-1$ . ", "page_idx": 14}]
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In fact, it has ", "type": "text", "cross_page": true}, {"bbox": [324, 202, 351, 211], "score": 0.92, "content": "a-1", "type": "inline_equation", "height": 9, "width": 27, "cross_page": true}, {"bbox": [352, 199, 441, 214], "score": 1.0, "content": " arcs, connecting ", "type": "text", "cross_page": true}, {"bbox": [442, 201, 454, 213], "score": 0.92, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12, "cross_page": true}, {"bbox": [455, 199, 481, 214], "score": 1.0, "content": " and ", "type": "text", "cross_page": true}, {"bbox": [481, 201, 496, 213], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [496, 199, 499, 214], "score": 1.0, "content": ",", "type": "text", "cross_page": true}], "index": 5}, {"bbox": [109, 214, 499, 228], "spans": [{"bbox": [109, 214, 198, 228], "score": 1.0, "content": "all oriented from ", "type": "text", "cross_page": true}, {"bbox": [198, 216, 210, 228], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 12, "cross_page": true}, {"bbox": [211, 214, 227, 228], "score": 1.0, "content": " to ", "type": "text", "cross_page": true}, {"bbox": [228, 216, 242, 228], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14, "cross_page": true}, {"bbox": [242, 214, 403, 228], "score": 1.0, "content": " and joining the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [404, 215, 499, 228], "score": 0.91, "content": "(a+1-(1+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 95, "cross_page": true}], "index": 6}, {"bbox": [109, 228, 500, 243], "spans": [{"bbox": [109, 228, 123, 243], "score": 1.0, "content": "of ", "type": "text", "cross_page": true}, {"bbox": [123, 230, 136, 242], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13, "cross_page": true}, {"bbox": [136, 228, 263, 243], "score": 1.0, "content": " with the vertex labelled ", "type": "text", "cross_page": true}, {"bbox": [263, 230, 359, 243], "score": 0.93, "content": "(a+1-(3+2k)r)^{\\prime}", "type": "inline_equation", "height": 13, "width": 96, "cross_page": true}, {"bbox": [360, 228, 375, 243], "score": 1.0, "content": " of ", "type": "text", "cross_page": true}, {"bbox": [375, 230, 390, 242], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15, "cross_page": true}, {"bbox": [390, 228, 414, 243], "score": 1.0, "content": ", for ", "type": "text", "cross_page": true}, {"bbox": [414, 231, 496, 242], "score": 0.93, "content": "k=1,\\dotsc,a-1", "type": "inline_equation", "height": 11, "width": 82, "cross_page": true}, {"bbox": [496, 228, 500, 243], "score": 1.0, "content": ".", "type": "text", "cross_page": true}], "index": 7}], "index": 36.5, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [110, 647, 499, 677]}]}
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Now, apply to the diagram a Singer move of type IC, cutting along the cycle (drawn in Figure 4) containing and and gluing the curve of the resulting disc with .
The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have a unique vertex ( and respectively). The cycle (resp. ) is connected with (resp. with ) by an arc joining (resp. ) with the vertex labelled (resp. , oriented as in Figure 5. The cycles and are joined by arcs, all oriented from to ; of them belong to and the other belong to . More precisely, for each , there is an arc of joining the vertex labelled of with the vertex labelled of ; while, for each , there is an arc of joining the vertex labelled of with the vertex labelled of .
Apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 5) containing and and gluing the curve of the resulting disc with .
The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have one vertex less. It is easy to see that the cycle has exactly the same arcs connecting and , all oriented from to ; if the labelling of the vertices of and is induced by the labelling of shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle instead has one arc less than in the previous step. In fact, it has arcs, connecting and , all oriented from to and joining the vertex labelled of with the vertex labelled of , for .
15
|
<div class="pdf-page">
<p>Now, apply to the diagram a Singer move of type IC, cutting along the cycle (drawn in Figure 4) containing and and gluing the curve of the resulting disc with .</p>
<p>The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have a unique vertex ( and respectively). The cycle (resp. ) is connected with (resp. with ) by an arc joining (resp. ) with the vertex labelled (resp. , oriented as in Figure 5. The cycles and are joined by arcs, all oriented from to ; of them belong to and the other belong to . More precisely, for each , there is an arc of joining the vertex labelled of with the vertex labelled of ; while, for each , there is an arc of joining the vertex labelled of with the vertex labelled of .</p>
<p>Apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 5) containing and and gluing the curve of the resulting disc with .</p>
<p>The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have one vertex less. It is easy to see that the cycle has exactly the same arcs connecting and , all oriented from to ; if the labelling of the vertices of and is induced by the labelling of shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle instead has one arc less than in the previous step. In fact, it has arcs, connecting and , all oriented from to and joining the vertex labelled of with the vertex labelled of , for .</p>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="499" data-width="652" data-height="54">Now, apply to the diagram a Singer move of type IC, cutting along the cycle (drawn in Figure 4) containing and and gluing the curve of the resulting disc with .</p>
<p class="pdf-text" data-x="182" data-y="554" data-width="654" data-height="224">The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have a unique vertex ( and respectively). The cycle (resp. ) is connected with (resp. with ) by an arc joining (resp. ) with the vertex labelled (resp. , oriented as in Figure 5. The cycles and are joined by arcs, all oriented from to ; of them belong to and the other belong to . More precisely, for each , there is an arc of joining the vertex labelled of with the vertex labelled of ; while, for each , there is an arc of joining the vertex labelled of with the vertex labelled of .</p>
<p class="pdf-text" data-x="184" data-y="779" data-width="652" data-height="54">Apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 5) containing and and gluing the curve of the resulting disc with .</p>
<p class="pdf-text" data-x="185" data-y="835" data-width="651" data-height="37">The resulting Heegaard diagram is shown in Figure 6. It contains the new cycles and , which are copies of the cutting cycle . These cycles replace and and they both have one vertex less. It is easy to see that the cycle has exactly the same arcs connecting and , all oriented from to ; if the labelling of the vertices of and is induced by the labelling of shown in Figure 5, these arcs join pairs of vertices with the same labelling of the previous step. The cycle instead has one arc less than in the previous step. In fact, it has arcs, connecting and , all oriented from to and joining the vertex labelled of with the vertex labelled of , for .</p>
<div class="pdf-discarded" data-x="500" data-y="893" data-width="20" data-height="14" style="opacity: 0.5;">15</div>
</div>
|
Now, apply to the diagram a Singer move of type IC, cutting along the cycle $E$ (drawn in Figure 4) containing $C_{1}^{\prime\prime}$ and $C_{2}^{\prime\prime}$ and gluing the curve $C_{2}^{\prime\prime}$ of the resulting disc with $C_{2}^{\prime}$ .
The new Heegaard diagram obtained in this way shown in Figure 5. It contains the new cycles $E^{\prime}$ and $E^{\prime\prime}$ , which are copies of the cutting cycle $E$ . These cycles replace $C_{2}^{\prime}$ and $C_{2}^{\prime\prime}$ and they both have a unique vertex ( $w^{\prime}$ and $w^{\prime\prime}$ respectively). The cycle $E^{\prime}$ (resp. $E^{\prime\prime}$ ) is connected with $C_{1}^{\prime}$ (resp. with $C_{1}^{\prime\prime}$ ) by an arc joining $w^{\prime}$ (resp. $w^{\prime\prime}$ ) with the vertex labelled $(a+1)^{\prime}$ (resp. $(a+1-r)^{\prime})$ , oriented as in Figure 5. The cycles $C_{1}^{\prime}$ and $C_{1}^{\prime\prime}$ are joined by $3a+1$ arcs, all oriented from $C_{1}^{\prime}$ to $C_{1}^{\prime\prime}$ ; $2a+1$ of them belong to $D_{2}^{\prime}$ and the other $a$ belong to $D_{1}^{\prime}$ . More precisely, for each $i=1,\dots,2a+1$ , there is an arc of $D_{2}^{\prime}$ joining the vertex labelled $i$ of $C_{1}^{\prime}$ with the vertex labelled $i-2r$ of $C_{1}^{\prime\prime}$ ; while, for each $k=0,\dotsc,a-1$ , there is an arc of $D_{1}^{\prime}$ joining the vertex labelled $(a+1-(1+2k)r)^{\prime}$ of $C_{1}^{\prime}$ with the vertex labelled $(a+1-(3+2k)r)^{\prime}$ of $C_{1}^{\prime\prime}$ .
Apply again a Singer move of type IC, cutting along the cycle $F_{1}$ (drawn in Figure 5) containing $C_{1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $C_{1}^{\prime\prime}$ of the resulting disc with $C_{1}^{\prime}$ .
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"cycle E (drawn in Figure 4) containing C_{1}^{\\prime\\prime} and C_{2}^{\\prime\\prime} and gluing the curve C_{2}^{\\prime\\prime}",
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"The resulting Heegaard diagram is shown in Figure 6. It contains the",
"new cycles F_{1}^{\\prime} and F_{1}^{\\prime\\prime} , which are copies of the cutting cycle F_{1} . These cycles",
"replace C_{1}^{\\prime} and C_{1}^{\\prime\\prime} and they both have one vertex less. It is easy to see",
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0003042v1
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"Now, apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 6) containing and and gluing the curve of the resulting disc with .",
"The new Heegaard diagram only differs from the previous one for con- taining one arc less in the cycle . By inductive application of Singer moves of type IC, cutting along the cycle (drawn in Figure 7) containing and and gluing the curve of the resulting disc with , we obtain, for , the situation shown in Figure 8, where the cycle contains only two arcs, none of which connects with .",
"After the move of type IC corresponding to , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of com- plementary handles given by the pair of cycles and by the cycle , composed by a unique arc connecting with . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Hee- gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space .",
"References",
"[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot . Proc. Edinb. Math. Soc. 42 (1999), 575–587 [2] Berge, J.: The knots in with non-trivial Dehn surgery yielding . Topology Appl. 38 (1991), 1–19 [3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad- sky groups. Algebra Colloq. 5 (1998), 203–217 [4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the mini- semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49–56 [5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701–717 [6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea ’94, Walter de Gruyter, Berlin-New York (1995), 47–55 [7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1–6 [8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221– 235 [9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165–179 [10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53–65 [11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169–5206 [12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23 [13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976 [14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60–96 [15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980 [16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231–235 [17] Kim, G., Kim, Y., Vesnin, A.: The knot and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961–980 [18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter, Berlin-New York, 2000 [19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261–284 [20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Trans- form. Groups 2 (1997), 165–182 [21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1–68 [22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109–123 [23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143–167 [24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. Knot Theory Ramifications 5 (1996), 239–263 [25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613 [26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471 [27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243 [28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251 [29] Reni, M., Zimmermann, B.,: Extending finite group actions from sur- faces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887 [30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170",
"16"
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[{"type": "text", "text": "", "page_idx": 15}, {"type": "text", "text": "Now, apply again a Singer move of type IC, cutting along the cycle $F_{2}$ (drawn in Figure 6) containing $F_{1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $F_{1}^{\\prime\\prime}$ of the resulting disc with $F_{1}^{\\prime}$ . ", "page_idx": 15}, {"type": "text", "text": "The new Heegaard diagram only differs from the previous one for containing one arc less in the cycle $D_{1}^{\\prime}$ . By inductive application of Singer moves of type IC, cutting along the cycle $F_{h}$ (drawn in Figure 7) containing $F_{h-1}^{\\prime\\prime}$ and $E^{\\prime\\prime}$ and gluing the curve $F_{h-1}^{\\prime\\prime}$ of the resulting disc with $F_{h-1}^{\\prime}$ , we obtain, for $h=a$ , the situation shown in Figure 8, where the cycle $D_{1}^{\\prime}$ contains only two arcs, none of which connects $F_{a}^{\\prime}$ with $F_{a}^{\\prime\\prime}$ . ", "page_idx": 15}, {"type": "text", "text": "After the move of type IC corresponding to $h\\,=\\,a\\,+\\,1$ , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of complementary handles given by the pair of cycles $E^{\\prime},E^{\\prime\\prime}$ and by the cycle $D_{1}^{\\prime}$ , composed by a unique arc connecting $E^{\\prime}$ with $E^{\\prime\\prime}$ . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Heegaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space $L(2a+1,2r)$ . ", "page_idx": 15}, {"type": "text", "text": "References ", "text_level": 1, "page_idx": 15}, {"type": "text", "text": "[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot $5_{2}$ . Proc. Edinb. Math. Soc. 42 (1999), 575\u2013587 \n[2] Berge, J.: The knots in $D^{2}\\times S^{1}$ with non-trivial Dehn surgery yielding $D^{2}\\times S^{1}$ . Topology Appl. 38 (1991), 1\u201319 \n[3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sieradsky groups. Algebra Colloq. 5 (1998), 203\u2013217 \n[4] Cavicchioli, A., Hegenbarth F., Repovs\u02c7, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the minisemester, Warsaw, Poland, July 13\u2013August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49\u201356 \n[5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701\u2013717 \n[6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea \u201994, Walter de Gruyter, Berlin-New York (1995), 47\u201355 \n[7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1\u20136 \n[8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221\u2013 235 \n[9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165\u2013179 \n[10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53\u201365 \n[11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169\u20135206 \n[12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23 \n[13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976 \n[14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60\u201396 \n[15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980 \n[16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231\u2013235 \n[17] Kim, G., Kim, Y., Vesnin, A.: The knot $5_{2}$ and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961\u2013980 \n[18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea \u201998, Walter de Gruyter, Berlin-New York, 2000 \n[19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261\u2013284 \n[20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Transform. Groups 2 (1997), 165\u2013182 \n[21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1\u201368 \n[22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109\u2013123 \n[23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143\u2013167 \n[24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. Knot Theory Ramifications 5 (1996), 239\u2013263 \n[25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603\u2013613 \n[26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454\u2013471 \n[27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213\u2013243 \n[28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245\u2013251 \n[29] Reni, M., Zimmermann, B.,: Extending finite group actions from surfaces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877\u20132887 \n[30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170 ", "page_idx": 15}]
|
{"preproc_blocks": [{"type": "text", "bbox": [110, 124, 500, 240], "lines": [{"bbox": [110, 127, 501, 142], "spans": [{"bbox": [110, 127, 151, 142], "score": 1.0, "content": "replace ", "type": "text"}, {"bbox": [151, 129, 164, 141], "score": 0.93, "content": "C_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [165, 127, 194, 142], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [194, 129, 208, 141], "score": 0.93, "content": "C_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [209, 127, 501, 142], "score": 1.0, "content": " and they both have one vertex less. It is easy to see", "type": "text"}], "index": 0}, {"bbox": [109, 141, 501, 157], "spans": [{"bbox": [109, 141, 183, 157], "score": 1.0, "content": "that the cycle ", "type": "text"}, {"bbox": [183, 144, 198, 155], "score": 0.93, "content": "D_{2}^{\\prime}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [198, 141, 310, 157], "score": 1.0, "content": " has exactly the same ", "type": "text"}, {"bbox": [311, 144, 342, 154], "score": 0.93, "content": "2a+1", "type": "inline_equation", "height": 10, "width": 31}, {"bbox": [342, 141, 426, 157], "score": 1.0, "content": " arcs connecting ", "type": "text"}, {"bbox": [427, 144, 439, 155], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 11, "width": 12}, {"bbox": [440, 141, 465, 157], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [465, 144, 480, 155], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 11, "width": 15}, {"bbox": [480, 141, 501, 157], "score": 1.0, "content": ", all", "type": "text"}], "index": 1}, {"bbox": [110, 157, 500, 170], "spans": [{"bbox": [110, 157, 181, 170], "score": 1.0, "content": "oriented from ", "type": "text"}, {"bbox": [181, 158, 194, 170], "score": 0.92, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [194, 157, 210, 170], "score": 1.0, "content": " to ", "type": "text"}, {"bbox": [210, 158, 225, 170], "score": 0.92, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [225, 157, 393, 170], "score": 1.0, "content": "; if the labelling of the vertices of ", "type": "text"}, {"bbox": [393, 158, 406, 170], "score": 0.93, "content": "F_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 13}, {"bbox": [406, 157, 430, 170], "score": 1.0, "content": " and ", "type": "text"}, {"bbox": [430, 158, 445, 170], "score": 0.93, "content": "F_{1}^{\\prime\\prime}", "type": "inline_equation", "height": 12, "width": 15}, {"bbox": [445, 157, 500, 170], "score": 1.0, "content": " is induced", "type": "text"}], "index": 2}, {"bbox": [110, 171, 500, 185], "spans": [{"bbox": [110, 171, 204, 185], "score": 1.0, "content": "by the labelling of ", "type": "text"}, {"bbox": [205, 173, 217, 183], "score": 0.92, "content": "F_{1}", "type": "inline_equation", "height": 10, "width": 12}, {"bbox": [217, 171, 500, 185], "score": 1.0, "content": " shown in Figure 5, these arcs join pairs of vertices with", "type": "text"}], "index": 3}, {"bbox": [109, 185, 500, 199], "spans": [{"bbox": [109, 185, 363, 199], "score": 1.0, "content": "the same labelling of the previous step. The cycle ", "type": "text"}, {"bbox": [364, 187, 378, 199], "score": 0.93, "content": "D_{1}^{\\prime}", "type": "inline_equation", "height": 12, "width": 14}, {"bbox": [378, 185, 500, 199], "score": 1.0, "content": " instead has one arc less", "type": "text"}], "index": 4}, {"bbox": [109, 199, 499, 214], "spans": [{"bbox": [109, 199, 324, 214], "score": 1.0, "content": "than in the previous step. 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|
Now, apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 6) containing and and gluing the curve of the resulting disc with .
The new Heegaard diagram only differs from the previous one for con- taining one arc less in the cycle . By inductive application of Singer moves of type IC, cutting along the cycle (drawn in Figure 7) containing and and gluing the curve of the resulting disc with , we obtain, for , the situation shown in Figure 8, where the cycle contains only two arcs, none of which connects with .
After the move of type IC corresponding to , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of com- plementary handles given by the pair of cycles and by the cycle , composed by a unique arc connecting with . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Hee- gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space .
# References
- [1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot . Proc. Edinb. Math. Soc. 42 (1999), 575–587 [2] Berge, J.: The knots in with non-trivial Dehn surgery yielding . Topology Appl. 38 (1991), 1–19 [3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad- sky groups. Algebra Colloq. 5 (1998), 203–217 [4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the mini- semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49–56 [5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701–717 [6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea ’94, Walter de Gruyter, Berlin-New York (1995), 47–55 [7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1–6 [8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221– 235 [9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165–179 [10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53–65 [11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169–5206 [12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23 [13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976 [14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60–96 [15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980 [16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231–235 [17] Kim, G., Kim, Y., Vesnin, A.: The knot and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961–980 [18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter, Berlin-New York, 2000 [19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261–284 [20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Trans- form. Groups 2 (1997), 165–182 [21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1–68 [22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109–123 [23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143–167 [24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. Knot Theory Ramifications 5 (1996), 239–263 [25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613 [26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471 [27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243 [28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251 [29] Reni, M., Zimmermann, B.,: Extending finite group actions from sur- faces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887 [30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170
16
|
<div class="pdf-page">
<p>Now, apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 6) containing and and gluing the curve of the resulting disc with .</p>
<p>The new Heegaard diagram only differs from the previous one for con- taining one arc less in the cycle . By inductive application of Singer moves of type IC, cutting along the cycle (drawn in Figure 7) containing and and gluing the curve of the resulting disc with , we obtain, for , the situation shown in Figure 8, where the cycle contains only two arcs, none of which connects with .</p>
<p>After the move of type IC corresponding to , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of com- plementary handles given by the pair of cycles and by the cycle , composed by a unique arc connecting with . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Hee- gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space .</p>
<h1>References</h1>
<ul>
<li>[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot . Proc. Edinb. Math. Soc. 42 (1999), 575–587 [2] Berge, J.: The knots in with non-trivial Dehn surgery yielding . Topology Appl. 38 (1991), 1–19 [3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad- sky groups. Algebra Colloq. 5 (1998), 203–217 [4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the mini- semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49–56 [5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701–717 [6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea ’94, Walter de Gruyter, Berlin-New York (1995), 47–55 [7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1–6 [8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221– 235 [9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165–179 [10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53–65 [11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169–5206 [12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23 [13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976 [14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60–96 [15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980 [16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231–235 [17] Kim, G., Kim, Y., Vesnin, A.: The knot and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961–980 [18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter, Berlin-New York, 2000 [19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261–284 [20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Trans- form. Groups 2 (1997), 165–182 [21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1–68 [22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109–123 [23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143–167 [24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. Knot Theory Ramifications 5 (1996), 239–263 [25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613 [26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471 [27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243 [28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251 [29] Reni, M., Zimmermann, B.,: Extending finite group actions from sur- faces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887 [30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170</li>
</ul>
</div>
|
<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="311" data-width="652" data-height="56">Now, apply again a Singer move of type IC, cutting along the cycle (drawn in Figure 6) containing and and gluing the curve of the resulting disc with .</p>
<p class="pdf-text" data-x="184" data-y="368" data-width="652" data-height="111">The new Heegaard diagram only differs from the previous one for con- taining one arc less in the cycle . By inductive application of Singer moves of type IC, cutting along the cycle (drawn in Figure 7) containing and and gluing the curve of the resulting disc with , we obtain, for , the situation shown in Figure 8, where the cycle contains only two arcs, none of which connects with .</p>
<p class="pdf-text" data-x="184" data-y="480" data-width="652" data-height="130">After the move of type IC corresponding to , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of com- plementary handles given by the pair of cycles and by the cycle , composed by a unique arc connecting with . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Hee- gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space .</p>
<h1 class="pdf-title" data-x="182" data-y="637" data-width="154" data-height="24">References</h1>
<ul class="pdf-list" data-x="182" data-y="676" data-width="656" data-height="190">
<li>[1] Bandieri, P., Kim, A.C., Mulazzani, M.,: On the cyclic coverings of the knot . Proc. Edinb. Math. Soc. 42 (1999), 575–587 [2] Berge, J.: The knots in with non-trivial Dehn surgery yielding . Topology Appl. 38 (1991), 1–19 [3] Cavicchioli, A., Hegenbarth F., Kim, A.C.: A geometric study of Sierad- sky groups. Algebra Colloq. 5 (1998), 203–217 [4] Cavicchioli, A., Hegenbarth F., Repovsˇ, D.: On manifold spines and cyclic presentations of groups. In: Knot theory. Proceedings of the mini- semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42 (1998), 49–56 [5] Doll, H.: A generalized bridge number for links in 3-manifold. Math. Ann. 294 (1992), 701–717 [6] Dunwoody, M.J.: Cyclic presentations and 3-manifolds. In: Proc. Inter. Conf., Groups-Korea ’94, Walter de Gruyter, Berlin-New York (1995), 47–55 [7] Gabai, D.: Surgery on knots in solid tori. Topology 28 (1989), 1–6 [8] Gabai, D.: 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221– 235 [9] Grasselli, L.: 3-manifold spines and bijoins. Rev. Mat. Univ. Complutense Madr. 3 (1990), 165–179 [10] Hayashi, C.: Genus one 1-bridge positions for the trivial knot and cabled knots. Math. Proc. Camb. Philos. Soc. 125 (1999), 53–65 [11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic 3-space. Commun. Algebra 23 (1995), 5169–5206 [12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci groups. J. Lie Theory 8 (1998), 1-23 [13] J. Hempel, J.: 3-manifolds. Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, N.J., 1976 [14] Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and manifolds - Vancouver 1983, Lect. Notes Math. 1144, Springer (1985), 60–96 [15] Johnson, D.L.: Topics in the theory of group presentations. London Math. Soc. Lect. Note Ser., vol. 42, Cambridge Univ. Press, Cambridge, U.K., 1980 [16] Kim, A.C.,: On the Fibonacci group and related topics. Contemp. Math. 184 (1995), 231–235 [17] Kim, G., Kim, Y., Vesnin, A.: The knot and cyclically presented groups. J. Korean Math. Soc. 35 (1998), 961–980 [18] Kim, A.C., Vesnin, A.: On a class of cyclically presented groups. To appear in: Proc. Inter. Conf., Groups-Korea ’98, Walter de Gruyter, Berlin-New York, 2000 [19] Lins, S., Mandel, A.: Graph-encoded 3-manifolds. Discrete Math. 57 (1985), 261–284 [20] Maclachlan, C., Reid, A.W.: Generalised Fibonacci manifolds. Trans- form. Groups 2 (1997), 165–182 [21] Minkus, J.: The branched cyclic coverings of 2 bridge knots and links. Mem. Am. Math. Soc. 35 Nr. 255 (1982), 1–68 [22] Montesinos, J.M.: Representing 3-manifolds by a universal branching set. Math. Proc. Camb. Philos. Soc. 94 (1983), 109–123 [23] Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. 289 (1991), 143–167 [24] Mulazzani, M.: All Lins-Mandel spaces are branched cyclic coverings of S . J. Knot Theory Ramifications 5 (1996), 239–263 [25] Neuwirth, L.: An algorithm for the construction of 3-manifolds from 2-complexes. Proc. Camb. Philos. Soc. 64 (1968), 603–613 [26] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds I. Amer. J. Math. 96 (1974), 454–471 [27] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds II. Trans. Amer. Math. Soc. 234 (1977), 213–243 [28] Osborne, R.P., Stevens, R.S.: Group presentations corresponding to spines of 3-manifolds III. Trans. Amer. Math. Soc. 234 (1977), 245–251 [29] Reni, M., Zimmermann, B.,: Extending finite group actions from sur- faces to handlebodies. Proc. Am. Math. Soc. 124 (1996), 2877–2887 [30] Schubert, H.: Knoten mit zwei Bricken. Math. Z. 65 (1956), 133-170</li>
</ul>
<div class="pdf-discarded" data-x="500" data-y="894" data-width="20" data-height="13" style="opacity: 0.5;">16</div>
</div>
|
Now, apply again a Singer move of type IC, cutting along the cycle $F_{2}$ (drawn in Figure 6) containing $F_{1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $F_{1}^{\prime\prime}$ of the resulting disc with $F_{1}^{\prime}$ .
The new Heegaard diagram only differs from the previous one for containing one arc less in the cycle $D_{1}^{\prime}$ . By inductive application of Singer moves of type IC, cutting along the cycle $F_{h}$ (drawn in Figure 7) containing $F_{h-1}^{\prime\prime}$ and $E^{\prime\prime}$ and gluing the curve $F_{h-1}^{\prime\prime}$ of the resulting disc with $F_{h-1}^{\prime}$ , we obtain, for $h=a$ , the situation shown in Figure 8, where the cycle $D_{1}^{\prime}$ contains only two arcs, none of which connects $F_{a}^{\prime}$ with $F_{a}^{\prime\prime}$ .
After the move of type IC corresponding to $h\,=\,a\,+\,1$ , we obtain the situation of Figure 9 in which the Heegaard diagram contains a pair of complementary handles given by the pair of cycles $E^{\prime},E^{\prime\prime}$ and by the cycle $D_{1}^{\prime}$ , composed by a unique arc connecting $E^{\prime}$ with $E^{\prime\prime}$ . The deletion of this pair of complementary handles (Singer move of type III) leads to the genus one Heegaard diagram drawn in Figure 10, which is the canonical Heegaard diagram of the lens space $L(2a+1,2r)$ .
# References
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"Now, apply again a Singer move of type IC, cutting along the cycle F_{2}",
"(drawn in Figure 6) containing F_{1}^{\\prime\\prime} and E^{\\prime\\prime} and gluing the curve F_{1}^{\\prime\\prime} of the",
"resulting disc with F_{1}^{\\prime} .",
"The new Heegaard diagram only differs from the previous one for con-",
"taining one arc less in the cycle D_{1}^{\\prime} . By inductive application of Singer moves",
"of type IC, cutting along the cycle F_{h} (drawn in Figure 7) containing F_{h-1}^{\\prime\\prime}",
"and E^{\\prime\\prime} and gluing the curve F_{h-1}^{\\prime\\prime} of the resulting disc with F_{h-1}^{\\prime} , we obtain,",
"for h=a , the situation shown in Figure 8, where the cycle D_{1}^{\\prime} contains only",
"two arcs, none of which connects F_{a}^{\\prime} with F_{a}^{\\prime\\prime} .",
"After the move of type IC corresponding to h\\,=\\,a\\,+\\,1 , we obtain the",
"situation of Figure 9 in which the Heegaard diagram contains a pair of com-",
"plementary handles given by the pair of cycles E^{\\prime},E^{\\prime\\prime} and by the cycle D_{1}^{\\prime} ,",
"composed by a unique arc connecting E^{\\prime} with E^{\\prime\\prime} . The deletion of this pair of",
"complementary handles (Singer move of type III) leads to the genus one Hee-",
"gaard diagram drawn in Figure 10, which is the canonical Heegaard diagram",
"of the lens space L(2a+1,2r) .",
"References",
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"[11] Helling, H., Kim, A.C., Mennicke, J.L.: Some honey-combs in hyperbolic",
"3-space. Commun. Algebra 23 (1995), 5169–5206",
"[12] Helling, H., Kim, A.C., Mennicke, J.L.: A geometric study of Fibonacci",
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[31] Singer, J.: Three-dimensional manifolds and their Heegaard diagrams. Trans. Am. Math. Soc. 35 (1933), 88–111
[32] Song, H.J., Kim, S.H.: Dunwoody 3-manifolds and -decomposible knots. Preprint, 1999
[34] Vesnin, A., Kim, A.C.: The fractional Fibonacci groups and manifolds. Sib. Math. J. 39 (1998), 655–664
[35] Wu, Y-Q.,: Incompressibility of surfaces in surgered 3-manifold. Topol- ogy 31 (1992), 271–279
[36] Wu, Y-Q.,: -reducing Dehn surgeries and 1-bridge-knots. Math. Ann. 295 (1992), 319–331
[37] Wu, Y-Q.,: Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebody. Math. Proc. Camb. Philos. Soc. 120 (1996), 687–696
[38] Zimmermann, B.: Genus actions of finite groups on 3-manifolds. Mich. Math. J. 43 (1996), 593–610
LUIGI GRASSELLI, Department of Sciences and Methods for Engineer- ing, University of Modena and Reggio Emilia, 42100 Reggio Emilia, ITALY. E-mail: [email protected]
MICHELE MULAZZANI, Department of Mathematics, University of Bologna, I-40127 Bologna, ITALY, and C.I.R.A.M., Bologna, ITALY. E-mail: [email protected]
19
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<p>[31] Singer, J.: Three-dimensional manifolds and their Heegaard diagrams. Trans. Am. Math. Soc. 35 (1933), 88–111</p>
<p>[32] Song, H.J., Kim, S.H.: Dunwoody 3-manifolds and -decomposible knots. Preprint, 1999</p>
<p>[34] Vesnin, A., Kim, A.C.: The fractional Fibonacci groups and manifolds. Sib. Math. J. 39 (1998), 655–664</p>
<p>[35] Wu, Y-Q.,: Incompressibility of surfaces in surgered 3-manifold. Topol- ogy 31 (1992), 271–279</p>
<p>[36] Wu, Y-Q.,: -reducing Dehn surgeries and 1-bridge-knots. Math. Ann. 295 (1992), 319–331</p>
<p>[37] Wu, Y-Q.,: Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebody. Math. Proc. Camb. Philos. Soc. 120 (1996), 687–696</p>
<p>[38] Zimmermann, B.: Genus actions of finite groups on 3-manifolds. Mich. Math. J. 43 (1996), 593–610</p>
<p>LUIGI GRASSELLI, Department of Sciences and Methods for Engineer- ing, University of Modena and Reggio Emilia, 42100 Reggio Emilia, ITALY. E-mail: [email protected]</p>
<p>MICHELE MULAZZANI, Department of Mathematics, University of Bologna, I-40127 Bologna, ITALY, and C.I.R.A.M., Bologna, ITALY. E-mail: [email protected]</p>
</div>
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<div class="pdf-page">
<p class="pdf-text" data-x="184" data-y="160" data-width="654" data-height="39">[31] Singer, J.: Three-dimensional manifolds and their Heegaard diagrams. Trans. Am. Math. Soc. 35 (1933), 88–111</p>
<p class="pdf-text" data-x="184" data-y="210" data-width="654" data-height="39">[32] Song, H.J., Kim, S.H.: Dunwoody 3-manifolds and -decomposible knots. Preprint, 1999</p>
<p class="pdf-text" data-x="184" data-y="311" data-width="654" data-height="39">[34] Vesnin, A., Kim, A.C.: The fractional Fibonacci groups and manifolds. Sib. Math. J. 39 (1998), 655–664</p>
<p class="pdf-text" data-x="184" data-y="360" data-width="654" data-height="40">[35] Wu, Y-Q.,: Incompressibility of surfaces in surgered 3-manifold. Topol- ogy 31 (1992), 271–279</p>
<p class="pdf-text" data-x="184" data-y="411" data-width="654" data-height="40">[36] Wu, Y-Q.,: -reducing Dehn surgeries and 1-bridge-knots. Math. Ann. 295 (1992), 319–331</p>
<p class="pdf-text" data-x="184" data-y="461" data-width="652" data-height="40">[37] Wu, Y-Q.,: Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebody. Math. Proc. Camb. Philos. Soc. 120 (1996), 687–696</p>
<p class="pdf-text" data-x="184" data-y="513" data-width="650" data-height="37">[38] Zimmermann, B.: Genus actions of finite groups on 3-manifolds. Mich. Math. J. 43 (1996), 593–610</p>
<p class="pdf-text" data-x="182" data-y="585" data-width="654" data-height="58">LUIGI GRASSELLI, Department of Sciences and Methods for Engineer- ing, University of Modena and Reggio Emilia, 42100 Reggio Emilia, ITALY. E-mail: [email protected]</p>
<p class="pdf-text" data-x="182" data-y="660" data-width="656" data-height="58">MICHELE MULAZZANI, Department of Mathematics, University of Bologna, I-40127 Bologna, ITALY, and C.I.R.A.M., Bologna, ITALY. E-mail: [email protected]</p>
<div class="pdf-discarded" data-x="498" data-y="893" data-width="24" data-height="15" style="opacity: 0.5;">19</div>
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[31] Singer, J.: Three-dimensional manifolds and their Heegaard diagrams. Trans. Am. Math. Soc. 35 (1933), 88–111
[32] Song, H.J., Kim, S.H.: Dunwoody 3-manifolds and $(1,1)$ -decomposible knots. Preprint, 1999
[34] Vesnin, A., Kim, A.C.: The fractional Fibonacci groups and manifolds. Sib. Math. J. 39 (1998), 655–664
[35] Wu, Y-Q.,: Incompressibility of surfaces in surgered 3-manifold. Topology 31 (1992), 271–279
[36] Wu, Y-Q.,: $\Dot{O}$ -reducing Dehn surgeries and 1-bridge-knots. Math. Ann. 295 (1992), 319–331
[37] Wu, Y-Q.,: Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebody. Math. Proc. Camb. Philos. Soc. 120 (1996), 687–696
[38] Zimmermann, B.: Genus actions of finite groups on 3-manifolds. Mich. Math. J. 43 (1996), 593–610
LUIGI GRASSELLI, Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, 42100 Reggio Emilia, ITALY. E-mail: [email protected]
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![Image]()
![Image]()
Figure 3:
![Image]() *Figure 3:*
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<div class="pdf-page">
<em>Figure 3:</em>
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![Image]()
![Image]()
Figure 4:
![Image]() *Figure 4:*
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<div class="pdf-page">
<em>Figure 4:</em>
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![Image]()
![Image]()
Figure 6:
![Image]() *Figure 6:*
22
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![Image]()
![Image]()
Figure 7:
![Image]() *Figure 7:*
![Image]()
![Image]()
Figure 8:
![Image]() *Figure 8:*
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<em>Figure 7:</em>
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![Image]()
![Image]()
Figure 9:
![Image]() *Figure 9:*
![Image]()
![Image]()
Figure 10:
![Image]() *Figure 10:*
24
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<div class="pdf-page">
<em>Figure 9:</em>
<em>Figure 10:</em>
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Figure 9:
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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 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.",
"1. Introduction.",
"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.",
"To describe our results, we need the following definition.",
"Definition 1.1. The corank of the representation where the are the standard generators of the group",
"Remark 1.1. Because the are conjugate to each other ([2], p.655), the number does not depend on , 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 representa- tion is 1.",
"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.",
"1. A one-dimensional representation ,",
"Date: November 4, 2018.",
"arXiv:math/0003047v1 [math.GR] 7 Mar 2000"
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Introduction.", "type": "text"}], "index": 10}], "index": 10, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [137, 668, 481, 684]}]}
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# IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO
arXiv:math/0003047v1 [math.GR] 7 Mar 2000
INNA SYSOEVA
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.
# 1. Introduction.
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.
To describe our results, we need the following definition.
Definition 1.1. The corank of the representation where the are the standard generators of the group
Remark 1.1. Because the are conjugate to each other ([2], p.655), the number does not depend on , 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 representa- tion is 1.
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.
1. A one-dimensional representation ,
Date: November 4, 2018.
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<div class="pdf-page">
<h1>IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO</h1>
<p>INNA SYSOEVA</p>
<p>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.</p>
<h1>1. Introduction.</h1>
<p>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.</p>
<p>To describe our results, we need the following definition.</p>
<p>Definition 1.1. The corank of the representation where the are the standard generators of the group</p>
<p>Remark 1.1. Because the are conjugate to each other ([2], p.655), the number does not depend on , which justifies the above definition.</p>
<p>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.</p>
<p>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.</p>
<p>1. A one-dimensional representation ,</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="249" data-y="182" data-width="524" data-height="37">IRREDUCIBLE REPRESENTATIONS OF BRAID GROUPS OF CORANK TWO</h1>
<div class="pdf-discarded" data-x="21" data-y="212" data-width="42" data-height="509" style="opacity: 0.5;">arXiv:math/0003047v1 [math.GR] 7 Mar 2000</div>
<p class="pdf-text" data-x="446" data-y="241" data-width="129" data-height="16">INNA SYSOEVA</p>
<p class="pdf-text" data-x="269" data-y="279" data-width="483" data-height="110">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.</p>
<h1 class="pdf-title" data-x="428" data-y="433" data-width="165" data-height="18">1. Introduction.</h1>
<p class="pdf-text" data-x="209" data-y="460" data-width="604" data-height="72">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.</p>
<p class="pdf-text" data-x="227" data-y="533" data-width="489" data-height="17">To describe our results, we need the following definition.</p>
<p class="pdf-text" data-x="210" data-y="559" data-width="603" data-height="55">Definition 1.1. The corank of the representation where the are the standard generators of the group</p>
<p class="pdf-text" data-x="209" data-y="632" data-width="604" data-height="55">Remark 1.1. Because the are conjugate to each other ([2], p.655), the number does not depend on , which justifies the above definition.</p>
<p class="pdf-text" data-x="209" data-y="695" data-width="604" data-height="54">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.</p>
<p class="pdf-text" data-x="209" data-y="751" data-width="604" data-height="107">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.</p>
<p class="pdf-text" data-x="225" data-y="862" data-width="583" data-height="19">1. A one-dimensional representation ,</p>
<div class="pdf-discarded" data-x="229" data-y="889" data-width="184" data-height="14" style="opacity: 0.5;">Date: November 4, 2018.</div>
</div>
|
# 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.
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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.\u201d 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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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.\u201d 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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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.\u201d 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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [124, 562, 488, 702]}]}
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|
I.SYSOEVA
2
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
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]:
$$
\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 , where is the identity matrix.
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 .
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 .
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.
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.
|
<div class="pdf-page">
<p>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</p>
<p>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]:</p>
<p>for , where is the identity matrix.</p>
<p>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 .</p>
<p>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 .</p>
<p>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.</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="465" data-y="116" data-width="85" data-height="14" style="opacity: 0.5;">I.SYSOEVA</div>
<div class="pdf-discarded" data-x="207" data-y="117" data-width="13" data-height="12" style="opacity: 0.5;">2</div>
<p class="pdf-text" data-x="227" data-y="142" data-width="586" data-height="73">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</p>
<p class="pdf-text" data-x="207" data-y="224" data-width="607" data-height="94">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]:</p>
<p class="pdf-text" data-x="207" data-y="452" data-width="519" data-height="19">for , where is the identity matrix.</p>
<p class="pdf-text" data-x="207" data-y="473" data-width="606" data-height="89">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 .</p>
<p class="pdf-text" data-x="207" data-y="562" data-width="606" data-height="90">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 .</p>
<p class="pdf-text" data-x="209" data-y="652" data-width="605" data-height="72">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.</p>
<p class="pdf-text" data-x="209" data-y="725" data-width="605" data-height="180">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.</p>
</div>
|
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.
|
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"2. An irreducible (n-1)- dimensional specialization of the reduced",
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"3. An irreducible (n-2)- dimensional specialization of the compo-",
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"The main goal of this paper is to classify all the irreducible complex",
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"the following representation \\rho:B_{n}\\to G L_{n}(\\mathbb{C}[u^{\\pm1}]) , first discovered by",
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"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",
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"Using the braid relations, we investigate the structure of the friend-",
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"if and only if \\vert i-j\\vert=1 .",
"For a given friendship graph it is relatively easy to classify all ir-",
"reducible 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 se-",
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"complex representation of B_{n} of dimension at most n-1 is a tensor",
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"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 .",
"Based on the above result we would like to make the following two conjectures.",
"Conjecture 1. For every for large enough there are no irre- ducible complex representations of of corank .",
"Conjecture 2. For every for large enough there are no irre- ducible complex representations of of dimension .",
"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.",
"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.",
"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.",
"2. Notation and preliminary results",
"Let be the braid group on strings. It has a presentation",
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"Lemma 2.1. For the braid group set",
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182], "score": 1.0, "content": "conjectures.", "type": "text"}], "index": 4}], "index": 3.5, "page_num": "page_2", "page_size": [612.0, 792.0], "bbox_fs": [125, 155, 486, 182]}, {"type": "text", "bbox": [125, 187, 485, 215], "lines": [{"bbox": [126, 189, 486, 204], "spans": [{"bbox": [126, 189, 263, 204], "score": 1.0, "content": "Conjecture 1. 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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_2", "page_size": [612.0, 792.0], "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, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_2", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_2", "page_size": [612.0, 792.0]}]}
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|
BRAID GROUP REPRESENTATIONS
3
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 .
Based on the above result we would like to make the following two conjectures.
Conjecture 1. For every for large enough there are no irre- ducible complex representations of of corank .
Conjecture 2. For every for large enough there are no irre- ducible complex representations of of dimension .
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.
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.
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.
# 2. Notation and preliminary results
Let be the braid group on 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
$$
Lemma 2.1. For the braid group set
$$
\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},
$$
|
<div class="pdf-page">
<p>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 .</p>
<p>Based on the above result we would like to make the following two conjectures.</p>
<p>Conjecture 1. For every for large enough there are no irre- ducible complex representations of of corank .</p>
<p>Conjecture 2. For every for large enough there are no irre- ducible complex representations of of dimension .</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<h1>2. Notation and preliminary results</h1>
<p>Let be the braid group on strings. It has a presentation</p>
<p>Lemma 2.1. For the braid group set</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="371" data-y="116" data-width="279" data-height="14" style="opacity: 0.5;">BRAID GROUP REPRESENTATIONS</div>
<div class="pdf-discarded" data-x="801" data-y="117" data-width="12" data-height="12" style="opacity: 0.5;">3</div>
<p class="pdf-text" data-x="209" data-y="143" data-width="604" data-height="53">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 .</p>
<p class="pdf-text" data-x="207" data-y="197" data-width="606" data-height="35">Based on the above result we would like to make the following two conjectures.</p>
<p class="pdf-text" data-x="209" data-y="241" data-width="602" data-height="36">Conjecture 1. For every for large enough there are no irre- ducible complex representations of of corank .</p>
<p class="pdf-text" data-x="209" data-y="294" data-width="602" data-height="36">Conjecture 2. For every for large enough there are no irre- ducible complex representations of of dimension .</p>
<p class="pdf-text" data-x="207" data-y="340" data-width="606" data-height="72">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.</p>
<p class="pdf-text" data-x="209" data-y="413" data-width="604" data-height="161">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.</p>
<p class="pdf-text" data-x="209" data-y="575" data-width="604" data-height="72">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.</p>
<h1 class="pdf-title" data-x="322" data-y="660" data-width="377" data-height="17">2. Notation and preliminary results</h1>
<p class="pdf-text" data-x="225" data-y="686" data-width="532" data-height="18">Let be the braid group on strings. It has a presentation</p>
<p class="pdf-text" data-x="207" data-y="743" data-width="360" data-height="19">Lemma 2.1. For the braid group set</p>
</div>
|
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
$$
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"",
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"",
"for all where indices are taken modulo .",
"Remark 2.2. Taking into account the above lemma, we also have the following presentation of :",
"1τ 1 >",
"for all where indices are taken modulo and is defined as above.",
"Let be a matrix representation of with",
"",
"and",
"",
"Then for any (indices are modulo ), the relation",
"",
"implies that",
"",
"Hence all the are conjugate to each other, so they have the same rank, spectrum and Jordan normal form.",
"Lemma 2.3. For a representation of with",
"",
"we have:",
"1) , for ; for all , where indices are taken modulo .",
"Proof. This follows easily from the relations on the generators of .",
"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 . 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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Taking into account the above lemma, we also have the", "type": "text"}], "index": 4}, {"bbox": [125, 207, 278, 221], "spans": [{"bbox": [125, 207, 255, 221], "score": 1.0, "content": "following presentation of ", "type": "text"}, {"bbox": [255, 209, 270, 219], "score": 0.9, "content": "B_{n}", "type": "inline_equation", "height": 10, "width": 15}, {"bbox": [271, 207, 278, 221], "score": 1.0, "content": " :", "type": "text"}], "index": 5}], "index": 4.5, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [125, 192, 486, 221]}, {"type": "text", "bbox": [125, 224, 563, 241], "lines": [{"bbox": [125, 226, 561, 244], "spans": [{"bbox": [125, 227, 527, 241], "score": 0.82, "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}", "type": "inline_equation"}, {"bbox": [527, 226, 561, 244], "score": 1.0, 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367], "score": 0.86, "content": "i", "type": "inline_equation", "height": 8, "width": 4}, {"bbox": [201, 357, 308, 370], "score": 1.0, "content": " (indices are modulo ", "type": "text"}, {"bbox": [309, 362, 316, 367], "score": 0.85, "content": "n", "type": "inline_equation", "height": 5, "width": 7}, {"bbox": [316, 357, 387, 370], "score": 1.0, "content": "), the relation", "type": "text"}], "index": 12}], "index": 12, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [127, 357, 387, 370]}, {"type": "interline_equation", "bbox": [269, 376, 341, 390], "lines": [{"bbox": [269, 376, 341, 390], "spans": [{"bbox": [269, 376, 341, 390], "score": 0.92, "content": "\\tau\\sigma_{i}\\tau^{-1}=\\sigma_{i+1}", "type": "interline_equation"}], "index": 13}], "index": 13, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"type": "text", "bbox": [124, 395, 189, 409], "lines": [{"bbox": [125, 396, 189, 411], "spans": [{"bbox": [125, 396, 189, 411], "score": 1.0, "content": "implies 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"rank, spectrum and Jordan normal form.", "type": "text"}], "index": 17}], "index": 16.5, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0], "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, "page_num": "page_3", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_3", "page_size": [612.0, 792.0], "bbox_fs": [124, 631, 487, 701]}]}
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|
I.SYSOEVA
4
$$
\sigma_{i+1}=\tau\sigma_{i}\tau^{-1},
$$
and
$$
\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},|i-j|\geq2
$$
for all where indices are taken modulo .
Remark 2.2. Taking into account the above lemma, we also have the following presentation of :
1τ 1 >
for all where indices are taken modulo and is defined as above.
Let be a matrix representation of with
$$
\rho(\sigma_{i})=1+A_{i},
$$
and
$$
\rho(\tau)=T\in G L_{r}(\mathbb{C}).
$$
Then for any (indices are modulo ), the relation
$$
\tau\sigma_{i}\tau^{-1}=\sigma_{i+1}
$$
implies that
$$
T A_{i}T^{-1}=A_{i+1}.
$$
Hence all the are conjugate to each other, so they have the same rank, spectrum and Jordan normal form.
Lemma 2.3. For a representation of with
$$
\rho(\sigma_{i})=1+A_{i},
$$
we have:
1) , for ; for all , where indices are taken modulo .
Proof. This follows easily from the relations on the generators of .
# 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 . 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.
|
<div class="pdf-page">
<p>and</p>
<p>for all where indices are taken modulo .</p>
<p>Remark 2.2. Taking into account the above lemma, we also have the following presentation of :</p>
<p>1τ 1 ></p>
<p>for all where indices are taken modulo and is defined as above.</p>
<p>Let be a matrix representation of with</p>
<p>and</p>
<p>Then for any (indices are modulo ), the relation</p>
<p>implies that</p>
<p>Hence all the are conjugate to each other, so they have the same rank, spectrum and Jordan normal form.</p>
<p>Lemma 2.3. For a representation of with</p>
<p>we have:</p>
<p>1) , for ; for all , where indices are taken modulo .</p>
<p>Proof. This follows easily from the relations on the generators of .</p>
<h1>3. The friendship graph.</h1>
<p>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.</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="465" data-y="115" data-width="85" data-height="15" style="opacity: 0.5;">I.SYSOEVA</div>
<div class="pdf-discarded" data-x="207" data-y="116" data-width="13" data-height="13" style="opacity: 0.5;">4</div>
<p class="pdf-text" data-x="209" data-y="165" data-width="38" data-height="18">and</p>
<p class="pdf-text" data-x="207" data-y="209" data-width="385" data-height="18">for all where indices are taken modulo .</p>
<p class="pdf-text" data-x="207" data-y="245" data-width="607" data-height="38">Remark 2.2. Taking into account the above lemma, we also have the following presentation of :</p>
<p class="pdf-text" data-x="209" data-y="289" data-width="733" data-height="22">1τ 1 ></p>
<p class="pdf-text" data-x="209" data-y="316" data-width="604" data-height="18">for all where indices are taken modulo and is defined as above.</p>
<p class="pdf-text" data-x="225" data-y="362" data-width="524" data-height="19">Let be a matrix representation of with</p>
<p class="pdf-text" data-x="209" data-y="415" data-width="36" data-height="18">and</p>
<p class="pdf-text" data-x="209" data-y="457" data-width="441" data-height="20">Then for any (indices are modulo ), the relation</p>
<p class="pdf-text" data-x="207" data-y="510" data-width="109" data-height="18">implies that</p>
<p class="pdf-text" data-x="209" data-y="553" data-width="605" data-height="36">Hence all the are conjugate to each other, so they have the same rank, spectrum and Jordan normal form.</p>
<p class="pdf-text" data-x="209" data-y="598" data-width="418" data-height="20">Lemma 2.3. For a representation of with</p>
<p class="pdf-text" data-x="210" data-y="652" data-width="77" data-height="16">we have:</p>
<p class="pdf-text" data-x="227" data-y="669" data-width="520" data-height="55">1) , for ; for all , where indices are taken modulo .</p>
<p class="pdf-text" data-x="207" data-y="733" data-width="607" data-height="37">Proof. This follows easily from the relations on the generators of .</p>
<h1 class="pdf-title" data-x="383" data-y="787" data-width="254" data-height="18">3. The friendship graph.</h1>
<p class="pdf-text" data-x="207" data-y="814" data-width="607" data-height="89">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.</p>
</div>
|
# 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},
$$
$$
\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.
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"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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"",
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"\\sigma_{i+1}=\\tau\\sigma_{i}\\tau^{-1},",
"\\sigma_{i}\\sigma_{j}=\\sigma_{j}\\sigma_{i},|i-j|\\geq2",
"\\rho(\\sigma_{i})=1+A_{i},",
"\\rho(\\tau)=T\\in G L_{r}(\\mathbb{C}).",
"\\tau\\sigma_{i}\\tau^{-1}=\\sigma_{i+1}",
"T A_{i}T^{-1}=A_{i+1}.",
"\\rho(\\sigma_{i})=1+A_{i},"
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"content": [
"We assume throughout this section that we have a representation",
"",
"with",
"",
"Definition 3.1. 1) , are neighbors (indices modulo ). 2) , are friends if",
"",
"3) , are true friends if either",
"and are not neighbors, and",
"",
"or",
"(b) and are neighbors, and",
"",
"Lemma 3.1. If are true friends, then they are friends.",
"Proof. 1) If and are not neighbors, then , so,",
"",
"2) If and are neighbors, then",
"and again",
"",
"Definition 3.2. The full friendship graph (associated with the rep- resentation ) is the simple-edged graph with n vertices and an edge joining and ) if and only if and are friends.",
"The friendship graph is the subgraph with vertices obtained from the full friendship graph by deleting 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.",
"BRAID GROUP REPRESENTATIONS",
"5"
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[{"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$$", "text_format": "latex", "page_idx": 4}, {"type": "text", "text": "Definition 3.2. The full friendship graph (associated with the representation $\\rho:B_{n}\\to G L_{n}(\\mathbb{C})$ ) is the simple-edged graph with n vertices $A_{0},A_{1},\\ldots,A_{n-1}$ and an edge joining $A_{i}$ and $A_{j}$ $\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}^{\\mathrm{\\textit{\\cdot}}}\\mathrm{\\Delta}j,$ ) if and only if $A_{i}$ and $A_{j}$ are friends. ", "page_idx": 4}, {"type": "text", "text": "The friendship graph is the subgraph with vertices $A_{1},\\dotsc,A_{n-1}$ obtained from the full friendship graph by deleting $A_{0}$ and all edges incident to it. ", "page_idx": 4}, {"type": "text", "text": "Our main interest is the friendship graph, but it is convenient to introduce the full friendship graph as a tool, because of the following lemma. ", "page_idx": 4}]
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|
BRAID GROUP REPRESENTATIONS
5
We assume throughout this section that we have a representation
$$
\rho:B_{n}\to G L_{r}(\mathbb{C}),
$$
with
$$
\rho(\sigma_{i})=1+A_{i},\;\;(i=0,1,\ldots,n-1).
$$
Definition 3.1. 1) , are neighbors (indices modulo ). 2) , are friends if
$$
I m(A_{i})\cap I m(A_{j})\neq\{0\}.
$$
3) , are true friends if either
and are not neighbors, and
$$
A_{i}A_{j}=A_{j}A_{i}\neq0;
$$
or
(b) and are neighbors, and
$$
A_{i}+A_{i}^{2}+A_{i}A_{j}A_{i}=A_{j}+A_{j}^{2}+A_{j}A_{i}A_{j}\neq0.
$$
Lemma 3.1. If are true friends, then they are friends.
Proof. 1) If and are not neighbors, then , so,
$$
I m(A)\cap I m(B)\supseteq I m(A B)\cap I m(B A)=I m(A B)\neq\{0\}.
$$
2) If and are neighbors, then
and again
$$
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 rep- resentation ) is the simple-edged graph with n vertices and an edge joining and ) if and only if and are friends.
The friendship graph is the subgraph with vertices obtained from the full friendship graph by deleting 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.
|
<div class="pdf-page">
<p>We assume throughout this section that we have a representation</p>
<p>with</p>
<p>Definition 3.1. 1) , are neighbors (indices modulo ). 2) , are friends if</p>
<p>3) , are true friends if either</p>
<p>and are not neighbors, and</p>
<p>or</p>
<p>(b) and are neighbors, and</p>
<p>Lemma 3.1. If are true friends, then they are friends.</p>
<p>Proof. 1) If and are not neighbors, then , so,</p>
<p>2) If and are neighbors, then</p>
<p>and again</p>
<p>Definition 3.2. The full friendship graph (associated with the rep- resentation ) is the simple-edged graph with n vertices and an edge joining and ) if and only if and are friends.</p>
<p>The friendship graph is the subgraph with vertices obtained from the full friendship graph by deleting and all edges incident to it.</p>
<p>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>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="369" data-y="115" data-width="283" data-height="15" style="opacity: 0.5;">BRAID GROUP REPRESENTATIONS</div>
<div class="pdf-discarded" data-x="801" data-y="116" data-width="12" data-height="13" style="opacity: 0.5;">5</div>
<p class="pdf-text" data-x="229" data-y="142" data-width="565" data-height="19">We assume throughout this section that we have a representation</p>
<p class="pdf-text" data-x="209" data-y="197" data-width="41" data-height="16">with</p>
<p class="pdf-text" data-x="210" data-y="252" data-width="561" data-height="37">Definition 3.1. 1) , are neighbors (indices modulo ). 2) , are friends if</p>
<p class="pdf-text" data-x="229" data-y="324" data-width="321" data-height="18">3) , are true friends if either</p>
<p class="pdf-text" data-x="229" data-y="343" data-width="328" data-height="19">and are not neighbors, and</p>
<p class="pdf-text" data-x="210" data-y="400" data-width="22" data-height="13">or</p>
<p class="pdf-text" data-x="227" data-y="415" data-width="293" data-height="19">(b) and are neighbors, and</p>
<p class="pdf-text" data-x="207" data-y="492" data-width="524" data-height="20">Lemma 3.1. If are true friends, then they are friends.</p>
<p class="pdf-text" data-x="227" data-y="521" data-width="564" data-height="19">Proof. 1) If and are not neighbors, then , so,</p>
<p class="pdf-text" data-x="227" data-y="575" data-width="296" data-height="19">2) If and are neighbors, then</p>
<p class="pdf-text" data-x="207" data-y="601" data-width="606" data-height="48">and again</p>
<p class="pdf-text" data-x="207" data-y="712" data-width="609" data-height="74">Definition 3.2. The full friendship graph (associated with the rep- resentation ) is the simple-edged graph with n vertices and an edge joining and ) if and only if and are friends.</p>
<p class="pdf-text" data-x="209" data-y="786" data-width="605" data-height="54">The friendship graph is the subgraph with vertices obtained from the full friendship graph by deleting and all edges incident to it.</p>
<p class="pdf-text" data-x="207" data-y="849" data-width="607" data-height="56">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>
</div>
|
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.
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"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,",
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"The friendship graph is the subgraph with vertices A_{1},\\dotsc,A_{n-1}",
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"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.",
"Proof. This follows immediately from the fact that conjugation by permutes cyclically (Lemma 2.1).",
"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 .",
"",
"Proof. By lemma 3.1, and are true not friends, because they are not friends, that is",
"",
"Consider such that ( exists because and are friends). Then",
"",
"because and is invertible.",
"So, ; that is, and are true friends.",
"Theorem 3.4. Let 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 and for all .",
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[{"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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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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "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, "page_num": "page_5", "page_size": [612.0, 792.0], "bbox_fs": [126, 673, 487, 702]}]}
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I.SYSOEVA
6
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.
Proof. This follows immediately from the fact that conjugation by permutes cyclically (Lemma 2.1).
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 .
![Image]()
![Image]()
Proof. By lemma 3.1, and are true not friends, because they are not friends, that is
$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$
Consider such that ( exists because and are friends). Then
$$
B A C y=B A B z=-(B+B^{2})z=-(1+B)B z\neq0
$$
because and is invertible.
So, ; that is, and are true friends.
Theorem 3.4. Let 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 and for all .
(c) The full friendship graph has an edge between and whenever and are not neighbors.
|
<div class="pdf-page">
<p>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.</p>
<p>Proof. This follows immediately from the fact that conjugation by permutes cyclically (Lemma 2.1).</p>
<p>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 .</p>
<p>Proof. By lemma 3.1, and are true not friends, because they are not friends, that is</p>
<p>Consider such that ( exists because and are friends). Then</p>
<p>because and is invertible.</p>
<p>So, ; that is, and are true friends.</p>
<p>Theorem 3.4. Let be a representation. Then one of the following holds.</p>
<p>(a) The full friendship graph is totally disconnected (no friends at all).</p>
<p>(b) The full friendship graph has an edge between and for all .</p>
<p>(c) The full friendship graph has an edge between and whenever and are not neighbors.</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="465" data-y="116" data-width="85" data-height="14" style="opacity: 0.5;">I.SYSOEVA</div>
<div class="pdf-discarded" data-x="207" data-y="117" data-width="13" data-height="12" style="opacity: 0.5;">6</div>
<p class="pdf-text" data-x="207" data-y="142" data-width="607" data-height="73">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.</p>
<p class="pdf-text" data-x="207" data-y="224" data-width="606" data-height="56">Proof. This follows immediately from the fact that conjugation by permutes cyclically (Lemma 2.1).</p>
<p class="pdf-text" data-x="207" data-y="290" data-width="607" data-height="56">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 .</p>
<p class="pdf-text" data-x="207" data-y="545" data-width="606" data-height="36">Proof. By lemma 3.1, and are true not friends, because they are not friends, that is</p>
<p class="pdf-text" data-x="207" data-y="619" data-width="606" data-height="37">Consider such that ( exists because and are friends). Then</p>
<p class="pdf-text" data-x="207" data-y="694" data-width="360" data-height="18">because and is invertible.</p>
<p class="pdf-text" data-x="227" data-y="713" data-width="455" data-height="18">So, ; that is, and are true friends.</p>
<p class="pdf-text" data-x="207" data-y="758" data-width="607" data-height="37">Theorem 3.4. Let be a representation. Then one of the following holds.</p>
<p class="pdf-text" data-x="207" data-y="797" data-width="607" data-height="34">(a) The full friendship graph is totally disconnected (no friends at all).</p>
<p class="pdf-text" data-x="209" data-y="832" data-width="607" data-height="35">(b) The full friendship graph has an edge between and for all .</p>
<p class="pdf-text" data-x="210" data-y="868" data-width="604" data-height="37">(c) The full friendship graph has an edge between and whenever and are not neighbors.</p>
</div>
|
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$ .
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"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 , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.",
"Remark 3.6. For there is a friendship graph which is neither totally disconnected nor connected:",
"",
"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.",
"Now consider the case when the friendship graph is totally discon- nected (that is, statement of theorem 3.4 holds).",
"Lemma 3.7. If and are neighbors and not friends then: (a) ; . If , then and .",
"Proof. (a). By lemma 3.1, and are not true friends, so",
"",
"BRAID GROUP REPRESENTATIONS",
"7"
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[{"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}]
|
{"preproc_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}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0], "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, "page_num": "page_6", "page_size": [612.0, 792.0]}]}
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BRAID GROUP REPRESENTATIONS
7
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.
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 , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.
Remark 3.6. For there is a friendship graph which is neither totally disconnected nor connected:
![Image]()
![Image]()
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.
Now consider the case when the friendship graph is totally discon- nected (that is, statement of theorem 3.4 holds).
- Lemma 3.7. If and are neighbors and not friends then: (a) ; . If , then and .
Proof. (a). By lemma 3.1, and are not true friends, so
$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$
|
<div class="pdf-page">
<p>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.</p>
<p>Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors.</p>
<p>Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph.</p>
<p>Corollary 3.5. For , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.</p>
<p>Remark 3.6. For there is a friendship graph which is neither totally disconnected nor connected:</p>
<p>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.</p>
<p>Now consider the case when the friendship graph is totally discon- nected (that is, statement of theorem 3.4 holds).</p>
<ul>
<li>Lemma 3.7. If and are neighbors and not friends then: (a) ; . If , then and .</li>
</ul>
<p>Proof. (a). By lemma 3.1, and are not true friends, so</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="371" data-y="116" data-width="279" data-height="14" style="opacity: 0.5;">BRAID GROUP REPRESENTATIONS</div>
<div class="pdf-discarded" data-x="801" data-y="117" data-width="12" data-height="12" style="opacity: 0.5;">7</div>
<p class="pdf-text" data-x="209" data-y="142" data-width="605" data-height="128">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.</p>
<p class="pdf-text" data-x="207" data-y="299" data-width="606" data-height="38">Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors.</p>
<p class="pdf-text" data-x="227" data-y="349" data-width="470" data-height="37">Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph.</p>
<p class="pdf-text" data-x="209" data-y="415" data-width="604" data-height="38">Corollary 3.5. For , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.</p>
<p class="pdf-text" data-x="207" data-y="477" data-width="606" data-height="37">Remark 3.6. For there is a friendship graph which is neither totally disconnected nor connected:</p>
<p class="pdf-text" data-x="207" data-y="665" data-width="607" data-height="39">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.</p>
<p class="pdf-text" data-x="207" data-y="720" data-width="606" data-height="38">Now consider the case when the friendship graph is totally discon- nected (that is, statement of theorem 3.4 holds).</p>
<ul class="pdf-list" data-x="207" data-y="769" data-width="606" data-height="75">
<li>Lemma 3.7. If and are neighbors and not friends then: (a) ; . If , then and .</li>
</ul>
<p class="pdf-text" data-x="225" data-y="853" data-width="527" data-height="21">Proof. (a). By lemma 3.1, and are not true friends, so</p>
</div>
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"Proof. Suppose neither (a) nor (b) holds. Since the graph is not",
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[{"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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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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "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, "page_num": "page_7", "page_size": [612.0, 792.0], "bbox_fs": [137, 673, 420, 700]}]}
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I.SYSOEVA
8
Multiplying the left hand side on the right by and the right hand side on the left by gives
$$
A B+A^{2}B+A B A B=0=A B+A B^{2}+A B A B.
$$
Thus, ; by a symmetric argument .
$$
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.
$$
Thus, .
Theorem 3.8. Let , ) be an irreducible rep- resentation, whose associated friendship graph is totally disconnected. Then .
Proof. If , is a trivial representation and
If , choose an eigenvalue for and a non-zero vector
$$
x_{1}\in I m(A_{1})\cap K e r(A_{1}-\lambda I).
$$
Set By induction and lemma 3.7 (b) .
Let . Then by lemma 3.7 (b) and the fact that , if and are not neighbors,
$$
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 is invariant under . Hence , since is irreducible.
Corollary 3.9. Let be irreducible, where , .
Then the associated friendship graph is connected.
Proof. By corollary 3.5 the friendship graph of is either totally disconnected or connected. By theorem 3.8 it is not disconnected.
Corollary 3.10. Let be irreducible, where , . Suppose , where rank .
- Then . In particular, for , , where .
|
<div class="pdf-page">
<p>Multiplying the left hand side on the right by and the right hand side on the left by gives</p>
<p>Thus, ; by a symmetric argument .</p>
<p>and</p>
<p>Thus, .</p>
<p>Theorem 3.8. Let , ) be an irreducible rep- resentation, whose associated friendship graph is totally disconnected. Then .</p>
<p>Proof. If , is a trivial representation and</p>
<p>If , choose an eigenvalue for and a non-zero vector</p>
<p>Set By induction and lemma 3.7 (b) .</p>
<p>Let . Then by lemma 3.7 (b) and the fact that , if and are not neighbors,</p>
<p>and</p>
<p>Thus is invariant under . Hence , since is irreducible.</p>
<p>Corollary 3.9. Let be irreducible, where , .</p>
<p>Then the associated friendship graph is connected.</p>
<p>Proof. By corollary 3.5 the friendship graph of is either totally disconnected or connected. By theorem 3.8 it is not disconnected.</p>
<p>Corollary 3.10. Let be irreducible, where , . Suppose , where rank .</p>
<ul>
<li>Then . In particular, for , , where .</li>
</ul>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="465" data-y="116" data-width="85" data-height="14" style="opacity: 0.5;">I.SYSOEVA</div>
<div class="pdf-discarded" data-x="207" data-y="117" data-width="13" data-height="12" style="opacity: 0.5;">8</div>
<p class="pdf-text" data-x="205" data-y="142" data-width="609" data-height="36">Multiplying the left hand side on the right by and the right hand side on the left by gives</p>
<p class="pdf-text" data-x="209" data-y="208" data-width="510" data-height="19">Thus, ; by a symmetric argument .</p>
<p class="pdf-text" data-x="209" data-y="277" data-width="36" data-height="16">and</p>
<p class="pdf-text" data-x="207" data-y="324" data-width="219" data-height="19">Thus, .</p>
<p class="pdf-text" data-x="209" data-y="368" data-width="604" data-height="54">Theorem 3.8. Let , ) be an irreducible rep- resentation, whose associated friendship graph is totally disconnected. Then .</p>
<p class="pdf-text" data-x="230" data-y="431" data-width="481" data-height="17">Proof. If , is a trivial representation and</p>
<p class="pdf-text" data-x="225" data-y="451" data-width="544" data-height="17">If , choose an eigenvalue for and a non-zero vector</p>
<p class="pdf-text" data-x="207" data-y="497" data-width="653" data-height="36">Set By induction and lemma 3.7 (b) .</p>
<p class="pdf-text" data-x="209" data-y="535" data-width="605" data-height="35">Let . Then by lemma 3.7 (b) and the fact that , if and are not neighbors,</p>
<p class="pdf-text" data-x="209" data-y="643" data-width="36" data-height="17">and</p>
<p class="pdf-text" data-x="207" data-y="682" data-width="609" data-height="36">Thus is invariant under . Hence , since is irreducible.</p>
<p class="pdf-text" data-x="207" data-y="725" data-width="617" data-height="37">Corollary 3.9. Let be irreducible, where , .</p>
<p class="pdf-text" data-x="234" data-y="762" data-width="426" data-height="18">Then the associated friendship graph is connected.</p>
<p class="pdf-text" data-x="207" data-y="787" data-width="606" data-height="37">Proof. By corollary 3.5 the friendship graph of is either totally disconnected or connected. By theorem 3.8 it is not disconnected.</p>
<p class="pdf-text" data-x="207" data-y="831" data-width="606" data-height="37">Corollary 3.10. Let be irreducible, where , . Suppose , where rank .</p>
<ul class="pdf-list" data-x="227" data-y="868" data-width="479" data-height="38">
<li>Then . In particular, for , , where .</li>
</ul>
</div>
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"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 .",
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"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",
"",
"",
"",
"Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following",
"Theorem 3.11. Let be irreducible, where , . Suppose , where rank .",
"Then and one of the following holds.",
"(a) The full friendship graph has an edge between and for all .",
"(b) The full friendship graph has an edge between and whenever and 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",
"",
"where , and",
"",
"Theorem 4.1. Let be an irreducible representation, where and . Let .",
"Then for ; that is the friendship graph of contains the chain graph.",
"Proof. Suppose not. Then by Theorem 0 whenever and are not neighbors. Consider",
"",
"Since , .",
"For , let be, respectively, nonzero elements of and . Since ,",
"BRAID GROUP REPRESENTATIONS",
"9"
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[{"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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{"preproc_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}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_8", "page_size": [612.0, 792.0], "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, "page_num": "page_8", "page_size": [612.0, 792.0], "bbox_fs": [126, 673, 487, 702]}]}
|
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BRAID GROUP REPRESENTATIONS
9
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
$$
\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.
$$
Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following
Theorem 3.11. Let be irreducible, where , . Suppose , where rank .
Then and one of the following holds.
(a) The full friendship graph has an edge between and for all .
(b) The full friendship graph has an edge between and whenever and are not neighbors.
# 4. For corank 2 the friendship graph is a chai
In this section, we assume throughout that we have an irreducible representation
$$
\rho:B_{n}\to G L_{r}(\mathbb{C}),
$$
where , and
$$
\rho(\sigma_{i})=1+A_{i},\;\;r a n k(A_{i})=2,\;\;1\leq i\leq n-1.
$$
Theorem 4.1. Let be an irreducible representation, where and . Let .
Then for ; that is the friendship graph of contains the chain graph.
Proof. Suppose not. Then by Theorem 0 whenever and are not neighbors. Consider
$$
U=I m(A_{1})+I m(A_{2})+I m(A_{3}).
$$
Since , .
For , let be, respectively, nonzero elements of and . Since ,
|
<div class="pdf-page">
<p>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</p>
<p>Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following</p>
<p>Theorem 3.11. Let be irreducible, where , . Suppose , where rank .</p>
<p>Then and one of the following holds.</p>
<p>(a) The full friendship graph has an edge between and for all .</p>
<p>(b) The full friendship graph has an edge between and whenever and are not neighbors.</p>
<h1>4. For corank 2 the friendship graph is a chai</h1>
<p>In this section, we assume throughout that we have an irreducible representation</p>
<p>where , and</p>
<p>Theorem 4.1. Let be an irreducible representation, where and . Let .</p>
<p>Then for ; that is the friendship graph of contains the chain graph.</p>
<p>Proof. Suppose not. Then by Theorem 0 whenever and are not neighbors. Consider</p>
<p>Since , .</p>
<p>For , let be, respectively, nonzero elements of and . Since ,</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="369" data-y="115" data-width="281" data-height="15" style="opacity: 0.5;">BRAID GROUP REPRESENTATIONS</div>
<div class="pdf-discarded" data-x="799" data-y="117" data-width="14" data-height="12" style="opacity: 0.5;">9</div>
<p class="pdf-text" data-x="205" data-y="142" data-width="608" data-height="75">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</p>
<p class="pdf-text" data-x="207" data-y="367" data-width="607" data-height="37">Combining Theorem 3.4 and Corollaries 3.9 and 3.10, we get the following</p>
<p class="pdf-text" data-x="207" data-y="412" data-width="606" data-height="37">Theorem 3.11. Let be irreducible, where , . Suppose , where rank .</p>
<p class="pdf-text" data-x="229" data-y="449" data-width="370" data-height="17">Then and one of the following holds.</p>
<p class="pdf-text" data-x="209" data-y="468" data-width="605" data-height="34">(a) The full friendship graph has an edge between and for all .</p>
<p class="pdf-text" data-x="210" data-y="504" data-width="604" data-height="36">(b) The full friendship graph has an edge between and whenever and are not neighbors.</p>
<h1 class="pdf-title" data-x="262" data-y="554" data-width="480" data-height="18">4. For corank 2 the friendship graph is a chai</h1>
<p class="pdf-text" data-x="209" data-y="581" data-width="605" data-height="35">In this section, we assume throughout that we have an irreducible representation</p>
<p class="pdf-text" data-x="209" data-y="641" data-width="152" data-height="18">where , and</p>
<p class="pdf-text" data-x="207" data-y="699" data-width="606" data-height="36">Theorem 4.1. Let be an irreducible representation, where and . Let .</p>
<p class="pdf-text" data-x="207" data-y="736" data-width="606" data-height="37">Then for ; that is the friendship graph of contains the chain graph.</p>
<p class="pdf-text" data-x="207" data-y="780" data-width="606" data-height="38">Proof. Suppose not. Then by Theorem 0 whenever and are not neighbors. Consider</p>
<p class="pdf-text" data-x="207" data-y="849" data-width="348" data-height="19">Since , .</p>
<p class="pdf-text" data-x="209" data-y="868" data-width="605" data-height="38">For , let be, respectively, nonzero elements of and . Since ,</p>
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"Proof. By corollary 3.9, the friendship graph of the representa-",
"tion 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",
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"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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"U=I m(A_{1})+I m(A_{2})+I m(A_{3})."
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"and are linearly independent, so they are a basis for , and . Thus",
"",
"which is invariant under . Thus , by the irreducibility of , a contradiction with .",
"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:",
"",
"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).",
"Lemma 4.3. Let be an irreducible representation, where , , and . Suppose that the associated friendship graph contains the chain.",
"Then and the associated friendship graph is the chain (that is, the only edges are between neighbors).",
"Proof. By corollary 3.10, . Consider the full friendship graph of . Then",
"",
"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.",
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[{"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$$", "text_format": "latex", "page_idx": 9}, {"type": "text", "text": "which is invariant under $\\rho(B_{n})$ . Thus $r\\leq5$ , by the irreducibility of $\\rho$ , a contradiction with $r\\geq n\\geq6$ . ", "page_idx": 9}, {"type": "text", "text": "Remark 4.2. For $n\\,=\\,5$ and $\\rho$ satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case: ", "page_idx": 9}, {"type": "image", "img_path": "images/37aca974d99beb835af9d8617adf99dee9ba68cb20de9282570299f927538ced.jpg", "img_caption": [], "img_footnote": [], "page_idx": 9}, {"type": "text", "text": "By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to $B_{5}$ of the Jones\u2019 representation (see [3], p. 296). ", "page_idx": 9}, {"type": "text", "text": "Lemma 4.3. Let $\\rho:B_{n}\\,\\to\\,G L_{r}(\\mathbb{C})$ be an irreducible representation, where $r\\,\\geq\\,n$ , $n\\,\\geq\\,5$ , and $r a n k(A_{1})\\,=\\,2$ . Suppose that the associated friendship graph contains the chain. ", "page_idx": 9}, {"type": "text", "text": "Then $r=n$ and the associated friendship graph is the chain (that is, the only edges are between neighbors). ", "page_idx": 9}, {"type": "text", "text": "Proof. By corollary 3.10, $r=n$ . Consider the full friendship graph of $\\rho$ . Then ", "page_idx": 9}, {"type": "equation", "text": "$$\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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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\u2019 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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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\u2019 representation (see [3], p. 296).", "type": "text"}], "index": 12}], "index": 11, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_9", "page_size": [612.0, 792.0], "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, "page_num": "page_9", "page_size": [612.0, 792.0], "bbox_fs": [126, 659, 486, 703]}]}
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|
I.SYSOEVA
10
and are linearly independent, so they are a basis for , and . Thus
$$
U=I m(A_{1})+I m(A_{2})+\cdot\cdot\cdot+I m(A_{n-1}),
$$
which is invariant under . Thus , by the irreducibility of , a contradiction with .
Remark 4.2. For and satisfying the hypothesis of theorem 4.1 there are two possible friendship graphs: 1) all neighbors are friends and 2) an exceptional case:
![Image]()
![Image]()
By [5], Theorem 7.1, part 2, every irreducible representation with the above friendship graph is equivalent to the restriction to of the Jones’ representation (see [3], p. 296).
Lemma 4.3. Let be an irreducible representation, where , , and . Suppose that the associated friendship graph contains the chain.
Then and the associated friendship graph is the chain (that is, the only edges are between neighbors).
Proof. By corollary 3.10, . Consider the full friendship graph of . Then
$$
I m(A_{i})\cap I m(A_{i+1})\neq\{0\}
$$
for any where indices are taken modulo . If is two-dimensional, then , and is a two- dimensional invariant subspace, contradicting the irreducibility of . Hence are one-dimensional.
For any , , , we have that
$$
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 . Moreover, because is invertible.
Choose to be a basis vector for . Define for . Then is a basis vector for .
|
<div class="pdf-page">
<p>and are linearly independent, so they are a basis for , and . Thus</p>
<p>which is invariant under . Thus , by the irreducibility of , a contradiction with .</p>
<p>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:</p>
<p>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).</p>
<p>Lemma 4.3. Let be an irreducible representation, where , , and . Suppose that the associated friendship graph contains the chain.</p>
<p>Then and the associated friendship graph is the chain (that is, the only edges are between neighbors).</p>
<p>Proof. By corollary 3.10, . Consider the full friendship graph of . Then</p>
<p>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.</p>
<p>For any , , , we have that</p>
<p>for . Moreover, because is invertible.</p>
<p>Choose to be a basis vector for . Define for . Then is a basis vector for .</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="465" data-y="116" data-width="85" data-height="14" style="opacity: 0.5;">I.SYSOEVA</div>
<div class="pdf-discarded" data-x="210" data-y="117" data-width="19" data-height="12" style="opacity: 0.5;">10</div>
<p class="pdf-text" data-x="207" data-y="142" data-width="606" data-height="37">and are linearly independent, so they are a basis for , and . Thus</p>
<p class="pdf-text" data-x="207" data-y="213" data-width="606" data-height="37">which is invariant under . Thus , by the irreducibility of , a contradiction with .</p>
<p class="pdf-text" data-x="207" data-y="258" data-width="606" data-height="56">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:</p>
<p class="pdf-text" data-x="207" data-y="471" data-width="606" data-height="56">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).</p>
<p class="pdf-text" data-x="207" data-y="544" data-width="606" data-height="55">Lemma 4.3. Let be an irreducible representation, where , , and . Suppose that the associated friendship graph contains the chain.</p>
<p class="pdf-text" data-x="207" data-y="601" data-width="604" data-height="36">Then and the associated friendship graph is the chain (that is, the only edges are between neighbors).</p>
<p class="pdf-text" data-x="207" data-y="646" data-width="606" data-height="36">Proof. By corollary 3.10, . Consider the full friendship graph of . Then</p>
<p class="pdf-text" data-x="207" data-y="707" data-width="606" data-height="72">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.</p>
<p class="pdf-text" data-x="230" data-y="780" data-width="435" data-height="19">For any , , , we have that</p>
<p class="pdf-text" data-x="207" data-y="831" data-width="482" data-height="19">for . Moreover, because is invertible.</p>
<p class="pdf-text" data-x="207" data-y="850" data-width="607" data-height="56">Choose to be a basis vector for . Define for . Then is a basis vector for .</p>
</div>
|
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},
$$
we have:
1) $A_{i}A_{j}=A_{j}A_{i}$ , for $|i-j|\geq2$ ; $\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\end{array}$ for all $i=0,1,\dotsc,n-1$ , where indices are taken modulo $n$ .
Proof. This follows easily from the relations on the generators of $B_{n}$ .
# 3. The friendship graph.
In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of $B_{n}$ . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself.
We assume throughout this section that we have a representation
$$
\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;
$$
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"content": [
"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 two-",
"dimensional 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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"U=I m(A_{1})+I m(A_{2})+\\cdot\\cdot\\cdot+I m(A_{n-1}),",
"I m(A_{i})\\cap I m(A_{i+1})\\neq\\{0\\}",
"T x=T A_{i}y=T A_{i}T^{-1}(T y)=A_{i+1}(T y)\\in I m(A_{i+1})"
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"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",
"",
"and",
"",
"So, if then . But this means that is an invariant subspace and the representation is not irreducible.",
"So, if the representation is irreducible, then for any , . From this follows that for any",
"",
"and the vectors form a basis of . Then for any two non-neighbors and",
"",
"Now, we have the following",
"Theorem 4.4. Let be irreducible, where . Suppose that for any generator , , where .",
"1) If , then and has a friendship graph which is a chain.",
"2) If , then and either has a friendship graph which is a chain or has the exceptional friendship graph (see Remark 4.2).",
"3) If , then either and has a friendship graph which is a chain; or has one of the following exceptional friendship graphs:",
"",
"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 .",
"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",
"BRAID GROUP REPRESENTATIONS",
"11"
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[{"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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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": ". 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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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_10", "page_size": [612.0, 792.0], "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, "page_num": "page_10", "page_size": [612.0, 792.0], "bbox_fs": [126, 674, 487, 702]}]}
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|
BRAID GROUP REPRESENTATIONS
11
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
$$
A_{i}A_{j}=A_{j}A_{i}
$$
and
$$
A_{i}A_{j+1}=A_{j+1}A_{i}.
$$
So, if then . But this means that is an invariant subspace and the representation is not irreducible.
So, if the representation is irreducible, then for any , . From this follows that for any
$$
I m(A_{i})=s p a n\{x_{i-1},x_{i}\}
$$
and the vectors form a basis of . Then for any two non-neighbors and
$$
I m(A_{i})\cap I m(A_{j})=\{0\}.
$$
Now, we have the following
- Theorem 4.4. Let be irreducible, where . Suppose that for any generator , , where .
1) If , then and has a friendship graph which is a chain.
2) If , then and either has a friendship graph which is a chain or has the exceptional friendship graph (see Remark 4.2).
3) If , then either and has a friendship graph which is a chain; or has one of the following exceptional friendship graphs:
![Image]()
![Image]()
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 .
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
|
<div class="pdf-page">
<p>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</p>
<p>and</p>
<p>So, if then . But this means that is an invariant subspace and the representation is not irreducible.</p>
<p>So, if the representation is irreducible, then for any , . From this follows that for any</p>
<p>and the vectors form a basis of . Then for any two non-neighbors and</p>
<p>Now, we have the following</p>
<ul>
<li>Theorem 4.4. Let be irreducible, where . Suppose that for any generator , , where .</li>
</ul>
<p>1) If , then and has a friendship graph which is a chain.</p>
<p>2) If , then and either has a friendship graph which is a chain or has the exceptional friendship graph (see Remark 4.2).</p>
<p>3) If , then either and has a friendship graph which is a chain; or has one of the following exceptional friendship graphs:</p>
<p>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 .</p>
<p>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</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="369" data-y="115" data-width="281" data-height="15" style="opacity: 0.5;">BRAID GROUP REPRESENTATIONS</div>
<div class="pdf-discarded" data-x="794" data-y="116" data-width="17" data-height="13" style="opacity: 0.5;">11</div>
<p class="pdf-text" data-x="207" data-y="142" data-width="607" data-height="73">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</p>
<p class="pdf-text" data-x="209" data-y="250" data-width="36" data-height="18">and</p>
<p class="pdf-text" data-x="207" data-y="296" data-width="607" data-height="53">So, if then . But this means that is an invariant subspace and the representation is not irreducible.</p>
<p class="pdf-text" data-x="207" data-y="350" data-width="609" data-height="36">So, if the representation is irreducible, then for any , . From this follows that for any</p>
<p class="pdf-text" data-x="207" data-y="422" data-width="606" data-height="38">and the vectors form a basis of . Then for any two non-neighbors and</p>
<p class="pdf-text" data-x="227" data-y="495" data-width="241" data-height="18">Now, we have the following</p>
<ul class="pdf-list" data-x="207" data-y="523" data-width="606" data-height="36">
<li>Theorem 4.4. Let be irreducible, where . Suppose that for any generator , , where .</li>
</ul>
<p class="pdf-text" data-x="209" data-y="561" data-width="604" data-height="33">1) If , then and has a friendship graph which is a chain.</p>
<p class="pdf-text" data-x="210" data-y="596" data-width="604" data-height="36">2) If , then and either has a friendship graph which is a chain or has the exceptional friendship graph (see Remark 4.2).</p>
<p class="pdf-text" data-x="210" data-y="632" data-width="604" data-height="37">3) If , then either and has a friendship graph which is a chain; or has one of the following exceptional friendship graphs:</p>
<p class="pdf-text" data-x="205" data-y="814" data-width="608" data-height="53">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 .</p>
<p class="pdf-text" data-x="207" data-y="868" data-width="607" data-height="37">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</p>
</div>
|
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\}.
$$
Definition 3.2. The full friendship graph (associated with the representation $\rho:B_{n}\to G L_{n}(\mathbb{C})$ ) is the simple-edged graph with n vertices $A_{0},A_{1},\ldots,A_{n-1}$ and an edge joining $A_{i}$ and $A_{j}$ $\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}^{\mathrm{\textit{\cdot}}}\mathrm{\Delta}j,$ ) if and only if $A_{i}$ and $A_{j}$ are friends.
The friendship graph is the subgraph with vertices $A_{1},\dotsc,A_{n-1}$ obtained from the full friendship graph by deleting $A_{0}$ and all edges incident to it.
Our main interest is the friendship graph, but it is convenient to introduce the full friendship graph as a tool, because of the following lemma.
Lemma 3.2. There is an edge between $A_{i}$ and $A_{j}$ in the full friendship graph if and only if there is an edge between $A_{i+k}$ and $A_{j+k}$ where indices are taken modulo $n$ . In other words, $\mathbb{Z}_{n}$ acts on the full friendship graph by permuting the vertices cyclically.
Proof. This follows immediately from the fact that conjugation by $T=\rho(\tau)=\rho(\sigma_{1}\dots\sigma_{n-1})$ permutes $\sigma_{0},\sigma_{1},\ldots,\sigma_{n-1}$ cyclically (Lemma 2.1).
Lemma 3.3 (Lemma about friends). Let $A$ and $B$ be neighbors which are not friends. If $C$ is not a neighbor of $A$ and $C$ is a friend of $B$ then $C$ is a true friend of $A$ .
Proof. By lemma 3.1, $A$ and $B$ are true not friends, because they are not friends, that is
$$
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
$$
because $B z\neq0$ and $(1+B)$ is invertible.
So, $A C=C A\neq0$ ; that is, $A$ and $C$ are true friends.
Theorem 3.4. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be a representation. Then one of the following holds.
(a) The full friendship graph is totally disconnected (no friends at all).
(b) The full friendship graph has an edge between $A_{i}$ and $A_{i+1}$ for all $i$ .
(c) The full friendship graph has an edge between $A_{i}$ and $A_{j}$ whenever $A_{i}$ and $A_{j}$ are not neighbors.
Proof. Suppose neither (a) nor (b) holds. Since the graph is not totally disconnected, there is an edge joining some vertices $B$ and $C$ . Since (b) does not hold, no neighbors are joined by an edge. Lemma 3.3 implies that there is an edge between $C$ and any neighbor of $B$ which is not a neighbor of $C$ . It follows inductively that there is an edge joining $C$ to every vertex which is not a neighbor of $C$ . Then (c) holds, because the full friendship graph is a $\mathbb{Z}_{n}$ -graph.
Definition 3.3. The friendship graph (the full friendship graph) is a chain, if the only edges are between neighbors.
Case (b) of the above theorem can be restated as (b) The full friendship graph contains the chain graph.
Corollary 3.5. For $n\neq4$ , the friendship graph and the full friendship graph are either totally disconnected (no edges) or connected.
Remark 3.6. For $n=4$ there is a friendship graph which is neither totally disconnected nor connected:
By [5], Lemmas 6.2 and 6.3, every representation of $B_{4}$ of corank 2 and dimension at least 4, which has this friendship graph, is reducible.
Now consider the case when the friendship graph is totally disconnected (that is, statement $(a)$ of theorem 3.4 holds).
Lemma 3.7. If $A$ and $B$ are neighbors and not friends then: (a) $A^{2}B=A B^{2}$ ; $B A^{2}=B^{2}A$ . $(b)$ If $x\in I m(A)\cap K e r(A-\lambda I)$ , then $B(B x)=\lambda(B x)$ and $A B x=$
$-(1+\lambda)x$ .
Proof. (a). By lemma 3.1, $A$ and $B$ are not true friends, so
$$
A+A^{2}+A B A=B+B^{2}+B A B=0.
$$
Multiplying the left hand side on the right by $B$ and the right hand side on the left by $A$ gives
$$
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.
$$
Thus $U$ is invariant under $B_{n}$ . Hence $r=d i m U\le n-1$ , since $\rho$ is irreducible.
Corollary 3.9. Let $\rho:B_{n}\to G L_{r}(\mathbb{C})$ be irreducible, where $r=d i m V\ge$ $n$ , $n\neq4$ .
Then the associated friendship graph is connected.
Proof. By corollary 3.5 the friendship graph of $\rho$ is either totally disconnected or connected. By theorem 3.8 it is not disconnected.
Corollary 3.10. Let $\rho~:~B_{n}~\to~G L_{r}(\mathbb{C})$ be irreducible, where $r=$ $d i m V\geq n$ , $n\neq4$ . Suppose $\rho(\sigma_{i})=1+A_{i}$ , where rank $:(A_{i})=k$ .
Then $r=d i m V\leq(n-1)(k-1)+1$ .
In particular, for $k=2$ , $r=d i m V=n$ , where $V=\mathbb{C}^{n}$ .
Proof. By corollary 3.9, the friendship graph of the representation is connected. Arrange the vertices of the graph in a sequence $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{n-1}}$ such that each term $A_{i_{j}}$ , $2\leq j\leq n-1$ , is a friend of one the terms $A_{i_{1}},A_{i_{2}},\ldots,A_{i_{j-1}}$ . Then
$$
\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}).
$$
Since $I m(A_{1})\cap I m(A_{3})\neq0$ , $d i m U\leq5$ .
For $i=4,\dots,n-1$ , let $a_{i},\ b_{i}$ be, respectively, nonzero elements of $I m(A_{1})\cap I m(A_{i})$ and $I m(A_{2})\cap I m(A_{i})$ . Since $I m(A_{1})\cap I m(A_{2})=0$ ,
$a_{i}$ and $b_{i}$ are linearly independent, so they are a basis for $I m(A_{i})$ , and $I m(A_{i})\subseteq I m(A_{1})+I m(A_{2})$ . Thus
$$
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})
$$
for $T=\rho(\tau)$ . Moreover, $T x\neq0$ because $T$ is invertible.
Choose $x_{1}~\neq~0$ to be a basis vector for $I m(A_{1})\cap I m(A_{2})$ . Define $x_{i+1}=T^{i}x_{1}$ for $1\leq i\leq n-1$ . Then $x_{i}$ is a basis vector for $I m(A_{i})\cap$ $I m(A_{i+1})$ .
If for some $i$ , $x_{i}$ is proportional to $x_{i+1}$ then, because a full friendship graph is a $\mathbb{Z}_{n}$ -graph, all the $x_{j}$ are proportional to $x_{1}$ . Then, because we have 5 or more vertices in the full friendship graph, for any $A_{i}$ there exists $j$ such that both $A_{j}$ and $A_{j+1}$ are not neighbors of $A_{i}$ . Then
$$
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$ .
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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 , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible graphs on 4 vertices.",
"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.",
"5. Representations whose friendship graph is a chain",
"Definition 5.1. The standard representation is the representa- tion",
"",
"defined by",
"",
"for , where is the identity matrix.",
"Theorem 5.1. Let be an irreducible representation, where . Suppose that , where , and the associated friendship graph of is a chain.",
"Then is equivalent to a specialization of the standard repre- sentation for some .",
"",
"Before proving the theorem, we will need the following technical lemma:",
"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 :",
"I.SYSOEVA",
"12"
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|
[{"type": "text", "text": "4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case. ", "page_idx": 11}, {"type": "text", "text": "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. ", "page_idx": 11}, {"type": "text", "text": "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. ", "page_idx": 11}, {"type": "text", "text": "5. Representations whose friendship graph is a chain ", "page_idx": 11}, {"type": "text", "text": "Definition 5.1. The standard representation is the representation ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\tau_{n}:B_{n}\\to G L_{n}(\\mathbb{Z}[t^{\\pm1}]\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "defined by ", "page_idx": 11}, {"type": "equation", "text": "$$\n\\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),\n$$", "text_format": "latex", "page_idx": 11}, {"type": "text", "text": "for $i=1,2,\\dots,n-1$ , where $I_{k}$ is the $k\\times k$ identity matrix. ", "page_idx": 11}, {"type": "text", "text": "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. ", "page_idx": 11}, {"type": "text", "text": "Then $\\rho$ is equivalent to a specialization $\\tau_{n}(u)$ of the standard representation for some $u\\in\\mathbb{C}^{*}$ . ", "page_idx": 11}, {"type": "image", "img_path": "images/46455dd8a7a1417e8b22d93b52df66f3bfc7847444aac8ea343f3f447329fc1e.jpg", "img_caption": [], "img_footnote": [], "page_idx": 11}, {"type": "text", "text": "Before proving the theorem, we will need the following technical lemma: ", "page_idx": 11}, {"type": "text", "text": "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$ : ", "page_idx": 11}]
|
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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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_11", "page_size": [612.0, 792.0], "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, "page_num": "page_11", "page_size": [612.0, 792.0], "bbox_fs": [125, 674, 487, 702]}]}
|
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I.SYSOEVA
12
4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case.
3) If , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible graphs on 4 vertices.
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.
5. Representations whose friendship graph is a chain
Definition 5.1. The standard representation is the representa- tion
$$
\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 , where is the identity matrix.
Theorem 5.1. Let be an irreducible representation, where . Suppose that , where , and the associated friendship graph of is a chain.
Then is equivalent to a specialization of the standard repre- sentation for some .
![Image]()
![Image]()
Before proving the theorem, we will need the following technical lemma:
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 :
|
<div class="pdf-page">
<p>4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case.</p>
<p>3) If , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible graphs on 4 vertices.</p>
<p>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.</p>
<p>5. Representations whose friendship graph is a chain</p>
<p>Definition 5.1. The standard representation is the representa- tion</p>
<p>defined by</p>
<p>for , where is the identity matrix.</p>
<p>Theorem 5.1. Let be an irreducible representation, where . Suppose that , where , and the associated friendship graph of is a chain.</p>
<p>Then is equivalent to a specialization of the standard repre- sentation for some .</p>
<p>Before proving the theorem, we will need the following technical lemma:</p>
<p>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 :</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="465" data-y="116" data-width="85" data-height="14" style="opacity: 0.5;">I.SYSOEVA</div>
<div class="pdf-discarded" data-x="210" data-y="116" data-width="17" data-height="13" style="opacity: 0.5;">12</div>
<p class="pdf-text" data-x="205" data-y="142" data-width="609" data-height="36">4.3, it has no other edges. If it does not contain a chain graph, we obtain the exceptional case.</p>
<p class="pdf-text" data-x="207" data-y="179" data-width="606" data-height="55">3) If , then by theorem 3.8 the friendship graph is not totally disconnected. Hence, we have only three possible graphs on 4 vertices.</p>
<p class="pdf-text" data-x="207" data-y="241" data-width="607" data-height="56">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.</p>
<p class="pdf-text" data-x="234" data-y="318" data-width="554" data-height="16">5. Representations whose friendship graph is a chain</p>
<p class="pdf-text" data-x="207" data-y="342" data-width="606" data-height="36">Definition 5.1. The standard representation is the representa- tion</p>
<p class="pdf-text" data-x="209" data-y="405" data-width="92" data-height="20">defined by</p>
<p class="pdf-text" data-x="207" data-y="571" data-width="519" data-height="21">for , where is the identity matrix.</p>
<p class="pdf-text" data-x="209" data-y="610" data-width="605" data-height="55">Theorem 5.1. Let be an irreducible representation, where . Suppose that , where , and the associated friendship graph of is a chain.</p>
<p class="pdf-text" data-x="210" data-y="667" data-width="601" data-height="35">Then is equivalent to a specialization of the standard repre- sentation for some .</p>
<p class="pdf-text" data-x="205" data-y="821" data-width="609" data-height="37">Before proving the theorem, we will need the following technical lemma:</p>
<p class="pdf-text" data-x="207" data-y="867" data-width="607" data-height="38">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 :</p>
</div>
|
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),
$$
for $i=1,2,\dots,n-1$ , where $I_{k}$ is the $k\times k$ identity matrix.
Theorem 5.1. Let $\rho:B_{n}\to G L_{n}(\mathbb{C})$ be an irreducible representation, where $n\geq4$ . Suppose that $\rho(\sigma_{1})=1+A_{1}$ , where $r a n k(A_{1})=2$ , and the associated friendship graph of $\rho$ is a chain.
Then $\rho$ is equivalent to a specialization $\tau_{n}(u)$ of the standard representation for some $u\in\mathbb{C}^{*}$ .
Before proving the theorem, we will need the following technical lemma:
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"4.3, it has no other edges. If it does not contain a chain graph, we",
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"3) If n=4 , then by theorem 3.8 the friendship graph is not totally",
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"Remark 4.5. It is proven in [5], Chapter 6, that any representation",
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"5. Representations whose friendship graph is a chain",
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"",
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"BRAID GROUP REPRESENTATIONS",
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[{"type": "text", "text": "Let $a~\\ne~0$ be such that s $\\L)a n\\{a\\}\\;=\\;I m(A)\\cap I m(B)$ , and let $b\\,=$ $(1+B)a$ . Then: ", "page_idx": 12}, {"type": "text", "text": "1) $s p a n\\{b\\}=I m(C)\\cap I m(B)$ . \n$\\mathcal{Q}_{g}$ ) $(1+B)b\\in s p a n\\{a\\}$ and $(1+B)b\\neq0$ . \n3) The vectors $a$ and $b$ are linearly independent. ", "page_idx": 12}, {"type": "text", "text": "Proof. First of all, notice that the vector $b$ is non-zero, because $1+B$ is invertible and $a\\ne0$ . ", "page_idx": 12}, {"type": "text", "text": "1) $b=(1+B)a\\in I m(B)$ , because $a\\in I m(B)$ . ", "page_idx": 12}, {"type": "text", "text": "$A$ and $C$ are not friends, that is $C A=0$ , so $C a=0$ . Let $a=B a_{1}$ . Then ", "page_idx": 12}, {"type": "equation", "text": "$$\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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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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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, "page_num": "page_12", "page_size": [612.0, 792.0], "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, "page_num": "page_12", "page_size": [612.0, 792.0]}, {"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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|
BRAID GROUP REPRESENTATIONS
13
Let be such that s , and let . Then:
- 1) . ) and . 3) The vectors and are linearly independent.
Proof. First of all, notice that the vector is non-zero, because is invertible and .
1) , because .
and are not friends, that is , so . Let . 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);
$$
that is, , and because is one- dimensional and ,
$$
s p a n\{b\}=I m(C)\cap I m(B).
$$
2) Clearly, .
Note, that , as by the above, and . Let . Then
$$
(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).
$$
3) , by part 1), and by the hypothesis of the lemma.
Proof of Theorem 5.1 We include the redundant generator , and indices are modulo . Consider , which is , , or dimensional. It is nonzero, because of the hypothesis that the friendship graph is a chain. It is not 2-dimensional, for then
$$
I m(A_{0})=I m(A_{1})=\cdots=I m(A_{n-1})
$$
would be a dimensional invariant subspace, contradicting the irre- ducibility of . Hence, is one-dimensional.
Let be a basis vector for . Let
$$
a_{1}=(1+A_{1})a_{0},\;\;a_{2}=(1+A_{2})a_{1},\;\;.\;.\;.\;,\;\;a_{n-1}=(1+A_{n-1})a_{n-2}.
$$
|
<div class="pdf-page">
<p>Let be such that s , and let . Then:</p>
<ul>
<li>1) . ) and . 3) The vectors and are linearly independent.</li>
</ul>
<p>Proof. First of all, notice that the vector is non-zero, because is invertible and .</p>
<p>1) , because .</p>
<p>and are not friends, that is , so . Let . Then</p>
<p>that is, , and because is one- dimensional and ,</p>
<p>2) Clearly, .</p>
<p>Note, that , as by the above, and . Let . Then</p>
<p>3) , by part 1), and by the hypothesis of the lemma.</p>
<p>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</p>
<p>would be a dimensional invariant subspace, contradicting the irre- ducibility of . Hence, is one-dimensional.</p>
<p>Let be a basis vector for . Let</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="369" data-y="116" data-width="281" data-height="14" style="opacity: 0.5;">BRAID GROUP REPRESENTATIONS</div>
<div class="pdf-discarded" data-x="794" data-y="116" data-width="19" data-height="13" style="opacity: 0.5;">13</div>
<p class="pdf-text" data-x="207" data-y="214" data-width="607" data-height="38">Let be such that s , and let . Then:</p>
<ul class="pdf-list" data-x="227" data-y="252" data-width="417" data-height="54">
<li>1) . ) and . 3) The vectors and are linearly independent.</li>
</ul>
<p class="pdf-text" data-x="210" data-y="316" data-width="603" data-height="36">Proof. First of all, notice that the vector is non-zero, because is invertible and .</p>
<p class="pdf-text" data-x="227" data-y="352" data-width="405" data-height="20">1) , because .</p>
<p class="pdf-text" data-x="207" data-y="372" data-width="604" data-height="35">and are not friends, that is , so . Let . Then</p>
<p class="pdf-text" data-x="207" data-y="473" data-width="604" data-height="36">that is, , and because is one- dimensional and ,</p>
<p class="pdf-text" data-x="227" data-y="546" data-width="261" data-height="19">2) Clearly, .</p>
<p class="pdf-text" data-x="205" data-y="566" data-width="609" data-height="36">Note, that , as by the above, and . Let . Then</p>
<p class="pdf-text" data-x="207" data-y="638" data-width="607" data-height="38">3) , by part 1), and by the hypothesis of the lemma.</p>
<p class="pdf-text" data-x="207" data-y="712" data-width="607" data-height="72">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</p>
<p class="pdf-text" data-x="207" data-y="822" data-width="607" data-height="36">would be a dimensional invariant subspace, contradicting the irre- ducibility of . Hence, is one-dimensional.</p>
<p class="pdf-text" data-x="225" data-y="859" data-width="442" data-height="18">Let be a basis vector for . Let</p>
</div>
|
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},
$$
we have:
1) $A_{i}A_{j}=A_{j}A_{i}$ , for $|i-j|\geq2$ ; $\begin{array}{r}{2)A_{i}+A_{i}^{2}+A_{i}A_{i+1}A_{i}=A_{i+1}+A_{i+1}^{2}+A_{i+1}A_{i}A_{i+1}}\end{array}$ for all $i=0,1,\dotsc,n-1$ , where indices are taken modulo $n$ .
Proof. This follows easily from the relations on the generators of $B_{n}$ .
# 3. The friendship graph.
In this section we define and prove some properties of the friendship graph which is a finite graph associated with a representation of $B_{n}$ . Our graphs are simple-edged, which means that there is at most one unoriented edge joining two vertices, and no edges joining a vertex to itself.
We assume throughout this section that we have a representation
$$
\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;
$$
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"\\mathcal{Q}_{g} ) (1+B)b\\in s p a n\\{a\\} and (1+B)b\\neq0 .",
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"the hypothesis of the lemma.",
"Proof of Theorem 5.1 We include the redundant generator \\sigma_{0} ,",
"and indices are modulo n . Consider I m(A_{i})\\cap I m(A_{i+1}) , which is \\boldsymbol{0} , 1 ,",
"or 2- dimensional. It is nonzero, because of the hypothesis that the",
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"By induction and lemma 5.2, part 1), is a basis vector for , for . By lemma 5.2, part 3), and are linearly independent. Thus is a basis for .",
"Since",
"",
"is invariant under and is an dimensional irreducible represen- tation, is a basis for .",
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"",
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"",
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"Now let us consider when the standard representation is irreducible.",
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"Proof. If then the vector is a fixed vector.",
"Lemma 5.4. If then is irreducible.",
"I.SYSOEVA",
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[{"type": "text", "text": "By induction and lemma 5.2, part 1), $a_{i}$ is a basis vector for $I m(A_{i})\\cap$ $I m(A_{i+1})$ , for $0\\leq\\,i\\leq n-1$ . By lemma 5.2, part 3), $a_{i}$ and $a_{i+1}$ are linearly independent. Thus $\\{a_{i},a_{i+1}\\}$ is a basis for $I m(A_{i})$ . ", "page_idx": 13}, {"type": "text", "text": "Since ", "page_idx": 13}, {"type": "equation", "text": "$$\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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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": ". 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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, "page_num": "page_13", "page_size": [612.0, 792.0], "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, "page_num": "page_13", "page_size": [612.0, 792.0], "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. 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|
I.SYSOEVA
14
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 .
Since
$$
s p a n\{a_{0},\ldots a_{n-1}\}=I m(A_{1})+\cdots+I m(A_{n-1})
$$
is invariant under and is an dimensional irreducible represen- tation, is a basis for .
We now wish to determine the action of on this basis.
Consider . If , then is not a neighbor of one of , (since ), say , and then , so , and
$$
\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}
$$
for .
By lemma 5.2, part 2),
$$
\rho(\sigma_{i})a_{i}=(1+A_{i})a_{i}=u_{i}a_{i-1},
$$
for , where .
By the above calculations the matrices of with respect to the basis are
$$
\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 , where is the identity matrix, and . Since are conjugate in , the are all equal, and we have the standard representation.
Now let us consider when the standard representation is irreducible.
Lemma 5.3. If then is reducible.
Proof. If then the vector is a fixed vector.
Lemma 5.4. If then is irreducible.
|
<div class="pdf-page">
<p>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 .</p>
<p>Since</p>
<p>is invariant under and is an dimensional irreducible represen- tation, is a basis for .</p>
<p>We now wish to determine the action of on this basis.</p>
<p>Consider . If , then is not a neighbor of one of , (since ), say , and then , so , and</p>
<p>By our construction</p>
<p>for .</p>
<p>By lemma 5.2, part 2),</p>
<p>for , where .</p>
<p>By the above calculations the matrices of with respect to the basis are</p>
<p>for , where is the identity matrix, and . Since are conjugate in , the are all equal, and we have the standard representation.</p>
<p>Now let us consider when the standard representation is irreducible.</p>
<p>Lemma 5.3. If then is reducible.</p>
<p>Proof. If then the vector is a fixed vector.</p>
<p>Lemma 5.4. If then is irreducible.</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="465" data-y="116" data-width="85" data-height="14" style="opacity: 0.5;">I.SYSOEVA</div>
<div class="pdf-discarded" data-x="209" data-y="117" data-width="18" data-height="12" style="opacity: 0.5;">14</div>
<p class="pdf-text" data-x="207" data-y="142" data-width="607" data-height="55">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 .</p>
<p class="pdf-text" data-x="229" data-y="197" data-width="50" data-height="16">Since</p>
<p class="pdf-text" data-x="205" data-y="246" data-width="606" data-height="35">is invariant under and is an dimensional irreducible represen- tation, is a basis for .</p>
<p class="pdf-text" data-x="207" data-y="283" data-width="606" data-height="33">We now wish to determine the action of on this basis.</p>
<p class="pdf-text" data-x="207" data-y="318" data-width="607" data-height="54">Consider . If , then is not a neighbor of one of , (since ), say , and then , so , and</p>
<p class="pdf-text" data-x="229" data-y="403" data-width="175" data-height="17">By our construction</p>
<p class="pdf-text" data-x="207" data-y="451" data-width="156" data-height="18">for .</p>
<p class="pdf-text" data-x="229" data-y="470" data-width="197" data-height="17">By lemma 5.2, part 2),</p>
<p class="pdf-text" data-x="207" data-y="518" data-width="289" data-height="18">for , where .</p>
<p class="pdf-text" data-x="207" data-y="537" data-width="606" data-height="37">By the above calculations the matrices of with respect to the basis are</p>
<p class="pdf-text" data-x="205" data-y="714" data-width="609" data-height="56">for , where is the identity matrix, and . Since are conjugate in , the are all equal, and we have the standard representation.</p>
<p class="pdf-text" data-x="225" data-y="787" data-width="586" data-height="19">Now let us consider when the standard representation is irreducible.</p>
<p class="pdf-text" data-x="207" data-y="814" data-width="398" data-height="19">Lemma 5.3. If then is reducible.</p>
<p class="pdf-text" data-x="227" data-y="840" data-width="586" data-height="21">Proof. If then the vector is a fixed vector.</p>
<p class="pdf-text" data-x="207" data-y="885" data-width="415" data-height="21">Lemma 5.4. If then is irreducible.</p>
</div>
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"Now let us consider when the standard representation is irreducible.",
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"Lemma 5.4. If u\\ne1 then \\tau_{n}(u) is irreducible."
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"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",
"",
"where and . By a direct calculation . Because the vector is a non-zero multiple of .",
"Now, we have the main result of this paper:",
"Theorem 5.5 (The Main Theorem). Let be an ir- reducible representation of for . Let , and let with .",
"Then and is equivalent to the following representation :",
"",
"",
"for , where is the identity matrix, and , . These representations are non-equivalent for different values of .",
"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 .",
"Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following",
"Corollary 5.6. Let be an irreducible representation of for . Let .",
"Then is equivalent to a specialization of the standard representation , for some .",
"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.",
"BRAID GROUP REPRESENTATIONS",
"15"
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[{"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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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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0], "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, "page_num": "page_14", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_14", "page_size": [612.0, 792.0], "bbox_fs": [131, 665, 486, 701]}]}
|
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|
BRAID GROUP REPRESENTATIONS
15
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
$$
H=A+A^{2}+A B A=B+B^{2}+B A B,
$$
where and . By a direct calculation . Because the vector is a non-zero multiple of .
Now, we have the main result of this paper:
Theorem 5.5 (The Main Theorem). Let be an ir- reducible representation of for . Let , and let with .
Then and is equivalent to the following representation :
$$
\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 , where is the identity matrix, and , . These representations are non-equivalent for different values of .
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 .
Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following
Corollary 5.6. Let be an irreducible representation of for . Let .
Then is equivalent to a specialization of the standard representation , for some .
# 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.
|
<div class="pdf-page">
<p>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</p>
<p>where and . By a direct calculation . Because the vector is a non-zero multiple of .</p>
<p>Now, we have the main result of this paper:</p>
<p>Theorem 5.5 (The Main Theorem). Let be an ir- reducible representation of for . Let , and let with .</p>
<p>Then and is equivalent to the following representation :</p>
<p>for , where is the identity matrix, and , . These representations are non-equivalent for different values of .</p>
<p>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 .</p>
<p>Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following</p>
<p>Corollary 5.6. Let be an irreducible representation of for . Let .</p>
<p>Then is equivalent to a specialization of the standard representation , for some .</p>
<h1>References</h1>
<p>[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>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="369" data-y="116" data-width="281" data-height="14" style="opacity: 0.5;">BRAID GROUP REPRESENTATIONS</div>
<div class="pdf-discarded" data-x="794" data-y="117" data-width="19" data-height="12" style="opacity: 0.5;">15</div>
<p class="pdf-text" data-x="207" data-y="142" data-width="607" data-height="72">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</p>
<p class="pdf-text" data-x="207" data-y="246" data-width="606" data-height="38">where and . By a direct calculation . Because the vector is a non-zero multiple of .</p>
<p class="pdf-text" data-x="227" data-y="302" data-width="380" data-height="18">Now, we have the main result of this paper:</p>
<p class="pdf-text" data-x="209" data-y="328" data-width="605" data-height="56">Theorem 5.5 (The Main Theorem). Let be an ir- reducible representation of for . Let , and let with .</p>
<p class="pdf-text" data-x="229" data-y="384" data-width="550" data-height="18">Then and is equivalent to the following representation :</p>
<p class="pdf-text" data-x="207" data-y="579" data-width="609" data-height="54">for , where is the identity matrix, and , . These representations are non-equivalent for different values of .</p>
<p class="pdf-text" data-x="207" data-y="642" data-width="606" data-height="56">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 .</p>
<p class="pdf-text" data-x="207" data-y="698" data-width="606" data-height="37">Combining Theorem 5.5 and the classification theorem of Formanek (see [3], Theorem 23), we get the following</p>
<p class="pdf-text" data-x="207" data-y="742" data-width="606" data-height="37">Corollary 5.6. Let be an irreducible representation of for . Let .</p>
<p class="pdf-text" data-x="207" data-y="780" data-width="606" data-height="35">Then is equivalent to a specialization of the standard representation , for some .</p>
<h1 class="pdf-title" data-x="451" data-y="831" data-width="121" data-height="18">References</h1>
<p class="pdf-text" data-x="214" data-y="855" data-width="599" data-height="51">[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>
</div>
|
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
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"(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 ir-",
"reducible 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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- [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 polynomi- als, 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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<li>[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 polynomi- als, 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.</li>
</ul>
<p>Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected]</p>
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<li>[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 polynomi- als, 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.</li>
</ul>
<p class="pdf-text" data-x="209" data-y="281" data-width="600" data-height="48">Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected]</p>
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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.
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"IMAGINARY QUADRATIC FIELDS WITH AND RANK",
"E. BENJAMIN, F. LEMMERMEYER, C. SNYDER",
"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.",
"1. Introduction",
"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",
"",
"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.",
"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.",
"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:",
"1991 Mathematics Subject Classification. Primary 11R37.",
"arXiv:math/0003244v1 [math.NT] 27 Mar 2000"
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[{"type": "text", "text": "IMAGINARY QUADRATIC FIELDS $k$ WITH $\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})$ AND RANK $\\mathrm{Cl}_{2}(k^{1})=2$ ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "E. BENJAMIN, F. LEMMERMEYER, C. SNYDER ", "page_idx": 0}, {"type": "text", "text": "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. ", "page_idx": 0}, {"type": "text", "text": "1. Introduction ", "text_level": 1, "page_idx": 0}, {"type": "text", "text": "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 ", "page_idx": 0}, {"type": "equation", "text": "$$\nk^{0}\\subseteq k^{1}\\subseteq k^{2}\\subseteq\\cdots\\subseteq k^{n}\\subseteq\\cdot\\cdot.\n$$", "text_format": "latex", "page_idx": 0}, {"type": "text", "text": "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. ", "page_idx": 0}, {"type": "text", "text": "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. ", "page_idx": 0}, {"type": "text", "text": "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: ", "page_idx": 0}, {"type": "text", "text": "1991 Mathematics Subject Classification. Primary 11R37. ", "page_idx": 0}]
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We give a characterization of those ", "type": "text"}, {"bbox": [361, 231, 365, 237], "score": 0.73, "content": "k", "type": "inline_equation", "height": 6, "width": 4}, {"bbox": [366, 230, 387, 240], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [388, 231, 449, 239], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 8, "width": 61}], "index": 4}, {"bbox": [161, 240, 294, 250], "spans": [{"bbox": [161, 240, 199, 250], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [199, 240, 227, 249], "score": 0.87, "content": "\\mathrm{Cl}_{2}(k^{1})", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [227, 240, 294, 250], "score": 1.0, "content": " has 2 generators.", "type": "text"}], "index": 5}], "index": 4}, {"type": "title", "bbox": [265, 282, 346, 294], "lines": [{"bbox": [264, 284, 348, 296], "spans": [{"bbox": [264, 284, 348, 296], "score": 1.0, "content": "1. 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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. 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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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BENJAMIN, F. LEMMERMEYER, C. SNYDER", "type": "text"}], "index": 2}], "index": 2, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [205, 192, 406, 201]}, {"type": "text", "bbox": [160, 218, 450, 248], "lines": [{"bbox": [162, 220, 450, 230], "spans": [{"bbox": [162, 220, 222, 230], "score": 1.0, "content": "Abstract. Let", "type": "text"}, {"bbox": [223, 222, 227, 227], "score": 0.88, "content": "k", "type": "inline_equation", "height": 5, "width": 4}, {"bbox": [228, 220, 396, 230], "score": 1.0, "content": " be an imaginary quadratic number field and ", "type": "text"}, {"bbox": [396, 220, 405, 227], "score": 0.89, "content": "k^{1}", "type": "inline_equation", "height": 7, "width": 9}, {"bbox": [405, 220, 450, 230], "score": 1.0, "content": " the Hilbert", "type": "text"}], "index": 3}, {"bbox": [161, 230, 449, 240], "spans": [{"bbox": [161, 230, 216, 240], "score": 1.0, "content": "2-class field of", "type": "text"}, {"bbox": [217, 231, 222, 237], "score": 0.85, "content": "k", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [222, 230, 360, 240], "score": 1.0, "content": ". We give a characterization of those ", "type": "text"}, {"bbox": [361, 231, 365, 237], "score": 0.73, "content": "k", "type": "inline_equation", "height": 6, "width": 4}, {"bbox": [366, 230, 387, 240], "score": 1.0, "content": " with ", "type": "text"}, {"bbox": [388, 231, 449, 239], "score": 0.92, "content": "\\mathrm{Cl}_{2}(k)\\simeq(2,2^{m})", "type": "inline_equation", "height": 8, "width": 61}], "index": 4}, {"bbox": [161, 240, 294, 250], "spans": [{"bbox": [161, 240, 199, 250], "score": 1.0, "content": "such that ", "type": "text"}, {"bbox": [199, 240, 227, 249], "score": 0.87, "content": "\\mathrm{Cl}_{2}(k^{1})", "type": "inline_equation", "height": 9, "width": 28}, {"bbox": [227, 240, 294, 250], "score": 1.0, "content": " has 2 generators.", "type": "text"}], "index": 5}], "index": 4, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [161, 220, 450, 250]}, {"type": "title", "bbox": [265, 282, 346, 294], "lines": [{"bbox": [264, 284, 348, 296], "spans": [{"bbox": [264, 284, 348, 296], "score": 1.0, "content": "1. Introduction", "type": "text"}], "index": 6}], "index": 6, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0]}, {"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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "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, "page_num": "page_0", "page_size": [612.0, 792.0], "bbox_fs": [138, 691, 353, 701]}]}
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# IMAGINARY QUADRATIC FIELDS WITH AND RANK
arXiv:math/0003244v1 [math.NT] 27 Mar 2000
E. BENJAMIN, F. LEMMERMEYER, C. SNYDER
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.
# 1. Introduction
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
$$
k^{0}\subseteq k^{1}\subseteq k^{2}\subseteq\cdots\subseteq k^{n}\subseteq\cdot\cdot.
$$
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.
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.
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:
1991 Mathematics Subject Classification. Primary 11R37.
|
<div class="pdf-page">
<h1>IMAGINARY QUADRATIC FIELDS WITH AND RANK</h1>
<p>E. BENJAMIN, F. LEMMERMEYER, C. SNYDER</p>
<p>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.</p>
<h1>1. Introduction</h1>
<p>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</p>
<p>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.</p>
<p>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.</p>
<p>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:</p>
<p>1991 Mathematics Subject Classification. Primary 11R37.</p>
</div>
|
<div class="pdf-page">
<h1 class="pdf-title" data-x="215" data-y="183" data-width="591" data-height="38">IMAGINARY QUADRATIC FIELDS WITH AND RANK</h1>
<div class="pdf-discarded" data-x="21" data-y="204" data-width="42" data-height="521" style="opacity: 0.5;">arXiv:math/0003244v1 [math.NT] 27 Mar 2000</div>
<p class="pdf-text" data-x="343" data-y="244" data-width="336" data-height="13">E. BENJAMIN, F. LEMMERMEYER, C. SNYDER</p>
<p class="pdf-text" data-x="267" data-y="281" data-width="485" data-height="39">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.</p>
<h1 class="pdf-title" data-x="443" data-y="364" data-width="135" data-height="16">1. Introduction</h1>
<p class="pdf-text" data-x="209" data-y="387" data-width="602" data-height="62">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</p>
<p class="pdf-text" data-x="209" data-y="475" data-width="604" data-height="47">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.</p>
<p class="pdf-text" data-x="210" data-y="523" data-width="603" data-height="261">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.</p>
<p class="pdf-text" data-x="210" data-y="786" data-width="603" data-height="76">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:</p>
<p class="pdf-text" data-x="229" data-y="890" data-width="361" data-height="15">1991 Mathematics Subject Classification. Primary 11R37.</p>
</div>
|
# 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:
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"IMAGINARY QUADRATIC FIELDS k WITH \\mathrm{Cl}_{2}(k)\\simeq(2,2^{m}) AND",
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"E. BENJAMIN, F. LEMMERMEYER, C. SNYDER",
"Abstract. Let k be an imaginary quadratic number field and k^{1} the Hilbert",
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"1. Introduction",
"Let k be an algebraic number field with \\mathrm{Cl_{2}}(k) , the Sylow 2-subgroup of its ideal",
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"At present there is no known decision procedure to determine whether or not",
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"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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"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.",
"The main result of the paper is that rank only occurs for fields of type B); more precisely, we prove the following",
"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 .",
"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]).",
"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 .",
"Table 1",
"",
"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].",
"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.",
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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\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</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 -22x\u00b2+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x\u00b2 - 171 x4 -30x\u00b2- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x\u00b2-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>\u22653 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\u2019s 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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"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": 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[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\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</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 -22x\u00b2+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x\u00b2 - 171 x4 -30x\u00b2- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x\u00b2-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>\u22653 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\u2019s 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\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</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 -22x\u00b2+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x\u00b2 - 171 x4 -30x\u00b2- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x\u00b2-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>\u22653 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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "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\u00b729 -31.8.5 -3\u00b713\u00b737 -11\u00b75\u00b729 -19\u00b75\u00b717 -7.8.29 -4\u00b75.89 -11\u00b75\u00b737 -3.13.53 -7.8\u00b737 -3\u00b713.61 -23\u00b78\u00b713</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 -22x\u00b2+261 x4 - 6x2 - 31 x4 - 86xc2 - 75 x4 + 26x2 + 1445 x4 + 26x\u00b2 - 171 x4 -30x\u00b2- 7 x4 + 6x2 + 89 4 - 54x2 - 11 x4 + x2 + 637 x4 + 34x2 - 7 x4 + 18x\u00b2-23</td><td>(4) (2) (2) (4) (2) (2) (4) (2,2) (2,2) (2,2) (2)</td><td>\u22653 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, "page_num": "page_1", "page_size": [612.0, 792.0]}, {"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\u2019s 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, "page_num": "page_1", "page_size": [612.0, 792.0], "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, "page_num": "page_1", "page_size": [612.0, 792.0], "bbox_fs": [125, 652, 487, 702]}]}
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2
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.
The main result of the paper is that rank only occurs for fields of type B); more precisely, we prove the following
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 .
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]).
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 .
Table 1
```
Table 1
```
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].
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.
|
<div class="pdf-page">
<p>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.</p>
<p>The main result of the paper is that rank only occurs for fields of type B); more precisely, we prove the following</p>
<p>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 .</p>
<p>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]).</p>
<p>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 .</p>
<h3>Table 1</h3>
<p>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].</p>
<p>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.</p>
</div>
|
<div class="pdf-page">
<div class="pdf-discarded" data-x="209" data-y="117" data-width="10" data-height="11" style="opacity: 0.5;">2</div>
<p class="pdf-text" data-x="209" data-y="164" data-width="604" data-height="76">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.</p>
<p class="pdf-text" data-x="209" data-y="241" data-width="605" data-height="31">The main result of the paper is that rank only occurs for fields of type B); more precisely, we prove the following</p>
<p class="pdf-text" data-x="209" data-y="280" data-width="605" data-height="94">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 .</p>
<p class="pdf-text" data-x="209" data-y="382" data-width="604" data-height="62">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]).</p>
<p class="pdf-text" data-x="207" data-y="444" data-width="606" data-height="47">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 .</p>
<caption class="pdf-table-caption" data-x="476" data-y="506" data-width="67" data-height="15">Table 1</caption>
<p class="pdf-text" data-x="209" data-y="779" data-width="604" data-height="62">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].</p>
<p class="pdf-text" data-x="207" data-y="841" data-width="606" data-height="62">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.</p>
</div>
|
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].
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"Table 1",
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