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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the equations and multidegrees of the biprojective variety that parametrizes pairs of symmetric matrices that are inverse to each other. As a consequence of our work, we provide an alternative proof for a result of \textit{L. Manivel} et al. [``Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming'', Preprint, \url{arXiv:2011.08791}] that settles a previous conjecture of \textit{B. Sturmfels} and \textit{C. Uhler} [Ann. Inst. Stat. Math. 62, No. 4, 603--638 (2010; Zbl 1440.62255)] regarding the polynomiality of maximum likelihood degree. symmetric matrix; multidegrees; maximum likelihood degree; rational map; Rees algebra; symmetric algebra Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Multiplicity theory and related topics, Algebraic statistics, Syzygies, resolutions, complexes and commutative rings, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Equations and multidegrees for inverse symmetric matrix pairs
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of the article is to give resolution of the singularities of a Schubert cycle in the Grassmannian \(G(r,n)\) by blowing-up certain sub- Schubert cycles.
Let \(K\) a field, \(r\) and \(n\) positive integers with \(r<n\), and \(G(r,n)\) the Grassmannian of \(r\)-dimensional subspaces of \(K^ n\). We call \(T\) the set of \(r\)-uples \((t) = (t_ 1, \dots, t_ r)\) of integers satisfying \(1 \leq t_ 1 < \cdots < t_ r \leq n\), and for each \((t)\) in \(T\) we introduce \(r(n - r)\) variables \(X^{(t)}_{i,j}\), \(1 \leq i \leq r\), \(1 \leq j \leq n\) and \(j \neq t_ 1, \dots, t_ r\), and we define an affine scheme \(U^{(t)} = \text{Spec} K[X^{(t)}_{i,j}]\). The scheme structure on the Grassmannian \(G(r,n)\) is obtained by gluing the affine sets \(U^{(t)}\). For any \((l)\) in \(T\), we define the Schubert cycle \(S_{(l)}\). For any \((t)\) and \((l)\) in \(T\), we define a closed subscheme \(S^{(t)}_{(l)}\) of \(U^{(t)}\) whose equations are the \((r - i + 1)\) minors of some matrix. The structure scheme on the Schubert cycle \(S_{(l)}\) is obtained by gluing the sets \(S^{(t)}_{(l)}\). The scheme \(S_{(l)}\) is not necessarily smooth and its dimension is equal to \(\sum^ r_{i = 1} (l_ i - i)\). We want to study the blowing-up of a Schubert cycle \(S_{(l)}\) in \(G(r,n)\), more precisely if we order the sub-Schubert cycles \(\sigma_ 1 \leq \cdots \leq \sigma_ k\) of \(S_{(l)}\) by dimension, we want to determine their strict transforms by this blowing-up. We put an order on \(T\) (inverse lexicographic): \((k) > (l)\) if and only if there exists \(s \geq 0\) so that \(k_ i = l_ i\) for \(i \leq s\) and \(k_{s + 1} < l_{s + 1}\).
Proposition: For any \((l)\) in \(T\), \(S^{(l)}_{(l)}\) is the first nontrivial Schubert cycle in \(S^{(l)}\), i.e. \(S^{(l)}_{(l)} \neq \emptyset\) and \(S^{(l)}_{(t)} = \emptyset\) for \((t) < (l)\).
Blowing-up \(S_{(l)}\), the strict transforms of the Schubert cycles in \(U^{(l)} = \text{Spec} K [X_{i,j}^{(l)}]\) are isomorphic to the Schubert cycles in \(U^{(l')}\), where \((l) > (l')\). More precisely, we can cover the blowing-up \(\widetilde {U^{(l)}}\) of \(U^{(l)}\) by affine chart \(B_{(l)} (s,t)\), with \(1 \leq s \leq r\), \(t \geq l_ s + 1\) and \(t \neq l_ 1, \dots, l_ r\). For any \((s,t)\), we can construct \((l')\) such that \((l) > (l')\) and the chart \(B_{(l)} (s,t)\) is isomorphic to \(U^{(l')}\), then the strict transform of a Schubert cycle in this chart is isomorphic to the Schubert cycle in \(U^{(l')}\). -- From the proposition, the author deduces the principal result:
Corollary: Let \(\sigma\) be a Schubert cycle in \(G(r,n)\), let \(\{\sigma_ i\}\) be the Schubert cycles contained in \(\sigma\) ordered so that \(\dim \sigma_ i \leq \dim \sigma_{i + 1}\). Then \(\sigma\) can be desingularized by blowing up \(\sigma_ 1\), then by blowing up the strict transform of \(\sigma_ 2\), and so on. resolution of the singularities of a Schubert cycle; Grassmannian; strict transforms of the Schubert cycles Boudhraa, Z.: Resolution of singularities of Schubert cycles. J. pure appl. Algebra 90, No. 2, 105-113 (1993) Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Resolution of singularities of Schubert cycles
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors provide a short explicit description for the (universal) Coble quartic. The universal Coble quartic is a bihomogeneous polynomial of bidegree (128,4) an equation in \(16 = 8 + 8\) variables, such that its partial derivatives with respect to the second variables give the defining equations for the universal family of Kummer varieties. Knowing the Coble quartic explicitly could thus be of further use in investigating the Hitchin integrable system and in particular the Hitchin connection for this case. The Coble quartic was recently determined completely in [\textit{Q. Ren} et al., Exp. Math. 22, No. 3, 327--362 (2013; Zbl 1312.14103)].
From the abstract: ``Our expression is in terms of products of theta constants with characteristics corresponding to Göpel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by \textit{A. B. Coble} [Algebraic geometry and theta functions. New York, NY: American Mathematical Society (1929; JFM 55.0808.02)] and further investigated by \textit{I. Dolgachev} and \textit{D. Ortland} [Point sets in projective spaces and theta functions. Paris: Société Mathématique de France; Centre National de la Recherche Scientifique (1988; Zbl 0685.14029)], and highlights the geometry and combinatorics of syzygetic octets of characteristics, the GIT quotient for 7 points in \(\mathbb{P}^2\), and the corresponding representations of \(\mathrm{Sp}(6,\mathbb{F}_2)\). One new ingredient is the relationship of Göpel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on \(\mathbb{P}^1\). universal Coble quartic; Kummer varieties; Goepel systems Grushevsky, [Grushevsky and Salvati Manni 12] S. and Salvati Manni, R. 2012. ''On the Coble Quartic''. arXiv:1212.1895v1 Theta functions and curves; Schottky problem, Jacobians, Prym varieties, Projective techniques in algebraic geometry On the Coble quartic
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One of the major open problems regarding the Macdonald polynomials \(J_{\mu}(x;q,t)\), \(\mu\) being a partition of \(n\), is the conjecture that, expanded in terms of certain modified Schur symmetric functions, the coefficients of \(J_{\mu}(x;q,t)\) are polynomials in the two parameters \(q,t\) with nonnegative integer coefficients. \textit{A. M. Garsia} and \textit{M. Haiman} [Proc. Natl. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)] introduced some bigraded \(S_n\)-module \({\mathbf H}_{\mu}\) and conjectured that the dimension of \({\mathbf H}_{\mu}\) is \(n!\) and this is known as the \(n!\) conjecture. It is known that the \(n!\) conjecture implies the Macdonald positivity conjecture. Many computer experimensts and some partial results support the validity of the \(n!\) conjecture. With the intention of proving the \(n!\) conjecture by induction \textit{F. Bergeron} and \textit{A. M. Garsia} [CRM Proc. Lect. Notes. 22, 1-52 (1999; Zbl 0947.20009)] formulated several conjectures concerning intersections of modules \({\mathbf M}_{\nu}\) for partitions lying below a given partition \(\mu\) of \(n+1\). The paper under review gives an explanation of these conjectures of Bergeron and Garsia in an algebraic-geometric setting, interpreting them in the context of the Hilbert scheme of \(n\) points in the plane. The author constructs a coherent sheaf \({\mathcal P}\) on the Hilbert scheme and shows that the \(n!\) conjecture is true if and only if \({\mathcal P}\) is a locally free sheaf, i.e. a vector bundle. Then the author studies the restriction of \({\mathcal P}\) to subvarieties isomorphic to \(\mathbb{P}^{k-1}\) contained in the Hilbert scheme and reduces the series of conjectures of Bergeron and Garsia to one conjecture on the structure of this vector bundle restricted to a projective space \(\mathbb{P}^k\) embedded in the Hilbert scheme. Finally the author reinterprets geometric statements combinatorially. Macdonald polynomials; partitions; Hilbert scheme; projective varieties; vector bundles Combinatorial aspects of representation theory, Symmetric functions and generalizations, Parametrization (Chow and Hilbert schemes) The \(n\)! conjecture and a vector bundle on the Hilbert scheme of \(n\) points in the plane
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple algebraic group over an algebraically closed field of arbitrary characteristic, and \(B\subset G\) a Borel subgroup. Recall that for every element \(w\) of the Weyl group of \(G\), the Schubert variety \(X(w)\subset G/B\) is defined as the Zariski closure of the Bruhat cell \(BwB/B\subset G/B\). A connection between Schubert varieties and toric varieties has been studied extensively, in particular, flat toric degenerations of Schubert varieties were constructed in [\textit{P. Caldero}, Transform. Groups 7, No. 1, 51--60 (2002; Zbl 1050.14040)].
The present paper describes all Schubert varieties that are already toric. Namely, the author proves that \(X(w)\) is a toric variety if and only if \(w\) is a product of pairwise distinct simple reflections (in particular, the dimension of \(X(w)\) does not exceed the rank of \(G\)). Schubert variety; toric variety Karuppuchamy, P., On Schubert varieties, Commun. Algebra, 41, 4, 1365-1368, (2013) Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies On Schubert varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe an isomorphism of categories conjectured by Kontsevich. If \(M\) and \(\widetilde{M}\) are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on \(M\) and a suitable version of Fukaya's category of Lagrangian submanifolds on \(\widetilde{M}.\) We prove this equivalence when \(M\) is an elliptic curve and \(\widetilde{M}\) is its dual curve, exhibiting the dictionary in detail. Polishchuk, A., Zaslow, E.: Categorical mirror symmetry in the elliptic curve. AMS/IP Stud. Adv. Math., vol. 23, pp. 275--295. AMS, Providence (1998) Calabi-Yau manifolds (algebro-geometric aspects), Elliptic curves Categorical mirror symmetry in the elliptic curve.
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A polynomial \(p(z,\bar{w})\) (\(z,w\in\mathbb{C}^n\)) is Hermitian when \(p(z,\bar{z})\) is real valued. In this case \(p(z,\bar{w})={\sum}c_{\alpha,\beta} z^\alpha\bar{w}^\beta\) for a Hermitian symmetric matrix \((c_{\alpha,\beta})\). Its signature \(\mathbf{s}(p)=(A,B)\) is uniquely determined and there are polynomial mappings \(f:\mathbb{C}^n\to\mathbb{C}^A\) and \(g:\mathbb{C}^n\to\mathbb{C}^B\) such that \(p(z,\bar{z})=\|f(z)\|^2-\|g(z)\|^2\). To investigate the relationship between the zero set of \(p\) and the matrix \((c_{\alpha,\beta})\) it is important to consider the Hermitian matrices arising from operations on the Hermitian polynomials. Here the authors investigate the signature of a product. They show that there are indefinite Hermitian polynomials \(p_1,p_2\) with \(\mathbf{s}(p_1p_2)\) matching any possible \((A,B)\), with the exception of the trivial cases \((0,0)\), \((0,1)\), \((1,0)\). They also show that one can get \(\mathbf{s}(p_1p_2)=(2,0)\) for \(p_1\) and \(p_2\) of arbitrarily large rank.
Application to \(CR\) geometry requires to analyze the signature and the rank of all possible multiples of a fixed polynomial. Hermitian symmetric polynomials are related to \(CR\) maps of hyperquadrics, e.g. to a rational map \(S^{2n-1}\to{S}^{2N-1}\), written as the quotient \(f/g\) of a polynomial map \(f\) and a polynomial \(g\) corresponds the Hermitian symmetric polynomial \(\|f\|^2-|g|^2\). Exploiting this correspondence, the authors investigate \(CR\) maps between hyperquadrics, getting fairly complete results for maps from \(S^3\) and proving some stability result in higher dimension. Hermitian forms; embeddings of CR manifolds; hyperquadrics; signature pairs; CR complexity theory; proper holomorphic mappings D'Angelo, J. P.; Lebl, J., Hermitian symmetric polynomials and CR complexity, J. Geom. Anal., 21, 3, 599-619, (2011) Embeddings of CR manifolds, Hermitian, skew-Hermitian, and related matrices, Proper holomorphic mappings, finiteness theorems, Real algebraic sets Hermitian symmetric polynomials and CR complexity
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One interesting question in the relationship between topology and the theory of integrable systems is the following: Given a smooth projective variety \(X\) can one construct a hierarchy of integrable PDEs such that the generating function of certain Gromov-Witten invariants of \(X\) are given by the tau-function of a particular solution to the hierarchy? This question is motivated by the KdV hierarchy and its relation to the topology of the moduli spaces of stable curves started by Witten. The answer to this question turns out to be yes by the work of the author of the paper under review and Zhang in the case when the quantum cohomology of \(X\) is semisimple.
Two methods were suggested to answer this question through a sequence of papers by the author and collaborators. One starts with a genus-0 hierarchy and applies the ``quasitriviality'' substitution. To find such a substitution in the first method, one uses a topological construction of the total descendent GW potential based on identities found by Getzler; and in the second method one uses the expressions for the higher-genus total GW potential via the genus-0 quantities derived from Virasoro constraints.
The main result of the paper under review is a comparison of these two methods. An outcome of this comparison implies a constraint on the Chern numbers of smooth projective varieties with semisimple quantum cohomology. KdV hierarchy; integrable system; semisimple quantum cohomology; Gromov-Witten invariants; Virasoro constraints Dubrovin, Boris, Gromov--{W}itten invariants and integrable hierarchies of topological type, Topology, Geometry, Integrable Systems, and Mathematical Physics, Amer. Math. Soc. Transl. Ser.~2, 234, 141-171, (2014), Amer. Math. Soc., Providence, RI Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Gromov-Witten invariants and integrable hierarchies of topological type
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper concerns an example of Landau Ginzburg mirror symmetry related to Grassmannians and cluster algebras. In this set up, the mirror to a Grassmannian \(X\) is given by a cluster variety \(\breve{X}\), together with a ``superpotential'' \(W:\breve{X} \to \mathbf{C}\).
The author introduces two spaces: ``the decorated Grassmannian'', and the ``decorated configuration space''. In rough terms, the decorated Grassmannian is a complement of an ample divisor in the affine cone over the ordinary Grassmannian. On the other hand the decorated configuration space parametrizes particular configuration of lines in a vector space. The main result of the paper shows that both of these spaces have natural cluster structures. More precisely, the decorated Grassmannian is an \(\mathcal{A}\)-cluster variety, whereas the decorated configuration space is an \(\mathcal{X}\)-cluster variety in the sense of Fock-Goncharov [\textit{V. V. Fock} and \textit{A. B. Goncharov}, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865--930 (2009; Zbl 1180.53081)]. To show this, author applies the key result obtained by Gross-Hacking-Keel-Kontsevich [\textit{M. Gross} et al., J. Am. Math. Soc. 31, No. 2, 497--608 (2018; Zbl 1446.13015)], showing that there is a canonical basis for the coordinate ring of the decorated Grassmannian, parametrized by integral tropical points of the decorated configuration space, to describe this canonical basis and obtain the superpotential \(W\) on the decorated configuration space. Moreover, a comparision of this potential to the superpotential introduced by Rietsch-Williams as mirror of the Grassmannian is provided [\textit{K. Rietsch} and \textit{L. Williams}, Duke Math. J. 168, No. 18, 3437--3527 (2019; Zbl 1439.14142)], and it is shown that they are compatible.
The author also proves a purely combinatorial result about plane partitions, called the ``cyclic sieving phenomenon'', by using the fact that the canonical basis for the coordinate ring of the decorated Grassmannian can be parametrized by integral tropical points. cluster algebra; cluster duality; mirror symmetry; Grassmannian; cyclic sieving phenomenon Cluster algebras, Combinatorial aspects of representation theory, Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Tropical geometry, Applications of deformations of analytic structures to the sciences Cyclic sieving and cluster duality of Grassmannian
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider the ``anti-dominant'' variants \(\Theta^-_\lambda\) of the elements \(\Theta_\lambda\) occurring in the Bernstein presentation of an affine Hecke algebra \(\mathcal H\). We find explicit formulae for \(\Theta^-_\lambda\) in terms of the Iwahori-Matsumoto generators \(T_w\) (\(w\) ranging over the extended affine Weyl group of the root system \(R\)), in the case (i) \(R\) is arbitrary and \(\lambda\) is a `minuscule' coweight, or (ii) \(R\) is attached to \(\text{GL}_n\) and \(\lambda=me_k\), where \(e_k\) is a standard basis vector and \(m\geq 1\). In the above cases, certain `minimal expressions' for \(\Theta^-_\lambda\) play a crucial role. Such minimal expressions exist in fact for any coweight \(\lambda\) for \(\text{GL}_n\). We give a sheaf-theoretic interpretation of the existence of a minimal expression for \(\Theta^-_\lambda\): the corresponding perverse sheaf on the affine Schubert variety \(X(t_\lambda)\) is the push-forward of an explicit perverse sheaf on the Demazure resolution \(m\colon\widetilde X(t_\lambda)\to X(t_\lambda)\). This approach yields, for a minuscule coweight \(\lambda\) of any \(R\), or for an arbitrary coweight \(\lambda\) of \(\text{GL}_n\), a conceptual albeit less explicit expression for the coefficient \(\Theta^-_\lambda(w)\) of the basis element \(T_w\) in terms of the cohomology of a fiber of the Demazure resolution. Bernstein presentations; affine Hecke algebras; Iwahori-Matsumoto generators; extended affine Weyl groups; root systems; minuscule coweights; standard basis vectors; minimal expressions; perverse sheaves; affine Schubert varieties; Demazure resolutions Haines, T., Pettet, A.: Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra. J. Algebra 252(1), 127--149 (2002) Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra.
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this paper is to study the symplectic properties and Wentzel-Kramers-Brillouin (WKB) expansion of a linear second-order equation on a Riemann surface \(\mathcal{C}\) of genus \(g\). The authors study symplectic properties of the monodromy map of second-order equations on \(\mathcal{C}\) whose potential is meromorphic with double poles. They show that the Poisson bracket defined in terms of periods of the meromorphic quadratic differential implies the Goldman Poisson structure on the monodromy manifold. They apply these results to a WKB analysis of this equation and show that the leading term in the WKB expansion of the generating function of the monodromy symplectomorphism (the Yang-Yang function introduced by \textit{N. A. Nekrasov} et al. [Theor. Math. Phys. 181, No. 1, 1206--1234 (2014; Zbl 1308.81133); translation from Teor. Mat. Fiz. 181, No. 1, 86--120 (2014); erratum ibid. 182, No. 2, 311 (2014)]) is determined by the Bergman tau function on the moduli space of meromorphic quadratic differentials.
This paper is organized as follows: Section 1 is an introduction to the subject and summarizes the main results. Section 2 deals with quadratic differentials with second-order poles. Section 3 is devoted to monodromy map, monodromy symplectomorphism and its generating function. In Section 4, the authors consider the notion of the Bergman tau function on moduli spaces of quadratic differentials with second-order poles and fixed biresidues. Section 5 is devoted to the WKB expansion of the generating function of the monodromy symplectomorphism. The authors find the WKB expansion of the homological shear coordinates. In Section 6, the authors list and discuss a few open problems related to this paper. The paper is supported by an appendix about topological double covers and Darboux coordinates for the \(\mathrm{PSL}(2)\) Goldman bracket. Riemann surface; monodromy map; symplectic map generating function; tau function; Goldman bracket Special algebraic curves and curves of low genus, Relationships between algebraic curves and physics WKB expansion for a Yang-Yang generating function and the Bergman tau function
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a Schubert cycle of a Grassmannian \(G(r,n)\). Here the author gives a combinatorial proof of Hodge's postulation formula giving the Hilbert function of \(X\) with respect to the Plücker embedding of \(G(r,n)\). The main point of the paper is to introduce algebraists to some combinatorial techniques which seem to be important in this area. Hilbert polynomial; Schubert cycle; Grassmannian; Hilbert function Ghorpade, S. R.: A note on Hodge's postulation formula for Schubert varieties. Geometric and combinatorial aspects of commutative algebra (Messina, 1999), 211-220 (2001) Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series A note on Hodge's postulation formula for Schubert varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0667.00008.]
In his book ``Kalkül der abzählenden Geometrie'' (1879; reprint 1979; Zbl 0417.51008), \textit{H. C. H. Schubert} discussed the enumerative theory of plane cuspidal cubics. Schubert's calculations rely on the method of degenerations, i.e. boundary components of the space of complete cuspidal plane cubics. The authors give detailed discussion of this space (for an algebraically closed underlying field of characteristic \(\neq 2, 3)\) and base upon this and other geometric results their enumerative computations. It turns out that there are 620 non-zero fundamental numbers, but only 391 of them were already given by Schubert.
[Unfortunately the reviewer has no access to a repeatedly quoted paper of the authors: ``On Schubert's degenerations of cuspidal cubics'', Preprint, Univ. Barcelona 1987.] enumerative theory of plane cuspidal cubics; fundamental numbers Miret, J. and Xambó-Descamps, S. Geometry of complete cuspidal plane curves. Trento 1988 Conference. Algebraic Curves and Projective Geometry, Edited by: Ballico and Ciliberto. pp.195--234. Springer. Volume 1389 of LNM Enumerative problems (combinatorial problems) in algebraic geometry, Projective analytic geometry Geometry of complete cuspidal plane cubics
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be the symplectic group \(\text{Sp}(2m,F)\) where \(F=\overline{\mathbf F}_q\), and \(P\) a parabolic subgroup of \(G\) with Levi subgroup \(L\) isomorphic to \(\text{GL}(m,F)\). Let \(p:P\to L\) be the natural projection. Let \(u\), \(w\) be fixed unipotent elements of \(G\), \(L\) respectively. The author studies the subvariety \(X_{u,w}\) of \(G/P\) defined by \(X_{u,w}=\{xP\mid uxP=xP\) and \(p(x^{-1}ux)\) is conjugate to \(w\) in \(L\}\). Let \(\lambda\), \(\mu\) be the partitions of \(2m\) and \(m\) determined by the Jordan forms of \(u\), \(w\) respectively. The polynomial \(g^\lambda_\mu(q)\) defined as the number of \({\mathbf F}_q\)- rational points of \(X_{u,w}\) is a symplectic analogue of a Hall polynomial for \(\text{GL}(n,q)\), and is important in the character theory of \(G\). Using the symplectic geometry afforded by the natural representation of \(G\) on \(F^m\), the author computes the polynomials \(g^\lambda_\mu(q)\) and gives a closed formula for them when \(\lambda=r^d\) for some \(r\) and \(\mu\) is arbitrary. partitions determined by Jordan forms; rational points; symplectic groups; parabolic subgroups; Levi subgroups; unipotent elements; Hall polynomials; symplectic geometry; natural representation; closed formula Representation theory for linear algebraic groups, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical groups (algebro-geometric aspects), Symmetric functions and generalizations Hall polynomials for symplectic groups. I
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution of two topological vertices and the HOMFLY-PT invariant of the 4-component link \(L_{8 n 8}\), which both depend on four arbitrary representations. The key point is that both of these are related to the Hopf polynomials in \textit{composite} representations, which are in turn expressed through composite Schur functions. The latter are further expressed through the skew Schur polynomials via the remarkable Koike formula. It is this decomposition that breaks under the Macdonald deformation and gets restored only in the (large \(N)\) limit of \(A^{\pm 1} \longrightarrow 0\). Another problem is that the Hopf polynomials in the composite representations in the refined case are ``chiral bilinears'' of Macdonald polynomials, while convolutions of topological vertices involve ``non-chiral combinations'' with one of the Macdonald polynomials entering with permuted \(t\) and \(q\). There are also other mismatches between the Hopf polynomials in the composite representation and the topological 4-point function in the refined case, which we discuss. Knot polynomials, Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.), Formal methods and deformations in algebraic geometry, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Can tangle calculus be applicable to hyperpolynomials?
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Matrix Schubert varieties are the orbit closures of \(B \times B\) acting on all \(n \times n\) matrices, where \(B\) is the group of invertible lower triangular matrices. We define skew-symmetric matrix Schubert varieties to be the orbit closures of \(B\) acting on all \(n \times n\) skew-symmetric matrices. In analogy with Knutson and Miller's work on matrix Schubert varieties, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gröbner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams, analogous to the pipe dreams of Bergeron and Billey. We show that these initial ideals are the Stanley-Reisner ideals of shellable simplicial complexes. As an application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials. Schubert varieties; Gröbner bases; Grothendieck polynomials; simplicial complexes Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of simplicial complexes, Exact enumeration problems, generating functions Gröbner geometry for skew-symmetric matrix Schubert varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The context of the research is as follows. Let \(G\) be a semisimple complex algebraic group, \(K\) be a closed subgroup of \(G\). Let \(P\) be a parabolic subgroup of \(G\) and \(B_K\) is a Borel subgroup of \(K\). Then \(G/P \times K/B_K\) is a double flag variety. In this paper, the author considers three cases:
\begin{itemize}
\item[(I)] \(G=SL(2n)\), \(K=(GL(n) \times GL(n)) \cap G\), \(G/P\) is the Grassmannian \(Gr(n,2n)\),
\item[(II)] \(G=Sp(2n)\), \(K=GL(n)\), \(G/P\) is the Lagrangian Grassmannian \(LG(2n)\),
\item[(III)] \(G=SO(2n)\), \(K=GL(n)\), \(G/P\) is the orthogonal Grassmannian \(OG(2n)\).
\end{itemize}
The main result of the paper is:
\begin{itemize}
\item[1.] Descriptions of \(K\)-orbits on \(G/P \times K/B_K\) intersecting certain subset of \(G/P\) in terms of rank condition (Proposition 13 for (I), Proposition 34 for (II), (II))
\item[2.] Statements that the cohomology classes of the \(K\)-orbits are represented by the back-stable double Schubert polynomials (Theorem for (I)), back-stable involution Schubert polynomials (Theorem 41 for (II) and Theorem 43 for (III)).
\end{itemize}
Then, the author obtains some corollaries:
\begin{itemize}
\item[a.] Geometric interpretations of the involution Stanley symmetric functions \(2^{cyc(y)}F_y^{O}\), \(F_z^{Sp}\) (Theorem 44).
\item[b.] A new proof that \(2^{cyc(y)}F_y^{O}\) is Schur \(Q\) positive and \(F_z^{Sp}\) is Schur \(P\) positive (Corollary 45), which are proved in [\textit{Z. Hamaker} et al., Int. Math. Res. Not. 2019, No. 17, 5389--5440 (2019; Zbl 1459.05338); J. Comb. 11, No. 1, 65--110 (2020; Zbl 1427.05226)].
\item[c.]A generalization of Pfaffian formulas of involution Schubert polynomials and Stanley symmetric functions for I-Grassmannian, fpf-I-Grassmannian involution in [\textit{Z. Hamaker} et al., Int. Math. Res. Not. 2019, No. 17, 5389--5440 (2019; Zbl 1459.05338); J. Comb. 11, No. 1, 65--110 (2020; Zbl 1427.05226)] to vexillary involution (Theorems 39,49,54,57).
\end{itemize}
The key point used to obtain the main results is the interpretation graph Schubert varieties of the as certain degeneracy when \(w,z\) are vexillary.
The structure of the paper is as follows. Section 2 prepares background about cohomology rings of flag varieties, degeneracy loci, Schubert varieties, and vexillary permutations. Sections 3 shows the main result 1. 2. for (I). Sections 4 prepares the background for involution graph Schubert varieties. Section 5 shows the main results 1. 2. for (II), (III), and then corollaries a. b. Section 6 shows corollaries c. The last section gives tableau formulas for back-stable involution Schubert polynomials in terms of shifted tableaux in vexillary settings. flag manifolds; Schubert varieties; Grassmannian; Lagrangian Grassmannian; orthogonal Grassmannian; Shubert polynomials; Stanley symmetric functions Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial aspects of commutative algebra Universal graph Schubert varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters, and Demazure atoms; the quasi-key, fundamental, and monomial slide bases introduced in [\textit{S. Assaf} and \textit{D. Searles}, Adv. Math. 306, 89--122 (2017; Zbl 1356.14039)]; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and that an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial. This rule simultaneously extends \textit{J. Haglund} et al. [Trans. Am. Math. Soc. 363, No. 3, 1665--1686 (2011; Zbl 1229.05269)] Littlewood-Richardson rule for quasi-Schur polynomials and refines their Littlewood-Richardson rule for Demazure characters. We also establish bijections connecting combinatorial models for these polynomials including semi-skyline fillings and quasi-key tableaux. Schur polynomials; Demazure atoms; quasi-key polynomials; slide polynomials Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Polynomial bases: positivity and Schur multiplication
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let $G=\mathrm{GL}_n(\mathbb{K})$ be the general linear group over an algebraically closed field $\mathbb{K}$. Let $G/B$ be the full flag variety consisting of complete flags
\[
F_\bullet:\quad \{0\}= F_0\subset F_1\subset \cdots \subset F_n=\mathbb{K}^n
\]
of subspaces of $\mathbb{K}^n$ so that $\dim F_i=i$. Fix a Hessenberg function $h:\{1, \dots, n\} \to
\{1, \dots, n\}$ which is, by definition, a non-decreasing function such that $h(i) \geq i$ for all $i$. Fix a linear operator $X:\mathbb{K}^n \to \mathbb{K}^n$. Recall that the
corresponding Hessenberg variety is defined by:
\[
\mathcal{Y}_{X, h}:= \{F_\bullet: X(F_i)\subset F_{h(i)} \text{ for all } i\}\subset G/B.
\]
In the paper under review, the authors give an explicit expression for the class of $ \mathcal{Y}_{X, h}$ in the $K$-theory $K(G/B)$ as well as in the cohomology $H^*(G/B)$ in terms of certain Grothendieck (resp. Schubert) polynomial if $X$ is a regular operator. In fact, they generalize the result for any $X$ such that $ \mathcal{Y}_{X, h}$ has the expected dimension $\sum_i(h(i)-i)$.
We recall that a different formula for the cohomology class of $ \mathcal{Y}_{X, h}$ was given by \textit{D. Anderson} and the second author [J. Algebra 323, No. 10, 2605--2623 (2010; Zbl 1218.14041)] using degeneracy arguments. Also, a different formula for the $K$-theory class of $ \mathcal{Y}_{X, h}$ was given by \textit{H. Abe}, \textit{N. Fujita} and \textit{H. Zeng} [``Geometry of regular Hessenberg varieties'', Transform. Groups (to appear; \url{doi:10.1007/s00031-020-09554-8})]. But the formulae in this paper differ from the ones given by Abe-Fujita-Zeng and Anderson-the second author viewed as polynomials (but of course not as $K$-theory or cohomology classes). The authors also reprove that $ \mathcal{Y}_{X, h}$ is Cohen-Macaulay (hence equi-dimensional). Hessenberg variety; $K$-theory class; cohomology class; Cohen-Macaulay schemes Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Computational aspects and applications of commutative rings, Algebraic combinatorics A formula for the cohomology and \(K\)-class of a regular Hessenberg variety
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal H\) be the Hecke algebra associated with a Coxeter system \((W,R)\) (i.e. \(W\) is a Coxeter group and \(R\) its set of distinguished involutory generators). \textit{D. Kazhdan} and \textit{G. Lusztig} [Invent. Math. 53, 165--184 (1979; Zbl 0499.20035)] introduced new bases of \(\mathcal H\), with respect to which the structure constants (which are Laurent polynomials) satisfy remarkable positivity properties. These properties were explained by interpreting the structure constants geometrically as Poincaré series. The authors of this paper point out some further positivity properties for the structure constants of \(\mathcal H\), prove some of them formally from existing results, others geometrically using intersection complexes on flag varieties, and state the unproved cases as conjectures. Hecke algebra; Coxeter system; Coxeter group; bases; structure constants; Laurent polynomials; positivity; Poincaré series; intersection complexes; flag varieties Dyer, M; Lusztig, G, Appendix to ``quantum groups at roots of 1{'', Geom Ded, 35, 112-113, (1990)} Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Representation theory of associative rings and algebras, Conditions on elements, Group actions on varieties or schemes (quotients) On positivity in Hecke algebras
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \noindent Let \(\text{Fl}_n\) be the flag variety i.e. the variety of complete flags \(0=V_0\subset V_1\subset \cdots \subset V_n=\mathbb C^n.\) It is well known that both the integral cohomology ring \(H^{*}(\text{Fl}_n)\) and \(K^{0}(\text{Fl}_n)\) are isomorphic to \({\mathbb Z}[x_1, \dots ,x_n]/I_{n},\) where \(I_{n}\) is the ideal generated by symmetric polynomials in \(x_1, \dots ,x_n\) with constant term equal to zero. There are natural \({\mathbb Z}\)-bases of \(H^{*}(\text{Fl}_n)\) and \(K^{0}(\text{Fl}_n)\) corresponding to Schubert varieties \(X_w\), \(w\in S_n.\) \noindent The Schubert and Grothendieck polynomials \({\mathfrak S}_w(x)\) and \({\mathfrak G}_w(x)\) are polynomial representatives of the bases elements in \(H^{*}(\text{Fl}_n)\) and \(K^{0}(\text{Fl}_n)\) corresponding to \(X_w.\) \noindent The author extends the work of \textit{S. Fomin} and \textit{A. Kirillov} in the cohomological case [in: Advances in geometry. Prog. Math. 172, 147--182 (1999; Zbl 0940.05070)]) to the \(K\)-theoretical situation. He defines a \(K\)-theoretic version of the Dunkl elements and uses them to describe the structure constants with respect to the above mentioned basis. flag variety; Schubert basis; \(K\)-theory Lenart, C, The \(K\)-theory of the flag variety and the Fomin-Kirillov quadratic algebra, J. Algebra, 285, 120-135, (2005) Grassmannians, Schubert varieties, flag manifolds, Miscellaneous applications of \(K\)-theory The \(K\)-theory of the flag variety and the Fomin--Kirillov quadratic algebra
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affine analogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation of the Kostka-Foulkes polynomials in terms of solvable lattice models by \textit{A. Nakayashiki} and \textit{Y. Yamada} [Sel. Math., New Ser. 3, No. 4, 547--599 (1997; Zbl 0915.17016)] to the affine setting. charge statistic; Pieri rule; \(k\)-Schur functions; energy function; affine Schubert calculus Morse, J.; Schilling, A., Affine charge and the k-bounded Pieri rule, (DMTCS Proceedings, (2015)) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Affine charge and the \(k\)-bounded Pieri rule
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The singular suppports of the simple perverse sheaves studied are shown to be contained in certain varieties of \(\Lambda\) similar to Lusztig's nilpotent quiver varieties in the negative part of the universal enveloping algebra of a generalized Kac-Moody Lie algebra is realized inside the algebra of constructible functions on \(\Lambda\).
For Part I, see Nagoya Math. J. 194, 169--193 (2009; Zbl 1194.17007). singular suppports; simple perverse sheaves; universal enveloping algebra; generalized Kac-Moody Lie algebra Universal enveloping (super)algebras, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Canonical bases of Cartan-Borcherds type. II: Constructible functions on singular supports
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Perhaps the nicest multivariate orthogonal polynomials are the Macdonald and Koornwinder polynomials, respectively 2-parameter deformations of Schur functions and 6-parameter deformations of orthogonal and symplectic characters, satisfying a trio of nice properties (evaluation, norm and symmetry) known as the Macdonald ``conjectures''.
In recent work, the author has constructed elliptic analogues: a family of multivariate functions on an elliptic curve satisfying analogues of Macdonald conjectures, and degenerating to Macdonald and Koornwinder polynomials under suitable limits. This article will discuss the two main constructions for these functions. While the first construction is intrinsically complex analytic in nature, the second one is much more combinatorial and algebraic. This paper will focus on the second construction, modified somewhat to make the arguments self-contained. Macdonald polynomials; Koornwinder polynomials; elliptic curves; special functions Rains, Eric M., The noncommutative geometry of elliptic difference equations, (None) Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Elliptic curves Elliptic analogues of the Macdonald and Koornwinder polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review studies the monodromy conjecture for symplectic resolutions as formulated by Braverman-Maulik-Okounkov. This conjecture roughly says that the monodromy of the quantum connection for two birationally equivalent smooth symplectic Deligne-Mumford stacks which are symplectic resolutions of a singular symplectic stack and they correspond to the two large radius points in the compactified Kähler moduli spaces is the same as the monodromy given by their equivalence in the \(K\)-theory.
This paper introduces the notion of ``extended stacky hyperplane arrangements'' and defines the hypertoric DM stacks associated with them. There is a one-to-one correspondence between such arrangements and the GIT data and in particular the wall crossing of hypertoric DM stacks gives the Mukai type flops.
Let \(X_+ \to X_-\) be a crepant birational map between two smooth Lawrence toric DM stacks given by a single wall crossing. It is known that \(X_+\) and \(X_-\) are derived equivalent and their equivalence is given by the Fourier-Mukai transformation, which in the level of \(K\)-theory matches the analytic continuation of the \(I\)-function, and so also the analytic continuation of the quantum connections which are determined by the \(I\)-function. The wall crossing above implies that the associated birational transformation \(Y_+\to Y_-\) for the hypertoric DM stacks is crepant, and the Fourier-Mukai functor gives an equivalence of derived categories of \(Y_+\) and \(Y_-\). The paper under review proves that this Fourier-Mukai transformation matches the analytic continuation of quantum connections of the hypertoric DM stacks, which is induced by the analytic continuation
of the associated Lawrence toric DM stacks.
Viewing crepant birational transformation of hypertoric DM stacks as the local model of stratified Mukai type flops for general symplectic DM stacks, one can say that the construction in this paper will play a role in the study for general Mukai type flops by degeneration method to the local models. crepant transformation conjecture; monodromy conjecture Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Generalizations (algebraic spaces, stacks), McKay correspondence The crepant transformation conjecture implies the monodromy conjecture
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a survey of some more recent work. In the second half of the article, we give a definition of a Grassmann polytope motivated by work of physicists on the amplituhedron. We propose to use Schubert calculus and canonical bases to replace linear algebra and convexity in the theory of polytopes. total nonnegativity; Grassmann polytope; positroid variety; dimers; Grassmann matroid; scattering amplitudes Grassmannians, Schubert varieties, flag manifolds, Planar graphs; geometric and topological aspects of graph theory, Special polytopes (linear programming, centrally symmetric, etc.), Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) Totally nonnegative Grassmannian and Grassmann polytopes
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The fusion potential for \(su(N)_K\) can be obtained as a perturbation of the Landau-Ginzberg potential that generates the cohomology ring of the Grassmannian. The connection between the fusion ring of \(su(N)_K\) and the cohomology ring is based on the fact that the rules for multiplying the Schubert cycles which generate the cohomology ring of the Grassmannian formally coincide with the rules for multiplying Schur functions. On the other hand, the irreducible characters of \(su(N)\) are also given by Schur functions with some constrain which is exactly the mentioned perturbation. The potential that generates the cohomology ring of the Grassmannian turns out to be given in terms of a power sum symmetric function. The author uses the Waring formula for presenting the power sums in terms of elementary symmetric polynomials and expresses the cohomology potential of the Grassmannian and the fusion potentials for \(su(N)_K\) and \(sp(N)_K\) by the coefficients of some algebraic equations. In the formulation of the paper, the fusion potentials for \(su(N)_K\) and \(sp(N)_K\) are generalizations in several variables of Chebyshev polynomials in one variable. As a consequence, the author obtains relations with generalized Fibonacci and Lucas numbers in the same way as the one-variable Chebyshev polynomials of the first kind and the second kind are related with the ordinary Lucas and Fibonacci numbers, respectively. fusion potentials; Fibonacci numbers; cohomology ring of the Grassmannian; Schur functions; symmetric function; Waring formula; Chebyshev polynomials; Lucas numbers N. Chair, The Waring formula and fusion rings, J. Geom. Phys. 37 (2001) 216--228. Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Fibonacci and Lucas numbers and polynomials and generalizations The Waring formula and fusion rings
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by \textit{M. Gross} et al. [J. Am. Math. Soc. 31, No. 2, 497--608 (2018; Zbl 1446.13015)]: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary.
We prove that the two cones are unimodularly equivalent. As a corollary of our combinatorial result we realize Caldero's toric degenerations of Schubert varieties as GHKK-degeneration using cluster theory. Schubert variety; flag variety; superpotential; string cone; mirror symmetry; pseudoline arrangement; combinatorics Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Cluster algebras String cone and superpotential combinatorics for flag and Schubert varieties in type A
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This chapter concerns the concept of \textit{edge labeled} Young tableaux, introduced by \textit{H. Thomas} and the third author [Ann. Inst. Fourier 68, No. 1, 275--318 (2018; Zbl 1400.05273)]. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences from combinatorics, algebraic and symplectic geometry, linear algebra, and computational complexity. We report on a new shifted analogue of edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule for the structure constants of the Anderson-Fulton ring, which is related to the equivariant cohomology of isotropic Grassmannians. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry, Equivariant algebraic topology of manifolds Equivariant cohomology, Schubert calculus, and edge labeled tableaux
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(O_q(G)\) be the algebra of quantized functions on an algebraic group \(G\) and \(O_q(B)\) its quotient algebra corresponding to a Borel subgroup \(B\) of \(G\). We define the category of sheaves on the ``quantum flag variety of \(G\)'' to be the \(O_q(B)\)-equivariant \(O_q(G)\)-modules and prove that this is a proj-category. We construct a category of equivariant quantum \({\mathcal D}\)-modules on this quantized flag variety and prove the Beilinson-Bernstein's localization theorem for this category in the case when \(q\) is transcendental. quantum groups; localization; noncommutative geometry Backelin, E.; Kremnizer, K., Quantum flag varieties, equivariant quantum \(\mathcal{D}\)-modules, and localization of quantum groups, Adv. Math., 203, 408-429, (2006) Quantum groups (quantized enveloping algebras) and related deformations, Geometry of quantum groups, Noncommutative algebraic geometry Quantum flag varieties, equivariant quantum \(\mathcal D\)-modules, and localization of quantum groups
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article, we prove a tableau formula for the double Grothendieck polynomials associated to 321-avoiding permutations. The proof is based on the compatibility of the formula with the \(K\)-theoretic divided difference operators. Our formula specializes to the one obtained by \textit{W. Y. C. Chen} et al. [Eur. J. Comb. 25, No. 8, 1181--1196 (2004; Zbl 1055.05149)] for the (double) skew Schur polynomials. symmetric polynomials; Grothendieck polynomials; \(K\)-theory; set-valued tableaux; 321-avoiding permutations Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices A tableau formula of double Grothendieck polynomials for 321-avoiding permutations
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The closure of a generic torus orbit in the flag variety \(G/B\) of type \(A\) is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in \(G/B\). When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth. Poincaré polynomials; projective toric varieties Toric varieties, Newton polyhedra, Okounkov bodies, Grassmannians, Schubert varieties, flag manifolds, Permutations, words, matrices Poincaré polynomials of generic torus orbit closures in Schubert varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author introduces a polynomial \(P^W\) associated with the \(d\)-web \(W\) given in \(\mathbb{C}^2\). If \(d \geq 4\), the web \(W\) is linearizable iff \(\text{deg }P^W\leq 3\) and its coefficients satisfy a nonlinear differential system. If \(W\) is a maximum rank web then only the condition \(\text{deg }P^W \leq 3\) is essential. In terms of the polynomial \(P^W\) the Darboux-Blaschke map is characterized. Besides, the author shows how \(D\)-modules theory is used to study some problems in \(d\)-web geometry in \(\mathbb{C}^2\). three-weg; linearizable web; maximum rank web Differential geometry of webs, Pencils, nets, webs in algebraic geometry On the linearization problem and some questions for webs in \(\mathbb{C}^ 2\)
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathcal F}=\widetilde{G}/P_{I}\) be an affine flag variety. Here \(G\) is a simply connected complex algebraic group with simple Lie algebra, \(\widetilde{G}=G({\mathbb C}[z,z^{-1}])\) is the corresponding affine group and \(P_{I}\) is the parabolic subgroup associated to a subset \(I\) of Coxeter generators \(\widetilde{S}\) of the affine Weyl group \(\widetilde{W}\). Then \({\mathcal F}\) has two stratifications: the Schubert or Bruhat cell decomposition:
\[
{\mathcal F}= \bigsqcup_{\lambda\in\widetilde{W}/\widetilde{W}_{I}} { e_{\lambda} },
\]
and the Birkhoff stratification:
\[
{\mathcal F}=\bigsqcup_{\lambda\in \widetilde{W}/\widetilde{W}_{I}} {S_{\lambda}}.
\]
The Schubert cells \(e_{\lambda}\) are the orbits of the Iwahori subgroup \(\widetilde{B}\), while the Birkhoff strata \(S_{\lambda}\) are the orbits of the opposite Iwahori subgroup \(\widetilde{B}^{-1}\).
The authors consider a finite union \({\mathcal Z}_{I}\) of Birkhoff subvarieties in the affine flag variety \(\mathcal F\). In their main theorem they first prove that \({\mathcal Z}_{I}\) is a strong deformation retract of \({\mathcal F}\). They further construct analogues of tubular neighborhoods for \({\mathcal Z}_{I}\) in \(\mathcal F\). In particular, they show that the inclusion \({\mathcal Z}_{I}\subset {\mathcal F}\) induces isomorphisms on ordinary and equivariant cohomology with any coefficients. Birkhoff varieties; affine flag varieties; homotopy equivalence L. Gutzwiller and S. A. Mitchell, The topology of Birkhoff varieties, Transform. Groups 14 (2009), no. 3, 541 -- 556. Grassmannians, Schubert varieties, flag manifolds, Homotopy equivalences in algebraic topology The topology of Birkhoff varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools.
Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.
The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to combinatorics, Other designs, configurations, Combinatorial identities, bijective combinatorics, Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Grassmannians, Schubert varieties, flag manifolds, Proceedings, conferences, collections, etc. pertaining to group theory, Collections of articles of miscellaneous specific interest Recent trends in algebraic combinatorics
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials First the authors introduce the notion of an enhanced Burnside ring \(B(G)\) associated with a finite group \(G\) and then they give an enhanced version of the equivariant Saito duality. Thus, in the case of a complex analytic \(G\)-manifold endowed with a \(G\)-equivariant transformation there is an enhanced equivariant Euler characteristic with values in a completion of \(B(G)\). This characteristic, for the Milnor fiber of a quasihomogeneous polynomial with the classical monodromy transformation on it, defines both the usual monodromy zeta function and the orbifold one. Among other things, the authors also prove that the reduced enhanced characteristics of the Milnor fibers of Berglund-Hübsch dual invertible polynomials (see [\textit{P. Berglund} and \textit{M. Henningson}, Nucl. Phys., B 433, No. 2, 311--332 (1995; Zbl 0899.58068)]) are enhanced dual to each other up to sign. As ``a by-product'' of their considerations, the authors obtain their own earlier result about the orbifold zeta functions of Berglund-Hübsch-Henningson dual pairs from [\textit{W. Ebeling} and \textit{S. M. Gusein-Zade}, Bull. Lond. Math. Soc. 44, No. 4, 814--822 (2012; Zbl 1298.32017)]. group action; monodromy; Burnside ring; invertible polynomials; Arnold's strange duality; Saito duality; Berglund-Hübsch-Henningson dual pairs; Berglund-Hübsch mirror symmetry; orbifold zeta functions Group actions on affine varieties, Frobenius induction, Burnside and representation rings, Mirror symmetry (algebro-geometric aspects), Topology and geometry of orbifolds, Monodromy on manifolds Enhanced equivariant Saito duality
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author, [in J. Pure Appl. Algebra 163, No. 2, 193-207 (2001; Zbl 0988.16026)], classified monogenic Hopf algebras (Hopf algebras which are generated as algebras by one element), which are commutative and cocommutative (Abelian) and are local and have a local dual (local-local) over a finite field \(k\) or the Witt ring \(W(k)\) of \(k\).
In this paper, the author continues this theme, and describes the ring of Hopf algebra endomorphisms of a monogenic Abelian local-local Hopf algebra \(H\) over a finite or algebraically closed field of positive characteristic. As in the paper classifying monogenic Hopf algebras mentioned above, key to this study is the fact that \(\text{End}(H)\) is isomorphic to \(\text{End}(M)\) where \(M\) is the Dieudonné module corresponding to the group scheme represented by \(H\). Here \(M\) is a cyclic \(E\)-module of the form \(M=E/E(F^n,\alpha F^r-V)\) where \(E\) is the noncommutative polynomial ring \(W[F,V]\) over \(W=W(k)\). There is a one-to-one map from \(\text{End}(M)\) to \(P_n\), the noncommutative truncated polynomial ring \(k[F]/(F^n)\), and the author thus can calculate within this polynomial ring. The author goes on to study \(\text{End}(H)\) for \(H\) a Hopf algebra over \(W(k)\). Here it is key to note the one-to-one correspondence between isomorphism classes of finite Honda systems (these are pairs \((M,L)\) with \(M\) a Dieudonné module and \(L\) a \(W(k)\) submodule of \(M\)) and isomorphism classes of monogenic \(W(k)\)-Hopf algebras. Then \(\text{End}(H)\cong\text{End}(M,L)\); here a general classification of \(\text{End}(H)\) is not obtained but several specific cases are covered. The final section of the paper discusses possible (or impossible) generalizations of this work. Hopf algebra endomorphisms; monogenic Hopf algebras; Witt rings; local-local Hopf algebras; Dieudonné modules; group schemes; polynomial rings; finite Honda systems Koch, A.: Endomorphisms of monogenic Hopf algebras. Communications in algebra 35, 747-758 (2007) Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Automorphisms and endomorphisms, Group schemes, Formal groups, \(p\)-divisible groups, Endomorphism rings; matrix rings Endomorphisms of monogenic Hopf algebras.
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The second author introduced [in J. Indian Math. Soc., New Ser. 40, 299- 349 (1976; Zbl 0447.14014)] the name Kempf varieties for certain Schubert varieties in G/B. Here B is a Borel subgroup of the semi-simple algebraic group G and it is assumed that G is classical. It is shown in this paper that for G of type A, B or C, the Kempf varieties are exactly the Schubert varieties on which the notion ''weakly standard Young diagrams'' [introduced in the joint work of the second author, \textit{C. Musili} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Sect A, Part III 88, No.4, 279-362 (1979; Zbl 0447.14013)] coincides with ''standard Young diagrams'' (loc. cit.).
To any Schubert variety the authors attach a poset of elements in the Weyl group. They prove that the corresponding order complex is shellable and then use this to deduce that the multicones over Kempf varieties (in G/P, P any parabolic subgroup) are normal and Cohen-Macaulay.
[It has recently been proved by a quite different method that multicones over any Schubert variety always have rational singularities, see \textit{G. Kempf} and \textit{A. Ramanathan}, Invent. Math. 87, 353-363 (1987).] standard monomials; Schubert varieties; weakly standard Young diagrams; multicones over Kempf varieties C. Huneke and V. Lakshmibai,A characterization of Kempf varieties by means of standard monomials and the geometric consequences, J. Alg.94 (1985), 52--105. Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields A characterization of Kempf varieties by means of standard monomials and the geometric consequences
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Motivated by a result of \textit{P. Fiebig} [Adv. Math. 217, No. 2, 683--712 (2008; Zbl 1140.14044)], we categorify some properties of Kazhdan-Lusztig polynomials via sheaves on Bruhat moment graphs. In order to do this, we develop new techniques and apply them to the combinatorial data encoded in these moment graphs. Bruhat graphs; moment graphs; Kazhdan-Lusztig polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Graph theory, Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Moment graphs and Kazhdan-Lusztig polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Double Hurwitz numbers \(H_g(\mu,\nu)\) count the number of maps from a connected complex curve of genus \(g\) to \(\mathbb{P}^1\) with the partition profile \(\mu\) over zero, \(\nu\) over infinity, and simple ramification over other fixed points (the contributions are normalized by the size of the automorphism group). The main result of the paper is an algorithm for computing \(H_g(\mu,\nu)\), which provides a closed form expression for the generating function \(H_{\mu,\nu}(z):=\sum_{g=0}^\infty\frac{z^r}{r!}\,H_g(\mu,\nu)\), where \(r:=2g-2+l(\mu)+l(\nu)\) and \(l\) is the number of parts in a partition. It is obtained from \textit{A. Okounkov}'s [Math. Res. Lett. 7, No. 4, 447--453 (2000; Zbl 0969.37033)] representation of \(H_{\mu,\nu}(z)\) as vacuum expectation values of operators on the infinite wedge.
A novelty is the application of this technique to proving the strong piecewise polynomiality conjecture of Goulden-Jackson-Vakil [\textit{I. P. Goulden} et al., Adv. Math. 198, No. 1, 43--92 (2005; Zbl 1086.14022)] for \(H_g(\mu,\nu)\), which represents them as alternating sums of positive polynomials over each chamber of \((\mu,\nu)\). The formula lends support to the geometric insight behind the conjecture: that the compactifications of the universal Picard variety exhibit a chamber structure with respect to some stability condition, which matches the chamber structure of the double Hurwitz numbers. Two other consequences of the main result are that the lower degree terms of the genus \(g\) positive polynomials are determined by the polynomials with lower genus, and a wall-crossing formula for \(H_{\mu,\nu}(z)\). In some special chambers the closed form expression reduces to a product formula for the generating function, and the authors find all such special chambers, which include the ``totally negative'' ones of Shadrin-Shapiro-Vainshtein in genus zero [\textit{S. Shadrin} et al., Adv. Math. 217, No. 1, 79--96 (2008; Zbl 1138.14018)].
The proofs are purely algebraic, and use the classical result that monodromy covers reduce the count to counting sets of elements of the symmetric group, which, following Burnside, is done via the group character theory. The same method was previously used by \textit{M. Roth} [``Counting covers of an elliptic curve'', Preprint, \url{http://www.mast.queens.ca/~mikeroth/notes/covers.pdf}] to count covers of an elliptic curve. The passage to the closed form expression for the generating function follows the infinite wedge techniques of \textit{A. Okounkov} and \textit{R. Pandharipande} [Ann. Math. (2) 163, No. 2, 561--605 (2006; Zbl 1105.14077); Ann. Math. (2) 163, No. 2, 517--560 (2006; Zbl 1105.14076)]. double Hurwitz numbers; infinite wedge; vacuum expectation values; strong piecewise polynomiality conjecture; wall-crossing formula P. Johnson, ''Double Hurwitz numbers via the infinite wedge,'' arXiv:1008.3266. Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Double Hurwitz numbers via the infinite wedge
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth projective variety over \(\mathbb{C}\). The quantum cohomology of \(X\) satisfies the axioms of Froebenius manifolds formulated by Dubrovin. In particular the third derivative of the Frobenius potential which is the generating function of the genus zero Gromov-Witten invariants of \(X\) give the structure constants of the product in the quantum cohomology. Dubrovin formulated a conjecture relating the monodromy data of the Frobenius manifold associated to the quantum cohomology of \(X\) and the structure of the bounded derived category of coherent sheaves on \(X\). In particular, the conjecture asserts that the Frobenius manifold is semisimple if and only if the derived category admits a full exceptional collection of \(\sum h^{p,p}(X)\) elements, and moreover, for any semisimple point of the Frobenius manifold, the Stokes matrix of the first structure connection is identified with the matrix of the Euler pairing of an exceptional collection. The conjecture is proven in some examples.
The paper under review generalizes Dubrovin's conjecture for a class of orbifolds given by the weighted projective line \(\mathbb{P}^1(a_1,a_2,a_3)\) such that \(\sum 1/a_i>1\). The existence of an exceptional collection for the derived category and the semisimplicity of the Frobenius manifold is already known for this class of orbifolds. The second part of the conjecture regarding the equivalence of the Stokes and Euler matrices is proven in this paper. The proof is based on the homological mirror symmetry. Stokes Matrices; quantum cohomology; Frobenius manifolds; mirror symmetry Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Stokes matrices for the quantum cohomologies of a class of orbifold projective lines
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{E. Gorsky} and \textit{A. Negut} [J. Math. Pures Appl. (9) 104, No. 3, 403--435 (2015; Zbl 1349.14012)] introduced operators \(Q_{m,n}\) on symmetric functions and conjectured that, in the case where \(m\) and \(n\) are relatively prime, the expansion of \(Q_{m,n}(-1)^n\) in terms of the fundamental quasi-symmetric functions are given by polynomials introduced by \textit{T. Hikita} [Adv. Math. 263, 88--122 (2014; Zbl 1302.14041)]. Later \textit{F. Bergeron} et al. [``Compositional \((km,kn)\)-shuffle conjectures'', Preprint, \url{arXiv:1404.4616}] extended and refined the conjectures of Gorsky and Negut to give a combinatorial interpretation of the coefficients that arise in expansion of \(Q_{m,n}(-1)^n\) in terms of the fundamental quasi-symmetric functions for arbitrary \(m\) and \(n\) which we will call the rational shuffle conjecture. The rational shuffle conjecture was later proved by \textit{A. Mellit} [``Toric braids and \((m, n)\)-parking functions'', Preprint, \url{arXiv:1604.07456}]. The main goal of this paper is to study the combinatorics of the coefficients that arise in the Schur function expansion of \(Q_{m,n}(-1)^n\) in the case where \(m\) or \(n\) equals 3. Macdonald polynomials; parking functions; Dyck paths; shuffle conjecture Symmetric functions and generalizations, Combinatorial aspects of representation theory, Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Schur function expansions and the rational shuffle conjecture
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Vénéreau polynomials are a sequence of polynomials
\[
b_{m}=y+x^{m}(xz+y(yu+z^{2}))\in\mathbb{C}[x][y,z,u],
\]
\(m\geq1\), which were proposed by S. Vénéreau as potential counterexamples to important conjectures in affine geometry: the Abhyankar-Sathaye embedding conjecture which asserts that every closed embedding of an affine space into another is equivalent to an embedding as a linear subspace, and the Dolgachev-Weisfeiler conjecture which asks whether every flat fibration from an affine space to another with all fibers also isomorphic to affine spaces is a trivial affine bundle.
It is known that the level hypersurfaces of Vénéreau polynomials are isomorphic to affine spaces of dimension \(3\) and that for every \(m\geq1\), the fibration \((b_{m},x):\mathbb{A}^{4}\rightarrow\mathbb{A}^{2}\) is flat with all fibers isomorphic to affine spaces \(\mathbb{A}^{2}\). So \(b_{m}\) provides a counterexample to the Abhyankar-Sathaye embedding conjecture unless it is a \textit{variable} of the polynomial ring \(\mathbb{C}[x,y,z,u]\), i. e., there exists polynomials \(f_{1},f_{2},f_{3}\) such that \(\mathbb{C}[x,y,z,u]=\mathbb{C}[b_{m},f_{1},f_{2},f_{3}]\), and a counterexample to the Dolgachev-Weisfeiler conjecture unless it has the stronger property to be a \(\mathbb{C}[x]\)-\textit{variable} of \(\mathbb{C}[x,y,z,u]\) in the sense that there exists polynomials \(g_{1},g_{2}\) such that \(\mathbb{C}[x,y,z,u]=\mathbb{C}[x,b_{m},g_{1},g_{2}]\). It was established by S. Vénéreau that \(b_{m}\) is indeed a \(\mathbb{C}[x]\)-coordinate for every \(m\geq3\) but the question for \(m=1,2\) remained open.
In the article under review, the author introduces a more general class of \textit{Vénéreau -type polynomials} of the form \(f_{Q}=y+xQ\) with \(Q\in\mathbb{C}[x][v,w]\) and he proves that \(b_{m}\) is a coordinate if and only if so is \(f_{Q}\) for \(Q=x^{2m-1}w\). This is applied to recover the fact that \(b_{m}\) is a \(\mathbb{C}[x]\)-coordinate for \(m\geq3\) and to prove the new result that \(b_{2}\) is a \(\mathbb{C}[x]\)-coordinate too. Other properties of polynomials \(f_{Q}\) in relation with the aforementioned conjectures are established. polynomial rings; Vénéreau polynomials; coordinates; Dolgachev-Weisfeiler conjecture Lewis, D., Vénéreau-type polynomials as potential counterexamples, J. Pure Appl. Algebra, 217, 5, 946-957, (2013) Polynomials over commutative rings, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Vénéreau-type polynomials as potential counterexamples
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by \textit{I. P. Goulden} et al. [Ann. Math. Blaise Pascal 21, No. 1, 71--89 (2014; Zbl 1296.05202)]. Generalizing a result of \textit{A. Okounkov} [Math. Res. Lett. 7, No.4, 447--453 (2000; Zbl 0969.37033)], we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of \textit{I. P. Goulden} et al. [Adv. Math. 198, No. 1, 43--92 (2005; Zbl 1086.14022)]. Hurwitz numbers; Toda lattice Goulden, IP; Guay-Paquet, M; Novak, J, Toda equations and piecewise polynomiality for mixed double Hurwitz numbers, SIGMA Symmetry Integrability Geom. Methods Appl., 12, 1-10, (2016) Permutations, words, matrices, Relationships between algebraic curves and integrable systems, Families, moduli of curves (algebraic) Toda equations and piecewise polynomiality for mixed double Hurwitz numbers
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the first half of this article, we review the Steinberg theory for double flag varieties for symmetric pairs. For a special case of the symmetric space of type AIII, we will consider \(X=\mathrm{GL}_{2n}/P_{(n,n)}\times\mathrm{GL}_n/B_n^+\times\mathrm{GL}_n/B_n^-\) on which \(K=\mathrm{GL}_n\times \mathrm{GL}_n\) acts diagonally. We give a classification of \(K\)-orbits in \(X\), and explicit combinatorial description of the Steinberg maps.
In the latter half, we develop the theory of embedding of a double flag variety into a larger one. This embedding is a powerful tool to study different types of double flag varieties in terms of the known ones. We prove an embedding theorem of orbits in full generality and give an example of type CI which is embedded into type AIII. Grassmannians, Schubert varieties, flag manifolds, Coadjoint orbits; nilpotent varieties, Differential geometry of symmetric spaces, Exact enumeration problems, generating functions Orbit embedding for double flag varieties and Steinberg maps
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Q\) be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid \(\mathcal{CM}(Q)\), as introduced by M. Reineke, and the generic composition algebra \(\mathcal C_q(Q)\), as introduced by C. M. Ringel, specialised at \(q=0\). In this thesis we continue their work. We show that if \(Q\) is a Dynkin quiver or an oriented cycle, then \(\mathcal C_0(Q)\) is isomorphic to the monoid algebra of \(\mathbb Q\mathcal{CM}(Q)\). Moreover, if \(Q\) is an acyclic, extended Dynkin quiver, we show that there exists a surjective homomorphism \(\Phi\colon\mathcal C_0(Q)\to\mathbb Q\mathcal{CM}(Q)\), and we describe its non-trivial kernel.
Our main tool is a geometric version of BGP reflection functors on quiver Grassmannians and quiver flags, that is varieties consisting of filtrations of a fixed representation by subrepresentations of fixed dimension vectors. These functors enable us to calculate various structure constants of the composition algebra.
Moreover, we investigate geometric properties of quiver flags and quiver Grassmannians, and show that under certain conditions, quiver flags are irreducible and smooth. If, in addition, we have a counting polynomial, these properties imply the positivity of the Euler characteristic of the quiver flag. representations of quivers; Hall polynomials; Hall algebras; Schur roots; composition monoids; extended Dynkin quivers; quiver flag varieties; composition algebras; reflection functors; quiver Grassmannians Wolf, S.: The Hall Algebra and the Composition Monoid. arXiv:0907.1106 Representations of quivers and partially ordered sets, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds The Hall algebra and the composition monoid.
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathfrak{g}_k\) be the Lie algebra of a connected semisimple algebraic group over an algebraically closed field \(k\) of positive characteristic. Two important results concerning the sheaf \(\mathcal{D}\) of twisted differential operators on the corresponding flag manifold, which are Beilinson-Bernstein type derived equivalence between the category of certain representations of \(\mathfrak{g}_k\) and that of \(\mathcal{D}\)-modules, and the split Azumaya property of \(\mathcal{D}\) over a certain central subalgebra.
The author gives an analogue using quantized flag manifolds and quantized enveloping algebras at roots of unity instead of ordinary flag manifolds and ordinary enveloping algebras in positive characteristics. More specifically, the author describes the cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power. For the De Concini-Kac type quantized enveloping algebra, where the parameter \(q\) is specialized to a root of unity whose order is a prime power, it follows that the number of irreducible modules with a certain specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber, giving a weak version of a conjecture of Lusztig concerning non-restricted representations of the quantized enveloping algebra. quantized enveloping algebra; flag manifold; differential operator Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations Differential operators on quantized flag manifolds at roots of unity. III
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the symmetric adjacency matrices of simply laced and extended Dynkin diagrams. These diagrams play an important role in Lie-theory (classification of root-systems), they also appear in singularity theory (linked to Kleinian singularies) and in representation theory of algebras (classification of symmetric algebras of radical cube zero of finite and tame representation type).
This last mentioned classification motivated the study of Chebyshev polynomials of the second kind, evaluated at the adjacency matrices of the aforementioned Dynkin diagrams.
The layout of the paper is as follows: Section 2: definition of the Chebyshev polynomials, Section 3: the values of the polynomials are periodic and grow linearly, Section 4: the values govern the minimal projective resolution of the symmetric algebras of radical cube zero, Section 5: Chebyshev polynomials evaluated at positive, symmetric matrices.
It is quite interesting that the main tools, the Chebyshev polynomials of the second kind, are introduced through their recurrence relation and as \(\det(2xI_n-A_n)\) (where \(A_n\) is the \(n\times n\) matrix whose elements are zero, except for the first super- and sub-diagonal, where the entries are one). There is no mentioning at all of their orthogonality properties (which also imply certain minimality and projection properties). Chebyshev polynomials; symmetric matrices; Dynkin diagrams; symmetric algebras; projective resolutions Erdmann, K.; Schroll, S., Chebyshev polynomials on symmetric matrices, Linear Algebra Appl., 434, 12, 2475-2496, (2011) Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Matrix equations and identities, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Homological methods in associative algebras, Deformations of singularities Chebyshev polynomials on symmetric matrices
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a semisimple simply connected algebraic group, \(T\) a maximal torus in \(G,W\) its Weyl group, and \(B \supset T\) a Borel subgroup. For any \(w \in W\) let \(e_w\) be the point of \(G/B\) and \(X(W)\) the Schubert variety associated to \(W\). Let \(\tau \leq w\) in \(W\), where \(\leq\) is the Bruhat order. In this note the author gives a recursive formula for \(m_\tau (w)\), the multiplicity of \(X(W)\) in \(e_w\). Complete proofs will appear in the author's forthcoming paper ``Tangent cones at singular points on a Schubert variety''. multiplicity of points; maximal torus; Schubert variety Grassmannians, Schubert varieties, flag manifolds Multiplicities of points on a Schubert variety
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei-Thoma theorem. In this paper, we study positive specializations of symmetric Grothendieck polynomials, \(K\)-theoretic deformations of Schur polynomials. Grothendieck polynomials; total positivity; symmetric functions; combinatorial \(K\)-theory Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Positive specializations of symmetric Grothendieck polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \((W, S)\) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let \(J\subseteq S\). Let \(W^J\) denote the set of minimal coset representatives modulo the parabolic subgroup \(W_J\). For \(w\in W^J\), let \(f^{w,J}_i\) denote the number of elements of length \(i\) below \(w\) in Bruhat order on \(W^J\) (with notation simplified to \(f^w_i\) in the case when \(W^J= W\)). We show that
\[
0\leq i< j\leq\ell(w)- i\quad\text{implies}\quad f^{w,J}_i\leq f^{w,J}_j.
\]
Also, the case of equalities \(f^w_i= f^w_{\ell(w)-i}\) for \(i= 1,\dots,k\) is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial \(P_{e,w}(q)\).
We show that if \(W\) is finite then the number sequence \(f^w_0, f^w_1,\dots, f^w_{\ell(w)}\) cannot grow too rapidly. Further, in the finite case, for any given \(k\geq 1\) and any \(w\in W\) of sufficiently great length (with respect to \(k\)), we show
\[
f^w_{\ell(w)-k}\geq f^w_{\ell(w)- k+1}\geq\cdots\geq f^w_{\ell(w)}.
\]
The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if \(\overline X_w\) is a Schubert variety of dimension \(d=\ell(w)\), and \(\lambda= c_1({\mathcal L})\in H^2(\overline X_w)\) is the restriction to \(\overline X_w\) of the Chem class of an ample line bundle, then
\[
(\lambda^k)\cdot: H^{d-k}(\overline X_w)\to H^{d+k}(\overline X_w)
\]
is injective for all \(k\geq 0\). crystallographic Coxeter group; Weyl group; Bruhat order; Schubert variety; \(\ell \)-adic cohomology; intersection cohomology; Kazhdan-Lusztig polynomial Björner, A; Ekedahl, T, On the shape of Bruhat intervals, Ann. Math., 170, 799-817, (2009) Algebraic combinatorics, Algebraic aspects of posets, Étale and other Grothendieck topologies and (co)homologies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Permutations, words, matrices On the shape of Bruhat intervals
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use the Thom polynomial theory developed by \textit{L. Fehér} and \textit{R. Rimányi} [Duke Math. J. 114, No.2, 193--213 (2002; Zbl 1054.14010)] to prove the component formula for quiver varieties conjectured by \textit{A. Knutson, E. Miller} and \textit{M. Shimozono} [Four positive formulae for type A quiver polynomials, preprint, \texttt{http://arxiv.org/abs/math.AG/0308142}]. This formula expresses the cohomology class of a quiver variety as a sum of products of Schubert polynomials indexed by minimal lace diagrams, and implies that the quiver coefficients of \textit{A. Buch} and \textit{W. Fulton} [Invent. Math. 135, 665--687 (1999; Zbl 0942.14027)] are non-negative. We also apply our methods to give a new proof of the component formula from the Gröbner degeneration of quiver varieties, and to give generating moves for the KMS-factorizations that form the index set in \(K\)-theoretic versions of the component formula. Buch, Anders S.; Fehér, László M.; Rimányi, Richárd, Positivity of quiver coefficients through Thom polynomials, Adv. Math., 197, 1, 306-320, (2005) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Singularities of differentiable mappings in differential topology, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Positivity of quiver coefficients through Thom polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this paper is to present explicit closed algebraic formulas for Orlov-Scherbin \(n\)-point functions. More precisely, the authors give an explicit closed formula for \[H_{g,n}=\sum_{m_1\cdots m_n=1}^\infty h_{g,(m_1,\dots,m_n)}X_1^{m_1}\cdots X_n^{m_n},\] where \(h_{g,(m_1,\dots,m_n)}\) are weighted double Hurwitz numbers for each pair \((g, n)\). They derive a new explicit formula in terms of sums over graphs for the \(n\)-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions). They use a change of variables deduced from the structure of the associated spectral curve, and obtain a formula that turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.
The paper is organized as follows. The first section is an introduction to the subject. In Section 2, the authors give some notations and recall the basic formalism of operators in the bosonic Fock space that they use throughout the paper. In Section 3 they compute \(H_{g,n}\) as a series in \(X_1,\dots,X_n\), which, in particular, leads to a formula giving each particular formal weighted double Hurwitz number \(h_{g,(m_1,\dots,m_n)}\) in a closed form. Strictly speaking, this section is not necessary for the rest of the paper, but it sets up the notation and clarifies the logic of the calculations. In Section 4 the authors derive an explicit closed formula for \(D_1\cdots D_nH_{g,n}\), where \(D_i=X_i\partial_{X_i}\). In Section 5 they prove the main result of the paper, which explicitly represents \(H_{g,n}\) for given \(g\) and \(n\) in a closed form. Section 6 deals with the slightly exceptional cases of \(n=1\) for any \(g\) and \((g, n)=(0, 2)\). Section 7 is devoted to some applications of main general formula obtained in this paper, thus deriving explicit expressions for \(H_{g,n}\) for small \(g\) and \(n\) in terms of small number of basic functions. Hurwitz numbers; KP tau functions; Fock space Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Coverings of curves, fundamental group, Enumerative problems (combinatorial problems) in algebraic geometry, Exact enumeration problems, generating functions Explicit closed algebraic formulas for Orlov-Scherbin \(n\)-point functions
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We exploit the doubled formalism to study comprehensive relations among T-duality, complex and bi-hermitian structures \(( J_+, J_-)\) in two-dimensional \(\mathcal{N} = (2, 2)\) sigma models with/without twisted chiral multiplets. The bi-hermitian structures \(( J_+, J_-)\) embedded in generalized Kähler structures \((\mathcal{J}_+, \mathcal{J}_-)\) are organized into the algebra of the tri-complex numbers. We write down an analogue of the Buscher rule by which the T-duality transformation of the bi-hermitian and Kähler structures are apparent. We also study the bi-hypercomplex and hyperkähler cases in \(\mathcal{N} = (4, 4)\) theories. They are expressed, as a T-duality covariant fashion, in the generalized hyperkähler structures and form the split-bi-quaternion algebras. As a concrete example, we show the explicit T-duality relation between the hyperkähler structures of the KK-monopole (Taub-NUT space) and the bi-hypercomplex structures of the H-monopole (smeared NS5-brane). Utilizing this result, we comment on a T-duality relation for the worldsheet instantons in these geometries. Correspondence, duality, holography (AdS/CFT, gauge/gravity, etc.), Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Holomorphic symplectic varieties, hyper-Kähler varieties Hyperkähler, bi-hypercomplex, generalized hyperkähler structures and T-duality
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper can be viewed as part of the quantum geometric Langlands program. One considers a connected reductive group \(G\) over \(k = \overline{\mathbb{F}_p}\) and \(F = k(\!(t)\!)\). We are given a central extension \(\mathbb{E}\) of \(G(F)\) by \(\mathbb{G}_m\), seen as group ind-schemes over \(k\), and assume that the extension splits over \(G(k[\![t]\!])\). Fix an embedding \(\zeta: \mu_N(k) \to \overline{\mathbb{Q}_\ell}^\times\). One can then consider various ``twisted'' objects in geometric Langlands program, such as \(\widetilde{\mathrm{Gr}}_G\) and \(\widetilde{\mathrm{Bun}}_G\) in the global case over a smooth projective \(k\)-curve \(X\). They are the geometric counterparts of the classical theory of metaplectic coverings.
Inspired by [\textit{D. Gaitsgory}, Sel. Math., New Ser. 13, No. 4, 617--659 (2007; Zbl 1160.17009)], the main result (Theorem 4.11.5) establishes an equivalence \(\overline{\mathbb{F}}\) from the twisted Whittaker category with \(n\) points \(\mathrm{Whit}_n^\kappa\) to the category \(\widetilde{\mathrm{FS}}^\kappa_n\) of factorizable sheaves; here \(\kappa\) stands for the quadratic form attached to \(\mathbb{E}\). The theorem requires a local assumption that \(\kappa\) satisfies the ``subtop cohomology property'', which is proven in the paper for all semi-simple simply connected \(G\) and for most \(\kappa\). Moreover, this functor is \(t\)-exact and commutes with Verdier duality.
In the \S 5 of the paper, the author defines an action of \(\mathrm{Rep}(\check{G}_\zeta)\) by Hecke functors on the twisted derived category \(D_\zeta(\widetilde{\mathrm{Bun}}_G)\) and on \(D\mathrm{Whit}_x^\kappa\), where \(\check{G}_\zeta\) is the metaplectic dual group attached to \(\mathbb{E}\), \(N\) and \(\zeta\) defined via the twisted geometric Satake correspondence. Another main result (Theorem 10.1.2) asserts that \(\mathbb{F}\) commutes with Hecke-actions. quantum geometric Langlands program; metaplectic group Lysenko, Sergey, Twisted Whittaker models for metaplectic groups, Geom. Funct. Anal., 27, 2, 289-372, (2017) Geometric Langlands program: representation-theoretic aspects, Geometric Langlands program (algebro-geometric aspects) Twisted Whittaker models for metaplectic groups
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We provide a generators and relation description of the deformed \(W_{1+\infty^-}\)-algebra introduced in previous joint work of E. Vasserot and the second author. This gives a presentation of the (spherical) cohomological Hall algebra of the one-loop quiver, or alternatively of the spherical degenerate double affine Hecke algebra of \(GL(\infty)\). presentations; deformed \(W\)-algebras; cohomological Hall algebras; double affine Hecke algebras N. Arbesfeld and O. Schiffmann, \textit{A presentation of the deformed W}\_{}\{1+\(\infty\)\}\textit{algebra}, in \textit{Symmetries, integrable systems and representations}, Springer, London, U.K., (2013), pg. 1. Deformations of associative rings, Quantum groups (quantized enveloping algebras) and related deformations, Vertex operators; vertex operator algebras and related structures, Noncommutative algebraic geometry, Hecke algebras and their representations A presentation of the deformed \(W_{1+\infty}\) algebra.
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The present paper considers Grothendieck dessins in the plane looking like flower trees, i.e. with graph diameter \(4\), a vertex of valency \(r\) as midpoint whose \(k+l\) neighbour vertices have \(k\) times valency \(m\) and \(l\) times valency \(n\not=m\,\). The author gives a nice and explicit method to determine the Belyi function belonging to this dessin and proves that the moduli field of the dessin is a field of definition for the Belyi function. The result generalises earlier contributions by \textit{L. Schneps} [in: The Grothendieck theory of dessins d'enfants. Lond. Math. Soc. Lect. Note Ser. 200, 47--77 (1994; Zbl 0823.14017)], \textit{G. Shabat} and \textit{A. Zvonkin} [Contemp. Math. 178, 233--275 (1994; Zbl 0816.05024)] and \textit{L. Zapponi} [Compos. Math. 122, No. 2, 113--133 (2000; Zbl 0968.14011)]. Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group Belyi function whose Grothendieck dessin is a flower tree with two ramification indices
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The introduction discusses various motivations for the following chapters of the
thesis, and their relation to questions around mirror symmetry.
The main theorem of chapter 2 says that if the quantum cohomology of a smooth
projective variety \(V\) yields a generically semisimple product, then the same holds true
for its blow-up at any number of points (theorem 3.1.1). This is a positive test for a
conjecture by Dubrovin, which claims that quantum cohomology of \(V\) is generically
semisimple if and only if its bounded derived category \(Db(V)\) has a complete exceptional collection.
Chapter 3 generalizes Bridgeland's notion of stability condition on a triangulated
category. The generalization, a polynomial stability condtion (definition 2.1.4), allows
the central charge to have values in polynomials \(\mathbb{C}[N ]\) instead of complex numbers \(\mathbb{C}\).
We show that polynomial stability conditions have the same deformation properties as
Bridgeland's stability conditions (theorem 3.2.5). In section 4, it is shown that every
projective variety has a canonical family of polynomial stability conditions.
In chapter 4, we define and study the notion of a weighted stable map (definition
2.1.2), depending on a set of weights. We show the existence of moduli spaces of
weighted stable maps as proper Deligne-Mumford stacks of finite type (theorem 2.1.4),
and study in detail their birational behaviour under changes of weights (section 4).
We introduce a category of weighted marked graphs in section 6, and show that it
keeps track of the boundary components of the moduli spaces, and natural morphisms
between them. We introduce weighted Gromov-Witten invariants by defining virtual
fundamental classes, and prove that these satisfy all properties to be expected (sections
5 and 7). In particular, we show that Gromov-Witten invariants without gravitational
descendants do not depend on the choice of weights. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Derived categories, triangulated categories Semisimple quantum cohomology, deformations of stability conditions and the derived category
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Deodhar decomposition of the Grassmannian is a refinement of the Schubert, Richardson, and positroid stratifications of the Grassmannian. Go-diagrams are certain fillings of Ferrers diagrams with black stones, white stones, and pluses which index Deodhar components in the Grassmannian. We provide a series of corrective flips on diagrams which may be used to transform arbitrary fillings of Ferrers shapes with black stones, white stones, and pluses into a Go-diagram. This provides an extension of Lam and Williams' Le-moves for transforming reduced diagrams into Le-diagrams to the context of non-reduced diagrams. Next, we address the question of describing when the closure of one Deodhar component is contained in the closure of another. We show that if one Go-diagram \(D\) is obtained from another \(D'\) by replacing certain stones with pluses, then applying corrective flips, that there is a containment of closures of the associated Deodhar components, \(\overline{\mathcal{D}'} \subset \overline{\mathcal{D}}\). Finally, we address the question of verifying whether an arbitrary filling of a Ferrers shape with black stones, white stones, and pluses is a Go-diagram. We show that no reasonable description of the class of Go-diagrams in terms of forbidden subdiagrams can exist by providing an injection from the set of valid Go-diagrams to the set of minimal forbidden subdiagrams for the class of Go-diagrams. In lieu of such a description, we offer an inductive characterization of the class of Go-diagrams. Deodhar decomposition; Go-diagram; Grassmannian Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Combinatorial aspects of block designs Combinatorics of the Deodhar decomposition of the Grassmannian
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An unpublished result of D. Peterson states that the complexified quantum cohomology of a homogeneous manifold \(G/P\) (where \(P\) is an parabolic subgroup in \(G\)) is isomorphic to the coordinate ring of a subvariety in \(G^\vee / B^\vee\) (where \(G^\vee\) is the Langlands dual group and \(B^\vee\) is a Borel subgroup in \(G^\vee\)). This paper verifies Peterson's result for the Lagrangian and orthogonal Grassmannians, using explicit presentations of their quantum cohomology rings given by \textit{A. Kresch} and \textit{H. Tamvakis} [Algebr. Geom. 12, No. 4, 777--810 (2003; Zbl 1051.53070)] and [Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083)]. In \S 3 and \S 4, explicit construction of the corresponding subvarieties and descriptions of their elements are given. \S 5 applies these explicit formulas to prove a Vafa-Intriligator type formula for the \(m\)-pointed Gromov-Witten invariants of these Grassmannians. It also verifies that the quantum Euler classes [\textit{L. Abrams}, Isr. J. Math. 117, 335--352 (2000; Zbl 0954.53048)] of these Grassmannians are invertible, which implies that the quantum cohomology rings are semisimple when restricted some \(q \in \mathbb C^*\). Vafa-intriligator; quantum cohomology; quantum Euler class Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations Vafa-intriligator type formulas and quantum Euler classes for Lagrangian and orthogonal Grassmannianns
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W\) be a polynomial or power series in several variables, or, more generally, a nonzero element in some regular commutative ring. A matrix factorization of \(W\) consists of a pair of square matrices \(X\) and \(Y\) of the same size, with entries in the given ring, such that the matrix product \(XY\) is \(W\) multiplied by the identity matrix. For example, if \(X\) is a matrix whose determinant is \(W\) and \(Y\) is its adjoint matrix, then \((X, Y)\) is a matrix factorization of \(W\). Such matrix factorizations are nowadays ubiquitous in several different fields in physics and mathematics, including String Theory, Commutative Algebra, Algebraic Geometry, both in its classical and its noncommutative version, Singularity Theory, Representation Theory, Topology, there in particular in Knot Theory. The workshop has brought together leading researchers and young colleagues from the various input fields; it was the first workshop on this topic in Oberwolfach. For some leading researchers from neighboring fields, this was their first visit to Oberwolfach. Collections of abstracts of lectures, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to category theory, Proceedings, conferences, collections, etc. pertaining to commutative algebra, Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Categorical structures, Derived categories, triangulated categories, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homological methods in commutative ring theory, Representation theory of associative rings and algebras Matrix factorizations in algebra, geometry, and physics. Abstracts from the workshop held September 1--7, 2013.
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G(r,n)\) be the Grassmannian variety of \(r\)-subspaces in an \(n\)-dimensional vector space over an algebraically closed field \(k\) of characteristic zero, \(k_q=k[q,q^{-1}]\). The paper under review contains a quite general extension of results of previous work [\textit{R.~Fioresi}, Rev. Math. Phys. 9, No. 4, 453-465 (1997; Zbl 0888.17008)] where the author studied deformations of the big cell of \(G(2,4)\) together with the coaction of the parabolic subgroup of \(\text{SL}_4(k)\). In particular, he considers the big cell \(U_l\) inside the Grassmannian manifold \(G(r,2r)\) and shows that the deformation \(k_q[U_l]\) of the big cell coordinate ring is isomorphic to the matrix bialgebra \(k_q[M_r]\), and that the former coincides with a quantum homogeneous space for the deformation \(k_q[P_l]\) of the lower maximal parabolic subgroup \(P_l\). Some properties of the holomorphic de Rham complex on \(k_q[U_l]\) are also discussed. quantum Grassmannians; big cells; parabolic subgroups; Minkowski spaces; conformal groups; quantum de Rham complexes; deformations; matrix bialgebras Fioresi, R., A deformation of the big cell inside the Grassmannian manifold \(G(r, n)\), Rev. Math. Phys., 11, 25-40, (1999) , Grassmannians, Schubert varieties, flag manifolds, Noncommutative geometry in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Geometry of quantum groups, Deformations of associative rings A deformation of the big cell inside the Grassmannian manifold \(G(r,n)\)
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux's interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this way, we obtain a direct proof of the result of Lenart and Postnikov and then prove that the set of permutations occurring in the result is actually an interval of the Bruhat order. double Grothendieck polynomials; key polynomials; 0-Hecke algebra; sorting operators; Bruhat order; Pieri formula Pons, V.: Interval structure of the Pieri formula for Grothendieck polynomials, Internat. J. Algebra comput. 23, 123-146 (2013) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Interval structure of the Pieri formula for Grothendieck polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the article under review, the author continues his study on the arithmetic intersection theory of projective arithmetic schemes whose generic fiber is a compact Hermitian homogeneous space of a compact Lie group. He determines the structure of the Arakelov Chow ring of the even orthogonal Grassmannian \(\text{OG}=\text{OG}(n+1,2n+2)\), which parameterizes isotopic subspaces of dimension \(n+1\) in a \((2n+2)\)-dimensional vector space equipped with a smooth quadratic form. He identifies the Arakelov Chow ring \(\text{CH}(\overline{\text{OG}})\otimes_{\mathbb Z}\mathbb Z[\frac 12]\) with a quotient of \(\mathbb Z[\frac 12][\widehat{x}_1,\dots,\widehat{x}_{n+1}]^{S_{n+1}} \oplus\mathbb R[x_1,\dots,x_n]^{S_n}\), where he gives the relations generating the kernel. In spite of the similarity to the construction for the Lagrangian Grassmannian case established previously by the author in [J. Reine Angew. Math. 516, 207--223 (1999; Zbl 0934.14018)], the situation here is more complicated. Already in the standard presentation of \(\text{CH}(\text{OG})\), an extra relation appears. Moreover, in the relations defining \(\text{CH}(\overline{\text{OG}})\), there is a constant \(r_n\) which the author keeps inexplicit for a moment.
The presentation of \(\text{CH}(\overline{\text{OG}})\) given, the author develops an arithmetic Schubert calculus in \(\text{CH}(\overline{\text{OG}})\), by using the technic of \(\widetilde{P}\)-polynomial introduced by \textit{P. Pragacz} and \textit{J. Ratajski} [Compos. Math. 107, No. 1, 11--87 (1997; Zbl 0916.14026)]. As an application, the author calculates the Faltings height of \(\text{OG}\) with respect to its fundamental embedding in projective space. The comparison to the computation effectuated previously by the author in [Mich. Math. J. 48, Spec. Vol., 593--610 (2000; Zbl 1077.14527)] gives the value of \(r_n\), which completes the proof of the structure theorem for \(\text{CH}(\overline{\text{OG}})\). Arakelov Chow group; orthogonal Grassmannian; height Tamvakis H.: Arakelov theory of even orthogonal Grassmannians. Comment. Math. Helv. 82, 455--475 (2007) Arithmetic varieties and schemes; Arakelov theory; heights, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Heights Arakelov theory of even orthogonal Grassmannians
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Associated to the quantum multiplication in the quantum cohomology of, say, a Nakajima quiver variety there is a flat connection called the quantum connection (a.k.a. Dubrovin connection or quantum differential equation). The paper under review constructs and studies the \(K\)-theoretic analog of the quantum connection. Namely, the authors describe the quantum differential equations that arise in the enumerative K-theory of ``quasimap'' counts of curves in Nakajima quiver varieties.
The centrality of this concept is illustrated by the following relations. (a) For certain choices of the quiver the varieties are Hilbert schemes of points on a surface, and the resulting quantum connection plays a key role in comparing the corresponding Gromov-Witten and Donaldson-Thomas theories. (b) The quantum connection should also be interpreted as the (K-theoretic version of the) generalized Casimir connection of the Maulik-Okounkov Yangian associated to the quiver. (c) The quantum differential equations commute with another set of differential equations, the quantum Knizhnik-Zamolodchikov equations.
The actual path the authors follow to describe the quantum difference equations is also remarkable. First they define a ``quantum dynamical Weyl group(oid)'', acting on the torus equivariant \(K\)-theory. This Weyl group naturally contains a lattice and the action of this lattice is identified with the sought after quantum differential equations. \(K\)-theory of Nakajima quiver varieties; quasimap count; quantum connection Projective and enumerative algebraic geometry, Surfaces and higher-dimensional varieties, \(K\)-theory, Quantum theory Quantum difference equation for Nakajima varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use operators to reformulate the Andersen-Jantzen-Soergel/Billey formula for the point restrictions of equivariant Schubert classes of the cohomology of \(G/B\). We introduce new operators whose coefficients compute Schubert structure constants (in a manifestly polynomial, but not positive, way), resulting in a formula much like and generalizing the positive AJS/Billey formula. Our proof involves Bott-Samelson manifolds, and in particular, the cohomology basis dual to the homology basis of classes of sub-Bott-Samelson manifolds. Schubert calculus; nil Hecke algebra Classical problems, Schubert calculus Schubert structure operators
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0553.00001.]
This report is devoted to the modern algebro-geometrical viewpoint on the representations of real reductive groups which connects the theory of such representations with the algebro-geometrical properties of Schubert- like varieties; this approach and appropriate techniques have been developed by the author, J. Bernstein, P. Deligne, J.-L. Brylinski, M. Kashiwara, D. Kazhdan, G. Lusztig, T. Springer, I. Verdier, D. Vogan, W. Fulton, R. MacPherson and others.
The following topics are included: A. Affine spaces and localization, B. Twisted rings of differential operators, C. Main construction, D. Functorial properties of D-modules, E. Holonomic modules and Harish- Chandra modules, F. Perverse sheaves, G. Motivic language: mixed perverse sheaves, H. The Kazhdan-Lusztig algorithm, I. Jantzen's filtration. representations of real reductive groups; rings of differential operators; Harish-Chandra modules; mixed perverse sheaves; Kazhdan- Lusztig algorithm; Jantzen's filtration A. A. Beilinson ''Localization of Representations of Reductive Lie Algebras,'' in Proc. IMC (Warsaw, 1983) (PWN, Warsaw, 1984), pp. 699--710. Representation theory for linear algebraic groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Simple, semisimple, reductive (super)algebras, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Classical groups (algebro-geometric aspects), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Localization of representations of reductive Lie algebras
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the framework of differential Galois theory we treat the classical spectral problem \(\Psi''- u(x)\Psi= \lambda\Psi\) and its finite-gap potentials as exactly solvable in quadratures by Picard-Vessiot without involving special functions (the ideology goes back to the 1919 works by \textit{J. Drach} [C. R. 168, 47--50 (1919; JFM 47.0411.03); C. R. 168, 337--340 (1919; JFM 47.0412.01)]). We show that duality between spectral and quadrature approaches is realized through the Weierstrass permutation theorem for a logarithmic abelian integral. From this standpoint we inspect known facts and obtain new ones: an important formula for the \(\Psi\)-function and \(\Theta\)-function extensions of Picard-Vessiot fields. In particular, extensions by Jacobi's \(\theta\)-functions lead to the (quadrature) algebraically integrable equations for the \(\theta\)-functions themselves. Schrödinger equation; finite-gap potentials; Picard-Vessiot theory; quadratures; Liouvillian extensions; Abelian integrals; theta-functions Yurii V. Brezhnev, Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials, Algebraic aspects of Darboux transformations, quantum integrable systems and supersymmetric quantum mechanics, Contemp. Math., vol. 563, Amer. Math. Soc., Providence, RI, 2012, pp. 1 -- 31. Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Selfadjoint operator theory in quantum theory, including spectral analysis, Exactly and quasi-solvable systems arising in quantum theory, Subvarieties of abelian varieties, Differential algebra, Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The non-negative integer cocharge statistic on words was introduced in [\textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. A 286, 323--324 (1978; Zbl 0374.20010); Quad. Ric. Sci. 109, 129--156 (1981; Zbl 0517.20036)] to combinatorially characterize the Hall-Littlewood polynomials. Cocharge has since been used to explain phenomena ranging from the graded decomposition of Garsia-Procesimodules to the cohomology structure of the Grassmann variety. Although its application to contemporary variations of these problems had been deemed intractable, we prove that the two-parameter, symmetric Macdonald polynomials are generating functions of a distinguished family of colored words. Cocharge adorns one parameter and the second measure its deviation from cocharge on words without color. We use the same framework to expand the plactic monoid, apply Kashiwara's crystal theory to various Garsia-Haiman modules, and to address problems in \(K\)-theoretic Schubert calculus. non-negative integer cocharge statistic on words; Kashiwara's crystal theory; Garsia-Haiman modules; \(K\)-theoretic Schubert calculus Combinatorial aspects of representation theory, Exact enumeration problems, generating functions, Representations of finite symmetric groups, Classical problems, Schubert calculus Colorful combinatorics and Macdonald polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The present paper focuses on pairings on the Hochschild cohomology ring of smooth complex projective varieties. There are mainly two such pairings in the literature, namely:
-- The Shklyarov pairing, introduced in the DG-framework in the preprint [\textit{D. Shklyarov}, ``Hirzebruch-Riemann-Roch theorems for DG-algebras'', \url{arXiv:0710.1937}].
-- The Mukai pairing, defined by \textit{A. Căldăraru} and \textit{S. Willerton} in [New York J. Math. 16, 61--98 (2010; Zbl 1214.14013)] via Serre duality.
Results of \textit{N. Markarian} [J. Lond. Math. Soc., II. Ser. 79, No. 1, 129--143 (2009; Zbl 1167.14005)] and the author [New York J. Math. 14, 643--717 (2008; Zbl 1158.19002)] imply that the Mukai pairing on a variety \(X\) is given (up to sign) via the Hochschild-Kostant-Rosenberg isomorphism by the formula \(<a, b>=\int_X a \wedge b \wedge \mathrm{Td} (X)\). Later on, it has been proved by the author [Mosc. Math. J. 10, No. 3, 629--645 (2010; Zbl 1208.14013)] that the two aforementioned pairings were the same up to signs.
The aim of the current paper is to obtain directly the expression of the Shklyarov pairing. The method relies on the theory of deformation quantization as developed in [\textit{M. Kashiwara} and \textit{P. Schapira}, Deformation quantization modules. Astérisque 345. Paris: Société Mathématique de France (2012; Zbl 1260.32001)] as well as on the index theorem of \textit{P. Bressler, R. Nest} and \textit{B. Tsygan} [Adv. Math. 167, No. 1, 1--25 (2002; Zbl 1021.53064); ibid. 167, No. 1, 26--73 (2002; Zbl 1021.53065)]. Using this, the proof reduces to prove that the Euler class of the structural sheaf \(\mathcal{O}_X\) is the Todd class of \(X\). This fact has been conjectured by Kashiwara in 1991 and proved by the reviewer in [J. Differ. Geom. 90, No. 2, 267--275 (2012; Zbl 1247.32013)]. A completely different proof is presented here. Hochschild homology; Mukai pairing; Riemann-Roch theorem; deformation quantization Riemann-Roch theorems, Chern characters, Riemann-Roch theorems, Deformation quantization, star products A variant of the Mukai pairing via deformation quantization
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K\) be a compact Lie group, \(G=K^{\mathbb{C}}\) a semisimple connected Lie group and \(P\subset G\) a parabolic subgroup. Let \(X\subset G/P\) be a Schubert variety being canonically embedded into a projective space via the identification of \(G/P\) with a co-adjoint orbit of \(K\). The maximal torus \(T\) of \(K\) acts linearly on the projective space leaving \(X\) invariant. Let \(\psi:X\to \text{Lie}(T)^\ast\) be the restriction of the moment map relative to the Fubini--Study symplectic form. It is proved that all pre-images \(\psi^{-1}(\mu), \mu\in \psi(X)\), are connected subspaces of \(X\).
Let \(S\subset T\) be a one-dimensional subtorus. Let \(f:X\to\mathbb{R}\) be the restriction of the \(S\) moment map to \(X\). Quotients of the form \(f^{-1}(r)/S, r\in \mathbb{R}\), are studied. It is proved that one obtains examples for which the Kirwan surjectivity theorem and Tolman and Weitsman's presentation of the kernel of the Kirwan map hold. The singular Schubert variety in the Grassmannian of planes in \(\mathbb{C}^4\) is discussed in detail. symplectic quotient; Schubert variety; semisimple Lie group Momentum maps; symplectic reduction, Geometric invariant theory On some symplectic quotients of Schubert varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Following the idea of \textit{M. Aganagic} and \textit{A. Okounkov} [J. Am. Math. Soc. 34, No. 1, 79--133 (2021; Zbl 07304878)]], we study vertex functions for hypertoric varieties, defined by \(K\)-theoretic counting of quasimaps from \(\mathbb{P}^1\). We prove the 3d mirror symmetry statement that the two sets of \(q\)-difference equations of a 3d hypertoric mirror pairs are equivalent to each other, with Kähler and equivariant parameters exchanged, and the opposite choice of polarization. Vertex functions of a 3d mirror pair, as solutions to the \(q\)-difference equations, satisfying particular asymptotic conditions, are related by the elliptic stable envelopes. Various notions of quantum \(K\)-theory for hypertoric varieties are also discussed. 3d mirror symmetry; hypertoric varieties; quantum \(K\)-theory String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Equivariant \(K\)-theory, Toric varieties, Newton polyhedra, Okounkov bodies, Mirror symmetry (algebro-geometric aspects), Elliptic cohomology 3d mirror symmetry and quantum \(K\)-theory of hypertoric varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K\) be a field (often assumed in the book to be algebraically closed) and \(V\) be an \(n\)-dimensional \(K\)-vector space. A \textit{flag} in \(V\) is a chain of subspaces: \(\{0\} = V_0 \subset V_1 \subset V_2 \subset \cdots \subset V_n = V\) where the dimension of \(V_i\) is \(i\). The set of all flags in \(V\) is called the \textit{flag variety}. Consider the general linear group \(GL_n(K)\). The flag variety can be identified with the quotient of \(GL_n(K)\) by the Borel subgroup of upper triangular matrices. Equivalently, one can take the special linear group \(SL_n(K)\) modulo upper triangular matrices. For an arbitrary semisimple algebraic group \(G\) and Borel subgroup \(B\), the quotient \(G/B\) may be considered as a \textit{generalized flag variety}. The (classical) flag variety can also be considered as a subvariety of the \textit{Grassmannian variety} \(G_{d,n}\) where \(1 \leq d \leq n - 1\). The Grassmannian variety \(G_{d,n}\) is by definition the set of all \(d\)-dimensional subspaces of \(V\). Using the Plücker map, the variety \(G_{d,n}\) can be identified with a subvariety of the projective space on the \(d\)th exterior power of \(V\); in fact, as the zero set of the so-called \textit{Plücker relations}. The Grassmannians can also be placed into a general algebraic group context since \(G_{d,n}\) can be identified with the quotient of \(GL_n(K)\) by a parabolic subgroup. For either the flag variety or a Grassmannian variety, an important collection of varieties are the Schubert subvarieties. For the flag variety, considered as \(G/B\), given an element \(w\) of the Weyl group, the Schubert variety associated to \(w\) is the Zariski closure of the \(B\)-orbit of the coset \(wB\) inside \(G/B\). For the Grassmannian \(G_{d,n}\), let \(I_{d,n}\) denote the set of \(d\)-tuples \((i_1, i_2, \dots, i_d)\) such that \(1 \leq i_1 < i_2 < \cdots < i_d \leq n\). Classically and more combinatorially, a Schubert variety can be associated to each such tuple by using a fixed basis for \(V\). These varieties can also be identified using \(B\)-orbits (for \(B\) being the upper triangular matrices inside \(GL_n(K)\)).
Flag varieties, Grassmannian varieties, and Schubert varieties have played a key role in the development of algebraic geometry and related subjects (such as algebraic groups). The goal of this book is to provide an introduction to these objects, presented (as suggested by the title) from the perspective that flag varieties involve the interplay of algebraic geometry, algebraic groups, combinatorics, and representation theory. This book stems from a series of lectures given by the first author at the Institute for Advanced Study in 2007 and is aimed at an introductory level. The authors are quite thorough in presenting the necessary background material to develop the subject. In principle, with a solid background in graduate algebra, a student or researcher could learn the basic ideas from this book, although some prior knowledge of algebraic geometry and algebraic groups would be beneficial. The discussion of flag varieties is focused on the classical case (working with the general or special linear group) in order to bring the reader more quickly to the key ideas. Sufficient references are provided to direct the interested reader to the more general case and further developments. To help the reader gain understanding of the ideas, a number of examples are given as well as proofs of most results. References are provided for the results (typically more significant) which are stated without proof. If used as a textbook, all but the first chapter contains a few exercises, which could be supplemented with problems from standard texts. We now briefly describe the contents of the book, from which one can see that it could serve as a concise reference for a number of topics.
The first chapter provides a brief overview of key ideas from commutative algebra (e.g., Noetherian, localization, radicals, Krull dimension, regular, and Cohen-Macauley) and algebraic geometry (e.g., affine and projective varieties, schemes, sheaves of modules, vector and line bundles, the Picard group and tangent spaces). Chapter Two presents a basic discussion of semisimple modules and rings, including the Artin-Molien-Wedderburn Structure Theorem. Brauer groups, central simple algebras, and group algebras (including Maschke's Theorem) are also introduced. Chapter Three presents some basic ideas in the representation theory of finite groups over fields of characteristic zero or prime to the order of the group, including characters, irreducible characters, tensor products, restriction and induction. Chapter Four then focuses on the representation theory of the finite symmetric group \(S_n\) over fields of characteristic zero or prime larger than \(n\). Young tableaux are introduced, which appear throughout the remainder of the book. The simple (or irreducible) \(S_n\)-modules correspond to partitions of \(n\), and a tableaux can be formed from a partition. Two constructions of the simple modules are presented: Frobenius-Young modules (arising as left ideals within the group algebra) and Specht modules (arising from \(S_n\) acting on a polynomial ring in \(n\) variables). The chapter ends with a discussion of the representation theory of the alternating group \(A_n\). In Chapter Five, the discussion continues with computations on the characters and dimensions of Young modules. The chapter contains some general discussion of symmetric polynomials and various bases for them.
Beginning with Chapter Six, the focus shifts toward algebraic groups. The chapter begins with a discussion of endomorphism algebras and then discusses Schur-Weyl Duality: the isomorphism of the group algebra of \(GL_n(\mathbb{C})\) with an endomorphism algebra over \(S_d\) and vice versa. This duality allows one to obtain some of the simple \(GL_n(\mathbb{C})\), the so-called Schur modules, from Young modules. The characters of the Schur modules are discussed. Over \(SL_n(\mathbb{C})\), all of the finite dimensional simple modules arise as certain Schur modules. From these, one can then give a complete description of the finite dimensional simple \(GL_n(\mathbb{C})\)-modules. Before beginning a general discussion of algebraic groups, Chapters Seven and Eight discuss Lie algebras. Chapter Seven provides a brief introduction to Lie algebras and the structure of semisimple Lie algebras, including root systems. Precise details are given only for the special linear Lie algebra \(\mathfrak{sl}_n(\mathbb{C})\). Chapter Eight turns to the representation theory of semisimple Lie algebras (in characteristic zero). The universal enveloping algebra and weight theory are introduced, along with the construction of irreducible highest weight modules. The chapter ends with Weyl's character and dimension formulas, and some specific discussion of the \(\mathfrak{sl}_n(\mathbb{C})\) case.
Chapters Nine through Eleven discuss algebraic groups. The basic concepts are presented in Chapter Nine, including the associated Lie algebra, tori, Borel subgroups, parabolic subgroups, Jordan decomposition, semisimple groups, reductive groups, and group actions. The flag variety is introduced along with its connection to \(G/B\) (as discussed above). Finally, the collection of Borel subgroups is studied and seen to be identifiable with the flag variety \(G/B\). In Chapter Ten, the basic structure theories of reductive and semisimple algebraic groups are presented, including a discussion of groups of adjoint and universal type for a given root system. Schubert varieties are then introduced along with the Bruhat Decomposition of \(G/B\) into a union of \(B\)-orbits. In Chapter Eleven, the representation theory of semisimple algebraic groups is discussed, including weight theory and the correspondence between simple modules and dominant weights, and the simple modules are constructed (algebraically and geometrically) in characteristic zero. Weyl modules are introduced as duals to sections of a line bundle on \(G/B\), and their characters and dimensions are given (analogous to the Lie algebra case) by Weyl's character formula. In characteristic zero, the Weyl modules are simple, but that is generally not true in prime characteristic. Related to the algebraic group discussion, the book contains an Appendix on Chevalley groups.
In Chapter 12, the reader arrives in a sense at the goal of the book. Grassmannian varieties and their Schubert subvarieties are introduced, including a thorough discussion of the Plücker map, coordinates, and relations. A Plücker coordinate \(p_{\tau}\) for \(G_{d,n}\) corresponds to a \(d\)-tuple \(\tau \in I_{d,n}\). The set \(I_{d_n}\) of \(d\)-tuples is partially ordered by \((i_1,i_2,\dots,i_d) \leq (j_1,j_2,\dots, j_d)\) if and only if \(i_s \leq j_s\) for each \(s\). A monomial \(p_{\tau_1}p_{\tau_2}\dots p_{\tau_m}\) in Plücker coordinates is said to be \textit{standard} if \(\tau_1 \geq \tau_2 \geq \cdots \geq \tau_m\). For a Schubert variety \(X(\omega)\) associated to a tuple \(\omega\), such a monomial is standard \textit{on \(X(\omega)\)} if in addition \(\omega \geq \tau_1\). The notion of \textit{Standard Monomial Theory} is developed, with the key result being that the monomials of degree \(m\) standard on \(X(\omega)\) give a \(K\)-basis for the degree \(m\) homogeneous coordinates of \(X(\omega)\). Further discussion is also given on unions and intersections of Schubert varieties. Next, the well-known sheaf cohomology vanishing theorem is presented. Specifically, for a Schubert variety \(X\), the higher cohomology of powers of \(\mathcal{O}_X(1)\) vanishes, and the zeroth cohomology (global sections) has a basis given by standard monomials. The remainder of the chapter presents an analogous development of standard monomial theory (and the vanishing theorem) for the flag variety. Here, one works with tableaux and must consider \textit{standard tableaux} (which extends the classical notion of standard).
In Chapter Thirteen, computations of the singular locus (the non-smooth points) of Schubert varieties for \(SL_n(K)/B\) are presented. Information on the singular locus can be obtained from combinatorial information in the Weyl group. Finally, in Chapter Fourteen, two key applications are presented. The first application involves classical invariant theory. With \(V\) as above, let \(X\) be the sum of \(m\)-copies of \(V\) along with \(q\)-copies of the dual module \(V^*\) for \(m, q > n\). Let \(GL_n(K)\) act diagonally on \(X\). Of interest is the subring of \(GL_n(K)\)-fixed points of \(K[X]\). This can be identified with the coordinate algebra of a determinantal variety. The determinantal variety can be identified with an open subset of a Schubert variety, which allows one to use the standard monomial theory for Schubert varieties to identify generators and relations for the ring of invariants. The second application is the degeneration of Schubert varieties (for the Grassmannian or flag variety) to toric varieties. The toric varieties are associated to distributive lattices within \(I_{d,n}\) (or a union of such). flag variety; Grassmannian variety; Schubert variety; symmetric group; algebraic group; Lie algebra; Schur-Weyl duality; singular locus; Chevalley group; standard monomial theory; general linear group; special linear group; Borel subgroup; parabolic subgroup; determinantal variety; commutative algebra; algebraic geometry; character; semisimple ring; finite group; toric variety; invariant theory; tableaux; irreducible module; Bruhat decomposition; sheaf cohomology Lakshmibai, V., Brown, J.: Flag varieties: an interplay of geometry, combinatorics, and representation theory, Texts and Readings in Mathematics, vol. 53. Hindustan Book Agency, New Delhi (2009) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, General commutative ring theory, Algebraic groups, Special varieties, Lie algebras and Lie superalgebras, Ordinary representations and characters, Representations of finite symmetric groups, Representation theory for linear algebraic groups, Linear algebraic groups over arbitrary fields Flag varieties. An interplay of geometry, combinatorics, and representation theory
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main result of this paper says that the canonical basis of (the modified form of) Letzter's coideal subalgebra of quantum \({sl}_2\), constructed via a geometric approach in [Bull. Inst. Math., Acad. Sin. (N.S.) 13, No. 2, 143-198 (2018; Zbl 1440.17012)] by the author and \textit{W. Wang}, coincides with the algebraic basis conjectured by \textit{H. Bao} and \textit{W. Wang} [A new approach to Kazhdan-Lusztig theory of type \(B\) via quantum symmetric pairs. Paris: SMF (2018; Zbl 1411.17001)] and proved in [J. Pure Appl. Algebra 222, No. 9, 2667--2702 (2018; Zbl 1388.17004)] by \textit{C. Berman} and \textit{W. Wang}. canonical basis; Letzter's coideal subalgebra Quantum groups (quantized enveloping algebras) and related deformations, Classical groups (algebro-geometric aspects) On canonical bases for the Letzter algebra \(\mathbf{U}^\imath(\mathfrak{sl}_2)\)
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce rectangular elements in the symmetric group. In the framework of PBW degenerations, we show that in type A the degenerate Schubert variety associated with a rectangular element is indeed a Schubert variety in a partial flag variety of the same type with larger rank. Moreover, the degenerate Demazure module associated with a rectangular element is isomorphic to the Demazure module for this particular Schubert variety of larger rank. This generalises previous results by \textit{G. Cerulli Irelli} et al. for the PBW degenerate flag variety in [Pac. J. Math. 284, No. 2, 283--308 (2016; Zbl 1433.17008)]. Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Degenerate Schubert varieties in type A
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Recall that a resolution \(p:\widetilde X \to X\) of an irreducible complex projective variety \(X\) is said to be small if, for all \(i>0\), \(\text{codim}_X \{x\in X : \dim p^{-1} (x)\geq i\}>2i\). Let \(G=SO(2n)\) or \(Sp(2n)\) and let \(P_n\) be the maximal parabolic subgroup obtained by deleting the right end root (following the Bourbaki convention). In an earlier work, ``Small resolutions of Schubert varieties in symplectic and orthogonal Grassmannians'' in Publ. Res. Inst. Math. Sci. 30, No. 3, 443-458 (1994), the authors exhibited `Bott-Samleson type' small resolutions \(p:\widetilde X(\lambda) \to X(\lambda)\) of certain Schubert varieties \(X(\lambda)\) in \(G/P_n\).
The authors, in the paper under review, give an inductive formula (following similar works of Zelevinskii in the case of Grassmannian Schubert varieties) to determine the Poincaré polynomials of the fibres of \(p\) over \(T\)-fixed points (where \(T\) is the maximal torus of \(G\) acting on \(G/P_n\) via the left multiplication). They use this result (and some results of Zelevinskii) to show that the Kazhdan-Lusztig (KL for short) polynomials \(P_{\theta, \lambda_0}\) (for the Weyl group associated to \(G)\), for certain pairs \(\theta\leq \lambda_0\), are equal to the KL-polynomials \(P_{\theta', \lambda_0'}'\) (where \(P'\) denotes the KL-polynomials for \(SL(M)\), for certain integer \(M\) and certain \(\theta'\leq \lambda_0'\) in the Weyl group of \(SL(M))\).
The authors also exhibit small resolutions for certain Schubert varieties in \(E_6/P_6\) and calculate the Poincaré polynomials of the fibres over \(T\)-fixed points for them (where \(P_6\) is again the maximal parabolic subgroup obtained by deleting the right end root). In addition, they explicitly determine the singular locus for most of the Schubert varieties in \(E_6/P_6\). Kazhdan-Lusztig polynomials; Schubert varieties; small resolutions Grassmannians, Schubert varieties, flag manifolds, Global theory and resolution of singularities (algebro-geometric aspects) Small resolutions of Schubert varieties and Kazhdan-Lusztig polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce new, combinatorially defined subvarieties of isotropic Grassmannians called symplectic restriction varieties. We study their geometric properties and compute their cohomology classes. In particular, we give a positive, combinatorial, geometric branching rule for computing the map in cohomology induced by the inclusion \(i:SG(k,n)\to G(k,n)\). This rule has many applications in algebraic geometry, symplectic geometry, combinatorics, and representation theory. In the final section of the paper, we discuss the rigidity of Schubert classes in the cohomology of \(SG(k,n)\). Symplectic restriction varieties, in certain instances, give explicit deformations of Schubert varieties, thereby showing that the corresponding classes are not rigid. isotropic Grassmannians; symplectic restriction varieties Coskun, I., Symplectic restriction varieties and geometric branching rules, \textit{Clay Math. Proc.}, 18, 205-239, (2013) Grassmannians, Schubert varieties, flag manifolds, Symplectic geometry, contact geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homogeneous complex manifolds Symplectic restriction varieties and geometric branching rules
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review the author studies the so-called \(\mathfrak{g}\)-endomorphism algebras, associated with irreducible representations of a connected semi-simple algebraic group over an algebraically closed field of characteristic zero. These algebras were introduced by \textit{A.~A.~Kirillov} in [Family algebras. Electron. Res. Announc. Am. Math. Soc. 6, No. 2, 7--20 (2000; Zbl 0946.16019)] under the name ``family algebras''. The main result of the present paper asserts that every commutative \(\mathfrak{g}\)-endomorphism algebra is Gorenstein. During the proof a connection between \(\mathfrak{g}\)-endomorphism algebras and Dynkin polynomials is established. The latter appear as the numerators of the Poincaré series of the \(\mathfrak{g}\)-endomorphism algebra. It is worth mentioning that the Poincaré series of \(\mathfrak{g}\)-endomorphism algebras are described explicitly in the paper. The paper is finished with a discussion of a connection between \(\mathfrak{g}\)-endomorphism algebras and equivariant cohomology. algebraic group; \(\mathfrak{g}\)-endomorphism algebra; module; weight; equivariant cohomology; Dynkin polynomial D. Panyushev, Weight multiplicity free representations,
\[
\mathfrak{g}
\]
-endomorphism algebras, and Dynkin polynomials. J. Lond. Math. Soc. 69, Part 2, 273--290 (2004) Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Group actions on varieties or schemes (quotients), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) Weight multiplicity free representations, \(\mathfrak g\)-endomorphism algebras, and Dynkin polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors prove that Lusztig's Frobenius map (for quantum groups at roots of unity) can be, after dualizing, viewed as a characteristic zero lift of the geometric Frobenius splitting of \(G/B\) (in char \(p>0\)) introduced by \textit{V. B. Mehta} and \textit{A. Ramanathan} [Ann. Math. (2) 122, 27-40 (1985; Zbl 0601.14043)]. Frobenius map; Schubert variety; quantum group; geometric Frobenius splitting Kumar, S., Littelmann, P.: Frobenius splitting in characteristic zero and the quantum Frobenius map. J. Pure Appl. Algebra 152, 201--216 (2000) Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized function algebras) and their representations, Cohomology theory for linear algebraic groups Frobenius splitting in characteristic zero and the quantum Frobenius map
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{A. S. Buch} et al. [Math. Ann. 340, No. 2, 359--382 (2008; Zbl 1157.14036)] defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials \(G_\pi\) indexed by permutations in the basis of stable Grothendieck polynomials \(G_\lambda\) indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on orbit closures of the orthogonal group acting on the complete flag variety. We construct a generalization of shifted Hecke insertion for maximal chains in an analogous weak order on orbit closures of the symplectic group. As an application, we identify a combinatorial rule for the expansion of ``orthogonal'' and ``symplectic'' shifted analogues of \(G_\pi\) in Ikeda and Naruse's basis of \(K\)-theoretic Schur \(P\)-functions. symmetric groups; Grothendieck polynomials; Hecke insertion; Schur \(P\)-functions; flag varieties Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds A symplectic refinement of shifted Hecke insertion
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal{O}_{\Lambda}\) be any block of category \(\mathcal{O}\) over a symmetrizable Kac-Moody algebra outside the critical hyperplanes, \(\mathcal{G}_{\Lambda}\) the associated moment graph, and \(\mathcal{M}_{\Lambda}\) the subcategory of modules admitting a Verma flag. The space of global sections of a sheaf on \(\mathcal{G}_{\Lambda}\) is a module for the algebra of global sections of the structure sheaf of \(\mathcal{G}_{\Lambda}\) and the corresponding categories are equivalent, giving two ways of viewing these objects. The main result of the paper is to use this dual viewpoint to establish an equivalence of exact categories between \(\mathcal{M}_{\Lambda}\) and a subcategory of sheaves on \(\mathcal{G}_{\Lambda}\). The projectives then correspond to the equivariant intersection cohomologies of the Schubert varieties of the Kac-Moody group via a Koszul-dual approach. sheaves on moment graphs; Koszul duality; modules with Verma flag Fiebig P.: Sheaves on moment graphs and a localization of Verma flags. Adv. Math. 217(2), 683--712 (2008) Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Sheaves on moment graphs and a localization of Verma flags
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0665.00008.]
Let \(X\) be a complex analytic manifold, and \(Y\) a closed hypersurface of \(X\). The author considers the sheaf of holomorphic functions (resp. holomorphic differential operators) on \(X\), and the sheaf of meromorphic functions with poles. Let \(M\) be a holonomic left module. Kashiwara proved the existence of a b-function for \((M,Y)\) and got some consequences. The purpose of this paper is to develop further this type of argument to classify certain holonomic modules. The result obtained can be considered as a generalization of a theorem by PacPherson and Vilonen on extension of perverse sheaves. Actually, their result is equivalent to the special case where the holonomic modules to be considered are ``regular''. This follows from the following double (non- trivial) translation: ``perverse sheaves'' correspond to ``holonomic regular modules'' by the so-called ``Riemann-Hilbert correspondence''; and ``vanishing cycles'' correspond to ``b-functions''. The point of view used by the author was developed more or less independently by C. Sabbah and M. Saito: the author points out that very close results are also known to A. Beilinson (unpublished).
The author points out that M. Sato was the first to introduce and study b-functions, and that he is pleased to publish these results in a volume dedicated to him. holonomic D-modules; holomorphic functions; holomorphic differential operators; meromorphic functions with poles; b-function B. Malgrange, Extension of holonomic \(\scr D\)-modules , Algebraic analysis, Vol. I, Academic Press, Boston, MA, 1988, pp. 403-411. Sheaves of differential operators and their modules, \(D\)-modules, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Holomorphic functions of several complex variables, Meromorphic functions of several complex variables, Other special functions Extension of holonomic \(\mathcal D\)-modules
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous \(K\)-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model. Schur polynomials; directed last-passage percolation model Symmetric functions and generalizations, Interacting random processes; statistical mechanics type models; percolation theory, Combinatorial probability, Grassmannians, Schubert varieties, flag manifolds Dual Grothendieck polynomials via last-passage percolation
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We explain some remarkable connections between the two-parameter symmetric polynomials discovered in 1988 by Macdonald, and the geometry of certain algebraic varieties, notably the Hilbert scheme \(\text{Hilb}^n(\mathbb{C}^2)\) of points in the plane, and the variety \(C_n\) of pairs of commuting \(n\times n\) matrices. symmetric functions; \(n!\) conjecture; Frobenius series; diagonal harmonics; commuting variety; symmetric polynomials; algebraic varieties; Hilbert scheme M. Haiman, ''Macdonald polynomials and geometry'' in New Perspectives in Algebraic Combinatorics (Berkeley, Calif., 1996--97) , Math. Sci. Res. Inst. Publ. 38 , Cambridge Univ. Press, Cambridge, 1999, 207--254. Symmetric functions and generalizations, Research exposition (monographs, survey articles) pertaining to combinatorics, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Macdonald polynomials and geometry
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give an explicit combinatorial description of cluster structures in Schubert varieties of the Grassmannian in terms of (target labelings of) Postnikov's plabic graphs. This description is a natural generalization of the description given by \textit{J. S. Scott} [Proc. Lond. Math. Soc. (3) 92, No. 2, 345--380 (2006; Zbl 1088.22009)] for the Grassmannian and has been believed by experts essentially since J. S. Scott [loc. cit.], though the statement was not formally written down until \textit{G. Muller} and \textit{D. E. Speyer} [Proc. Lond. Math. Soc. (3) 115, No. 5, 1014--1071 (2017; Zbl 1408.14154)]. To prove this conjecture we use a result of \textit{B. Leclerc} [Adv. Math. 300, 190--228 (2016; Zbl 1375.13036)], who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety admit cluster structures. We also adapt a construction of \textit{R. Karpman} [J. Comb. Theory, Ser. A 142, 113--146 (2016; Zbl 1337.05114)] to build cluster seeds associated to reduced expressions. Further, we explicitly describe cluster structures in skew Schubert varieties using plabic graphs whose boundary vertices need not be labeled in cyclic order. cluster algebra; Schubert variety; positroid; plabic graph Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Cluster algebras, Semisimple Lie groups and their representations Combinatorics of cluster structures in Schubert varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors go back to the original motivation of Yang for his introduction of the Yang-Baxter equation. He had been led to study certain elements of the group algebra of the symmetric group, and to expand them on the basis of permutations. They obtain some new results in the study of these objects by replacing elementary transpositions with generators of a Hecke algebra. They define a new basis of the Hecke algebra and a bilinear form on it. In some cases they give formulas for the coefficients of this basis when expanded in the usual basis. This involves Schubert and Grothendieck polynomials which were originally defined as canonical bases of the cohomology and Grothendieck rings of flag manifolds. Yang-Baxter equations; Hecke algebra; flag varieties; Schubert polynomials; Grothendieck polynomials Lascoux, A.; Leclerc, B.; Thibon, J. -Y: Flag varieties and the Yang--Baxter equation. Lett. math. Phys. 40, No. 1, 75-90 (1997) Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Flag varieties and the Yang-Baxter equation
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A geometric categorification is given for arbitrary-large-finite- dimensional quotients of quantum \( \mathfrak{osp}(1| 2)\) and tensor products of its simple modules. The modified quantum \( \mathfrak{osp}(1| 2)\) of Clark-Wang, a new version in this paper and the modified quantum \( \mathfrak{sl}(2)\) are shown to be isomorphic to each other over a field containing \( \mathbb{Q}(v)\) and \( \sqrt {-1}\). quantum \(\mathfrak{osp}(1\|2)\); quantum modified algebra; tensor product module; categorification; perverse sheaf Fan, Z., Li, Y.: A geometric setting for quantum \({osp(1|2)}\). Trans. Am. Math. Soc. \textbf{367}, 7895-7916 (2015). arXiv:1305.0710 Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Schur and \(q\)-Schur algebras A geometric setting for quantum \(\mathfrak{osp}(1|2)\)
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper provides a geometric framework within which to study the Landweber-Novikov algebra and its dual. The dual is realized as a subring of double complex cobordism, which is the cobordism theory of manifolds for which the normal bundle is the Whitney sum of two complex vector bundles. The subring is generated by a collection of toric varieties which are called bounded flag manifolds. In analogy with the classical Schubert decomposition, the cell structure of these flag manifolds is given by a collection of varieties which are nonsingular. In turn, these subvarieties give a base for the ring. Thus, one has a rather complicated collection of interesting nonsingular varieties having double complex structure. The behavior of the Landweber-Novikov algebra is expressible by formulae in these manifolds. V.\ M. Buchstaber and N. Ray, Flag manifolds and the Landweber-Novikov algebra, Geom. Topol.\textbf{2} (1998), 79-101. Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism), Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Flag manifolds and the Landweber-Novikov algebra
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a very basic introduction to the theory of compact hyper-Kähler manifolds, in particular describing the process of moving from the hyper-Kähler structure to the structure of a holomorphic symplectic manifold and vice versa. Then the examples constructed by Beauville are presented. The chapter on the geometry of compact hyper-Kähler manifolds finishes with a detailed study of the Atiyah class, the algebraic analogue of the Riemannian curvature tensor used in the construction of the Rozansky-Witten invariants.
Having described the ``\(X\)'' in the theory of Rozansky-Witten invariants of pairs \((X,\Gamma)\), we turn to the ``\(\Gamma\)'', the Jacobi diagrams, in the second chapter. After introducing basic notions like the one of a Jacobi diagram and the space of graph homology -- that is basically the space of Jacobi diagrams modulo the IHX relation -- we study metric Lie algebra objects. These objects are generalisations of ordinary Lie algebras with ad-invariant inner product in arbitrary symmetric monoidal categories. Graph homology classes are then interpreted as universal morphisms in this category. We have learned this point of view from J. Sawon and articles by J. Roberts and V. Hinich and A. Vaintrob. Using this interpretation, it is quite straight forward to construct invariants for metric Lie algebra objects from graph homology classes. These invariants are usually called weight systems. At the end of the chapter, some of the rich algebraic structure of the graph homology space is studied and the Wheeling theorem is stated.
In the derived category of coherent sheaves, the tangent sheaf of a holomorphic symplectic manifold can (after an adequate shift) be viewed as a metric Lie algebra object. This leads immediately to a weight system, in this case the Rozansky-Witten weight system. This is shown in chapter three. After discussing basic properties of this weight system, e.g., the relation between algebraic structures on the space of graph homology with algebraic structures on the cohomology ring of a holomorphic symplectic manifold, we exploit the Wheeling theorem and use it to give proofs for three major theorems: one theorem is a result of N. Hitchin and J. Sawon who were able to relate a topological invariant (a topological genus) to the \(L_2\)-norm of the Riemannian curvature tensor on a compact hyper-Kähler manifold. The second application is the proof of the existence of the so-called Beauville-Bogomolov form on a compact hyper-Kähler manifold. This quadratic form was invented long before L. Rozansky and E. Witten introduced the invariants named after them but we show how these invariants can be used to give a non-standard proof. Lastly, the third theorem is a result by the author concretising an observation made by D. Huybrechts on the specific form the Euler characteristic of a holomorphic line bundle on a holomorphic symplectic manifold has. We also see that all Chern numbers of a holomorphic symplectic manifold are Rozansky-Witten in variants so in a sense the Rozansky-Witten theory is a generalisation of the Chern-Weil theory of characteristic classes.
In the last chapter of the book, effective methods to calculate Chern numbers and Rozansky-Witten invariants for A. Beauville's examples of compact hyper-Kähler manifolds are given. In order to do so, we first have to study the geometry of the Hilbert schemes of points on a surface a little bit more in detail. For this, we follow G. Ellingsrud, L. Göttsche and M. Lehn and also give their method for calculation of Chern numbers by means of Bott's residue theorem. Finally, we show how a huge class of Rozansky-Witten invariants of the Beauville examples can be computed just from the Chern numbers. It is still an open question if this is true for every Rozansky-Witten invariant and every holomorphic symplectic manifold. Jacobi diagrams; Rozansky-Witten invariants; Wheeling theorem; Atiyah class; graph homology; weight systems; Beauville-Bogomolov form Nieper-Wißkirchen M., Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds, World Sci. Publ. Co. Inc., Singapore (2004). Research exposition (monographs, survey articles) pertaining to differential geometry, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Riemann-Roch theorems, Monoidal categories (= multiplicative categories) [See also 19D23], Structured objects in a category (group objects, etc.), Compact Kähler manifolds: generalizations, classification, Characteristic classes and numbers in differential topology Chern numbers and Rozansky-Witten invariants of compact hyper-Kähler manifolds
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe a geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of Schubert varieties. There are no restrictions on the base field, and all multiplicities arising are \(1\); this is important for applications. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. It has straightforward bijections to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. Geometric consequences are described here and in our other works. For example, the rule also has an interpretation in \(K\)-theory, suggested by Buch, which gives an extension of puzzles to \(K\)-theory.'' R. Vakil, A geometric Littlewood-Richardson rule. Ann. Math. 164, 371--422 (2006) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Combinatorial aspects of representation theory, Classical problems, Schubert calculus A geometric Littlewood-Richardson rule
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The focus of the paper is to present a sort of Positivstellensatz for univariate trace polynomials. And as a consequence a characterisation of positive univariate trace polynomials restricted to symmetric matrices.
The authors first present some interesting basic results on trace polynomials which are almost immediate from a much older and more abstract result in the 70's which dealt with multivariate trace polynomials [\textit{C. Procesi}, Adv. Math. 19, 306--381 (1976; Zbl 0331.15021)]. For instance, we have a nice characterization of when a function between symmetric matrices is actually a univariate trace polynomial (see Prop. 2.1 of the paper). These are not directly used in the main result but are of interest to readers interested in trace polynomials.
The proof of the main result can be found in Section 3 where the authors characterise trace polynomials that are positive when restricted to symmetric matrices in a positivity set defined by a finite set of pure trace polynomials. They also give examples and remarks why some of these conditions are indispensible in this characterization. trace polynomial; Positivstellensatz; Hankel matrix; real algebraic geometry Real algebra, Semialgebraic sets and related spaces, Trace rings and invariant theory (associative rings and algebras) Positive univariate trace polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors consider tangent cones of Schubert varieties in the complete flag variety \(\mathcal{F}\), and investigate the problem when the tangent cones of two different Schubert varieties coincide.
One can write \(\mathcal{F}=\mathrm{GL}(n,\mathbb{C})/B\) where \(B\) is the Borel subgroup stabilizing the standard flag \(F_0\).
The variety \(\mathcal{F}\) is a disjoint union of \(B\)-orbits called Schubert cells.
These cells are parametrized by elements of the symmetric group \(S_n\), acting naturally in \(\mathbb{C}^n\), and hence in \(\mathcal{F}\).
To each Schubert variety \(X_w\) (namely the closure of a Schubert cell \(C_w=x\cdot F_0\), avec \(w\in S_n\)), one can associate two subsets of the tangent space \(T_{F_0}\mathcal{F}\): \begin{itemize} \item the tangent cone \(T_w\), which is the set of vectors tangent to \(X_w\) at \(F_0\); \item the Zariski tangent space \(Z_w\) which is spanned by \(T_w\). \end{itemize}
The tangent cones \(T_w\) are algebraic subvarieties of \(T_{F_0}\mathcal{F}\) which have the same dimensions as \(X_w\). The tangent cone \(T_w\) and tangent space \(Z_w\) coincide if and only if \(X_w\) is not singular point.
Given two distinct the Schubert varieties \(X_w\) and \(X_{w'}\), the associated spaces may be equal: \(Z_w = Z_{w'}\) or \(T_w = T_{w'}\) (the second equality implying the first one).
[\textit{V. Lakshmibai}, Math. Res. Lett. 2, No. 4, 473--477 (1995; Zbl 0857.14027)] provides an explicit description of the Zariski tangent spaces \(Z_w\) using Chevalley basis. In particular, this results answers the question under which condition two different Schubert varieties have the same Zariski tangent space. On the contrary, the structure of tangent cones \(T_w\), is, by the authors knowledge, not well understood, in particular, the problem of their coincidence is mostly open.
The authors study the structure of the tangent cones \(T_w\) with the emphasis on the problem of their coincidence.
It was already known that \(T_w = T_{w^{-1}}\) for every permutation \(w\) (this fact was conjectured in [\textit{D. Yu. Eliseev} and \textit{A. N. Panov}, J. Math. Sci., New York 188, No. 5, 596--600 (2013; Zbl 1274.14060); translation from Zap. Nauchn. Semin. POMI 394, 218--225 (2011)]). Moreover all the Schubert varieties corresponding to a Coxeter element of \(S_n\) have the same tangent cone. Note that \(S_n\) has \(2^{n-2}\) Coxeter elements.
The authors' main tool is the notion of pillar entries in the rank matrix. Every Schubert cell \(C_w\) is determined by the \((n + 1) \times (n + 1)\) matrix formed by the dimensions \(r_{i, j}\) of the intersections \(V_i \cap V_j^0\) of the subspaces of the flag with the subspace of the standard flag \(F_0\); the corresponding Schubert variety \(W_w\) is determined by inequalities \(\dim(V_i \cap V_j^0 ) \geq r_{i, j}\).
The authors prove that the whole rank matrix \((r_{i, j})\) is determined by a relatively small set of entries, which they call pillar entries. Note that the notion of pillar entry is very close (yet different from) Fulton's notion of essential set (see [\textit{W. Fulton}, Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)]).
For example, if \((r_{i, j})\) is the rank matrix corresponding to a permutation \(w\), then the rank matrix corresponding to \(w^{-1}\) is obtained from \((r_{i, j})\) by a transposition.
The authors conjecture that if \(T_w = T_{w'}\), then the pillar entries for \(w\) are obtained from pillar entries for \(w\) by a partial transposition. This means that there is a one-to-one correspondence between pillar entries \(r_{i, j}\) and \(r'_{i, j}\) for \(w\) and \(w'\) such that the pillar entry corresponding to \(r_{i, j}\) is either \(r'_{i, j}=r_{i, j}\) or \(r'_{j,i}=r_{i, j}\).
However, the converse of this conjecture is false: examples show that a partial transposition of the set of pillar entries may lead to a set of entries which is not the set of pillar entries for any transposition, or is a set of pillar entries coming from a variety of different dimension.
Some pillar entries are ``linked'', that is, they can be transposed or not transposed only simultaneously. The authors give a precise definition of a linkage, and hence of ``admissible partial transposition''. Their main result states that an admissible partial transposition of pillars entries of \(w\) provides a set of pillar entries of some \(w'\), and that in this case \(T_w = T_{w'}\). However, examples show that their definition of linkage is not sufficient: there are partial transpositions of pillar entries, which are not admissible in their sense, but which still preserve the tangent cone.
The authors present also a formula of (co)dimension of a Schubert variety in terms of the pillar entries of the corresponding rank matrix. Then they present an algorithm that reconstructs a given permutation from the corresponding pillar entries.
Finally, the authors, led by the numeric examples, state the following ``2m-conjecture'' which is also closely related with the earlier cited result on Coxeter elements: the number of Schubert varieties with an identical tangent cone is always a power of 2. Schubert variety; singularity; tangent cone; rank matrix; essential set Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus On tangent cones of Schubert varieties
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There is a two-part programme to prove the existence of degenerations of Schubert varieties of \(SL(n)\) into toric varieties, see for instance [\textit{N. Gonciulea}, and \textit{V. Lakshmibai}, Transform. Groups 1, 215-248 (1996; Zbl 0909.14028)]. Here the authors present results concerning the first of the two parts. They prove that for a certain class of elements \(w\) of the Weyl group of \(sl_{n+1}\), there exists a lattice polytope \(\Delta_i^w \subset {\mathbb{R}}^{l(w)}\), such that for every dominant weight \(\lambda = \sum_{i = 1}^na_iw_i\) where the \(w_i\) are fundamental weights the number of lattice points in the Minkowski sum \(\Delta_\lambda^w = \sum_{i=1}^na_i\Delta_i^w\) is equal to the dimension of the Demazure module \(E_w(\lambda)\) (for its definition see [\textit{M. Demazure}, Bull. Sci. Math., II. Ser. 98, 163-172 (1975; Zbl 0365.17005)]. They also present an explicit formula which gives the character of \(E_w(\lambda)\).
For \(w = w_0\) the longest element of the Weyl group other polytopes satisfying the same condition have been constructed by \textit{A. D. Berenstein} and \textit{A. Zelevinsky} [J. Geom. Phys. 5, 453-472 (1988; Zbl 0712.17006)] and \textit{P. Littelmann} [Transform. Groups 3, 145-179 (1998; Zbl 0908.17010)]. The authors of the present paper believe that their construction can be generalized to every simple algebraic group. lattice polytope; representations of \(sl_{n}\); Demazure module; Minkowski sum; character formula; Schubert varieties; Weyl group Dehy, R.; Yu, R.: Polytopes associated to certain Demazure modules of \(sl(n)\). J. algebraic combin. 10, 149-172 (1999) Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Lattice polytopes associated to certain Demazure modules of \(sl_{n+1}\)
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth complex quasi-projective variety and \(\text{Hilb}^n (X)\) its Hilbert scheme of zero dimensional subschemes of length \(n\). The author expresses the virtual Hodge polynomials of \(\text{Hilb}^n (X)\) -- defined by cohomology with compact support -- in terms of those of \(X\) and the Hilbert scheme of subschemes of length \(n\) supported at a point of \(X\). -- The proof proceeds by comparison with the \(n\)-fold symmetric product of \(X\) and related spaces and uses a lemma on point Hilbert schemes from \(L\). Göttsche's 1991 Bonn thesis [see \textit{L. Göttsche}, ``Hilbertschemata nulldimensionaler Unterschemata glatter Varietäten'', Bonner Math. Schr. 243 (1991; Zbl 0846.14002)]. The key properties of the virtual Hodge polynomial used in the proof are its additivity over stratifications and multiplicativity for fibrations. The results extend those found for the Poincaré and Hodge polynomials of surfaces in a paper by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007)]. Hilbert scheme; virtual Hodge polynomials; symmetric product J. Cheah, ''On the Cohomology of Hilbert Schemes of Points,'' J. Algebr. Geom. 5, 479--511 (1996). Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the cohomology of Hilbert schemes of points
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author determines a factorization of a double specialization of Schubert polynomials from which he derives a factorization of a specialization of \(q\)-factorial Schur functions. Schubert polynomials; factorial and \(q\)-factorial Schur functions; factorization Prosper, V.: Factorization properties of the q-specialization of Schubert polynomials, Ann. comb. 4, 91-107 (2000) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Factorization properties of the \(q\)-specialization of Schubert polynomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to an anti-dominant minuscule fundamental weight \(- \varpi_k\) in the torus-equivariant quantum \(K\)-group of the partial flag manifold \(G / P_J\) (where \(J = I \setminus \{k \})\) corresponding to the maximal (standard) parabolic subgroup \(P_J\) of minuscule type in type \(A\), \(D\), \(E\), or \(B\). This result is obtained by proving a similar formula in a torus-equivariant \(K\)-group of the semi-infinite partial flag manifold \(\mathbf{Q}_J\) of minuscule type, and then by making use of the isomorphism between the torus-equivariant quantum \(K\)-group of \(G / P_J\) and the torus-equivariant \(K\)-group of \(\mathbf{Q}_J\), recently established by Kato. quantum Chevalley formula; quantum LS path; semi-infinite flag manifold; Grassmannian; (quantum) Schubert calculus Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Equivariant \(K\)-theory, Quantum groups and related algebraic methods applied to problems in quantum theory Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum \(K\)-group of partial flag manifolds
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Fix a simple Lie algebra \(\mathfrak{g}\) over \(\mathbb{C}\), and let \(T\) be a maximal torus in the associated simply-connected complex algebraic group \(G\).
The nonsymmetric Macdonald polynomials \(\{E_\lambda(q,t)\}_{\lambda\in P}\) form a distinguished basis for \(R(T)\otimes\mathbb{Q}(q,t),\) where \(R(T)\) is the representation ring of \(T\), \(P=\Hom(T,\mathbb{C}^\times)\) is its weight lattice, and \(q\) and \(t\) are indeterminates. (After choosing a basis for \(P\), one can identify \(R(T)\) with a Laurent polynomial ring over \(\mathbb{Z}\).)
Guided by one of the original motivations for the study of Macdonald polynomials -- namely, to interpolate between the major families of orthogonal polynomials in representation theory -- many efforts have been devoted to understanding the images of \(E_\lambda(q,t)\) under various specializations of \(q\) and \(t\). In this paper, the authors study, from the viewpoints of representation theory and geometry, the polynomials \(E_\lambda(q^{-1},\infty)\) obtained by sending \(t\to\infty\).
By an earlier combinatorial result of [the reviewer and \textit{M. Shimozono}, J. Algebr. Comb. 47, No. 1, 91--127 (2018; Zbl 1381.05089)], one knows that \[E_\lambda(q^{-1},\infty)\in R(T)\otimes \mathbb{Z}_{\ge 0}[q],\] i.e., the specialized polynomials have positive integral coefficients. A marvelous result of [\textit{S. Kato}, Math. Ann. 371, No. 3--4, 1769--1801 (2018; Zbl 1398.14053)] explains this positivity by realizing these (and related) polynomials in terms of graded characters of spaces of sections of sheaves on the semi-infinite flag manifold associated to \(G\), parallel to the classical Demazure character formula.
The first main result of the present paper (Theorem 1.2; cf. Corollary 3.27) gives, by explicit presentation similar in spirit of [\textit{E. Feigin} and \textit{I. Makedonskyi}, Sel. Math., New Ser. 23, No. 4, 2863--2897 (2017; Zbl 1407.17028)], a cyclic module \(U_{\lambda}\) over the Iwahori subalgebra in \(\mathfrak{g}\otimes_{\mathbb{C}}\mathbb{C}[z]\) such that its graded character \(\mathrm{gch}\,U_\lambda\) coincides with \(E_\lambda(q^{-1},\infty)\), up to some twists by the long element \(w_0\). This ``local'' result is upgraded to a ``global'' geometric statement in Theorem 1.3 (cf. Theorem 4.7), which gives presentations, as cyclic Iwahori modules \(\mathbb{U}_\lambda\), for the spaces of the sections mentioned in the previous paragraph.
Finally, under the additional assumptions that \(\mathfrak{g}\) is simply-laced and not of type \(E_8\), the third main result of this paper (Theorem 1.5; cf. Theorem 5.12) asserts that the modules \(\{\mathbb{U}_{-\lambda}\}\) are dual under an Ext-pairing to the family of level-one affine Demazure modules associated to \(\mathfrak{g}\). (The need for an affine Dynkin diagram automorphism in the proof excludes type \(E_8\) from consideration; however, Theorem 1.5 is conjectured to hold in type \(E_8\)). In Appendix A, which is quite informative, it is explained how Theorem 1.5 lifts the well-known orthogonality between nonsymmetric Macdonald polynomials (at \(t=0\) and \(t=\infty)\) to the level of representation categories. Finally, we note that Theorem 1.5 is closely related to the work [\textit{V. Chari} and \textit{B. Ion}, Compos. Math. 151, No. 7, 1265--1287 (2015; Zbl 1337.17016)] pertaining to specialized \textit{symmetric} Macdonald polynomials. Macdonald polynomials; current algebra; semi-infinite flag manifold Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Representation theory for linear algebraic groups, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Representation theoretic realization of non-symmetric Macdonald polynomials at infinity
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Quantum cohomology and \(K\)-theory algebras of geometrically relevant spaces, for example Nakajima quiver varieties, are important objects of equivariant geometry. The paper under review defines a version of such quantum \(K\)-theory algebra, not based on the customary approach of counting stable maps, but based on counting quasi-maps. With this approach the authors define some distinguished classes (``quantum tautological classes''), and a deformed product operation.
Motivated by relations to quantum integrable systems the authors consider certain functions, called partition functions. In fact these partition functions come in two flavors, and an operator mapping one to the other (capping operator) satisfies quantum difference equations of the style of qKZ equations.
The first main result of the paper is that -- for quiver varieties with discrete fixed point set -- the eigenvalues of the quantum product by quantum tautological classes can be expressed via a certain asymptotic of the vertex function.
The second main result concern the special case of the cotangent bundle of a partial flag variety of type A. In this case the \(K\)-theory algebra can be identified with the Hilbert space of a quantum integrable system called XXZ model. Using this correspondence the authors calculate the vertex functions, and -- through the correspondence mentioned above -- obtain formulas for the eigenvalues of the quantum multiplication operators. The interesting degeneration to the compact 0-section of the cotangent bundle is given in detail.
The third main topic of the paper concerns the further special case of cotangent bundle over the full flag variety. The paper presents theorems similar in spirit to Giventhal-Kim theorems in quantum cohomology. Namely, the quantum \(K\)-theory algebra is presented as the algebra of functions on a Lagrangian subvariety of the phase space of an integrable model, called trigonometric Ruijsenaars-Schneider model. Again, the interesting degeneration to the 0-section is spelled out in detail, where now the Lagrangian variety lives in the phase space of the relativistic Toda lattice. quantum \(K\)-theory of quiver varieties; quantum tautological classes; spectrum of the quantum multiplication; asymptotic of vertex functions Equivariant \(K\)-theory, Classical groups (algebro-geometric aspects), Representations of quivers and partially ordered sets, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations, Exactly solvable models; Bethe ansatz, Supersymmetric field theories in quantum mechanics Quantum \(K\)-theory of quiver varieties and many-body systems
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper the author continues his interest in Galois groups and splitting fields of some polynomials over function field \(k(X)\) of positive characteristic \(p\). Let \(q\) be a power of \(p.\) The first sections of the paper are devoted to the trinomials \(F(Y)=Y^{1+q}+Y+X\), \(\Phi(Y)=Y^{q^2-1}+Y^{q-1}+X\) and \(\widehat{\Phi}(Y)=Y^{q^2}+Y^q+XY\) for which the splitting fields are examined. These polynomials have been considered in the earlier papers of the author [Isr. J. Math. 88, 1--23 (1994; Zbl 0828.14014) and Proc. Am. Math. Soc. 125, 1643--1650 (1997; Zbl 0912.12004)].
The second part of the paper concentrates on the Galois group of the \(n\)th iterate \(\widehat{\Phi}^{[[n]]}(Y)\) of the vectorial \(q\)-polynomial of \(q\)-degree \(m\) defined as \(\widehat{\Phi}(Y)=\sum_{i=0}^m\,a_iY^{q^{m-i}}.\) The paper ends with some generalisation of the results for the trinomial \(F(Y)\), \(\Phi(Y)\) and \(\widehat{\Phi}(Y)\) to the case of these trinomials slightly modified. Galois group; projective polynomial; vectorial polynomial; splitting field Separable extensions, Galois theory, Coverings of curves, fundamental group Galois theory of special trinomials
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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(w : \mathbb C^N \to \mathbb C\) be a quasi-homogeneous polynomial whose total degree \(d\) is equal to the sum of the weights of each variable. Let \(G \le SL_N (\mathbb C)\) be a diagonal subgroup of automorphisms of \(w\). In the article [\textit{Y.-P. Lee} et al., Ann. Sci. Éc. Norm. Supér. (4) 49, No. 6, 1403--1443 (2016; Zbl 1360.14133)] the commutativity of a diagram called LG/CY square is proven. The top row vertices of this square are \(GWT_0([\mathbb C^N /G])\), the genus zero Gromov-Witten theory of \([\mathbb C^N /G]\), and \(GWT_0(\mathrm{tot}(\mathcal O_{\mathbb P(G)}(-d)))\), the genus zero Gromov-Witten theory of a partial crepant resolution of \([\mathbb C^N /G]\). The bottom row vertices are \(FJRW_0(w, G)\), the genus zero FJRW theory of the Landau-Ginzburg model given by the pair \((w, G)\), and \(GWT_0(\mathcal Z)\), the genus zero Gromov-Witten theory of a hypersurface \(\mathcal Z\) defined as the vanishing locus of \(w\) in an appropriate finite quotient of weighted projective space \(\mathbb P(G)\). The arrows in the square are the crepant transformation conjecture, quantum Serre duality, the LG/CY correspondence, and the local GW/FJRW correspondence.
The goal of the paper under review is to relate each of the above correspondences to an integral transform between appropriate derived categories, i.e. to lift the LG/CY square to the derived category to obtain a cube of relations. It is known that the crepant transformation conjecture (the top horizontal arrow of the LG/CY square) is compatible with a natural Fourier-Mukai transform. A similar result is known for the bottom horizontal arrow at least when \(G\) is cyclic.
The paper under review shows that there are natural derived functors corresponding to both of the vertical arrows of LG/CY square as well, after restricting to subcategories of \(D([\mathbb C^N /G])\) and \(D(\mathrm{tot}(\mathcal O(-d)))\) with proper support. This requires a reformulation of both the local GW/FJRW correspondence and quantum Serre duality in terms of narrow quantum \(D\)-modules, which turns out to be a more natural way of describing these correspondences.
Even though the corresponding square of derived functors does not commute in general the paper shows that the induced maps on \(K\)-theory commute. Gromov-Witten theory; FJRW theory; crepant resolution conjecture; LG/CY correspondence; wall crossing; Fourier-Mukai Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Integral transforms and quantum correspondences
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