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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 Gaudin model associated to a point \(z\in\mathbb{C}^n\) with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional \(sl_2\)-representations, [G]. The Bethe equations of this model provide the critical point system of a remarkable rational symmetric function. Any critical orbit determines a common eigenvector of the Gaudin hamiltonians called a Bethe vector. In [\textit{N. Reshetikhin} and \textit{A. Varchenko} in Geometry, Topology, and Physics. For Raoul Bott, Internat. Press. Cambridge, MA, 203--322 (1994)], it was shown that for generic \(z\) the Bethe vectors span the space of singular vectors, i.e. that the number of critical orbits is bounded from below by the dimension of this space. The upper bound by the same number is one of the main results of [\textit{I. Scherbak} and \textit{A. Varchenko}, Mosc. Math. J. 3, No. 2, 621--645 (2003; Zbl 1039.34077)].
In the present paper we get this upper bound in another, ``less technical'', way. The crucial observation is that the symmetric function defining the Bethe equations can be interpreted as the generating function of the map sending a pair of complex polynomials into their Wroński determinant: the critical orbits determine the preimage of a given polynomial under this map. Within the framework of the Schubert calculus, the number of critical orbits can be estimated by the intersection number of special Schubert classes. Relations to the \(sl_2\) representation theory [\textit{W. Fulton}, Young Tableaux, Cambridge Univ. Press, (1997; Zbl 0878.14034)] imply that this number is the dimension of the space of singular vectors. We prove also that the spectrum of the Gaudin hamiltonians is simple for generic \(z\). Bethe-Salpeter and other integral equations arising in quantum theory, Computational aspects of algebraic curves, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Exactly solvable models; Bethe ansatz, Linear ordinary differential equations and systems, Many-body theory; quantum Hall effect, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Groups and algebras in quantum theory and relations with integrable systems Gaudin's model and the generating function of the Wroński map | 0 |
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 \(G(k,n)\) denote the Grassmannian space of all \(k\)-dimensional subspaces of an \(n\)-dimensional vector space \(V\). It can be also interpreted as the space of all \(k-1\) dimensional projective spaces in the projective space \(\mathbb P^{n-1}\). From this point of view, \(G(k,n)\) is denoted by \(\mathbb G(k-1,n-1)\).
Let us choose a complete flag \(F.\)
\[
0 = F_0 \subset F_1 \subset \dots \subset F_n = V.
\]
Let \(\lambda\) be a partition with \(k\) parts satisfying \(n-k \geq \lambda_1 \geq \dots \geq \lambda_k \geq 0\). The Schubert variety \(\Omega_{\lambda}(F.)\) of type \(\lambda\) associated to the flag \(F.\) is defined by
\[
\Omega_{\lambda}(F.) = \{[W] \in G(k,n) : \dim (W \cap F_{n-k+i-\lambda_i})\geq i\}.
\]
The homology class of a Schubert variety is independent of the defining flag, and depends only on the partition. Let us denote the Poincaré dual of the class of \(\Omega_{\lambda}\) by \(\sigma_{\lambda}\). The Poincaré duals of the classes of Schubert varieties give a module basis for the integral cohomology of \(G(k,n)\).
Given two Schubert cycles \(\sigma_{\lambda},\sigma_{\eta}\), their cup product in the integral cohomology algebra is a linear combinations of Schubert cycles \(\sum_{\rho} c^{\rho}_{\lambda, \eta} \sigma_{\rho}\).
The coefficients \(c^{\rho}_{\lambda, \eta}\) in the integral cohomology algebra with respect to the Schubert basis are called Littlewood-Richardson coefficients. A positive combinatorial rule giving these coefficients is called a Littlewood-Richardson rule. The basic problem is to find positive geometric algorithms for computing the Littlewood-Richardson structure constants.
In this work the authors first describe recent research on positive descriptions of the structure constants in the integral cohomology algebra of homogeneous spaces such as Grassmannians and flag spaces, by degenerations and related techniques. Later they give extensions of these rules to \(K\)-theory, equivarant cohomology with torus action and quantum cohomology. Grassmannians; Littlewood-Richardson coefficients; Schubert calculus Coşkun, I.; Vakil, R., Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus, Algebraic Geometry--Seattle 2005, pp., (2009), American Mathematical Society, Providence, RI Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus | 0 |
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 two different one-parameter generalizations of Littlewood-Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions are closely related to puzzles, originally introduced by \textit{A. Knutson} and \textit{T. Tao} [Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)] in their work on the equivariant cohomology of the Grassmannian. symmetric functions; Littlewood-Richardson coefficients; integrable lattice models Wheeler, M.; Zinn-Justin, P., Hall polynomials, inverse kostka polynomials and puzzles Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Classical problems, Schubert calculus, Groups acting on specific manifolds Hall polynomials, inverse Kostka polynomials and puzzles | 0 |
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 \(M(n,k)\) be the \(n\times n\)-matrices with entries in an algebraically closed field \(k\) and let \(V\) be a vector space over \(k\) of dimension \(n\). \textit{N. Spaltenstein} [Nederl. Akad. Wet., Proc., Ser. A 79, 452--456 (1976; Zbl 0343.20029)] established a bijection between the irreducible components of the space of full flags \(\mathcal F_x\) fixed by a nilpotent element \(x\in M(n,k)\) and the standard tableaux associated to the Young diagram of \(x\).
The main result of the present article is to determine, when \(x\) is of hook type, for each irreducible component \(X\) of \(\mathcal F_x\), the unique Schubert cell \(\mathcal C_X\) of the full flag manifold \(\mathcal F(V)\), such that \(\mathcal C_X\cap X\) is a dense subspace of \(X\). flag manifolds; nilpotent element; standard tableaux; Young diagram; Schubert cell; hook type; irreducible components Pagnon, NGJ, On the spaltenstein correspondence, Indag. Mathem., 15, 101-114, (2004) Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields On the Spaltenstein correspondence. | 0 |
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 prove new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of factorial and Gorenstein varieties in terms of so called bivincular patterns. These are generalizations of classical patterns where conditions are placed on the location of an occurrence in a permutation, as well as on the values in the occurrence. This clarifies what happens when the requirement of smoothness is weakened to factoriality and further to Gorensteinness, extending work of \textit{M. Bousquet-Mélou} and \textit{S. Butler} [Ann. Comb. 11, No. 3--4, 335--354 (2007; Zbl 1141.05011)], and \textit{A. Woo} and \textit{A. Yong} [Adv. Math. 207, No. 1, 205--220 (2006; Zbl 1112.14058)]. We also prove results that translate some known patterns in the literature into bivincular patterns. patterns; permutations; Schubert varieties; singularities Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds A unification of permutation patterns related to Schubert varieties | 0 |
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 provides a survey of the authors' work on the relationship between birational geometry and Gromov-Witten theory. For an ordinary flop \(f: X \relbar \to X'\), the graph closure induces an isomorphism of the cohomology groups \(H^*(X) \to H^*(X')\), but this isomorphism does not generally preserve the ring structure. Based on a conjecture of Y. Ruan and C.-L. Wang, one expects an isomorphism of rings for quantum cohomology after analytic continuation of the Kähler parameters, and this has been established in various cases. The authors summarize the four main steps in their approach to proving invariance of the big quantum cohomology ring, and they include quantum corrections, divisorial reconstruction, Birkoff factorization and the generalized mirror transformation, and analytic continuation. The last two sections of the paper are devoted to a worked example which helps to illustrate the main ideas at work. quantum cohomology; Gromov-Witten invariants; ordinary flops; analytic continuation Lee, Yuan-Pin and Lin, Hui-Wen and Wang, Chin-Lung, Quantum cohomology under birational maps and transitions, (None) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays) Analytic continuations of quantum cohomology | 0 |
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 prove an explicit inverse Chevalley formula in the equivariant \(K\)-theory of semi-infinite flag manifolds of simply laced type. By an `inverse Chevalley formula' we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a \(\mathbb{Z}\left [q^{\pm 1}\right]\)-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars \(e^{\lambda}\), where \(\lambda\) is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type \(E_8\). The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric \(q\)-Toda operators for minuscule weights in ADE type. Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Equivariant \(K\)-theory, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations Inverse \(K\)-Chevalley formulas for semi-infinite flag manifolds. I: Minuscule weights in ADE type | 0 |
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 an orthogonal group defined over an algebraically closed field of characteristic different from 2, and let \(G/P\) be the orthogonal Grassmannian, i.e., the variety of maximal isotropic subspaces with respect to the symmetric bilinear form preserved by \(G\). For any Schubert variety \(X_w=\overline{BwP/B}\), the authors provide an explicit combinatorial description of the multiplicity and, more generally, the Hilbert function of the local ring at any point in \(X_w\). A similar problem was earlier solved for Grassmannians by \textit{V. Kreiman} and \textit{V. Lakshmibai} [Multiplicities of singular points in Schubert varieties of Grassmannians. Berlin: Springer, 553--563 (2003; Zbl 1092.14060)] and for symplectic Grassmannians by \textit{S. Ghorpade} and \textit{K. Raghavan} [Trans. Am. Math. Soc. 358, No. 12, 5401--5423 (2006; Zbl 1111.14046)].
For this, the geometric problem is reduced to a combinatorial one by means of standard monomial theory. The solution of this combinatorial problem forms the bulk of the paper. In particular, it turns out that the multiplicity of a point in a Schubert variety can be interpreted as the number of non-intersecting lattice paths of a certain kind.
For a special choice of a Schubert variety and a point on it, this problem specializes to a one about Pfaffian ideals, previously considered by \textit{E. De Negri} [Math. J. Toyama Univ. 24, 93--106 (2001; Zbl 1078.13508)] and \textit{J. Herzog} and \textit{N. V. Trung} [Adv. Math. 96, No. 1, 1--37 (1992; Zbl 0778.13022)]. orthogonal Grassmannian; Schubert variety; Hilbert function; multiplicity; Pfaffian ideal K. N. Raghavan and S. Upadhyay, Hilbert functions of points on Schubert varieties in orthogonal Grassmannians, preprint arXiv: 0704.0542 [math.CO] Grassmannians, Schubert varieties, flag manifolds, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Rings with straightening laws, Hodge algebras Hilbert functions of points on Schubert varieties in orthogonal grassmannians | 0 |
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 construct a certain solution to the Witten-Dijkgraf-Verlinde-Verlinde equation related to the small quantum cohomology ring of flag variety, and study the \(t\)-deformation of quantum Schubert polynomials corresponding to this solution. Kirillov, Anatol N., {\(t\)}-deformations of quantum {S}chubert polynomials, Funkcialaj Ekvacioj. Serio Internacia, 43, 1, 57-69, (2000) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds \(t\)-deformations of quantum Schubert polynomials. | 0 |
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 connected, simply connected, simple, complex Lie group and \(B\subset G\) a Borel subgroup. The author recovered in [Math. Res. Lett. 11, 35--48 (2004; Zbl 1062.14069)], in a purely combinatorial fashion, \textit{B.~Kim}'s presentation [Ann. Math. (2) 149, 129--148 (1999; Zbl 1054.14533)] of the quantum cohomology ring of the flag variety \(G/B\).
The main goal of this paper is to construct a combinatorial quantum product on \(H^*(G/B)\otimes\mathbb R[\{q_i\}_{i=1,\dots,b_2(G/B)}]\) which satisfies the usual properties of the quantum product (e.g. commutativity, associativity, Frobenius property). Next, Mare applies his result in [loc. cit.] to describe this ring in terms of relations and generators, and finds explicit quantum representatives for the Schubert classes. Finally, he proves that the combinatorial and the usual quantum product on \(H^*(G/B)\otimes\mathbb R[\{q_i\}]\) agree. flag manifolds; quantum Chevalley formula; quantum Giambelli formula Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds The combinatorial quantum cohomology ring of \(G/B\) | 0 |
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 Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac-Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type \(\tilde{A}_2\) on a Euclidean space \(V \cong \mathbb{R}^2\) to study the Bruhat order on \(W\). We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of \textit{B. D. Boe} and \textit{W. Graham} [Am. J. Math. 125, No. 2, 317--356 (2003; Zbl 1074.14045)] (which is a conjectural simplification of the Carrell-Peterson criterion (see [\textit{J. B. Carrell}, Proc. Symp. Pure Math. 56, 53--61 (1994; Zbl 0818.14020)]) for rational smoothness) to type \(\tilde{A}_2\). Computational evidence suggests that the only Schubert varieties in type \(\tilde{A}_2\) where the ``nontrivial'' case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety \(X ( w )\) of type \(\tilde{A}_2\). The proof uses descriptions we obtain of the elements \(x \leq w\) and of the rationally smooth locus of \(X ( w )\) in terms of the \(W\)-action on \(V\). As a consequence we describe the maximal nonrationally smooth points of \(X ( w )\). The results of this paper are used in a sequel to describe the smooth locus of \(X ( w )\), which is different from the rationally smooth locus. Schubert variety; rationally smooth; lookup conjecture Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds The Bruhat order, the lookup conjecture and spiral Schubert varieties of type \(\tilde{A}_2\) | 0 |
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. Neguţ} [J. Math. Pures Appl. (9) 104, No. 3, 403--435 (2015; Zbl 1349.14012)] introduced operators \(\text{Q}_{m, n}\) on symmetric functions and conjectured that, in the case where \(m\) and \(n\) are relatively prime, the expression \(\text{Q}_{m, n}(1)\) is given by the Hikita polynomial \(\text{H}_{m, n} [X; q, t]\). Later, \textit{F. Bergeron} et al. [Int. Math. Res. Not. 2016, No. 14, 4229--4270 (2016; Zbl 1404.05213)] extended and refined the conjectures of \(\text{Q}_{m, n}(1)\) for arbitrary \(m\) and \(n\) which we call the extended rational shuffle conjecture. In the special case \(\text{Q}_{n + 1, n}(1)\), the rational shuffle conjecture becomes the shuffle conjecture of \textit{J. Haglund} et al. [Duke Math. J. 126, No. 2, 195--232 (2005; Zbl 1069.05077)], which was proved in [\textit{E. Carlsson} and \textit{A. Mellit}, J. Am. Math. Soc. 31, No. 3, 661--697 (2018; Zbl 1387.05265)] as the shuffle theorem. The extended rational shuffle conjecture was later proved by \textit{A. Mellit} [``Toric braids and \((m, n)\)-parking functions'', Preprint, \url{arXiv:1604.07456}] as the extended rational shuffle theorem. The main goal of this paper is to study the combinatorics of the coefficients that arise in the Schur function expansion of \(\text{Q}_{m, n}(1)\) in certain special cases. \textit{E. Leven} [in: Proceedings of the 26th international conference on formal power series and algebraic combinatorics, FPSAC 2014, Chicago, IL, USA, June 29 -- July 3, 2014. Nancy: The Association. Discrete Mathematics \& Theoretical Computer Science (DMTCS). 789--800 (2014; Zbl 1393.05275)] gave a combinatorial proof of the Schur function expansion of \(\text{Q}_{2, 2 n + 1}(1)\) and \(\text{Q}_{2n + 1, 2}(1)\). In this paper, we explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of \(\text{Q}_{m, n}(1)\). Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of \(\text{Q}_{m, n}(1)\) in the case where \(m\) or \(n\) equals 3. Macdonald polynomials; parking functions; Dyck paths; rational shuffle theorem Symmetric functions and generalizations, Parametrization (Chow and Hilbert schemes), Combinatorial aspects of representation theory Schur function expansions and the rational shuffle theorem | 0 |
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 Frobenius algebra of functions on the critical set of the master function of a weighted arrangement of hyperplanes in \(\mathbb{C}^k\) with normal crossings. We construct two \textit{potential} functions (of first and second kind) of variables labeled by hyperplanes of the arrangement and prove that the matrix coefficients of the Grothendieck residue bilinear form on the algebra are given by the \(2k\)-th derivatives of the potential function of first kind and the matrix coefficients of the multiplication operators on the algebra are given by the \((2k+1)\)-st derivatives of the potential function of second kind. Thus the two potentials completely determine the Frobenius algebra. The presence of these potentials is a manifestation of a Frobenius like structure similar to the Frobenius manifold structure.
We introduce the notion of an elementary subarrangement of an arrangement with normal crossings. It turns out that our potential functions are local in the sense that the potential functions are sums of contributions from elementary subarrangements of the given arrangement. This is a new phenomenon of locality of the Grothendieck residue bilinear form and multiplication on the algebra.
It is known that this Frobenius algebra of functions on the critical set is isomorphic to the Bethe algebra of this arrangement. This Bethe algebra is an analog of the Bethe algebras in the theory of quantum integrable models. Thus our potential functions describe that Bethe algebra too. arrangement of hyperplanes; Grothendieck residue form; master function; critical set Relations with arrangements of hyperplanes, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Configurations and arrangements of linear subspaces Potentials of a family of arrangements of hyperplanes and elementary subarrangements | 0 |
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 \(WP\)-matroids are combinatorial structures, vastly generalizing matroids, that were introduced by \textit{I. M. Gel'fand} and \textit{V. V. Serganova} [Combinatorial geometries and torus strata on homogeneous compact manifolds, Russ. Math. Surv. 42, No. 2, 133-168 (1987); translation from Usp. Mat. Nauk 42, No. 2, 107-134 (1987; Zbl 0629.14035)]: if \(G\) is a semisimple Lie group with Weyl group \(W\), and if \(G_ P\) is a parabolic subgroup with corresponding parabolic \(P \subset W\), then the Schubert cells of \(G/G_ P\) are indexed by the set \(W/P\) of left cosets. The Schubert decomposition is, however, not invariant under \(W\). The \(W\)- invariant stratification of \(G/G_ P\) into thin Schubert cells is obtained as the common refinement of all \(W\)-images of the Schubert decomposition. \(WP\)-matroids are defined in such a way that very thin Schubert cell has a canonically associated \(WP\)-matroid.
(Ordinary matroids are obtained in the special case where \(G\) is a special linear group, \(W\) is a symmetric group, and \(P\) is a maximal parabolic subgroup. Already in this case the thin Schubert cells, corresponding to realization spaces of matroids, can be arbitrarily complicated real semialgebraic sets.)
The paper under review presents a new, equivalent definition of \(WP\)- matroids that is considerably simpler, and also works in situations where the Coxeter group \(W\) is infinite. The main result associates \(WP\)- matroids with the canonical retraction map \(\rho_{w,W}\) of an arbitrary thick building of type \(W\) to an appartment.
It is asked under what conditions \(WP\)-matroids have a ``geometric realization,'' that is, correspond to non-empty thin Schubert cells. A further generalization to the (very general) situation of ``thin chamber systems'' (e.g., barycentric subdivisions of triangulated pseudomanifolds) is also given. Tits systems; \(WP\)-matroids; Schubert cells; Schubert decomposition; Coxeter group A. V. Borovik and I. M. Gelfand,WP-matroids and thin Schubert cells on Tits systems. Adv. Math.103 (1994) 162--179. Combinatorial aspects of matroids and geometric lattices, Groups with a \(BN\)-pair; buildings, Grassmannians, Schubert varieties, flag manifolds, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) \(WP\)-matroids and thin Schubert cells on Tits systems | 0 |
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 Kac--Moody group, let \(P\) be its parabolic subgroup with the associated dominant weight \(\Lambda\), and let \(X=G/P\). Denote by \(W\) (resp., \(W_P\)) the Weyl group of \(G\) (resp.\;\(P\)), and let \(W^P\) be the set of minimal length representatives of \(W/W_P\). If \(\sigma^w\) denotes the Schubert class of \(X\) corresponding to \(w\in W^P\), then the Littlewood-Richardson coefficients are the constants \(c^{w}_{u, v}\) defined for \(u, v\in W^P\) by the formula \(\sigma^u\cup \sigma^v=\sum_{w\in W^P} c^{w}_{u, v}\sigma^w\). Following D. Peterson, the authors define special elements in \(W^P\) called \(\Lambda\)-minuscule. Given two elements \(u, v\in W\) smaller than a \(\Lambda\)-minuscule element \(w\), they define, combinatorially using jeu de taquin, an integer \(t^{w}_{u, v}\). Further, the authors extend these considerations introducing \(\Lambda\)-cominuscule elements by means of \(\Lambda\)-minuscule elements in the Langlands dual group, and defining some integer \(m^{w}_{u, v}\). The main result of this paper is the following
Theorem. Let \(u, v, w\in W\) be \(\Lambda\)-(co)minuscule. Assume that \(w\) is slant-finite-dimensional. Then \(c^{w}_{u, v}=m^{w}_{u, v}t^{w}_{u, v}\). Littelewood-Richardson rule; Schubert calculus; Kac-Moody homogeneous spaces; jeu de taquin P.-E. Chaput and N. Perrin. Towards a Littlewood-Richardson rule for KacMoody homogeneous spaces. J. Lie Theory, 22(1):17--80, 2012. Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Towards a Littlewood-Richardson rule for Kac-Moody homogeneous spaces | 0 |
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 \(\mathfrak g\) be a real reductive Lie algebra, \(G\) be the adjoint group of \(\mathfrak g\), \(K\) a maximal compact subgroup of \(G\) with the Lie algebra \(\mathfrak k\). Let \({\mathfrak g}={\mathfrak k}\oplus {\mathfrak p}\) be a Cartan decomposition of \({\mathfrak g}\). Then the subgroup \(K\) acts on \(\mathfrak p\). Let \(\mathfrak a\) be a maximal abelian subalgebra of \(\mathfrak p\) and \(W\) the Weyl group of the pair \((\mathfrak g,\mathfrak a)\). The classical Chevalley Theorem asserts that the restriction to \({\mathfrak a}\) is an isomorphism of the algebra \(S(\mathfrak g)^ K\) of \(K\)-invariant polynomial functions on \(\mathfrak p\) onto the algebra \(S(\mathfrak a)^ W\) of \(W\)-invariant polynomial functions on \({\mathfrak a}\). Further, \(S(\mathfrak a)^ W\) is the algebra generated by a system of algebraically independent homogeneous \(W\)-invariant polynomials called the elementary \(W\)-invariants. The corresponding elements of \(S(\mathfrak p)^ K\) are also called elementary \(K\)-invariants.
The theorem can be extended to the \(C^{\infty}\)-functions. Namely, the restriction is an isomorphism of the algebra \(C^{\infty}(\mathfrak p)^ K\) of \(K\)-invariant \(C^{\infty}\)-functions on \(\mathfrak p\) onto the algebra \(C^{\infty}(\mathfrak a)^ W\) of \(W\)-invariant \(C^{\infty}\)-functions on \(\mathfrak a\) [cf. \textit{J. Dadok}, Adv. Math. 44, 121--131 (1982; Zbl 0521.22009)].
The author extends the Theorem to \(C^ r\)-functions and proves that any \(W\)-invariant \(C^ r\)-functions on \(\mathfrak a\) can be extended to \(K\)-invariant \(C^ q\)-functions on \(\mathfrak p\) with \(q=[r/(1+d-s)]\), where \(d\) and \(s\) are numbers determined by \(W\). Further, the extended functions can be expressed as \(C^ q\)-functions with respect to elementary \(K\)-invariants on \(\mathfrak p\). Also he shows that the restriction to \(\mathfrak a\) induces an isomorphism of Fréchet spaces between \(C^ r(\mathfrak p)^ K\) and \(C^ r(\mathfrak a)^ W\). real reductive Lie algebra; adjoint group; Cartan decomposition; Weyl group; Chevalley Theorem; invariant polynomial functions; invariant \(C^ r\)-functions G. Barbançon, ''Invariants de classeCr des groupes finis engendres par des reflexions et théorème de Chevalley en classeCr,''Duke Math. J.,53, 563--584 (1986). Real-valued functions on manifolds, Differentiable maps on manifolds, \(L^p\)-spaces and other function spaces on groups, semigroups, etc., Group actions on varieties or schemes (quotients) Invariants of class \(C^r\) of finite groups generated by reflections and Chevalley's theorem in class \(C^r\) | 0 |
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 typical question of the real Schubert calculus is whether the intersection of real Schubert cells is totally real. It is, in fact, a very rare event. The Shapiro-Shapiro conjecture posed in 90s asserts that the osculating flags of a real normal curve always define real Schubert cells whose intersection is transversal and totally real. One of the formulations of the Shapiro-Shapiro conjecture is as follows: If \(f_1(z),\dots,f_d(z)\in{\mathbb C}[z]\) are linearly independent polynomials such that the Wronski determinant \(Wr(f_1,\dots,f_d)\) has only real roots, then \(f_1,\dots,f_d\) span a real subspace in \({\mathbb C}[z]\). There were found links between this conjecture and a variety of problems in combinatorics, representation theory, families over the moduli space of stable marked rations curves \({\mathcal M}_{0,n}\). The was settled in affirmative by \textit{E. Mukhin} et al. [Ann. Math. (2) 170, No. 2, 863--881 (2009; Zbl 1213.14101); J. Am. Math. Soc. 22, No. 4, 909--940 (2009; Zbl 1205.17026)], using heavy techniques from quantum integrable systems, Fuchsian differential equations, and representation theory. The paper under review suggests a completely different proof of the Shapiro-Shapiro conjecture based on geometric and topological methods. It is known that the Wronski map from a Schubert cell in the Grassmannian of linear subspaces spanned by polynomials \(f_1,\dots,f_d\) of given degrees to the projectivized space of polynomials of degree \(n=\sum_i\deg f_i-d(d-1)/2\) is finite of the degree equal to the number of standard Young tableaux of a certain shape. On the other side, the degree of the Wronski map equals \(\chi^\lambda(1^n)\), the value of the character of the symmetric group \(S_n\) determined by \(\deg f_1,\dots,\deg f_d\). The authors prove that the restriction of the Wronski map to the real part of the Grassmannian with the target set formed by polynomials having only real roots has the topological degree \(\chi^\lambda(1^n)\). This implies the Shapiro-Shapiro conjecture. The key ingredient of the proof is the so-called character orientation of the components of the source space of the restricted real Wronski map. By construction, the characted orientation is defined globally on each component of the source. Then one takes a real polynomial of degree \(n\) with \(n\) real roots, for which the fibre of the Wronski map can be explicitly described. The elements of the fibre are joined with paths along which the change of the character orientation obeys the Murnaghan-Nakayama rule, and this finally yields that the signed count of the points in the fibre amounts to the evaluation of \(\chi^\lambda(1^n)\). The same approach applied to the restrictions of the real Wronski map to the target spaces formed by real polynomials with \(n_1\le n\) real roots provides non-trivial lower bounds to the topological degree of these restrictions in terms of other values of the same character. real Schubert calculus; Wronski map; rational normal curves; characters of the symmetric group Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Real algebraic sets A topological proof of the Shapiro-Shapiro conjecture | 0 |
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 show that the irreducible components of the singular locus of a Schubert variety in \(\text{GL}_n/B\), associated to a covexillary permutation \(w\), are parametrized by some of the coessential points of the graph of \(w\). We give an explicit description of these components and we describe the singularity along each of them. Schubert varieties; vexillary permutations; generic singularities; singular loci Cortez, A.: Singularités génériques des variétés de Schubert covexillaires. Ann. inst. Fourier (Grenoble) 51, No. 2, 375-393 (2001) Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Reflection and Coxeter groups (group-theoretic aspects) Singularités génériques des variétés de Schubert covexillaires. (Generic singularities of covexillary Schubert varieties) | 0 |
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 reductive complex Lie group, with a chosen maximal torus \(T\), associated Weyl group \(W\), Borel subgroup \(B_+=T_+N_+\) and opposite Borel \(B_-=TN_-\). For \(w \in W\), \(X_w=\overline{B_-wB_+}/B_+\) is the associated Schubert variety. For \(v \in W\), \(X_o^v=N_+vB_+/B_+\) is the opposite Schubert cell, and \(X^v_{wo}=X_w \cap X^v_o\) is the corresponding Kazhdan--Lusztig variety.
Now suppose that \(v\) is expressed as a minimal length product of reflections associated to simple roots. The sequence \(Q\) of simple roots is called a reduced word for \(v\). Then if \(w \in W\), the author defines a simplicial complex \(\Delta(Q,w)\) called the subword complex. To any simplicial complex \(\Delta\) with vertex set \(V\), and any field \(F\), the Stanley--Reisner scheme \(SR(\Delta) \subseteq F^V\) is the union of the hyperplanes \(F^S\) for \(S \in \Delta\); there is a natural action of the the torus \((\mathbb G_m)^V\) on \(SR(\Delta)\). The author's main result asserts that for \(w \leq v \in W\) and reduced word \(Q\) for \(v\), there is a sequence of flat \(T\)--equivariant degenerations from \(X^v_{wo}\) to \(SR(\Delta(Q,w))\). This implies a number of previously known forumlae in equivariant \(K\)--theory of Schubert classes and geometry of Schubert cells. Schubert cell; Kazhdan-Lusztig variety; Stanley-Reisner scheme A. Knutson, Schubert patches degenerate to subword complexes, Transform. Groups 13 (2008), no. 3--4, 715--726. Grassmannians, Schubert varieties, flag manifolds, Fibrations, degenerations in algebraic geometry Schubert patches degenerate to subword complexes | 0 |
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 Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as \(\hslash \to 0\), and becomes non-commutative or ``quantum'' away from this limit. For a classical curve defined by the zero locus of a polynomial \(A(x, y)\), we provide a construction of its non-commutative counterpart \(\widehat{A}\left({\widehat{x},\widehat{y}} \right)\) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing \(\widehat{A}\) that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that come from knots or topological strings, their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be ``quantizable,'' and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices. matrix models; non-commutative geometry; Chern-Simons theories; topological strings Gukov, S.; Sułkowski, P., A-polynomial, B-model, and quantization, J. High Energy Phys., 1202, (2012) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Topological field theories in quantum mechanics, Noncommutative geometry methods in quantum field theory, Eta-invariants, Chern-Simons invariants, Quantization in field theory; cohomological methods, Relationships between algebraic curves and physics A-polynomial, B-model, and quantization | 0 |
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. mirror symmetry; elliptic curve; Fukaya's category A. Polishchuk and E. Zaslow, \textit{Categorical mirror symmetry: the elliptic curve}, \textit{Adv. Theor. Math. Phys.}\textbf{2} (1998) 443 [math/9801119] [INSPIRE]. Elliptic curves, Calabi-Yau manifolds (algebro-geometric aspects), Derived categories, triangulated categories Categorical mirror symmetry: The elliptic curve | 0 |
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 Wolf's domain \(\Omega_D\) of \(q\)-cycles in an open \(SL_n(\mathbb{R})\) orbit \(D\) in a classical flag manifold \(Z\) [see \textit{J. A. Wolf}, Bull. Am. Math. Soc. 75, 1121-1237 (1969; Zbl 0183.50901)] is studied in detail. Special Schubert varieties which are in particular transversal to the cycles are introduced. It is shown that \(\Omega_D\) is Stein with respect to special rational functions in the image of the Andreotti-Norguet integration transform \(AN:H^q(D, \Omega^P)\to O(\Omega_D)\). For \(n\) fixed with \(Z_1\) and \(Z_2\) any two flag manifolds containing open orbits \(D_1\) and \(D_2\) it is proved that \(\Omega_{D_1}\) and \(\Omega_{D_2}\) are naturally biholomorphic.
The basic results which are needed from the theory of cycle spaces are sketched in the appendix. domain of \(q\)-cycles; flag manifold; Schubert varieties; Andreotti-Norguet integration A.T. Huckleberry, A. Simon. On cycle spaces of flag domains of SL
\[
(n,{\mathbb{R}})
\]
( n , R ) (Appendix by D. Barlet). J. Reine Angew. Math. 541 (2001), 171--208. Stein spaces, Complex Lie groups, group actions on complex spaces, Grassmannians, Schubert varieties, flag manifolds, Semisimple Lie groups and their representations On cycle spaces of flag domains of \(SL_{n}{\mathbb R}\) | 0 |
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 connection between the cohomology of the homogeneous space \(G/H\) and the quantum field theory called the coset model \(G/H\) was given by Lerche. It was claimed that the cohomology ring \(H^*(G/H)\) is the Jacobi ring of some function which is called the potential. Later Gapner calculated the potential for the cohomology of any Grassmann manifold \(Gr(k,n)\) and discovered that the cohomology of Grassmann manifolds and the fusion rings of the unitary group are Jacobi rings of the same potential. Also Witten extended this claim for quantum cohomology of Grassmann manifolds. It gave a beautiful link between representation of loop groups, quantum Schubert calculus, integrable systems and singularity theory.
In this work the authors show that the cohomology of a Hermitian symmetric homogeneous manifold is a Jacobi ring of a certain potential function and the generalized cohomology theories such as equivariant quantum cohomology and \(K\)-theory are Jacobi rings of a particular deformation of this potential. The structure of the Jacobi ring of such a potential is related to singularity theory. Schubert calculus; homogeneous spaces; Jacobi rings; Landau; Ginzburg model; mirror symmetry; Frobenius manifolds Homology and cohomology of homogeneous spaces of Lie groups, Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Generalized (extraordinary) homology and cohomology theories in algebraic topology, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Schubert calculus and singularity theory | 0 |
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 thesis we establish several links between parametrizations and combinatorics of canonical bases occurring in representation theory and the theory of mirror dual cluster varieties. We establish a Crossing Formula for Lusztig's crystal models associated to arbitrary reduced words for the longest element of the Weyl group of \(\mathrm{SL}_n\). By passing to tropical cluster charts of the relevant double Bruhat cell we obtain a second version of the Crossing Formula in which the \(p\)-map of the mirror dual \(\mathcal A\)- and \(\mathcal X\)-cluster variety plays a prominent role. We establish a duality between Lusztig's models and the string model of crystals.
We obtain a partial generalisation of our results to the simply laced type. We explain the relations between the various cones and relate them to the potential function appearing in the work of Gross, Hacking, Keel and Kontsevich and to the decoration function appearing in the work of Berenstein and Kazhdan. crossing formula for Lusztig's crystal models; mirror dual cluster varieties Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Quantum groups (quantized enveloping algebras) and related deformations, Mirror symmetry (algebro-geometric aspects), Combinatorial aspects of representation theory Crystal combinatorics and mirror symmetry for cluster varieties | 0 |
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 embedding of the Lagrangian Grassmannian \(\mathrm{LG}(n)\) inside an ordinary Grassmannian that is well-behaved with respect to the Wronski map. As a consequence, we obtain an analogue of the Mukhin-Tarasov-Varchenko theorem for \(\mathrm{LG}(n)\). The restriction of the Wronski map to \(\mathrm{LG}(n)\) has degree equal to the number of shifted or unshifted tableaux of staircase shape. For special fibres one can define bijections, which, in turn, gives a bijection between these two classes of tableaux. The properties of these bijections lead to a geometric proof of a branching rule for the cohomological map \(H^\ast(\mathrm{Gr}(n, 2 n)) \otimes H^\ast(\mathrm{LG}(n)) \rightarrow H^\ast(\mathrm{LG}(n))\), induced by the diagonal inclusion \(\mathrm{LG}(n) \hookrightarrow \mathrm{LG}(n) \times \mathrm{Gr}(n, 2 n)\). We also discuss applications to the orbit structure of jeu de taquin promotion on staircase tableaux. Schubert calculus; Lagrangian Grassmannian; Wronski map; Young tableaux Combinatorial aspects of representation theory, Classical problems, Schubert calculus A marvellous embedding of the Lagrangian Grassmannian | 0 |
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 any field with \(\text{char }K\neq 2,3.\) The authors classify all cubic homogeneous polynomial mappings \(H\in K[x_1,\dots ,x_n]^m\) over \(K\) for which the rank \(r\) of the Jacobian matrix \(JH\) of \(H\) is \(\leq 2\). Namely, there exist non-singular linear mappings \(S\in \mathrm{GL}_m(K)\) and \(T\in\mathrm{GL}_n(K)\) such that for \(\widetilde{H}=S\circ H\circ T\) one of the following holds:
1. \(\widetilde{H}_{r+1}=\cdots =\widetilde{H}_m=0\),
2. \(r=2\) and \(\widetilde{H}\in K[x_1,x_2]^m\),
3. \(r=2\) and \(\mathrm{Lin}_K(\widetilde{H}_1,\dots ,\widetilde{H}_m)= \mathrm{Lin}_K(x_3x_1^2,x_3x_2x_1,x_3x_2^2)\)
As an application they prove a particular case of the Jacobian Conjecture. They show that for such an \(H\) with \(n=m,\) if \(F=x+H\) and \(\det JF=\mathrm{const}.\neq 0\) then \(F\) is invertible. polynomial map; cubic homogeneous polynomial map; Jacobian conjecture; Keller map; tame map de Bondt, M.; Sun, X., Classification of cubic homogeneous polynomial maps with Jacobian matrices of rank two, Bull. Aust. Math. Soc., 98, 1, 89-101, (2018) Jacobian problem Classification of cubic homogeneous polynomial maps with Jacobian matrices of rank two | 0 |
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 results of this article deal with the intersection of two or more Schubert subvarieties of \(G/B\) which are assumed to be in general position. A combinatorial procedure termed `root game' is devised to decide whether such an intersection is empty. Following the common practise in such investigations Kleiman's transversality theorem is employed to reduce to the corresponding transversality problem in the tangent space of \(G/B.\) Thus the results are confined to characteristic \(0.\) The root game is inconclusive in certain cases, i.e., a game which can not be won provides no information. The author convincingly argues that the natural density of such cases is asymptotically \(0.\) Of course, for determining the structure of the intersection ring these cases do matter. The root game is shown to generalize to the related problems of `branching Schubert calculus'. The article is well written and presents a good selection of examples to help the reader. Schubert varieties K. Purbhoo, \textit{Vanishing and nonvanishing criteria in Schubert calculus}, Inter. Math. Res. Not. (2006), Art. ID 24590, 38 pp. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Vanishing and nonvanishing criteria in Schubert calculus | 0 |
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 \(W_d\) denote the cycle on the Jacobian of a smooth projective curve of genus \(g\) given by the image of the \(d\)th symmetric product of the curve under the Abel-Jacobi map. Poincaré's formula says that \(W_{g-d}={W^d_{g-1}\over d!}\). The present paper gives an analogue of this formula for a nodal curve \(X\) of arithmetic genus \(g\) with \(k\) nodes in characteristic 0. Let \(\overline J^0(X)\) denote the moduli space of rank-1 torsion free sheaves of degree 0 on \(X\) and \(\widetilde J^0(X)\) its normalization. Using the symmetric powers of the smooth part of \(X\) the autors define cycles \(\widetilde W_d\) in \(\widetilde J^0(X)\) and proved the analogue of Poincaré's formula \(\widetilde W_{g-d}= {\widetilde W^d_{g-1}\over d!}\). Moreover, identifying \(\widetilde J^0(X)\) with the corresponding Brill-Noether locus \(\widetilde B_X(1,d,1)\), also an analogue of the Riemann singularity theorem is shown, namely:
(1) \(\widetilde B_X(1,d,1)\) is a normal projective variety for all \(d\), (2) \(\widetilde B_X(1,d,2)\) is the singular locus of \(\widetilde B_X(1,d,1)\) for all \(d< g\) and (3) for \(d< g\), \(r\geq 2\), \(\widetilde B_X(1,d,r)\) has codimension \(\geq r\) in \(\widetilde B_X(1,d,1)\).
The proofs use some results on compactifications of the Picard scheme as well as on parabolic bundles of the first author. nodal curve; Poincaré's theorem; Riemann's singularity theorem Bhosle Usha, N; Parameswaran, AJ, On the Poincaré formula and Riemann singularity theorem over nodal curves, Math. Ann., 342, 885-902, (2008) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Singularities of curves, local rings On the Poincaré formula and the Riemann singularity theorem over nodal curves | 0 |
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 Von den fünf Kapiteln des vorliegenden Lehrbuches der Sammlung \textit{Schubert} erfaßt das erste den Begriff der algebraischen Funktion als einer mehrwertigen Funktion mit polaren Unstetigkeiten und stellt die Lehre von der Fortsetzung solcher analytischen Funktionen auf Grund der \textit{Puiseux}schen Methode der Reihenentwicklungen dar. Das zweite Kapitel konstruiert die \textit{Riemann}sche Verzweigungsfläche, entwickelt den Begriff einer Klasse algebraischer Funktionen und sucht die zwischen den Funktionen der Klasse bestehenden algebraischen Beziehungen auf. Das dritte gibt eine Klassifizierung und Aufstellung der zur gegebenen \textit{Riemann}schen Fläche gehörigen Abelschen Integrale unter der Voraussetzung, daß\ die Grundkurve nur Doppelpunkte als Singularitäten besitzt, während das vierte sodann den \textit{Riemann}-\textit{Roch}schen Satz darlegt und das fünfte und letzte von den birationalen Transformationen des Gebildes und den Moduln der Klasse handelt. Die ersten beiden Kapitel halten im wesentlichen den in den bisherigen Lehrbüchern üblichen Entwicklungsgang inne, die drei letzten benutzen in der Hauptsache die von \textit{Christoffel} in Abhandlungen und Vorlesungen gegebenen Methoden. Referent ist der Meinung, daß\ diese weder mehr dem heutigen Stande der Wissenschaft entsprechen, noch zur Einführung in das betrachtete Gebiet geeignet erscheinen können. Außerdem findet sich auch im einzelnen in dem Buche eine Anzahl tatsächlicher Irrtümer. Die Begründung dieser Behauptungen findet man in einer im Archiv der Mathematik und Physik erscheinenden ausführlichen Besprechung des Werkes. Algebraic functions of one variable. Research exposition (monographs, survey articles) pertaining to algebraic geometry The Theory of Algebraic Functions and Their Integrals. | 0 |
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 show that general isotropic flags for odd-orthogonal and symplectic groups are general for Schubert calculus on the classical Grassmannian in that Schubert varieties defined by such flags meet transversally. This strengthens a result of \textit{P.
Belkale} and \textit{Sh. Kumar} [J. Algebr. Geom. 19, No. 2, 199--242 (2010; Zbl 1233.20040); \url{arXiv:0708.0398}]. T. Haines, \textit{Equidimensionality of convolution morphisms and applications to saturation problems} (with Appendix by T. Haines, M. Kapovich, J. J. Millson), Advances in Math. \textbf{207} (2006), 297-327. Classical problems, Schubert calculus General isotropic flags are general (for Grassmannian Schubert calculus) | 0 |
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 \(H({\mathbb X})=H(\text{Coh}({\mathbb X}))\) be the Hall algebra of the abelian category of coherent sheaves on the weighted projective line \(\mathbb X\) over a finite field. In the paper under review, the authors study the composition algebra \(U({\mathbb X})\) of \(H({\mathbb X})\) and its reduced Drinfeld double \(DU({\mathbb X})\).
The motivation comes from several places:
(i) The importance of Hall algebras of the category of nilpotent representations of finite quivers and their role in the theory of quantized Kac-Moody algebras;
(ii) The relation of the Hall algebra of the category of coherent sheaves on the classical projective line \({\mathbb P}^1\) with Drinfeld's new realization of the quantized enveloping algebra \(U_q(\widehat{{\mathfrak s}{\mathfrak l}}_2)\);
(iii) An attempt to generalize the results of the previous work by the authors [Glasg. Math. J. 54, No. 2, 283--307 (2012; Zbl 1247.17010)] to other quantized enveloping algebras.
The first result of the paper presents a new definition of the composition algebra \(U({\mathbb X})\) and gives that \(U({\mathbb X})\) is a topological bialgebra, i.e., a subalgebra of \(H({\mathbb X})\) closed under the Green comultiplication. Then the authors characterize the subalgebra \(\bar{U}({\mathbb X})_{\text{tor}}\) of \(U({\mathbb X})\) generated by the skyscraper sheaves as a tensor product of the Macdonald ring of symmetric functions and algebras \(U_q^+(\widehat{{\mathfrak s}{\mathfrak l}}_p)\). The authors also obtain other important properties of \(U({\mathbb X})\). In particular, for any weighted projective line \(\mathbb X\) the Drinfeld double \(DU({\mathbb X})\) of \(U({\mathbb X})\) contains a subalgebra isomorphic to \(U_q(\widehat{{\mathfrak s}{\mathfrak l}}_2)\). The same technique leads to the construction of new embeddings, e.g., for the quantized enveloping algebras \(U_q(\hat{A}_3)\to U_q(\hat{D}_4)\). Moreover, the approach of the paper allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types \(D^{(1,1)}_4,E^{(1,1)}_6,E^{(1,1)}_7\) and \(E^{(1,1)}_8\). In particular, the method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras. Hall algebra; Drinfeld double; coherent sheaves; weighted projective line; quantized enveloping algebra; PBW-type basis Burban, I.; Schiffmann, O.: The composition Hall algebra of a weighted projective line. J. reine angew. Math. 679, 75-124 (2013) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups (quantized enveloping algebras) and related deformations, Derived categories, triangulated categories The composition Hall algebra of a weighted projective line | 0 |
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 Our investigation in the present paper is based on three important results. (1) In [14], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [6], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [11], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula. quiver; perverse sheaf; restriction functor; Green's formula The parity of Lusztig's restriction functor and Green's formula | 0 |
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 semi-simple algebraic group and \(R\) its root system. Denote by \(\Theta\) (resp. \(\theta\)) the highest root (resp. the highest short root) of \(R\). A dominant weight is said to be adjoint if it is equal to \(\Theta\) and coadjoint if it is equal to \(\theta\). Similarly, if \(P\) is the parabolic subgroup of \(G\) associated to the adjoint (resp. coadjoint) weight, then \(G/P\) is called an adjoint (resp. coadjoint) homogeneous space. Examples are the orthogonal and symplectic Grassmannians of lines, odd-dimensional quadrics and projective spaces, the two-step flag variety \(F(1,n;n+1)\) as well as six exceptional homogeneous spaces.
The paper under review studies various aspects of the (small) quantum cohomology of (co)adjoint varieties, in line with previous work on minuscule homogeneous spaces by the same authors and \textit{L. Manivel} [Transform. Groups 13, No. 1, 47--89 (2008; Zbl 1147.14023)]. Let \(q\) denote the quantum parameter of a (co)adjoint variety \(X=G/P\). The main results are a simplified formula for the quantum multiplication by the hyperplane class of \(X\), a bound on the possible degrees in \(q\) of the quantum product of two Schubert classes of \(X\), and a presentation of the quantum cohomology ring for the six exceptional (co)adjoint varieties -- such a presentation being already known for the classical (co)adjoint varieties from [\textit{A.S. Buch}, \textit{A. Kresch} and \textit{H. Tamvakis}, Invent. Math. 178, No. 2, 345--405 (2009; Zbl 1193.14071)].
As a consequence of the presentation of the quantum cohomology ring, the authors are able to tell whether the (small) quantum cohomology ring of (co)adjoint varieties is semi-simple. Semi-simplicity of the quantum cohomology is an important problem. For instance, it appears in a conjecture of \textit{B. Dubrovin} [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 315--326 (1998 ; Zbl 0916.32018)] on Fano varieties, in relation with properties of their derived categories. Another consequence of semi-simplicity for homogeneous spaces \(G/P\) is a phenomenon known as strange duality, i.e. the existence of an involution of the localised quantum cohomology ring \(\text{QH}^*(G/P)_{q\neq 0}\). Strange duality for (co)minuscule homogeneous spaces has been studied in [\textit{P. E. Chaput}, \textit{L. Manivel} and \textit{N. Perrin}, Canadian J. Math. 62, No. 6, 1246--1263 (2010; Zbl 1219.14060)]. Here the authors prove that the quantum cohomology ring of coadjoint varieties is almost never semi-simple, while it always is for adjoint non-coadjoint varieties. In the latter case they give a partial description of strange duality. quantum cohomology; adjoint homogeneous spaces; Schubert calculus; strange duality Chaput, P.; Perrin, N., On the quantum cohomology of adjoint varieties, Proc. Lond. Math. Soc., 103, 294-330, (2011) Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) On the quantum cohomology of adjoint varieties | 0 |
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 preprojective cohomological Hall algebra (CoHA) introduced by the authors in [Proc. Lond. Math. Soc. (3) 116, No. 5, 1029--1074 (2018; Zbl 1431.17013)] for any quiver \(Q\) and any one-parameter formal group \({\mathbb {G}}\). In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra \(\mathfrak {g}_Q\) associated to \(Q\), whose quantization comes from the formal group \({\mathbb G}\). We prove, when the group \({\mathbb G}\) is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian. quantum group; shuffle algebra; Hall algebra; Yangian; Drinfeld double Quantum groups (quantized enveloping algebras) and related deformations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Bordism and cobordism theories and formal group laws in algebraic topology Cohomological Hall algebras and affine quantum groups | 0 |
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 \((X, X^!)\) be a pair of Nakajima quiver varieties dual to each other under 3d mirror symmetry. Following results in [\textit{M. Aganagic} and \textit{A. Okounkov}, J. Am. Math. Soc. 34, No. 1, 79--133 (2021; Zbl 07304878)], a general enumerative expectation first proposed by Okounkov is that, up to the exchange of equivariant and Kähler variables prescribed by 3d mirror symmetry,
\[
\mathsf{V}_{\mathrm{QM}}(X^!) = \mathrm{Stab}^{\mathsf{Ell}}(X) \mathsf{V}_{\mathrm{QM}}(X)
\]
where \(\mathsf{V}_{\mathrm{QM}}(X)\) is the equivariant K-theoretic quasimap vertex of \(X\) and \(\mathrm{Stab}^{\mathsf{Ell}}(X)\) is a certain normalization of the elliptic stable envelope of \(X\). The important and simplest case, when \(X = T^*\mathrm{GL}(n)/B\) is the cotangent bundle of the full flag variety, is known to be 3d mirror to itself ([\textit{R. Rimányi} et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 15, Paper 093, 22 p. (2019; Zbl 1451.53116)]) and the relation above was verified manually by Dinkins. The paper under review then takes the index limit ([\textit{N. Nekrasov} and \textit{A. Okounkov}, Algebr. Geom. 3, No. 3, 320--369 (2016; Zbl 1369.14069)]) of this relation, where:
\begin{itemize}
\item the right-hand side becomes K-theoretic stable envelopes of \(X\) or of certain subvarieties of \(X\) (see [\textit{Y. Kononov} and \textit{A. Smirnov}, Lett. Math. Phys. 112, No. 4, Paper No. 69, 25 p. (2022; Zbl 07569277)]);
\item the left-hand side becomes the so-called index vertex of \(X^!\), which can be viewed as a generalization of the refined topological vertex of Iqbal-Koczaz-Vafa.
\end{itemize}
This holds for index limits of any, not necessarily generic, slope. One consequence is that index vertices are always expansions of rational functions. The simplest example \(X = T^*\mathbb{P}^1\) is given explicitly. stable envelopes; 3d mirror symmetry; quasimaps; enumerative geometry Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Elliptic cohomology, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial aspects of algebraic geometry Euler characteristic of stable envelopes | 0 |
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 first prove an identity involving symmetric polynomials. This identity leads us into exploring the geometry of Lagrangian Grassmannians. As an insight applications, we obtain a formula for the integral over the Lagrangian Grassmannian of a characteristic class of the tautological sub-bundle. Moreover, a relation to that over the ordinary Grassmannian and its application to the degree formula for the Lagrangian Grassmannian are given. Finally, we present further applications to the computation of Schubert structure constants and three-point, degree 1, genus 0 Gromov-Witten invariants of the Lagrangian Grassmannian. Some examples together with explicit computations are presented. equivariant cohomology; Gromov-Witten invariant; Lagrangian Grassmannian; interpolation; Schubert structure constant; symmetric polynomial; quantum cohomology Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Symmetric functions and generalizations, Equivariant homology and cohomology in algebraic topology An identity involving symmetric polynomials and the geometry of Lagrangian Grassmannians | 0 |
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 Define
\[
\widetilde{\mathcal{C}_n}:=\{(X,Y)\in\mathrm{Mat}(n\times n, \mathbb{C})\times\mathrm{Mat}(n\times n, \mathbb{C}): \mathrm{rank}([X, Y]+{\mathrm{id}})=1\}.
\]
Let GL\(_n(\mathbb{C})\) act on \(\widetilde{\mathcal{C}_n}\) defined by
\[
G.(X,Y):=(GXG^{-1}, GYG^{-1})
\]
and
\[
\mathcal{C}_n:=\widetilde{\mathcal{C}_n}//\mathrm{GL}_n(\mathbb{C}),
\]
which is called \(n\)-th Calogero-Moser space. By G. Wilson's result, the space \(\mathcal{C}_n\) is diffeomorphic to the Hilbert scheme of \(n\) points in \(\mathbb{C}^2\).
Let \(M\) be complex affine-algebraic manifold. We say that \(M\) enjoys the algebraic density property if the Lie algebra that is generated by all complete polynomial vector fields on \(M\) agrees with the Lie algebra of all polynomial vector fields on \(M\). The main result of this article is that the Calogero-Moser space \(\mathcal{C}_n\) enjoys the algebraic density property. density property; holomorphic automorphisms; flexibility; infinite transitivity; Andersen-Lempert theory; Calogero-Moser space Automorphism groups of \(\mathbb{C}^n\) and affine manifolds, Group actions on affine varieties, Oka principle and Oka manifolds The density property for Calogero-Moser spaces | 0 |
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 Thom polynomials express invariants of singularities of a general map \(f:X\to Y\) between complex analytic manifolds in terms of invariants of \(X\) and \(Y\). Knowing the Thom polynomial of a singularity \(\eta\) one can compute the cohomology classes represented by \(\eta\)-points of \(f\). One of the most successful methods to compute Thom polynomials is the method of restriction equations developed mainly by Rimanyi which converts the problem to an algebraic one. Pragacz, in collaboration with Lascoux, combined the method with techniques of Schur functions and, with Weber, established that the coefficients of Schur function expansions of the Thom polynomials of stable singularities are nonnegative. In the paper under review, the author studies from this point of view the structure of the Thom polynomials for \(A_4(-)\) singularities. The Schur function expansions of these polynomials are analysed and it is shown that partitions indexing the Schur function expansions of Thom polynomials for \(A_4(-)\) singularities have at most four parts. The system of equations that determines these polynomials is simplified and a recursive description of Thom polynomials for \(A_4(-)\) singularities is given. Also, the author gives Thom polynomials for \(A_4(3)\) and \(A_4(4)\) singularities. Thom polynomials; singularities; global singularity theory; classes of degeneracy loci; Schur functions; resultants Symmetric functions and generalizations, Enumerative problems (combinatorial problems) in algebraic geometry, Singularities of differentiable mappings in differential topology On Thom polynomials for \(A_4(-)\) via Schur functions | 0 |
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 \({\mathbb P}^{d}\) denote projective space over an algebraically closed field \(k\). Let \(K_{0}( {\mathbb P}^{d} )\) denote the Grothendieck group of locally free sheaves on \({\mathbb P}^{d}\). For \(x \in K_{0}( {\mathbb P}^{d} )\) with Chern classes \(c_{i}(x)\) the Chern polynomial is defined by
\[
C_{x}(t) = 1 + c_{1}(x)t + c_{2}(x)t^{2} + \dots + c_{d}(x)t^{d}
\]
lying in \({\mathbb Z}[t]/(t^{d+1})\). On the other hand a graded, finitely generated module \(M\) over the graded ring \(k[x_{0}, \dots , x_{d}]\) gives a coherent sheaf on \({\mathbb P}^{d}\) and a class in the Grothendieck group \(G_{0}( {\mathbb P}^{d} ) \cong K_{0}( {\mathbb P}^{d} )\). The Hilbert polynomial \(P_{M}(t)\) is related to \(C_{M}(t)\) by the Hirzebruch-Riemann-Roch theorem [see \textit{D. Eisenbud}, ``Commutative algebra. With a view towards algebraic geometry'', Grad. Texts Math. 150 (1995; Zbl 0819.13001)].
The author shows that the homomorphism
\[
\xi : K_{0}( {\mathbb P}^{d} ) \rightarrow ({\mathbb Z}[t]/(t^{d+1}))^{*} \times {\mathbb Z}
\]
given by \(\xi(M) = (C_{M}(t) , \text{rank}M))\) is injective. As an application she shows that the classes of \(M\) and \(N\) in \(K_{0}({\mathbb P}^{d})\) are equal if and only if \(C_{M}(t) = C_{N}(t)\) and \( \text{rank}M) = \text{rank}N)\) or if and only if \(P_{M}(t) = P_{N}(t)\). The paper concludes with a section detailing the precise relation between Chern and Hilbert polynomials, which appears in [\textit{D. Eisenbud}, loc. cit., Exercise 19.18]. Chan, C. -Y.J.: A correspondence between Hilbert polynomials and Chern polynomials over projective spaces, Illinois J. Math. 48, No. 2, 451-462 (2004) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry A correspondence between {H}ilbert polynomials and {C}hern polynomials over projective spaces | 0 |
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 combinatorial connection between the real geometry and the \(K\)-theory of complex \textit{Schubert curves} \(S(\lambda_\bullet )\), which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In [the second author, ``One-dimensional Schubert problems with respect to osculating flags'', Can. J. Math. 69, No. 1, 143--177 (2017; \url{doi:10.4153/CJM-2015-061-1})], it was shown that the real geometry of these curves is described by the orbits of a map \(\omega \) on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of \(\mathbb{RP}^1\), with \(\omega \) as the monodromy operator. We provide a fast, local algorithm for computing \(\omega\) without rectifying the skew tableau and show that certain steps in our algorithm are in bijective correspondence with \textit{O. Pechenik} and \textit{A. Yong}'s genomic tableaux [``Genomic tableaux'', Preprint, \url{arXiv:1603.08490}), which enumerate the \(K\)-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the \(K\)-theory and real geometry of \(S(\lambda _\bullet )\). Schubert calculus; Young tableaux; \textit{jeu de taquin}; \(K\)-theory; monodromy; osculating flag Gillespie, M., Levinson, J.: Monodromy and \(K\)-theory of Schubert curves via generalized jeu de taquin. J. Algebraic Comb. (2016) (to appear). arXiv:1602.02375 Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Applications of methods of algebraic \(K\)-theory in algebraic geometry Monodromy and \(K\)-theory of Schubert curves via generalized \textit{jeu de taquin} | 0 |
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 the projective variety parametrizing the complete flags of vector subspaces of \(\mathbb{C}^n\). Let \(u_1,\dots, u_n\) be the standard basis of \(\mathbb{C}^n\). For each \(w\) in the symmetric group \(S_n\), the Schubert subvariety \(X_w\) of \(X\) consists of the flags \(V_1\subset V_2\subset\cdots\subset V_n= \mathbb{C}^n\) for which \(\dim(V_p\cap (u_1,\dots, u_q))\geq \text{card}\{i\leq p: w(i)\leq q\}\), for every \(p\), \(q\). It is the closure of \(Be_w\), where \(e_w\) is the flag \((u_{w(1)})\subset (u_{w(1)}, u_{w(2)})\subset\cdots\subset (u_{w(1)},\dots, u_{w(n)})\) and \(B\) is the Borel subgroup of \(\text{GL}(n)\) that leaves the standard flag \(e_{\text{id}}\) invariant. One defines the Bruhat order on \(S_n\) by \(v\leq w\) iff \(X_v\subseteq X_w\). It has several combinatorial characterizations.
In this paper, the authors give an explicit combinatorial description of the irreducible components of the singular locus of \(X_w\). Such a component must be of the form \(X_v\) for some \(v\leq w\). Moreover, \(X_v\) is contained in the singular locus of \(X_w\) iff \(e_v\) is a singular point of \(X_w\). \textit{V. Lakshmibai} and \textit{C. S. Seshadri} [Bull. Am. Math. Soc., New Ser. 11, 363--366 (1984; Zbl 0549.14016)] described the tangent space to \(X_w\) at \(e_v\). As a consequence of their description, \(e_v\) is a nonsingular point of \(X_w\) iff \(\text{card}\{i< j: v\cdot(i,j)\leq w\}= \text{card}\{i< j: w(i)> w(j)\}\). Then, \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, 45--52 (1990; Zbl 0714.14033)] proved that the singular locus of \(X_w\) is non-empty iff in \(w\) there is a pattern of type 4231 or 3412, i.e., there exist \(i< j< k< 1\) such that \(w(1)< w(j)< w(k)< w(i)\) or \(w(k)< w(1)< w(i)< w(j)\). They also gave a conjectural description of the singular locus of \(X_w\). \textit{V. Gasharov} [Compos. Math. 126, 47--56 (2001; Zbl 0983.14038)] verified that the components appearing in the conjecture of Lakshmibai and Sandhya are contained in the singular locus of \(X_w\).
Using their combinatorial description, the authors are able to show, in this paper, that the conjectured components are, indeed, the only components of the singular locus of \(X_w\). These irreducible components are indexed by permutations \(v\) which differ from \(w\) by a cycle depending naturally on a 4231 or 3412 pattern in \(w\). Furthermore, they give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points \(e_v\). variety of complete flags; Schubert variety; singular locus; symmetry group; Bruhat order Billey, Sara C.; Warrington, Gregory S., Maximal singular loci of Schubert varieties in \(\operatorname{SL}(n) / B\), Trans. Amer. Math. Soc., 355, 10, 3915-3945, (2003), MR 1990570 Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric groups, Singularities in algebraic geometry Maximal singular loci of Schubert varieties in \(SL(n)/B\) | 0 |
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 link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. Bruhat order; Schubert variety; rational smoothness; palidromic; hyperplanes; Coxeter arrangement S. Oh and H. Yoo. ''Bruhat order, rationally smooth Schubert varieties, and hyperplane ar rangements''. 22nd International Conference on Formal Power Series and Algebraic Combinatorics. DMTCS Proceedings, 2010, pp. 965--972.URL. Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements | 0 |
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 provides a systematic treatment of adjointness situations between compactly generated triangulated categories; this clarifies the relationship between Grothendieck-Neeman duality and the so-called Wirthmuller isomorphisms. The treatment is extremely elegant and leads to peculiar situations, as for example the fact that a string of adjoints between compactly generated, tensor triangulated categories is made by either three, five of infinitely many adjoints. Even though the main motivation lies in Grothendieck duality theory, there is plenty of examples where such a calculus is useful in stable homotopy theory; it is perhaps one of the most enticing suggestions of the paper that algebraic geometry à la Grothendieck and stable homotopy theory find in the theory of ``well-behaved triangulated categories'' a natural place to be developed on the same footing. It is also remarkable how the arguments employed in the paper are completely elementary, i.e. relying only on the classical theory of triangulated categories, making no use of explicitly \(\infty\)-categorical methods. Serre functor; Grothendieck duality; adjoints; compactly generated triangulated category; dualizing object; Wirthmuller isomorphism Derived categories, triangulated categories, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Abstract and axiomatic homotopy theory in algebraic topology Grothendieck-Neeman duality and the Wirthmüller isomorphism | 0 |
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 and \(Q_0\) be its set of vertices. Let \(M\) be a representation of \(Q\) having dimension vector \(\underline{m}=(m_p)_{p \in Q_0}\) and let \(\displaystyle{m=\sum_{p \in Q_0} m_p}\). The quiver Grassmannian \(\mathrm{Gr}_{\underline{e}}(M)\) of subrepresentations \(V\) of \(M\) with dimension vector \(\underline{e}\) is defined as a closed subscheme of the usual Grassmannian \(\mathrm{Gr}(e,m)\) where \(e = \sum_{p \in Q_0} e_p\). The intersection of \(\mathrm{Gr}_{\underline{e}}(M)\) with a Schubert decomposition of \(\mathrm{Gr}(e,m)\) defines a Schubert decomposition of \(\mathrm{Gr}_{\underline{e}}(M)\). In general, this is not a decomposition into affine spaces, and the isomorphism type of the Schubert cells is not independent of the choices that define the Schubert decomposition for \(\mathrm{Gr}(e,m)\).
The results of this paper concentrate on establishing cases of quiver Grassmannians that have a Schubert decomposition into affine spaces. The main result roughly says the following: Let \(S \subset T\) be an inclusion of quivers such that the quotient \(T/S\) is a tree and let \(M\) be a representation of \(T\). Let \(F : T \rightarrow Q\) be a morphism of quivers that satisfies a certain Hypothesis (\(H\)). Then the Schubert cell \(C_{\beta}^{F*M}\) of the push-forward \(F*M\) of \(M\) equals the product \(A^n \times C_{\beta}^{F*M}\) of an affine space with the corresponding Schubert cell for the push-forward of the restriction \(M_S\) of \(M\) to \(S\). While Hypothesis (H) is too technical to explain in brevity, it should be mentioned that this hypothesis is a purely combinatorial condition on the structure of the fibres of \(F : T \rightarrow Q\), which can be checked easily in examples, and which can be implemented in a computer algorithm. The following are some of its consequences and other results of this paper.
(i) Let \(M\) be an exceptional indecomposable representation of the Kronecker quiver and \(\underline{e}\) a dimension vector. Then \(\mathrm{Gr}_{\underline{e}}(M)\) has a Schubert decomposition into affine spaces.
(ii) Let \(T\) be a tree and \(M\) a representation of \(T\) whose linear maps are block matrices of the form \(\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\) where \(1\) is a square identity matrix. Then \(\mathrm{Gr}_{\underline{e}}(M)\) has a Schubert decomposition into affine spaces. If for all arrows \(\alpha\) of Q, the linear maps \(M_{\alpha}\) are isomorphisms, then the quiver Grassmannian decomposes into a series of fibre bundles whose fibres are usual Grassmannians.
(iii) The author re-obtains the Schubert decompositions of \textit{G. Cerulli Irelli} and \textit{F. Esposito} [Algebra Number Theory 5, No. 6, 777--801 (2011; Zbl 1267.13043)] and \textit{G. Cerulli Irelli} et al. [Algebra Number Theory 6, No. 1, 165--194 (2012; Zbl 1282.14083)].
(iv) If \(\displaystyle{\mathrm{Gr}_{\underline{e}}(M, \mathbb{C}) = \amalg_{i \in I} X_i(\mathbb{C})}\) is a decomposition into complex affine spaces \(X_i(\mathbb{C})\), then the Euler characteristic of \(\mathrm{Gr}_{\underline{e}}(M)\) equals the number of non-empty Schubert cells \(C_{\beta}^M\) where \(\beta\) is of type \(\underline{e}\). (Proposition 6.3). If \(\mathrm{Gr}_{\underline{e}}(M)\) is smooth, then the singular cohomology is concentrated in even degrees and is generated by the closure of the classes of the Schubert cells (Corollary 6.2). In particular, this reproduces the formulas in [\textit{G. Cerulli Irelli}, J. Algebr. Comb. 33, No. 2, 259--276 (2011; Zbl 1243.16013)] and [\textit{N. Haupt}, Algebr. Represent. Theory 15, No. 4, 755--793 (2012; Zbl 1275.16016)] (under assumption of Hypothesis (\(H\))) in terms of the combinatorics of the Schubert cells.
(v) If \(\mathrm{Gr}_{\underline{e}}(M, \mathbb{C}) = \amalg_{i \in I} X_i(\mathbb{C})\) is a regular decomposition into complex affine spaces, then the multiplication of \(H^*(\mathrm{Gr}_{\underline{e}}(M, \mathbb{C}))\) is determined by the cohomology rings of the irreducible components of \(\mathrm{Gr}_{\underline{e}}(M)\) (Lemma 6.5). quiver representations; Schubert decompositions; trees; singular cohomology; Euler characteristic; quiver Grassmannian Cluster algebras, Grassmannians, Schubert varieties, flag manifolds On Schubert decompositions of quiver Grassmannians | 0 |
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, simply connected algebraic group, \(B\) a Borel subgroup of \(G\) and \(T\) a maximal torus contained in \(B\). For any element \(w\) of the Weyl group \(W=N_G(T)/T\), denote by \(X(w)\) its Schubert variety, i.e. the closure of the \(B\)-orbit \(Bw\) in the flag variety \(G/B\). Let \(x\) be a \(T\)-fixed point in \(X(w)\) and \({\mathcal T}_x\bigl(X(w)\bigr)\) be the reduced tangent cone of \(X(w)\) at \(x\). The paper contains a proof of the earlier announced result, that there is a natural one to one correspondence between \(T\)-invariant curves in \(X(w)\) containing \(x\) and \(T\)-invariant lines in \({\mathcal T}_x\bigl(X(w)\bigr)\), if \(G\) is not of type \(G_2\). If \(G\) is of type \(ADE\), one can replace the tangent cone in this correspondence by its linear span \(\theta(x,w)\) in the Zariski tangent space of \(X(w)\) at \(x\). Furthermore, the weight structure of the \(T\)-module \(\theta(x,w)\) is investigated. In particular it is shown that \(\theta(x,w)\) is the \(B\)-module span of the set of \(T\)-invariant lines in \({\mathcal T}_x\bigl(X(w)\bigr)\) if \(x\) is the unique \(B\)-fixed point in \(G/B\). Schubert variety; tangent cone; Weyl group; flag variety; weight structure J. B. Carrell, The span of the tangent cone of a Schubert variety, in: Algebraic Groups and Lie Groups, Austral. Math. Soc. Lect. Ser., Vol. 9, Cambridge University Press, Cambridge, 1997, pp. 51--59. Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations The span of the tangent cone of a Schubert variety | 0 |
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 is a lovely article: It answers a concrete, explicit mathematical problem by concrete results from abstract theory. The question is the following: ``Is it possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian?'' The conjecture stated by the authors is then that a Lie algebra is finite dimensional if and only if the universal enveloping algebra \(U(L)\) is noetherian.
To help answering the question, and to strengthen the validity of the conjecture, the authors prove that the conjecture holds for the enveloping algebra \(U(W_+)\) of the positive Witt algebra \(W_+\), and as a consequence, for the full Witt algebra \(U(W)\). They also prove the conjecture true for the Virasoro algebra \(V\), and any infinite dimensional \(\mathbb Z\)-graded simple Lie algebra of polynomial growth. It is even so that all central factors of \(U(V)\) are non-noetherian. The authors use a very explicit representation of the Witt algebra by generators and relations. Then it is possible to give an explicitly given homomorphism \(\rho: U(W_+)\rightarrow K[t;\tau]\), which by definition makes the image \(R:=\text{im}(\rho)\) birationally commutative. Here \(K\) is a field and \(\tau\in\text{Aut}_k(K)\). Then one can use the classification of birationally commutative projective surfaces to prove that \(R\) is not noetherian.
The actual point schemes are not needed for the results of the article, but the homomorphism \(\rho\) is constructed using the truncated point schemes of \(U(W_+)\) which have geometric points parameterizing graded \(U(W_+)\)-modules with Hilbert series \(1+s+\dots+s^n\). As this is the classification of birationally commutative graded domains of Gelfand Kirillov dimension 3, it also follows that the GK dimension of \(R\) is 3, which is a nice bonus result of the article.
Every computation is explicitly given, and because Macaulay2 is used, however just to verify the computations, appendices with the scripts are given.
This article proves that the noncommutative moduli theory can be applied to explicit and concrete problems. birationally commutative algebra; centerless Virasoro algebra; infinite dimensional Lie algebra; non-noetherian universal enveloping algebra; Witt algebra; birationally commutative projective surfaces Sierra, S.S., Walton, Ch.: The universal enveloping algebra of Witt algebra is not noetherian. ArXiv:1304.0114 [math.RA] Noncommutative algebraic geometry, Universal enveloping algebras of Lie algebras, Rings arising from noncommutative algebraic geometry, Virasoro and related algebras The universal enveloping algebra of the Witt algebra is not Noetherian | 0 |
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 general context, the paper deals with, is the following: Let be given real algebraic sets \(W\) and \(X\subset W\times \mathbb{R}^n\) satisfying the condition \(W\times \{0\} \subset X\). The real algebraic set \(X\) may be given by a polynomial function \(F:W\times \mathbb{R}^n\to \mathbb{R}^m\) as \(X=F^{-1} (0)\). For \(w\in W\) let us write \(X_w:= \{y\in \mathbb{R}^n; (w;y) \in X\}\) and \(F_w:\mathbb{R}^n \to\mathbb{R}^m\) for the polynomial function defined by \(F_w(y): =F(w;y)\), where \(y\) is any point of \(\mathbb{R}^n\). Thus \((X_w)_{w\in W}\) and \((F_w)_{w\in W}\) are by the real algebraic variety \(W\) parametrized families of real algebraic sets and polynomial functions.
The paper addresses particular aspects of the following general problem: What can be said about the families \((X_w)_{w\in W}\) and \((F_w)_{w\in W}\) in terms of the (abstract) parameter variety \(W\) only? What are the ``invariant'' properties of these objects? The paper gives the following answers to these questions:
Let \(m=n\). Suppose that for any \(w\in W\) the point \(0\in \mathbb{R}^n\) is isolated in \(F_w^{-1}(0)\). For \(w\in W\) denote by \(\deg_0F_w\) the topological degree of \(F_w\) in the point \(0\in \mathbb{R}^n\). Then there exist polynomial functions \(g_1, \dots, g_s\) defined on the real algebraic variety \(W\), such that for any \(w\in W\) the identity \(\deg_0 F_w= \text{sign} g_1(w) +\cdots +\text{sign} g_s(w)\) holds (theorem 3.4).
For any point \(w\in W\) let us denote by \(\chi (X_\omega)\) the Euler characteristic of \(X_w\). Then there are polynomial functions \(g_1, \dots, g_s\), defined on the real algebraic variety \(W\), such that \(\chi (X_w)=\text{sign} g_1 (w)+ \dots+\text{sign} g_s(w)\) holds for any \(w\in W\). - In particular, if \(W\) is irreducible, there exist a proper real algebraic subset \(\Sigma \subset W\), an integer \(\mu\) and a polynomial function \(g:W\to \mathbb{R}\), with \(g\) nowhere vanishing on \(W \setminus\Sigma\), such that for any point \(w\in W\setminus \Sigma\) the congruence relations \(\chi (X_w) \equiv \mu+ \text{sign} g(w) \pmod 4\) and \(\chi (X_w) \equiv \mu+1 \pmod 2\) hold (theorem 5.3).
These results are applied to algebraically constructible functions define on the real algebraic variety \(W\) (i.e. to functions \(\varphi: W\to \mathbb{Z}\) which admit a presentation as a finite sum \(\varphi= \sum_{1\leq i\leq t} m_if_i* \mathbf{1}_{Z_i}\) where, for any \(1\leq i\leq t\), \(Z_i\) is a real algebraic set which characteristic function \(\mathbf{1}_{Z_i}\) is a proper regular morphism, \(f_i* \mathbf{1}_{Z_i}: =\int_{Z_i} f_i\) and \(m_1, \dots, m_t\) are integers. - In this context, the authors show the following results:
A given integer valued function \(\varphi: W\to \mathbb{Z}\) is algebraically constructible if and only if there exist polynomial functions \(g_1, \dots, g_s:W\to \mathbb{R}\) such that for any point \(w\in W\), the identity \(\varphi (w)= \text{sign} g_1(w) +\cdots +\text{sign} g_s (w)\) holds (theorem 6.1).
Let \(m=n\). Assume that for any \(w\in W\) the point \(0\in \mathbb{R}^n\) is isolated in \(F_w^{-1} (0)\). Then, the map which assigns to any point \(w\in W\) the value \(\deg_0F_w\) is algebraically constructible. -- Moreover, the function which assigns to any point \(w\in W\) the value \(\chi(X_w)\) is algebraically constructible (corollary 6.2).
Assume that the real algebraic set \(W\) is irreducible. Let \(\varphi: W\to \mathbb{Z}\) be an algebraic constructible function. Then there exist a proper real algebraic subset \(\Sigma \subset W\), an integer \(\mu\) and a polynomial function \(g:W\to \mathbb{R}\), nowhere vanishing on \(W \smallsetminus \Sigma\), such that the congruence relations \(\varphi(w) \equiv\mu +\text{sign} g(w) \pmod 4\) and \(\varphi(w) \equiv \mu+1 \pmod 2\) hold (corollary 6.3).
Let \(\varphi: W\to \mathbb{Z}\) be an algebraically constructible function. For \(w\in W\) let \(\Lambda \varphi (w): =\int_{S_{(w, \varepsilon)}}\varphi\), where \(S_{(w, \varepsilon)}\) is the \(\varepsilon\)-sphere centered at the point \(w\) for sufficient small \(\varepsilon>0\). Let \(D\varphi: =\varphi -\Lambda \varphi\). Then \(1\over 2\Lambda \varphi\) is integer valued and algebraically constructible (theorem 6.4). topology of real algebraic varieties; families of real algebraic sets; algebraically constructible functions Parusiński, A.; Szafraniec, Z., Algebraically constructible functions and signs of polynomials, Manuscr. Math., 93, 443-456, (1997) Topology of real algebraic varieties, Real algebraic sets Algebraically constructible functions and signs of polynomials | 0 |
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 is devoted to the study of the double affine Hecke algebra (DAHA) of type \(A_1\) and its degenerations. DAHA has two parameters \(q\) and \(t\), and has a natural polynomial representation. In a series of earlier papers, the first author constructed the non-symmetric generalizations of Macdonald polynomials defined as eigenvectors of a certain commutative subalgebra inside DAHA.
In present paper, the authors write explicit formulas for nonsymmetric Macdonald polynomials of type \(A_1\) and study their degenerations in the limit \(t\to 0\). In this limit, DAHA is transformed to nil-DAHA, which can be written explicitly by generators and relations, and nonsymmetric Macdonald polynomials are transformed to a certain nonsymmetric analogue of \(q\)-Whittaker functions. A possible connection to the work of \textit{A. Givental} and \textit{Y.-P. Lee} [Invent. Math. 151, No. 1, 193-219 (2003; Zbl 1051.14063)] on quantum \(K\)-theory of flag varieties is also discussed. double affine Hecke algebras; nil-DAHAs; nonsymmetric Macdonald polynomials; Whittaker functions I. Cherednik and D. Orr. ''One-dimensional nil-DAHA and Whittaker functions I''. Trans form. Groups 17 (2012), pp. 953--987.DOI. Hecke algebras and their representations, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds One-dimensional nil-DAHA and Whittaker functions. I. | 0 |
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 depicts a beautiful and powerful framework to deal with the multiplicative structure of the equivariant cohomology ring of Grassmannians. It contains also some foundational material intended for the full appreciation of another important paper on the same subject by the same author, ``Schubert Calculus and the Equivariant Cohomology of Grassmannians'' [Adv. Math. 217, No. 4, 1869--1888 (2008; Zbl 1136.14042)]. The latter, due to the laws of the editorial world, has already appeared two years ago, reversing both the logical and the chronological order. There are many reasons, however, to be happy for the publication, at last, of both papers: To speak frankly and with a direct language, the reviewer's opinion is that these are two important pieces of mathematics, showing that the depth of the results is not necessarily related with the technical sophistication used to get them. One should also add that these two papers are a sort of crowning of a relevant mathematical path the author began to walk a few years ago together with \textit{A.~Thorup} [Indiana Univ. Math. J. 56, No. 2, 825--845 (2007; Zbl 1121.14045) and Indiana Univ. Math. J. 58, No. 1, 283--300 (2009; Zbl 1198.14052)].
The approach taken in the paper under review makes truly clear that \textsl{equivariant Schubert calculus}, rather than being an extension of the classical theory is, instead, just a particular case of the classical picture. This becomes clear once one rephrases it via a framework, elaborated by Laksov and Thorup, based on the symmetric structure of the exterior powers of (quotients of) polynomial rings. Emphasizing the novelties of this approach, also through the comparison with pre-existent literature on the subject, will be the goal of the remaining part of this review.
Recall that classical Schubert calculus is concerned with the product structure of the intersection ring of Grassmannians, which are finite \({\mathbb Z}\)-modules freely generated by the so-called \textsl{Schubert cycles}: the task consists in determining the structural constants with respect to such a basis. The goal is achieved by means of the celebrated Pieri's and Giambelli's formulas: as the intersection ring is generated as a \({\mathbb Z}\)-algebra by certain \textsl{special Schubert cycles}, the former computes the structural constants of the product of any Schubert cycle with a special one and the latter expresses any Schubert cycle as an explicit polynomial in the algebra generators. The picture can be easily generalized to the intersection theory of Grassmann bundles. Laksov and Thorup show that the Chow group of a Grassmann scheme, thought of as a module over the Chow ring (Poincaré duality), is related with an amazing algebraic fact. Let \(A[X]\) be the polynomial ring in one indeterminate, \({\mathtt p}\in A[X]\) a monic polynomial (or zero), and \(R=A[X]/{\mathtt p}\). Then the \(k\)th exterior power of \(R\) possesses a unique module structure over \(S:=A[X_1,\dots, X_n]^{sym}\), the ring of symmetric polynomials, such that the natural projection \(\otimes^kR\rightarrow \bigwedge^kR\) is \(S\)-linear. This is the \textsl{symmetric structure} of \(\bigwedge^kR\). Moreover, the latter is generated by the class modulo \({\mathtt p}\) of \(X^{k-1}\wedge X^{k-2} \wedge \dots\wedge 1\) as an \(S\)-module (and is a free \(S\)-module if \({\mathtt p}=0\)). Pieri's formula comes out from cancellations, due to the \({\mathbb Z}_2\)-symmetry of the exterior algebra, of the module multiplication of an element of \(S\) with an element of \(\bigwedge^kR\). Furthermore if \(f_1,\dots, f_k\in A[X]\), a very general Giambelli's type formula expresses the product \(f_1(X)\wedge \dots\wedge f_k(X)\) as an \(S\)-multiple of \(X^{k-1}\wedge X^{k-2}\wedge \dots\wedge X\wedge 1\). The main point is that this framework can be applied verbatim also in the case of the (small quantum) equivariant cohomology of Grassmannians. To be more precise, if \({\mathbb P}^{n-1}\) is acted on diagonally by an \(n\)-dimensional torus, there is an obvious induced action on the Grassmannian \(G(k,n)\). Many important papers came to be concerned with \textsl{equivariant versions of Pieri's or Giambelli's formulas} -- see e.g [\textit{A. Knutson} and \textit{T. Tao}, Duke Math. J. 119, No. 2, 221--260 (2003; Zbl 1064.14063)] and [\textit{V. Lakshmibai, K. N. Raghavan} and \textit{P. Sankaran}, Pure Appl. Math. Q. 2, No. 3, 699--717 (2006; Zbl 1105.14065)]. Such formulas are supposed to be \textsl{equivariant corrections} of classical Pieri's and Giambelli's ones. Indeed, equivariant classes of Schubert varieties are identified with a list of polynomials satisfying certain (GKM) conditions, topologically established by \textit{M. Goresky, R. Kottwitz} and \textit{R. MacPherson} [Invent. Math. 131, No.1, 25--83 (1998; Zbl 0897.22009)].
In the paper under review the author shows that such conditions naturally arise from purely algebraic manipulations. In fact, Theorem 3.4, the central result tying the \textsl{general Schubert calculus} with the above way of spelling equivariant Schubert calculus, shows that there is a natural \(A\)-algebra isomorphism between the ring \(A[T_1,\dots, T_l]^{sym}\) of symmetric polynomials in \(l\) indeterminates and the \(A\)-algebra \(H(l)\) of all \(l\)-tuples of homogeneous polynomials, with respect to the componentwise sum and product, satisfying the GKM conditions. In other words the underlying topology of the GKM conditions has a strong algebraic root. If \(\xi\) is the equivariant first Chern class of the bundle \(O_{{\mathbb P}^{n-1}}(1)\), \(A:={\mathbb Z}[y_1,\dots,y_n]\) is the equivariant cohomology of a point and \({\mathtt p}\in A[X]\) is the monic polynomial of degree \(n\) such that \({\mathtt p}(\xi)=0\), then the general Schubert Calculus by Laksov and Thorup gives exactly, in this situation, the equivariant Schubert calculus for the Grassmannian \(G(k,n)\)
Laksov's paper culminates in fact (Theorem 4.5) with the computation of what in previous literature is referred to as an ``equivariant correction'' of Pieri's formula: indeed, it is the ``classical'' Pieri written in the basis of \(A[X]\) given by the factorial Schur functions as in [\textit{L. C. Mihalcea}, Adv. Math. 203, No. 1, 1--33 (2006; Zbl 1100.14045)]. Furthermore, if one chooses the natural basis \(X^i+{\mathtt p}\) of \(A[X]/{\mathtt p}\), then Giambelli's formula has no equivariant correction while Theorem 4.5 (equivariant Pieri's formula) answers to a question posed by Lakshmibai, Raghavan and Sankaran \textsl{(loc.cit.)}.
Laksov's theory, and the naturality with the GKM conditions come into play, shows not only what one should mean by Pieri's and Giambelli's formulas in Grassmann's cohomologies, but also that it is very suited for computations. Indeed, the reviewer quite disagrees with Laksov, as well as with others authors making similar claims, when he says, in the introduction, that the explicit Pieri's formula proved in Theorem 4.7 can be inferred as a particular case from the Pieri type formulas got in [\textit{S. Robinson}, J. Algebra 249, No. 1, 38--58 (2002; Zbl 1061.14060)]. The latter, clearly, are very important and have a strong theoretical impact; but it is hard even to guess, in the case of Grassmannians, how a full set of equivariant Pieri's formulas could look like or can be deduced by them. In the absence of any new reference, or without doing explicitly the job, any claim in this sense is just the vague consideration that, in mathematics, everything is related with everything else. Which, of course, is true.
In conclusion, it would really be a big surprise if, in the nearest future, the reviewed paper would not put itself among the unavoidable references for people working in equivariant cohomology of homogeneous varieties, especially Grassmannians, attracting all the attention it truly deserves. equivariant Pieri's formula; equivariant Giambelli's formula ----, A formalism for equivariant Schubert calculus Algebra Number Theory 3 (2009), 711-727. Classical problems, Schubert calculus, Equivariant algebraic topology of manifolds, Grassmannians, Schubert varieties, flag manifolds A formalism for equivariant Schubert calculus | 0 |
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 topic of the paper under review is the study of genus \(0\) double Hurwitz numbers. For two partitions \(\lambda,\mu\) of the same integer \(d\), the double Hurwitz number \(h_{0;\lambda,\mu}\) is the number of non-isomorphic degree \(d\) coverings of the sphere by a sphere, with branching corresponding to \(\lambda\) (resp., \(\mu\)) over \(0\) (resp., \(\infty)\) and an appropriate number of fixed simple branch points. Double Hurwitz numbers for every genus \(g\) were studied by \textit{I.~P.~Goulden, D.~M.~Jackson} and \textit{R.~Vakil} [Adv. Math. 198, No. 1, 43--92 (2005; Zbl 1086.14022)]. There it was proved that if the length \(m\) of \(\lambda\) and \(n\) of \(\mu\) is fixed, double Hurwitz numbers are piecewise polynomials in the variables \(\lambda_1,\dots,\lambda_m,\mu_1,\dots,\mu_n\). Moreover, for genus \(0\) this piecewise polynomial is homogeneous of degree \(m+n-3\).
In the article under review, it is explained how these polynomials can be computed for fixed \(m,n\). The authors consider the parameter space for \(h_{0;m,n}\), i.e. the subset of \(\mathbb R^{m+n}\) on which \(h_{0;m,n}\) is defined, and divide it into the polynomiality domains, called chambers. An explicit description of the hyperplanes separating the chambers is given, as well as a formula describing how \(h_{0;m,n}\) varies between neighbouring chambers. Together with an expression for a special chamber called totally negative, these formulas enable to compute all Hurwitz numbers \(h_{0;m,n}\) by recurrence. double Hurwitz number; piecewise polynomiality; chambers; wall crossing Shadrin, S., Shapiro, M., Vainshtein, A.: On double Hurwitz numbers in genus 0. In: Proceedings of FPSAC'07. Tianjin, China (2007) Families, moduli of curves (algebraic) Chamber behavior of double Hurwitz numbers in genus 0 | 0 |
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 \(w_0,w_1,\dots, w_n\) be a sequence of positive integers and let \(\mathbb P^w\) be the weighted projective space \(\mathbb P(w_0,w_1,\dots, w_n)\), i.e., the quotient \([(\mathbb{C}^{n+ 1}\setminus\{0\})/\mathbb{C}^\times]\), where \(\mathbb{C}^\times\) acts with weights \(-w_0,-w_1,\dots, -w_n\). The authors calculate the small quantum orbifold cohomology ring of the weighted projective space \(\mathbb P(w_0,w_1,\dots, w_n)\). Their approach is essentially due to \textit{A. B. Givental} [Sel. Math., New Ser. 1, No.~2, 325--345 (1995; Zbl 0920.14028) in: Proceedings of the international congress of mathematicians, ICM `94, August 3--11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 472--480 (1995; Zbl 0863.14021) and Int. Math. Res. Not. 1996, No.~13, 613--663 (1996; Zbl 0881.55006)].
They begin with a heuristic argument relating the quantum cohomology of \(\mathbb P^w\) to the \(S^1\)-equivariant Floer cohomology of the loop space \(L\mathbb P^w\), and from this they conjecture a formula for a certain generating function -- the small J-function for genus-zero Gromov-Witten invariants of \(\mathbb P^w\). The small J-function determines the small quantum orbifold cohomology of \(\mathbb P^w\). Next, the authors prove that their conjectural formula for the small J-function is correct by analyzing the relationship between two compactifications of the space of parameterized rational curves in \(\mathbb P^w\) -- a toric compactification and the space of genus-zero stable maps to \(\mathbb P^w\times\mathbb P(1, r)\) of degree \(1/r\) with respect to the second factor. These compactifications carry naturally a \(\mathbb{C}^\times\)-action, which one can think of as arising from the rotation of loops, and there is a map between them which is \(\mathbb{C}^\times\)-equivariant. The authors formula for the small J-function can be expressed in terms of integrals of \(\mathbb{C}^\times\)-equivariant cohomology classes on the toric compactification. Then, using results of \textit{A. Bertram} [Invent. Math. 142, No.~3, 487--512 (2000; Zbl 1031.14027)], and localization in equivariant cohomology, they transform these into integrals of classes on the stable map compactification. This establishes the authors formula for the small J-function, and so allows them to determine the small quantum orbifold cohomology ring of \(\mathbb P^w\). weighted projective space; quantum cohomology; rational curves; Gromov-Witten invariants Coates, T; Lee, Y-P; Corti, A; Tseng, HH, \textit{the quantum orbifold cohomology of weighted projective spaces}, Acta Math., 202, 139-193, (2009) Symplectic aspects of Floer homology and cohomology, Floer homology, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Differential geometric aspects of gerbes and differential characters, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Relationships between surfaces, higher-dimensional varieties, and physics The quantum orbifold cohomology of weighted projective spaces | 0 |
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 combinatorial connection between the real geometry and the \(K\)-theory of complex Schubert curves \(\mathcal{S}(\lambda_\bullet)\), which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. The second author [Can. J. Math. 69, No. 1, 143--177 (2017; Zbl 1390.14163)] showed that the real geometry of these curves is described by the orbits of a map \(\omega\) on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of \(\mathbb{RP}^1\), with \(\omega\) as the monodromy operator.
We provide a fast, local algorithm for computing \(\omega\) without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with \textit{O. Pechenik} and \textit{A. Yong}'s genomic tableaux [Forum Math. Pi 5, Paper No. e3, 128 p. (2017; Zbl 1369.14060)], which enumerate the \(K\)-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the \(K\)-theory and real geometry of \(\mathcal{S}(\lambda_\bullet)\). Young tableaux; monodromy; Schubert calculus; \(K\)-theory; osculating flag; jeu de taquin Classical problems, Schubert calculus, Combinatorial aspects of representation theory Monodromy and \(K\)-theory of Schubert curves via generalized jeu de taquin | 0 |
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 Grothendieck duality goes back to 1958, to the talk at the ICM in Edinburgh [\textit{A. Grothendieck}, ``The cohomology theory of abstract algebraic varieties'', in: Proc. Int. Congr. Math. 1958. New York: Cambridge Univ. Press. 103--118 (1960; Zbl 0119.36902)] announcing the result. Hochschild homology is even older, its roots can be traced back to the 1945 article [\textit{G. Hochschild}, Ann. Math. (2) 46, 58--67 (1945; Zbl 0063.02029)]. The fact that the two might be related is relatively recent. The first hint of a relationship came in 1987 in \textit{J. Lipman} [Contemp. Math. 61 (1987; Zbl 0606.14015)], and another was found in 1997 in \textit{M. Van den Bergh} [J. Algebra 195, No. 2, 662--679 (1997; Zbl 0894.16020)]. Each of these discoveries was interesting and had an impact, Lipman's mostly by giving another approach to the computations and van den Bergh's especially on the development of non-commutative versions of the subject. However in this survey we will almost entirely focus on a third, much more recent connection, discovered in 2008 by \textit{L. L. Avramov} and \textit{S. B. Iyengar} [Mich. Math. J. 57, 17--35 (2008; Zbl 1245.13011)] and later developed and extended in several papers, see for example [\textit{L. L. Avramov} et al., Adv. Math. 223, No. 2, 735--772 (2010; Zbl 1183.13021); \textit{S. B. Iyengar} et al., Compos. Math. 151, No. 4, 735--764 (2015; Zbl 1348.13022)].
There are two classical paths to the foundations of Grothendieck duality, one following Grothendieck and Hartshorne [\textit{R. Hartshorne}, Residues and duality. Berlin-Heidelberg-New York: Springer (1966; Zbl 0212.26101)] and (much later) \textit{B. Conrad} [Grothendieck duality and base change. Berlin: Springer (2000; Zbl 0992.14001)], and the other following \textit{P. Deligne} [``Cohomology à support propre en construction du foncteur \(f^!\)'', Lect. Notes Math. 20, 404--421 (1966)], \textit{J.-L. Verdier} [Algebr. Geom., Bombay Colloq. 1968, 393--408 (1969; Zbl 0202.19902)] and (much later) \textit{J. Lipman} [``Notes on derived functors and Grothendieck duality'', Lect. Notes Math. 1960, 1--259 (2009)]. The accepted view is that each of these has its drawbacks: the first approach (of Grothendieck, Hartshorne and Conrad) is complicated and messy to set up, while the second (of Deligne, Verdier and Lipman) might be cleaner to present but leads to a theory where it's not obvious how to compute anything.
The point of this article is that the recently-discovered connection with Hochschild homology and cohomology (the one due to Avramov and Iyengar) changes this. It renders clearly superior the highbrow approach to the subject, the one due to Deligne, Verdier and Lipman. Not only is it (relatively) easy to set up the machinery, the computations also become transparent. And in the process we learn that Grothendieck duality is not really about residues of meromorphic differential forms, it is about the local cohomology of the Hochschild homology. By a fortuitous accident, if \(f:X\to Y\) is a smooth map then the top Hochschild homology happens to be isomorphic to the relative canonical bundle, and its top local cohomology is represented by meromorphic differential forms. This is the reason that, as long as we stick to smooth maps, what comes up is residues of meromorphic forms. For non-smooth, flat maps it's Hochschild homology and maps from it that we need to study. Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Homotopical algebra, Quillen model categories, derivators, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) The relation between Grothendieck duality and Hochschild homology | 0 |
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 connected and simply connected complex semi-simple group, and let \(B\) be a Borel subgroup of \(G\) and \(H\) a Cartan subgroup of \(B\). Let \(X\) denote the flag variety \(G/B\) with the \(H\)-action by multiplication on the left. The aim of this paper is to give a formula (called a Chevalley formula) for the product of a complex line bundle and a Schubert class in the \(H\)-equivariant \(K\)-theory \(K(H, X)\) of \(X\). In the present case the Grothendieck group of \(H\)-equivariant complex vector bundles over \(X\) is isomorphic to that of \(H\)-equivariant coherent sheaves on \(X\), so that it is understood that \(K(H, X)\) represents these two groups which are identified. Hence one sees that \(K(H, X)\) has two kinds of canonical classes \([{\mathcal{L}}^X_\lambda]^H\) and \([{\mathcal{O}}_{\overline{X}_w}]^H\) where \(\lambda \in X(H)\) the group of characters of \(H\) and \(w \in W=N_G(H)/H\) the Weyl group of \(G\), which are defined in its respective underlying Grothendieck groups. Here \({\mathcal{L}}^X_\lambda\) denotes the associated line bundle to the principal \(B\)-bundle \(G \to X\) where \(B\) acts on \(G\) through the natural extension of \(\lambda\) to \(B\), and \({\mathcal{O}}_{\overline{X}_w}\) denotes the structure sheaf of the Schubert variety \(\overline{X}_w\) obtained by using the Bruhat decomposition of \(G=\bigsqcup_{w \in W}BwB\) where \(\overline{X}_w\) is the closure of the \(B\)-orbit of \(w \in W\) (in the Zariski topology).
Letting \(R(H)\) denote the ring of complex representations of \(H\), it is known that the classes \(\{ [{\mathcal{O}}_{\overline{X}_w}]^H : w \in W \}\) form an \(R(H)\)-basis of \(K(H, X)\) and on the other hand the classes \(\{ [{\mathcal{L}}^X_\lambda]^H : \lambda \in X(H) \}\) generate \(K(H, X)\) as an \(R(H)\)-algebra. From this it follows that the product \([{\mathcal{L}}^X_\lambda]^H[{\mathcal{O}}_{\overline{X}_w}]^H\) can be written in the form
\[
[{\mathcal{L}}^X_\lambda]^H[{\mathcal{O}}_{\overline{X}_w}]^H=\sum_{v \in W} q^\lambda_{w, v}[{\mathcal{O}}_{\overline{X}_v}]^H
\]
where \(q^\lambda_{w, v} \in R(H)\). The main result (Theorem 4) gives an algorithm to compute these coefficients \(q^\lambda_{w, v}\). The proof uses the \(H\)-equivariant \(K\)-theory of a Bott-Samelson variety \(\Gamma\). By observing restrictions to fixed points in equivariant \(K\)-theory, the author obtains a decomposition of the class \([{\mathcal{L}}^\Gamma_\lambda]^H\) in its \(R(H)\)-basis. This enables one to deduce the required formula via a standard map from \(\Gamma\) to \(X\). equivariant \(K\)-theory; flag varieties DOI: 10.1016/j.jalgebra.2006.07.031 \(K\)-theory of schemes, Equivariant \(K\)-theory, Algebraic \(K\)-theory of spaces, Grassmannians, Schubert varieties, flag manifolds A Chevalley formula in equivariant \(K\)-theory | 0 |
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 connect \(k\)-triangulations of a convex \(n\)-gon to the theory of Schubert polynomials. We use this connection to prove that the simplicial complex with \(k\)-triangulations as facets is a vertex-decomposable triangulated sphere, and we give a new proof of the determinantal formula for the number of \(k\)-triangulations. \(k\)-triangulation; triangulated sphere; enumerative combinatorics; pipe dream; Schubert polynomial Stump, Christian, A new perspective on \(k\)-triangulations, J. Comb. Theory, Ser. A, 118, 6, 1794-1800, (2011) Symmetric functions and generalizations, Classical problems, Schubert calculus A new perspective on \(k\)-triangulations | 0 |
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 survey article, which may also serve as background material while reading, for instance, [\textit{E. Arbarello}, Contemp. Math. 312, 9--69 (2002; Zbl 1056.14023), Section 4] and parts of [\textit{M. Mulase}, Perspectives in mathematical physics. Proceedings of the conference on interface between mathematics and physics, held in Taiwan in summer 1992 and the special session on topics in geometry and physics, held in Los Angeles, CA, USA in winter of 1992. Boston, MA: International Press. 151--217 (1994; Zbl 0837.35132), \textit{M. Sato}, Random systems and dynamical systems, Proc. Symp., Kyoto 1981, RIMS Kokyuroku 439, 30--46 (1981; Zbl 0507.58029)], has the purpose to advertise the notion of Schubert derivation on an exterior algebra, introduced by the first author [Asian J. Math. 9, No. 3, 315--322 (2005; Zbl 1099.14045)] (see also the authors [Hasse-Schmidt derivations on Grassmann algebras. With applications to vertex operators. Cham: Springer (2016; Zbl 1350.15001)]), by showing how it provides another approach to look at the quadratic equations describing the Plücker embedding of Grasmannians -- a very classical and widely studied subject.
In particular, it allows i) to ``discover'' the vertex operators generating the fermionic vertex superalgebra (in the sense of [\textit{S. Galkin} and \textit{D. Ben-Zvi}, Lond. Math. Soc. Lect. Note Ser. 308, 46--97 (2004; Zbl 1170.17303), Section 5.3]); ii) to compute their bosonic expressions as by \textit{V. G. Kac} et al. [Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. 2nd ed. Hackensack, NJ: World Scientific (2013; Zbl 1294.17021), Lecture 5]; iii) to interpret them in terms of Schubert derivations and iv) to provide an almost effortless deduction of the celebrated Hirota bilinear form of the KP hierarchy (after Kadomtsev and Petviashvilii) [Kac et al. loc. cit.]. Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Derivations and commutative rings On Plucker equations characterizing Grassmann cones | 0 |
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{S. K. Donaldson} [Topology 29, No. 3, 257-315 (1990; Zbl 0715.57007)] has introduced polynomials on the second cohomology of any simply connected closed real four-manifold with odd \(b^ +_ 2\). Their importance results from their diffeomorphism invariance and they are used to distinguish different differentiable structures with the same underlying topology. But they are not easy to compute.
Over a smooth projective surface these polynomials are closely related to moduli spaces of holomorphic rank-two vector bundles. Recently \textit{K. G. O'Grady} [Invent. Math. 107, No. 2, 351-395 (1992)] introduced algebro- geometric analogues of Donaldson's polynomials. More recently \textit{J. Morgan} [``Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues'', Preprint (1992)] has shown that these two kinds of polynomials coincide (under certain circumstances).
In the paper under review the author defines similar to O'Grady symmetric polynomials on the second cohomology in the case of a ruled surface \(X\) over a curve \(C\). He uses the explicit description of the moduli space of rank two bundles on ruled surfaces of his earlier paper [``Moduli spaces of stable rank-2 bundles on ruled surfaces''. I. (Preprint); see also J. Reine Angew. Math. 433, 201-219 (1992; Zbl 0753.14031)] and a universal rank-two sheaf parametrized by that. The polynomial is defined with the aid of the first Pontryagin class of the projective bundle associated to this universal sheaf. From the computation of Chern classes of a vector bundle over the Picard torus of \(C\) the author can explicitly compute his polynomials defined before. \(\mu\)-stable; Grothendieck-Riemann-Roch; Donaldson polynomials; moduli space of rank two bundles on ruled surfaces Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Characteristic classes and numbers in differential topology, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Fine and coarse moduli spaces Symmetric polynomials constructed from moduli of stable sheaves with odd \(c_ 1\) on ruled surfaces | 0 |
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 From the introduction: \textit{A. S. Buch} and \textit{W. Fulton} [Invent. Math. 135, 665--687 (1999; Zbl 0942.14027)] established a formula for the cohomology class of a quiver variety, which \textit{A. S. Buch} [Duke Math. J. 115, No.~1, 75--103 (2002; Zbl 1052.14056)] later extended to \(K\)-theory. The \(K\)-theory formula expresses a quiver class as an integer linear combination of products of stable Grothendieck polynomials. Quiver coefficients are the coefficients of this linear combination. The quiver coefficients were conjectured to be nonnegative in cohomology and to alternate in sign in \(K\)-theory. These conjectures were recently proved by \textit{A. Knutson, E. Miller} and \textit{M. Shimozono} [Invent. Math. 166, No. 2, 229--325 (2006; Zbl 1107.14046)], \textit{A. S. Buch} [J. Am. Math. Soc. 18, No. 1, 217--237 (2005; Zbl 1061.14050)] and \textit{E. Miller} [Duke Math. J. 128, No. 1, 1--17 (2005; Zbl 1099.05079)].
\textit{A. S. Buch, A. Kresch, H. Tamvakis} and \textit{A. Yong} [Am. J. Math. 127, No. 3, 551--567 (2005; Zbl 1084.14048)] gave combinatorial formulas for the decomposition coefficients expressing a Grothendieck polynomial as an integer linear combination of products of stable Grothendieck polynomials. In particular, it was proved that the decomposition coefficients alternate in sign.
Alternation in sign also occurs in the Schubert calculus of the flag variety; Brion proved that the \(K\)-theory Schubert structure constants alternate in sign.
We give explicit and natural equalities between the three aforementioned integers. \beginbarticle \bauthor\binitsA. S. \bsnmBuch, \bauthor\binitsF. \bsnmSottile and \bauthor\binitsA. \bsnmYong, \batitleQuiver coefficients are Schubert structure constants, \bjtitleMath. Res. Lett. \bvolume12 (\byear2005), no. \bissue4, page 567-\blpage574. \endbarticle \OrigBibText Anders Skovsted Buch, Frank Sottile, and Alexander Yong, Quiver coefficients are Schubert structure constants , Math. Res. Lett. 12 (2005), no. 4, 567-574. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quiver coefficients are Schubert structure constants | 0 |
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 description of the singular locus of a Schubert variety X, in the flag variety G/B, where G is a classical group and B is a Borel subgroup. The singular locus is determined by using standard monomial theory as developed in ''Geometry of G/B'' [by the authors and \textit{C. Musieli}; part I-IV in Collect. Publ. C. P. Ramanujan and Papers in his Mem., Tata Inst. Fundam. Res., Stud. Math. 8, 207-239 (1978); Proc. Indian Acad. Sci., Sect. A 87, No.2, 1-54 (1978); ibid. 88, No.2, 93-177 (1979); ibid. 88, No.4, 279-362 (1979; Zbl 0447.14010-14013); part V (to appear)]. A consequence of this theory is the determination of the ideal defining X in G/B, using which, we are able to write the Jacobian matrix Jw,\(\tau\) (here w is given by \(X=X(w)\) and \(e_{\tau}\) is the T-fixed point in G/B corresponding to \(\tau\), \(\tau\leq w\), T being a maximal torus contained in B) in the affine neighborhood \(U^-_{\tau}\cdot\tau \) of \(e_{\tau}\), where \(U^-_{\tau}=\tau U^-\tau^{-1}\), \(U^-\) being the unipotent part of the Borel subgroup of G opposite to B. Evaluating Jw,\(\tau\) at \(e_{\tau}\), we obtain the dimension of \(Z_{w,\tau}\), the Zariski tangent space to X(w) at \(e_{\tau}\). Denoting by \(\{\) \(p(\lambda\),\(\mu)\}\) the weight vectors as given by standard monomial theory, let \(R(w,\tau)=\{\beta\in \tau (\Delta^+)\quad there\quad exists\quad a\quad p(\lambda,\mu),\quad such\quad that\quad\quad w\ngeq\lambda \quad and\quad X_{- \beta}p(\lambda,\mu)=cp(\tau),c\in k^*\}.\) Then we have the main theorem \(\dim Z_{w,\tau}=N-\# R(w,\tau)\) where \(N=\# \Delta^+=\# \{positive\quad roots\}).\) In particular we have X(w) is smooth at \(e_{\tau}\) if and only if \(N-\# R(w,\tau)=\upharpoonright (w),\) the length of w. dimension of Zariski tangent space; singular locus of a Schubert variety; flag variety; standard monomial theory; Jacobian matrix; weight vectors C.S. Seshadri : Normality of Schubert variety . Proceeding de ''Algebraic Geometry'' (Bombay, Avril 1984). Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Singular locus of a Schubert variety | 0 |
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 note is an announcement of two results on Schubert supervarieties and is a continuation of the previous work by the author and \textit{Yu. I. Manin} [Funkt. Anal. Appl. 18, 329-330 (1984); translation from Funkts. Anal. Prilozh. 18, No.4, 75-76 (1984; Zbl 0608.14038)].
The first result is a combinatorial description of the ordering in the Weyl supergroup defined by the position of the Schubert subvarieties corresponding to the elements of the Weyl supergroups of the complex algebraic supergroups SL(m\(| n)\), OSp(m\(| n)\) (automorphisms of an even symmetric form), Sp(m) (automorphisms of odd skew form), Q(m) (automorphisms of an odd involution). In the latter ordering \(w_ 1<w_ 2\) iff \(X_{w_ 1}\subset \bar X_{w_ 2}.\)
The second result is a superanalog of Demazure's resolution of the Schubert varieties using Bott-Samelson schemes [\textit{M. Demazure}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)].
This brief note contains definitions and statements of the results. Schubert supervarieties; Weyl supergroup A. A. Voronov, ?Relative position of Schubert supervarieties and resolution of their singularities?, Funkts. Anal. Prilozhen.,21, No. 1, 72?73 (1987). Grassmannians, Schubert varieties, flag manifolds, Applications of global differential geometry to the sciences, Quantum field theory; related classical field theories Relative disposition of the Schubert supervarieties and resolution of their singularities | 0 |
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 concept of an algebraic loop group emerged in the context of conformal quantum field theory and its algebro-geometric framework, especially in connection with moduli spaces of bundles over curves, the theory of infinite Grassmannians, Krichever theory and infinite-dimensional Lie algebras. Whilst many basic properties of algebraic loop groups are well-explored in characteristic zero, comparatively little has been known in positive characteristic, at least so until recently.
The paper under review changes this situation radically, in that it starts an attempt to check the known facts on algebraic loop groups (in characteristic zero) in full generality. In the seven sections of this article, the author develops a general theory of algebraic loop groups, affine Grassmannians, generalized Schubert varieties, and moduli stacks of \(G\)-bundles over projective curves in characteristic \(p> 0\). The main results include a proof of the normality and Cohen-Macaulay property of the generalized Schubert varieties introduced here, the construction of line bundles on the affine Grassmannian, and a proof that the latter ones induce line bundles on the moduli stack of \(G\)-bundles. Throughout this important, generalizing and systematizing work, the author expertly points out the relations of his constructions to the various other ones (in characteristic zero) as well as to some linked recent results in positive characteristic. infinite Grassmannians; algebraic loop group; moduli stacks; generalized Schubert varieties Faltings, G.:Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. (JEMS) \textbf{5}(1), 41-68 (2003) \textbf{(MR 1961134)} Algebraic moduli problems, moduli of vector bundles, Other algebraic groups (geometric aspects), Grassmannians, Schubert varieties, flag manifolds Algebraic loop groups and moduli spaces of bundles | 0 |
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 [\textit{E. Mukhin} and \textit{A. Varchenko}, Commun. Contemp. Math. 6, No. 1, 111--163 (2004; Zbl 1050.17022)], some correspondences were defined between critical points of master functions associated to \(sl_{N+1}\) and subspaces of \(\mathbb{C}[x]\) with given ramification properties. In this paper we show that these correspondences are in fact scheme theoretic isomorphisms of appropriate schemes. This gives relations between multiplicities of critical point loci of the relevant master functions and multiplicities in Schubert calculus. Belkale P, Mukhin E, Varchenko A (2006) Multiplicity of critical points of master functions and Schubert calculus. In: Bertrand D, Enriquez B, Mitschi C, Sabbah C, Schäfke R (eds) Differential equations and quantum groups: Andrey A Bolibrukh memorial volume. IRMA Lect Math Theor Phys, vol 9. Europ Math Soc Publ, pp 59--84 Classical problems, Schubert calculus, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Exactly solvable models; Bethe ansatz Multiplicity of critical points of master functions and Schubert calculus | 0 |
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 generalize the computation of Feynman integrals of log divergent graphs in terms of the Kirchhoff polynomial to the case of graphs with both fermionic and bosonic edges, to which we assign a set of ordinary and Grassmann variables. This procedure gives a computation of the Feynman integrals in terms of a period on a supermanifold, for graphs admitting a basis of the first homology satisfying a condition generalizing the log divergence in this context. The analog in this setting of the graph hypersurfaces is a graph supermanifold given by the divisor of zeros and poles of the Berezinian of a matrix associated with the graph, inside a superprojective space. We introduce a Grothendieck group for supermanifolds and identify the subgroup generated by the graph supermanifolds. This can be seen as a general procedure for constructing interesting classes of supermanifolds with associated periods. DOI: 10.1088/1751-8113/41/31/315402 Feynman diagrams, Supermanifolds and graded manifolds, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Functional analysis on superspaces (supermanifolds) or graded spaces Supermanifolds from Feynman graphs | 0 |
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 any compact Kähler variety \(X\), let \(\text{Hdg}^{2p} (X, \mathbb{Q})\) denote the \(\mathbb{Q}\)-vector space of Hodge classes: \(\text{Hdg}^{2p}(X,\mathbb{Q})= H^{2p} (X, \mathbb{Q})\cap H^{p,p} (X)\). Furthermore let \(\text{Hdg}^{2p} (X))_{\text{an}}\) be the subspace of \(H^{2p}(X,\mathbb{Q})\) generated by the rational Chern classes \(c_p ({\mathcal F})\in H^{2p}(X,\mathbb{Q})\) for an analytic coherent sheaf \({\mathcal F}\), \(\text{Hdg}^{2p}(X)_{\text{lf}}\) generated by \(c_p({\mathcal F})\) with locally free sheaf \({\mathcal F}\) and \(\text{Cycle}^{2p}(X)\) generated by the classes which are Poincaré dual to closed analytic subsets of codimension \(p\). In the projective case, the latter three subspaces coincide, since any coherent sheaf has a finite resolution by locally free sheaves, and any globally generated locally free sheaf is the pullback of the tautological quotient bundle on a Grassmannian \(G\) by a morphism from \(X\) to \(G\). In this paper the author shows by constructing an explicit example (a complex 4-dimensional torus of Weil type) that, in the general Kähler case, none of the equalities between the four subspaces \(\text{Hdg}^{2p}(X,\mathbb{Q})\supset \text{Hdg}^{2p} (X)_{\text{lf}} \supset\text{Cycle}^{2p} (X)\) holds. Hodge class; Kähler variety Voisin, C., A counterexample to the Hodge conjecture extended to Kähler varieties, Int. math. res. not. IMRN, 20, 1057-1075, (2002) Transcendental methods, Hodge theory (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects), Compact Kähler manifolds: generalizations, classification A counterexample to the Hodge conjecture extended to Kähler varieties. | 0 |
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, the authors consider the GIT quotients of minuscule Schubert varieties for the action of a maximal torus. Let \(G\) be a semisimple simply connected complex algebraic group, \(T\) a maximal torus, \(B\supset T\) a Borel subgroup and \(P\supset B\) a parabolic subgroup. Let \(\Phi^+\) be the set of positive roots with respect to \(B\). A fundamental weight \(\omega\) is called \textit{minuscule} if \(\langle\omega,\check{\beta}\rangle\le1\) for all \(\beta\in\Phi^+\) (Definition 4.1). For \(\omega\) a minuscule weight and \(P:=P_\omega\) the associated parabolic subgroup, the flag variety \(G/P\) and the Schubert varieties in \(G/P\) are also called \textit{minuscule}. Let \({\mathcal L}_\omega\) be the homogeneous line bundle on \(G/P\) associated to \(\omega\) and \(W^P\) the associated Weyl group. For \(w\in W^P\), denote by \(X_P(w)^{ss}_T({\mathcal L}_\omega)\) the set of semistable points in the Schubert variety \(X_P(w)\) with respect to \({\mathcal L}_\omega\) for the action of \(T\). Recall also that there is a unique minimal Schubert variety \(X_P(v)\) admitting semistable points. For any \(w\in W^P\), let \(Q_w\) be the associated quiver variety (for relevant material on quivers, see section 4).
Given this setup, the authors prove that the semistable locus is contained in the smooth locus for \(X_P(w)\) if and only if \(Q_v\) contains all the essential holes of \(Q_w\) (Theorem 4.9). From now on, suppose that \(G=SL(n,{\mathbb C})\). If \(1<r<n-1\) with \(\gcd(n,r)=1\) and \(P_r\) is the maximal parabolic subgroup of \(G\) associated to the simple root \(\alpha_r\), then \(X_P(w)^{ss}_T({\mathcal L}_{\omega_r})//T\) is smooth if \(Q_v\) contains all the essential holes of \(Q_w\) (Corollary 3.5). Now let \(P\) be the parabolic subgroup of \(SL(n,{\mathbb C})\) corresonding to the highest root \(\alpha_0\). Let \({\mathcal M}\) denote the descent of \({\mathcal L}_{\alpha_0}\) to the quotient \(X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})//T\). Then (Theorem 5.1) the polarized variety \((X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})//T,{\mathcal M})\) is projectively normal and (Corollary 5.2) \(X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})\) is isomophic to a projective space. Theorem 5.1 is proved using standard monomials. Schubert varieties; GIT quotients; quivers; standard monomials Grassmannians, Schubert varieties, flag manifolds, Geometric invariant theory, Representation theory for linear algebraic groups, Representations of quivers and partially ordered sets On the torus quotients of Schubert varieties | 0 |
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 previous work with \textit{M. Khovanov} and \textit{A. Lauda} [``The odd nilHecke algebra and its diagrammatics'', \url{arXiv:1111.1320}] we introduced two odd analogues of the Schur functions: one via the combinatorics of Young tableaux (odd Kostka numbers) and one via an odd symmetrization operator. In this paper we introduce a third analogue, the plactic Schur functions. We show they coincide with both previously defined types of Schur function, confirming a conjecture. Using the plactic definition, we establish an odd Littlewood-Richardson rule. We also re-cast this rule in the language of polytopes, via the Knutson-Tao hive model. symmetric functions; Hopf algebras; supalgebra; odd symmetric functions; hives; Littlewood-Richardson; Schubert calculus Ellis, A.P.: The odd Littlewood-Richardson rule. J. Algebraic Comb. \textbf{37}(4), 777-799 (2013). arXiv:1111.3932 Symmetric functions and generalizations, Classical problems, Schubert calculus, Connections of Hopf algebras with combinatorics The odd Littlewood-Richardson rule | 0 |
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 \(\text{Gr}(l, {\mathbb C}^n)\) denote the Grassmannian of \(l\)-dimensional subspaces in \({\mathbb C}^n\). The cohomology ring \(H^*(\text{Gr}(l, {\mathbb C}^n), \mathbb Z)\) has an additive basis of Schubert classes \(\sigma_\lambda\), indexed by Young diagrams \(\lambda = (\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_l \geq 0)\) contained in the \(l \times k\) rectangle where \(k = n - l\) (which is denoted by \(\lambda \subset l\times k\)). The product of two Schubert classes in \(H^*({\text{Gr}}(l, \mathbb C^n), \mathbb Z)\) is given by
\[
\sigma_\lambda \cdot \sigma_\mu =\sum_{ \nu \subset l\times k} c^\nu_{\lambda,\mu}\sigma_\nu ,
\]
where \(c^\nu_{\lambda,\mu}\) is the classical Littlewood-Richardson coefficient. This expansion is \textit{multiplicity-free} if \(c^\nu_{\lambda,\mu}\in \{0, 1\}\) for all \(\nu\subset l\times k\). In this paper, the authors give a nonrecursive, combinatorial answer to the following question of W. Fulton.
\textbf{Question.} When is \(\sigma_\lambda \cdot \sigma_\mu\) multiplicity-free?
Their answer exploits the following considerations. For partitions \(\lambda, \mu \subset l\times k\), place \(\lambda\) against the upper left corner of the rectangle. Then rotate \(\mu\) 180 degrees and place it in the lower right corner. The resulting subshape of \(l\times k\) is referred to as \(\text{rotate}(\mu)\).
If \(\lambda \cap \text{rotate}(\mu)\neq \emptyset\), then the product \(\sigma_\lambda \cdot \sigma_\mu\) is zero, and the interesting part is to handle the case of empty intersection. A \textit{Richardson quadruple} is the datum \((\lambda,\mu, l\times k)\), where \(\lambda \cap \text{rotate}(\mu)= \emptyset\). If \(\lambda \cup \text{rotate}(\mu)\) does not contain a full \(l\)-column or \(k\)-row, call this Richardson quadruple \textit{basic}. The main result of the article, Theorem~1.2, gives a criterion for multiplicity-freeness in terms of combinatorial conditions imposed on the basic Richardson quadruples. Grassmannian; Richardson variety; Schubert class Thomas H., Yong A.: Multiplicity-free Schubert calculus. Canad. Math. Bull. 53, 171--186 (2010) Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Classical problems, Schubert calculus Multiplicity-free Schubert calculus | 0 |
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 show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of \textit{R. Vakil} [Ann. Math. (2) 164, No. 2, 489--512 (2006; Zbl 1115.14043)] and a special position argument due to \textit{H. Schubert} [Math. Ann. 26, 26--51 (1885; JFM 17.0668.01)], our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formulas to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality. Galois groups; Schubert calculus; Kostka numbers; enumerative geometry Brooks, C.; Martín del Campo, A.; Sottile, F., Galois groups of Schubert problems of lines are at least alternating, Trans. Amer. Math. Soc., 367, 4183-4206, (2015) Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Galois groups of Schubert problems of lines are at least alternating | 0 |
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 is the second edition of a work that first appeared in 1987. The first printing was reviewed extensively by \textit{V. L. Popov}, see Zbl 0654.20039. Let me just say that Jantzen's book has been an indispensable reference for anyone involved with representations of algebraic groups, in particular of reductive groups in positive characteristic.
The second edition differs substantially from the first. Again it is a book one must have. About one third is new. The old part has been sprinkled with new comments, but has been left intact as much as is feasible. New chapters at the end, identified with capital letters, take care of several developments in the intervening years.
The following topics are now also covered in the familiar clear manner. Chapter A: Truncated categories and Schur algebras. Here the truncations are those of Donkin, associated to a finite `saturated' set of dominant weights. Chapter B: Results over the integers. This globalizes several cohomological results. Chapter C: Lusztig's Conjecture and some consequences. Chapter D: Radical filtrations and Kazhdan-Lusztig polynomials. These two chapters belong together and discuss various aspects of this central conjecture concerning characters of certain simple modules. Chapter E: Tilting modules. Chapter F: Frobenius splitting. Chapter G: Frobenius splitting and good filtrations. This chapter gives the proof by Mathieu of the theorem about tensor products of modules with good filtration. Logically chapters F and G should be read early. Chapter H: Representations of quantum groups. This chapter is in a different style. It is just a survey.
It is a pity that the author has still not adopted certain widely used terminology. For instance, the usual form of `sum formula' is `Jantzen sum formula'. representation theory; reductive algebraic groups; simple modules; highest weights; character formulas; Weyl's character formula; affine group schemes; injective modules; injective resolutions; derived functors; Hochschild cohomology groups; hyperalgebra; split reductive group schemes; Steinberg's tensor product theorem; irreducible representations; Kempf's vanishing theorem; Borel-Bott-Weil theorem; characters; linkage principle; dominant weights; filtrations; Steinberg modules; cohomology rings; rings of regular functions; Schubert schemes; line bundles; Schur algebras; quantum groups; Kazhdan-Lusztig polynomials J. C. Jantzen, \textit{Representations of Algebraic Groups. Second edition}, Amer. Math. Soc., Providence (2003). Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Group schemes, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Affine algebraic groups, hyperalgebra constructions, Linear algebraic groups over arbitrary fields Representations of algebraic groups. | 0 |
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 example of complex Grassmannians, we demonstrate various techniques available for computing genus-0 \(K\)-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the \(q\)-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant \(K\)-theoretic mirrors. Gromov-Witten invariants; \(K\)-theory; Grassmannians; non-abelian localization Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry Quantum K-theory of Grassmannians and non-abelian localization | 0 |
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 studying the ordinary cohomology of an (isotropic) Grassmannian, a triple intersection number refers to the number of intersection points of three Schubert varieties in general position. By convention, this number is zero when the triple intersection has positive dimension. Algebraically this number is given as the pushforward of the product of three Schubert classes to the cohomology ring of a single point.
Given three Schubert varieties in general position, let \(Z\) denote their schemetheoretic triple intersection. The corresponding \(K\)-theoretic triple intersection number is the sheaf Euler characteristic of \(Z\); that is, the pushforward of the product of the three Schubert classes to the Grothendieck ring of a point. This number is denoted by \(\chi (Z)\). If \(Z\) is finite, then, just as in cohomology, \(\chi (Z)\) is equal to the number of points in \(Z\) (since these finitely many points are reduced, by Kleiman's transversality theorem). If \(Z\) has positive dimension, then \(\chi (Z)\) can be a nonzero (and possibly negative) integer.
The triple intersection numbers determine the structure constants for multiplication with respect to the Schubert basis. These structure constants are known as Littlewood-Richardson coefficients, and in ordinary cohomology they are equal to triple intersection numbers. In \(K\)-theory however, the Littlewood-Richardson coefficients are alternating sums of triple intersection numbers. An arbitrary Schubert class can be written as an integer polynomial in certain special Schubert classes, which (in cohomology) are closely related to the Chern classes of the tautological quotient bundle on the Grassmannian in question. A triple intersection number is said to be of Pieri-type if one of the three Schubert classes is a special Schubert class. Similarly, a Pieri coefficient refers to a Littlewood-Richardson coefficient occurring in the product of an arbitrary Schubert class and a special Schubert class.
In this paper, the author determine \(K\)-theoretic Pieri-type triple intersection numbers for all isotropic Grassmannians of types \(B, C\), and \(D\). triple intersection numbers; isotropic Grassmannian; Richardson variety; projected Richardson variety; Pieri formula; \(K\)-theoretic Pieri formula; \(K\)-theoretic triple intersection Ravikumar, V.: Triple intersection formulas for isotropic Grassmannians. Algebra Num. Theory (to appear, preprint). arXiv:1403.1741 [math.AG] Classical problems, Schubert calculus, \(K\)-theory of schemes, Grassmannians, Schubert varieties, flag manifolds Triple intersection formulas for isotropic Grassmannians | 0 |
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 introduce a partial order on the set of skew characters of the symmetric group which we use to classify the multiplicity-free skew characters. Furthermore, we give a short and easy proof that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the `inverse' partitions (Theorem 4.3). multiplicity-free; skew characters; symmetric group; skew Schur functions; Schubert calculus Gutschwager C.: On multiplicity-free skew characters and the Schubert calculus. Ann. Combin. 14(3), 339--353 (2010) Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups On multiplicity-free skew characters and the Schubert calculus | 0 |
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 \(V\) be a vector space of even dimension. The Lagrangian Grassmannian \(\text{LG}_n(V)\) parametrizes Lagrangian vector subspaces of a (even dimensional) vector space equipped with a bilinear symplectic form. A subspace of \(V\) is said to be Lagrangian if it coincides with its symplectic annihilator. The cohomology of \(\text{LG}_n\) is known to be freely generated by the classes of certain Schubert varieties \(X(\lambda)\), which are parametrized by sequences \(\lambda=(\lambda_1,\ldots, \lambda_n)\) of \(n\) integers such that \(n\geq \lambda_1> \ldots>\lambda_n\geq 1\). The symplectic group \(\text{Sp}(V)\) acts naturally on \(\text{LG}_n(V)\) and hence any maximal torus \(T\) does. As a consequence the Schubert varieties \(X(\lambda)\), as above, turn out to be \(T\)-invariant, reflecting the fact that the coordinate Lagrangian subspaces are \(T\)-fixed points of \(\text{LG}_n\). General standard facts about equivariant cohomology ensure that the equivariant cohomology \(H_T^*(\text{LG}_n)\) is generated by the \(T\)-equivariant classes \(\sigma(\lambda)\) of \(X(\lambda)\).
The main result of this beautiful and very well written paper is the statement, and the proof, of an explicit formula for the the restriction of the classes \(\sigma(\lambda)\) to the torus fixed points of \(\text{LG}_n\).
The paper is concerned with a very central topic related with many different areas of mathematics, attracting the interest of many mathematicians. The first part of the introduction takes a special care to describe the state of art of the subject, by quoting and discussing the contributions of many important authors. Section 2 collects some basic fundamental definitions, notation and facts which are needed in the rest of the paper, making the article almost self contained (at least for algebraic geometers). Furthermore, in Section 3, the author recalls the definition of Schubert varieties in the Lagrangian context. The main theorem is proven in section 6 by checking that a certain function satisfies a recurrence relation with a initial condition characterizing the factorial Schur \(Q\)-function. Section 8 is devoted to the presentation of the \(T\)-equivariant cohomology ring of the Lagrangian Grassmannian, while Section 5, together with a very useful appendix, recalls the properties of the Schur \(Q\)-functions. The paper ends with a comprehensive list of references, orienting the reader along the mazes of this beautiful part of mathematics. The paper under review, on the other hand, goes far beyond its mathematical contributions, candidating itself to be a new useful and important reference for people interested in the subject. restriction to fixed points; Schur \(Q\)-functions Ikeda T.: Schubert Classes in the Equivariant Cohomology of the Lagrangian Grassmannianr. Adv. Math. 215, 1--23 (2007) Classical problems, Schubert calculus, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian | 0 |
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 short and elementary survey, we aim to give a unified approach, via the standard monomial theory for classical groups (especially for the linear group), to an understanding of some families of affine varieties such as the determinantal/ladder determinantal varieties, variety of complexes, variety of idempotents, etc., that were previously studied by several authors using essentially combinatorial methods. The main observation is that all these objects are particular members of a big family of affine varieties, the so-called ``opposite big cells of Schubert varieties'' in the generalized flag varieties \(G/Q\) where \(G\) is a connected reductive algebraic group and \(Q\) a parabolic subgroup of \(G\). In \S2, we set the notation, explain the theme and recall some facts needed in the sequel. In \S3, we give a complete structure of the determinantal varieties \(\bigl\{ D_ P (\theta) \bigr\}\) in the Grassmannian \(GL_ n/P\), \(P\) a maximal parabolic subgroup of \(GL_ n\), using the classical determinantal varieties (the zeros of \(p \times p\) minors) as the building blocks. A similar result for the flag variety is given in \S7. In \S\S4 to 6, we give a brief summary of the main results of the original sources fitting into the theme of this exposition. We conclude by pointing out some other aspects of applications of standard monomial theory not touched upon here. opposite big cells of Schubert varieties; standard monomial theory for classical groups; generalized flag varieties Musili, C.: Applications of standard monomial theory. Proceedings Hyderabad conference on alg. Groups, 381-406 (1991) Classical groups (algebro-geometric aspects), Linear algebraic groups over arbitrary fields, Grassmannians, Schubert varieties, flag manifolds Applications of standard monomial theory | 0 |
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 \(V = V(x) \in k[x_1,\dots,x_m]\) is called a \textit{homogeneous potential} if is it homogeneous of degree \(2\) with respect to some given \(\mathbb Q_+\)-grading. Two such potentials \(V(x),W(y)\) (over the same base field \(k \subset \mathbb C\), but not necessarily the same number of variables \(m\)) are called \textit{orbifold equivalent} if there exists a finite-rank graded matrix factorization of \(W(y) - V(x) \in k[x_1,\dots, x_m,y_1,\dots y_n]\) whose left and right quantum dimensions are nonzero (concrete definitions, as well as a comparison with their more general counterparts in the context of pivotal categories, are given in the paper).
The main result of the paper is a classification of orbifold equivalence classes of potentials over \(k = \mathbb C\) which define ADE singularities. Denoting by \(d\) the central charge, equivalence classes are
\[
\begin{align*}{ &\{ A_{d-1} \}, \text{ for \(d\) odd},\cr &\{ A_{d-1}, D_{d/2+1} \} \text{ for \(d\) even, } d \not\in \{ 12,18,30 \},\cr &\{ A_{11}, D_7, E_6 \}, \{ A_{17}, D_{10}, E_7 \}, \{ A_{29}, D_{16}, E_8 \}.}\end{align*}
\]
The final section of the paper explains the relation of the above with the (conjectural) correspondence between \(\mathcal N = 2\) supersymmetric Landau--Ginzburg models and \(\mathcal N = 2\) superconformal field theories in two dimensions. The results of the paper also shed light on the relation between derived categories of matrix factorizations and Dynkin quiver representations. orbifold equivalence; matrix factorization; quantum dimension; ADE singularities N. Carqueville, A. Ros Camacho, and I. Runkel, Orbifold equivalent potentials. J. Pure Appl. Algebra 220(2016), no. 2, 759--781.MR 3399388 Double categories, \(2\)-categories, bicategories, hypercategories, Singularities in algebraic geometry, Derived categories, triangulated categories, Derived categories and commutative rings, Representations of quivers and partially ordered sets Orbifold equivalent potentials | 0 |
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 constructs and analyzes a certain manifold \(X\) associated with a generalized Kac-Moody algebra. In the case of a Kac-Moody algebra, \(X\) equals the flag manifold constructed by \textit{M. Kashiwara} [The flag manifold of Kac-Moody Lie algebra. Algebraic analysis. Geometry and number theory. Johns Hopkins Univ. Press., 161-190 (1990; Zbl 0764.17019)]. So this manifold is called the flag manifold of the generalized Kac-Moody algebra. The author also gives certain kinds of line bundles on the flag manifold \(X\), and determines the spaces of global sections of the bundles. The author emphasizes that the analysis of such spaces plays an important role in the highest weight representation theory in the case of a Kac-Moody algebra, especially in the proof of the Kazhdan-Lusztig conjecture. generalized Kac-Moody algebra; flag manifold; Kazhdan-Lusztig conjecture Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds Borel-Weil type theorem for the flag manifold of a generalized Kac-Moody algebra | 0 |
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 Based on Pieri's formula on Schubert varieties [see \textit{F. Sottile}, Can. J. Math. 49, 1281-1298 (1997; Zbl 0933.14031)], the Pieri homotopy algorithm was first proposed by \textit{B. Huber, F. Sottile}, and \textit{B. Sturmfels} [J. Symb. Comput. 26, 767-788 (1998; Zbl 1064.14508)]\ for numerical Schubert calculus to enumerate all \(p\)-planes in \({\mathbb C}^{m+p}\) that meet \(n\) given planes in general position. The algorithm has been improved by \textit{B. Huber} and \textit{J. Verschelde} [SIAM J. Control Optim. 38, 1265-1287 (2000; Zbl 0955.14038)]\ to be more intuitive and more suitable for computer implementations.
A different approach of employing the Pieri homotopy algorithm for numerical Schubert calculus is presented in this paper. A major advantage of our method is that the polynomial equations in the process are all square systems admitting the same number of equations and unknowns. Moreover, the degree of each polynomial equation is always 2, which warrants much better numerical stability when the solutions are being solved. Numerical results for a big variety of examples illustrate that a considerable advance in speed as well as much smaller storage requirements have been achieved by the resulting algorithm. enumerative geometry; Schubert variety; Pieri formula; Pieri homotopy algorithm; Pieri poset; algorithm Li D, Qi L, Zhou S (2002) Descent directions of quasi-Newton methods for symmetric nonlinear equations. SIAM J Numer Anal 40(5): 1763--1774 Grassmannians, Schubert varieties, flag manifolds, Computational aspects of higher-dimensional varieties, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation, Enumerative problems (combinatorial problems) in algebraic geometry Numerical Schubert calculus by the Pieri homotopy algorithm | 0 |
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 0665.00004.]
The paper is an overview of the author's generalization of the Kazhdan- Lusztig polynomials introduced for the study of Bruhat orderings on Coxeter groups. Details will be found in the author's article [J. Algebra 111, 483-506 (1987; Zbl 0656.22007)]. If (W,S) is a Coxeter group and J any subset of generators in S, consider the Bruhat ordering and let \(W_ J\) denote the subgroup generated by J and \(W^ J\) the set of minimal coset representatives of \(W/W_ J\). If \({\mathcal H}\) denotes the Hecke algebra of (W,S) with coefficients in the ring \({\mathcal R}={\mathbb{Z}}[\sqrt{q}, 1/\sqrt{q}]\) then the author shows the existence of an \({\mathcal H}\)-module \(M^ J\) with an \({\mathcal R}\)-basis \(\{m_{\sigma}|\) \(\sigma \in W^ J\}\), and of an involution \(m\mapsto \bar m\) on \(M^ J\) compatible with the action of \({\mathcal H}\), and, finally, of two families of polynomials \(\{R_{\tau \sigma}|\) \((\tau,\sigma)\in \Gamma \}\), \(\{P_{\tau \sigma}|\) \((\tau,\sigma)\in \Gamma \}\), where \(\Gamma =\{(\tau,\sigma)\in W^ J\times W^ J|\) \(\tau\leq \sigma \}\) which satisfy a number of properties. For instance, with the length \(\ell (\sigma)\) of an element of the Coxeter group one may express the involution on the basis elements as follows
\[
\overline{m_{\sigma}}=\sum_{(\tau,\sigma)\in \Gamma}(- 1)^{\ell (\sigma)+\ell (\tau)}\cdot q^{-\ell (\sigma)}*R_{\tau \sigma}\cdot m_{\tau}.
\]
The R- and P-polynomials are linked by a formula involving the involution, and the P-polynomials allow the explicit description of a basis for the fixed elements of the involution. There is a uniqueness statement for the P-polynomials. For \(J=\emptyset\) one obtains the Kazhdan-Lusztig polynomials as a special case. One of the author's results specifies that for the element w of maximum length in \(W^ J\) and for \((\tau,\sigma)\in \Gamma\), his polynomial \(P_{\tau \sigma}\) is the Kazhdan-Lusztig polynomial \(p_{\tau w,\sigma w}\). The modules \(M^ J\) are used for the construction of a complex giving a (essentially) resolution of \({\mathcal H}\) which is investigated.
The entire set-up is applied to Kac-Moody algebras and the associated groups G for the study of the geometry of the varieties \(G/P_ J\) where \(P_ J\) are generalizations of a Borel subgroup \(P_{\emptyset}\). Kazhdan-Lusztig polynomials; Bruhat orderings on Coxeter groups; Kac- Moody algebras Infinite-dimensional Lie (super)algebras, Other algebraic groups (geometric aspects), Representation theory for linear algebraic groups, Infinite-dimensional Lie groups and their Lie algebras: general properties, Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) An extension of Kazhdan-Lusztig theory | 0 |
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 To each finite subset of \(\mathbb{Z}^2\) (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). \textit{R. I. Liu} [Trans. Am. Math. Soc. 364, No. 2, 1089--1107 (2012; Zbl 1235.05147)] has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.
However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class \(\sigma\) is at least an upper bound on the actual class \(\tau\), in the sense that \(\sigma-\tau\) is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians. positroid varieties; Specht modules; Stanley symmetric functions Pawlowski, Brendan, Chromatic symmetric functions via the group algebra of s_{n}, (2018), arXiv preprint Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Symmetric groups Cohomology classes of interval positroid varieties and a conjecture of Liu | 0 |
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 \(q\)-Kostka polynomials \(K_{\lambda\mu}(q)\) expressing the Schur function \(S_{\lambda}(\mathbf{x})\) as a linear combination of Hall-Littlewood polynomials \(P_{\mu}(\mathbf{x},q)\) have been an object of intensive study in the last two decades at the crossroads of combinatorics, algebra and geometry. The topic of the paper under review comes from the work of Lascoux and Schützenberger presenting the variant polynomial \(\widetilde K_{\lambda\mu}(q)=q^{n(\mu)}K_{\lambda\mu}(1/q)\) as a sum of the atom polynomials \(R_{\lambda\nu}(q)\), \(\nu\geq\mu\), where \(R_{\nu\mu}(q)\) themselves have non-negative coefficients. The main purpose of the paper is to give a geometric interpretation of the atomic decomposition in the language of scheme-theoretic intersections of nilpotent orbit varieties introduced by Kraft and De Concini-Procesi in the early 80's. In particular, involving a recent result of Broer, and as a consequence of their approach, the authors obtain a new proof of the atomic decomposition of the \(q\)-Kostka polynomials. Kostka polynomials; atomic decomposition; nilpotent conjugacy classes; nilpotent orbit varieties William Brockman and Mark Haiman, Nilpotent orbit varieties and the atomic decomposition of the \?-Kostka polynomials, Canad. J. Math. 50 (1998), no. 3, 525 -- 537. Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Special varieties, Representation theory for linear algebraic groups Nilpotent orbit varieties and the atomic decomposition of the \(q\)-Kostka polynomials | 0 |
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 subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\mathrm{IG}(2, 2n)$. We show that these rings are regular. In particular, by ``generic smoothness'', we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\mathrm{IG}(2, 2n)$. Further, by a general result of \textit{C. Hertling} [Frobenius manifolds and moduli spaces for singularities. Cambridge: Cambridge University Press (2002; Zbl 1023.14018)], the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on $\mathrm{IG}(2, 2n)$. Such a collection is constructed in the appendix by Alexander Kuznetsov. semisimplicity of quantum cohomology; unfoldings of singularities; Lefschetz exceptional collections Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Mirror symmetry (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections | 0 |
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 Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted \(\mathcal{A}\) and \(\mathcal{B}\)) that are constructed from a non-degenerate quasihomogeneous polynomial \(W\) and a related group of symmetries \(G\). Duality between \(\mathcal{A}\) and \(\mathcal{B}\) models has been conjectured for particular choices of \(W\) and \(G\). These conjectures have been proven in many instances where \(W\) is restricted to having the same number of monomials as variables (called \textit{invertible}). Some conjectures have been made regarding isomorphisms between \(\mathcal{A}\) and \(\mathcal{B}\) models when \(W\) is allowed to have more monomials than variables. In this paper we show these conjectures are false; that is, the conjectured isomorphisms do not exist. Insight into this problem will not only generate new results for Landau-Ginzburg mirror symmetry, but will also be interesting from a purely algebraic standpoint as a result about groups acting on graded algebras. Mirror symmetry (algebro-geometric aspects) Transposing noninvertible polynomials | 0 |
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 Two Schubert problems on possibly different Grassmannians may be composed to obtain a Schubert problem on a larger Grassmannian whose number of solutions is the product of the numbers of the original problems. This generalizes a construction discovered while classifying Schubert problems on the Grassmannian of \(4\)-planes in \(\mathbb{C}^9\) with imprimitive Galois groups. We give an algebraic proof of the product formula. In a number of cases, we show that the Galois group of the composed Schubert problem is a subgroup of a wreath product of the Galois groups of the original problems, and is therefore imprimitive. We also present evidence for a conjecture that all composed Schubert problems have imprimitive Galois groups. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Galois groups of composed Schubert problems | 0 |
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 quantum groups of finite and affine type \(A\) have a realization in terms of partial flag varieties of finite and affine type \(A\). Recently, it was shown that ``quantum group associated to partial flag varieties of finite type \(B/C\) can be realized as coideal subalgebras of the quantum group of finite type \(A\). The goal of the book is to ``initiate the study of the Schur algebras and quantum groups arising from partial flag varieties of classical affine type beyond type \(A\), generalizing the constructions in finite type \(B/C\). In the book the authors focus focus on the affine type \(C\).
They show ``that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotent) coideal subalgebras of quantum groups of affine \(sl\) and \(gl\) types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotent coideal algebras of affine \(sl\) type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotent quantum affine \(gl\) and its canonical basis.''
The book consists of three parts.
In Part 1 the basic constructions of the affine Schur algebra and its distinguished Lusztig subalgebra (and their ȷȷ, ȷı, ıȷ, ıı-variants) are introduced.
In Part 2 the structures of the family of Lusztig algebras ( and their ȷȷ, ȷı, ıȷ, ıı-variants) is studied. It is shown that they lead to coideal subalgebras \(U^c(\widehat{sl}_n)\) in \(U(\widehat{sl}_n)\).
Part 3 focused on ``the study of the stabilization properties of the family of Schur algebras \(S_{n,d}^c\) ( and their ȷȷ, ȷı, ıȷ, ıı-variants) leading to stabilization algebras which are identified as idempotented coideal subalgebras \(U^c(\widehat{sl}_n)\) of quantum affine \(gl_n\); these stabilization algebras are shown to admit canonical bases (without positivity).'' affine flag variety; affine quantum symmetric pair; canonical basis Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Quantum groups (quantized enveloping algebras) and related deformations, Linear algebraic groups over local fields and their integers, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Affine flag varieties and quantum symmetric pairs | 0 |
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 explicitly compute all Littlewood-Richardson coefficients for semisimple and Kac-Moody groups \(G\), that is, the structure constants (also known as the Schubert structure constants) of the cohomology algebra \(H^*(G/P,\mathbb C)\), where \(P\) is a parabolic subgroup of \(G\). These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of \(G\). However, if some off-diagonal entries of the Cartan matrix are \(0\) or \(-1\), the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies \(a_{ij}a_{ji}\geq 4\) for all \(i\), \(j\), then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood-Richardson coefficients. We extend this and other results to the structure coefficients of the \(T\)-equivariant cohomology of flag varieties \(G/P\) and Bott-Samelson varieties \(\Gamma_{\mathbf i}(G)\). Littlewood-Richardson coefficients; flag varieties; Schubert varieties; semisimple groups; Kac-Moody groups; reflection groups; Cartan matrices; Weyl groups Berenstein, A; Richmond, E, Littlewood-Richardson coefficients for reflection groups, Adv. Math., 284, 54-111, (2015) Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Kac-Moody groups Littlewood-Richardson coefficients for reflection groups. | 0 |
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 quantum connection of a complex Fano variety \(X\) is a certain connection on the trivial bundle with fiber \(H^*(X,\mathbb{C})\) on \(\mathbb{P}^1\) defined in terms of quantum multiplication on \(H^*(X,\mathbb{C})\). It has two singular points, one of which is irregular and as such is subject to the Stokes phenomenon.
The classical quantum equation for projective space \(\mathbb{P}^{n-1}\) was studied by Dubrovin who in particular related the asymptotic of solutions at the regular singular point to a certain characteristic class of the tangent bundle, called the Gamma-class. On the other hand, by a result of Guzzeti [\textit{D. Guzzetti}, Commun. Math. Phys. 207, No. 2, 341--383 (1999; Zbl 0976.53094)], the Stokes matrix at the irregular singular point is related to the Gram matrix of \(\mathbb{P}^{n-1}\) defined in terms of a full exceptional collection for the derived category of coherent sheaves. For general Fano varieties this conjectural relation is known as Dubrovin's conjecture.
In this paper, the authors consider the action of the n-dimensional torus \(T=(\mathbb{C}^{\times})^{n}\) on \(\mathbb{P}^{n-1}\) and study an equivariant version of the quantum equation of \(\mathbb{P}^{n-1}\). A key point is the compatibility of the equivariant quantum equation with a system of qKZ-difference equations, see Theorem 3.1. This allows the authors to identify the space of solutions of the joint system of equations with the space of the equivariant K-theory algebra \(K_{T}(\mathbb{P}^{n-1},\mathbb{C})\) of \(\mathbb{P}^{n-1}\). This identification comes from earlier work of the authors in which they prove it for more general partial flag varieties [\textit{V. Tarasov} and \textit{A. Varchenko}, J. Geom. Phys. 142, 179--212 (2019; Zbl 1419.82020)].
Using this identification the authors study the asymptotic of solutions close to the singular points. They prove an equivariant Gamma theorem relating the asymptotics at the regular singular point of the equivariant quantum equation to an equivariant Gamma class, see Theorem 4.3., which is based on their earlier work [\textit{V. Tarasov} and \textit{A. Varchenko}, J. Geom. Phys. 142, 179--212 (2019; Zbl 1419.82020)] on partial flag varieties. At the irregular singular point the authors describe Stokes bases in terms of exceptional bases of \(K_{T}(\mathbb{P}^{n-1},\mathbb{C})\). equivariant quantum differential equation; equivariant \(K\)-theory; \(q\)-hypergeometric solutions; braid group action Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain, Applications of Lie algebras and superalgebras to integrable systems Equivariant quantum differential equation, Stokes bases, and \(K\)-theory for a projective space | 0 |
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 interpret the orthogonality relation of Kostka polynomials arising from complex reflection groups [\textit{T. Shoji}, Invent. Math. 74, 239--267 (1983; Zbl 0525.20027); J. Algebra 245, No. 2, 650--694 (2001; Zbl 0997.20044); \textit{G. Lusztig}, Adv. Math. 61, 103--155 (1986; Zbl 0602.20036)] in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counterpart of Kostka polynomials. Then, we show that every generalized Springer correspondence [\textit{G. Lusztig}, Invent. Math. 75, 205--272 (1984; Zbl 0547.20032)] in a good characteristic gives rise to a Kostka system. This enables us to see the top-term generation property of the (twisted) homology of generalized Springer fibers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type \(\mathsf{BC}\). The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [\textit{D. Ciubotaru} and the author, Adv. Math. 226, No. 2, 1538--1590 (2011; Zbl 1207.22010)] \S3 to its graded version. In the appendices, we present purely algebraic proofs that Kostka systems exist for type \(\mathsf{A}\) and asymptotic type \(\mathsf{BC}\) cases, and therefore one can skip geometric sections \S3--5 to see the key ideas and basic examples/techniques. generalized Springer correspondences; Kostka polynomials; Lusztig-Shoji algorithm; \(\mathrm{Ext}\)-orthogonal collections; Kostka systems Kato, S., A homological study of Green's polynomial, Ann. sci. éc. norm. supér. (4), 48, 5, 1035-1074, (2015) Reflection and Coxeter groups (group-theoretic aspects), Symmetric functions and generalizations, Combinatorial aspects of representation theory, Representation theory for linear algebraic groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Hecke algebras and their representations A homological study of Green polynomials | 0 |
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 \(X=G/P\) be a a rational homogeneous variety (or, equivalently, a complex flag manifold). The Schubert classes form an additive basis of its integral homology. \textit{A. Borel} and \textit{A. Haefliger} [Bull. Soc. Math. Fr. 89, 461--513 (1961; Zbl 0102.38502)] asked whether an algebraic subvariety of~\(X\) homologous to a Schubert class must be a \(G\)-translate of it. When this happens, it is said that the Schubert class is ``rigid''.
In case \(X\) is a compact Hermitian symmetric space, it follows form the work of \textit{B. Kostant} [Ann. Math. (2) 77, 72--144 (1963; Zbl 0134.03503)] that this problem can be approached via a system of differential equations known as the ``Schur system''. Associated to the differential system is a Lie algebra cohomology. This gives rise to a cohomological obstruction to rigidity. The goal of this paper is to show the converse, namely, the non-vanishing of the obstruction implies flexibility. Schubert varieties; Hermitian symmetric spaces; Schur flexibility C. Robles, \textit{Schur flexibility of cominuscule Schubert varieties}, Comm. Anal. Geom. \textbf{21}, no. 5, 979-1013. Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Grassmannians, Schubert varieties, flag manifolds, Differential geometry of homogeneous manifolds, Rigidity results Schur flexibility of cominuscule Schubert varieties | 0 |
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 set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grassmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear partial differential equations. Specifically, just as the Schur polynomials are used to expand tau-functions as a sum, it is shown that it is natural to expand a quotient of tau-functions in terms of these generalized Schur functions. The coefficients in this expansion are found to be constrained by the Plücker relations of a Grassmannian. Schur polynomials; Grassmannian manifolds; KP hierarchy; tau-functions KdV equations (Korteweg-de Vries equations), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Grassmannians, Schubert varieties, flag manifolds Grassmannians, nonlinear wave equations and generalized Schur functions | 0 |
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 prove that, analogous to the Hilbert-Kunz density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the Hibert-Kunz function), there exists a \(\beta \)-density function \(g_{R, \mathbf{m}}:[0,\infty )\longrightarrow{\mathbb{R}} \), where \((R, \mathbf{m})\) is the homogeneous coordinate ring associated with the toric pair \((X, D)\), such that
\[
\int_0^{\infty }g_{R, \mathbf{m}}(x)\text{d}x = \beta (R, \mathbf{m}),
\]
where \(\beta (R, \mathbf{m})\) is the second coefficient of the Hilbert-Kunz function for \((R, \mathbf{m})\), as constructed by \textit{C. Huneke} et al. [Math. Res. Lett. 11, No. 4, 539--546 (2004; Zbl 1099.13508)]. Moreover, we prove, (1) the function \(g_{R, \mathbf{m}}:[0, \infty )\longrightarrow{\mathbb{R}}\) is compactly supported and is continuous except at finitely many points, (2) the function \(g_{R, \mathbf{m}}\) is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by \textit{M. Henk} and \textit{E. Link} [Online J. Anal. Comb. 10, Article 2, 12 p. (2015; Zbl 1333.52016)]) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes. coefficients of Hilbert-Kunz function; projective toric variety; Hilbert-Kunz density function; rational Ehrhart quasi-polynomial; rational Minkowski sum Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Tilings in \(n\) dimensions (aspects of discrete geometry) Density function for the second coefficient of the Hilbert-Kunz function on projective toric varieties | 0 |
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 cohomology modules of line bundles on Schubert varieties given by non-dominant weights in the Kac-Moody setting.
The base field in the field of complex numbers. Let \(G\) be a Kac-Moody Lie algebra associated to a generalized symmetrizable Cartan matrix. The author also uses \(G\) for Kac-Moody group associated to \(G\). Let \(B\) be a Borel subgroup of \(G\), and, for any element \(w\) in the Weyl group \(W\), denote by \(X(w)\) the associated Schubert variety in \(G/B\). Consider a character \(\lambda\) of \(B\), and denote by \(L_{\lambda}\) the line bundle on \(X(w)\) corresponding to the 1-dimensional representation of \(B\) given by the character \(\lambda\). Let \(H^i(X(w),L_{\lambda})\) be the cohomology module.
When \(\lambda\) is non-dominant, \(X(w)=G/B\) and \(G\) is not finite-dimensional, the equivalent of the Borel-Weil-Bott theorem is known to be true and was proved independently by S.~Kumar and O.~Mathieu. When \(G\) is finite-dimensional, a systematic study of the cohomology modules of the line bundles on Schubert varieties given by non-dominant weights was done by \textit{V.~Balaji, S.~Senthamarai Kannan} and \textit{K. V. Subrahmanyam} [Transform. Groups 9, No.~2, 105--131 (2004; Zbl 1078.14071)]. The present paper is a continuation of this study. Besides, several results here are new even in the finite-dimensional case.
The author describes the indices of the top and the least non-vanishing cohomologies \(H^i(X(w),L_{\lambda})\). Also he proves certain surjectivity results for maps between some cohomology modules. The main technical tool is a delicate use of the Bott-Samelson inductive machinery that works in the Kac-Moody setting also. Schubert varieties; non-dominant weights; cohomology Kannan S S, Cohomology of line bundles on Schubert varieties in the Kac-Moody setting, J. Algebra 310 (2007) 88--107 Homogeneous spaces and generalizations, Vanishing theorems in algebraic geometry Cohomology of line bundles on Schubert varieties in the Kac-Moody setting | 0 |
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 Noting the discrepency between the number, 7, of Clay Mathematics Institute Millennium Problems and the number, 23, of problems posed by David Hilbert at the 1900 Paris ICM, the editors of the Australian Mathematics Society Gazette asked some leading Australian mathematicians to make up the difference with their own favourite `Millennium problem'. In this, the first installment of that series, Alexander Molev proposed the Littlewood-Richardson problem for Schubert polynomials. That problem, which is dear to this reviewer's heart, is to give a manifestly positive combinatorial formula for the product of two Schubert classes in the cohomology ring of a flag manifold, in terms of Schubert classes. For flag manifolds of type A, these classes are represented by Schubert polynomials, and the problem is to generalize the Littlewood-Richardson rule for Schur functions, which governs the multiplication of Schubert classes in the cohomology ring of the Grassmannian.
Like Hilbert's 3rd problem (partially solved in 1900) and perhaps the 5th Millennium problem (the Poincaré conjecture), this challenge problem may be solved soon after it was posed. In 2004, Izzet Coskun announced a solution generalizing Ravi Vakil's geometric Littlewood-Richardson rule for the Grassmannian. Coskun now (August 2005) has a manuscript with a proof of this rule in the important special case of two-step flag manifolds, which gives a Littlewood-Richardson rule for the quantum cohomology of a Grassmannian.
While this problem is not as deep as the Clay Mathematical Institute's Millennium Challenge Problems, it is one of the most vexing and important open problems in algebraic combinatorics. Having resisted the efforts of many strong mathematicians in the past 15 years, it is a fine beginning to the Australian Mathematical Society Gazette's light-hearted series of 16 challenges. flag manifolds; cohomology ring; Grassmannian; quantum cohomology Molev, A.: Littlewood-Richardson problem for Schubert polynomials. Austral. math. Soc. gaz. 31, No. 5, 295-297 (2004) Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds The 8th problem: Littlewood-Richardson problem for Schubert polynomials | 0 |
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 Hecke algebra \(H(W)\) of a Weyl group \(W\) is known to have some remarkable realizations in characteristic \(p>0\) (cf. work of Iwahori and more recently Beilinson, Brylinski and Lusztig-Vogan; in the work of the latter authors \(H(W)\) appears to be a \(\mathbb{Z}[q,q^{-1}]\)-algebra isomorphic with the Grothendieck group of a suitable category of \(\mathbb{Q}_{\ell}\)-sheaves on \(X\times X\) -- where \(X\) is the flag variety of a reductive group over a finite field -- viewed as an algebra in some ``natural'' way).
The paper under review uses M. Saito's theory of Hodge modules to give a realization of \(H(W)\) in characteristic zero (i.e. to prove that \(H(W)\) is a \(\mathbb{Z}[q,q^{-1}]\)-algebra isomorphic with the Grothendieck group of a certain category of Hodge modules on \(X\times X\) where \(X\) is the flag variety of a reductive group over \(\mathbb{C})\). A similar realization is given for \(H(W_a)\) \((W_a= \) affine Weyl group) in terms of equivariant \(K\)-theory. Hecke algebra of a Weyl group; Hodge modules; equivariant K-theory Tanisaki, Toshiyuki, Hodge modules, equivariant \(K\)-theory and Hecke algebras, Publ. Res. Inst. Math. Sci., 0034-5318, 23, 5, 841-879, (1987) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), \(K\)-theory of schemes Hodge modules, equivariant \(K\)-theory and Hecke algebras | 0 |
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 is arranged in the following way. In \S1 we introduce notations for Schubert cells and we will show what cells and their neighbourhoods give rise to a uniquely constructed Schubert stratification.
In \S2 we show how a flattening corresponds to a Schubert cell.
In \S3 we introduce the concept of stable equivalence of flattenings, which allows us to compare cascades that generally consist of a different number of curves lying in spaces of different dimensions.
The relation of equivalence of flattenings is constructed in \S4.
In \S5 we give a classification of the flattenings occurring in generic three-parameter families of cascades; relative to this equivalence, we study the singularities of their bifurcation diagrams, and give results of V. I. Arnol'd and O. P. Shcherbak on the connection of these singularities with the geometry of the swallowtail, tangential singularities, and the singularities of projections.
In \S6 we give lists of the simple flattenings of curves, and also of cascades, corresponding to complete flags. The methods used in the proof of the classification theorems of \S\S5 and 6 are validated in \S\S7-9. In \S7 we prove a generalization of the Frobenius theorem on integrable distributions to the case of distributions with singularities. In \S8 this result is carried over to the case of the infinite-dimensional space of germs of cascades. Using these results, we prove the finite determinacy and versality theorems in \S9.
Section 10 and 11 are devoted to applications of the theory of flattenings to the study of oscillatory properties of linear differential equations and to the decomposition of Weierstrass points of algebraic curves, respectively. Grassmannians; flag manifolds; Schubert cells; Schubert stratification; flattening; equivalence; singularities; bifurcation; Weierstrass points; algebraic curves DOI: 10.1070/RM1991v046n05ABEH002844 Deformations of complex singularities; vanishing cycles, Formal methods and deformations in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Differentiable maps on manifolds, Theory of singularities and catastrophe theory Flattenings of projective curves, singularities of Schubert stratifications of Grassmannians and flag varieties, and bifurcations of Weierstrass points of algebraic curves | 0 |
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 way to reformulate the celebrated theorem of Beilinson is that \((\mathcal{O}(-n),\dots , \mathcal{O})\) and \((\Omega^n(n), \dots , \Omega^1 (1), \mathcal{O})\) are strong complete exceptional sequences in \(D^b(Coh\,\mathbb{P}^n)\), the bounded derived category of coherent sheaves on \(\mathbb{P}^n\). In a series of papers M. M. Kapranov generalized this result to flag manifolds of type \(A_n\) and quadrics. In another direction, Y. Kawamata has recently proven existence of complete exceptional sequences on toric varieties. Starting point of the present work is a conjecture of F. Catanese which says that on every rational homogeneous manifold \(X=G/P\), where \(G\) is a connected complex semisimple Lie group and \(P\subset G\) a parabolic subgroup, there should exist a complete strong exceptional poset and a bijection of the elements of the poset with the Schubert varieties in \(X\) such that the partial order on the poset is the order induced by the Bruhat-Chevalley order. An answer to this question would also be of interest with regard to a conjecture of B. Dubrovin which has its source in considerations concerning a hypothetical mirror partner of a projective variety \(Y\): There is a complete exceptional sequence in \(D^b(Coh\, Y)\) if and only if the quantum cohomology of \(Y\) is generically semisimple (the complete form of the conjecture also makes a prediction about the Gram matrix of such a collection). A proof of this conjecture would also support M. Kontsevich's homological mirror conjecture, one of the most important open problems in applications of complex geometry to physics today. The goal of this work will be to provide further evidence for F. Catanese's conjecture, to clarify some aspects of it and to supply new techniques. In section 2 it is shown among other things that the length of every complete exceptional sequence on \(X\) must be the number of Schubert varieties in \(X\) and that one can find a complete exceptional sequence on the product of two varieties once one knows such sequences on the single factors, both of which follow from known methods developed by Rudakov, Gorodentsev, Bondal et al. Thus one reduces the problem to the case \(X=G/P\) with \(G\) simple. Furthermore it is shown that the conjecture holds true for the sequences given by Kapranov for Grassmannians and quadrics. One computes the matrix of the bilinear form on the Grothendieck \(K\)-group \(K_{\circ}(X)\) given by the Euler characteristic with respect to the basis formed by the classes of structure sheaves of Schubert varieties in \(X\); this matrix is conjugate to the Gram matrix of a complete exceptional sequence. Section 3 contains a proof of theorem 3.2.7 which gives complete exceptional sequences on quadric bundles over base manifolds on which such sequences are known. This enlarges substantially the class of varieties (in particular rational homogeneous manifolds) on which those sequences are known to exist. In the remainder of section 3 we consider varieties of isotropic flags in a symplectic resp. orthogonal vector space. By a theorem due to Orlov (thm. 3.1.5) one reduces the problem of finding complete exceptional sequences on them to the case of isotropic Grassmannians. For these, theorem 3.3.3 gives generators of the derived category which are homogeneous vector bundles; in special cases those can be used to construct complete exceptional collections. In subsection 3.4 it is shown how one can extend the preceding method to the orthogonal case with the help of theorem 3.2.7. In particular we prove theorem 3.4.1 which gives a generating set for the derived category of coherent sheaves on the Grassmannian of isotropic 3-planes in a 7-dimensional orthogonal vector space. Section 4 is dedicated to providing the geometric motivation of Catanese's conjecture and it contains an alternative approach to the construction of complete exceptional sequences on rational homogeneous manifolds which is based on a theorem of M. Brion (thm. 4.1.1) and cellular resolutions of monomial ideals a la Bayer/Sturmfels. We give a new proof of the theorem of Beilinson on \(\mathbb{P}^n\) in order to show that this approach might work in general. We also prove theorem 4.2.5 which gives a concrete description of certain functors that have to be investigated in this approach. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Derived categories, triangulated categories, Homogeneous spaces and generalizations Derived categories of coherent sheaves on rational homogeneous manifolds | 0 |
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 exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field \(\mathbb{K}\) of characteristic \(\neq 2\) from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a \(\mathbb{Z}\)-model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [\textit{S. Kato}, Math. Ann. 371, No. 3--4, 1769--1801 (2018; Zbl 1398.14053)]) when \(\mathsf{char}\, \mathbb{K} =0\) or \(\gg 0\), and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic \(\neq 2\). Some particular cases of these results play crucial roles in our proof [``Loop structure on equivariant \(K\)-theory of semi-infinite flag manifolds'', Preprint (2018); \url{arXiv:1805.01718v6}] of a conjecture by \textit{T. Lam} et al. [J. Algebra 513, 326--343 (2018; Zbl 1423.14278)] that describes an isomorphism between affine and quantum \(K\)-groups of a flag manifold. Kac-Moody groups, Classical problems, Schubert calculus, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Frobenius splitting of Schubert varieties of semi-infinite flag manifolds | 0 |
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 new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series [the authors, SIAM J. Discrete Math. 31, No. 3, 1953--1989 (2017; Zbl 1370.05007); J. Comb. Theory, Ser. A 154, 350--405 (2018; Zbl 1373.05026)]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications. skew standard tableaux; product formulas; hook length; lozenge tilings; Schubert polynomials Combinatorial aspects of representation theory, Permutations, words, matrices, Classical problems, Schubert calculus Hook formulas for skew shapes. III: Multivariate and product formulas | 0 |
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 Consider a semi-simple algebraic group \(G\) over \(\mathbb{C}\), \(B\) a Borel subgroup, and \(T\) the corresponding maximal torus. The affine Grassmannian of \(G\) is \(\mathrm{Gr}_G = G(\mathbb{C}((t))) / G(\mathbb{C}[[t]])\). A \(T\)-equivariant Schubert basis, indexed by the affine Weyl group \(W_{\text{af}}^0\), has been constructed by Peterson and appears in his 1997 M.I.T. lecture notes. The associated Schubert homology structure constants are precisely the Schubert structure constants for the \(T\)-equivariant cohomology of the flag variety \(G/B\), as proved by Peterson and \textit{T. Lam} and \textit{M. Shimozono} [Acta Math. 204, 49--90 (2010; Zbl 1216.14052)]. This sparked interest in the study of the homology of the affine Grassmannian.
Let \(\{ \xi_w \mid w \in W_{\text{af}}^0 \}\) be the Schubert basis of \(H_T(\mathrm{Gr}_G)\). The equivariant Schubert homology structure constants \(d_{u v}^w\) are defined by \(\xi_u \xi_v = \sum_w d_{u v}^w \xi_w\), where \(u,v \in W_{\text{af}}^0\). They are always non-negative, as proved in [\textit{L. Mihalcea}, Am. J. Math. 128, No. 3, 787--803 (2006; Zbl 1099.14047)]. In the paper under review, the authors prove an ``equivariant homology Chevalley formula'', that is, an explicit rule for the product of a Schubert class with a degree two class in \(H_T(\mathrm{Gr}_G)\). Moreover, they define generating classes for \(H_T(\mathrm{Gr}_G)\) (which they call ``special Schubert classes'' by analogy with the case of usual Grassmannians) in the case where \(G\) is \({SL}_n\), \({SO}_{2n+1}\), or \({Sp}_{2n}\). When \(G={SL}_n\), they prove a positive ``equivariant homology Pieri formula'' for the product of a Schubert class with a special Schubert class. In the last two cases, they give a conjecture for a (not manifestly positive) equivariant homology Pieri formula. Schubert calculus; affine Grassmannian; Pieri rule; quantum cohomology Lam, T; Shimozono, M, Equivariant Pieri rule for the homology of the affine Grassmannian, J. Algebr. Combin., 36, 623-648, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Equivariant Pieri rule for the homology of the affine Grassmannian | 0 |
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 a homogeous polynomial \(f\) of degree \(d\) in \(n+1\) variables with coefficients in \(\mathbb{C}\), which defines a holomorphic function germ at the origin of \(\mathbb{C}^{n+1}\), its monodromy map \(T\) is defined by \(x_j\mapsto e^{2\pi i/d}x_j\), \(j=0\), \(\ldots\), \(n\). When \(f\) is an isolated homogeneous singularity, several invariants such as the Milnor number, the characteristic polynomials of \(T^\ast\), the signature and Hodge numbers of the Milnor fibre can be computed by classical topological and algebraic methods as well as via mixed Hodge structures. The authors investigate the monodromy characteristic polynomials \(\Delta _l(t)\) as well as the relation between the monodromy zeta function and the Hodge spectrum of the singularity. In the case \(n=2\), they give a description of \(\Delta _C(t)=\Delta _1(t)\) in terms of the multiplier ideal. homogeneous singularity; log-resolution; local system; multiplier ideal; finite abelian cover; Hodge spectrum; spectrum multiplicity; monodromy zeta function Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Multiplier ideals, Global theory of complex singularities; cohomological properties, Mixed Hodge theory of singular varieties (complex-analytic aspects) On complex homogeneous singularities | 0 |
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 offer a conjecture on the relation between weighted Euler characteristics of Hilbert schemes of points of integral complex curves with at worst locally planar singularities, and the HOMFLY polynomials of the links of these singularities. The link of a singularity \(C\) is its intersection with a small surrounding \(3\)-sphere. For non-singular curves a formula for the weighted Euler characteristic is well-known, and the authors show that each singularity produces a multiplicative correction to it. Each correction is expressed by an explicit conjectural formula in terms of the singularity's HOMFLY polynomial. Since it is known how to express the weighted Euler characteristic of \(C\) in terms of the Gopakumar-Vafa invariants the conjecture, if true, will provide an indirect evidence in favor of the large \(N\) duality between the Gromov-Witten and the Chern-Simons theories. The latter gives rise to the HOMFLY polynomials along the lines explained by Witten.
Most of the paper is dedicated to proving some corollaries of the conjectured formula, and verifying its special cases. In particular, the limit case when the HOMFLY polynomial reduces to the Alexander polynomial is derived from a theorem of Campillo, Delgado and Gusein-Zade. Symmetries implied by the conjecture are verified for the integrals giving the weighted Euler characteristic. Finally, for \(y^k=x^n\) singularities with gcd\((k,n)=1\), whose links are \((k,n)\)-torus knots, and for a singularity, whose link is the \((2,13)\) cable of the right-handed trefoil, the two sides of the formula are matched by direct computations. Hilbert schemes of points; locally planar singularity; weighted Euler characteristics; link of singularity; Gopakumar-Vafa invariants; HOMFLY polynomial; Alexander polynomial A. Oblomkov, J. Rasmussen, and V. Shende, The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link, \textit{Duke Math. J.}, 161 (2012), no. 7, 1277--1303.Zbl 1256.14025 MR 2922375 Singularities of curves, local rings, Knots and links in the 3-sphere The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link | 0 |
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